Subgroups — C₇×#60827 #13238043

Order 1792.

2154 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3224	C₂	1	8	yes	maximal, complemented
4	C₇×#3219	C₂	1	4	yes	maximal, complemented
5	C₇×#3217	C₂	1	4	yes	maximal, complemented
6	C₇×#3153	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#496	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3138	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3135	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3043	C₂	1	2	yes	maximal, complemented
11	C₇×#3041	C₂	1	2	yes	maximal, complemented
12	C₇×#3009	C₂	1	2	yes	maximal, complemented
13	C₇×#3008	C₂	1	2	yes	maximal, complemented
14	C₇×#2923	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₇×H(5,𝔽₂)	C₂²	1	2	yes	—
16	C₂×C₇×#137	C₂²	1	2	yes	complemented
17	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
18	C₂²×C₇×SD₁₆	C₂²	1	2	yes	complemented
19	C₂²×C₇×D₈	C₂²	1	2	yes	complemented
20	C₇×#559	—	2	2	no	—
21	C₇×#556	—	2	2	no	—
22	C₇×#551	—	2	4	no	—
23	C₇×#550	—	2	2	no	—
24	C₇×D₄²	—	2	2	no	—
25	C₇×D₄²	—	2	4	no	—
26	C₇×#542	C₂²	1	4	yes	complemented
27	C₇×#536	C₂²	1	1	yes	characteristic
28	C₇×#535	C₂²	1	2	yes	—
29	C₇×#535	C₂²	1	1	yes	characteristic
30	C₇×#535	C₂²	1	4	yes	complemented
31	C₇×#533	C₂²	1	1	yes	characteristic, complemented
32	C₇×#532	C₂²	1	1	yes	characteristic
33	C₇×#532	C₂²	1	2	yes	complemented
34	C₂×C₇×#127	C₂²	1	1	yes	characteristic
35	C₂×C₇×#127	C₂²	1	2	yes	complemented
36	C₇×#529	C₂²	1	4	yes	complemented
37	C₂×C₇×#124	C₂²	1	1	yes	characteristic, complemented
38	C₇×#519	C₂²	1	1	yes	characteristic, complemented
39	C₇×#518	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₇×#41	C₂²	1	1	yes	characteristic
41	C₇×#497	C₂²	1	2	yes	complemented
42	C₇×#496	C₂²	1	2	yes	complemented
43	C₇×#496	C₂²	1	4	yes	complemented
44	C₇×#496	C₂²	1	4	yes	complemented
45	C₇×#495	C₂²	1	4	yes	complemented
46	C₇×#491	C₂²	1	2	yes	complemented
47	C₇×#490	C₂²	1	2	yes	complemented
48	C₇×#490	C₂²	1	4	yes	complemented
49	C₇×#489	C₂²	1	4	yes	complemented
50	C₇×#486	C₂²	1	4	yes	complemented
51	C₇×#464	C₂²	1	8	yes	complemented
52	C₇×#463	C₂²	1	8	yes	complemented
53	C₇×#460	C₂²	1	8	yes	complemented
54	C₇×#459	C₂²	1	8	yes	complemented
55	C₇×#452	C₂²	1	8	yes	complemented
56	C₇×#450	C₂²	1	8	yes	complemented
57	C₇×#449	C₂²	1	8	yes	complemented
58	C₇×#447	C₂²	1	8	yes	complemented
59	C₇×#433	C₂²	1	4	yes	complemented
60	C₇×#432	C₂²	1	4	yes	complemented
61	C₇×#424	C₂²	1	2	yes	complemented
62	C₇×#418	C₂²	1	2	yes	complemented
63	C₇×#418	C₂²	1	2	yes	complemented
64	C₇×#416	C₂²	1	2	yes	complemented
65	C₇×#416	C₂²	1	2	yes	complemented
66	C₇×#408	C₂²	1	2	yes	—
67	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic, complemented
68	C₂×C₇×#97	C₂²	1	1	yes	characteristic, complemented
69	#60827	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₇×H(5,𝔽₂)	—	4	4	no	—
71	C₇×H(5,𝔽₂)	—	4	4	no	—
72	C₂×C₇×#41	—	2	2	no	—
73	C₂×C₇×#41	C₂³	1	2	yes	—
74	C₂×C₇×#41	C₂³	1	4	yes	—
75	C₂×C₇×#41	C₂³	1	1	yes	characteristic
