Subgroups — C₇×#61479 #13238695

Order 1792.

1880 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3517	C₂	1	2	yes	maximal, complemented
2	C₇×#3391	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3383	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3368	C₂	1	2	yes	maximal, complemented
5	C₇×#3339	C₂	1	2	yes	maximal, complemented
6	C₇×#3334	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#3332	C₂	1	2	yes	maximal, complemented
8	C₇×#3309	C₂	1	4	yes	maximal, complemented
9	C₇×#3287	C₂	1	2	yes	maximal, complemented
10	C₇×#3287	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#3176	C₂	1	2	yes	maximal, complemented
12	C₇×#3169	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#3119	C₂	1	4	yes	maximal, complemented
14	C₇×#3115	C₂	1	2	yes	maximal, complemented
15	C₇×#2966	C₂	1	1	yes	maximal, characteristic, complemented
16	C₇×#2951	C₂	1	2	yes	maximal, complemented
17	C₄×C₇×#135	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×C₇×#135	C₂²	1	1	yes	characteristic
19	C₂×C₇×#135	C₂²	1	2	yes	—
20	C₇×#557	C₂²	1	2	yes	—
21	C₇×#553	—	2	4	no	—
22	C₇×#553	C₂²	1	4	yes	—
23	C₇×#550	—	2	2	no	—
24	C₇×#550	C₂²	1	4	yes	—
25	C₇×#550	C₂²	1	2	yes	—
26	C₇×#550	C₂²	1	2	yes	—
27	C₇×#547	—	2	2	no	—
28	C₇×#547	C₂²	1	2	yes	—
29	C₇×#540	—	2	4	no	—
30	C₇×#540	C₂²	1	4	yes	—
31	C₇×#532	—	2	2	no	—
32	C₇×#532	C₂²	1	2	yes	—
33	C₇×#529	—	2	2	no	—
34	C₇×#529	C₂²	1	1	yes	characteristic
35	C₇×#520	C₂²	1	4	yes	—
36	C₄×C₇×#41	C₂²	1	2	yes	—
37	C₇×#500	C₂²	1	1	yes	characteristic
38	C₇×#499	C₂²	1	2	yes	—
39	C₇×#493	C₂²	1	1	yes	characteristic
40	C₇×#493	C₂²	1	2	yes	—
41	C₇×#490	C₂²	1	4	yes	—
42	C₇×#486	C₂²	1	2	yes	—
43	C₇×#482	C₂²	1	4	yes	—
44	C₇×#481	C₂²	1	4	yes	—
45	C₇×#480	C₂²	1	4	yes	—
46	C₇×#479	C₂²	1	4	yes	—
47	C₇×#478	C₂²	1	4	yes	—
48	C₇×#475	C₂²	1	2	yes	—
49	C₇×#474	C₂²	1	2	yes	—
50	C₇×#469	C₂²	1	4	yes	—
51	C₇×#466	C₂²	1	2	yes	—
52	C₇×#466	C₂²	1	2	yes	—
53	C₇×#466	C₂²	1	2	yes	—
54	C₇×#466	C₂²	1	2	yes	—
55	C₇×#460	C₂²	1	2	yes	—
56	C₇×#460	C₂²	1	4	yes	—
57	C₇×#459	C₂²	1	4	yes	—
58	C₇×#459	C₂²	1	2	yes	—
59	C₇×#449	C₂²	1	4	yes	—
60	C₇×#449	C₂²	1	4	yes	—
61	C₇×#449	C₂²	1	4	yes	—
62	C₇×#446	C₂²	1	1	yes	characteristic
63	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
64	C₇×#442	C₂²	1	4	yes	—
65	C₇×#440	C₂²	1	4	yes	—
66	C₇×#440	C₂²	1	4	yes	—
67	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
68	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
69	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
70	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
71	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
72	C₄×C₇×D₈	C₂²	1	2	yes	—
73	C₄×C₇×D₈	C₂²	1	2	yes	—
74	C₄×C₇×D₈	C₂²	1	1	yes	characteristic
75	C₄×C₇×D₈	C₂²	1	1	yes	characteristic
76	C₄×C₇×D₈	C₂²	1	2	yes	—
77	C₇×#436	C₂²	1	1	yes	characteristic
78	C₇×#435	C₂²	1	2	yes	—
79	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
80	C₇×C₈×D₄	C₂²	1	2	yes	—
81	C₇×#433	C₂²	1	2	yes	—
82	C₇×#432	C₂²	1	1	yes	characteristic
83	C₇×#427	C₂²	1	1	yes	characteristic
84	C₂×C₇×#107	C₂²	1	2	yes	—
85	C₇×#417	C₂²	1	4	yes	—
86	C₇×#416	C₂²	1	2	yes	—
87	C₇×#405	C₂²	1	2	yes	—
88	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
