Subgroups — C₇×#60830 #13238046

Order 1792.

2150 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3512	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3222	C₂	1	4	yes	maximal, complemented
4	C₇×#3220	C₂	1	4	yes	maximal, complemented
5	C₇×#3219	C₂	1	4	yes	maximal, complemented
6	C₇×#3217	C₂	1	4	yes	maximal, complemented
7	C₇×#3151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3150	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3142	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×#490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#3045	C₂	1	4	yes	maximal, complemented
12	C₇×#3012	C₂	1	4	yes	maximal, complemented
13	C₇×#2923	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₇×H(5,𝔽₂)	C₂²	1	2	yes	—
15	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
16	C₂×C₇×#135	C₂²	1	4	yes	complemented
17	C₇×#553	—	2	4	no	—
18	C₇×#553	—	2	2	no	—
19	C₇×#552	—	2	2	no	—
20	C₇×#550	—	2	2	no	—
21	C₇×D₄²	—	2	4	no	—
22	C₇×D₄²	—	2	2	no	—
23	C₇×#541	C₂²	1	2	yes	complemented
24	C₇×#540	C₂²	1	2	yes	complemented
25	C₇×#537	C₂²	1	2	yes	—
26	C₇×#535	C₂²	1	4	yes	—
27	C₇×#535	C₂²	1	2	yes	—
28	C₇×#532	C₂²	1	2	yes	—
29	C₇×#532	C₂²	1	1	yes	characteristic
30	C₇×#532	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₇×#127	C₂²	1	2	yes	—
32	C₂×C₇×#127	C₂²	1	1	yes	characteristic
33	C₇×#529	C₂²	1	4	yes	complemented
34	C₂×C₇×#125	C₂²	1	1	yes	characteristic, complemented
35	C₇×#520	C₂²	1	2	yes	complemented
36	C₄×C₇×#41	C₂²	1	1	yes	characteristic
37	C₇×#497	C₂²	1	2	yes	complemented
38	C₇×#497	C₂²	1	2	yes	complemented
39	C₇×#496	C₂²	1	2	yes	complemented
40	C₇×#496	C₂²	1	4	yes	complemented
41	C₇×#496	C₂²	1	2	yes	complemented
42	C₇×#493	C₂²	1	2	yes	complemented
43	C₇×#492	C₂²	1	2	yes	complemented
44	C₇×#491	C₂²	1	2	yes	complemented
45	C₇×#490	C₂²	1	2	yes	complemented
46	C₇×#490	C₂²	1	4	yes	complemented
47	C₇×#490	C₂²	1	4	yes	complemented
48	C₇×#488	C₂²	1	4	yes	complemented
49	C₇×#464	C₂²	1	8	yes	complemented
50	C₇×#463	C₂²	1	8	yes	complemented
51	C₇×#460	C₂²	1	8	yes	complemented
52	C₇×#459	C₂²	1	8	yes	complemented
53	C₇×#452	C₂²	1	8	yes	complemented
54	C₇×#450	C₂²	1	8	yes	complemented
55	C₇×#449	C₂²	1	8	yes	complemented
56	C₇×#447	C₂²	1	8	yes	complemented
57	C₇×#433	C₂²	1	4	yes	complemented
58	C₇×#432	C₂²	1	4	yes	complemented
59	C₇×#424	C₂²	1	2	yes	complemented
60	C₇×#417	C₂²	1	4	yes	complemented
61	C₂×C₇×#102	C₂²	1	4	yes	complemented
62	C₇×#408	C₂²	1	2	yes	—
63	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic, complemented
64	C₂×C₇×#97	C₂²	1	1	yes	characteristic, complemented
65	#60830	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₇×H(5,𝔽₂)	—	4	8	no	—
67	C₂×C₇×#41	—	2	2	no	—
68	C₂×C₇×#41	C₂³	1	2	yes	—
