Subgroups — C₇×#10797 #13188013

Order 1792.

918 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#336	C₂	1	2	yes	maximal
3	C₂×C₇×#336	C₂	1	2	yes	maximal
4	C₂×C₇×#336	C₂	1	1	yes	maximal, characteristic
5	C₄×C₇×#95	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₇×C₈	C₂²	1	2	yes	—
7	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	—	2	4	no	—
12	C₂×C₄×C₇×C₈	C₄	1	4	yes	complemented
13	C₂×C₇×#95	C₄	1	2	yes	—
14	C₂×C₇×#95	C₄	1	2	yes	—
15	C₇×#336	—	2	4	no	—
16	C₇×#336	—	2	4	no	—
17	C₇×#336	C₄	1	4	yes	complemented
18	#10797	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂³×C₄×C₇	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂²×C₇×C₈	—	2	2	no	—
24	C₂²×C₇×C₈	—	2	2	no	—
25	C₂²×C₇×C₈	—	2	2	no	—
26	C₂²×C₇×C₈	—	2	1	no	—
27	C₂²×C₇×C₈	—	2	1	no	—
28	C₂²×C₇×C₈	—	2	2	no	—
29	C₂²×C₇×C₈	—	2	2	no	—
30	C₂²×C₇×C₈	—	2	2	no	—
31	C₂²×C₇×C₈	—	2	2	no	—
32	C₂²×C₇×C₈	—	2	2	no	—
33	C₂²×C₇×C₈	—	2	2	no	—
34	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
35	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
37	C₂²×C₇×C₈	D₄	1	4	yes	—
38	C₂²×C₇×C₈	D₄	1	2	yes	—
39	C₂²×C₇×C₈	Q₈	1	2	yes	—
40	C₂×C₄²×C₇	—	2	2	no	—
41	C₂×C₄²×C₇	—	2	2	no	—
42	C₂×C₄²×C₇	—	2	2	no	—
43	C₂×C₄²×C₇	—	2	2	no	—
44	C₂×C₄²×C₇	—	2	2	no	—
45	C₂×C₄²×C₇	—	2	2	no	—
46	C₂×C₄²×C₇	—	2	2	no	—
47	C₂×C₄²×C₇	—	2	2	no	—
48	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
50	C₄×C₇×C₈	—	2	8	no	—
51	C₇×#95	—	2	4	no	—
52	C₇×#95	C₈	1	8	yes	complemented
53	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
54	C₂×#336	C₂×C₇	1	2	yes	—
55	C₂×#336	C₂×C₇	1	2	yes	—
56	C₂×#336	C₂×C₇	1	1	yes	characteristic
57	C₄×#95	C₂×C₇	1	1	yes	characteristic
58	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	—	2	2	no	—
60	C₂²×C₄×C₇	—	2	2	no	—
61	C₂²×C₄×C₇	—	2	2	no	—
62	C₂²×C₄×C₇	—	2	2	no	—
63	C₂²×C₄×C₇	—	2	4	no	—
64	C₂²×C₄×C₇	—	2	4	no	—
65	C₂²×C₄×C₇	—	2	4	no	—
66	C₂²×C₄×C₇	—	2	4	no	—
67	C₂²×C₄×C₇	—	2	4	no	—
68	C₂²×C₄×C₇	—	2	4	no	—
69	C₂²×C₄×C₇	C₄²	1	2	yes	—
70	C₂²×C₄×C₇	C₄²	1	2	yes	—
71	C₂²×C₄×C₇	C₂×C₈	1	8	yes	—
72	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
73	C₂²×C₄×C₇	M₁₆	1	4	yes	—
74	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₇	#41	1	2	yes	—
80	C₂²×C₄×C₇	#41	1	2	yes	—
81	C₂²×C₄×C₇	#41	1	1	yes	characteristic
82	C₂²×C₄×C₇	#41	1	2	yes	—
83	C₂²×C₄×C₇	#41	1	1	yes	characteristic
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	2	2	no	—
87	C₂×C₇×C₈	—	2	2	no	—
88	C₂×C₇×C₈	—	2	4	no	—
89	C₂×C₇×C₈	—	2	4	no	—
90	C₂×C₇×C₈	—	2	4	no	—
91	C₂×C₇×C₈	—	2	4	no	—
92	C₂×C₇×C₈	—	2	4	no	—
93	C₂×C₇×C₈	—	2	4	no	—
94	C₂×C₇×C₈	—	2	4	no	—
95	C₂×C₇×C₈	—	4	2	no	—
96	C₂×C₇×C₈	—	4	2	no	—
