Subgroups — C₇×#60859 #13238075

Order 1792.

1654 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3458	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3455	C₂	1	1	yes	maximal, characteristic
3	C₇×#3244	C₂	1	4	yes	maximal, complemented
4	C₇×#3243	C₂	1	4	yes	maximal, complemented
5	C₇×#3242	C₂	1	4	yes	maximal, complemented
6	C₇×#3241	C₂	1	4	yes	maximal, complemented
7	C₇×#3162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3135	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3131	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#3091	C₂	1	4	yes	maximal, complemented
12	C₇×#3076	C₂	1	4	yes	maximal, complemented
13	C₂×C₇×#433	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#544	—	2	2	no	—
15	C₇×#544	C₂²	1	4	yes	—
16	C₇×#544	C₂²	1	4	yes	—
17	C₇×#543	—	2	2	no	—
18	C₇×#542	C₂²	1	4	yes	—
19	C₇×#541	C₂²	1	2	yes	—
20	C₇×#540	C₂²	1	2	yes	—
21	C₇×#539	—	2	2	no	—
22	C₇×#538	—	2	2	no	—
23	C₇×#533	—	2	4	no	—
24	C₇×#533	C₂²	1	1	yes	characteristic
25	C₇×#533	C₂²	1	1	yes	characteristic
26	C₇×#533	C₂²	1	1	yes	characteristic
27	C₇×#533	C₂²	1	2	yes	—
28	C₇×#532	C₂²	1	1	yes	characteristic
29	C₇×#529	—	2	4	no	—
30	C₇×#525	C₂²	1	2	yes	—
31	C₄×C₇×#41	C₂²	1	2	yes	—
32	C₂×C₇×#117	C₂²	1	2	yes	—
33	C₂×C₇×#117	C₂²	1	1	yes	characteristic
34	C₇×#501	C₂²	1	4	yes	complemented
35	C₇×#501	C₂²	1	4	yes	complemented
36	C₇×#500	C₂²	1	2	yes	complemented
37	C₇×#499	C₂²	1	4	yes	complemented
38	C₇×#498	C₂²	1	2	yes	complemented
39	C₇×#491	C₂²	1	2	yes	complemented
40	C₇×#490	C₂²	1	2	yes	complemented
41	C₇×#489	C₂²	1	4	yes	complemented
42	C₇×#488	C₂²	1	2	yes	complemented
43	C₇×#488	C₂²	1	2	yes	complemented
44	C₇×#487	C₂²	1	2	yes	complemented
45	C₇×#486	C₂²	1	2	yes	complemented
46	C₇×#485	C₂²	1	4	yes	—
47	C₇×#485	C₂²	1	4	yes	—
48	C₇×#484	C₂²	1	4	yes	—
49	C₇×#483	C₂²	1	4	yes	—
50	C₇×#482	C₂²	1	4	yes	—
51	C₇×#482	C₂²	1	4	yes	—
52	C₇×#481	C₂²	1	4	yes	—
53	C₇×#480	C₂²	1	4	yes	—
54	C₇×#479	C₂²	1	4	yes	complemented
55	C₇×#479	C₂²	1	4	yes	complemented
56	C₇×#478	C₂²	1	4	yes	complemented
57	C₇×#478	C₂²	1	4	yes	complemented
58	C₇×#477	C₂²	1	4	yes	complemented
59	C₇×#476	C₂²	1	4	yes	complemented
60	C₇×#475	C₂²	1	4	yes	complemented
61	C₇×#474	C₂²	1	4	yes	complemented
62	C₇×#433	C₂²	1	4	yes	—
63	C₇×#433	C₂²	1	2	yes	—
64	C₇×#433	C₂²	1	2	yes	—
65	C₇×#427	C₂²	1	4	yes	—
66	C₇×#427	C₂²	1	4	yes	—
67	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
68	C₇×#416	C₂²	1	4	yes	—
69	C₇×#416	C₂²	1	4	yes	—
70	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
71	C₂×C₇×#97	C₂²	1	1	yes	characteristic
72	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
73	#60859	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂×C₇×#41	C₂³	1	2	yes	—
75	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
76	C₂²×C₇×C₈	C₂³	1	2	yes	—
77	C₂²×C₇×C₈	C₂³	1	2	yes	—
78	C₇×C₄⋊Q₈	—	2	4	no	—
79	C₇×C₄⋊Q₈	—	2	4	no	—
