Subgroups — C₇×#60825 #13238041

Order 1792.

2146 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3221	C₂	1	4	yes	maximal, complemented
4	C₇×#3220	C₂	1	4	yes	maximal, complemented
5	C₇×#3214	C₂	1	4	yes	maximal, complemented
6	C₇×#3213	C₂	1	4	yes	maximal, complemented
7	C₇×#3151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#496	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3140	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3135	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#3042	C₂	1	2	yes	maximal, complemented
12	C₇×#3041	C₂	1	2	yes	maximal, complemented
13	C₇×#3010	C₂	1	2	yes	maximal, complemented
14	C₇×#3008	C₂	1	2	yes	maximal, complemented
15	C₇×#2923	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₇×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
17	C₂×C₇×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
18	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	2	yes	—
19	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
20	C₂²×C₇×SD₁₆	C₂²	1	2	yes	complemented
21	C₂²×C₇×D₈	C₂²	1	2	yes	—
22	C₇×#559	—	2	2	no	—
23	C₇×#550	—	2	2	no	—
24	C₇×#548	—	2	2	no	—
25	C₇×#546	—	2	4	no	—
26	C₇×#546	—	2	2	no	—
27	C₇×#546	—	2	4	no	—
28	C₇×#540	C₂²	1	4	yes	complemented
29	C₇×#536	C₂²	1	1	yes	characteristic
30	C₇×#536	C₂²	1	2	yes	complemented
31	C₇×#535	C₂²	1	1	yes	characteristic
32	C₇×#535	C₂²	1	2	yes	—
33	C₇×#535	C₂²	1	2	yes	complemented
34	C₇×#533	C₂²	1	1	yes	characteristic, complemented
35	C₇×#532	C₂²	1	1	yes	characteristic
36	C₇×#532	C₂²	1	2	yes	complemented
37	C₇×#532	C₂²	1	4	yes	complemented
38	C₂×C₇×#127	C₂²	1	1	yes	characteristic, complemented
39	C₂×C₇×#124	C₂²	1	1	yes	characteristic
40	C₂×C₇×#124	C₂²	1	2	yes	complemented
41	C₇×#518	C₂²	1	2	yes	complemented
42	C₄×C₇×#41	C₂²	1	1	yes	characteristic
43	C₇×#496	C₂²	1	2	yes	complemented
44	C₇×#496	C₂²	1	4	yes	complemented
45	C₇×#496	C₂²	1	2	yes	complemented
46	C₇×#496	C₂²	1	4	yes	complemented
47	C₇×#493	C₂²	1	2	yes	complemented
48	C₇×#492	C₂²	1	2	yes	complemented
49	C₇×#491	C₂²	1	2	yes	complemented
50	C₇×#490	C₂²	1	2	yes	complemented
51	C₇×#489	C₂²	1	2	yes	complemented
52	C₇×#489	C₂²	1	4	yes	complemented
53	C₇×#489	C₂²	1	2	yes	complemented
54	C₇×#488	C₂²	1	4	yes	complemented
55	C₇×#463	C₂²	1	8	yes	complemented
56	C₇×#463	C₂²	1	8	yes	complemented
57	C₇×#461	C₂²	1	8	yes	complemented
58	C₇×#459	C₂²	1	8	yes	complemented
59	C₇×#449	C₂²	1	8	yes	complemented
60	C₇×#449	C₂²	1	8	yes	complemented
61	C₇×#448	C₂²	1	8	yes	complemented
62	C₇×#447	C₂²	1	8	yes	complemented
63	C₇×#433	C₂²	1	4	yes	complemented
64	C₇×#432	C₂²	1	4	yes	complemented
65	C₇×#424	C₂²	1	2	yes	complemented
66	C₇×#419	C₂²	1	2	yes	complemented
67	C₇×#418	C₂²	1	2	yes	—
68	C₇×#416	C₂²	1	2	yes	—
69	C₇×#416	C₂²	1	2	yes	complemented
70	C₇×#408	C₂²	1	2	yes	—
71	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic, complemented
72	C₂×C₇×#97	C₂²	1	1	yes	characteristic, complemented
73	#60825	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₇×2₋¹⁺⁴	—	4	4	no	—
75	C₇×H(5,𝔽₂)	—	4	4	no	—
76	C₂×C₇×#41	—	2	1	no	—
77	C₂×C₇×#41	—	2	2	no	—
78	C₂×C₇×#41	—	2	1	no	—
79	C₂×C₇×#41	—	2	4	no	—
80	C₂×C₇×#41	C₂³	1	1	yes	characteristic
