Subgroups — #11737205

Order 896.

360 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#90260	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic
3	C₇×#440	C₂	1	1	yes	maximal, characteristic
4	#89936	C₂	1	1	yes	maximal, characteristic
5	SD₁₆×Dic₇	C₂	1	1	yes	maximal, characteristic
6	#89793	C₂	1	1	yes	maximal, characteristic, complemented
7	#89781	C₂	1	1	yes	maximal, characteristic
8	#89648	C₂	1	1	yes	maximal, characteristic, complemented
9	#89646	C₂	1	1	yes	maximal, characteristic, complemented
10	#89566	C₂	1	1	yes	maximal, characteristic, complemented
11	#89563	C₂	1	1	yes	maximal, characteristic, complemented
12	#89535	C₂	1	1	yes	maximal, characteristic
13	#89531	C₂	1	1	yes	maximal, characteristic
14	#89474	C₂	1	1	yes	maximal, characteristic, complemented
15	#89473	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₁₄	—	2	1	no	—
17	C₂×C₇×SD₁₆	C₂²	1	1	yes	characteristic
18	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
19	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
20	Q₈×Dic₇	C₂²	1	1	yes	characteristic
21	Q₈×Dic₇	C₂²	1	1	yes	characteristic
22	D₄×Dic₇	C₂²	1	1	yes	characteristic
23	D₄×Dic₇	C₂²	1	1	yes	characteristic
24	#6487	—	2	1	no	—
25	#6453	—	2	1	no	—
26	#6453	C₂²	1	1	yes	characteristic
27	#6451	—	2	1	no	—
28	#6451	C₂²	1	1	yes	characteristic
29	#6451	C₂²	1	1	yes	characteristic
30	#6451	C₂²	1	1	yes	characteristic
31	#6441	—	2	1	no	—
32	#6441	—	2	1	no	—
33	#6433	—	2	1	no	—
34	#6433	C₂²	1	1	yes	characteristic
35	#6432	C₂²	1	1	yes	characteristic
36	C₄×Dic₁₄	—	2	1	no	—
37	C₄×Dic₁₄	C₂²	1	1	yes	characteristic
38	C₇×#107	C₂²	1	1	yes	characteristic
39	C₇×#103	C₂²	1	1	yes	characteristic
40	C₇×#102	C₂²	1	1	yes	characteristic
41	C₇×#97	C₂²	1	1	yes	characteristic
42	#6409	C₂²	1	1	yes	characteristic
43	#6409	C₂²	1	1	yes	characteristic
44	#6406	C₂²	1	1	yes	characteristic
45	#6406	C₂²	1	1	yes	characteristic
46	#6392	C₂²	1	1	yes	characteristic
47	#6392	C₂²	1	1	yes	characteristic
48	#6391	C₂²	1	1	yes	characteristic
49	#6391	C₂²	1	1	yes	characteristic
50	#6389	C₂²	1	1	yes	characteristic
51	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
52	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
53	C₈×Dic₇	C₂²	1	1	yes	characteristic
54	#6382	C₂²	1	1	yes	characteristic
55	#6381	C₂²	1	1	yes	characteristic
56	#6381	C₂²	1	1	yes	characteristic
57	#6381	C₂²	1	1	yes	characteristic
58	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
59	#3386	—	7	1	no	maximal, Sylow
