Subgroups — C₇×#60712 #13237928

Order 1792.

1950 conjugacy classes of subgroups.

The subgroup lattice has 496 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×#3182	C₂	1	4	yes	maximal, complemented
4	C₇×#3174	C₂	1	8	yes	maximal, complemented
5	C₇×#3167	C₂	1	4	yes	maximal, complemented
6	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#492	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3139	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3057	C₂	1	4	yes	maximal, complemented
11	C₇×#2949	C₂	1	4	yes	maximal, complemented
12	C₄×C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×C₇×SD₁₆	C₂²	1	4	yes	—
14	C₂×C₇×#131	C₂²	1	2	yes	—
15	C₇×#559	—	2	2	no	—
16	C₇×#557	—	2	2	no	—
17	C₇×#556	—	2	2	no	—
18	C₇×#552	—	2	4	no	—
19	C₇×#550	—	2	2	no	—
20	C₇×#546	—	2	4	no	—
21	C₇×#541	C₂²	1	4	yes	complemented
22	C₇×#540	C₂²	1	4	yes	—
23	C₇×#536	C₂²	1	2	yes	—
24	C₇×#535	C₂²	1	2	yes	—
25	C₇×#533	C₂²	1	1	yes	characteristic
26	C₇×#533	C₂²	1	1	yes	characteristic
27	C₇×#533	C₂²	1	4	yes	complemented
28	C₇×#532	C₂²	1	4	yes	—
29	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
30	C₂×C₇×#127	C₂²	1	1	yes	characteristic
31	C₇×#519	C₂²	1	2	yes	—
32	C₇×#518	C₂²	1	2	yes	—
33	C₄×C₇×#41	C₂²	1	1	yes	characteristic
34	C₇×#500	C₂²	1	2	yes	complemented
35	C₇×#499	C₂²	1	4	yes	complemented
36	C₇×#498	C₂²	1	2	yes	complemented
37	C₇×#492	C₂²	1	2	yes	—
38	C₇×#492	C₂²	1	4	yes	—
39	C₇×#492	C₂²	1	2	yes	—
40	C₇×#491	C₂²	1	4	yes	complemented
41	C₇×#490	C₂²	1	4	yes	—
42	C₇×#487	C₂²	1	4	yes	complemented
43	C₇×#486	C₂²	1	4	yes	—
44	C₇×#468	C₂²	1	16	yes	complemented
45	C₇×#465	C₂²	1	16	yes	complemented
46	C₇×#440	C₂²	1	16	yes	complemented
47	C₄×C₇×SD₁₆	C₂²	1	16	yes	complemented
48	C₇×#427	C₂²	1	4	yes	—
49	C₇×#419	C₂²	1	4	yes	—
50	C₇×#418	C₂²	1	4	yes	—
51	C₇×#405	C₂²	1	2	yes	—
52	C₇×#405	C₂²	1	4	yes	complemented
53	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
54	C₄×C₇×M₁₆	C₂²	1	2	yes	complemented
55	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
56	C₂×C₄×C₇×C₈	C₂²	1	2	yes	complemented
57	#60712	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂×C₇×#41	—	2	4	no	—
59	C₂×C₇×#41	C₂³	1	1	yes	characteristic
60	C₂²×C₇×Q₈	C₂³	1	2	yes	—
61	C₂²×C₇×D₄	—	2	2	no	—
62	C₂²×C₇×D₄	C₂³	1	2	yes	—
63	C₂×C₇×SD₁₆	—	2	16	no	—
64	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
65	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
66	C₇×#131	—	2	8	no	—
67	C₂×C₇×M₁₆	C₂³	1	4	yes	—
68	C₂×C₇×M₁₆	C₂³	1	2	yes	—
69	C₂²×C₇×C₈	C₂³	1	4	yes	—
70	C₂²×C₇×C₈	C₂³	1	2	yes	—
71	C₇×C₄⋊Q₈	—	2	2	no	—
72	C₇×C₄⋊Q₈	—	2	2	no	—
73	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
74	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
75	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
76	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
77	C₇×#127	—	2	2	no	—
78	C₇×#127	—	2	2	no	—
79	C₇×#127	C₂³	1	2	yes	—
