Subgroups — C₇×#7891 #13185107

Order 1792.

770 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₇×#117	C₂	1	1	yes	maximal, characteristic
4	C₇×#1279	C₂	1	4	yes	maximal, complemented
5	C₂³×C₇×C₈	—	2	1	no	—
6	C₂×C₇×#117	C₄	1	2	yes	—
7	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₇×#431	—	2	2	no	—
9	C₇×#431	C₄	1	4	yes	complemented
10	C₂×C₇×C₄⋊C₈	—	2	1	no	—
11	C₂×C₇×C₄⋊C₈	—	2	2	no	—
12	C₂×C₇×C₄⋊C₈	—	2	1	no	—
13	C₂×C₇×C₄⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₇×C₂²⋊C₈	—	2	1	no	—
16	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
17	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
18	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₄×C₇×C₂²⋊C₄	—	2	1	no	—
20	C₄×C₇×C₂²⋊C₄	—	2	1	no	—
21	C₇×#376	—	2	1	no	—
22	C₄³×C₇	—	2	1	no	—
23	#7891	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	—	2	4	no	—
28	C₂²×C₇×C₈	—	2	2	no	—
29	C₂²×C₇×C₈	—	2	2	no	—
30	C₂²×C₇×C₈	—	4	4	no	—
31	C₇×#117	—	2	4	no	—
32	C₇×#117	C₈	1	8	yes	complemented
33	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
34	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
35	C₂×C₄²×C₇	—	2	2	no	—
36	C₂×C₄²×C₇	—	2	2	no	—
37	C₂×C₄²×C₇	—	2	2	no	—
38	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇	D₄	1	2	yes	—
44	C₂×C₄²×C₇	Q₈	1	2	yes	—
45	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
46	C₇×C₄⋊C₈	—	2	4	no	—
47	C₇×C₄⋊C₈	—	2	4	no	—
48	C₇×C₄⋊C₈	—	2	2	no	—
49	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
50	C₇×C₂²⋊C₈	—	2	4	no	—
51	C₄×C₇×C₈	—	2	2	no	—
52	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
53	C₇×#95	—	2	2	no	—
54	C₇×#95	—	2	2	no	—
55	C₂×#431	C₂×C₇	1	1	yes	characteristic
56	C₂²×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
57	C₄×#117	C₂×C₇	1	1	yes	characteristic
58	#1279	C₂×C₇	1	4	yes	complemented
59	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	—	2	2	no	—
61	C₂²×C₄×C₇	—	2	2	no	—
62	C₂²×C₄×C₇	—	2	4	no	—
63	C₂²×C₄×C₇	—	4	2	no	—
64	C₂²×C₄×C₇	C₄²	1	2	yes	—
65	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
74	C₂²×C₄×C₇	M₁₆	1	4	yes	—
75	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₇×C₈	—	2	4	no	—
79	C₂×C₇×C₈	—	2	4	no	—
80	C₂×C₇×C₈	—	4	4	no	—
81	C₂×C₇×C₈	—	4	8	no	—
82	C₂×C₇×C₈	—	4	4	no	—
83	C₇×C₄⋊C₄	—	2	4	no	—
84	C₇×C₄⋊C₄	C₂×C₈	1	8	yes	complemented
85	C₇×C₂²⋊C₄	—	2	8	no	—
86	C₄²×C₇	—	2	8	no	—
87	C₄²×C₇	—	4	4	no	—
88	C₄²×C₇	—	4	2	no	—
89	C₄²×C₇	—	4	2	no	—
90	C₄²×C₇	C₄²	1	2	yes	—
91	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₇	C₄⋊C₄	1	2	yes	—
94	C₄²×C₇	C₄⋊C₄	1	2	yes	—
95	C₄²×C₇	C₄⋊C₄	1	2	yes	—
96	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
100	C₂³×C₈	—	2	1	no	—
101	C₂×#117	C₄×C₇	1	2	yes	—
102	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
103	#431	—	2	2	no	—
104	#431	C₄×C₇	1	4	yes	complemented
105	C₂×C₄⋊C₈	—	2	1	no	—
106	C₂×C₄⋊C₈	—	2	2	no	—
107	C₂×C₄⋊C₈	—	2	1	no	—
108	C₂×C₄⋊C₈	C₄×C₇	1	4	yes	complemented
109	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
110	C₂×C₂²⋊C₈	—	2	1	no	—
111	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
112	C₄×C₄⋊C₄	C₄×C₇	1	2	yes	—
