Subgroups — C₇×#61012 #13238228

Order 1792.

1776 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3508	C₂	1	1	yes	maximal, characteristic
3	C₇×#3260	C₂	1	4	yes	maximal, complemented
4	C₇×#3231	C₂	1	2	yes	maximal, complemented
5	C₇×#3228	C₂	1	2	yes	maximal, complemented
6	C₇×#3208	C₂	1	2	yes	maximal, complemented
7	C₇×#3206	C₂	1	2	yes	maximal, complemented
8	C₇×#3200	C₂	1	4	yes	maximal, complemented
9	C₇×#3182	C₂	1	2	yes	maximal, complemented
10	C₇×#3167	C₂	1	2	yes	maximal, complemented
11	C₇×#3126	C₂	1	4	yes	maximal, complemented
12	C₇×#3123	C₂	1	4	yes	maximal, complemented
13	C₇×#2922	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#559	—	2	2	no	—
15	C₇×#557	—	2	2	no	—
16	C₇×#556	—	2	2	no	—
17	C₇×#552	—	2	2	no	—
18	C₇×#552	—	2	2	no	—
19	C₇×#550	—	2	2	no	—
20	C₇×#546	—	2	2	no	—
21	C₇×#546	—	2	2	no	—
22	C₇×#541	C₂²	1	2	yes	—
23	C₇×#541	C₂²	1	2	yes	complemented
24	C₇×#540	C₂²	1	2	yes	—
25	C₇×#540	C₂²	1	2	yes	—
26	C₇×#536	C₂²	1	2	yes	—
27	C₇×#535	C₂²	1	2	yes	—
28	C₇×#533	C₂²	1	2	yes	—
29	C₇×#533	C₂²	1	1	yes	characteristic
30	C₇×#533	C₂²	1	2	yes	complemented
31	C₇×#533	C₂²	1	1	yes	characteristic, complemented
32	C₇×#532	C₂²	1	2	yes	—
33	C₇×#532	C₂²	1	2	yes	—
34	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₇×#127	C₂²	1	1	yes	characteristic
36	C₇×#519	C₂²	1	2	yes	complemented
37	C₇×#518	C₂²	1	2	yes	—
38	C₄×C₇×#41	C₂²	1	1	yes	characteristic
39	C₇×#500	C₂²	1	2	yes	complemented
40	C₇×#498	C₂²	1	2	yes	complemented
41	C₇×#492	C₂²	1	2	yes	—
42	C₇×#492	C₂²	1	2	yes	—
43	C₇×#487	C₂²	1	2	yes	complemented
44	C₇×#487	C₂²	1	2	yes	complemented
45	C₇×#486	C₂²	1	2	yes	—
46	C₇×#486	C₂²	1	2	yes	—
47	C₇×#485	C₂²	1	8	yes	—
48	C₇×#482	C₂²	1	8	yes	complemented
49	C₇×#479	C₂²	1	8	yes	complemented
50	C₇×#478	C₂²	1	8	yes	complemented
51	C₇×#468	C₂²	1	8	yes	complemented
52	C₇×#465	C₂²	1	8	yes	complemented
53	C₇×#461	C₂²	1	8	yes	complemented
54	C₇×#460	C₂²	1	8	yes	—
55	C₇×#450	C₂²	1	8	yes	—
56	C₇×#448	C₂²	1	8	yes	complemented
57	C₇×#440	C₂²	1	8	yes	complemented
58	C₄×C₇×SD₁₆	C₂²	1	8	yes	complemented
59	C₇×#433	C₂²	1	2	yes	—
60	C₇×#433	C₂²	1	2	yes	complemented
61	C₇×#431	C₂²	1	2	yes	—
62	C₇×#431	C₂²	1	2	yes	complemented
63	C₇×#424	C₂²	1	2	yes	complemented
64	C₇×#408	C₂²	1	2	yes	—
65	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic, complemented
66	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
67	#61012	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₇×#41	—	2	2	no	—
69	C₂×C₇×#41	—	2	2	no	—
70	C₂×C₇×#41	C₂³	1	1	yes	characteristic
71	C₂²×C₇×Q₈	C₂³	1	2	yes	—
72	C₂²×C₇×D₄	—	2	2	no	—
73	C₂²×C₇×D₄	C₂³	1	2	yes	—
74	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