89	#61479	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂×C₇×#41	—	2	2	no	—
91	C₂×C₇×#41	—	2	2	no	—
92	C₂×C₇×#41	—	2	2	no	—
93	C₂×C₇×#41	C₂³	1	2	yes	—
94	C₂×C₇×#41	C₂³	1	4	yes	—
95	C₇×#135	—	2	4	no	—
96	C₇×#135	—	2	8	no	—
97	C₂×C₇×Q₁₆	C₂³	1	1	yes	characteristic
98	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
99	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
100	C₂×C₇×D₈	C₂³	1	2	yes	—
101	C₂×C₇×D₈	C₂³	1	1	yes	characteristic
102	C₂×C₇×D₈	C₂³	1	2	yes	—
103	C₂×C₇×D₈	C₂³	1	4	yes	—
104	C₂×C₇×M₁₆	C₂³	1	2	yes	—
105	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
106	C₂²×C₇×C₈	C₂³	1	2	yes	—
107	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
108	C₇×C₄⋊Q₈	—	2	2	no	—
109	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
110	C₇×#127	—	2	2	no	—
111	C₇×#127	—	2	4	no	—
112	C₇×#127	C₂³	1	2	yes	—
113	C₇×#127	C₂³	1	4	yes	—
114	C₇×#126	—	2	4	no	—
115	C₇×#126	D₄	1	2	yes	—
116	C₇×#125	—	2	4	no	—
117	C₇×#125	D₄	1	1	yes	characteristic
118	C₇×#125	C₂³	1	4	yes	—
119	C₇×#124	—	2	2	no	—
120	C₇×#124	D₄	1	1	yes	characteristic
121	C₇×#123	—	2	4	no	—
122	C₇×#123	—	2	4	no	—
123	C₇×#123	—	2	4	no	—
124	C₇×#123	—	2	4	no	—
125	C₇×#123	D₄	1	2	yes	—
126	C₇×C₂²⋊Q₈	—	2	2	no	—
127	C₇×C₂²⋊Q₈	D₄	1	1	yes	characteristic
128	C₇×#121	—	2	4	no	—
129	C₇×#121	—	2	2	no	—
130	C₇×#121	—	2	2	no	—
131	C₇×#121	—	2	4	no	—
132	C₇×#121	—	2	4	no	—
133	C₇×#121	—	2	4	no	—
134	C₇×#121	—	2	4	no	—
135	C₇×#121	—	2	4	no	—
136	C₇×#121	—	2	4	no	—
137	C₇×#121	D₄	1	1	yes	characteristic
138	C₇×#121	C₂³	1	4	yes	—
139	C₇×#121	C₂³	1	4	yes	—
140	C₇×#121	C₂³	1	4	yes	—
141	C₄×C₇×Q₈	—	2	2	no	—
142	C₄×C₇×Q₈	C₂³	1	2	yes	—
143	C₄×C₇×Q₈	C₂³	1	4	yes	—
144	C₄×C₇×Q₈	C₂³	1	2	yes	—
145	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
146	C₄×C₇×D₄	—	2	2	no	—
147	C₄×C₇×D₄	—	2	2	no	—
148	C₄×C₇×D₄	—	2	2	no	—
149	C₄×C₇×D₄	—	2	4	no	—
150	C₄×C₇×D₄	—	2	4	no	—
151	C₄×C₇×D₄	—	2	4	no	—
152	C₄×C₇×D₄	—	2	4	no	—
153	C₄×C₇×D₄	—	2	2	no	—
154	C₄×C₇×D₄	C₂³	1	2	yes	—
155	C₄×C₇×D₄	C₂³	1	4	yes	—
156	C₄×C₇×D₄	C₂³	1	2	yes	—
157	C₄×C₇×D₄	C₂³	1	4	yes	—
158	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
159	C₄×C₇×D₄	C₂³	1	2	yes	—
160	C₇×#117	—	2	4	no	—
161	C₇×#117	—	2	2	no	—
162	C₇×#117	C₂³	1	2	yes	—
163	C₇×#117	C₂³	1	2	yes	—
164	C₂×C₇×C₄⋊C₄	—	2	4	no	—
165	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
166	C₂×C₄²×C₇	—	2	2	no	—
167	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
168	C₇×#107	C₂³	1	1	yes	characteristic
169	C₇×#107	C₂³	1	2	yes	—
170	C₇×#107	C₂³	1	2	yes	—
171	C₇×#107	C₂³	1	4	yes	—
172	C₇×#107	C₂³	1	1	yes	characteristic
173	C₇×#106	C₂³	1	4	yes	—
174	C₇×#106	C₂³	1	2	yes	—
175	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
176	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
177	C₇×C₄⋊C₈	C₂³	1	2	yes	—
178	C₇×C₄⋊C₈	C₂³	1	2	yes	—
179	C₇×#103	C₂³	1	4	yes	—
180	C₇×#103	C₂³	1	2	yes	—
181	C₇×#103	C₂³	1	2	yes	—
182	C₇×#102	C₂³	1	4	yes	—
183	C₇×#102	C₂³	1	4	yes	—
184	C₇×#102	C₂³	1	4	yes	—
185	C₇×#102	C₂³	1	4	yes	—
186	C₇×#102	C₂³	1	2	yes	—
187	C₇×#102	C₂³	1	2	yes	—
188	C₇×#102	C₂³	1	4	yes	—
189	C₇×C₂²⋊C₈	C₂³	1	2	yes	—
190	C₇×C₂²⋊C₈	C₂³	1	2	yes	—
191	C₇×C₂²⋊C₈	C₂³	1	2	yes	—
192	C₇×#97	C₂³	1	2	yes	—
193	C₇×#97	C₂³	1	2	yes	—