76	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
77	C₂²×C₇×D₄	—	2	4	no	—
78	C₂²×C₇×D₄	—	2	2	no	—
79	C₂²×C₇×D₄	—	2	2	no	—
80	C₂²×C₇×D₄	—	2	4	no	—
81	C₂²×C₇×D₄	C₂³	1	2	yes	—
82	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
83	C₂²×C₇×D₄	C₂³	1	4	yes	—
84	C₇×#137	—	4	4	no	—
85	C₇×AGL(1,ℤ/8)	—	4	4	no	—
86	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
87	C₂×C₇×SD₁₆	—	4	4	no	—
88	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
89	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
90	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
91	C₂×C₇×D₈	—	4	4	no	—
92	C₂×C₇×D₈	C₂³	1	4	yes	—
93	C₂×C₇×D₈	C₂³	1	8	yes	—
94	C₂×C₇×M₁₆	C₂³	1	2	yes	—
95	C₂²×C₇×C₈	C₂³	1	2	yes	—
96	C₇×C₄⋊Q₈	—	2	1	no	—
97	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
98	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
99	C₇×#127	—	2	1	no	—
100	C₇×#127	—	2	2	no	—
101	C₇×#127	—	2	4	no	—
102	C₇×#127	C₂³	1	2	yes	—
103	C₇×#127	C₂³	1	2	yes	—
104	C₇×#127	C₂³	1	1	yes	characteristic
105	C₇×#127	C₂³	1	4	yes	—
106	C₇×#126	—	2	4	no	—
107	C₇×#126	—	2	4	no	—
108	C₇×#125	C₂³	1	2	yes	—
109	C₇×#124	—	2	1	no	—
110	C₇×#124	—	2	1	no	—
111	C₇×#124	—	2	4	no	—
112	C₇×#124	—	2	2	no	—
113	C₇×#124	C₂³	1	2	yes	—
114	C₇×#124	C₂³	1	2	yes	—
115	C₇×#124	C₂³	1	2	yes	—
116	C₇×#124	C₂³	1	4	yes	—
117	C₇×#123	—	2	2	no	—
118	C₇×#123	—	2	4	no	—
119	C₇×#123	—	2	4	no	—
120	C₇×C₂²⋊Q₈	—	2	2	no	—
121	C₇×C₂²⋊Q₈	—	2	4	no	—
122	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
123	C₇×#121	—	2	2	no	—
124	C₇×#121	—	2	4	no	—
125	C₇×#121	—	2	4	no	—
126	C₇×#121	—	2	8	no	—
127	C₇×#121	C₂³	1	4	yes	—
128	C₇×#120	—	2	2	no	—
129	C₇×#120	—	2	4	no	—
130	C₇×#120	—	2	8	no	—
131	C₄×C₇×Q₈	—	2	2	no	—
132	C₄×C₇×Q₈	—	2	2	no	—
133	C₄×C₇×Q₈	D₄	1	2	yes	—
134	C₄×C₇×Q₈	C₂³	1	4	yes	—
135	C₄×C₇×D₄	—	2	4	no	—
136	C₄×C₇×D₄	—	2	2	no	—
137	C₄×C₇×D₄	—	2	2	no	—
138	C₄×C₇×D₄	—	2	4	no	—
139	C₄×C₇×D₄	D₄	1	4	yes	—
140	C₄×C₇×D₄	D₄	1	2	yes	—
141	C₄×C₇×D₄	C₂³	1	4	yes	—
142	C₇×#117	—	2	2	no	—
143	C₇×#117	—	2	4	no	—
144	C₇×#117	C₂³	1	1	yes	characteristic
145	C₇×#117	C₂³	1	1	yes	characteristic
146	C₇×#117	C₂³	1	1	yes	characteristic
147	C₇×#117	C₂³	1	2	yes	—
148	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
149	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
150	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
151	C₂×C₄²×C₇	C₂³	1	2	yes	—
152	C₇×C₄⋊C₈	C₂³	1	8	yes	—
153	C₇×#103	C₂³	1	4	yes	—
154	C₇×#103	C₂³	1	4	yes	—
155	C₇×#102	C₂³	1	8	yes	—
156	C₇×#102	C₂³	1	8	yes	—
157	C₇×#102	C₂³	1	4	yes	—
158	C₇×#102	C₂³	1	4	yes	—
159	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
160	C₇×#97	C₂³	1	2	yes	—
161	C₇×#97	C₂³	1	2	yes	—
162	C₇×#97	C₂³	1	2	yes	—
163	C₄×C₇×C₈	C₂³	1	2	yes	—
164	#3510	C₂×C₇	1	1	yes	characteristic, complemented
165	#3498	C₂×C₇	1	1	yes	characteristic, complemented
166	#3224	C₂×C₇	1	8	yes	complemented
167	#3219	C₂×C₇	1	4	yes	complemented
168	#3217	C₂×C₇	1	4	yes	complemented
169	#3153	C₂×C₇	1	1	yes	characteristic, complemented
170	C₂×#496	C₂×C₇	1	1	yes	characteristic, complemented