81	C₂×C₇×#41	C₂³	1	1	yes	characteristic
82	C₂×C₇×#41	C₂³	1	2	yes	—
83	C₂×C₇×#41	C₂³	1	2	yes	—
84	C₂×C₇×#41	C₂³	1	1	yes	characteristic
85	C₂²×C₇×Q₈	—	2	1	no	—
86	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
87	C₂²×C₇×Q₈	C₂³	1	2	yes	—
88	C₂²×C₇×D₄	—	2	2	no	—
89	C₂²×C₇×D₄	—	2	1	no	—
90	C₂²×C₇×D₄	—	2	2	no	—
91	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
92	C₂²×C₇×D₄	C₂³	1	2	yes	—
93	C₂²×C₇×D₄	C₂³	1	2	yes	—
94	C₇×AGL(1,ℤ/8)	—	4	4	no	—
95	C₇×AGL(1,ℤ/8)	—	4	4	no	—
96	C₂×C₇×SD₁₆	—	4	4	no	—
97	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
98	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
99	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
100	C₂×C₇×D₈	—	4	4	no	—
101	C₂×C₇×D₈	C₂³	1	4	yes	—
102	C₂×C₇×D₈	C₂³	1	4	yes	—
103	C₂×C₇×D₈	C₂³	1	8	yes	—
104	C₂×C₇×M₁₆	C₂³	1	2	yes	—
105	C₂²×C₇×C₈	C₂³	1	2	yes	—
106	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
107	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
108	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
109	C₇×#127	—	2	2	no	—
110	C₇×#127	—	2	2	no	—
111	C₇×#127	C₂³	1	1	yes	characteristic
112	C₇×#127	C₂³	1	2	yes	—
113	C₇×#127	C₂³	1	2	yes	—
114	C₇×#127	C₂³	1	2	yes	—
115	C₇×#125	C₂³	1	2	yes	—
116	C₇×#124	—	2	2	no	—
117	C₇×#124	—	2	2	no	—
118	C₇×#124	—	2	4	no	—
119	C₇×#124	—	2	4	no	—
120	C₇×#124	C₂³	1	2	yes	—
121	C₇×#124	C₂³	1	2	yes	—
122	C₇×#124	C₂³	1	1	yes	characteristic
123	C₇×#124	C₂³	1	1	yes	characteristic
124	C₇×#124	C₂³	1	2	yes	—
125	C₇×#124	C₂³	1	2	yes	—
126	C₇×#123	—	2	8	no	—
127	C₇×#123	—	2	2	no	—
128	C₇×#123	—	2	4	no	—
129	C₇×C₂²⋊Q₈	—	2	2	no	—
130	C₇×C₂²⋊Q₈	—	2	2	no	—
131	C₇×C₂²⋊Q₈	—	2	4	no	—
132	C₇×#121	—	2	2	no	—
133	C₇×#121	—	2	4	no	—
134	C₇×#121	—	2	4	no	—
135	C₇×#121	—	2	2	no	—
136	C₇×#121	—	2	4	no	—
137	C₇×#121	—	2	8	no	—
138	C₇×#121	C₂³	1	8	yes	—
139	C₇×#120	—	2	4	no	—
140	C₇×#120	—	2	2	no	—
141	C₇×#120	—	2	4	no	—
142	C₄×C₇×Q₈	—	2	2	no	—
143	C₄×C₇×Q₈	D₄	1	2	yes	—
144	C₄×C₇×D₄	—	2	4	no	—
145	C₄×C₇×D₄	—	2	2	no	—
146	C₄×C₇×D₄	—	2	4	no	—
147	C₄×C₇×D₄	—	2	4	no	—
148	C₄×C₇×D₄	D₄	1	4	yes	—
149	C₄×C₇×D₄	D₄	1	2	yes	—
150	C₄×C₇×D₄	C₂³	1	8	yes	—
151	C₇×#117	C₂³	1	2	yes	—
152	C₇×#117	C₂³	1	1	yes	characteristic
153	C₇×#117	C₂³	1	2	yes	—
154	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
155	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
156	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
157	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
158	C₂×C₄²×C₇	C₂³	1	2	yes	—
159	C₇×C₄⋊C₈	C₂³	1	8	yes	—
160	C₇×#103	C₂³	1	8	yes	—
161	C₇×#103	C₂³	1	4	yes	—
162	C₇×#103	C₂³	1	4	yes	—
163	C₇×#102	C₂³	1	8	yes	—
164	C₇×#102	C₂³	1	4	yes	—
165	C₇×#102	C₂³	1	4	yes	—
166	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
167	C₇×#97	C₂³	1	2	yes	—
168	C₇×#97	C₂³	1	2	yes	—
169	C₇×#97	C₂³	1	2	yes	—
170	C₄×C₇×C₈	C₂³	1	2	yes	—
171	#3508	C₂×C₇	1	1	yes	characteristic, complemented
172	#3500	C₂×C₇	1	1	yes	characteristic, complemented
173	#3221	C₂×C₇	1	4	yes	complemented
174	#3220	C₂×C₇	1	4	yes	complemented
175	#3214	C₂×C₇	1	4	yes	complemented
176	#3213	C₂×C₇	1	4	yes	complemented
177	#3151	C₂×C₇	1	1	yes	characteristic, complemented
178	C₂×#496	C₂×C₇	1	1	yes	characteristic, complemented