80	C₇×#127	C₂³	1	2	yes	—
81	C₇×#126	—	2	8	no	—
82	C₇×#125	—	2	8	no	—
83	C₇×#125	—	2	4	no	—
84	C₇×#125	C₂³	1	2	yes	—
85	C₇×#125	C₂³	1	2	yes	—
86	C₇×#124	—	2	2	no	—
87	C₇×#124	—	2	2	no	—
88	C₇×#124	—	2	4	no	—
89	C₇×#123	—	2	4	no	—
90	C₇×#123	—	2	8	no	—
91	C₇×C₂²⋊Q₈	—	2	2	no	—
92	C₇×C₂²⋊Q₈	—	2	2	no	—
93	C₇×C₂²⋊Q₈	—	2	4	no	—
94	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
95	C₇×#121	—	2	4	no	—
96	C₇×#121	—	2	8	no	—
97	C₇×#121	C₂³	1	8	yes	—
98	C₇×#120	—	2	4	no	—
99	C₄×C₇×Q₈	—	2	4	no	—
100	C₄×C₇×Q₈	—	2	2	no	—
101	C₄×C₇×Q₈	—	2	2	no	—
102	C₄×C₇×Q₈	—	2	4	no	—
103	C₄×C₇×Q₈	D₄	1	2	yes	—
104	C₄×C₇×Q₈	C₂³	1	8	yes	—
105	C₄×C₇×D₄	—	2	4	no	—
106	C₄×C₇×D₄	—	2	4	no	—
107	C₄×C₇×D₄	D₄	1	4	yes	—
108	C₄×C₇×D₄	D₄	1	2	yes	—
109	C₄×C₇×D₄	C₂³	1	8	yes	—
110	C₇×#117	—	2	4	no	—
111	C₇×#117	—	2	4	no	—
112	C₇×#117	C₂³	1	2	yes	—
113	C₇×#117	C₂³	1	2	yes	—
114	C₇×#117	C₂³	1	1	yes	characteristic
115	C₇×#117	C₂³	1	2	yes	—
116	C₂×C₇×C₄⋊C₄	—	2	2	no	—
117	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
118	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₇	C₂³	1	2	yes	—
120	C₇×#107	C₂³	1	8	yes	—
121	C₇×#106	C₂³	1	8	yes	—
122	C₇×#103	C₂³	1	16	yes	—
123	C₇×#102	C₂³	1	16	yes	—
124	C₇×#97	C₂³	1	2	yes	—
125	C₇×#97	C₂³	1	4	yes	—
126	C₄×C₇×C₈	C₂³	1	2	yes	—
127	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
128	C₄×C₇×C₈	C₂³	1	2	yes	—
129	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
130	C₄×C₇×C₈	C₂³	1	4	yes	—
131	#3509	C₂×C₇	1	1	yes	characteristic, complemented
132	#3508	C₂×C₇	1	1	yes	characteristic
133	#3182	C₂×C₇	1	4	yes	complemented
134	#3174	C₂×C₇	1	8	yes	complemented
135	#3167	C₂×C₇	1	4	yes	complemented
136	#3160	C₂×C₇	1	1	yes	characteristic, complemented
137	C₂×#492	C₂×C₇	1	1	yes	characteristic, complemented
138	#3139	C₂×C₇	1	1	yes	characteristic, complemented
139	#3138	C₂×C₇	1	1	yes	characteristic, complemented
140	#3057	C₂×C₇	1	4	yes	complemented
141	#2949	C₂×C₇	1	4	yes	complemented
142	C₄×#131	C₂×C₇	1	1	yes	characteristic, complemented
143	C₂⁴×C₇	—	2	2	no	—
144	C₇×#41	—	2	4	no	—
145	C₇×#41	—	8	8	no	—
146	C₂×C₇×Q₈	—	2	2	no	—
147	C₂×C₇×Q₈	—	2	2	no	—
148	C₂×C₇×Q₈	—	2	2	no	—
149	C₂×C₇×Q₈	—	2	8	no	—
150	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
151	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
152	C₂×C₇×D₄	—	2	8	no	—
153	C₂×C₇×D₄	—	2	2	no	—
154	C₂×C₇×D₄	—	2	4	no	—
155	C₂×C₇×D₄	—	2	4	no	—
156	C₂×C₇×D₄	—	2	8	no	—
157	C₂×C₇×D₄	—	8	4	no	—
158	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
159	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
160	C₂×C₇×D₄	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₇	—	2	2	no	—
162	C₂²×C₄×C₇	—	2	8	no	—
163	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
165	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
167	C₇×SD₁₆	—	2	16	no	—
168	C₇×SD₁₆	—	2	16	no	—
169	C₇×M₁₆	—	2	4	no	—
170	C₇×M₁₆	—	2	8	no	—