194	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
195	C₄×C₇×C₈	C₂³	1	2	yes	—
196	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
197	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
198	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
199	#3517	C₂×C₇	1	2	yes	complemented
200	#3391	C₂×C₇	1	1	yes	characteristic, complemented
201	#3383	C₂×C₇	1	1	yes	characteristic, complemented
202	#3368	C₂×C₇	1	2	yes	complemented
203	#3339	C₂×C₇	1	2	yes	complemented
204	#3334	C₂×C₇	1	1	yes	characteristic, complemented
205	#3332	C₂×C₇	1	2	yes	complemented
206	#3309	C₂×C₇	1	4	yes	complemented
207	#3287	C₂×C₇	1	1	yes	characteristic, complemented
208	#3287	C₂×C₇	1	2	yes	complemented
209	#3176	C₂×C₇	1	2	yes	complemented
210	#3169	C₂×C₇	1	1	yes	characteristic, complemented
211	#3119	C₂×C₇	1	4	yes	complemented
212	#3115	C₂×C₇	1	2	yes	complemented
213	#2966	C₂×C₇	1	1	yes	characteristic, complemented
214	#2951	C₂×C₇	1	2	yes	complemented
215	C₄×#135	C₂×C₇	1	1	yes	characteristic, complemented
216	C₇×#41	—	2	4	no	—
217	C₇×#41	—	2	8	no	—
218	C₇×#41	—	4	2	no	—
219	C₇×#41	—	4	4	no	—
220	C₇×#41	—	8	4	no	—
221	C₇×#41	—	8	4	no	—
222	C₇×#41	—	8	4	no	—
223	C₂×C₇×Q₈	—	2	2	no	—
224	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
225	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
226	C₂×C₇×D₄	—	2	2	no	—
227	C₂×C₇×D₄	—	2	4	no	—
228	C₂×C₇×D₄	—	2	4	no	—
229	C₂×C₇×D₄	—	2	4	no	—
230	C₂×C₇×D₄	—	2	4	no	—
231	C₂×C₇×D₄	—	2	2	no	—
232	C₂×C₇×D₄	—	2	2	no	—
233	C₂×C₇×D₄	—	2	4	no	—
234	C₂×C₇×D₄	—	2	4	no	—
235	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
236	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
237	C₂×C₇×D₄	C₂⁴	1	4	yes	—
238	C₂×C₇×D₄	C₂⁴	1	2	yes	—
239	C₂²×C₄×C₇	—	2	4	no	—
240	C₂²×C₄×C₇	—	2	2	no	—
241	C₂²×C₄×C₇	—	2	2	no	—
242	C₂²×C₄×C₇	—	2	4	no	—
243	C₂²×C₄×C₇	—	2	2	no	—
244	C₂²×C₄×C₇	—	2	4	no	—
245	C₂²×C₄×C₇	—	2	4	no	—
246	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
247	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
248	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
249	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
250	C₇×Q₁₆	#41	1	4	yes	complemented
251	C₇×SD₁₆	—	2	4	no	—
252	C₇×SD₁₆	—	2	8	no	—
253	C₇×D₈	—	2	4	no	—
254	C₇×D₈	—	2	2	no	—
255	C₇×D₈	—	4	2	no	—
256	C₇×D₈	—	4	4	no	—
257	C₇×M₁₆	—	4	2	no	—
258	C₂×C₇×C₈	—	2	2	no	—
259	C₂×C₇×C₈	—	2	4	no	—
260	C₂×C₇×C₈	—	4	1	no	—
261	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
262	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
263	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
264	C₂×C₇×C₈	C₂⁴	1	2	yes	—
265	C₂×C₇×C₈	C₂⁴	1	2	yes	—
266	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
267	C₇×C₄⋊C₄	—	2	2	no	—
268	C₇×C₄⋊C₄	—	2	4	no	—
269	C₇×C₄⋊C₄	—	2	4	no	—
270	C₇×C₄⋊C₄	—	2	2	no	—
271	C₇×C₄⋊C₄	—	2	2	no	—
272	C₇×C₄⋊C₄	—	2	4	no	—
273	C₇×C₄⋊C₄	—	2	4	no	—
274	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
275	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
276	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
277	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
278	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
279	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
