Subgroups — #13508447

Order 1792.

598 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₅₆	C₂	1	1	yes	maximal, characteristic
3	#11729142	C₂	1	2	yes	maximal, complemented
4	#11729141	C₂	1	2	yes	maximal, complemented
5	#11729110	C₂	1	4	yes	maximal, complemented
6	#11728406	C₂	1	2	yes	maximal
7	#11728403	C₂	1	2	yes	maximal, complemented
8	#11727497	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#533	—	2	1	no	—
10	C₂²×D₅₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₂₈	—	2	1	no	—
12	C₇×#500	C₂²	1	1	yes	characteristic, complemented
13	C₇×#500	C₂²	1	1	yes	characteristic, complemented
14	C₇×#499	—	2	1	no	—
15	C₇×#499	C₂²	1	2	yes	complemented
16	C₇×#498	—	2	1	no	—
17	C₇×#427	—	2	1	no	—
18	C₇×#427	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₂×#6395	—	2	1	no	—
21	C₂×#6392	C₂²	1	1	yes	characteristic
22	C₂×#6386	C₂²	1	2	yes	—
23	C₄×D₅₆	—	2	1	no	—
24	C₄×D₅₆	—	2	1	no	—
25	C₄×D₅₆	C₂²	1	2	yes	—
26	C₄×D₅₆	C₂²	1	2	yes	—
27	#89286	C₂²	1	4	yes	—
28	#89286	C₂²	1	4	yes	—
29	#89285	C₂²	1	4	yes	complemented
30	#89284	C₂²	1	4	yes	complemented
31	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	complemented
32	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	complemented
33	#33627	—	7	1	no	maximal, Sylow
34	C₂²×D₂₈	—	2	1	no	—
35	C₂²×C₄×D₇	—	4	1	no	—
36	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
37	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
38	C₇×C₄⋊Q₈	—	2	1	no	—
39	C₇×C₄⋊Q₈	—	2	1	no	—
40	C₇×#125	—	2	2	no	—
41	C₇×C₂²⋊Q₈	—	4	1	no	—
42	C₇×C₂²⋊Q₈	—	4	1	no	—
43	C₄×C₇×Q₈	—	4	1	no	—
44	C₄×C₇×Q₈	—	4	1	no	—
45	C₇×#117	—	2	2	no	—
46	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
47	C₂×(C₂×D₇)⋊C₄	—	4	1	no	—
48	C₂×C₄⋊Dic₇	—	2	1	no	—
49	C₂×D₅₆	—	2	1	no	—
50	C₂×D₅₆	—	2	2	no	—
51	C₂×D₅₆	Q₈	1	4	yes	complemented
52	C₂×D₅₆	C₂³	1	2	yes	—
53	C₄×D₂₈	—	2	2	no	—
54	C₄×D₂₈	—	2	2	no	—
55	C₄×D₂₈	—	4	1	no	—
56	C₄×D₂₈	—	4	1	no	—
57	C₇×#107	—	2	1	no	—
58	C₇×#107	—	2	1	no	—
59	C₇×#107	C₂³	1	2	yes	—
60	C₇×#107	C₂³	1	2	yes	—
61	C₇×#106	—	2	2	no	—
62	C₇×#106	—	2	2	no	—
63	C₄×C₇×C₈	D₄	1	1	yes	characteristic
64	C₄×C₇×C₈	D₄	1	1	yes	characteristic
65	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
66	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
67	#6395	—	2	2	no	—
68	#6395	—	2	2	no	—
69	#6392	—	2	1	no	—
70	#6392	C₂³	1	2	yes	—
71	#6386	—	2	4	no	—
72	C₂×C₇⋊C₁₆	C₂³	1	4	yes	—
73	#3160	D₇	1	1	yes	characteristic, complemented
74	C₂×C₄×D₈	—	7	1	no	—
75	#2228	—	7	4	no	—
76	#2226	—	7	2	no	—
77	#2224	—	7	2	no	—
78	#2159	—	7	2	no	—
79	#2150	—	7	1	no	—
80	#2140	—	7	2	no	—
81	C₂³×D₇	—	4	1	no	—
82	C₂×C₇×Q₈	—	4	1	no	—
83	C₂×C₇×Q₈	—	4	1	no	—
84	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×Dic₇	—	4	1	no	—
88	C₂×D₂₈	—	2	4	no	—
89	C₂×D₂₈	—	2	2	no	—
90	C₂×D₂₈	—	4	1	no	—
91	C₂×D₂₈	—	4	2	no	—
92	C₂×C₄×D₇	—	4	2	no	—
93	C₂×C₄×D₇	—	4	2	no	—
94	C₂×C₄×D₇	—	8	4	no	—
95	C₂×C₇×C₈	—	2	2	no	—
96	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
100	C₂×C₇×C₈	#41	1	2	yes	—
101	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
102	C₇×C₄⋊C₄	—	2	2	no	—
103	C₇×C₄⋊C₄	—	2	2	no	—
104	C₇×C₄⋊C₄	—	4	2	no	—
105	C₇×C₄⋊C₄	—	4	2	no	—
106	C₇×C₄⋊C₄	—	4	1	no	—
107	C₇×C₄⋊C₄	—	4	1	no	—
108	C₇×C₂²⋊C₄	—	4	2	no	—
109	C₄²×C₇	—	4	2	no	—
110	C₄²×C₇	D₈	1	1	yes	characteristic
111	C₄²×C₇	D₈	1	1	yes	characteristic