179	#3140	C₂×C₇	1	1	yes	characteristic, complemented
180	#3135	C₂×C₇	1	1	yes	characteristic, complemented
181	#3042	C₂×C₇	1	2	yes	complemented
182	#3041	C₂×C₇	1	2	yes	complemented
183	#3010	C₂×C₇	1	2	yes	complemented
184	#3008	C₂×C₇	1	2	yes	complemented
185	#2923	C₂×C₇	1	1	yes	characteristic, complemented
186	C₂⁴×C₇	—	2	2	no	—
187	C₂⁴×C₇	—	2	1	no	—
188	C₂⁴×C₇	—	2	2	no	—
189	C₇×#41	—	2	4	no	—
190	C₇×#41	—	2	4	no	—
191	C₇×#41	—	2	4	no	—
192	C₇×#41	—	4	4	no	—
193	C₇×#41	—	4	4	no	—
194	C₇×#41	—	4	8	no	—
195	C₇×#41	—	4	4	no	—
196	C₇×#41	—	4	4	no	—
197	C₇×#41	—	8	8	no	—
198	C₂×C₇×Q₈	—	2	2	no	—
199	C₂×C₇×Q₈	—	2	2	no	—
200	C₂×C₇×Q₈	—	2	1	no	—
201	C₂×C₇×Q₈	—	2	4	no	—
202	C₂×C₇×Q₈	—	4	4	no	—
203	C₂×C₇×Q₈	—	4	4	no	—
204	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
205	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
206	C₂×C₇×Q₈	C₂⁴	1	1	yes	characteristic
207	C₂×C₇×Q₈	C₂⁴	1	1	yes	characteristic
208	C₂×C₇×D₄	—	2	2	no	—
209	C₂×C₇×D₄	—	2	1	no	—
210	C₂×C₇×D₄	—	2	2	no	—
211	C₂×C₇×D₄	—	2	1	no	—
212	C₂×C₇×D₄	—	2	8	no	—
213	C₂×C₇×D₄	—	2	2	no	—
214	C₂×C₇×D₄	—	2	4	no	—
215	C₂×C₇×D₄	—	2	2	no	—
216	C₂×C₇×D₄	—	2	1	no	—
217	C₂×C₇×D₄	—	2	4	no	—
218	C₂×C₇×D₄	—	2	2	no	—
219	C₂×C₇×D₄	—	2	4	no	—
220	C₂×C₇×D₄	—	2	4	no	—
221	C₂×C₇×D₄	—	2	4	no	—
222	C₂×C₇×D₄	—	2	4	no	—
223	C₂×C₇×D₄	—	4	4	no	—
224	C₂×C₇×D₄	—	4	8	no	—
225	C₂×C₇×D₄	—	4	4	no	—
226	C₂×C₇×D₄	—	8	4	no	—
227	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
228	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
229	C₂×C₇×D₄	C₂⁴	1	2	yes	—
230	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
231	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
232	C₂×C₇×D₄	C₂⁴	1	4	yes	—
233	C₂²×C₄×C₇	—	2	2	no	—
234	C₂²×C₄×C₇	—	2	1	no	—
235	C₂²×C₄×C₇	—	2	2	no	—
236	C₂²×C₄×C₇	—	2	1	no	—
237	C₂²×C₄×C₇	—	2	2	no	—
238	C₂²×C₄×C₇	—	2	1	no	—
239	C₂²×C₄×C₇	—	2	8	no	—
240	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
241	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
242	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
243	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
244	C₇×SD₁₆	—	4	8	no	—
245	C₇×SD₁₆	—	4	4	no	—
246	C₇×SD₁₆	—	4	4	no	—
247	C₇×D₈	—	4	8	no	—
248	C₇×D₈	—	4	4	no	—
249	C₇×D₈	—	4	4	no	—
250	C₇×M₁₆	—	2	4	no	—
251	C₂×C₇×C₈	—	2	4	no	—
252	C₂×C₇×C₈	C₂⁴	1	4	yes	—
253	C₂×C₇×C₈	C₂⁴	1	4	yes	—
254	C₇×C₄⋊C₄	—	2	2	no	—
255	C₇×C₄⋊C₄	—	2	4	no	—
256	C₇×C₄⋊C₄	—	2	2	no	—
257	C₇×C₄⋊C₄	—	2	4	no	—
258	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
259	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
260	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
261	C₇×C₂²⋊C₄	—	2	2	no	—
262	C₇×C₂²⋊C₄	—	2	4	no	—
263	C₇×C₂²⋊C₄	—	2	2	no	—
264	C₇×C₂²⋊C₄	—	2	8	no	—
265	C₇×C₂²⋊C₄	—	2	4	no	—
266	C₇×C₂²⋊C₄	—	2	4	no	—
267	C₇×C₂²⋊C₄	—	2	4	no	—
268	C₇×C₂²⋊C₄	—	2	2	no	—
269	C₇×C₂²⋊C₄	—	2	8	no	—
270	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
271	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
272	C₄²×C₇	—	2	2	no	—
273	C₄²×C₇	C₂×D₄	1	4	yes	—
274	C₄²×C₇	C₂×D₄	1	2	yes	—
275	C₄²×C₇	C₂⁴	1	1	yes	characteristic
276	C₄²×C₇	C₂⁴	1	1	yes	characteristic
277	C₄²×C₇	C₂⁴	1	2	yes	—
278	C₂×2₋¹⁺⁴	C₂²×C₇	1	1	yes	characteristic