60	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
61	C₂×C₇×D₄	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₇	—	2	1	no	—
63	C₂²×Dic₇	—	2	1	no	—
64	C₂²×Dic₇	—	2	1	no	—
65	C₂×Dic₁₄	—	2	1	no	—
66	C₂×Dic₁₄	—	2	1	no	—
67	C₂×Dic₁₄	—	2	1	no	—
68	C₂×Dic₁₄	—	2	1	no	—
69	C₂×Dic₁₄	—	4	4	no	—
70	C₂×Dic₁₄	D₄	1	1	yes	characteristic
71	C₇×SD₁₆	Q₈	1	4	yes	complemented
72	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
74	C₇×C₄⋊C₄	—	2	1	no	—
75	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
76	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₇×C₂²⋊C₄	—	2	1	no	—
78	C₄²×C₇	—	2	1	no	—
79	C₄²×C₇	C₂³	1	1	yes	characteristic
80	C₂²⋊Dic₇	—	2	1	no	—
81	C₂²⋊Dic₇	—	2	1	no	—
82	C₄⋊Dic₇	—	2	1	no	—
83	C₄⋊Dic₇	—	2	1	no	—
84	C₄⋊Dic₇	—	2	1	no	—
85	C₄⋊Dic₇	—	2	1	no	—
86	C₄⋊Dic₇	D₄	1	1	yes	characteristic
87	C₄⋊Dic₇	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₇	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₇	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₇	C₂³	1	1	yes	characteristic
91	Dic₇⋊C₄	—	2	1	no	—
92	Dic₇⋊C₄	—	2	1	no	—
93	Dic₇⋊C₄	—	2	1	no	—
94	Dic₇⋊C₄	—	2	1	no	—
95	Dic₇⋊C₄	—	2	1	no	—
96	Dic₇⋊C₄	—	2	1	no	—
97	Dic₇⋊C₄	D₄	1	1	yes	characteristic
98	Dic₇⋊C₄	D₄	1	1	yes	characteristic
99	C₄×Dic₇	—	2	1	no	—
100	C₄×Dic₇	—	2	1	no	—
101	C₄×Dic₇	C₂³	1	1	yes	characteristic
102	C₄×Dic₇	C₂³	1	1	yes	characteristic
103	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
105	#557	—	7	1	no	—
106	D₄×Q₈	—	7	1	no	—
107	#501	—	7	1	no	—
108	#501	—	7	1	no	—
109	#500	—	7	1	no	—
110	#478	—	7	1	no	—
111	#478	—	7	1	no	—
112	#477	—	7	1	no	—
113	#477	—	7	1	no	—
114	#475	—	7	1	no	—
115	#474	—	7	1	no	—
116	#446	—	7	1	no	—
117	#440	—	7	1	no	—
118	#440	D₇	1	1	yes	characteristic
119	C₄×SD₁₆	—	7	1	no	—
120	C₂³×C₇	—	2	1	no	—
121	C₇×Q₈	—	2	1	no	—
122	C₇×Q₈	C₂×Q₈	1	2	yes	complemented
123	C₇×D₄	—	2	1	no	—
124	C₇×D₄	C₂×Q₈	1	2	yes	—
125	C₂×C₄×C₇	—	2	1	no	—
126	C₂×C₄×C₇	—	2	1	no	—
127	C₂×C₄×C₇	—	2	1	no	—
128	C₂×C₄×C₇	—	4	2	no	—
129	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
132	C₂×Dic₇	—	2	1	no	—
133	C₂×Dic₇	—	2	1	no	—
134	C₂×Dic₇	—	2	1	no	—
135	C₂×Dic₇	—	2	1	no	—
136	C₂×Dic₇	—	4	2	no	—
137	C₂×Dic₇	—	4	2	no	—
138	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
142	Dic₁₄	—	2	2	no	—
143	Dic₁₄	—	4	2	no	—
144	Dic₁₄	—	4	2	no	—
145	Dic₁₄	—	4	2	no	—
146	Dic₁₄	—	4	2	no	—
147	C₇×C₈	—	2	1	no	—
148	C₇×C₈	C₂×Q₈	1	2	yes	complemented
149	C₇⋊C₈	—	2	1	no	—
150	C₇⋊C₈	—	2	1	no	—