80	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
82	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
83	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
84	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
85	C₇×#127	C₂³	1	1	yes	characteristic
86	C₇×#126	—	2	4	no	—
87	C₇×#126	—	2	4	no	—
88	C₇×#125	—	2	4	no	—
89	C₇×#125	—	2	4	no	—
90	C₇×#125	—	2	4	no	—
91	C₇×#125	C₂³	1	4	yes	—
92	C₇×#125	C₂³	1	2	yes	—
93	C₇×#125	C₂³	1	2	yes	—
94	C₇×#125	C₂³	1	2	yes	—
95	C₇×#124	—	2	4	no	—
96	C₇×#124	C₂³	1	2	yes	—
97	C₇×#123	—	2	2	no	—
98	C₇×#123	—	2	4	no	—
99	C₇×#123	—	2	4	no	—
100	C₇×C₂²⋊Q₈	—	2	2	no	—
101	C₇×C₂²⋊Q₈	—	2	4	no	—
102	C₇×C₂²⋊Q₈	—	2	2	no	—
103	C₇×C₂²⋊Q₈	—	2	2	no	—
104	C₇×C₂²⋊Q₈	—	2	4	no	—
105	C₇×C₂²⋊Q₈	—	2	4	no	—
106	C₇×C₂²⋊Q₈	—	2	4	no	—
107	C₇×C₂²⋊Q₈	—	2	4	no	—
108	C₇×C₂²⋊Q₈	—	2	4	no	—
109	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
110	C₇×#121	—	2	2	no	—
111	C₇×#121	—	2	4	no	—
112	C₇×#121	C₂³	1	4	yes	—
113	C₇×#120	—	2	2	no	—
114	C₄×C₇×Q₈	—	2	2	no	—
115	C₄×C₇×Q₈	—	2	4	no	—
116	C₄×C₇×Q₈	—	2	2	no	—
117	C₄×C₇×Q₈	—	2	2	no	—
118	C₄×C₇×Q₈	C₂³	1	4	yes	—
119	C₄×C₇×D₄	—	2	4	no	—
120	C₄×C₇×D₄	—	2	4	no	—
121	C₄×C₇×D₄	—	2	2	no	—
122	C₄×C₇×D₄	C₂³	1	4	yes	—
123	C₇×#117	—	2	2	no	—
124	C₇×#117	—	2	2	no	—
125	C₇×#117	—	2	2	no	—
126	C₇×#117	D₄	1	4	yes	—
127	C₇×#117	C₂³	1	4	yes	—
128	C₇×#117	C₂³	1	2	yes	—
129	C₇×#117	C₂³	1	2	yes	—
130	C₇×#117	C₂³	1	2	yes	—
131	C₇×#117	C₂³	1	2	yes	—
132	C₇×#117	C₂³	1	2	yes	—
133	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
134	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
135	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
136	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
137	C₂×C₄²×C₇	—	2	2	no	—
138	C₂×C₄²×C₇	—	2	2	no	—
139	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
140	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
141	C₇×#107	C₂³	1	4	yes	—
142	C₇×#107	C₂³	1	4	yes	—
143	C₇×#107	C₂³	1	4	yes	—
144	C₇×#107	C₂³	1	4	yes	—
145	C₇×#106	C₂³	1	4	yes	—
146	C₇×#106	C₂³	1	4	yes	—
147	C₇×#106	C₂³	1	4	yes	—
148	C₇×#106	C₂³	1	4	yes	—
149	C₇×C₄⋊C₈	C₂³	1	4	yes	—
150	C₇×C₄⋊C₈	C₂³	1	4	yes	—
151	C₇×#103	C₂³	1	4	yes	—
152	C₇×#103	C₂³	1	4	yes	—
153	C₇×#103	C₂³	1	4	yes	—
154	C₇×#103	C₂³	1	4	yes	—
155	C₇×#102	C₂³	1	4	yes	—
156	C₇×#102	C₂³	1	4	yes	—
157	C₇×#102	C₂³	1	4	yes	—
158	C₇×#102	C₂³	1	4	yes	—
159	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
160	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
161	C₇×#97	C₂³	1	2	yes	—
162	C₇×#97	C₂³	1	2	yes	—
163	C₄×C₇×C₈	C₂³	1	2	yes	—
164	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
165	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
166	#3458	C₂×C₇	1	1	yes	characteristic, complemented
167	#3455	C₂×C₇	1	1	yes	characteristic
168	#3244	C₂×C₇	1	4	yes	complemented
169	#3243	C₂×C₇	1	4	yes	complemented