279	C₂×H(5,𝔽₂)	C₂²×C₇	1	1	yes	characteristic
280	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	2	yes	—
281	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	2	yes	complemented
282	C₂²×SD₁₆	C₂²×C₇	1	2	yes	complemented
283	C₂²×D₈	C₂²×C₇	1	2	yes	—
284	#559	—	2	2	no	—
285	#550	—	2	2	no	—
286	#548	—	2	2	no	—
287	#546	—	2	4	no	—
288	#546	—	2	2	no	—
289	#546	—	2	4	no	—
290	#540	C₂²×C₇	1	4	yes	complemented
291	#536	C₂²×C₇	1	1	yes	characteristic
292	#536	C₂²×C₇	1	2	yes	complemented
293	#535	C₂²×C₇	1	1	yes	characteristic
294	#535	C₂²×C₇	1	2	yes	—
295	#535	C₂²×C₇	1	2	yes	complemented
296	#533	C₂²×C₇	1	1	yes	characteristic, complemented
297	#532	C₂²×C₇	1	1	yes	characteristic
298	#532	C₂²×C₇	1	2	yes	complemented
299	#532	C₂²×C₇	1	4	yes	complemented
300	C₂×#127	C₂²×C₇	1	1	yes	characteristic, complemented
301	C₂×#124	C₂²×C₇	1	1	yes	characteristic
302	C₂×#124	C₂²×C₇	1	2	yes	complemented
303	#518	C₂²×C₇	1	2	yes	complemented
304	C₄×#41	C₂²×C₇	1	1	yes	characteristic
305	#496	C₂²×C₇	1	4	yes	complemented
306	#496	C₂²×C₇	1	4	yes	complemented
307	#496	C₂²×C₇	1	2	yes	complemented
308	#496	C₂²×C₇	1	2	yes	complemented
309	#493	C₂²×C₇	1	2	yes	complemented
310	#492	C₂²×C₇	1	2	yes	complemented
311	#491	C₂²×C₇	1	2	yes	complemented
312	#490	C₂²×C₇	1	2	yes	complemented
313	#489	C₂²×C₇	1	2	yes	complemented
314	#489	C₂²×C₇	1	2	yes	complemented
315	#489	C₂²×C₇	1	4	yes	complemented
316	#488	C₂²×C₇	1	4	yes	complemented
317	#463	C₂²×C₇	1	8	yes	complemented
318	#463	C₂²×C₇	1	8	yes	complemented
319	#461	C₂²×C₇	1	8	yes	complemented
320	#459	C₂²×C₇	1	8	yes	complemented
321	#449	C₂²×C₇	1	8	yes	complemented
322	#449	C₂²×C₇	1	8	yes	complemented
323	#448	C₂²×C₇	1	8	yes	complemented
324	#447	C₂²×C₇	1	8	yes	complemented
325	#433	C₂²×C₇	1	4	yes	complemented
326	#432	C₂²×C₇	1	4	yes	complemented
327	#424	C₂²×C₇	1	2	yes	complemented
328	#419	C₂²×C₇	1	2	yes	complemented
329	#418	C₂²×C₇	1	2	yes	—
330	#416	C₂²×C₇	1	2	yes	—
331	#416	C₂²×C₇	1	2	yes	complemented
332	#408	C₂²×C₇	1	2	yes	—
333	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
334	C₂×#97	C₂²×C₇	1	1	yes	characteristic, complemented
335	C₂³×C₇	—	2	2	no	—
336	C₂³×C₇	—	2	1	no	—
337	C₂³×C₇	—	2	4	no	—
338	C₂³×C₇	—	2	1	no	—
339	C₂³×C₇	—	4	4	no	—
340	C₂³×C₇	—	4	2	no	—
341	C₂³×C₇	—	4	4	no	—
342	C₂³×C₇	—	8	4	no	—
343	C₂³×C₇	—	8	2	no	—
344	C₂³×C₇	—	8	4	no	—
345	C₂³×C₇	C₂²×D₄	1	2	yes	—
346	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
347	C₇×Q₈	—	2	2	no	—
348	C₇×Q₈	—	2	4	no	—
349	C₇×Q₈	—	2	2	no	—
350	C₇×Q₈	—	2	2	no	—
351	C₇×Q₈	—	4	4	no	—
352	C₇×Q₈	—	4	4	no	—
353	C₇×Q₈	—	4	2	no	—
354	C₇×D₄	—	2	4	no	—
355	C₇×D₄	—	2	2	no	—
356	C₇×D₄	—	2	4	no	—
357	C₇×D₄	—	2	4	no	—
358	C₇×D₄	—	2	2	no	—
359	C₇×D₄	—	2	2	no	—
360	C₇×D₄	—	2	4	no	—
361	C₇×D₄	—	4	4	no	—
362	C₇×D₄	—	4	8	no	—
363	C₇×D₄	—	4	4	no	—
364	C₇×D₄	—	4	4	no	—
365	C₇×D₄	—	4	2	no	—
366	C₇×D₄	—	4	2	no	—
367	C₇×D₄	—	4	4	no	—
368	C₇×D₄	—	4	2	no	—
369	C₇×D₄	—	8	4	no	—
370	C₇×D₄	—	8	2	no	—
371	C₇×D₄	—	8	4	no	—
372	C₇×D₄	—	8	8	no	—
373	C₂×C₄×C₇	—	2	4	no	—
374	C₂×C₄×C₇	—	2	4	no	—
375	C₂×C₄×C₇	—	2	2	no	—
376	C₂×C₄×C₇	—	2	1	no	—
377	C₂×C₄×C₇	—	2	1	no	—
378	C₂×C₄×C₇	—	4	2	no	—
379	C₂×C₄×C₇	—	4	4	no	—