69	C₂×C₇×#41	C₂³	1	2	yes	—
70	C₂×C₇×#41	C₂³	1	4	yes	—
71	C₂×C₇×#41	C₂³	1	1	yes	characteristic
72	C₂²×C₇×D₄	—	2	4	no	—
73	C₂²×C₇×D₄	—	2	2	no	—
74	C₂²×C₇×D₄	—	2	2	no	—
75	C₂²×C₇×D₄	—	2	4	no	—
76	C₂²×C₇×D₄	C₂³	1	2	yes	—
77	C₂²×C₇×D₄	C₂³	1	4	yes	—
78	C₇×AGL(1,ℤ/8)	—	4	8	no	—
79	C₇×#135	—	4	8	no	—
80	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
81	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
82	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
83	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
84	C₂×C₇×D₈	C₂³	1	8	yes	—
85	C₂×C₇×D₈	C₂³	1	4	yes	—
86	C₂×C₇×M₁₆	C₂³	1	2	yes	—
87	C₂²×C₇×C₈	C₂³	1	2	yes	—
88	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
90	C₇×#127	—	2	4	no	—
91	C₇×#127	—	2	2	no	—
92	C₇×#127	C₂³	1	2	yes	—
93	C₇×#127	C₂³	1	1	yes	characteristic
94	C₇×#127	C₂³	1	2	yes	—
95	C₇×#127	C₂³	1	1	yes	characteristic
96	C₇×#127	C₂³	1	4	yes	—
97	C₇×#126	—	2	4	no	—
98	C₇×#126	—	2	4	no	—
99	C₇×#125	—	2	4	no	—
100	C₇×#125	—	2	2	no	—
101	C₇×#125	C₂³	1	2	yes	—
102	C₇×#125	C₂³	1	2	yes	—
103	C₇×#124	C₂³	1	2	yes	—
104	C₇×#124	C₂³	1	2	yes	—
105	C₇×#124	C₂³	1	4	yes	—
106	C₇×#123	—	2	4	no	—
107	C₇×#123	—	2	4	no	—
108	C₇×C₂²⋊Q₈	—	2	2	no	—
109	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
110	C₇×#121	—	2	2	no	—
111	C₇×#121	—	2	4	no	—
112	C₇×#121	—	2	2	no	—
113	C₇×#121	—	2	4	no	—
114	C₇×#121	—	2	2	no	—
115	C₇×#121	—	2	4	no	—
116	C₇×#121	—	2	8	no	—
117	C₇×#121	—	2	4	no	—
118	C₇×#121	C₂³	1	4	yes	—
119	C₇×#120	—	2	4	no	—
120	C₇×#120	—	2	8	no	—
121	C₄×C₇×Q₈	D₄	1	2	yes	—
122	C₄×C₇×Q₈	C₂³	1	4	yes	—
123	C₄×C₇×D₄	—	2	4	no	—
124	C₄×C₇×D₄	—	2	4	no	—
125	C₄×C₇×D₄	—	2	4	no	—
126	C₄×C₇×D₄	—	2	2	no	—
127	C₄×C₇×D₄	—	2	2	no	—
128	C₄×C₇×D₄	D₄	1	4	yes	—
129	C₄×C₇×D₄	D₄	1	2	yes	—
130	C₄×C₇×D₄	C₂³	1	4	yes	—
131	C₇×#117	—	2	2	no	—
132	C₇×#117	—	2	4	no	—
133	C₇×#117	C₂³	1	1	yes	characteristic
134	C₇×#117	C₂³	1	2	yes	—
135	C₂×C₇×C₄⋊C₄	—	2	2	no	—
136	C₂×C₇×C₄⋊C₄	—	2	2	no	—
137	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
138	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₇	C₂³	1	2	yes	—
140	C₇×C₄⋊C₈	C₂³	1	8	yes	—
141	C₇×#103	C₂³	1	8	yes	—
142	C₇×#102	C₂³	1	8	yes	—
143	C₇×#102	C₂³	1	8	yes	—
144	C₇×#102	C₂³	1	8	yes	—
145	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
146	C₇×#97	C₂³	1	2	yes	—
147	C₇×#97	C₂³	1	2	yes	—
148	C₇×#97	C₂³	1	2	yes	—
149	C₄×C₇×C₈	C₂³	1	2	yes	—
150	#3512	C₂×C₇	1	1	yes	characteristic, complemented
151	#3498	C₂×C₇	1	1	yes	characteristic, complemented
152	#3222	C₂×C₇	1	4	yes	complemented
153	#3220	C₂×C₇	1	4	yes	complemented
154	#3219	C₂×C₇	1	4	yes	complemented
155	#3217	C₂×C₇	1	4	yes	complemented
156	#3151	C₂×C₇	1	1	yes	characteristic, complemented
