Subgroups — #11736770

Order 896.

380 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#90170	C₂	1	1	yes	maximal, characteristic
2	C₇×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#89610	C₂	1	1	yes	maximal, characteristic, complemented
5	#89606	C₂	1	1	yes	maximal, characteristic, complemented
6	#89603	C₂	1	1	yes	maximal, characteristic, complemented
7	#89602	C₂	1	1	yes	maximal, characteristic, complemented
8	#89502	C₂	1	2	yes	maximal, complemented
9	(C₂²⋊D₇)⋊C₈	C₂	1	2	yes	maximal, complemented
10	#89481	C₂	1	2	yes	maximal, complemented
11	#89480	C₂	1	2	yes	maximal, complemented
12	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
13	#6493	C₄	1	1	yes	characteristic
14	#6492	C₄	1	2	yes	—
15	C₄×C₂²⋊D₇	C₂²	1	2	yes	—
16	#6487	C₄	1	1	yes	characteristic
17	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×#1131	C₂²	1	2	yes	—
20	C₂×C₈×D₇	C₂²	1	2	yes	—
21	#6439	C₄	1	2	yes	—
22	#6438	C₄	1	1	yes	characteristic
23	C₄×D₂₈	C₂²	1	1	yes	characteristic
24	#6435	C₂²	1	2	yes	—
25	#6433	C₄	1	1	yes	characteristic
26	C₄×Dic₁₄	C₂²	1	1	yes	characteristic
27	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₇×C₂²⋊C₈	C₂²	1	2	yes	—
30	C₇×#97	C₂²	1	2	yes	—
31	(C₂×D₇)⋊C₈	C₂²	1	2	yes	—
32	(C₂×D₇)⋊C₈	C₂²	1	2	yes	—
33	#6389	C₂²	1	2	yes	—
34	Dic₇⋊C₈	C₂²	1	2	yes	—
35	Dic₇⋊C₈	C₂²	1	2	yes	—
36	C₈×Dic₇	C₂²	1	2	yes	—
37	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
38	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
39	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
41	#2966	—	7	1	no	maximal, Sylow
42	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₇	C₂×C₄	1	2	yes	—
46	C₂×D₂₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₇	C₂³	1	2	yes	—
48	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
49	C₂×C₇×C₈	C₂³	1	2	yes	—
50	C₂×C₇×C₈	C₂³	1	2	yes	—
51	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₇	C₂³	1	1	yes	characteristic
54	C₄²×C₇	C₂³	1	1	yes	characteristic
55	C₂²⋊Dic₇	C₂×C₄	1	2	yes	—
56	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
57	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
58	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
62	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
63	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
64	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
65	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
66	C₄×Dic₇	C₂³	1	2	yes	—
67	C₂×C₇⋊C₈	—	2	2	no	—
68	C₂×C₇⋊C₈	—	2	2	no	—
69	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
73	#1131	—	2	4	no	—
74	C₈×D₇	—	2	4	no	—
75	#529	—	7	1	no	—
76	#446	—	7	1	no	—
77	C₈×Q₈	—	7	1	no	—
78	#436	—	7	1	no	—
79	#435	—	7	2	no	—
80	C₈×D₄	—	7	1	no	—
81	C₈×D₄	—	7	2	no	—
82	#433	—	7	2	no	—
83	#432	D₇	1	1	yes	characteristic, complemented
84	#405	—	7	2	no	—
85	C₂×C₄×C₈	—	7	1	no	—
86	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₇	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	2	1	no	—
91	C₂×C₄×C₇	—	2	1	no	—
92	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₇	—	2	4	no	—
99	C₂×Dic₇	C₂²×C₄	1	2	yes	—
100	C₂×Dic₇	C₂²×C₄	1	2	yes	—
101	C₂×Dic₇	C₂²×C₄	1	2	yes	—
102	D₂₈	—	2	2	no	—
103	C₄×D₇	—	2	2	no	—
104	C₄×D₇	—	2	2	no	—
105	Dic₁₄	—	2	2	no	—
106	C₇×C₈	—	2	2	no	—
107	C₇×C₈	—	2	2	no	—
108	C₇⋊C₈	—	2	1	no	—
109	C₇⋊C₈	—	2	1	no	—
110	C₇⋊C₈	#41	1	2	yes	—
111	C₇⋊C₈	#41	1	2	yes	—
112	C₂×M₁₆	—	7	2	no	—
113	C₂²×C₈	—	7	1	no	—
114	C₂²×C₈	—	7	1	no	—
115	C₂²×C₈	—	7	2	no	—
116	#126	—	7	2	no	—
117	#125	—	7	1	no	—
118	#124	—	7	1	no	—
119	#123	—	7	2	no	—
120	C₂²⋊Q₈	—	7	1	no	—
121	#121	—	7	1	no	—
122	C₄×Q₈	—	7	1	no	—
123	C₄×D₄	—	7	2	no	—
124	C₄×D₄	—	7	1	no	—
125	#117	—	7	2	no	—
126	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