151	C₂²×Q₈	—	14	1	no	—
152	C₂×SD₁₆	C₂×D₇	1	1	yes	characteristic
153	C₄⋊Q₈	—	7	1	no	—
154	C₄⋊Q₈	—	7	1	no	—
155	C₄⋊Q₈	—	7	1	no	—
156	C₄⋊Q₈	—	7	1	no	—
157	C₄⋊Q₈	—	14	1	no	—
158	#125	—	7	1	no	—
159	#125	—	7	1	no	—
160	#125	—	14	1	no	—
161	#125	—	14	1	no	—
162	C₂²⋊Q₈	—	14	1	no	—
163	C₂²⋊Q₈	—	14	1	no	—
164	C₂²⋊Q₈	—	14	1	no	—
165	C₄×Q₈	—	7	1	no	—
166	C₄×Q₈	—	7	1	no	—
167	C₄×Q₈	—	7	1	no	—
168	C₄×Q₈	—	14	1	no	—
169	C₄×Q₈	C₂×D₇	1	1	yes	characteristic
170	C₄×D₄	—	7	1	no	—
171	C₄×D₄	—	7	1	no	—
172	C₄×D₄	C₂×D₇	1	1	yes	characteristic
173	#107	—	7	1	no	—
174	#107	—	7	1	no	—
175	#107	—	7	1	no	—
176	#107	—	7	1	no	—
177	#107	—	7	1	no	—
178	#107	C₂×D₇	1	1	yes	characteristic
179	#106	—	7	1	no	—
180	#106	—	7	1	no	—
181	#106	—	7	1	no	—
182	C₄⋊C₈	—	7	1	no	—
183	C₄⋊C₈	—	7	1	no	—
184	C₄⋊C₈	—	7	1	no	—
185	#103	—	7	1	no	—
186	#103	—	7	1	no	—
187	#103	C₂×D₇	1	1	yes	characteristic
188	#102	—	7	1	no	—
189	#102	—	7	1	no	—
190	#102	C₂×D₇	1	1	yes	characteristic
191	#97	—	7	1	no	—
192	#97	C₂×D₇	1	1	yes	characteristic
193	C₄×C₈	—	7	1	no	—
194	C₂²×C₇	—	2	2	no	—
195	C₂²×C₇	—	4	1	no	—
196	C₂²×C₇	—	4	1	no	—
197	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
198	C₄×C₇	—	2	2	no	—
199	C₄×C₇	—	2	1	no	—
200	C₄×C₇	—	2	1	no	—
201	C₄×C₇	—	4	1	no	—
202	C₄×C₇	—	4	1	no	—
203	C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
204	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
205	Dic₇	—	2	1	no	—
206	Dic₇	—	2	1	no	—
207	Dic₇	—	2	1	no	—
208	Dic₇	—	4	1	no	—
209	Dic₇	—	4	1	no	—
210	Dic₇	—	4	1	no	—
211	Dic₇	—	4	1	no	—
212	Dic₇	AGL(1,ℤ/8)	1	2	yes	—
213	C₂×Q₈	—	7	1	no	—
214	C₂×Q₈	—	14	1	no	—
215	C₂×Q₈	—	14	1	no	—
216	C₂×Q₈	—	14	1	no	—
217	C₂×Q₈	—	14	1	no	—
218	C₂×Q₈	—	28	4	no	—
219	C₂×Q₈	C₂²×D₇	1	1	yes	characteristic
220	C₂×D₄	C₂²×D₇	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	14	1	no	—
223	C₂²×C₄	—	14	1	no	—
224	SD₁₆	Dic₁₄	1	4	yes	complemented
225	C₂×C₈	—	7	1	no	—
226	C₂×C₈	—	7	1	no	—
227	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
228	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	7	1	no	—
231	C₄⋊C₄	—	7	1	no	—
232	C₄⋊C₄	—	7	1	no	—
233	C₄⋊C₄	—	7	1	no	—
234	C₄⋊C₄	—	7	1	no	—
235	C₄⋊C₄	—	7	1	no	—
236	C₄⋊C₄	—	7	1	no	—
237	C₄⋊C₄	—	14	1	no	—
238	C₄⋊C₄	—	14	1	no	—
239	C₄⋊C₄	—	14	1	no	—
240	C₄⋊C₄	—	14	1	no	—
241	C₄⋊C₄	—	14	1	no	—
242	C₄⋊C₄	—	14	1	no	—
243	C₄⋊C₄	—	14	1	no	—
244	C₄⋊C₄	—	14	1	no	—
245	C₄⋊C₄	—	14	1	no	—
246	C₄⋊C₄	—	14	1	no	—
247	C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	14	1	no	—