113	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
114	C₄×C₂²⋊C₄	—	2	1	no	—
115	C₄×C₂²⋊C₄	—	2	1	no	—
116	#376	—	2	1	no	—
117	C₄³	—	2	1	no	—
118	C₂³×C₇	—	2	2	no	—
119	C₂³×C₇	—	2	2	no	—
120	C₂³×C₇	#95	1	2	yes	—
121	C₂³×C₇	C₂²⋊C₈	1	4	yes	—
122	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇	—	2	2	no	—
124	C₂×C₄×C₇	—	2	2	no	—
125	C₂×C₄×C₇	—	2	2	no	—
126	C₂×C₄×C₇	—	2	2	no	—
127	C₂×C₄×C₇	—	2	2	no	—
128	C₂×C₄×C₇	—	2	4	no	—
129	C₂×C₄×C₇	—	2	8	no	—
130	C₂×C₄×C₇	—	4	2	no	—
131	C₂×C₄×C₇	—	4	2	no	—
132	C₂×C₄×C₇	—	4	4	no	—
133	C₂×C₄×C₇	—	4	4	no	—
134	C₂×C₄×C₇	#95	1	2	yes	—
135	C₂×C₄×C₇	#95	1	2	yes	—
136	C₂×C₄×C₇	#95	1	2	yes	—
137	C₂×C₄×C₇	C₄×C₈	1	4	yes	—
138	C₂×C₄×C₇	#97	1	4	yes	—
139	C₂×C₄×C₇	C₂²⋊C₈	1	4	yes	—
140	C₂×C₄×C₇	C₄⋊C₈	1	8	yes	—
141	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
148	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
149	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
150	C₂×C₄×C₇	#137	1	2	yes	—
151	C₇×C₈	—	4	4	no	—
152	C₇×C₈	—	4	4	no	—
153	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
154	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
155	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	4	4	no	—
160	#117	—	2	4	no	—
161	#117	C₇×C₈	1	8	yes	complemented
162	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
163	C₂×C₂²⋊C₄	—	2	2	no	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
168	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
169	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
170	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
171	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
172	C₂×C₄²	C₇×D₄	1	2	yes	—
173	C₂×C₄²	C₇×Q₈	1	2	yes	—
174	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
175	C₄⋊C₈	—	2	4	no	—
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
179	C₂²⋊C₈	—	2	4	no	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
182	#95	—	2	2	no	—
183	#95	—	2	2	no	—
184	C₂²×C₇	—	2	8	no	—
185	C₂²×C₇	—	2	2	no	—
186	C₂²×C₇	—	2	2	no	—
187	C₂²×C₇	#336	1	4	yes	—
188	C₂²×C₇	C₂×#95	1	1	yes	characteristic
189	C₂²×C₇	C₂×C₂²⋊C₈	1	2	yes	—
190	C₂²×C₇	#417	1	2	yes	—
191	C₂²×C₇	#421	1	2	yes	—
192	C₄×C₇	—	2	1	no	—
193	C₄×C₇	—	2	2	no	—
194	C₄×C₇	—	2	1	no	—
195	C₄×C₇	—	2	2	no	—
196	C₄×C₇	—	4	8	no	—
197	C₄×C₇	#336	1	4	yes	—
198	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
199	C₄×C₇	C₂×#97	1	1	yes	characteristic
200	C₄×C₇	#418	1	1	yes	characteristic
201	C₄×C₇	#419	1	1	yes	characteristic
202	C₄×C₇	C₂×C₄⋊C₈	1	2	yes	—
203	C₄×C₇	#428	1	2	yes	—
204	C₂⁴	C₇×C₂²⋊C₄	1	1	yes	characteristic
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₄²×C₇	1	2	yes	—
210	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
213	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
214	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
215	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
216	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
217	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
218	C₂²×C₄	C₂×C₇×C₈	1	4	yes	—
219	C₂²×C₄	C₇×M₁₆	1	4	yes	—