171	#3138	C₂×C₇	1	1	yes	characteristic, complemented
172	#3135	C₂×C₇	1	1	yes	characteristic, complemented
173	#3043	C₂×C₇	1	2	yes	complemented
174	#3041	C₂×C₇	1	2	yes	complemented
175	#3009	C₂×C₇	1	2	yes	complemented
176	#3008	C₂×C₇	1	2	yes	complemented
177	#2923	C₂×C₇	1	1	yes	characteristic, complemented
178	C₂⁴×C₇	—	2	1	no	—
179	C₂⁴×C₇	—	2	4	no	—
180	C₂⁴×C₇	—	2	2	no	—
181	C₇×#41	—	2	4	no	—
182	C₇×#41	—	2	4	no	—
183	C₇×#41	—	2	4	no	—
184	C₇×#41	—	4	4	no	—
185	C₇×#41	—	4	4	no	—
186	C₇×#41	—	4	8	no	—
187	C₂×C₇×Q₈	—	2	1	no	—
188	C₂×C₇×Q₈	—	2	2	no	—
189	C₂×C₇×Q₈	—	2	4	no	—
190	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
191	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
192	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
193	C₂×C₇×D₄	—	2	4	no	—
194	C₂×C₇×D₄	—	2	2	no	—
195	C₂×C₇×D₄	—	2	4	no	—
196	C₂×C₇×D₄	—	2	1	no	—
197	C₂×C₇×D₄	—	2	2	no	—
198	C₂×C₇×D₄	—	2	4	no	—
199	C₂×C₇×D₄	—	2	4	no	—
200	C₂×C₇×D₄	—	2	2	no	—
201	C₂×C₇×D₄	—	2	8	no	—
202	C₂×C₇×D₄	—	2	4	no	—
203	C₂×C₇×D₄	—	2	4	no	—
204	C₂×C₇×D₄	—	2	4	no	—
205	C₂×C₇×D₄	—	2	4	no	—
206	C₂×C₇×D₄	—	4	4	no	—
207	C₂×C₇×D₄	—	4	4	no	—
208	C₂×C₇×D₄	—	4	16	no	—
209	C₂×C₇×D₄	—	4	8	no	—
210	C₂×C₇×D₄	—	8	4	no	—
211	C₂×C₇×D₄	—	8	8	no	—
212	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
213	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
214	C₂×C₇×D₄	C₂⁴	1	4	yes	—
215	C₂×C₇×D₄	C₂⁴	1	2	yes	—
216	C₂×C₇×D₄	C₂⁴	1	2	yes	—
217	C₂²×C₄×C₇	—	2	1	no	—
218	C₂²×C₄×C₇	—	2	4	no	—
219	C₂²×C₄×C₇	—	2	2	no	—
220	C₂²×C₄×C₇	—	2	4	no	—
221	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
222	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
223	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
224	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
225	C₇×Q₁₆	—	4	4	no	—
226	C₇×SD₁₆	—	4	8	no	—
227	C₇×SD₁₆	—	4	4	no	—
228	C₇×SD₁₆	—	4	4	no	—
229	C₇×D₈	—	4	8	no	—
230	C₇×D₈	—	4	4	no	—
231	C₇×M₁₆	—	2	4	no	—
232	C₂×C₇×C₈	—	2	4	no	—
233	C₂×C₇×C₈	C₂⁴	1	4	yes	—
234	C₂×C₇×C₈	C₂⁴	1	4	yes	—
235	C₇×C₄⋊C₄	—	2	2	no	—
236	C₇×C₄⋊C₄	—	2	4	no	—
237	C₇×C₄⋊C₄	—	2	4	no	—
238	C₇×C₄⋊C₄	—	2	2	no	—
239	C₇×C₄⋊C₄	—	2	2	no	—
240	C₇×C₄⋊C₄	—	2	2	no	—
241	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
242	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
243	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
244	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
245	C₇×C₂²⋊C₄	—	2	1	no	—
246	C₇×C₂²⋊C₄	—	2	1	no	—
247	C₇×C₂²⋊C₄	—	2	4	no	—
248	C₇×C₂²⋊C₄	—	2	4	no	—
249	C₇×C₂²⋊C₄	—	2	4	no	—
250	C₇×C₂²⋊C₄	—	2	4	no	—
251	C₇×C₂²⋊C₄	—	2	4	no	—
252	C₇×C₂²⋊C₄	—	2	8	no	—
253	C₇×C₂²⋊C₄	—	2	4	no	—
254	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
255	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
256	C₄²×C₇	—	2	4	no	—
257	C₄²×C₇	—	2	1	no	—
258	C₄²×C₇	—	2	1	no	—
259	C₄²×C₇	C₂×D₄	1	4	yes	—
260	C₄²×C₇	C₂×D₄	1	2	yes	—
261	C₄²×C₇	C₂⁴	1	1	yes	characteristic
262	C₄²×C₇	C₂⁴	1	1	yes	characteristic
263	C₄²×C₇	C₂⁴	1	2	yes	—
264	C₂×H(5,𝔽₂)	C₂²×C₇	1	2	yes	—