280	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
281	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
282	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
283	C₇×C₂²⋊C₄	—	2	2	no	—
284	C₇×C₂²⋊C₄	—	2	4	no	—
285	C₇×C₂²⋊C₄	—	2	2	no	—
286	C₇×C₂²⋊C₄	—	2	2	no	—
287	C₇×C₂²⋊C₄	—	2	4	no	—
288	C₇×C₂²⋊C₄	—	2	4	no	—
289	C₇×C₂²⋊C₄	—	2	4	no	—
290	C₇×C₂²⋊C₄	—	2	4	no	—
291	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
292	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
293	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
294	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
295	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
296	C₄²×C₇	—	2	2	no	—
297	C₄²×C₇	—	2	2	no	—
298	C₄²×C₇	—	2	2	no	—
299	C₄²×C₇	—	2	4	no	—
300	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
301	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
302	C₄²×C₇	C₂⁴	1	1	yes	characteristic
303	C₄²×C₇	C₂⁴	1	1	yes	characteristic
304	C₄²×C₇	C₂⁴	1	2	yes	—
305	C₂×#135	C₂²×C₇	1	1	yes	characteristic
306	C₂×#135	C₂²×C₇	1	2	yes	—
307	#557	C₂²×C₇	1	2	yes	—
308	#553	—	2	4	no	—
309	#553	C₂²×C₇	1	4	yes	—
310	#550	—	2	2	no	—
311	#550	C₂²×C₇	1	2	yes	—
312	#550	C₂²×C₇	1	4	yes	—
313	#550	C₂²×C₇	1	2	yes	—
314	#547	—	2	2	no	—
315	#547	C₂²×C₇	1	2	yes	—
316	#540	—	2	4	no	—
317	#540	C₂²×C₇	1	4	yes	—
318	#532	—	2	2	no	—
319	#532	C₂²×C₇	1	2	yes	—
320	#529	—	2	2	no	—
321	#529	C₂²×C₇	1	1	yes	characteristic
322	#520	C₂²×C₇	1	4	yes	—
323	C₄×#41	C₂²×C₇	1	2	yes	—
324	#500	C₂²×C₇	1	1	yes	characteristic
325	#499	C₂²×C₇	1	2	yes	—
326	#493	C₂²×C₇	1	1	yes	characteristic
327	#493	C₂²×C₇	1	2	yes	—
328	#490	C₂²×C₇	1	4	yes	—
329	#486	C₂²×C₇	1	2	yes	—
330	#482	C₂²×C₇	1	4	yes	—
331	#481	C₂²×C₇	1	4	yes	—
332	#480	C₂²×C₇	1	4	yes	—
333	#479	C₂²×C₇	1	4	yes	—
334	#478	C₂²×C₇	1	4	yes	—
335	#475	C₂²×C₇	1	2	yes	—
336	#474	C₂²×C₇	1	2	yes	—
337	#469	C₂²×C₇	1	4	yes	—
338	#466	C₂²×C₇	1	2	yes	—
339	#466	C₂²×C₇	1	2	yes	—
340	#466	C₂²×C₇	1	2	yes	—
341	#466	C₂²×C₇	1	2	yes	—
342	#460	C₂²×C₇	1	2	yes	—
343	#460	C₂²×C₇	1	4	yes	—
344	#459	C₂²×C₇	1	2	yes	—
345	#459	C₂²×C₇	1	4	yes	—
346	#449	C₂²×C₇	1	4	yes	—
347	#449	C₂²×C₇	1	4	yes	—
348	#449	C₂²×C₇	1	4	yes	—
349	#446	C₂²×C₇	1	1	yes	characteristic
350	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic
351	#442	C₂²×C₇	1	4	yes	—
352	#440	C₂²×C₇	1	4	yes	—
353	#440	C₂²×C₇	1	4	yes	—
354	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
355	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
356	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
357	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
358	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
359	C₄×D₈	C₂²×C₇	1	1	yes	characteristic
360	C₄×D₈	C₂²×C₇	1	2	yes	—
361	C₄×D₈	C₂²×C₇	1	2	yes	—
362	C₄×D₈	C₂²×C₇	1	2	yes	—
363	C₄×D₈	C₂²×C₇	1	1	yes	characteristic
364	#436	C₂²×C₇	1	1	yes	characteristic
365	#435	C₂²×C₇	1	2	yes	—
366	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
367	C₈×D₄	C₂²×C₇	1	2	yes	—
368	#433	C₂²×C₇	1	2	yes	—
369	#432	C₂²×C₇	1	1	yes	characteristic
370	#427	C₂²×C₇	1	1	yes	characteristic
371	C₂×#107	C₂²×C₇	1	2	yes	—
372	#417	C₂²×C₇	1	4	yes	—
373	#416	C₂²×C₇	1	2	yes	—