157	#3150	C₂×C₇	1	1	yes	characteristic, complemented
158	#3142	C₂×C₇	1	1	yes	characteristic, complemented
159	C₂×#490	C₂×C₇	1	1	yes	characteristic, complemented
160	#3045	C₂×C₇	1	4	yes	complemented
161	#3012	C₂×C₇	1	4	yes	complemented
162	#2923	C₂×C₇	1	1	yes	characteristic, complemented
163	C₂⁴×C₇	—	2	4	no	—
164	C₂⁴×C₇	—	2	2	no	—
165	C₇×#41	—	2	4	no	—
166	C₇×#41	—	2	8	no	—
167	C₇×#41	—	2	8	no	—
168	C₇×#41	—	4	8	no	—
169	C₇×#41	—	4	8	no	—
170	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
171	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
172	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
173	C₂×C₇×D₄	—	2	2	no	—
174	C₂×C₇×D₄	—	2	4	no	—
175	C₂×C₇×D₄	—	2	2	no	—
176	C₂×C₇×D₄	—	2	4	no	—
177	C₂×C₇×D₄	—	2	4	no	—
178	C₂×C₇×D₄	—	2	2	no	—
179	C₂×C₇×D₄	—	2	4	no	—
180	C₂×C₇×D₄	—	2	2	no	—
181	C₂×C₇×D₄	—	2	2	no	—
182	C₂×C₇×D₄	—	2	8	no	—
183	C₂×C₇×D₄	—	2	4	no	—
184	C₂×C₇×D₄	—	2	8	no	—
185	C₂×C₇×D₄	—	4	8	no	—
186	C₂×C₇×D₄	—	4	16	no	—
187	C₂×C₇×D₄	—	4	8	no	—
188	C₂×C₇×D₄	—	8	8	no	—
189	C₂×C₇×D₄	—	8	4	no	—
190	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
191	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
192	C₂×C₇×D₄	C₂⁴	1	4	yes	—
193	C₂×C₇×D₄	C₂⁴	1	2	yes	—
194	C₂×C₇×D₄	C₂⁴	1	2	yes	—
195	C₂²×C₄×C₇	—	2	2	no	—
196	C₂²×C₄×C₇	—	2	2	no	—
197	C₂²×C₄×C₇	—	2	4	no	—
198	C₂²×C₄×C₇	—	2	2	no	—
199	C₂²×C₄×C₇	—	2	4	no	—
200	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
201	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
202	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
203	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
204	C₇×Q₁₆	—	4	4	no	—
205	C₇×SD₁₆	—	4	4	no	—
206	C₇×SD₁₆	—	4	4	no	—
207	C₇×SD₁₆	—	4	8	no	—
208	C₇×D₈	—	4	4	no	—
209	C₇×D₈	—	4	8	no	—
210	C₇×M₁₆	—	2	4	no	—
211	C₂×C₇×C₈	—	2	4	no	—
212	C₂×C₇×C₈	C₂⁴	1	4	yes	—
213	C₂×C₇×C₈	C₂⁴	1	4	yes	—
214	C₇×C₄⋊C₄	—	2	2	no	—
215	C₇×C₄⋊C₄	—	2	4	no	—
216	C₇×C₄⋊C₄	—	2	4	no	—
217	C₇×C₄⋊C₄	—	2	4	no	—
218	C₇×C₄⋊C₄	—	2	4	no	—
219	C₇×C₄⋊C₄	—	2	2	no	—
220	C₇×C₄⋊C₄	—	2	2	no	—
221	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
222	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
223	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
224	C₇×C₂²⋊C₄	—	2	4	no	—
225	C₇×C₂²⋊C₄	—	2	2	no	—
226	C₇×C₂²⋊C₄	—	2	4	no	—
227	C₇×C₂²⋊C₄	—	2	4	no	—
228	C₇×C₂²⋊C₄	—	2	2	no	—
229	C₇×C₂²⋊C₄	—	2	8	no	—
230	C₇×C₂²⋊C₄	—	2	4	no	—
231	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
232	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
233	C₄²×C₇	—	2	4	no	—
234	C₄²×C₇	C₂×D₄	1	4	yes	—
235	C₄²×C₇	C₂×D₄	1	2	yes	—
236	C₄²×C₇	C₂⁴	1	1	yes	characteristic
237	C₄²×C₇	C₂⁴	1	1	yes	characteristic
238	C₄²×C₇	C₂⁴	1	2	yes	—
239	C₂×H(5,𝔽₂)	C₂²×C₇	1	2	yes	—