75	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
76	C₂×C₇×M₁₆	C₂³	1	2	yes	—
77	C₂²×C₇×C₈	C₂³	1	2	yes	—
78	C₇×C₄⋊Q₈	—	2	2	no	—
79	C₇×C₄⋊Q₈	—	2	2	no	—
80	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
81	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
82	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
83	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
84	C₇×#127	—	2	2	no	—
85	C₇×#127	—	2	2	no	—
86	C₇×#127	C₂³	1	2	yes	—
87	C₇×#127	C₂³	1	2	yes	—
88	C₇×#126	—	2	4	no	—
89	C₇×#126	—	2	4	no	—
90	C₇×#125	—	2	4	no	—
91	C₇×#125	—	2	4	no	—
92	C₇×#125	—	2	2	no	—
93	C₇×#125	—	2	2	no	—
94	C₇×#125	C₂³	1	2	yes	—
95	C₇×#125	C₂³	1	2	yes	—
96	C₇×#124	—	2	2	no	—
97	C₇×#124	—	2	2	no	—
98	C₇×#124	—	2	2	no	—
99	C₇×#124	—	2	2	no	—
100	C₇×#123	—	2	2	no	—
101	C₇×#123	—	2	2	no	—
102	C₇×#123	—	2	4	no	—
103	C₇×#123	—	2	4	no	—
104	C₇×C₂²⋊Q₈	—	2	2	no	—
105	C₇×C₂²⋊Q₈	—	2	2	no	—
106	C₇×C₂²⋊Q₈	—	2	2	no	—
107	C₇×C₂²⋊Q₈	—	2	2	no	—
108	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
109	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
110	C₇×#121	—	2	2	no	—
111	C₇×#121	—	2	2	no	—
112	C₇×#121	—	2	4	no	—
113	C₇×#121	—	2	4	no	—
114	C₇×#121	C₂³	1	4	yes	—
115	C₇×#121	C₂³	1	4	yes	—
116	C₇×#120	—	2	2	no	—
117	C₇×#120	—	2	2	no	—
118	C₄×C₇×Q₈	—	2	4	no	—
119	C₄×C₇×Q₈	—	2	2	no	—
120	C₄×C₇×Q₈	—	2	2	no	—
121	C₄×C₇×Q₈	—	2	2	no	—
122	C₄×C₇×Q₈	—	2	2	no	—
123	C₄×C₇×Q₈	D₄	1	2	yes	—
124	C₄×C₇×Q₈	C₂³	1	4	yes	—
125	C₄×C₇×Q₈	C₂³	1	4	yes	—
126	C₄×C₇×D₄	—	2	4	no	—
127	C₄×C₇×D₄	—	2	2	no	—
128	C₄×C₇×D₄	—	2	2	no	—
129	C₄×C₇×D₄	D₄	1	2	yes	—
130	C₄×C₇×D₄	D₄	1	2	yes	—
131	C₄×C₇×D₄	D₄	1	2	yes	—
132	C₄×C₇×D₄	C₂³	1	4	yes	—
133	C₄×C₇×D₄	C₂³	1	4	yes	—
134	C₇×#117	—	2	2	no	—
135	C₇×#117	—	2	2	no	—
136	C₇×#117	—	2	4	no	—
137	C₇×#117	C₂³	1	2	yes	—
138	C₇×#117	C₂³	1	2	yes	—
139	C₇×#117	C₂³	1	1	yes	characteristic
140	C₇×#117	C₂³	1	1	yes	characteristic
141	C₇×#117	C₂³	1	1	yes	characteristic
142	C₂×C₇×C₄⋊C₄	—	2	2	no	—
143	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
144	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
145	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
146	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
147	C₇×#107	C₂³	1	8	yes	—
148	C₇×#106	C₂³	1	8	yes	—
149	C₇×C₄⋊C₈	C₂³	1	4	yes	—
150	C₇×C₄⋊C₈	C₂³	1	4	yes	—
151	C₇×#103	C₂³	1	8	yes	—
152	C₇×#103	C₂³	1	8	yes	—
153	C₇×#102	C₂³	1	8	yes	—
154	C₇×#102	C₂³	1	8	yes	—
155	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
156	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
157	C₇×#97	C₂³	1	2	yes	—
158	C₄×C₇×C₈	C₂³	1	2	yes	—
159	C₄×C₇×C₈	C₂³	1	2	yes	—
160	C₄×C₇×C₈	C₂³	1	2	yes	—
161	#3509	C₂×C₇	1	1	yes	characteristic, complemented