171	C₂×C₇×C₈	—	2	8	no	—
172	C₂×C₇×C₈	#41	1	8	yes	complemented
173	C₂×C₇×C₈	C₂⁴	1	4	yes	—
174	C₂×C₇×C₈	C₂⁴	1	2	yes	—
175	C₂×C₇×C₈	C₂⁴	1	2	yes	—
176	C₇×C₄⋊C₄	—	2	4	no	—
177	C₇×C₄⋊C₄	—	2	4	no	—
178	C₇×C₄⋊C₄	—	2	2	no	—
179	C₇×C₄⋊C₄	—	2	8	no	—
180	C₇×C₄⋊C₄	—	2	2	no	—
181	C₇×C₄⋊C₄	—	2	4	no	—
182	C₇×C₄⋊C₄	—	2	4	no	—
183	C₇×C₄⋊C₄	—	2	8	no	—
184	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
185	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
186	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
187	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
188	C₇×C₂²⋊C₄	—	2	2	no	—
189	C₇×C₂²⋊C₄	—	2	2	no	—
190	C₇×C₂²⋊C₄	—	2	8	no	—
191	C₇×C₂²⋊C₄	—	2	8	no	—
192	C₇×C₂²⋊C₄	—	2	8	no	—
193	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
194	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
195	C₄²×C₇	—	2	8	no	—
196	C₄²×C₇	—	2	2	no	—
197	C₄²×C₇	—	2	2	no	—
198	C₄²×C₇	C₂×D₄	1	4	yes	—
199	C₄²×C₇	C₂×D₄	1	2	yes	—
200	C₄²×C₇	C₂⁴	1	1	yes	characteristic
201	C₄²×C₇	C₂⁴	1	2	yes	—
202	C₄²×C₇	C₂⁴	1	1	yes	characteristic
203	C₂²×SD₁₆	C₂²×C₇	1	4	yes	—
204	C₂×#131	C₂²×C₇	1	2	yes	—
205	#559	—	2	2	no	—
206	#557	—	2	2	no	—
207	#556	—	2	2	no	—
208	#552	—	2	4	no	—
209	#550	—	2	2	no	—
210	#546	—	2	4	no	—
211	#541	C₂²×C₇	1	4	yes	complemented
212	#540	C₂²×C₇	1	4	yes	—
213	#536	C₂²×C₇	1	2	yes	—
214	#535	C₂²×C₇	1	2	yes	—
215	#533	C₂²×C₇	1	1	yes	characteristic
216	#533	C₂²×C₇	1	1	yes	characteristic
217	#533	C₂²×C₇	1	4	yes	complemented
218	#532	C₂²×C₇	1	4	yes	—
219	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
220	C₂×#127	C₂²×C₇	1	1	yes	characteristic
221	#519	C₂²×C₇	1	2	yes	—
222	#518	C₂²×C₇	1	2	yes	—
223	C₄×#41	C₂²×C₇	1	1	yes	characteristic
224	#500	C₂²×C₇	1	2	yes	complemented
225	#499	C₂²×C₇	1	4	yes	complemented
226	#498	C₂²×C₇	1	2	yes	complemented
227	#492	C₂²×C₇	1	2	yes	—
228	#492	C₂²×C₇	1	4	yes	—
229	#492	C₂²×C₇	1	2	yes	—
230	#491	C₂²×C₇	1	4	yes	complemented
231	#490	C₂²×C₇	1	4	yes	—
232	#487	C₂²×C₇	1	4	yes	complemented
233	#486	C₂²×C₇	1	4	yes	—
234	#468	C₂²×C₇	1	16	yes	complemented
235	#465	C₂²×C₇	1	16	yes	complemented
236	#440	C₂²×C₇	1	16	yes	complemented
237	C₄×SD₁₆	C₂²×C₇	1	16	yes	complemented
238	#427	C₂²×C₇	1	4	yes	—
239	#419	C₂²×C₇	1	4	yes	—
240	#418	C₂²×C₇	1	4	yes	—
241	#405	C₂²×C₇	1	2	yes	—
242	#405	C₂²×C₇	1	4	yes	complemented
243	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
244	C₄×M₁₆	C₂²×C₇	1	2	yes	complemented
245	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
246	C₂×C₄×C₈	C₂²×C₇	1	2	yes	complemented
247	C₂³×C₇	—	2	4	no	—
248	C₂³×C₇	—	4	4	no	—
249	C₂³×C₇	—	8	4	no	—
250	C₂³×C₇	C₂²×D₄	1	2	yes	—
251	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
252	C₇×Q₈	—	2	2	no	—
253	C₇×Q₈	—	2	8	no	—
254	C₇×Q₈	—	4	4	no	—
255	C₇×Q₈	—	4	4	no	—
256	C₇×Q₈	—	4	4	no	—
257	C₇×D₄	—	2	4	no	—
258	C₇×D₄	—	2	2	no	—
259	C₇×D₄	—	2	8	no	—
260	C₇×D₄	—	4	4	no	—