380	C₂×C₄×C₇	—	4	4	no	—
381	C₂×C₄×C₇	—	4	4	no	—
382	C₂×C₄×C₇	—	4	4	no	—
383	C₂×C₄×C₇	—	4	2	no	—
384	C₂×C₄×C₇	—	4	2	no	—
385	C₂×C₄×C₇	—	4	2	no	—
386	C₂×C₄×C₇	—	8	8	no	—
387	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
388	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
389	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
390	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
391	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
392	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
393	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
394	C₇×C₈	—	2	4	no	—
395	C₇×C₈	—	2	4	no	—
396	2₋¹⁺⁴	—	4	4	no	—
397	H(5,𝔽₂)	—	4	4	no	—
398	C₂×#41	—	2	1	no	—
399	C₂×#41	—	2	2	no	—
400	C₂×#41	—	2	1	no	—
401	C₂×#41	—	2	4	no	—
402	C₂×#41	C₂³×C₇	1	1	yes	characteristic
403	C₂×#41	C₂³×C₇	1	1	yes	characteristic
404	C₂×#41	C₂³×C₇	1	2	yes	—
405	C₂×#41	C₂³×C₇	1	2	yes	—
406	C₂×#41	C₂³×C₇	1	1	yes	characteristic
407	C₂²×Q₈	—	2	1	no	—
408	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
409	C₂²×Q₈	C₂³×C₇	1	2	yes	—
410	C₂²×D₄	—	2	2	no	—
411	C₂²×D₄	—	2	1	no	—
412	C₂²×D₄	—	2	2	no	—
413	C₂²×D₄	C₂³×C₇	1	2	yes	—
414	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
415	C₂²×D₄	C₂³×C₇	1	2	yes	—
416	AGL(1,ℤ/8)	—	4	4	no	—
417	AGL(1,ℤ/8)	—	4	4	no	—
418	C₂×SD₁₆	—	4	4	no	—
419	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
420	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
421	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
422	C₂×D₈	—	4	4	no	—
423	C₂×D₈	C₂³×C₇	1	8	yes	—
424	C₂×D₈	C₂³×C₇	1	4	yes	—
425	C₂×D₈	C₂³×C₇	1	4	yes	—
426	C₂×M₁₆	C₂³×C₇	1	2	yes	—
427	C₂²×C₈	C₂³×C₇	1	2	yes	—
428	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
429	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
430	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
431	#127	—	2	2	no	—
432	#127	—	2	2	no	—
433	#127	C₂³×C₇	1	2	yes	—
434	#127	C₂³×C₇	1	2	yes	—
435	#127	C₂³×C₇	1	1	yes	characteristic
436	#127	C₂³×C₇	1	2	yes	—
437	#125	C₂³×C₇	1	2	yes	—
438	#124	—	2	2	no	—
439	#124	—	2	2	no	—
440	#124	—	2	4	no	—
441	#124	—	2	4	no	—
442	#124	C₂³×C₇	1	2	yes	—
443	#124	C₂³×C₇	1	2	yes	—
444	#124	C₂³×C₇	1	1	yes	characteristic
445	#124	C₂³×C₇	1	2	yes	—
446	#124	C₂³×C₇	1	2	yes	—
447	#124	C₂³×C₇	1	1	yes	characteristic
448	#123	—	2	8	no	—
449	#123	—	2	2	no	—
450	#123	—	2	4	no	—
451	C₂²⋊Q₈	—	2	2	no	—
452	C₂²⋊Q₈	—	2	2	no	—
453	C₂²⋊Q₈	—	2	4	no	—
454	#121	—	2	2	no	—
455	#121	—	2	4	no	—
456	#121	—	2	4	no	—
457	#121	—	2	2	no	—
458	#121	—	2	4	no	—
459	#121	—	2	8	no	—
460	#121	C₂³×C₇	1	8	yes	—
461	#120	—	2	4	no	—
462	#120	—	2	2	no	—
463	#120	—	2	4	no	—
464	C₄×Q₈	—	2	2	no	—
465	C₄×Q₈	C₇×D₄	1	2	yes	—
466	C₄×D₄	—	2	4	no	—
467	C₄×D₄	—	2	2	no	—
468	C₄×D₄	—	2	4	no	—
469	C₄×D₄	—	2	4	no	—
470	C₄×D₄	C₇×D₄	1	4	yes	—
471	C₄×D₄	C₇×D₄	1	2	yes	—
472	C₄×D₄	C₂³×C₇	1	8	yes	—
473	#117	C₂³×C₇	1	2	yes	—
474	#117	C₂³×C₇	1	1	yes	characteristic
475	#117	C₂³×C₇	1	2	yes	—
476	C₂×C₂²⋊C₄	—	2	2	no	—
477	C₂×C₂²⋊C₄	—	2	4	no	—
478	C₂×C₂²⋊C₄	—	2	4	no	—
479	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
480	C₂×C₄²	C₂³×C₇	1	2	yes	—
481	C₄⋊C₈	C₂³×C₇	1	8	yes	—
482	#103	C₂³×C₇	1	4	yes	—
483	#103	C₂³×C₇	1	4	yes	—