265	C₂×#137	C₂²×C₇	1	2	yes	complemented
266	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	2	yes	complemented
267	C₂²×SD₁₆	C₂²×C₇	1	2	yes	complemented
268	C₂²×D₈	C₂²×C₇	1	2	yes	complemented
269	#559	—	2	2	no	—
270	#556	—	2	2	no	—
271	#551	—	2	4	no	—
272	#550	—	2	2	no	—
273	D₄²	—	2	2	no	—
274	D₄²	—	2	4	no	—
275	#542	C₂²×C₇	1	4	yes	complemented
276	#536	C₂²×C₇	1	1	yes	characteristic
277	#535	C₂²×C₇	1	2	yes	—
278	#535	C₂²×C₇	1	1	yes	characteristic
279	#535	C₂²×C₇	1	4	yes	complemented
280	#533	C₂²×C₇	1	1	yes	characteristic, complemented
281	#532	C₂²×C₇	1	1	yes	characteristic
282	#532	C₂²×C₇	1	2	yes	complemented
283	C₂×#127	C₂²×C₇	1	1	yes	characteristic
284	C₂×#127	C₂²×C₇	1	2	yes	complemented
285	#529	C₂²×C₇	1	4	yes	complemented
286	C₂×#124	C₂²×C₇	1	1	yes	characteristic, complemented
287	#519	C₂²×C₇	1	1	yes	characteristic, complemented
288	#518	C₂²×C₇	1	1	yes	characteristic, complemented
289	C₄×#41	C₂²×C₇	1	1	yes	characteristic
290	#497	C₂²×C₇	1	2	yes	complemented
291	#496	C₂²×C₇	1	4	yes	complemented
292	#496	C₂²×C₇	1	4	yes	complemented
293	#496	C₂²×C₇	1	2	yes	complemented
294	#495	C₂²×C₇	1	4	yes	complemented
295	#491	C₂²×C₇	1	2	yes	complemented
296	#490	C₂²×C₇	1	2	yes	complemented
297	#490	C₂²×C₇	1	4	yes	complemented
298	#489	C₂²×C₇	1	4	yes	complemented
299	#486	C₂²×C₇	1	4	yes	complemented
300	#464	C₂²×C₇	1	8	yes	complemented
301	#463	C₂²×C₇	1	8	yes	complemented
302	#460	C₂²×C₇	1	8	yes	complemented
303	#459	C₂²×C₇	1	8	yes	complemented
304	#452	C₂²×C₇	1	8	yes	complemented
305	#450	C₂²×C₇	1	8	yes	complemented
306	#449	C₂²×C₇	1	8	yes	complemented
307	#447	C₂²×C₇	1	8	yes	complemented
308	#433	C₂²×C₇	1	4	yes	complemented
309	#432	C₂²×C₇	1	4	yes	complemented
310	#424	C₂²×C₇	1	2	yes	complemented
311	#418	C₂²×C₇	1	2	yes	complemented
312	#418	C₂²×C₇	1	2	yes	complemented
313	#416	C₂²×C₇	1	2	yes	complemented
314	#416	C₂²×C₇	1	2	yes	complemented
315	#408	C₂²×C₇	1	2	yes	—
316	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
317	C₂×#97	C₂²×C₇	1	1	yes	characteristic, complemented
318	C₂³×C₇	—	2	2	no	—
319	C₂³×C₇	—	2	4	no	—
320	C₂³×C₇	—	2	2	no	—
321	C₂³×C₇	—	4	8	no	—
322	C₂³×C₇	—	4	4	no	—
323	C₂³×C₇	—	4	2	no	—
324	C₂³×C₇	—	8	4	no	—
325	C₂³×C₇	—	8	8	no	—
326	C₂³×C₇	—	8	2	no	—
327	C₂³×C₇	C₂²×D₄	1	2	yes	—
328	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
329	C₇×Q₈	—	2	2	no	—
330	C₇×Q₈	—	2	4	no	—
331	C₇×Q₈	—	2	2	no	—
332	C₇×Q₈	—	2	2	no	—
333	C₇×Q₈	—	4	2	no	—
334	C₇×Q₈	—	8	2	no	—
335	C₇×D₄	—	2	4	no	—
336	C₇×D₄	—	2	2	no	—
337	C₇×D₄	—	2	4	no	—
338	C₇×D₄	—	2	4	no	—
339	C₇×D₄	—	2	2	no	—
340	C₇×D₄	—	2	4	no	—
341	C₇×D₄	—	2	2	no	—
342	C₇×D₄	—	4	8	no	—
343	C₇×D₄	—	4	4	no	—
344	C₇×D₄	—	4	8	no	—
345	C₇×D₄	—	4	8	no	—
346	C₇×D₄	—	4	4	no	—
347	C₇×D₄	—	4	2	no	—
348	C₇×D₄	—	4	4	no	—
349	C₇×D₄	—	8	8	no	—
350	C₇×D₄	—	8	4	no	—
351	C₇×D₄	—	8	4	no	—
352	C₂×C₄×C₇	—	2	4	no	—
353	C₂×C₄×C₇	—	2	2	no	—
354	C₂×C₄×C₇	—	2	2	no	—
355	C₂×C₄×C₇	—	2	2	no	—
356	C₂×C₄×C₇	—	2	2	no	—
357	C₂×C₄×C₇	—	4	4	no	—
358	C₂×C₄×C₇	—	4	2	no	—
359	C₂×C₄×C₇	—	4	8	no	—