162	#3508	C₂×C₇	1	1	yes	characteristic
163	#3260	C₂×C₇	1	4	yes	complemented
164	#3231	C₂×C₇	1	2	yes	complemented
165	#3228	C₂×C₇	1	2	yes	complemented
166	#3208	C₂×C₇	1	2	yes	complemented
167	#3206	C₂×C₇	1	2	yes	complemented
168	#3200	C₂×C₇	1	4	yes	complemented
169	#3182	C₂×C₇	1	2	yes	complemented
170	#3167	C₂×C₇	1	2	yes	complemented
171	#3126	C₂×C₇	1	4	yes	complemented
172	#3123	C₂×C₇	1	4	yes	complemented
173	#2922	C₂×C₇	1	1	yes	characteristic, complemented
174	C₂⁴×C₇	—	2	2	no	—
175	C₇×#41	—	4	2	no	—
176	C₇×#41	—	8	4	no	—
177	C₇×#41	—	8	4	no	—
178	C₂×C₇×Q₈	—	2	2	no	—
179	C₂×C₇×Q₈	—	2	2	no	—
180	C₂×C₇×Q₈	—	2	2	no	—
181	C₂×C₇×Q₈	—	4	4	no	—
182	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
183	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
184	C₂×C₇×D₄	—	2	4	no	—
185	C₂×C₇×D₄	—	2	4	no	—
186	C₂×C₇×D₄	—	2	2	no	—
187	C₂×C₇×D₄	—	2	2	no	—
188	C₂×C₇×D₄	—	2	2	no	—
189	C₂×C₇×D₄	—	2	4	no	—
190	C₂×C₇×D₄	—	4	4	no	—
191	C₂×C₇×D₄	—	8	4	no	—
192	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
193	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
194	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
195	C₂×C₇×D₄	C₂⁴	1	4	yes	—
196	C₂²×C₄×C₇	—	2	2	no	—
197	C₂²×C₄×C₇	—	2	4	no	—
198	C₂²×C₄×C₇	—	2	4	no	—
199	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
200	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
201	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
202	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
203	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
204	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
205	C₇×SD₁₆	—	4	8	no	—
206	C₇×SD₁₆	—	4	8	no	—
207	C₇×M₁₆	—	4	2	no	—
208	C₂×C₇×C₈	—	4	2	no	—
209	C₂×C₇×C₈	C₂⁴	1	4	yes	—
210	C₂×C₇×C₈	C₂⁴	1	4	yes	—
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	2	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	—	2	4	no	—
219	C₇×C₄⋊C₄	—	2	4	no	—
220	C₇×C₄⋊C₄	—	2	4	no	—
221	C₇×C₄⋊C₄	—	2	4	no	—
222	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
223	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
224	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
225	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
226	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
227	C₇×C₂²⋊C₄	—	2	2	no	—
228	C₇×C₂²⋊C₄	—	2	2	no	—
229	C₇×C₂²⋊C₄	—	2	4	no	—
230	C₇×C₂²⋊C₄	—	2	4	no	—
231	C₇×C₂²⋊C₄	—	2	4	no	—
232	C₇×C₂²⋊C₄	—	2	4	no	—
233	C₇×C₂²⋊C₄	—	2	4	no	—
234	C₇×C₂²⋊C₄	—	2	4	no	—
235	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
236	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
237	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