240	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	4	yes	complemented
241	C₂×#135	C₂²×C₇	1	4	yes	complemented
242	#553	—	2	4	no	—
243	#553	—	2	2	no	—
244	#552	—	2	2	no	—
245	#550	—	2	2	no	—
246	D₄²	—	2	4	no	—
247	D₄²	—	2	2	no	—
248	#541	C₂²×C₇	1	2	yes	complemented
249	#540	C₂²×C₇	1	2	yes	complemented
250	#537	C₂²×C₇	1	2	yes	—
251	#535	C₂²×C₇	1	4	yes	—
252	#535	C₂²×C₇	1	2	yes	—
253	#532	C₂²×C₇	1	1	yes	characteristic
254	#532	C₂²×C₇	1	2	yes	—
255	#532	C₂²×C₇	1	1	yes	characteristic, complemented
256	C₂×#127	C₂²×C₇	1	1	yes	characteristic
257	C₂×#127	C₂²×C₇	1	2	yes	—
258	#529	C₂²×C₇	1	4	yes	complemented
259	C₂×#125	C₂²×C₇	1	1	yes	characteristic, complemented
260	#520	C₂²×C₇	1	2	yes	complemented
261	C₄×#41	C₂²×C₇	1	1	yes	characteristic
262	#497	C₂²×C₇	1	2	yes	complemented
263	#497	C₂²×C₇	1	2	yes	complemented
264	#496	C₂²×C₇	1	2	yes	complemented
265	#496	C₂²×C₇	1	2	yes	complemented
266	#496	C₂²×C₇	1	4	yes	complemented
267	#493	C₂²×C₇	1	2	yes	complemented
268	#492	C₂²×C₇	1	2	yes	complemented
269	#491	C₂²×C₇	1	2	yes	complemented
270	#490	C₂²×C₇	1	4	yes	complemented
271	#490	C₂²×C₇	1	4	yes	complemented
272	#490	C₂²×C₇	1	2	yes	complemented
273	#488	C₂²×C₇	1	4	yes	complemented
274	#464	C₂²×C₇	1	8	yes	complemented
275	#463	C₂²×C₇	1	8	yes	complemented
276	#460	C₂²×C₇	1	8	yes	complemented
277	#459	C₂²×C₇	1	8	yes	complemented
278	#452	C₂²×C₇	1	8	yes	complemented
279	#450	C₂²×C₇	1	8	yes	complemented
280	#449	C₂²×C₇	1	8	yes	complemented
281	#447	C₂²×C₇	1	8	yes	complemented
282	#433	C₂²×C₇	1	4	yes	complemented
283	#432	C₂²×C₇	1	4	yes	complemented
284	#424	C₂²×C₇	1	2	yes	complemented
285	#417	C₂²×C₇	1	4	yes	complemented
286	C₂×#102	C₂²×C₇	1	4	yes	complemented
287	#408	C₂²×C₇	1	2	yes	—
288	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
289	C₂×#97	C₂²×C₇	1	1	yes	characteristic, complemented
290	C₂³×C₇	—	2	2	no	—
291	C₂³×C₇	—	2	4	no	—
292	C₂³×C₇	—	2	2	no	—
293	C₂³×C₇	—	4	8	no	—
294	C₂³×C₇	—	4	4	no	—
295	C₂³×C₇	—	8	4	no	—
296	C₂³×C₇	—	8	8	no	—
297	C₂³×C₇	C₂²×D₄	1	2	yes	—
298	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
299	C₇×Q₈	—	2	2	no	—
300	C₇×Q₈	—	2	4	no	—
301	C₇×Q₈	—	2	4	no	—
302	C₇×D₄	—	2	4	no	—
303	C₇×D₄	—	2	2	no	—
304	C₇×D₄	—	2	4	no	—
305	C₇×D₄	—	2	8	no	—
306	C₇×D₄	—	2	4	no	—
307	C₇×D₄	—	4	8	no	—
308	C₇×D₄	—	4	8	no	—
309	C₇×D₄	—	4	4	no	—
310	C₇×D₄	—	4	8	no	—
311	C₇×D₄	—	4	4	no	—
312	C₇×D₄	—	4	4	no	—
313	C₇×D₄	—	4	4	no	—
314	C₇×D₄	—	8	2	no	—
315	C₇×D₄	—	8	8	no	—
316	C₇×D₄	—	8	4	no	—
317	C₇×D₄	—	8	4	no	—
318	C₂×C₄×C₇	—	2	4	no	—
319	C₂×C₄×C₇	—	2	2	no	—
320	C₂×C₄×C₇	—	2	4	no	—
321	C₂×C₄×C₇	—	2	2	no	—
322	C₂×C₄×C₇	—	4	4	no	—
323	C₂×C₄×C₇	—	4	2	no	—
324	C₂×C₄×C₇	—	4	8	no	—
325	C₂×C₄×C₇	—	4	4	no	—
326	C₂×C₄×C₇	—	4	4	no	—