112	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
114	(C₂×D₇)⋊C₄	—	4	2	no	—
115	(C₂×D₇)⋊C₄	—	4	2	no	—
116	C₄⋊Dic₇	—	2	2	no	—
117	C₄⋊Dic₇	—	4	1	no	—
118	D₅₆	—	2	4	no	—
119	D₅₆	—	2	2	no	—
120	D₅₆	C₂×Q₈	1	4	yes	—
121	C₇⋊C₁₆	—	2	4	no	—
122	C₂²×D₈	—	7	1	no	—
123	#533	—	2	1	no	—
124	C₂×C₄×D₄	—	14	1	no	—
125	C₂×M₃₂	—	7	2	no	—
126	#500	C₂×D₇	1	1	yes	characteristic, complemented
127	#500	C₂×D₇	1	1	yes	characteristic, complemented
128	#499	—	2	1	no	—
129	#499	C₂×D₇	1	2	yes	complemented
130	#498	—	2	1	no	—
131	C₄×D₈	—	7	2	no	—
132	C₄×D₈	—	7	2	no	—
133	C₄×D₈	—	14	1	no	—
134	C₄×D₈	—	14	1	no	—
135	#427	—	2	1	no	—
136	#427	C₂×D₇	1	2	yes	—
137	C₂×#107	—	7	1	no	—
138	C₂×#102	—	14	1	no	—
139	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
140	#367	—	7	4	no	—
141	#366	—	7	4	no	—
142	C₄⋊C₁₆	—	7	2	no	—
143	C₄⋊C₁₆	—	7	2	no	—
144	#357	—	7	4	no	—
145	#357	—	7	4	no	—
146	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
147	C₂²×D₇	—	4	2	no	—
148	C₂²×D₇	—	4	4	no	—
149	C₂²×D₇	—	8	2	no	—
150	C₂²×D₇	—	8	2	no	—
151	C₇×Q₈	—	8	2	no	—
152	C₇×Q₈	—	8	2	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	—	4	1	no	—
155	C₂×C₄×C₇	—	4	2	no	—
156	C₂×C₄×C₇	—	4	2	no	—
157	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
158	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₇	#137	1	2	yes	—
164	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
165	C₂×Dic₇	—	4	2	no	—
166	C₂×Dic₇	—	8	2	no	—
167	D₂₈	—	2	4	no	—
168	D₂₈	—	4	2	no	—
169	D₂₈	—	4	4	no	—
170	C₄×D₇	—	8	4	no	—
171	C₄×D₇	—	8	4	no	—
172	C₇×C₈	—	2	1	no	—
173	C₇×C₈	—	2	1	no	—
174	C₇×C₈	C₂²⋊Q₈	1	2	yes	—
175	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
176	C₇×C₈	C₂×#41	1	1	yes	characteristic
177	C₂²×D₄	—	14	1	no	—
178	C₂³×C₄	—	28	1	no	—
179	C₂×D₈	—	7	4	no	—
180	C₂×D₈	—	7	2	no	—
181	C₂×D₈	—	14	2	no	—
182	C₂×D₈	—	14	1	no	—
183	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
185	C₄⋊Q₈	—	2	1	no	—
186	C₄⋊Q₈	—	2	1	no	—
187	#125	—	2	2	no	—
188	C₂²⋊Q₈	—	4	1	no	—
189	C₂²⋊Q₈	—	4	1	no	—
190	C₄×Q₈	—	4	1	no	—
191	C₄×Q₈	—	4	1	no	—
192	C₄×D₄	—	14	2	no	—
193	C₄×D₄	—	14	2	no	—
194	C₄×D₄	—	28	1	no	—
195	C₄×D₄	—	28	1	no	—
196	#117	—	2	2	no	—
197	C₂×C₄⋊C₄	—	14	1	no	—
198	C₂×C₂²⋊C₄	—	28	1	no	—
199	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
200	M₃₂	—	14	4	no	—
201	C₂×C₁₆	—	7	4	no	—
202	#107	—	2	1	no	—
203	#107	—	2	1	no	—
204	#107	—	7	2	no	—
205	#107	—	14	1	no	—
206	#107	C₂²×D₇	1	2	yes	—
207	#107	C₂²×D₇	1	2	yes	—
208	#106	—	2	2	no	—
209	#106	—	2	2	no	—
210	#102	—	14	2	no	—
211	#102	—	14	2	no	—
212	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
213	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
216	C₂²×C₇	—	2	1	no	—
217	C₂²×C₇	—	2	1	no	—
218	C₂²×C₇	#474	1	2	yes	—
219	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
220	C₂×D₇	—	4	4	no	—
221	C₂×D₇	—	8	8	no	—
222	C₂×D₇	—	8	2	no	—
223	C₂×D₇	—	8	2	no	—
224	C₄×C₇	—	2	1	no	—
225	C₄×C₇	—	2	1	no	—
226	C₄×C₇	—	2	1	no	—
227	C₄×C₇	—	8	2	no	—
228	C₄×C₇	—	8	2	no	—
229	C₄×C₇	#474	1	2	yes	—
230	C₄×C₇	#509	1	2	yes	—
231	C₄×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
232	C₄×C₇	C₂×#137	1	1	yes	characteristic
233	Dic₇	—	8	2	no	—
234	C₂⁴	—	28	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×D₄	—	14	4	no	—
238	C₂×D₄	—	14	2	no	—
239	C₂×D₄	—	28	1	no	—
240	C₂×D₄	—	28	2	no	—
241	C₂²×C₄	—	28	1	no	—
242	C₂²×C₄	—	28	2	no	—
243	C₂²×C₄	—	28	2	no	—