238	C₄²×C₇	—	2	4	no	—
239	C₄²×C₇	—	2	4	no	—
240	C₄²×C₇	—	2	2	no	—
241	C₄²×C₇	—	2	2	no	—
242	C₄²×C₇	C₂×D₄	1	2	yes	—
243	C₄²×C₇	C₂×D₄	1	2	yes	—
244	C₄²×C₇	C₂×D₄	1	2	yes	—
245	C₄²×C₇	C₂⁴	1	1	yes	characteristic
246	C₄²×C₇	C₂⁴	1	1	yes	characteristic
247	C₄²×C₇	C₂⁴	1	1	yes	characteristic
248	C₄²×C₇	C₂⁴	1	1	yes	characteristic
249	#559	—	2	2	no	—
250	#557	—	2	2	no	—
251	#556	—	2	2	no	—
252	#552	—	2	2	no	—
253	#552	—	2	2	no	—
254	#550	—	2	2	no	—
255	#546	—	2	2	no	—
256	#546	—	2	2	no	—
257	#541	C₂²×C₇	1	2	yes	—
258	#541	C₂²×C₇	1	2	yes	complemented
259	#540	C₂²×C₇	1	2	yes	—
260	#540	C₂²×C₇	1	2	yes	—
261	#536	C₂²×C₇	1	2	yes	—
262	#535	C₂²×C₇	1	2	yes	—
263	#533	C₂²×C₇	1	2	yes	—
264	#533	C₂²×C₇	1	1	yes	characteristic
265	#533	C₂²×C₇	1	2	yes	complemented
266	#533	C₂²×C₇	1	1	yes	characteristic, complemented
267	#532	C₂²×C₇	1	2	yes	—
268	#532	C₂²×C₇	1	2	yes	—
269	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
270	C₂×#127	C₂²×C₇	1	1	yes	characteristic
271	#519	C₂²×C₇	1	2	yes	complemented
272	#518	C₂²×C₇	1	2	yes	—
273	C₄×#41	C₂²×C₇	1	1	yes	characteristic
274	#500	C₂²×C₇	1	2	yes	complemented
275	#498	C₂²×C₇	1	2	yes	complemented
276	#492	C₂²×C₇	1	2	yes	—
277	#492	C₂²×C₇	1	2	yes	—
278	#487	C₂²×C₇	1	2	yes	complemented
279	#487	C₂²×C₇	1	2	yes	complemented
280	#486	C₂²×C₇	1	2	yes	—
281	#486	C₂²×C₇	1	2	yes	—
282	#485	C₂²×C₇	1	8	yes	—
283	#482	C₂²×C₇	1	8	yes	complemented
284	#479	C₂²×C₇	1	8	yes	complemented
285	#478	C₂²×C₇	1	8	yes	complemented
286	#468	C₂²×C₇	1	8	yes	complemented
287	#465	C₂²×C₇	1	8	yes	complemented
288	#461	C₂²×C₇	1	8	yes	complemented
289	#460	C₂²×C₇	1	8	yes	—
290	#450	C₂²×C₇	1	8	yes	—
291	#448	C₂²×C₇	1	8	yes	complemented
292	#440	C₂²×C₇	1	8	yes	complemented
293	C₄×SD₁₆	C₂²×C₇	1	8	yes	complemented
294	#433	C₂²×C₇	1	2	yes	—
295	#433	C₂²×C₇	1	2	yes	complemented
296	#431	C₂²×C₇	1	2	yes	—
297	#431	C₂²×C₇	1	2	yes	complemented
298	#424	C₂²×C₇	1	2	yes	complemented
299	#408	C₂²×C₇	1	2	yes	—
300	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
301	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic, complemented
302	C₂³×C₇	—	2	4	no	—
303	C₂³×C₇	—	4	4	no	—
304	C₂³×C₇	—	8	4	no	—
305	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
306	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
307	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
308	C₇×Q₈	—	2	8	no	—
309	C₇×Q₈	—	4	1	no	—
310	C₇×Q₈	—	4	4	no	—
311	C₇×Q₈	—	4	4	no	—
312	C₇×Q₈	—	4	4	no	—
313	C₇×D₄	—	2	8	no	—
314	C₇×D₄	—	4	1	no	—
315	C₇×D₄	—	4	1	no	—
316	C₇×D₄	—	4	1	no	—
317	C₇×D₄	—	4	4	no	—
318	C₇×D₄	—	4	4	no	—
319	C₇×D₄	—	4	4	no	—
320	C₇×D₄	—	8	4	no	—
321	C₇×D₄	—	8	4	no	—
322	C₇×D₄	—	8	4	no	—
323	C₂×C₄×C₇	—	2	4	no	—