327	C₂×C₄×C₇	—	8	4	no	—
328	C₂×C₄×C₇	—	8	2	no	—
329	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
330	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
331	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
332	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
333	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
334	C₂×C₄×C₇	H(5,𝔽₂)	1	4	yes	—
335	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
336	C₇×C₈	—	2	4	no	—
337	C₇×C₈	—	2	4	no	—
338	H(5,𝔽₂)	—	4	8	no	—
339	C₂×#41	—	2	2	no	—
340	C₂×#41	C₂³×C₇	1	2	yes	—
341	C₂×#41	C₂³×C₇	1	2	yes	—
342	C₂×#41	C₂³×C₇	1	4	yes	—
343	C₂×#41	C₂³×C₇	1	1	yes	characteristic
344	C₂²×D₄	—	2	4	no	—
345	C₂²×D₄	—	2	2	no	—
346	C₂²×D₄	—	2	2	no	—
347	C₂²×D₄	—	2	4	no	—
348	C₂²×D₄	C₂³×C₇	1	2	yes	—
349	C₂²×D₄	C₂³×C₇	1	4	yes	—
350	AGL(1,ℤ/8)	—	4	8	no	—
351	#135	—	4	8	no	—
352	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
353	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
354	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
355	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
356	C₂×D₈	C₂³×C₇	1	4	yes	—
357	C₂×D₈	C₂³×C₇	1	8	yes	—
358	C₂×M₁₆	C₂³×C₇	1	2	yes	—
359	C₂²×C₈	C₂³×C₇	1	2	yes	—
360	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
361	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
362	#127	—	2	4	no	—
363	#127	—	2	2	no	—
364	#127	C₂³×C₇	1	2	yes	—
365	#127	C₂³×C₇	1	1	yes	characteristic
366	#127	C₂³×C₇	1	4	yes	—
367	#127	C₂³×C₇	1	2	yes	—
368	#127	C₂³×C₇	1	1	yes	characteristic
369	#126	—	2	4	no	—
370	#126	—	2	4	no	—
371	#125	—	2	4	no	—
372	#125	—	2	2	no	—
373	#125	C₂³×C₇	1	2	yes	—
374	#125	C₂³×C₇	1	2	yes	—
375	#124	C₂³×C₇	1	2	yes	—
376	#124	C₂³×C₇	1	4	yes	—
377	#124	C₂³×C₇	1	2	yes	—
378	#123	—	2	4	no	—
379	#123	—	2	4	no	—
380	C₂²⋊Q₈	—	2	2	no	—
381	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
382	#121	—	2	2	no	—
383	#121	—	2	4	no	—
384	#121	—	2	2	no	—
385	#121	—	2	4	no	—
386	#121	—	2	2	no	—
387	#121	—	2	4	no	—
388	#121	—	2	8	no	—
389	#121	—	2	4	no	—
390	#121	C₂³×C₇	1	4	yes	—
391	#120	—	2	4	no	—
392	#120	—	2	8	no	—
393	C₄×Q₈	C₇×D₄	1	2	yes	—
394	C₄×Q₈	C₂³×C₇	1	4	yes	—
395	C₄×D₄	—	2	4	no	—
396	C₄×D₄	—	2	4	no	—
397	C₄×D₄	—	2	4	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	C₇×D₄	1	4	yes	—
401	C₄×D₄	C₇×D₄	1	2	yes	—
402	C₄×D₄	C₂³×C₇	1	4	yes	—
403	#117	—	2	2	no	—
404	#117	—	2	4	no	—
405	#117	C₂³×C₇	1	1	yes	characteristic
406	#117	C₂³×C₇	1	2	yes	—
407	C₂×C₄⋊C₄	—	2	2	no	—
408	C₂×C₄⋊C₄	—	2	2	no	—
409	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
410	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
411	C₂×C₄²	C₂³×C₇	1	2	yes	—
412	C₄⋊C₈	C₂³×C₇	1	8	yes	—
413	#103	C₂³×C₇	1	8	yes	—
414	#102	C₂³×C₇	1	8	yes	—
415	#102	C₂³×C₇	1	8	yes	—