170	#3242	C₂×C₇	1	4	yes	complemented
171	#3241	C₂×C₇	1	4	yes	complemented
172	#3162	C₂×C₇	1	1	yes	characteristic, complemented
173	#3160	C₂×C₇	1	1	yes	characteristic, complemented
174	#3135	C₂×C₇	1	1	yes	characteristic, complemented
175	#3131	C₂×C₇	1	1	yes	characteristic, complemented
176	#3091	C₂×C₇	1	4	yes	complemented
177	#3076	C₂×C₇	1	4	yes	complemented
178	C₂×#433	C₂×C₇	1	1	yes	characteristic, complemented
179	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
180	C₇×#41	—	2	8	no	—
181	C₂×C₇×Q₈	—	2	2	no	—
182	C₂×C₇×Q₈	—	2	4	no	—
183	C₂×C₇×Q₈	—	2	2	no	—
184	C₂×C₇×Q₈	—	2	2	no	—
185	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
186	C₂×C₇×D₄	—	2	2	no	—
187	C₂×C₇×D₄	—	2	4	no	—
188	C₂×C₇×D₄	—	2	2	no	—
189	C₂×C₇×D₄	C₂⁴	1	2	yes	—
190	C₂²×C₄×C₇	—	2	4	no	—
191	C₂²×C₄×C₇	—	2	4	no	—
192	C₂²×C₄×C₇	—	2	2	no	—
193	C₂²×C₄×C₇	—	2	4	no	—
194	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
195	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
196	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
197	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
198	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
200	C₂×C₇×C₈	—	2	4	no	—
201	C₂×C₇×C₈	—	2	4	no	—
202	C₂×C₇×C₈	C₂⁴	1	2	yes	—
203	C₂×C₇×C₈	C₂⁴	1	2	yes	—
204	C₂×C₇×C₈	C₂⁴	1	4	yes	—
205	C₇×C₄⋊C₄	—	2	4	no	—
206	C₇×C₄⋊C₄	—	2	4	no	—
207	C₇×C₄⋊C₄	—	2	2	no	—
208	C₇×C₄⋊C₄	—	2	4	no	—
209	C₇×C₄⋊C₄	—	2	4	no	—
210	C₇×C₄⋊C₄	—	2	4	no	—
211	C₇×C₄⋊C₄	—	2	2	no	—
212	C₇×C₄⋊C₄	—	2	2	no	—
213	C₇×C₄⋊C₄	—	2	4	no	—
214	C₇×C₄⋊C₄	—	2	4	no	—
215	C₇×C₄⋊C₄	—	2	4	no	—
216	C₇×C₄⋊C₄	—	2	4	no	—
217	C₇×C₄⋊C₄	—	2	2	no	—
218	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
219	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
220	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
221	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
222	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
223	C₇×C₂²⋊C₄	—	2	4	no	—
224	C₇×C₂²⋊C₄	—	2	2	no	—
225	C₇×C₂²⋊C₄	—	2	2	no	—
226	C₇×C₂²⋊C₄	—	2	2	no	—
227	C₇×C₂²⋊C₄	—	2	4	no	—
228	C₇×C₂²⋊C₄	—	2	4	no	—
229	C₇×C₂²⋊C₄	—	2	4	no	—
230	C₇×C₂²⋊C₄	—	2	4	no	—
231	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
232	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
233	C₄²×C₇	—	2	4	no	—
234	C₄²×C₇	—	2	4	no	—
235	C₄²×C₇	—	2	2	no	—
236	C₄²×C₇	—	2	2	no	—
237	C₄²×C₇	—	2	2	no	—
238	C₄²×C₇	C₂×D₄	1	2	yes	—
239	C₄²×C₇	C₂×D₄	1	2	yes	—
240	C₄²×C₇	C₂⁴	1	1	yes	characteristic
241	C₄²×C₇	C₂⁴	1	1	yes	characteristic
242	C₄²×C₇	C₂⁴	1	2	yes	—
243	#544	—	2	2	no	—
244	#544	C₂²×C₇	1	4	yes	—
245	#544	C₂²×C₇	1	4	yes	—
246	#543	—	2	2	no	—
247	#542	C₂²×C₇	1	4	yes	—
248	#541	C₂²×C₇	1	2	yes	—
249	#540	C₂²×C₇	1	2	yes	—
250	#539	—	2	2	no	—
251	#538	—	2	2	no	—
252	#533	—	2	4	no	—
253	#533	C₂²×C₇	1	2	yes	—
254	#533	C₂²×C₇	1	1	yes	characteristic
255	#533	C₂²×C₇	1	1	yes	characteristic
256	#533	C₂²×C₇	1	1	yes	characteristic