251	C₂²⋊C₄	—	14	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	7	1	no	—
254	C₄²	—	7	1	no	—
255	C₄²	—	14	1	no	—
256	C₄²	—	14	1	no	—
257	C₄²	C₂²×D₇	1	1	yes	characteristic
258	C₂×C₇	—	4	2	no	—
259	C₂×C₇	D₄×Q₈	1	1	yes	characteristic
260	C₂×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
261	C₂×C₇	#578	1	1	yes	characteristic
262	C₂³	—	2	1	no	—
263	Q₈	—	2	1	no	—
264	Q₈	—	14	2	no	—
265	Q₈	—	28	2	no	—
266	Q₈	—	28	2	no	—
267	Q₈	—	28	2	no	—
268	Q₈	—	28	2	no	—
269	Q₈	C₂×Dic₁₄	1	2	yes	complemented
270	D₄	—	2	1	no	—
271	D₄	C₂×Dic₁₄	1	2	yes	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	7	1	no	—
277	C₂×C₄	—	7	1	no	—
278	C₂×C₄	—	7	1	no	—
279	C₂×C₄	—	7	1	no	—
280	C₂×C₄	—	14	1	no	—
281	C₂×C₄	—	14	1	no	—
282	C₂×C₄	—	14	1	no	—
283	C₂×C₄	—	14	1	no	—
284	C₂×C₄	—	28	2	no	—
285	C₂×C₄	—	28	2	no	—
286	C₂×C₄	D₄×D₇	1	1	yes	characteristic
287	C₂×C₄	D₄×D₇	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	14	1	no	—
291	C₈	—	14	1	no	—
292	C₈	C₂×Dic₁₄	1	2	yes	complemented
293	C₇	#3386	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	2	2	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₂×D₄×D₇	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	2	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	7	2	no	—
304	C₄	—	14	1	no	—
305	C₄	—	14	1	no	—
306	C₄	—	14	1	no	—
307	C₄	—	28	1	no	—
308	C₄	—	28	1	no	—
309	C₄	—	28	1	no	—
310	C₄	—	28	1	no	—
311	C₄	C₂²×Dic₁₄	1	1	yes	characteristic
312	C₄	#6554	1	1	yes	characteristic
313	C₂	—	4	2	no	—
314	C₂	D₄×Dic₁₄	1	1	yes	characteristic
315	C₂	D₇×AGL(1,ℤ/8)	1	1	yes	characteristic
316	C₂	#90470	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₇×#440	C₂
socle	C₂²×C₇	C₂²×D₄
prime radical	C₂×C₇×D₄	C₂³
hypercenter	#440	D₇
hypocenter	C₇	#3386
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #90260	C₂	semidirect
6. #89793	C₂	semidirect
8. #89648	C₂	semidirect
9. #89646	C₂	semidirect
10. #89566	C₂	semidirect
11. #89563	C₂	semidirect
14. #89474	C₂	semidirect
15. #89473	C₂	semidirect
71. C₇×SD₁₆	Q₈	semidirect
122. C₇×Q₈	C₂×Q₈	semidirect
148. C₇×C₈	C₂×Q₈	semidirect
224. SD₁₆	Dic₁₄	semidirect
269. Q₈	C₂×Dic₁₄	semidirect
292. C₈	C₂×Dic₁₄	semidirect
293. C₇	#3386	semidirect

Maximal subgroups

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

subgroup	index
#90260	2
D₄×Dic₁₄	2
C₇×#440	2
#89936	2
SD₁₆×Dic₇	2
#89793	2
#89781	2
#89648	2
#89646	2
#89566	2
#89563	2
#89535	2
#89531	2
#89474	2
#89473	2
#3386	7