360	C₂×C₄×C₇	—	4	2	no	—
361	C₂×C₄×C₇	—	4	4	no	—
362	C₂×C₄×C₇	—	8	4	no	—
363	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
364	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
365	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
366	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
367	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
368	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
369	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
370	C₇×C₈	—	2	4	no	—
371	C₇×C₈	—	2	4	no	—
372	H(5,𝔽₂)	—	4	4	no	—
373	H(5,𝔽₂)	—	4	4	no	—
374	C₂×#41	—	2	2	no	—
375	C₂×#41	C₂³×C₇	1	2	yes	—
376	C₂×#41	C₂³×C₇	1	4	yes	—
377	C₂×#41	C₂³×C₇	1	1	yes	characteristic
378	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
379	C₂²×D₄	—	2	4	no	—
380	C₂²×D₄	—	2	2	no	—
381	C₂²×D₄	—	2	2	no	—
382	C₂²×D₄	—	2	4	no	—
383	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
384	C₂²×D₄	C₂³×C₇	1	2	yes	—
385	C₂²×D₄	C₂³×C₇	1	4	yes	—
386	#137	—	4	4	no	—
387	AGL(1,ℤ/8)	—	4	4	no	—
388	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
389	C₂×SD₁₆	—	4	4	no	—
390	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
391	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
392	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
393	C₂×D₈	—	4	4	no	—
394	C₂×D₈	C₂³×C₇	1	8	yes	—
395	C₂×D₈	C₂³×C₇	1	4	yes	—
396	C₂×M₁₆	C₂³×C₇	1	2	yes	—
397	C₂²×C₈	C₂³×C₇	1	2	yes	—
398	C₄⋊Q₈	—	2	1	no	—
399	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
400	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
401	#127	—	2	1	no	—
402	#127	—	2	2	no	—
403	#127	—	2	4	no	—
404	#127	C₂³×C₇	1	2	yes	—
405	#127	C₂³×C₇	1	2	yes	—
406	#127	C₂³×C₇	1	1	yes	characteristic
407	#127	C₂³×C₇	1	4	yes	—
408	#126	—	2	4	no	—
409	#126	—	2	4	no	—
410	#125	C₂³×C₇	1	2	yes	—
411	#124	—	2	1	no	—
412	#124	—	2	1	no	—
413	#124	—	2	4	no	—
414	#124	—	2	2	no	—
415	#124	C₂³×C₇	1	2	yes	—
416	#124	C₂³×C₇	1	2	yes	—
417	#124	C₂³×C₇	1	2	yes	—
418	#124	C₂³×C₇	1	4	yes	—
419	#123	—	2	2	no	—
420	#123	—	2	4	no	—
421	#123	—	2	4	no	—
422	C₂²⋊Q₈	—	2	2	no	—
423	C₂²⋊Q₈	—	2	4	no	—
424	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
425	#121	—	2	2	no	—
426	#121	—	2	4	no	—
427	#121	—	2	4	no	—
428	#121	—	2	8	no	—
429	#121	C₂³×C₇	1	4	yes	—
430	#120	—	2	2	no	—
431	#120	—	2	4	no	—
432	#120	—	2	8	no	—
433	C₄×Q₈	—	2	2	no	—
434	C₄×Q₈	—	2	2	no	—
435	C₄×Q₈	C₇×D₄	1	2	yes	—
436	C₄×Q₈	C₂³×C₇	1	4	yes	—
437	C₄×D₄	—	2	4	no	—
438	C₄×D₄	—	2	2	no	—
439	C₄×D₄	—	2	2	no	—
440	C₄×D₄	—	2	4	no	—
441	C₄×D₄	C₇×D₄	1	4	yes	—
442	C₄×D₄	C₇×D₄	1	2	yes	—
443	C₄×D₄	C₂³×C₇	1	4	yes	—
444	#117	—	2	2	no	—
445	#117	—	2	4	no	—
446	#117	C₂³×C₇	1	1	yes	characteristic
447	#117	C₂³×C₇	1	1	yes	characteristic
448	#117	C₂³×C₇	1	1	yes	characteristic
449	#117	C₂³×C₇	1	2	yes	—
450	C₂×C₂²⋊C₄	—	2	2	no	—
451	C₂×C₂²⋊C₄	—	2	2	no	—
452	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
453	C₂×C₄²	C₂³×C₇	1	2	yes	—
454	C₄⋊C₈	C₂³×C₇	1	8	yes	—
455	#103	C₂³×C₇	1	4	yes	—
456	#103	C₂³×C₇	1	4	yes	—
457	#102	C₂³×C₇	1	4	yes	—