244	C₂²×C₄	—	56	4	no	—
245	C₂²×C₄	#1141	1	1	yes	characteristic
246	C₂²×C₄	#1141	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
248	D₈	—	7	4	no	—
249	D₈	—	14	4	no	—
250	D₈	—	14	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	Q₈×D₇	1	2	yes	—
253	C₂×C₈	#1161	1	2	yes	—
254	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
255	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
256	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
257	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	14	2	no	—
265	C₄⋊C₄	—	28	1	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	28	2	no	—
268	C₂²⋊C₄	—	28	2	no	—
269	C₄²	—	4	2	no	—
270	C₄²	#1141	1	1	yes	characteristic
271	C₄²	#1141	1	1	yes	characteristic
272	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
273	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
274	C₁₆	—	14	4	no	—
275	C₂×C₇	—	2	2	no	—
276	C₂×C₇	C₂×#474	1	1	yes	characteristic
277	C₂×C₇	C₂×#509	1	1	yes	characteristic
278	C₂×C₇	#3414	1	1	yes	characteristic
279	D₇	—	8	4	no	—
280	C₂³	—	28	4	no	—
281	C₂³	—	28	2	no	—
282	C₂³	—	56	2	no	—
283	C₂³	—	56	2	no	—
284	C₂³	C₂×#1141	1	1	yes	characteristic
285	Q₈	—	8	2	no	—
286	Q₈	—	8	2	no	—
287	D₄	—	14	4	no	—
288	D₄	—	28	4	no	—
289	D₄	—	28	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	28	2	no	—
295	C₂×C₄	—	56	4	no	—
296	C₂×C₄	—	56	2	no	—
297	C₂×C₄	—	56	4	no	—
298	C₂×C₄	C₂×#1141	1	1	yes	characteristic
299	C₂×C₄	C₂×#1141	1	1	yes	characteristic
300	C₂×C₄	C₂×#1141	1	1	yes	characteristic
301	C₂×C₄	C₂×#1141	1	1	yes	characteristic
302	C₂×C₄	C₂×#1141	1	1	yes	characteristic
303	C₂×C₄	#6505	1	2	yes	—
304	C₂×C₄	#6509	1	2	yes	—
305	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	#6509	1	2	yes	—
309	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
310	C₈	C₂×#1161	1	1	yes	characteristic
311	C₇	#33627	1	1	yes	characteristic, complemented, Sylow
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	28	4	no	—
315	C₂²	—	56	8	no	—
316	C₂²	—	56	2	no	—
317	C₂²	—	56	2	no	—
318	C₂²	#89859	1	2	yes	—
319	C₂²	C₂²×#1141	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	56	2	no	—
326	C₄	#89859	1	2	yes	—
327	C₄	#89965	1	2	yes	—
328	C₄	C₂×#6505	1	1	yes	characteristic
329	C₄	C₂×#6509	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	56	4	no	—
332	C₂	C₂×#89859	1	1	yes	characteristic
333	C₂	C₂×#89965	1	1	yes	characteristic
334	C₂	#11741339	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#1141
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#3160	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂²×D₅₆	C₂²
hypercenter	#3160	D₇
hypocenter	C₇	#33627
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
1. C₇×#3160	C₂	semidirect
3. #11729142	C₂	semidirect
4. #11729141	C₂	semidirect
5. #11729110	C₂	semidirect
7. #11728403	C₂	semidirect
8. #11727497	C₂	semidirect
12. C₇×#500	C₂²	semidirect
13. C₇×#500	C₂²	semidirect
15. C₇×#499	C₂²	semidirect
29. #89285	C₂²	semidirect
30. #89284	C₂²	semidirect
31. C₄⋊(C₇⋊C₁₆)	C₂²	semidirect
32. C₄⋊(C₇⋊C₁₆)	C₂²	semidirect
51. C₂×D₅₆	Q₈	semidirect
73. #3160	D₇	semidirect
126. #500	C₂×D₇	semidirect
127. #500	C₂×D₇	semidirect
129. #499	C₂×D₇	semidirect
311. C₇	#33627	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3160	2
C₂×C₄×D₅₆	2
#11729142	2
#11729141	2
#11729110	2
#11728406	2
#11728403	2
#11727497	2
#33627	7