257	#532	C₂²×C₇	1	1	yes	characteristic
258	#529	—	2	4	no	—
259	#525	C₂²×C₇	1	2	yes	—
260	C₄×#41	C₂²×C₇	1	2	yes	—
261	C₂×#117	C₂²×C₇	1	2	yes	—
262	C₂×#117	C₂²×C₇	1	1	yes	characteristic
263	#501	C₂²×C₇	1	4	yes	complemented
264	#501	C₂²×C₇	1	4	yes	complemented
265	#500	C₂²×C₇	1	2	yes	complemented
266	#499	C₂²×C₇	1	4	yes	complemented
267	#498	C₂²×C₇	1	2	yes	complemented
268	#491	C₂²×C₇	1	2	yes	complemented
269	#490	C₂²×C₇	1	2	yes	complemented
270	#489	C₂²×C₇	1	4	yes	complemented
271	#488	C₂²×C₇	1	2	yes	complemented
272	#488	C₂²×C₇	1	2	yes	complemented
273	#487	C₂²×C₇	1	2	yes	complemented
274	#486	C₂²×C₇	1	2	yes	complemented
275	#485	C₂²×C₇	1	4	yes	—
276	#485	C₂²×C₇	1	4	yes	—
277	#484	C₂²×C₇	1	4	yes	—
278	#483	C₂²×C₇	1	4	yes	—
279	#482	C₂²×C₇	1	4	yes	—
280	#482	C₂²×C₇	1	4	yes	—
281	#481	C₂²×C₇	1	4	yes	—
282	#480	C₂²×C₇	1	4	yes	—
283	#479	C₂²×C₇	1	4	yes	complemented
284	#479	C₂²×C₇	1	4	yes	complemented
285	#478	C₂²×C₇	1	4	yes	complemented
286	#478	C₂²×C₇	1	4	yes	complemented
287	#477	C₂²×C₇	1	4	yes	complemented
288	#476	C₂²×C₇	1	4	yes	complemented
289	#475	C₂²×C₇	1	4	yes	complemented
290	#474	C₂²×C₇	1	4	yes	complemented
291	#433	C₂²×C₇	1	4	yes	—
292	#433	C₂²×C₇	1	2	yes	—
293	#433	C₂²×C₇	1	2	yes	—
294	#427	C₂²×C₇	1	4	yes	—
295	#427	C₂²×C₇	1	4	yes	—
296	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
297	#416	C₂²×C₇	1	4	yes	—
298	#416	C₂²×C₇	1	4	yes	—
299	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
300	C₂×#97	C₂²×C₇	1	1	yes	characteristic
301	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
302	C₂³×C₇	—	2	2	no	—
303	C₂³×C₇	—	2	2	no	—
304	C₂³×C₇	—	4	2	no	—
305	C₂³×C₇	C₂²×D₄	1	2	yes	—
306	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
307	C₇×Q₈	—	2	4	no	—
308	C₇×Q₈	—	8	4	no	—
309	C₇×Q₈	—	8	2	no	—
310	C₇×Q₈	—	8	2	no	—
311	C₇×Q₈	—	8	2	no	—
312	C₇×D₄	—	2	4	no	—
313	C₇×D₄	—	4	4	no	—
314	C₇×D₄	—	8	2	no	—
315	C₇×D₄	—	8	4	no	—
316	C₂×C₄×C₇	—	2	2	no	—
317	C₂×C₄×C₇	—	2	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	4	no	—
322	C₂×C₄×C₇	—	2	4	no	—
323	C₂×C₄×C₇	—	4	2	no	—
324	C₂×C₄×C₇	—	4	2	no	—
325	C₂×C₄×C₇	—	4	4	no	—
326	C₂×C₄×C₇	—	8	4	no	—
327	C₂×C₄×C₇	—	8	4	no	—
328	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
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	1	yes	characteristic
333	C₂×C₄×C₇	2₋¹⁺⁴	1	4	yes	—
334	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
335	C₇×C₈	—	2	2	no	—
336	C₇×C₈	—	2	2	no	—
337	C₇×C₈	—	2	4	no	—
338	C₂×#41	C₂³×C₇	1	2	yes	—
339	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
340	C₂²×C₈	C₂³×C₇	1	2	yes	—
341	C₂²×C₈	C₂³×C₇	1	2	yes	—
342	C₄⋊Q₈	—	2	4	no	—
343	C₄⋊Q₈	—	2	4	no	—
344	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
345	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
346	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
347	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