458	#102	C₂³×C₇	1	4	yes	—
459	#102	C₂³×C₇	1	8	yes	—
460	#102	C₂³×C₇	1	8	yes	—
461	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
462	#97	C₂³×C₇	1	2	yes	—
463	#97	C₂³×C₇	1	2	yes	—
464	#97	C₂³×C₇	1	2	yes	—
465	C₄×C₈	C₂³×C₇	1	2	yes	—
466	C₂²×C₇	—	2	2	no	—
467	C₂²×C₇	—	2	4	no	—
468	C₂²×C₇	—	2	2	no	—
469	C₂²×C₇	—	4	4	no	—
470	C₂²×C₇	—	4	8	no	—
471	C₂²×C₇	—	4	4	no	—
472	C₂²×C₇	—	4	4	no	—
473	C₂²×C₇	—	4	2	no	—
474	C₂²×C₇	—	4	2	no	—
475	C₂²×C₇	—	16	2	no	—
476	C₂²×C₇	—	16	4	no	—
477	C₂²×C₇	—	16	8	no	—
478	C₂²×C₇	#535	1	2	yes	—
479	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
480	C₄×C₇	—	2	2	no	—
481	C₄×C₇	—	2	4	no	—
482	C₄×C₇	—	2	2	no	—
483	C₄×C₇	—	2	2	no	—
484	C₄×C₇	—	4	2	no	—
485	C₄×C₇	—	4	2	no	—
486	C₄×C₇	—	4	2	no	—
487	C₄×C₇	—	4	2	no	—
488	C₄×C₇	#535	1	2	yes	—
489	C₄×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
490	C₂⁴	—	2	1	no	—
491	C₂⁴	—	2	4	no	—
492	C₂⁴	—	2	2	no	—
493	#41	—	2	4	no	—
494	#41	—	2	4	no	—
495	#41	—	2	4	no	—
496	#41	—	4	4	no	—
497	#41	—	4	4	no	—
498	#41	—	4	8	no	—
499	C₂×Q₈	—	2	1	no	—
500	C₂×Q₈	—	2	2	no	—
501	C₂×Q₈	—	2	4	no	—
502	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
503	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
504	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
505	C₂×D₄	—	2	4	no	—
506	C₂×D₄	—	2	2	no	—
507	C₂×D₄	—	2	4	no	—
508	C₂×D₄	—	2	1	no	—
509	C₂×D₄	—	2	2	no	—
510	C₂×D₄	—	2	4	no	—
511	C₂×D₄	—	2	4	no	—
512	C₂×D₄	—	2	2	no	—
513	C₂×D₄	—	2	8	no	—
514	C₂×D₄	—	2	4	no	—
515	C₂×D₄	—	2	4	no	—
516	C₂×D₄	—	2	4	no	—
517	C₂×D₄	—	2	4	no	—
518	C₂×D₄	—	4	4	no	—
519	C₂×D₄	—	4	4	no	—
520	C₂×D₄	—	4	16	no	—
521	C₂×D₄	—	4	8	no	—
522	C₂×D₄	—	8	4	no	—
523	C₂×D₄	—	8	8	no	—
524	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
525	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
526	C₂×D₄	C₂⁴×C₇	1	4	yes	—
527	C₂×D₄	C₂⁴×C₇	1	2	yes	—
528	C₂×D₄	C₂⁴×C₇	1	2	yes	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	2	4	no	—
531	C₂²×C₄	—	2	2	no	—
532	C₂²×C₄	—	2	4	no	—
533	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
534	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
535	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
536	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
537	Q₁₆	—	4	4	no	—
538	SD₁₆	—	4	8	no	—
539	SD₁₆	—	4	4	no	—
540	SD₁₆	—	4	4	no	—
541	D₈	—	4	8	no	—
542	D₈	—	4	4	no	—
543	M₁₆	—	2	4	no	—
544	C₂×C₈	—	2	4	no	—
545	C₂×C₈	C₂⁴×C₇	1	4	yes	—
546	C₂×C₈	C₂⁴×C₇	1	4	yes	—
547	C₄⋊C₄	—	2	2	no	—
548	C₄⋊C₄	—	2	4	no	—
549	C₄⋊C₄	—	2	4	no	—
550	C₄⋊C₄	—	2	2	no	—
551	C₄⋊C₄	—	2	2	no	—
552	C₄⋊C₄	—	2	2	no	—
553	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
554	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
555	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
556	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
557	C₂²⋊C₄	—	2	1	no	—
558	C₂²⋊C₄	—	2	1	no	—
559	C₂²⋊C₄	—	2	4	no	—
560	C₂²⋊C₄	—	2	4	no	—
561	C₂²⋊C₄	—	2	4	no	—
562	C₂²⋊C₄	—	2	4	no	—
563	C₂²⋊C₄	—	2	4	no	—
564	C₂²⋊C₄	—	2	8	no	—
565	C₂²⋊C₄	—	2	4	no	—