484	#103	C₂³×C₇	1	8	yes	—
485	#102	C₂³×C₇	1	4	yes	—
486	#102	C₂³×C₇	1	4	yes	—
487	#102	C₂³×C₇	1	8	yes	—
488	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
489	#97	C₂³×C₇	1	2	yes	—
490	#97	C₂³×C₇	1	2	yes	—
491	#97	C₂³×C₇	1	2	yes	—
492	C₄×C₈	C₂³×C₇	1	2	yes	—
493	C₂²×C₇	—	2	2	no	—
494	C₂²×C₇	—	2	4	no	—
495	C₂²×C₇	—	2	2	no	—
496	C₂²×C₇	—	4	8	no	—
497	C₂²×C₇	—	4	2	no	—
498	C₂²×C₇	—	4	4	no	—
499	C₂²×C₇	—	4	2	no	—
500	C₂²×C₇	—	4	1	no	—
501	C₂²×C₇	—	4	2	no	—
502	C₂²×C₇	—	4	1	no	—
503	C₂²×C₇	—	4	4	no	—
504	C₂²×C₇	—	16	4	no	—
505	C₂²×C₇	—	16	2	no	—
506	C₂²×C₇	—	16	4	no	—
507	C₂²×C₇	#535	1	2	yes	—
508	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
509	C₄×C₇	—	2	2	no	—
510	C₄×C₇	—	2	4	no	—
511	C₄×C₇	—	2	2	no	—
512	C₄×C₇	—	2	2	no	—
513	C₄×C₇	—	4	4	no	—
514	C₄×C₇	—	4	1	no	—
515	C₄×C₇	—	4	2	no	—
516	C₄×C₇	—	4	1	no	—
517	C₄×C₇	#535	1	2	yes	—
518	C₄×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
519	C₂⁴	—	2	2	no	—
520	C₂⁴	—	2	1	no	—
521	C₂⁴	—	2	2	no	—
522	#41	—	2	4	no	—
523	#41	—	2	4	no	—
524	#41	—	2	4	no	—
525	#41	—	4	4	no	—
526	#41	—	4	4	no	—
527	#41	—	4	8	no	—
528	#41	—	4	4	no	—
529	#41	—	4	4	no	—
530	#41	—	8	8	no	—
531	C₂×Q₈	—	2	2	no	—
532	C₂×Q₈	—	2	2	no	—
533	C₂×Q₈	—	2	1	no	—
534	C₂×Q₈	—	2	4	no	—
535	C₂×Q₈	—	4	4	no	—
536	C₂×Q₈	—	4	4	no	—
537	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
538	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
539	C₂×Q₈	C₂⁴×C₇	1	1	yes	characteristic
540	C₂×Q₈	C₂⁴×C₇	1	1	yes	characteristic
541	C₂×D₄	—	2	2	no	—
542	C₂×D₄	—	2	1	no	—
543	C₂×D₄	—	2	2	no	—
544	C₂×D₄	—	2	1	no	—
545	C₂×D₄	—	2	8	no	—
546	C₂×D₄	—	2	2	no	—
547	C₂×D₄	—	2	4	no	—
548	C₂×D₄	—	2	2	no	—
549	C₂×D₄	—	2	1	no	—
550	C₂×D₄	—	2	4	no	—
551	C₂×D₄	—	2	2	no	—
552	C₂×D₄	—	2	4	no	—
553	C₂×D₄	—	2	4	no	—
554	C₂×D₄	—	2	4	no	—
555	C₂×D₄	—	2	4	no	—
556	C₂×D₄	—	4	4	no	—
557	C₂×D₄	—	4	8	no	—
558	C₂×D₄	—	4	4	no	—
559	C₂×D₄	—	8	4	no	—
560	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
561	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
562	C₂×D₄	C₂⁴×C₇	1	2	yes	—
563	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
564	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
565	C₂×D₄	C₂⁴×C₇	1	4	yes	—
566	C₂²×C₄	—	2	2	no	—
567	C₂²×C₄	—	2	1	no	—
568	C₂²×C₄	—	2	2	no	—
569	C₂²×C₄	—	2	1	no	—
570	C₂²×C₄	—	2	2	no	—
571	C₂²×C₄	—	2	1	no	—
572	C₂²×C₄	—	2	8	no	—
573	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
574	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
575	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
576	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
577	SD₁₆	—	4	8	no	—
578	SD₁₆	—	4	4	no	—
579	SD₁₆	—	4	4	no	—
580	D₈	—	4	8	no	—
581	D₈	—	4	4	no	—
582	D₈	—	4	4	no	—
583	M₁₆	—	2	4	no	—
584	C₂×C₈	—	2	4	no	—
585	C₂×C₈	C₂⁴×C₇	1	4	yes	—
586	C₂×C₈	C₂⁴×C₇	1	4	yes	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	—	2	4	no	—
589	C₄⋊C₄	—	2	2	no	—
590	C₄⋊C₄	—	2	4	no	—
591	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
592	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
593	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
594	C₂²⋊C₄	—	2	2	no	—
595	C₂²⋊C₄	—	2	4	no	—
596	C₂²⋊C₄	—	2	2	no	—
597	C₂²⋊C₄	—	2	8	no	—
598	C₂²⋊C₄	—	2	4	no	—