416	#102	C₂³×C₇	1	8	yes	—
417	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
418	#97	C₂³×C₇	1	2	yes	—
419	#97	C₂³×C₇	1	2	yes	—
420	#97	C₂³×C₇	1	2	yes	—
421	C₄×C₈	C₂³×C₇	1	2	yes	—
422	C₂²×C₇	—	2	2	no	—
423	C₂²×C₇	—	2	4	no	—
424	C₂²×C₇	—	2	2	no	—
425	C₂²×C₇	—	4	2	no	—
426	C₂²×C₇	—	4	8	no	—
427	C₂²×C₇	—	4	4	no	—
428	C₂²×C₇	—	4	2	no	—
429	C₂²×C₇	—	4	4	no	—
430	C₂²×C₇	—	4	2	no	—
431	C₂²×C₇	—	4	2	no	—
432	C₂²×C₇	—	16	4	no	—
433	C₂²×C₇	—	16	8	no	—
434	C₂²×C₇	#535	1	2	yes	—
435	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
436	C₄×C₇	—	2	2	no	—
437	C₄×C₇	—	2	4	no	—
438	C₄×C₇	—	2	2	no	—
439	C₄×C₇	—	2	2	no	—
440	C₄×C₇	—	4	4	no	—
441	C₄×C₇	—	4	2	no	—
442	C₄×C₇	—	4	2	no	—
443	C₄×C₇	#535	1	2	yes	—
444	C₄×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
445	C₂⁴	—	2	4	no	—
446	C₂⁴	—	2	2	no	—
447	#41	—	2	4	no	—
448	#41	—	2	8	no	—
449	#41	—	2	8	no	—
450	#41	—	4	8	no	—
451	#41	—	4	8	no	—
452	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
453	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
454	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	4	no	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	4	no	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	—	2	4	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	—	2	2	no	—
464	C₂×D₄	—	2	8	no	—
465	C₂×D₄	—	2	4	no	—
466	C₂×D₄	—	2	8	no	—
467	C₂×D₄	—	4	8	no	—
468	C₂×D₄	—	4	16	no	—
469	C₂×D₄	—	4	8	no	—
470	C₂×D₄	—	8	8	no	—
471	C₂×D₄	—	8	4	no	—
472	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
473	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
474	C₂×D₄	C₂⁴×C₇	1	4	yes	—
475	C₂×D₄	C₂⁴×C₇	1	2	yes	—
476	C₂×D₄	C₂⁴×C₇	1	2	yes	—
477	C₂²×C₄	—	2	2	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	4	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	4	no	—
482	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
483	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
484	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
485	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
486	Q₁₆	—	4	4	no	—
487	SD₁₆	—	4	4	no	—
488	SD₁₆	—	4	4	no	—
489	SD₁₆	—	4	8	no	—
490	D₈	—	4	4	no	—
491	D₈	—	4	8	no	—
492	M₁₆	—	2	4	no	—
493	C₂×C₈	—	2	4	no	—
494	C₂×C₈	C₂⁴×C₇	1	4	yes	—
495	C₂×C₈	C₂⁴×C₇	1	4	yes	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	4	no	—
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	—	2	4	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
504	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
505	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	8	no	—
512	C₂²⋊C₄	—	2	4	no	—
513	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
514	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