261	C₇×D₄	—	4	8	no	—
262	C₇×D₄	—	8	4	no	—
263	C₇×D₄	—	8	8	no	—
264	C₂×C₄×C₇	—	2	4	no	—
265	C₂×C₄×C₇	—	2	8	no	—
266	C₂×C₄×C₇	—	2	4	no	—
267	C₂×C₄×C₇	—	2	4	no	—
268	C₂×C₄×C₇	—	2	4	no	—
269	C₂×C₄×C₇	—	2	4	no	—
270	C₂×C₄×C₇	—	4	4	no	—
271	C₂×C₄×C₇	—	8	8	no	—
272	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
273	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
274	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
275	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
277	C₂×C₄×C₇	C₂×#41	1	4	yes	—
278	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
279	C₇×C₈	—	2	4	no	—
280	C₇×C₈	C₂×#41	1	4	yes	—
281	C₇×C₈	C₂×#41	1	4	yes	—
282	C₂×#41	—	2	4	no	—
283	C₂×#41	C₂³×C₇	1	1	yes	characteristic
284	C₂²×Q₈	C₂³×C₇	1	2	yes	—
285	C₂²×D₄	—	2	2	no	—
286	C₂²×D₄	C₂³×C₇	1	2	yes	—
287	C₂×SD₁₆	—	2	16	no	—
288	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
289	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
290	#131	—	2	8	no	—
291	C₂×M₁₆	C₂³×C₇	1	2	yes	—
292	C₂×M₁₆	C₂³×C₇	1	4	yes	—
293	C₂²×C₈	C₂³×C₇	1	4	yes	—
294	C₂²×C₈	C₂³×C₇	1	2	yes	—
295	C₄⋊Q₈	—	2	2	no	—
296	C₄⋊Q₈	—	2	2	no	—
297	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
298	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
299	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
300	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
301	#127	—	2	2	no	—
302	#127	—	2	2	no	—
303	#127	C₂³×C₇	1	2	yes	—
304	#127	C₂³×C₇	1	2	yes	—
305	#126	—	2	8	no	—
306	#125	—	2	8	no	—
307	#125	—	2	4	no	—
308	#125	C₂³×C₇	1	2	yes	—
309	#125	C₂³×C₇	1	2	yes	—
310	#124	—	2	2	no	—
311	#124	—	2	2	no	—
312	#124	—	2	4	no	—
313	#123	—	2	4	no	—
314	#123	—	2	8	no	—
315	C₂²⋊Q₈	—	2	2	no	—
316	C₂²⋊Q₈	—	2	2	no	—
317	C₂²⋊Q₈	—	2	4	no	—
318	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
319	#121	—	2	4	no	—
320	#121	—	2	8	no	—
321	#121	C₂³×C₇	1	8	yes	—
322	#120	—	2	4	no	—
323	C₄×Q₈	—	2	4	no	—
324	C₄×Q₈	—	2	2	no	—
325	C₄×Q₈	—	2	2	no	—
326	C₄×Q₈	—	2	4	no	—
327	C₄×Q₈	C₇×D₄	1	2	yes	—
328	C₄×Q₈	C₂³×C₇	1	8	yes	—
329	C₄×D₄	—	2	4	no	—
330	C₄×D₄	—	2	4	no	—
331	C₄×D₄	C₇×D₄	1	4	yes	—
332	C₄×D₄	C₇×D₄	1	2	yes	—
333	C₄×D₄	C₂³×C₇	1	8	yes	—
334	#117	—	2	4	no	—
335	#117	—	2	4	no	—
336	#117	C₂³×C₇	1	2	yes	—
337	#117	C₂³×C₇	1	2	yes	—
338	#117	C₂³×C₇	1	1	yes	characteristic
339	#117	C₂³×C₇	1	2	yes	—
340	C₂×C₄⋊C₄	—	2	2	no	—
341	C₂×C₂²⋊C₄	—	2	4	no	—
342	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
343	C₂×C₄²	C₂³×C₇	1	2	yes	—
344	#107	C₂³×C₇	1	8	yes	—
345	#106	C₂³×C₇	1	8	yes	—
346	#103	C₂³×C₇	1	16	yes	—
347	#102	C₂³×C₇	1	16	yes	—
348	#97	C₂³×C₇	1	2	yes	—
349	#97	C₂³×C₇	1	4	yes	—
350	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
351	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
352	C₄×C₈	C₂³×C₇	1	2	yes	—
353	C₄×C₈	C₂³×C₇	1	2	yes	—
354	C₄×C₈	C₂³×C₇	1	4	yes	—
355	C₂²×C₇	—	2	2	no	—
356	C₂²×C₇	—	2	2	no	—
357	C₂²×C₇	—	2	4	no	—