348	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
349	#127	C₂³×C₇	1	1	yes	characteristic
350	#126	—	2	4	no	—
351	#126	—	2	4	no	—
352	#125	—	2	4	no	—
353	#125	—	2	4	no	—
354	#125	—	2	4	no	—
355	#125	C₂³×C₇	1	2	yes	—
356	#125	C₂³×C₇	1	4	yes	—
357	#125	C₂³×C₇	1	2	yes	—
358	#125	C₂³×C₇	1	2	yes	—
359	#124	—	2	4	no	—
360	#124	C₂³×C₇	1	2	yes	—
361	#123	—	2	2	no	—
362	#123	—	2	4	no	—
363	#123	—	2	4	no	—
364	C₂²⋊Q₈	—	2	2	no	—
365	C₂²⋊Q₈	—	2	4	no	—
366	C₂²⋊Q₈	—	2	2	no	—
367	C₂²⋊Q₈	—	2	2	no	—
368	C₂²⋊Q₈	—	2	4	no	—
369	C₂²⋊Q₈	—	2	4	no	—
370	C₂²⋊Q₈	—	2	4	no	—
371	C₂²⋊Q₈	—	2	4	no	—
372	C₂²⋊Q₈	—	2	4	no	—
373	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
374	#121	—	2	2	no	—
375	#121	—	2	4	no	—
376	#121	C₂³×C₇	1	4	yes	—
377	#120	—	2	2	no	—
378	C₄×Q₈	—	2	2	no	—
379	C₄×Q₈	—	2	4	no	—
380	C₄×Q₈	—	2	2	no	—
381	C₄×Q₈	—	2	2	no	—
382	C₄×Q₈	C₂³×C₇	1	4	yes	—
383	C₄×D₄	—	2	4	no	—
384	C₄×D₄	—	2	4	no	—
385	C₄×D₄	—	2	2	no	—
386	C₄×D₄	C₂³×C₇	1	4	yes	—
387	#117	—	2	2	no	—
388	#117	—	2	2	no	—
389	#117	—	2	2	no	—
390	#117	C₇×D₄	1	4	yes	—
391	#117	C₂³×C₇	1	2	yes	—
392	#117	C₂³×C₇	1	2	yes	—
393	#117	C₂³×C₇	1	4	yes	—
394	#117	C₂³×C₇	1	2	yes	—
395	#117	C₂³×C₇	1	2	yes	—
396	#117	C₂³×C₇	1	2	yes	—
397	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
398	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
399	C₂×C₂²⋊C₄	—	2	2	no	—
400	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
401	C₂×C₄²	—	2	2	no	—
402	C₂×C₄²	—	2	2	no	—
403	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
404	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
405	#107	C₂³×C₇	1	4	yes	—
406	#107	C₂³×C₇	1	4	yes	—
407	#107	C₂³×C₇	1	4	yes	—
408	#107	C₂³×C₇	1	4	yes	—
409	#106	C₂³×C₇	1	4	yes	—
410	#106	C₂³×C₇	1	4	yes	—
411	#106	C₂³×C₇	1	4	yes	—
412	#106	C₂³×C₇	1	4	yes	—
413	C₄⋊C₈	C₂³×C₇	1	4	yes	—
414	C₄⋊C₈	C₂³×C₇	1	4	yes	—
415	#103	C₂³×C₇	1	4	yes	—
416	#103	C₂³×C₇	1	4	yes	—
417	#103	C₂³×C₇	1	4	yes	—
418	#103	C₂³×C₇	1	4	yes	—
419	#102	C₂³×C₇	1	4	yes	—
420	#102	C₂³×C₇	1	4	yes	—
421	#102	C₂³×C₇	1	4	yes	—
422	#102	C₂³×C₇	1	4	yes	—
423	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
424	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
425	#97	C₂³×C₇	1	2	yes	—
426	#97	C₂³×C₇	1	2	yes	—
427	C₄×C₈	C₂³×C₇	1	2	yes	—
428	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
429	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
430	C₂²×C₇	—	2	2	no	—
431	C₂²×C₇	—	2	4	no	—
432	C₂²×C₇	—	2	2	no	—
433	C₂²×C₇	—	4	4	no	—
434	C₂²×C₇	—	4	2	no	—
435	C₂²×C₇	—	8	2	no	—
436	C₂²×C₇	#536	1	2	yes	—
437	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
438	C₄×C₇	—	2	4	no	—
439	C₄×C₇	—	2	2	no	—
440	C₄×C₇	—	2	2	no	—
441	C₄×C₇	—	2	2	no	—
442	C₄×C₇	—	4	4	no	—
443	C₄×C₇	—	4	4	no	—
444	C₄×C₇	—	4	4	no	—
445	C₄×C₇	—	4	2	no	—
446	C₄×C₇	#536	1	2	yes	—
447	C₄×C₇	C₂×2₋¹⁺⁴	1	2	yes	—