374	#405	C₂²×C₇	1	2	yes	—
375	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
376	C₂³×C₇	—	2	2	no	—
377	C₂³×C₇	—	2	4	no	—
378	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
379	C₂³×C₇	C₂²×D₄	1	2	yes	—
380	C₇×Q₈	—	2	2	no	—
381	C₇×Q₈	—	2	2	no	—
382	C₇×Q₈	—	4	4	no	—
383	C₇×Q₈	C₂×#41	1	4	yes	—
384	C₇×D₄	—	2	2	no	—
385	C₇×D₄	—	2	2	no	—
386	C₇×D₄	—	2	4	no	—
387	C₇×D₄	—	2	4	no	—
388	C₇×D₄	—	2	2	no	—
389	C₇×D₄	—	4	2	no	—
390	C₇×D₄	—	4	4	no	—
391	C₇×D₄	—	4	8	no	—
392	C₇×D₄	—	4	4	no	—
393	C₇×D₄	—	8	2	no	—
394	C₇×D₄	—	8	4	no	—
395	C₇×D₄	—	8	4	no	—
396	C₇×D₄	—	8	4	no	—
397	C₇×D₄	—	8	4	no	—
398	C₇×D₄	—	8	4	no	—
399	C₂×C₄×C₇	—	2	2	no	—
400	C₂×C₄×C₇	—	2	2	no	—
401	C₂×C₄×C₇	—	2	1	no	—
402	C₂×C₄×C₇	—	2	2	no	—
403	C₂×C₄×C₇	—	2	1	no	—
404	C₂×C₄×C₇	—	2	2	no	—
405	C₂×C₄×C₇	—	2	2	no	—
406	C₂×C₄×C₇	—	2	2	no	—
407	C₂×C₄×C₇	—	2	4	no	—
408	C₂×C₄×C₇	—	4	1	no	—
409	C₂×C₄×C₇	—	4	4	no	—
410	C₂×C₄×C₇	—	4	8	no	—
411	C₂×C₄×C₇	—	4	4	no	—
412	C₂×C₄×C₇	—	8	2	no	—
413	C₂×C₄×C₇	—	8	4	no	—
414	C₂×C₄×C₇	—	8	4	no	—
415	C₂×C₄×C₇	—	8	4	no	—
416	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
417	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
418	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
419	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
420	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
421	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
422	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
423	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
424	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
425	C₇×C₈	—	2	1	no	—
426	C₇×C₈	—	2	2	no	—
427	C₇×C₈	—	2	1	no	—
428	C₇×C₈	—	2	1	no	—
429	C₇×C₈	—	2	2	no	—
430	C₇×C₈	C₂×#41	1	2	yes	—
431	C₂×#41	—	2	2	no	—
432	C₂×#41	—	2	2	no	—
433	C₂×#41	—	2	2	no	—
434	C₂×#41	C₂³×C₇	1	2	yes	—
435	C₂×#41	C₂³×C₇	1	4	yes	—
436	#135	—	2	4	no	—
437	#135	—	2	8	no	—
438	C₂×Q₁₆	C₂³×C₇	1	1	yes	characteristic
439	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
440	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
441	C₂×D₈	C₂³×C₇	1	1	yes	characteristic
442	C₂×D₈	C₂³×C₇	1	2	yes	—
443	C₂×D₈	C₂³×C₇	1	2	yes	—
444	C₂×D₈	C₂³×C₇	1	4	yes	—
445	C₂×M₁₆	C₂³×C₇	1	2	yes	—
446	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
447	C₂²×C₈	C₂³×C₇	1	2	yes	—
448	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
449	C₄⋊Q₈	—	2	2	no	—
450	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
451	#127	—	2	2	no	—
452	#127	—	2	4	no	—
453	#127	C₂³×C₇	1	2	yes	—
454	#127	C₂³×C₇	1	4	yes	—
455	#126	—	2	4	no	—
456	#126	C₇×D₄	1	2	yes	—
457	#125	—	2	4	no	—
458	#125	C₇×D₄	1	1	yes	characteristic
459	#125	C₂³×C₇	1	4	yes	—
460	#124	—	2	2	no	—
461	#124	C₇×D₄	1	1	yes	characteristic
462	#123	—	2	4	no	—
463	#123	—	2	4	no	—
464	#123	—	2	4	no	—
465	#123	—	2	4	no	—
466	#123	C₇×D₄	1	2	yes	—
467	C₂²⋊Q₈	—	2	2	no	—
468	C₂²⋊Q₈	C₇×D₄	1	1	yes	characteristic
469	#121	—	2	4	no	—
470	#121	—	2	2	no	—
471	#121	—	2	2	no	—
472	#121	—	2	4	no	—
473	#121	—	2	4	no	—
474	#121	—	2	4	no	—
475	#121	—	2	4	no	—
476	#121	—	2	4	no	—
477	#121	—	2	4	no	—
478	#121	C₇×D₄	1	1	yes	characteristic
479	#121	C₂³×C₇	1	4	yes	—
480	#121	C₂³×C₇	1	4	yes	—
481	#121	C₂³×C₇	1	4	yes	—
482	C₄×Q₈	—	2	2	no	—