324	C₂×C₄×C₇	—	2	4	no	—
325	C₂×C₄×C₇	—	2	4	no	—
326	C₂×C₄×C₇	—	2	2	no	—
327	C₂×C₄×C₇	—	2	2	no	—
328	C₂×C₄×C₇	—	2	2	no	—
329	C₂×C₄×C₇	—	2	4	no	—
330	C₂×C₄×C₇	—	2	4	no	—
331	C₂×C₄×C₇	—	2	4	no	—
332	C₂×C₄×C₇	—	4	4	no	—
333	C₂×C₄×C₇	—	8	4	no	—
334	C₂×C₄×C₇	—	8	4	no	—
335	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
336	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
337	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
338	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
339	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
340	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
341	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
342	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
343	C₇×C₈	—	2	4	no	—
344	C₇×C₈	—	2	4	no	—
345	C₂×#41	—	2	2	no	—
346	C₂×#41	—	2	2	no	—
347	C₂×#41	C₂³×C₇	1	1	yes	characteristic
348	C₂²×Q₈	C₂³×C₇	1	2	yes	—
349	C₂²×D₄	—	2	2	no	—
350	C₂²×D₄	C₂³×C₇	1	2	yes	—
351	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
352	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
353	C₂×M₁₆	C₂³×C₇	1	2	yes	—
354	C₂²×C₈	C₂³×C₇	1	2	yes	—
355	C₄⋊Q₈	—	2	2	no	—
356	C₄⋊Q₈	—	2	2	no	—
357	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
358	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
359	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
360	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
361	#127	—	2	2	no	—
362	#127	—	2	2	no	—
363	#127	C₂³×C₇	1	2	yes	—
364	#127	C₂³×C₇	1	2	yes	—
365	#126	—	2	4	no	—
366	#126	—	2	4	no	—
367	#125	—	2	4	no	—
368	#125	—	2	4	no	—
369	#125	—	2	2	no	—
370	#125	—	2	2	no	—
371	#125	C₂³×C₇	1	2	yes	—
372	#125	C₂³×C₇	1	2	yes	—
373	#124	—	2	2	no	—
374	#124	—	2	2	no	—
375	#124	—	2	2	no	—
376	#124	—	2	2	no	—
377	#123	—	2	2	no	—
378	#123	—	2	2	no	—
379	#123	—	2	4	no	—
380	#123	—	2	4	no	—
381	C₂²⋊Q₈	—	2	2	no	—
382	C₂²⋊Q₈	—	2	2	no	—
383	C₂²⋊Q₈	—	2	2	no	—
384	C₂²⋊Q₈	—	2	2	no	—
385	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
386	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
387	#121	—	2	2	no	—
388	#121	—	2	2	no	—
389	#121	—	2	4	no	—
390	#121	—	2	4	no	—
391	#121	C₂³×C₇	1	4	yes	—
392	#121	C₂³×C₇	1	4	yes	—
393	#120	—	2	2	no	—
394	#120	—	2	2	no	—
395	C₄×Q₈	—	2	4	no	—
396	C₄×Q₈	—	2	2	no	—
397	C₄×Q₈	—	2	2	no	—
398	C₄×Q₈	—	2	2	no	—
399	C₄×Q₈	—	2	2	no	—
400	C₄×Q₈	C₇×D₄	1	2	yes	—
401	C₄×Q₈	C₂³×C₇	1	4	yes	—
402	C₄×Q₈	C₂³×C₇	1	4	yes	—
403	C₄×D₄	—	2	4	no	—
404	C₄×D₄	—	2	2	no	—
405	C₄×D₄	—	2	2	no	—
406	C₄×D₄	C₇×D₄	1	2	yes	—
407	C₄×D₄	C₇×D₄	1	2	yes	—
408	C₄×D₄	C₇×D₄	1	2	yes	—
409	C₄×D₄	C₂³×C₇	1	4	yes	—
410	C₄×D₄	C₂³×C₇	1	4	yes	—
411	#117	—	2	2	no	—
412	#117	—	2	2	no	—