515	C₄²	—	2	4	no	—
516	C₄²	C₂×C₇×D₄	1	4	yes	—
517	C₄²	C₂×C₇×D₄	1	2	yes	—
518	C₄²	C₂⁴×C₇	1	1	yes	characteristic
519	C₄²	C₂⁴×C₇	1	1	yes	characteristic
520	C₄²	C₂⁴×C₇	1	2	yes	—
521	C₂×C₇	—	2	2	no	—
522	C₂×C₇	—	4	2	no	—
523	C₂×C₇	—	8	2	no	—
524	C₂×C₇	—	8	4	no	—
525	C₂×C₇	—	8	2	no	—
526	C₂×C₇	C₂×#535	1	1	yes	characteristic
527	C₂×C₇	#3585	1	2	yes	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	4	no	—
530	C₂³	—	2	2	no	—
531	C₂³	—	4	8	no	—
532	C₂³	—	4	4	no	—
533	C₂³	—	8	4	no	—
534	C₂³	—	8	8	no	—
535	C₂³	C₂²×C₇×D₄	1	2	yes	—
536	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
537	Q₈	—	2	2	no	—
538	Q₈	—	2	4	no	—
539	Q₈	—	2	4	no	—
540	D₄	—	2	4	no	—
541	D₄	—	2	4	no	—
542	D₄	—	2	8	no	—
543	D₄	—	2	4	no	—
544	D₄	—	2	2	no	—
545	D₄	—	4	8	no	—
546	D₄	—	4	8	no	—
547	D₄	—	4	4	no	—
548	D₄	—	4	8	no	—
549	D₄	—	4	4	no	—
550	D₄	—	4	4	no	—
551	D₄	—	4	4	no	—
552	D₄	—	8	2	no	—
553	D₄	—	8	8	no	—
554	D₄	—	8	4	no	—
555	D₄	—	8	4	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	4	4	no	—
561	C₂×C₄	—	4	2	no	—
562	C₂×C₄	—	4	8	no	—
563	C₂×C₄	—	4	4	no	—
564	C₂×C₄	—	4	4	no	—
565	C₂×C₄	—	8	4	no	—
566	C₂×C₄	—	8	2	no	—
567	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
568	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
569	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
570	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
571	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
572	C₂×C₄	C₇×H(5,𝔽₂)	1	4	yes	—
573	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
574	C₈	—	2	4	no	—
575	C₈	—	2	4	no	—
576	C₇	#60830	1	1	yes	characteristic, direct product factor, Sylow
577	C₂²	—	2	2	no	—
578	C₂²	—	2	4	no	—
579	C₂²	—	2	2	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	4	8	no	—
582	C₂²	—	4	4	no	—
583	C₂²	—	4	2	no	—
584	C₂²	—	4	4	no	—
585	C₂²	—	4	2	no	—
586	C₂²	—	4	2	no	—
587	C₂²	—	16	4	no	—
588	C₂²	—	16	8	no	—
589	C₂²	C₇×#535	1	2	yes	—
590	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
591	C₄	—	2	2	no	—
592	C₄	—	2	4	no	—
593	C₄	—	2	2	no	—
594	C₄	—	2	2	no	—
595	C₄	—	4	4	no	—
596	C₄	—	4	2	no	—
597	C₄	—	4	2	no	—
598	C₄	C₇×#535	1	2	yes	—
599	C₄	C₂×C₇×H(5,𝔽₂)	1	2	yes	—
600	C₂	—	2	2	no	—
601	C₂	—	4	2	no	—
602	C₂	—	8	4	no	—
603	C₂	—	8	2	no	—
604	C₂	—	8	2	no	—
605	C₂	C₂×C₇×#535	1	1	yes	characteristic
606	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×#3512	C₂	semidirect
2. C₇×#3498	C₂	semidirect
3. C₇×#3222	C₂	semidirect
4. C₇×#3220	C₂	semidirect
5. C₇×#3219	C₂	semidirect
6. C₇×#3217	C₂	semidirect
7. C₇×#3151	C₂	semidirect
8. C₇×#3150	C₂	semidirect
9. C₇×#3142	C₂	semidirect
10. C₂×C₇×#490	C₂	semidirect
11. C₇×#3045	C₂	semidirect
12. C₇×#3012	C₂	semidirect
13. C₇×#2923	C₂	semidirect
15. C₂×C₇×AGL(1,ℤ/8)	C₂²	semidirect