448	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
449	#41	—	2	8	no	—
450	C₂×Q₈	—	2	2	no	—
451	C₂×Q₈	—	2	4	no	—
452	C₂×Q₈	—	2	2	no	—
453	C₂×Q₈	—	2	2	no	—
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	C₂⁴×C₇	1	2	yes	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	4	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
464	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
465	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
466	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
467	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
468	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
469	C₂×C₈	—	2	4	no	—
470	C₂×C₈	—	2	4	no	—
471	C₂×C₈	C₂⁴×C₇	1	2	yes	—
472	C₂×C₈	C₂⁴×C₇	1	2	yes	—
473	C₂×C₈	C₂⁴×C₇	1	4	yes	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	2	no	—
477	C₄⋊C₄	—	2	4	no	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	4	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	4	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	4	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
488	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
489	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
490	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
491	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	2	2	no	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	—	2	4	no	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	—	2	4	no	—
500	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
501	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
502	C₄²	—	2	2	no	—
503	C₄²	—	2	2	no	—
504	C₄²	—	2	2	no	—
505	C₄²	—	2	4	no	—
506	C₄²	—	2	4	no	—
507	C₄²	C₂×C₇×D₄	1	2	yes	—
508	C₄²	C₂×C₇×D₄	1	2	yes	—
509	C₄²	C₂⁴×C₇	1	1	yes	characteristic
510	C₄²	C₂⁴×C₇	1	1	yes	characteristic
511	C₄²	C₂⁴×C₇	1	2	yes	—
512	C₂×C₇	—	2	2	no	—
513	C₂×C₇	—	4	2	no	—
514	C₂×C₇	—	8	2	no	—
515	C₂×C₇	C₂×#536	1	1	yes	characteristic
516	C₂×C₇	#3584	1	2	yes	—
517	C₂³	—	2	2	no	—
518	C₂³	—	2	2	no	—
519	C₂³	—	4	2	no	—
520	C₂³	C₂²×C₇×D₄	1	2	yes	—
521	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
522	Q₈	—	2	4	no	—
523	Q₈	—	8	4	no	—
524	Q₈	—	8	2	no	—
525	Q₈	—	8	2	no	—
526	Q₈	—	8	2	no	—
527	D₄	—	2	4	no	—
528	D₄	—	4	4	no	—
529	D₄	—	8	2	no	—
530	D₄	—	8	4	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	4	2	no	—
539	C₂×C₄	—	4	2	no	—
540	C₂×C₄	—	4	4	no	—
541	C₂×C₄	—	8	4	no	—
542	C₂×C₄	—	8	4	no	—
543	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
544	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
545	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
546	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
547	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
548	C₂×C₄	C₇×2₋¹⁺⁴	1	4	yes	—
549	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
550	C₈	—	2	2	no	—
551	C₈	—	2	2	no	—
552	C₈	—	2	4	no	—