413	#117	—	2	4	no	—
414	#117	C₂³×C₇	1	2	yes	—
415	#117	C₂³×C₇	1	2	yes	—
416	#117	C₂³×C₇	1	1	yes	characteristic
417	#117	C₂³×C₇	1	1	yes	characteristic
418	#117	C₂³×C₇	1	1	yes	characteristic
419	C₂×C₄⋊C₄	—	2	2	no	—
420	C₂×C₂²⋊C₄	—	2	4	no	—
421	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
423	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
424	#107	C₂³×C₇	1	8	yes	—
425	#106	C₂³×C₇	1	8	yes	—
426	C₄⋊C₈	C₂³×C₇	1	4	yes	—
427	C₄⋊C₈	C₂³×C₇	1	4	yes	—
428	#103	C₂³×C₇	1	8	yes	—
429	#103	C₂³×C₇	1	8	yes	—
430	#102	C₂³×C₇	1	8	yes	—
431	#102	C₂³×C₇	1	8	yes	—
432	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
433	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
434	#97	C₂³×C₇	1	2	yes	—
435	C₄×C₈	C₂³×C₇	1	2	yes	—
436	C₄×C₈	C₂³×C₇	1	2	yes	—
437	C₄×C₈	C₂³×C₇	1	2	yes	—
438	C₂²×C₇	—	2	1	no	—
439	C₂²×C₇	—	2	1	no	—
440	C₂²×C₇	—	2	1	no	—
441	C₂²×C₇	—	2	2	no	—
442	C₂²×C₇	—	2	2	no	—
443	C₂²×C₇	—	2	2	no	—
444	C₂²×C₇	—	4	4	no	—
445	C₂²×C₇	—	4	4	no	—
446	C₂²×C₇	—	4	4	no	—
447	C₂²×C₇	—	16	4	no	—
448	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
449	C₄×C₇	—	2	1	no	—
450	C₄×C₇	—	2	1	no	—
451	C₄×C₇	—	2	1	no	—
452	C₄×C₇	—	2	2	no	—
453	C₄×C₇	—	2	2	no	—
454	C₄×C₇	—	2	2	no	—
455	C₄×C₇	—	4	4	no	—
456	C₄×C₇	—	4	4	no	—
457	C₄×C₇	—	4	4	no	—
458	C₄×C₇	#577	1	4	yes	—
459	C₄×C₇	#585	1	2	yes	—
460	C₂⁴	—	2	2	no	—
461	#41	—	4	2	no	—
462	#41	—	8	4	no	—
463	#41	—	8	4	no	—
464	C₂×Q₈	—	2	2	no	—
465	C₂×Q₈	—	2	2	no	—
466	C₂×Q₈	—	2	2	no	—
467	C₂×Q₈	—	4	4	no	—
468	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
469	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
470	C₂×D₄	—	2	4	no	—
471	C₂×D₄	—	2	4	no	—
472	C₂×D₄	—	2	2	no	—
473	C₂×D₄	—	2	2	no	—
474	C₂×D₄	—	2	2	no	—
475	C₂×D₄	—	2	4	no	—
476	C₂×D₄	—	4	4	no	—
477	C₂×D₄	—	8	4	no	—
478	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
479	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
480	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
481	C₂×D₄	C₂⁴×C₇	1	4	yes	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
486	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
487	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
488	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
489	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
490	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
491	SD₁₆	—	4	8	no	—
492	SD₁₆	—	4	8	no	—
493	M₁₆	—	4	2	no	—
494	C₂×C₈	—	4	2	no	—
495	C₂×C₈	C₂⁴×C₇	1	4	yes	—
496	C₂×C₈	C₂⁴×C₇	1	4	yes	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	4	no	—
508	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
509	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