599	C₂²⋊C₄	—	2	4	no	—
600	C₂²⋊C₄	—	2	4	no	—
601	C₂²⋊C₄	—	2	2	no	—
602	C₂²⋊C₄	—	2	8	no	—
603	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
604	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
605	C₄²	—	2	2	no	—
606	C₄²	C₂×C₇×D₄	1	4	yes	—
607	C₄²	C₂×C₇×D₄	1	2	yes	—
608	C₄²	C₂⁴×C₇	1	1	yes	characteristic
609	C₄²	C₂⁴×C₇	1	1	yes	characteristic
610	C₄²	C₂⁴×C₇	1	2	yes	—
611	C₂×C₇	—	2	2	no	—
612	C₂×C₇	—	4	2	no	—
613	C₂×C₇	—	8	1	no	—
614	C₂×C₇	—	8	4	no	—
615	C₂×C₇	—	8	2	no	—
616	C₂×C₇	—	8	1	no	—
617	C₂×C₇	C₂×#535	1	1	yes	characteristic
618	C₂×C₇	#3585	1	2	yes	—
619	C₂³	—	2	2	no	—
620	C₂³	—	2	1	no	—
621	C₂³	—	2	4	no	—
622	C₂³	—	2	1	no	—
623	C₂³	—	4	4	no	—
624	C₂³	—	4	2	no	—
625	C₂³	—	4	4	no	—
626	C₂³	—	8	4	no	—
627	C₂³	—	8	2	no	—
628	C₂³	—	8	4	no	—
629	C₂³	C₂²×C₇×D₄	1	2	yes	—
630	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
631	Q₈	—	2	2	no	—
632	Q₈	—	2	4	no	—
633	Q₈	—	2	2	no	—
634	Q₈	—	2	2	no	—
635	Q₈	—	4	4	no	—
636	Q₈	—	4	4	no	—
637	Q₈	—	4	2	no	—
638	D₄	—	2	4	no	—
639	D₄	—	2	4	no	—
640	D₄	—	2	4	no	—
641	D₄	—	2	2	no	—
642	D₄	—	2	2	no	—
643	D₄	—	2	4	no	—
644	D₄	—	2	2	no	—
645	D₄	—	4	4	no	—
646	D₄	—	4	8	no	—
647	D₄	—	4	4	no	—
648	D₄	—	4	4	no	—
649	D₄	—	4	2	no	—
650	D₄	—	4	2	no	—
651	D₄	—	4	4	no	—
652	D₄	—	4	2	no	—
653	D₄	—	8	4	no	—
654	D₄	—	8	2	no	—
655	D₄	—	8	4	no	—
656	D₄	—	8	8	no	—
657	C₂×C₄	—	2	4	no	—
658	C₂×C₄	—	2	4	no	—
659	C₂×C₄	—	2	2	no	—
660	C₂×C₄	—	2	1	no	—
661	C₂×C₄	—	2	1	no	—
662	C₂×C₄	—	4	2	no	—
663	C₂×C₄	—	4	4	no	—
664	C₂×C₄	—	4	4	no	—
665	C₂×C₄	—	4	4	no	—
666	C₂×C₄	—	4	4	no	—
667	C₂×C₄	—	4	2	no	—
668	C₂×C₄	—	4	2	no	—
669	C₂×C₄	—	4	2	no	—
670	C₂×C₄	—	8	8	no	—
671	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
672	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
673	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
674	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
675	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
676	C₂×C₄	C₇×H(5,𝔽₂)	1	4	yes	—
677	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
678	C₈	—	2	4	no	—
679	C₈	—	2	4	no	—
680	C₇	#60825	1	1	yes	characteristic, direct product factor, Sylow
681	C₂²	—	2	2	no	—
682	C₂²	—	2	4	no	—
683	C₂²	—	2	2	no	—
684	C₂²	—	4	8	no	—
685	C₂²	—	4	2	no	—
686	C₂²	—	4	4	no	—
687	C₂²	—	4	2	no	—
688	C₂²	—	4	1	no	—
689	C₂²	—	4	2	no	—
690	C₂²	—	4	1	no	—
691	C₂²	—	4	4	no	—
692	C₂²	—	16	2	no	—
693	C₂²	—	16	4	no	—
694	C₂²	—	16	4	no	—
695	C₂²	C₇×#535	1	2	yes	—
696	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
697	C₄	—	2	2	no	—
698	C₄	—	2	4	no	—
699	C₄	—	2	2	no	—
700	C₄	—	2	2	no	—
701	C₄	—	4	4	no	—
702	C₄	—	4	1	no	—
703	C₄	—	4	2	no	—
704	C₄	—	4	1	no	—
705	C₄	C₇×#535	1	2	yes	—
706	C₄	C₂×C₇×H(5,𝔽₂)	1	2	yes	—
707	C₂	—	2	2	no	—
708	C₂	—	4	2	no	—
709	C₂	—	8	2	no	—
710	C₂	—	8	1	no	—
711	C₂	—	8	4	no	—
712	C₂	—	8	1	no	—
713	C₂	C₂×C₇×#535	1	1	yes	characteristic
714	C₂	C₇×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×#41 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₇×#3508	C₂	semidirect
2. C₇×#3500	C₂	semidirect
3. C₇×#3221	C₂	semidirect
4. C₇×#3220	C₂	semidirect
5. C₇×#3214	C₂	semidirect
6. C₇×#3213	C₂	semidirect
7. C₇×#3151	C₂	semidirect
8. C₂×C₇×#496	C₂	semidirect