483	C₄×Q₈	C₂³×C₇	1	2	yes	—
484	C₄×Q₈	C₂³×C₇	1	2	yes	—
485	C₄×Q₈	C₂³×C₇	1	4	yes	—
486	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
487	C₄×D₄	—	2	2	no	—
488	C₄×D₄	—	2	2	no	—
489	C₄×D₄	—	2	2	no	—
490	C₄×D₄	—	2	4	no	—
491	C₄×D₄	—	2	4	no	—
492	C₄×D₄	—	2	4	no	—
493	C₄×D₄	—	2	4	no	—
494	C₄×D₄	—	2	2	no	—
495	C₄×D₄	C₂³×C₇	1	2	yes	—
496	C₄×D₄	C₂³×C₇	1	2	yes	—
497	C₄×D₄	C₂³×C₇	1	4	yes	—
498	C₄×D₄	C₂³×C₇	1	4	yes	—
499	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
500	C₄×D₄	C₂³×C₇	1	2	yes	—
501	#117	—	2	4	no	—
502	#117	—	2	2	no	—
503	#117	C₂³×C₇	1	2	yes	—
504	#117	C₂³×C₇	1	2	yes	—
505	C₂×C₄⋊C₄	—	2	4	no	—
506	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
507	C₂×C₄²	—	2	2	no	—
508	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
509	#107	C₂³×C₇	1	1	yes	characteristic
510	#107	C₂³×C₇	1	2	yes	—
511	#107	C₂³×C₇	1	4	yes	—
512	#107	C₂³×C₇	1	1	yes	characteristic
513	#107	C₂³×C₇	1	2	yes	—
514	#106	C₂³×C₇	1	2	yes	—
515	#106	C₂³×C₇	1	4	yes	—
516	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
517	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
518	C₄⋊C₈	C₂³×C₇	1	2	yes	—
519	C₄⋊C₈	C₂³×C₇	1	2	yes	—
520	#103	C₂³×C₇	1	2	yes	—
521	#103	C₂³×C₇	1	2	yes	—
522	#103	C₂³×C₇	1	4	yes	—
523	#102	C₂³×C₇	1	2	yes	—
524	#102	C₂³×C₇	1	2	yes	—
525	#102	C₂³×C₇	1	4	yes	—
526	#102	C₂³×C₇	1	4	yes	—
527	#102	C₂³×C₇	1	4	yes	—
528	#102	C₂³×C₇	1	4	yes	—
529	#102	C₂³×C₇	1	4	yes	—
530	C₂²⋊C₈	C₂³×C₇	1	2	yes	—
531	C₂²⋊C₈	C₂³×C₇	1	2	yes	—
532	C₂²⋊C₈	C₂³×C₇	1	2	yes	—
533	#97	C₂³×C₇	1	2	yes	—
534	#97	C₂³×C₇	1	2	yes	—
535	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
536	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
537	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
538	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
539	C₄×C₈	C₂³×C₇	1	2	yes	—
540	C₂²×C₇	—	2	1	no	—
541	C₂²×C₇	—	2	2	no	—
542	C₂²×C₇	—	2	1	no	—
543	C₂²×C₇	—	2	2	no	—
544	C₂²×C₇	—	2	1	no	—
545	C₂²×C₇	—	2	2	no	—
546	C₂²×C₇	—	4	2	no	—
547	C₂²×C₇	—	4	4	no	—
548	C₂²×C₇	—	4	2	no	—
549	C₂²×C₇	—	4	4	no	—
550	C₂²×C₇	—	4	2	no	—
551	C₂²×C₇	—	4	4	no	—
552	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
553	C₄×C₇	—	2	1	no	—
554	C₄×C₇	—	2	1	no	—
555	C₄×C₇	—	2	2	no	—
556	C₄×C₇	—	2	1	no	—
557	C₄×C₇	—	2	2	no	—
558	C₄×C₇	—	2	4	no	—
559	C₄×C₇	—	2	1	no	—
560	C₄×C₇	—	2	2	no	—
561	C₄×C₇	—	4	2	no	—
562	C₄×C₇	—	4	2	no	—
563	C₄×C₇	—	4	4	no	—
564	C₄×C₇	#576	1	2	yes	—
565	C₄×C₇	C₂²×#41	1	1	yes	characteristic
566	C₄×C₇	#585	1	1	yes	characteristic
567	#41	—	2	4	no	—
568	#41	—	2	8	no	—
569	#41	—	4	2	no	—
570	#41	—	4	4	no	—
571	#41	—	8	4	no	—
572	#41	—	8	4	no	—
573	#41	—	8	4	no	—
574	C₂×Q₈	—	2	2	no	—
575	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
576	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
577	C₂×D₄	—	2	2	no	—
578	C₂×D₄	—	2	4	no	—
579	C₂×D₄	—	2	4	no	—
580	C₂×D₄	—	2	4	no	—
581	C₂×D₄	—	2	4	no	—
582	C₂×D₄	—	2	2	no	—
583	C₂×D₄	—	2	2	no	—
584	C₂×D₄	—	2	4	no	—
585	C₂×D₄	—	2	4	no	—
586	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
587	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
588	C₂×D₄	C₂⁴×C₇	1	2	yes	—
589	C₂×D₄	C₂⁴×C₇	1	4	yes	—
590	C₂²×C₄	—	2	4	no	—
591	C₂²×C₄	—	2	2	no	—
592	C₂²×C₄	—	2	2	no	—