510	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
511	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
512	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
513	C₂²⋊C₄	—	2	2	no	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	4	no	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	—	2	4	no	—
519	C₂²⋊C₄	—	2	4	no	—
520	C₂²⋊C₄	—	2	4	no	—
521	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
522	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
523	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
524	C₄²	—	2	4	no	—
525	C₄²	—	2	4	no	—
526	C₄²	—	2	2	no	—
527	C₄²	—	2	2	no	—
528	C₄²	C₂×C₇×D₄	1	2	yes	—
529	C₄²	C₂×C₇×D₄	1	2	yes	—
530	C₄²	C₂×C₇×D₄	1	2	yes	—
531	C₄²	C₂⁴×C₇	1	1	yes	characteristic
532	C₄²	C₂⁴×C₇	1	1	yes	characteristic
533	C₄²	C₂⁴×C₇	1	1	yes	characteristic
534	C₄²	C₂⁴×C₇	1	1	yes	characteristic
535	C₂×C₇	—	4	1	no	—
536	C₂×C₇	—	4	1	no	—
537	C₂×C₇	—	4	1	no	—
538	C₂×C₇	—	8	4	no	—
539	C₂×C₇	#3449	1	1	yes	characteristic
540	C₂×C₇	C₂×#577	1	2	yes	—
541	C₂³	—	2	4	no	—
542	C₂³	—	4	4	no	—
543	C₂³	—	8	4	no	—
544	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
545	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
546	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
547	Q₈	—	2	8	no	—
548	Q₈	—	4	1	no	—
549	Q₈	—	4	4	no	—
550	Q₈	—	4	4	no	—
551	Q₈	—	4	4	no	—
552	D₄	—	2	8	no	—
553	D₄	—	4	1	no	—
554	D₄	—	4	1	no	—
555	D₄	—	4	1	no	—
556	D₄	—	4	4	no	—
557	D₄	—	4	4	no	—
558	D₄	—	4	4	no	—
559	D₄	—	8	4	no	—
560	D₄	—	8	4	no	—
561	D₄	—	8	4	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	4	4	no	—
572	C₂×C₄	—	8	4	no	—
573	C₂×C₄	—	8	4	no	—
574	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
575	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
576	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
577	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
578	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
579	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
580	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
581	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
582	C₈	—	2	4	no	—
583	C₈	—	2	4	no	—
584	C₇	#61012	1	1	yes	characteristic, direct product factor, Sylow
585	C₂²	—	2	2	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	1	no	—
589	C₂²	—	2	1	no	—
590	C₂²	—	2	1	no	—
591	C₂²	—	4	4	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	4	4	no	—
594	C₂²	—	16	4	no	—
595	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
596	C₄	—	2	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	2	2	no	—
599	C₄	—	2	1	no	—
600	C₄	—	2	1	no	—
601	C₄	—	2	1	no	—
602	C₄	—	4	4	no	—
603	C₄	—	4	4	no	—
604	C₄	—	4	4	no	—
605	C₄	C₇×#577	1	4	yes	—
