Finite Group Databasebeta

Subgroups — #11726765

Order 896.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#89905	C₂	1	1	yes	maximal, characteristic
2	#89901	C₂	1	1	yes	maximal, characteristic
3	#89738	C₂	1	1	yes	maximal, characteristic
4	C₂×#6382	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#343	C₂	1	1	yes	maximal, characteristic, complemented
6	#89353	C₂	1	1	yes	maximal, characteristic, complemented
7	#89349	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#1157	C₂²	1	1	yes	characteristic
9	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
13	C₂×#1136	C₂²	1	1	yes	characteristic
14	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
15	#6395	—	2	1	no	—
16	(C₂×D₇)⋊C₈	—	2	2	no	—
17	#6390	—	2	1	no	—
18	#6384	—	2	1	no	—
19	#6383	—	2	1	no	—
20	#6382	—	2	2	no	—
21	#2079	—	7	1	no	maximal, Sylow
22	C₂²×C₄×C₇	—	2	1	no	—
23	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂×C₂²⋊D₇	—	2	1	no	—
25	C₂²×Dic₇	—	2	1	no	—
26	#1157	—	2	1	no	—
27	#1157	—	2	1	no	—
28	#1157	—	2	1	no	—
29	#1157	—	2	1	no	—
30	C₂×D₂₈	D₄	1	1	yes	characteristic
31	C₂×C₄×D₇	—	2	1	no	—
32	C₂×Dic₁₄	D₄	1	1	yes	characteristic
33	C₇×M₁₆	—	2	1	no	—
34	C₇×M₁₆	—	2	1	no	—
35	C₇×M₁₆	—	2	1	no	—
36	C₇×M₁₆	—	2	1	no	—
37	C₂×C₇×C₈	—	2	1	no	—
38	C₂×C₇×C₈	—	2	1	no	—
39	C₇×C₄⋊C₄	—	2	1	no	—
40	C₇×C₄⋊C₄	—	2	1	no	—
41	C₄⋊Dic₇	—	2	1	no	—
42	C₄⋊Dic₇	—	2	1	no	—
43	#1136	—	2	1	no	—
44	#1136	—	2	1	no	—
45	C₂×C₇⋊C₈	—	2	1	no	—
46	C₂×C₇⋊C₈	—	2	1	no	—
47	C₂×C₇⋊C₈	—	2	1	no	—
48	C₂×C₇⋊C₈	D₄	1	2	yes	—
49	C₂×#106	—	7	1	no	—
50	#417	—	7	1	no	—
51	#417	—	7	1	no	—
52	#408	—	7	1	no	—
53	#343	—	7	1	no	—
54	#343	D₇	1	1	yes	characteristic, complemented
55	#341	—	7	1	no	—
56	C₂³×C₇	#41	1	1	yes	characteristic
57	C₂²×D₇	—	2	1	no	—
58	C₂×C₄×C₇	—	4	1	no	—
59	C₂×C₄×C₇	—	8	1	no	—
60	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇	#41	1	1	yes	characteristic
63	C₂×C₄×C₇	#41	1	1	yes	characteristic
64	C₂×C₄×C₇	#41	1	1	yes	characteristic
65	C₂×C₄×C₇	#41	1	1	yes	characteristic
66	C₂²⋊D₇	—	4	1	no	—
67	C₂²⋊D₇	—	4	1	no	—
68	C₂×Dic₇	—	2	1	no	—
69	C₂×Dic₇	—	4	1	no	—
70	C₂×Dic₇	—	8	1	no	—
71	D₂₈	—	2	1	no	—
72	D₂₈	—	2	1	no	—
73	C₄×D₇	—	4	1	no	—
74	C₄×D₇	—	4	1	no	—
75	Dic₁₄	—	2	1	no	—
76	Dic₁₄	—	2	1	no	—
77	C₇×C₈	—	4	1	no	—
78	C₇×C₈	—	4	1	no	—
79	C₇⋊C₈	—	2	2	no	—
80	C₇⋊C₈	—	4	1	no	—
81	C₂×#41	—	7	1	no	—
82	C₂×M₁₆	—	7	1	no	—
83	C₂×M₁₆	C₂×D₇	1	1	yes	characteristic
84	C₂×M₁₆	C₂×D₇	1	1	yes	characteristic
85	C₂²×C₈	—	7	1	no	—
86	C₂×C₄⋊C₄	—	7	1	no	—
87	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
88	#106	—	14	2	no	—
89	#103	—	14	1	no	—
90	#103	—	14	1	no	—
91	#102	—	14	1	no	—
92	#102	—	14	1	no	—
93	C₂²⋊C₈	—	14	2	no	—
94	C₂²×C₇	—	2	1	no	—
95	C₂²×C₇	—	2	1	no	—
96	C₂²×C₇	#121	1	1	yes	characteristic
97	C₂²×C₇	#126	1	1	yes	characteristic