553	C₇	#60859	1	1	yes	characteristic, direct product factor, Sylow
554	C₂²	—	2	2	no	—
555	C₂²	—	2	4	no	—
556	C₂²	—	2	2	no	—
557	C₂²	—	4	4	no	—
558	C₂²	—	4	2	no	—
559	C₂²	—	8	2	no	—
560	C₂²	C₇×#536	1	2	yes	—
561	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
562	C₄	—	2	4	no	—
563	C₄	—	2	2	no	—
564	C₄	—	2	2	no	—
565	C₄	—	2	2	no	—
566	C₄	—	4	4	no	—
567	C₄	—	4	4	no	—
568	C₄	—	4	4	no	—
569	C₄	—	4	2	no	—
570	C₄	C₇×#536	1	2	yes	—
571	C₄	C₂×C₇×2₋¹⁺⁴	1	2	yes	—
572	C₂	—	2	2	no	—
573	C₂	—	4	2	no	—
574	C₂	—	8	2	no	—
575	C₂	C₂×C₇×#536	1	1	yes	characteristic
576	C₂	C₇×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle	C₂²×C₇	C₂³×D₄
prime radical	C₇×#3455	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3458	C₂	semidirect
3. C₇×#3244	C₂	semidirect
4. C₇×#3243	C₂	semidirect
5. C₇×#3242	C₂	semidirect
6. C₇×#3241	C₂	semidirect
7. C₇×#3162	C₂	semidirect
8. C₇×#3160	C₂	semidirect
9. C₇×#3135	C₂	semidirect
10. C₇×#3131	C₂	semidirect
11. C₇×#3091	C₂	semidirect
12. C₇×#3076	C₂	semidirect
13. C₂×C₇×#433	C₂	semidirect
34. C₇×#501	C₂²	semidirect
35. C₇×#501	C₂²	semidirect
36. C₇×#500	C₂²	semidirect
37. C₇×#499	C₂²	semidirect
38. C₇×#498	C₂²	semidirect
39. C₇×#491	C₂²	semidirect
40. C₇×#490	C₂²	semidirect
41. C₇×#489	C₂²	semidirect
42. C₇×#488	C₂²	semidirect
43. C₇×#488	C₂²	semidirect
44. C₇×#487	C₂²	semidirect
45. C₇×#486	C₂²	semidirect
54. C₇×#479	C₂²	semidirect
55. C₇×#479	C₂²	semidirect
56. C₇×#478	C₂²	semidirect
57. C₇×#478	C₂²	semidirect
58. C₇×#477	C₂²	semidirect
59. C₇×#476	C₂²	semidirect
60. C₇×#475	C₂²	semidirect
61. C₇×#474	C₂²	semidirect
73. #60859	C₇	direct (N × H)
166. #3458	C₂×C₇	semidirect
168. #3244	C₂×C₇	semidirect
169. #3243	C₂×C₇	semidirect
170. #3242	C₂×C₇	semidirect
171. #3241	C₂×C₇	semidirect
172. #3162	C₂×C₇	semidirect
173. #3160	C₂×C₇	semidirect
174. #3135	C₂×C₇	semidirect
175. #3131	C₂×C₇	semidirect
176. #3091	C₂×C₇	semidirect
177. #3076	C₂×C₇	semidirect
178. C₂×#433	C₂×C₇	semidirect
263. #501	C₂²×C₇	semidirect
264. #501	C₂²×C₇	semidirect
265. #500	C₂²×C₇	semidirect
266. #499	C₂²×C₇	semidirect
267. #498	C₂²×C₇	semidirect
268. #491	C₂²×C₇	semidirect
269. #490	C₂²×C₇	semidirect
270. #489	C₂²×C₇	semidirect
271. #488	C₂²×C₇	semidirect
272. #488	C₂²×C₇	semidirect
273. #487	C₂²×C₇	semidirect
274. #486	C₂²×C₇	semidirect
283. #479	C₂²×C₇	semidirect
284. #479	C₂²×C₇	semidirect
285. #478	C₂²×C₇	semidirect
286. #478	C₂²×C₇	semidirect
287. #477	C₂²×C₇	semidirect
288. #476	C₂²×C₇	semidirect
289. #475	C₂²×C₇	semidirect
290. #474	C₂²×C₇	semidirect
553. C₇	#60859	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×#3458	2
C₇×#3455	2
C₇×#3244	2
C₇×#3243	2
C₇×#3242	2
C₇×#3241	2
C₇×#3162	2
C₇×#3160	2
C₇×#3135	2
C₇×#3131	2
C₇×#3091	2
C₇×#3076	2
C₂×C₇×#433	2
#60859	7

Subgroups — C7×#60859 #13238075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60859 #13238075