593	C₂²×C₄	—	2	4	no	—
594	C₂²×C₄	—	2	2	no	—
595	C₂²×C₄	—	2	4	no	—
596	C₂²×C₄	—	2	4	no	—
597	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
598	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
599	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
600	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
601	Q₁₆	C₇×#41	1	4	yes	complemented
602	SD₁₆	—	2	4	no	—
603	SD₁₆	—	2	8	no	—
604	D₈	—	2	2	no	—
605	D₈	—	2	4	no	—
606	D₈	—	4	2	no	—
607	D₈	—	4	4	no	—
608	M₁₆	—	4	2	no	—
609	C₂×C₈	—	2	2	no	—
610	C₂×C₈	—	2	4	no	—
611	C₂×C₈	—	4	1	no	—
612	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
613	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
614	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
615	C₂×C₈	C₂⁴×C₇	1	2	yes	—
616	C₂×C₈	C₂⁴×C₇	1	2	yes	—
617	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
618	C₄⋊C₄	—	2	2	no	—
619	C₄⋊C₄	—	2	4	no	—
620	C₄⋊C₄	—	2	4	no	—
621	C₄⋊C₄	—	2	2	no	—
622	C₄⋊C₄	—	2	2	no	—
623	C₄⋊C₄	—	2	4	no	—
624	C₄⋊C₄	—	2	4	no	—
625	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
626	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
627	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
628	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
629	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
630	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
631	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
632	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
633	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
634	C₂²⋊C₄	—	2	2	no	—
635	C₂²⋊C₄	—	2	4	no	—
636	C₂²⋊C₄	—	2	2	no	—
637	C₂²⋊C₄	—	2	2	no	—
638	C₂²⋊C₄	—	2	4	no	—
639	C₂²⋊C₄	—	2	4	no	—
640	C₂²⋊C₄	—	2	4	no	—
641	C₂²⋊C₄	—	2	4	no	—
642	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
643	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
644	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
645	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
646	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
647	C₄²	—	2	2	no	—
648	C₄²	—	2	2	no	—
649	C₄²	—	2	2	no	—
650	C₄²	—	2	4	no	—
651	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
652	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
653	C₄²	C₂⁴×C₇	1	1	yes	characteristic
654	C₄²	C₂⁴×C₇	1	1	yes	characteristic
655	C₄²	C₂⁴×C₇	1	2	yes	—
656	C₂×C₇	—	4	1	no	—
657	C₂×C₇	—	4	2	no	—
658	C₂×C₇	—	8	2	no	—
659	C₂×C₇	—	8	4	no	—
660	C₂×C₇	D₄×#41	1	1	yes	characteristic
661	C₂×C₇	C₂×#576	1	1	yes	characteristic
662	C₂×C₇	#3584	1	1	yes	characteristic
663	C₂³	—	2	2	no	—
664	C₂³	—	2	4	no	—
665	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
666	C₂³	C₂²×C₇×D₄	1	2	yes	—
667	Q₈	—	2	2	no	—
668	Q₈	—	2	2	no	—
669	Q₈	—	4	4	no	—
670	Q₈	C₂×C₇×#41	1	4	yes	—
671	D₄	—	2	2	no	—
672	D₄	—	2	4	no	—
673	D₄	—	2	4	no	—
674	D₄	—	2	2	no	—
675	D₄	—	2	2	no	—
676	D₄	—	4	2	no	—
677	D₄	—	4	4	no	—
678	D₄	—	4	8	no	—
679	D₄	—	4	4	no	—
680	D₄	—	8	2	no	—
681	D₄	—	8	4	no	—
682	D₄	—	8	4	no	—
683	D₄	—	8	4	no	—
684	D₄	—	8	4	no	—
685	D₄	—	8	4	no	—
686	C₂×C₄	—	2	2	no	—
687	C₂×C₄	—	2	1	no	—
688	C₂×C₄	—	2	2	no	—
689	C₂×C₄	—	2	1	no	—
690	C₂×C₄	—	2	2	no	—
691	C₂×C₄	—	2	2	no	—
692	C₂×C₄	—	2	2	no	—
693	C₂×C₄	—	2	2	no	—
694	C₂×C₄	—	2	4	no	—