9. C₇×#3140	C₂	semidirect
10. C₇×#3135	C₂	semidirect
11. C₇×#3042	C₂	semidirect
12. C₇×#3041	C₂	semidirect
13. C₇×#3010	C₂	semidirect
14. C₇×#3008	C₂	semidirect
15. C₇×#2923	C₂	semidirect
19. C₂×C₇×AGL(1,ℤ/8)	C₂²	semidirect
20. C₂²×C₇×SD₁₆	C₂²	semidirect
28. C₇×#540	C₂²	semidirect
30. C₇×#536	C₂²	semidirect
33. C₇×#535	C₂²	semidirect
34. C₇×#533	C₂²	semidirect
36. C₇×#532	C₂²	semidirect
37. C₇×#532	C₂²	semidirect
38. C₂×C₇×#127	C₂²	semidirect
40. C₂×C₇×#124	C₂²	semidirect
41. C₇×#518	C₂²	semidirect
43. C₇×#496	C₂²	semidirect
44. C₇×#496	C₂²	semidirect
45. C₇×#496	C₂²	semidirect
46. C₇×#496	C₂²	semidirect
47. C₇×#493	C₂²	semidirect
48. C₇×#492	C₂²	semidirect
49. C₇×#491	C₂²	semidirect
50. C₇×#490	C₂²	semidirect
51. C₇×#489	C₂²	semidirect
52. C₇×#489	C₂²	semidirect
53. C₇×#489	C₂²	semidirect
54. C₇×#488	C₂²	semidirect
55. C₇×#463	C₂²	semidirect
56. C₇×#463	C₂²	semidirect
57. C₇×#461	C₂²	semidirect
58. C₇×#459	C₂²	semidirect
59. C₇×#449	C₂²	semidirect
60. C₇×#449	C₂²	semidirect
61. C₇×#448	C₂²	semidirect
62. C₇×#447	C₂²	semidirect
63. C₇×#433	C₂²	semidirect
64. C₇×#432	C₂²	semidirect
65. C₇×#424	C₂²	semidirect
66. C₇×#419	C₂²	semidirect
69. C₇×#416	C₂²	semidirect
71. C₄×C₇×M₁₆	C₂²	semidirect
72. C₂×C₇×#97	C₂²	semidirect
73. #60825	C₇	direct (N × H)
171. #3508	C₂×C₇	semidirect
172. #3500	C₂×C₇	semidirect
173. #3221	C₂×C₇	semidirect
174. #3220	C₂×C₇	semidirect
175. #3214	C₂×C₇	semidirect
176. #3213	C₂×C₇	semidirect
177. #3151	C₂×C₇	semidirect
178. C₂×#496	C₂×C₇	semidirect
179. #3140	C₂×C₇	semidirect
180. #3135	C₂×C₇	semidirect
181. #3042	C₂×C₇	semidirect
182. #3041	C₂×C₇	semidirect
183. #3010	C₂×C₇	semidirect
184. #3008	C₂×C₇	semidirect
185. #2923	C₂×C₇	semidirect
281. C₂×AGL(1,ℤ/8)	C₂²×C₇	semidirect
282. C₂²×SD₁₆	C₂²×C₇	semidirect
290. #540	C₂²×C₇	semidirect
292. #536	C₂²×C₇	semidirect
295. #535	C₂²×C₇	semidirect
296. #533	C₂²×C₇	semidirect
298. #532	C₂²×C₇	semidirect
299. #532	C₂²×C₇	semidirect
300. C₂×#127	C₂²×C₇	semidirect
302. C₂×#124	C₂²×C₇	semidirect
303. #518	C₂²×C₇	semidirect
305. #496	C₂²×C₇	semidirect
306. #496	C₂²×C₇	semidirect
307. #496	C₂²×C₇	semidirect
308. #496	C₂²×C₇	semidirect
309. #493	C₂²×C₇	semidirect
310. #492	C₂²×C₇	semidirect
311. #491	C₂²×C₇	semidirect
312. #490	C₂²×C₇	semidirect
313. #489	C₂²×C₇	semidirect
314. #489	C₂²×C₇	semidirect
315. #489	C₂²×C₇	semidirect
316. #488	C₂²×C₇	semidirect
317. #463	C₂²×C₇	semidirect
318. #463	C₂²×C₇	semidirect
319. #461	C₂²×C₇	semidirect
320. #459	C₂²×C₇	semidirect
321. #449	C₂²×C₇	semidirect
322. #449	C₂²×C₇	semidirect
323. #448	C₂²×C₇	semidirect
324. #447	C₂²×C₇	semidirect
325. #433	C₂²×C₇	semidirect
326. #432	C₂²×C₇	semidirect
327. #424	C₂²×C₇	semidirect
328. #419	C₂²×C₇	semidirect
331. #416	C₂²×C₇	semidirect
333. C₄×M₁₆	C₂²×C₇	semidirect
334. C₂×#97	C₂²×C₇	semidirect
680. C₇	#60825	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#3508	2
C₇×#3500	2
C₇×#3221	2
C₇×#3220	2
C₇×#3214	2
C₇×#3213	2
C₇×#3151	2
C₂×C₇×#496	2
C₇×#3140	2
C₇×#3135	2
C₇×#3042	2
C₇×#3041	2
C₇×#3010	2
C₇×#3008	2
C₇×#2923	2
#60825	7

Subgroups — C7×#60825 #13238041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60825 #13238041