97	C₂×C₇×C₈	—	4	2	no	—
98	C₂×C₇×C₈	—	4	2	no	—
99	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
100	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
101	C₄²×C₇	—	2	2	no	—
102	C₄²×C₇	—	2	2	no	—
103	C₄²×C₇	—	2	4	no	—
104	C₄²×C₇	—	4	2	no	—
105	C₄²×C₇	—	4	2	no	—
106	C₄²×C₇	—	4	2	no	—
107	C₄²×C₇	—	4	2	no	—
108	C₂³×C₈	C₂²×C₇	1	2	yes	—
109	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
110	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
111	C₂²×C₄²	C₂²×C₇	1	2	yes	—
112	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
113	C₂×C₄×C₈	—	2	4	no	—
114	C₂×C₄×C₈	C₄×C₇	1	4	yes	complemented
115	C₂×#95	C₄×C₇	1	2	yes	—
116	C₂×#95	C₄×C₇	1	2	yes	—
117	#336	—	2	4	no	—
118	#336	—	2	4	no	—
119	#336	C₄×C₇	1	4	yes	complemented
120	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
121	C₂³×C₇	#117	1	2	yes	—
122	C₂³×C₇	C₄×D₄	1	2	yes	—
123	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
124	C₂³×C₇	C₄×Q₈	1	1	yes	characteristic
125	C₂³×C₇	C₂²×C₈	1	2	yes	—
126	C₂³×C₇	C₂×M₁₆	1	2	yes	—
127	C₂³×C₇	#131	1	2	yes	—
128	C₂³×C₇	#131	1	2	yes	—
129	C₂×C₄×C₇	—	2	2	no	—
130	C₂×C₄×C₇	—	2	2	no	—
131	C₂×C₄×C₇	—	2	2	no	—
132	C₂×C₄×C₇	—	2	2	no	—
133	C₂×C₄×C₇	—	2	8	no	—
134	C₂×C₄×C₇	—	2	8	no	—
135	C₂×C₄×C₇	—	2	8	no	—
136	C₂×C₄×C₇	—	2	8	no	—
137	C₂×C₄×C₇	—	2	8	no	—
138	C₂×C₄×C₇	—	4	4	no	—
139	C₂×C₄×C₇	—	4	4	no	—
140	C₂×C₄×C₇	—	4	4	no	—
141	C₂×C₄×C₇	—	4	4	no	—
142	C₂×C₄×C₇	C₄×C₈	1	4	yes	—
143	C₂×C₄×C₇	#97	1	4	yes	—
144	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₇	#117	1	1	yes	characteristic
147	C₂×C₄×C₇	#117	1	1	yes	characteristic
148	C₂×C₄×C₇	#117	1	1	yes	characteristic
149	C₂×C₄×C₇	#117	1	1	yes	characteristic
150	C₂×C₄×C₇	#117	1	2	yes	—
151	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
152	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₇	#121	1	2	yes	—
155	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
156	C₂×C₄×C₇	#123	1	2	yes	—
157	C₂×C₄×C₇	#123	1	2	yes	—
158	C₂×C₄×C₇	#124	1	2	yes	—
159	C₂×C₄×C₇	#125	1	2	yes	—
160	C₂×C₄×C₇	#126	1	2	yes	—
161	C₂×C₄×C₇	#126	1	2	yes	—
162	C₇×C₈	—	2	4	no	—
163	C₇×C₈	—	2	4	no	—
164	C₇×C₈	—	4	4	no	—
165	C₂³×C₄	C₂×C₄×C₇	1	4	yes	—
166	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
167	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
168	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
181	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
182	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
183	C₂²×C₈	C₇×D₄	1	4	yes	—
184	C₂²×C₈	C₇×D₄	1	2	yes	—
185	C₂²×C₈	C₇×Q₈	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
196	C₄×C₈	—	2	8	no	—
197	#95	—	2	4	no	—