16. C₂×C₇×#135	C₂²	semidirect
23. C₇×#541	C₂²	semidirect
24. C₇×#540	C₂²	semidirect
30. C₇×#532	C₂²	semidirect
33. C₇×#529	C₂²	semidirect
34. C₂×C₇×#125	C₂²	semidirect
35. C₇×#520	C₂²	semidirect
37. C₇×#497	C₂²	semidirect
38. C₇×#497	C₂²	semidirect
39. C₇×#496	C₂²	semidirect
40. C₇×#496	C₂²	semidirect
41. C₇×#496	C₂²	semidirect
42. C₇×#493	C₂²	semidirect
43. C₇×#492	C₂²	semidirect
44. C₇×#491	C₂²	semidirect
45. C₇×#490	C₂²	semidirect
46. C₇×#490	C₂²	semidirect
47. C₇×#490	C₂²	semidirect
48. C₇×#488	C₂²	semidirect
49. C₇×#464	C₂²	semidirect
50. C₇×#463	C₂²	semidirect
51. C₇×#460	C₂²	semidirect
52. C₇×#459	C₂²	semidirect
53. C₇×#452	C₂²	semidirect
54. C₇×#450	C₂²	semidirect
55. C₇×#449	C₂²	semidirect
56. C₇×#447	C₂²	semidirect
57. C₇×#433	C₂²	semidirect
58. C₇×#432	C₂²	semidirect
59. C₇×#424	C₂²	semidirect
60. C₇×#417	C₂²	semidirect
61. C₂×C₇×#102	C₂²	semidirect
63. C₄×C₇×M₁₆	C₂²	semidirect
64. C₂×C₇×#97	C₂²	semidirect
65. #60830	C₇	direct (N × H)
150. #3512	C₂×C₇	semidirect
151. #3498	C₂×C₇	semidirect
152. #3222	C₂×C₇	semidirect
153. #3220	C₂×C₇	semidirect
154. #3219	C₂×C₇	semidirect
155. #3217	C₂×C₇	semidirect
156. #3151	C₂×C₇	semidirect
157. #3150	C₂×C₇	semidirect
158. #3142	C₂×C₇	semidirect
159. C₂×#490	C₂×C₇	semidirect
160. #3045	C₂×C₇	semidirect
161. #3012	C₂×C₇	semidirect
162. #2923	C₂×C₇	semidirect
240. C₂×AGL(1,ℤ/8)	C₂²×C₇	semidirect
241. C₂×#135	C₂²×C₇	semidirect
248. #541	C₂²×C₇	semidirect
249. #540	C₂²×C₇	semidirect
255. #532	C₂²×C₇	semidirect
258. #529	C₂²×C₇	semidirect
259. C₂×#125	C₂²×C₇	semidirect
260. #520	C₂²×C₇	semidirect
262. #497	C₂²×C₇	semidirect
263. #497	C₂²×C₇	semidirect
264. #496	C₂²×C₇	semidirect
265. #496	C₂²×C₇	semidirect
266. #496	C₂²×C₇	semidirect
267. #493	C₂²×C₇	semidirect
268. #492	C₂²×C₇	semidirect
269. #491	C₂²×C₇	semidirect
270. #490	C₂²×C₇	semidirect
271. #490	C₂²×C₇	semidirect
272. #490	C₂²×C₇	semidirect
273. #488	C₂²×C₇	semidirect
274. #464	C₂²×C₇	semidirect
275. #463	C₂²×C₇	semidirect
276. #460	C₂²×C₇	semidirect
277. #459	C₂²×C₇	semidirect
278. #452	C₂²×C₇	semidirect
279. #450	C₂²×C₇	semidirect
280. #449	C₂²×C₇	semidirect
281. #447	C₂²×C₇	semidirect
282. #433	C₂²×C₇	semidirect
283. #432	C₂²×C₇	semidirect
284. #424	C₂²×C₇	semidirect
285. #417	C₂²×C₇	semidirect
286. C₂×#102	C₂²×C₇	semidirect
288. C₄×M₁₆	C₂²×C₇	semidirect
289. C₂×#97	C₂²×C₇	semidirect
576. C₇	#60830	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×#3512	2
C₇×#3498	2
C₇×#3222	2
C₇×#3220	2
C₇×#3219	2
C₇×#3217	2
C₇×#3151	2
C₇×#3150	2
C₇×#3142	2
C₂×C₇×#490	2
C₇×#3045	2
C₇×#3012	2
C₇×#2923	2
#60830	7

Subgroups — C7×#60830 #13238046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60830 #13238046