127	C₄⋊C₈	—	7	2	no	—
128	C₄⋊C₈	—	7	2	no	—
129	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
130	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
131	C₂²⋊C₈	—	7	2	no	—
132	C₂²⋊C₈	—	7	2	no	—
133	C₂²⋊C₈	C₂×D₇	1	2	yes	—
134	#97	—	7	2	no	—
135	#97	C₂×D₇	1	2	yes	—
136	C₄×C₈	—	7	2	no	—
137	C₄×C₈	—	7	1	no	—
138	C₄×C₈	—	7	1	no	—
139	C₄×C₈	—	7	1	no	—
140	C₄×C₈	—	7	1	no	—
141	C₂²×C₇	—	2	1	no	—
142	C₂²×C₇	—	2	1	no	—
143	C₂²×C₇	#131	1	2	yes	—
144	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
145	C₂×D₇	—	2	2	no	—
146	C₂×D₇	—	2	2	no	—
147	C₂×D₇	—	2	2	no	—
148	C₄×C₇	—	2	1	no	—
149	C₄×C₇	—	2	1	no	—
150	C₄×C₇	—	2	1	no	—
151	C₄×C₇	—	2	1	no	—
152	C₄×C₇	#131	1	2	yes	—
153	C₄×C₇	C₂×#41	1	1	yes	characteristic
154	C₄×C₇	C₂×#41	1	1	yes	characteristic
155	Dic₇	—	2	2	no	—
156	Dic₇	—	2	2	no	—
157	Dic₇	—	2	2	no	—
158	C₂×Q₈	—	7	1	no	—
159	C₂×D₄	—	7	2	no	—
160	C₂×D₄	—	7	1	no	—
161	C₂²×C₄	—	7	2	no	—
162	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
163	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
164	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
165	M₁₆	—	14	4	no	—
166	C₂×C₈	—	7	1	no	—
167	C₂×C₈	—	7	1	no	—
168	C₂×C₈	—	7	1	no	—
169	C₂×C₈	—	7	1	no	—
170	C₂×C₈	—	14	4	no	—
171	C₂×C₈	—	14	2	no	—
172	C₂×C₈	—	14	2	no	—
173	C₂×C₈	C₂²×D₇	1	2	yes	—
174	C₂×C₈	C₂²×D₇	1	2	yes	—
175	C₄⋊C₄	—	7	2	no	—
176	C₄⋊C₄	—	7	2	no	—
177	C₄⋊C₄	—	7	2	no	—
178	C₄⋊C₄	—	7	2	no	—
179	C₄⋊C₄	—	7	1	no	—
180	C₄⋊C₄	—	7	1	no	—
181	C₂²⋊C₄	—	7	2	no	—
182	C₂²⋊C₄	—	7	2	no	—
183	C₂²⋊C₄	—	7	2	no	—
184	C₂²⋊C₄	—	7	2	no	—
185	C₂²⋊C₄	—	7	2	no	—
186	C₄²	—	7	2	no	—
187	C₄²	C₄×D₇	1	1	yes	characteristic
188	C₄²	C₄×D₇	1	1	yes	characteristic
189	C₄²	C₂²×D₇	1	1	yes	characteristic
190	C₄²	C₂²×D₇	1	1	yes	characteristic
191	C₂×C₇	—	2	2	no	—
192	C₂×C₇	C₄×#41	1	1	yes	characteristic
193	C₂×C₇	C₂×#131	1	1	yes	characteristic
194	C₂×C₇	C₂×#131	1	1	yes	characteristic
195	D₇	—	4	2	no	—
196	C₂³	—	7	2	no	—
197	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
198	Q₈	—	14	2	no	—
199	D₄	—	14	4	no	—
200	D₄	—	14	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	7	2	no	—
206	C₂×C₄	—	7	2	no	—
207	C₂×C₄	—	7	2	no	—
208	C₂×C₄	—	14	2	no	—
209	C₂×C₄	—	14	2	no	—
210	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
215	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	2	2	no	—
218	C₈	—	7	2	no	—
219	C₈	—	7	2	no	—
220	C₈	—	14	1	no	—
221	C₈	—	14	1	no	—
222	C₇	#2966	1	1	yes	characteristic, complemented, Sylow
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	14	2	no	—
226	C₂²	—	14	2	no	—
227	C₂²	—	14	2	no	—
228	C₂²	#6470	1	2	yes	—
229	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	14	2	no	—
235	C₄	—	14	2	no	—
236	C₄	—	14	2	no	—
237	C₄	#6470	1	2	yes	—
238	C₄	D₇×#41	1	1	yes	characteristic
239	C₄	D₇×#41	1	1	yes	characteristic
240	C₂	—	2	2	no	—
241	C₂	—	28	2	no	—
242	C₂	#90213	1	1	yes	characteristic
243	C₂	C₂×#6470	1	1	yes	characteristic
244	C₂	C₂×#6470	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₇×#432	C₂
prime radical	#6493	C₄
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	#432	D₇
hypocenter	C₇	#2966
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. C₇×#432	C₂	semidirect
3. C₂×C₄×C₇⋊C₈	C₂	semidirect
4. #89610	C₂	semidirect
5. #89606	C₂	semidirect
6. #89603	C₂	semidirect
7. #89602	C₂	semidirect
8. #89502	C₂	semidirect
9. (C₂²⋊D₇)⋊C₈	C₂	semidirect
10. #89481	C₂	semidirect
11. #89480	C₂	semidirect
83. #432	D₇	semidirect
222. C₇	#2966	semidirect

Maximal subgroups

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

subgroup	index
#90170	2
C₇×#432	2
C₂×C₄×C₇⋊C₈	2
#89610	2
#89606	2
#89603	2
#89602	2
#89502	2
(C₂²⋊D₇)⋊C₈	2
#89481	2
#89480	2
#2966	7