695	C₂×C₄	—	4	1	no	—
696	C₂×C₄	—	4	4	no	—
697	C₂×C₄	—	4	8	no	—
698	C₂×C₄	—	4	4	no	—
699	C₂×C₄	—	8	2	no	—
700	C₂×C₄	—	8	4	no	—
701	C₂×C₄	—	8	4	no	—
702	C₂×C₄	—	8	4	no	—
703	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
704	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
705	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
706	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
707	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
708	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
709	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
710	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
711	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
712	C₈	—	2	1	no	—
713	C₈	—	2	2	no	—
714	C₈	—	2	1	no	—
715	C₈	—	2	1	no	—
716	C₈	—	2	2	no	—
717	C₈	C₂×C₇×#41	1	2	yes	—
718	C₇	#61479	1	1	yes	characteristic, direct product factor, Sylow
719	C₂²	—	2	1	no	—
720	C₂²	—	2	2	no	—
721	C₂²	—	2	1	no	—
722	C₂²	—	2	2	no	—
723	C₂²	—	2	1	no	—
724	C₂²	—	2	2	no	—
725	C₂²	—	4	2	no	—
726	C₂²	—	4	4	no	—
727	C₂²	—	4	2	no	—
728	C₂²	—	4	4	no	—
729	C₂²	—	4	2	no	—
730	C₂²	—	4	4	no	—
731	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
732	C₄	—	2	1	no	—
733	C₄	—	2	2	no	—
734	C₄	—	2	1	no	—
735	C₄	—	2	1	no	—
736	C₄	—	2	2	no	—
737	C₄	—	2	1	no	—
738	C₄	—	2	2	no	—
739	C₄	—	2	4	no	—
740	C₄	—	4	2	no	—
741	C₄	—	4	4	no	—
742	C₄	—	4	2	no	—
743	C₄	C₇×#576	1	2	yes	—
744	C₄	C₂²×C₇×#41	1	1	yes	characteristic
745	C₄	C₇×#585	1	1	yes	characteristic
746	C₂	—	4	1	no	—
747	C₂	—	4	2	no	—
748	C₂	—	8	2	no	—
749	C₂	—	8	4	no	—
750	C₂	C₇×D₄×#41	1	1	yes	characteristic
751	C₂	C₂×C₇×#576	1	1	yes	characteristic
752	C₂	C₇×#3584	1	1	yes	characteristic

	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 ◁ C₇×#529 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3517	C₂	semidirect
2. C₇×#3391	C₂	semidirect
3. C₇×#3383	C₂	semidirect
4. C₇×#3368	C₂	semidirect
5. C₇×#3339	C₂	semidirect
6. C₇×#3334	C₂	semidirect
7. C₇×#3332	C₂	semidirect
8. C₇×#3309	C₂	semidirect
9. C₇×#3287	C₂	semidirect
10. C₇×#3287	C₂	semidirect
11. C₇×#3176	C₂	semidirect
12. C₇×#3169	C₂	semidirect
13. C₇×#3119	C₂	semidirect
14. C₇×#3115	C₂	semidirect
15. C₇×#2966	C₂	semidirect
16. C₇×#2951	C₂	semidirect
17. C₄×C₇×#135	C₂	semidirect
89. #61479	C₇	direct (N × H)
199. #3517	C₂×C₇	semidirect
200. #3391	C₂×C₇	semidirect
201. #3383	C₂×C₇	semidirect
202. #3368	C₂×C₇	semidirect
203. #3339	C₂×C₇	semidirect
204. #3334	C₂×C₇	semidirect
205. #3332	C₂×C₇	semidirect
206. #3309	C₂×C₇	semidirect
207. #3287	C₂×C₇	semidirect
208. #3287	C₂×C₇	semidirect
209. #3176	C₂×C₇	semidirect
210. #3169	C₂×C₇	semidirect
211. #3119	C₂×C₇	semidirect
212. #3115	C₂×C₇	semidirect
213. #2966	C₂×C₇	semidirect
214. #2951	C₂×C₇	semidirect
215. C₄×#135	C₂×C₇	semidirect
250. C₇×Q₁₆	#41	semidirect
601. Q₁₆	C₇×#41	semidirect
718. C₇	#61479	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×#3517	2
C₇×#3391	2
C₇×#3383	2
C₇×#3368	2
C₇×#3339	2
C₇×#3334	2
C₇×#3332	2
C₇×#3309	2
C₇×#3287	2
C₇×#3176	2
C₇×#3169	2
C₇×#3119	2
C₇×#3115	2
C₇×#2966	2
C₇×#2951	2
C₄×C₇×#135	2
#61479	7

Subgroups — C7×#61479 #13238695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61479 #13238695