566	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
567	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
568	C₄²	—	2	4	no	—
569	C₄²	—	2	1	no	—
570	C₄²	—	2	1	no	—
571	C₄²	C₂×C₇×D₄	1	4	yes	—
572	C₄²	C₂×C₇×D₄	1	2	yes	—
573	C₄²	C₂⁴×C₇	1	1	yes	characteristic
574	C₄²	C₂⁴×C₇	1	1	yes	characteristic
575	C₄²	C₂⁴×C₇	1	2	yes	—
576	C₂×C₇	—	2	2	no	—
577	C₂×C₇	—	4	2	no	—
578	C₂×C₇	—	8	2	no	—
579	C₂×C₇	—	8	4	no	—
580	C₂×C₇	—	8	2	no	—
581	C₂×C₇	C₂×#535	1	1	yes	characteristic
582	C₂×C₇	#3585	1	2	yes	—
583	C₂³	—	2	2	no	—
584	C₂³	—	2	4	no	—
585	C₂³	—	2	2	no	—
586	C₂³	—	4	8	no	—
587	C₂³	—	4	4	no	—
588	C₂³	—	4	2	no	—
589	C₂³	—	8	4	no	—
590	C₂³	—	8	8	no	—
591	C₂³	—	8	2	no	—
592	C₂³	C₂²×C₇×D₄	1	2	yes	—
593	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
594	Q₈	—	2	2	no	—
595	Q₈	—	2	4	no	—
596	Q₈	—	2	2	no	—
597	Q₈	—	2	2	no	—
598	Q₈	—	4	2	no	—
599	Q₈	—	8	2	no	—
600	D₄	—	2	4	no	—
601	D₄	—	2	4	no	—
602	D₄	—	2	4	no	—
603	D₄	—	2	2	no	—
604	D₄	—	2	4	no	—
605	D₄	—	2	2	no	—
606	D₄	—	2	2	no	—
607	D₄	—	4	8	no	—
608	D₄	—	4	4	no	—
609	D₄	—	4	8	no	—
610	D₄	—	4	8	no	—
611	D₄	—	4	4	no	—
612	D₄	—	4	2	no	—
613	D₄	—	4	4	no	—
614	D₄	—	8	8	no	—
615	D₄	—	8	4	no	—
616	D₄	—	8	4	no	—
617	C₂×C₄	—	2	4	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	4	4	no	—
623	C₂×C₄	—	4	2	no	—
624	C₂×C₄	—	4	8	no	—
625	C₂×C₄	—	4	2	no	—
626	C₂×C₄	—	4	4	no	—
627	C₂×C₄	—	8	4	no	—
628	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
629	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
630	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
631	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
632	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
633	C₂×C₄	C₇×H(5,𝔽₂)	1	4	yes	—
634	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
635	C₈	—	2	4	no	—
636	C₈	—	2	4	no	—
637	C₇	#60827	1	1	yes	characteristic, direct product factor, Sylow
638	C₂²	—	2	2	no	—
639	C₂²	—	2	4	no	—
640	C₂²	—	2	2	no	—
641	C₂²	—	4	4	no	—
642	C₂²	—	4	8	no	—
643	C₂²	—	4	4	no	—
644	C₂²	—	4	4	no	—
645	C₂²	—	4	2	no	—
646	C₂²	—	4	2	no	—
647	C₂²	—	16	4	no	—
648	C₂²	—	16	2	no	—
649	C₂²	—	16	8	no	—
650	C₂²	C₇×#535	1	2	yes	—
651	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
652	C₄	—	2	2	no	—
653	C₄	—	2	4	no	—
654	C₄	—	2	2	no	—
655	C₄	—	2	2	no	—
656	C₄	—	4	2	no	—
657	C₄	—	4	2	no	—
658	C₄	—	4	2	no	—
659	C₄	—	4	2	no	—
660	C₄	C₇×#535	1	2	yes	—
661	C₄	C₂×C₇×H(5,𝔽₂)	1	2	yes	—
662	C₂	—	2	2	no	—
663	C₂	—	4	2	no	—
664	C₂	—	8	4	no	—
665	C₂	—	8	2	no	—
666	C₂	—	8	2	no	—
667	C₂	C₂×C₇×#535	1	1	yes	characteristic
668	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×#3510	C₂	semidirect
2. C₇×#3498	C₂	semidirect
3. C₇×#3224	C₂	semidirect
4. C₇×#3219	C₂	semidirect
5. C₇×#3217	C₂	semidirect
6. C₇×#3153	C₂	semidirect
7. C₂×C₇×#496	C₂	semidirect
8. C₇×#3138	C₂	semidirect
9. C₇×#3135	C₂	semidirect
10. C₇×#3043	C₂	semidirect
11. C₇×#3041	C₂	semidirect
12. C₇×#3009	C₂	semidirect
13. C₇×#3008	C₂	semidirect
14. C₇×#2923	C₂	semidirect
16. C₂×C₇×#137	C₂²	semidirect
17. C₂×C₇×AGL(1,ℤ/8)	C₂²	semidirect