606	C₄	C₇×#585	1	2	yes	—
607	C₂	—	4	1	no	—
608	C₂	—	4	1	no	—
609	C₂	—	4	1	no	—
610	C₂	—	8	4	no	—
611	C₂	C₇×#3449	1	1	yes	characteristic
612	C₂	C₂×C₇×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₇×#3508	C₂
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×#3509	C₂	semidirect
3. C₇×#3260	C₂	semidirect
4. C₇×#3231	C₂	semidirect
5. C₇×#3228	C₂	semidirect
6. C₇×#3208	C₂	semidirect
7. C₇×#3206	C₂	semidirect
8. C₇×#3200	C₂	semidirect
9. C₇×#3182	C₂	semidirect
10. C₇×#3167	C₂	semidirect
11. C₇×#3126	C₂	semidirect
12. C₇×#3123	C₂	semidirect
13. C₇×#2922	C₂	semidirect
23. C₇×#541	C₂²	semidirect
30. C₇×#533	C₂²	semidirect
31. C₇×#533	C₂²	semidirect
34. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
36. C₇×#519	C₂²	semidirect
39. C₇×#500	C₂²	semidirect
40. C₇×#498	C₂²	semidirect
43. C₇×#487	C₂²	semidirect
44. C₇×#487	C₂²	semidirect
48. C₇×#482	C₂²	semidirect
49. C₇×#479	C₂²	semidirect
50. C₇×#478	C₂²	semidirect
51. C₇×#468	C₂²	semidirect
52. C₇×#465	C₂²	semidirect
53. C₇×#461	C₂²	semidirect
56. C₇×#448	C₂²	semidirect
57. C₇×#440	C₂²	semidirect
58. C₄×C₇×SD₁₆	C₂²	semidirect
60. C₇×#433	C₂²	semidirect
62. C₇×#431	C₂²	semidirect
63. C₇×#424	C₂²	semidirect
65. C₄×C₇×M₁₆	C₂²	semidirect
66. C₂×C₄×C₇×C₈	C₂²	semidirect
67. #61012	C₇	direct (N × H)
161. #3509	C₂×C₇	semidirect
163. #3260	C₂×C₇	semidirect
164. #3231	C₂×C₇	semidirect
165. #3228	C₂×C₇	semidirect
166. #3208	C₂×C₇	semidirect
167. #3206	C₂×C₇	semidirect
168. #3200	C₂×C₇	semidirect
169. #3182	C₂×C₇	semidirect
170. #3167	C₂×C₇	semidirect
171. #3126	C₂×C₇	semidirect
172. #3123	C₂×C₇	semidirect
173. #2922	C₂×C₇	semidirect
258. #541	C₂²×C₇	semidirect
265. #533	C₂²×C₇	semidirect
266. #533	C₂²×C₇	semidirect
269. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
271. #519	C₂²×C₇	semidirect
274. #500	C₂²×C₇	semidirect
275. #498	C₂²×C₇	semidirect
278. #487	C₂²×C₇	semidirect
279. #487	C₂²×C₇	semidirect
283. #482	C₂²×C₇	semidirect
284. #479	C₂²×C₇	semidirect
285. #478	C₂²×C₇	semidirect
286. #468	C₂²×C₇	semidirect
287. #465	C₂²×C₇	semidirect
288. #461	C₂²×C₇	semidirect
291. #448	C₂²×C₇	semidirect
292. #440	C₂²×C₇	semidirect
293. C₄×SD₁₆	C₂²×C₇	semidirect
295. #433	C₂²×C₇	semidirect
297. #431	C₂²×C₇	semidirect
298. #424	C₂²×C₇	semidirect
300. C₄×M₁₆	C₂²×C₇	semidirect
301. C₂×C₄×C₈	C₂²×C₇	semidirect
584. C₇	#61012	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×#3509	2
C₇×#3508	2
C₇×#3260	2
C₇×#3231	2
C₇×#3228	2
C₇×#3208	2
C₇×#3206	2
C₇×#3200	2
C₇×#3182	2
C₇×#3167	2
C₇×#3126	2
C₇×#3123	2
C₇×#2922	2
#61012	7

Subgroups — C7×#61012 #13238228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61012 #13238228