220	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	4	4	no	—
226	C₂×C₈	—	4	8	no	—
227	C₂×C₈	—	4	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	C₂×C₇×C₈	1	8	yes	complemented
230	C₂²⋊C₄	—	2	8	no	—
231	C₄²	—	2	8	no	—
232	C₄²	—	4	4	no	—
233	C₄²	—	4	2	no	—
234	C₄²	—	4	2	no	—
235	C₄²	C₄²×C₇	1	2	yes	—
236	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
237	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
238	C₄²	C₇×C₄⋊C₄	1	2	yes	—
239	C₄²	C₇×C₄⋊C₄	1	2	yes	—
240	C₄²	C₇×C₄⋊C₄	1	2	yes	—
241	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
242	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
243	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
244	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
245	C₂×C₇	—	2	4	no	—
246	C₂×C₇	#1477	1	4	yes	—
247	C₂×C₇	C₂×#336	1	1	yes	characteristic
248	C₂×C₇	#1733	1	1	yes	characteristic
249	C₂×C₇	#1736	1	1	yes	characteristic
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	C₇×#95	1	2	yes	—
253	C₂³	C₇×C₂²⋊C₈	1	4	yes	—
254	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	C₇×#95	1	2	yes	—
267	C₂×C₄	C₇×#95	1	2	yes	—
268	C₂×C₄	C₇×#95	1	2	yes	—
269	C₂×C₄	C₄×C₇×C₈	1	4	yes	—
270	C₂×C₄	C₇×#97	1	4	yes	—
271	C₂×C₄	C₇×C₂²⋊C₈	1	4	yes	—
272	C₂×C₄	C₇×C₄⋊C₈	1	8	yes	—
273	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
274	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
277	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
280	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
281	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
282	C₂×C₄	C₇×#137	1	2	yes	—
283	C₈	—	4	4	no	—
284	C₈	—	4	4	no	—
285	C₇	#7891	1	1	yes	characteristic, direct product factor, Sylow
286	C₂²	—	2	8	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	C₇×#336	1	4	yes	—
290	C₂²	C₂×C₇×#95	1	1	yes	characteristic
291	C₂²	C₂×C₇×C₂²⋊C₈	1	2	yes	—
292	C₂²	C₇×#417	1	2	yes	—
293	C₂²	C₇×#421	1	2	yes	—
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	8	no	—
299	C₄	C₇×#336	1	4	yes	—
300	C₄	C₂×C₄×C₇×C₈	1	1	yes	characteristic
301	C₄	C₂×C₇×#97	1	1	yes	characteristic
302	C₄	C₇×#418	1	1	yes	characteristic
303	C₄	C₇×#419	1	1	yes	characteristic
304	C₄	C₂×C₇×C₄⋊C₈	1	2	yes	—
305	C₄	C₇×#428	1	2	yes	—
306	C₂	—	2	4	no	—
307	C₂	C₇×#1477	1	4	yes	—
308	C₂	C₂×C₇×#336	1	1	yes	characteristic
309	C₂	C₇×#1733	1	1	yes	characteristic
310	C₂	C₇×#1736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₄	C₂×C₄×C₇×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂³×C₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1279	C₂	semidirect
9. C₇×#431	C₄	semidirect
13. C₂×C₇×C₄⋊C₈	C₄	semidirect
23. #7891	C₇	direct (N × H)
32. C₇×#117	C₈	semidirect
58. #1279	C₂×C₇	semidirect
84. C₇×C₄⋊C₄	C₂×C₈	semidirect
104. #431	C₄×C₇	semidirect
108. C₂×C₄⋊C₈	C₄×C₇	semidirect
161. #117	C₇×C₈	semidirect
229. C₄⋊C₄	C₂×C₇×C₈	semidirect
285. C₇	#7891	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#431	2
C₂²×C₇×C₄⋊C₈	2
C₄×C₇×#117	2
C₇×#1279	2
#7891	7

Subgroups — C7×#7891 #13185107

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7891 #13185107