358	C₂²×C₇	—	4	4	no	—
359	C₂²×C₇	—	4	8	no	—
360	C₂²×C₇	—	16	4	no	—
361	C₂²×C₇	#532	1	2	yes	—
362	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
363	C₄×C₇	—	2	2	no	—
364	C₄×C₇	—	2	4	no	—
365	C₄×C₇	—	2	2	no	—
366	C₄×C₇	—	4	4	no	—
367	C₄×C₇	—	4	4	no	—
368	C₄×C₇	—	4	4	no	—
369	C₄×C₇	#532	1	2	yes	—
370	C₄×C₇	#577	1	4	yes	—
371	C₄×C₇	C₂²×#41	1	2	yes	—
372	C₂⁴	—	2	2	no	—
373	#41	—	2	4	no	—
374	#41	—	8	8	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	2	2	no	—
377	C₂×Q₈	—	2	2	no	—
378	C₂×Q₈	—	2	8	no	—
379	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
380	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
381	C₂×D₄	—	2	8	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	4	no	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	2	8	no	—
386	C₂×D₄	—	8	4	no	—
387	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
388	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
389	C₂×D₄	C₂⁴×C₇	1	4	yes	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	8	no	—
392	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
393	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
394	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
395	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
396	SD₁₆	—	2	16	no	—
397	SD₁₆	—	2	16	no	—
398	M₁₆	—	2	4	no	—
399	M₁₆	—	2	8	no	—
400	C₂×C₈	—	2	8	no	—
401	C₂×C₈	C₇×#41	1	8	yes	complemented
402	C₂×C₈	C₂⁴×C₇	1	4	yes	—
403	C₂×C₈	C₂⁴×C₇	1	2	yes	—
404	C₂×C₈	C₂⁴×C₇	1	2	yes	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	8	no	—
413	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
414	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
415	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
416	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
417	C₂²⋊C₄	—	2	2	no	—
418	C₂²⋊C₄	—	2	2	no	—
419	C₂²⋊C₄	—	2	8	no	—
420	C₂²⋊C₄	—	2	8	no	—
421	C₂²⋊C₄	—	2	8	no	—
422	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
423	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
424	C₄²	—	2	8	no	—
425	C₄²	—	2	2	no	—
426	C₄²	—	2	2	no	—
427	C₄²	C₂×C₇×D₄	1	4	yes	—
428	C₄²	C₂×C₇×D₄	1	2	yes	—
429	C₄²	C₂⁴×C₇	1	1	yes	characteristic
430	C₄²	C₂⁴×C₇	1	2	yes	—
431	C₄²	C₂⁴×C₇	1	1	yes	characteristic
432	C₂×C₇	—	2	2	no	—
433	C₂×C₇	—	4	2	no	—
434	C₂×C₇	—	8	4	no	—
435	C₂×C₇	C₂×#532	1	1	yes	characteristic
436	C₂×C₇	C₂×#577	1	2	yes	—
437	C₂³	—	2	4	no	—
438	C₂³	—	4	4	no	—
439	C₂³	—	8	4	no	—
440	C₂³	C₂²×C₇×D₄	1	2	yes	—
441	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
442	Q₈	—	2	2	no	—
443	Q₈	—	2	8	no	—
444	Q₈	—	4	4	no	—
445	Q₈	—	4	4	no	—
446	Q₈	—	4	4	no	—
447	D₄	—	2	4	no	—
448	D₄	—	2	8	no	—
449	D₄	—	2	2	no	—
450	D₄	—	4	4	no	—
451	D₄	—	4	8	no	—
452	D₄	—	8	4	no	—
453	D₄	—	8	8	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	8	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	4	4	no	—
461	C₂×C₄	—	8	8	no	—