18. C₂²×C₇×SD₁₆	C₂²	semidirect
19. C₂²×C₇×D₈	C₂²	semidirect
26. C₇×#542	C₂²	semidirect
30. C₇×#535	C₂²	semidirect
31. C₇×#533	C₂²	semidirect
33. C₇×#532	C₂²	semidirect
35. C₂×C₇×#127	C₂²	semidirect
36. C₇×#529	C₂²	semidirect
37. C₂×C₇×#124	C₂²	semidirect
38. C₇×#519	C₂²	semidirect
39. C₇×#518	C₂²	semidirect
41. C₇×#497	C₂²	semidirect
42. C₇×#496	C₂²	semidirect
43. C₇×#496	C₂²	semidirect
44. C₇×#496	C₂²	semidirect
45. C₇×#495	C₂²	semidirect
46. C₇×#491	C₂²	semidirect
47. C₇×#490	C₂²	semidirect
48. C₇×#490	C₂²	semidirect
49. C₇×#489	C₂²	semidirect
50. C₇×#486	C₂²	semidirect
51. C₇×#464	C₂²	semidirect
52. C₇×#463	C₂²	semidirect
53. C₇×#460	C₂²	semidirect
54. C₇×#459	C₂²	semidirect
55. C₇×#452	C₂²	semidirect
56. C₇×#450	C₂²	semidirect
57. C₇×#449	C₂²	semidirect
58. C₇×#447	C₂²	semidirect
59. C₇×#433	C₂²	semidirect
60. C₇×#432	C₂²	semidirect
61. C₇×#424	C₂²	semidirect
62. C₇×#418	C₂²	semidirect
63. C₇×#418	C₂²	semidirect
64. C₇×#416	C₂²	semidirect
65. C₇×#416	C₂²	semidirect
67. C₄×C₇×M₁₆	C₂²	semidirect
68. C₂×C₇×#97	C₂²	semidirect
69. #60827	C₇	direct (N × H)
164. #3510	C₂×C₇	semidirect
165. #3498	C₂×C₇	semidirect
166. #3224	C₂×C₇	semidirect
167. #3219	C₂×C₇	semidirect
168. #3217	C₂×C₇	semidirect
169. #3153	C₂×C₇	semidirect
170. C₂×#496	C₂×C₇	semidirect
171. #3138	C₂×C₇	semidirect
172. #3135	C₂×C₇	semidirect
173. #3043	C₂×C₇	semidirect
174. #3041	C₂×C₇	semidirect
175. #3009	C₂×C₇	semidirect
176. #3008	C₂×C₇	semidirect
177. #2923	C₂×C₇	semidirect
265. C₂×#137	C₂²×C₇	semidirect
266. C₂×AGL(1,ℤ/8)	C₂²×C₇	semidirect
267. C₂²×SD₁₆	C₂²×C₇	semidirect
268. C₂²×D₈	C₂²×C₇	semidirect
275. #542	C₂²×C₇	semidirect
279. #535	C₂²×C₇	semidirect
280. #533	C₂²×C₇	semidirect
282. #532	C₂²×C₇	semidirect
284. C₂×#127	C₂²×C₇	semidirect
285. #529	C₂²×C₇	semidirect
286. C₂×#124	C₂²×C₇	semidirect
287. #519	C₂²×C₇	semidirect
288. #518	C₂²×C₇	semidirect
290. #497	C₂²×C₇	semidirect
291. #496	C₂²×C₇	semidirect
292. #496	C₂²×C₇	semidirect
293. #496	C₂²×C₇	semidirect
294. #495	C₂²×C₇	semidirect
295. #491	C₂²×C₇	semidirect
296. #490	C₂²×C₇	semidirect
297. #490	C₂²×C₇	semidirect
298. #489	C₂²×C₇	semidirect
299. #486	C₂²×C₇	semidirect
300. #464	C₂²×C₇	semidirect
301. #463	C₂²×C₇	semidirect
302. #460	C₂²×C₇	semidirect
303. #459	C₂²×C₇	semidirect
304. #452	C₂²×C₇	semidirect
305. #450	C₂²×C₇	semidirect
306. #449	C₂²×C₇	semidirect
307. #447	C₂²×C₇	semidirect
308. #433	C₂²×C₇	semidirect
309. #432	C₂²×C₇	semidirect
310. #424	C₂²×C₇	semidirect
311. #418	C₂²×C₇	semidirect
312. #418	C₂²×C₇	semidirect
313. #416	C₂²×C₇	semidirect
314. #416	C₂²×C₇	semidirect
316. C₄×M₁₆	C₂²×C₇	semidirect
317. C₂×#97	C₂²×C₇	semidirect
637. C₇	#60827	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×#3510	2
C₇×#3498	2
C₇×#3224	2
C₇×#3219	2
C₇×#3217	2
C₇×#3153	2
C₂×C₇×#496	2
C₇×#3138	2
C₇×#3135	2
C₇×#3043	2
C₇×#3041	2
C₇×#3009	2
C₇×#3008	2
C₇×#2923	2
#60827	7

Subgroups — C7×#60827 #13238043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60827 #13238043