198	#95	C₇×C₈	1	8	yes	complemented
199	C₂²×C₇	#376	1	1	yes	characteristic
200	C₂²×C₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
201	C₂²×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
202	C₂²×C₇	#387	1	2	yes	—
203	C₂²×C₇	#388	1	2	yes	—
204	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
205	C₂²×C₇	C₂×#97	1	1	yes	characteristic
206	C₂²×C₇	#405	1	1	yes	characteristic
207	C₂²×C₇	#405	1	1	yes	characteristic
208	C₂²×C₇	#431	1	4	yes	—
209	C₂²×C₇	#433	1	4	yes	—
210	C₂²×C₇	C₈×D₄	1	4	yes	—
211	C₂²×C₇	C₈×D₄	1	2	yes	—
212	C₂²×C₇	#435	1	4	yes	—
213	C₂²×C₇	#436	1	2	yes	—
214	C₂²×C₇	C₈×Q₈	1	2	yes	—
215	C₂²×C₇	#446	1	2	yes	—
216	C₄×C₇	—	2	4	no	—
217	C₄×C₇	—	2	4	no	—
218	C₄×C₇	—	4	8	no	—
219	C₄×C₇	#380	1	1	yes	characteristic
220	C₄×C₇	#381	1	1	yes	characteristic
221	C₄×C₇	#382	1	1	yes	characteristic
222	C₄×C₇	#383	1	1	yes	characteristic
223	C₄×C₇	#387	1	2	yes	—
224	C₄×C₇	#388	1	2	yes	—
225	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	C₄²×C₇	1	2	yes	—
237	C₂²×C₄	C₄²×C₇	1	2	yes	—
238	C₂²×C₄	C₂×C₇×C₈	1	8	yes	—
239	C₂²×C₄	C₂×C₇×C₈	1	4	yes	—
240	C₂²×C₄	C₇×M₁₆	1	4	yes	—
241	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
242	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
244	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
246	C₂²×C₄	C₇×#41	1	2	yes	—
247	C₂²×C₄	C₇×#41	1	2	yes	—
248	C₂²×C₄	C₇×#41	1	1	yes	characteristic
249	C₂²×C₄	C₇×#41	1	2	yes	—
250	C₂²×C₄	C₇×#41	1	1	yes	characteristic
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	4	2	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	C₇×C₂²⋊C₄	1	4	yes	—
267	C₂×C₈	C₇×C₄⋊C₄	1	4	yes	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	4	no	—
271	C₄²	—	4	2	no	—
272	C₄²	—	4	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	—	4	2	no	—
275	C₂×C₇	#1434	1	1	yes	characteristic
276	C₂×C₇	#1744	1	1	yes	characteristic
277	C₂×C₇	#1746	1	1	yes	characteristic
278	C₂×C₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₇	#1752	1	1	yes	characteristic
280	C₂×C₇	C₈×C₄⋊C₄	1	1	yes	characteristic
281	C₂×C₇	#1770	1	1	yes	characteristic
282	C₂×C₇	#1916	1	4	yes	—
283	C₂×C₇	#1923	1	4	yes	—
284	C₂³	C₂×C₄²×C₇	1	1	yes	characteristic
285	C₂³	C₇×#117	1	2	yes	—
286	C₂³	C₄×C₇×D₄	1	2	yes	—
287	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
288	C₂³	C₄×C₇×Q₈	1	1	yes	characteristic
289	C₂³	C₂²×C₇×C₈	1	2	yes	—
290	C₂³	C₂×C₇×M₁₆	1	2	yes	—
291	C₂³	C₇×#131	1	2	yes	—
292	C₂³	C₇×#131	1	2	yes	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	2	8	no	—
301	C₂×C₄	—	2	8	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	C₄×C₇×C₈	1	4	yes	—