462	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
463	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
464	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
465	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
466	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
467	C₂×C₄	C₂×C₇×#41	1	4	yes	—
468	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
469	C₈	—	2	4	no	—
470	C₈	C₂×C₇×#41	1	4	yes	—
471	C₈	C₂×C₇×#41	1	4	yes	—
472	C₇	#60712	1	1	yes	characteristic, direct product factor, Sylow
473	C₂²	—	2	2	no	—
474	C₂²	—	2	4	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	4	8	no	—
477	C₂²	—	4	4	no	—
478	C₂²	—	16	4	no	—
479	C₂²	C₇×#532	1	2	yes	—
480	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
481	C₄	—	2	4	no	—
482	C₄	—	2	2	no	—
483	C₄	—	2	2	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	4	no	—
486	C₄	—	4	4	no	—
487	C₄	C₇×#532	1	2	yes	—
488	C₄	C₇×#577	1	4	yes	—
489	C₄	C₂²×C₇×#41	1	2	yes	—
490	C₂	—	2	2	no	—
491	C₂	—	4	2	no	—
492	C₂	—	8	4	no	—
493	C₂	C₂×C₇×#532	1	1	yes	characteristic
494	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×#3182	C₂	semidirect
4. C₇×#3174	C₂	semidirect
5. C₇×#3167	C₂	semidirect
6. C₇×#3160	C₂	semidirect
7. C₂×C₇×#492	C₂	semidirect
8. C₇×#3139	C₂	semidirect
9. C₇×#3138	C₂	semidirect
10. C₇×#3057	C₂	semidirect
11. C₇×#2949	C₂	semidirect
12. C₄×C₇×#131	C₂	semidirect
21. C₇×#541	C₂²	semidirect
27. C₇×#533	C₂²	semidirect
34. C₇×#500	C₂²	semidirect
35. C₇×#499	C₂²	semidirect
36. C₇×#498	C₂²	semidirect
40. C₇×#491	C₂²	semidirect
42. C₇×#487	C₂²	semidirect
44. C₇×#468	C₂²	semidirect
45. C₇×#465	C₂²	semidirect
46. C₇×#440	C₂²	semidirect
47. C₄×C₇×SD₁₆	C₂²	semidirect
52. C₇×#405	C₂²	semidirect
54. C₄×C₇×M₁₆	C₂²	semidirect
56. C₂×C₄×C₇×C₈	C₂²	semidirect
57. #60712	C₇	direct (N × H)
131. #3509	C₂×C₇	semidirect
133. #3182	C₂×C₇	semidirect
134. #3174	C₂×C₇	semidirect
135. #3167	C₂×C₇	semidirect
136. #3160	C₂×C₇	semidirect
137. C₂×#492	C₂×C₇	semidirect
138. #3139	C₂×C₇	semidirect
139. #3138	C₂×C₇	semidirect
140. #3057	C₂×C₇	semidirect
141. #2949	C₂×C₇	semidirect
142. C₄×#131	C₂×C₇	semidirect
172. C₂×C₇×C₈	#41	semidirect
211. #541	C₂²×C₇	semidirect
217. #533	C₂²×C₇	semidirect
224. #500	C₂²×C₇	semidirect
225. #499	C₂²×C₇	semidirect
226. #498	C₂²×C₇	semidirect
230. #491	C₂²×C₇	semidirect
232. #487	C₂²×C₇	semidirect
234. #468	C₂²×C₇	semidirect
235. #465	C₂²×C₇	semidirect
236. #440	C₂²×C₇	semidirect
237. C₄×SD₁₆	C₂²×C₇	semidirect
242. #405	C₂²×C₇	semidirect
244. C₄×M₁₆	C₂²×C₇	semidirect
246. C₂×C₄×C₈	C₂²×C₇	semidirect
401. C₂×C₈	C₇×#41	semidirect
472. C₇	#60712	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×#3182	2
C₇×#3174	2
C₇×#3167	2
C₇×#3160	2
C₂×C₇×#492	2
C₇×#3139	2
C₇×#3138	2
C₇×#3057	2
C₇×#2949	2
C₄×C₇×#131	2
#60712	7

Subgroups — C7×#60712 #13237928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60712 #13237928