307	C₂×C₄	C₇×#97	1	4	yes	—
308	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
309	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
310	C₂×C₄	C₇×#117	1	1	yes	characteristic
311	C₂×C₄	C₇×#117	1	1	yes	characteristic
312	C₂×C₄	C₇×#117	1	1	yes	characteristic
313	C₂×C₄	C₇×#117	1	1	yes	characteristic
314	C₂×C₄	C₇×#117	1	2	yes	—
315	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
316	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
317	C₂×C₄	C₄×C₇×Q₈	1	1	yes	characteristic
318	C₂×C₄	C₇×#121	1	2	yes	—
319	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
320	C₂×C₄	C₇×#123	1	2	yes	—
321	C₂×C₄	C₇×#123	1	2	yes	—
322	C₂×C₄	C₇×#124	1	2	yes	—
323	C₂×C₄	C₇×#125	1	2	yes	—
324	C₂×C₄	C₇×#126	1	2	yes	—
325	C₂×C₄	C₇×#126	1	2	yes	—
326	C₈	—	2	4	no	—
327	C₈	—	2	4	no	—
328	C₈	—	4	4	no	—
329	C₇	#10797	1	1	yes	characteristic, direct product factor, Sylow
330	C₂²	C₇×#376	1	1	yes	characteristic
331	C₂²	C₄×C₇×C₂²⋊C₄	1	1	yes	characteristic
332	C₂²	C₄×C₇×C₄⋊C₄	1	1	yes	characteristic
333	C₂²	C₇×#387	1	2	yes	—
334	C₂²	C₇×#388	1	2	yes	—
335	C₂²	C₂×C₄×C₇×C₈	1	1	yes	characteristic
336	C₂²	C₂×C₇×#97	1	1	yes	characteristic
337	C₂²	C₇×#405	1	1	yes	characteristic
338	C₂²	C₇×#405	1	1	yes	characteristic
339	C₂²	C₇×#431	1	4	yes	—
340	C₂²	C₇×#433	1	4	yes	—
341	C₂²	C₇×C₈×D₄	1	4	yes	—
342	C₂²	C₇×C₈×D₄	1	2	yes	—
343	C₂²	C₇×#435	1	4	yes	—
344	C₂²	C₇×#436	1	2	yes	—
345	C₂²	C₇×C₈×Q₈	1	2	yes	—
346	C₂²	C₇×#446	1	2	yes	—
347	C₄	—	2	4	no	—
348	C₄	—	2	4	no	—
349	C₄	—	4	8	no	—
350	C₄	C₇×#380	1	1	yes	characteristic
351	C₄	C₇×#381	1	1	yes	characteristic
352	C₄	C₇×#382	1	1	yes	characteristic
353	C₄	C₇×#383	1	1	yes	characteristic
354	C₄	C₇×#387	1	2	yes	—
355	C₄	C₇×#388	1	2	yes	—
356	C₂	C₇×#1434	1	1	yes	characteristic
357	C₂	C₇×#1744	1	1	yes	characteristic
358	C₂	C₇×#1746	1	1	yes	characteristic
359	C₂	C₇×C₈×C₂²⋊C₄	1	1	yes	characteristic
360	C₂	C₇×#1752	1	1	yes	characteristic
361	C₂	C₇×C₈×C₄⋊C₄	1	1	yes	characteristic
362	C₂	C₇×#1770	1	1	yes	characteristic
363	C₂	C₇×#1916	1	4	yes	—
364	C₂	C₇×#1923	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₇	C₂³
commutator	C₂²	C₂×C₄×C₇×C₈
Frattini, cosocle	C₂³×C₄	C₂³×C₇
socle, prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂×C₄×C₇×C₈	C₄	semidirect
17. C₇×#336	C₄	semidirect
18. #10797	C₇	direct (N × H)
52. C₇×#95	C₈	semidirect
114. C₂×C₄×C₈	C₄×C₇	semidirect
119. #336	C₄×C₇	semidirect
198. #95	C₇×C₈	semidirect
329. C₇	#10797	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄×C₇×C₈	2
C₂×C₇×#336	2
C₄×C₇×#95	2
#10797	7

Subgroups — C7×#10797 #13188013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10797 #13188013