98	C₂²×C₇	#126	1	1	yes	characteristic
99	C₂×D₇	—	4	1	no	—
100	C₂×D₇	—	4	1	no	—
101	C₂×D₇	—	4	1	no	—
102	C₄×C₇	—	8	1	no	—
103	C₄×C₇	#123	1	1	yes	characteristic
104	C₄×C₇	#123	1	1	yes	characteristic
105	C₄×C₇	#123	1	1	yes	characteristic
106	C₄×C₇	#124	1	1	yes	characteristic
107	Dic₇	—	4	1	no	—
108	Dic₇	—	8	1	no	—
109	#41	—	14	1	no	—
110	#41	—	14	1	no	—
111	#41	—	14	1	no	—
112	#41	—	14	1	no	—
113	C₂×Q₈	—	7	1	no	—
114	C₂×D₄	—	7	1	no	—
115	C₂×D₄	—	14	1	no	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	14	1	no	—
118	C₂²×C₄	—	14	1	no	—
119	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
120	M₁₆	—	2	1	no	—
121	M₁₆	—	2	1	no	—
122	M₁₆	—	2	1	no	—
123	M₁₆	—	2	1	no	—
124	M₁₆	—	14	1	no	—
125	M₁₆	—	14	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	7	2	no	—
129	C₂×C₈	—	14	1	no	—
130	C₂×C₈	—	14	1	no	—
131	C₂×C₈	—	14	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	14	1	no	—
135	C₄⋊C₄	—	14	1	no	—
136	C₂×C₇	—	2	1	no	—
137	C₂×C₇	—	2	1	no	—
138	C₂×C₇	#397	1	1	yes	characteristic
139	C₂×C₇	#471	1	1	yes	characteristic
140	C₂×C₇	#471	1	1	yes	characteristic
141	D₇	—	8	1	no	—
142	C₂³	—	14	1	no	—
143	C₂³	#1157	1	1	yes	characteristic
144	Q₈	—	14	1	no	—
145	Q₈	—	14	1	no	—
146	D₄	—	14	1	no	—
147	D₄	—	14	1	no	—
148	D₄	—	28	1	no	—
149	D₄	—	28	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	8	1	no	—
152	C₂×C₄	—	14	1	no	—
153	C₂×C₄	—	28	1	no	—
154	C₂×C₄	—	28	1	no	—
155	C₂×C₄	—	28	1	no	—
156	C₂×C₄	—	56	1	no	—
157	C₂×C₄	#1157	1	1	yes	characteristic
158	C₂×C₄	#1157	1	1	yes	characteristic
159	C₂×C₄	D₄×D₇	1	1	yes	characteristic
160	C₂×C₄	D₄×D₇	1	1	yes	characteristic
161	C₂×C₄	#1159	1	1	yes	characteristic
162	C₂×C₄	#1161	1	1	yes	characteristic
163	C₈	—	4	1	no	—
164	C₈	—	4	1	no	—
165	C₈	—	14	2	no	—
166	C₈	—	28	1	no	—
167	C₇	#2079	1	1	yes	characteristic, complemented, Sylow
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	28	1	no	—
171	C₂²	—	28	1	no	—
172	C₂²	—	28	1	no	—
173	C₂²	#6439	1	1	yes	characteristic
174	C₂²	#6447	1	1	yes	characteristic
175	C₂²	#6461	1	1	yes	characteristic
176	C₄	—	8	1	no	—
177	C₄	—	28	1	no	—
178	C₄	—	56	1	no	—
179	C₄	#6446	1	1	yes	characteristic
180	C₄	#6448	1	1	yes	characteristic
181	C₄	#6457	1	1	yes	characteristic
182	C₄	#6457	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	—	56	1	no	—
186	C₂	#89447	1	1	yes	characteristic
187	C₂	#89520	1	1	yes	characteristic
188	C₂	#89526	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#6447
commutator, cosocle	C₂²×C₄×C₇	C₂³
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#343	C₂
socle	C₂²×C₇	#121
prime radical	C₂×#1157	C₂²
hypercenter	#343	D₇
hypocenter	C₇	#2079
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

Complements

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

N	H ≅ G/N	type
4. C₂×#6382	C₂	semidirect
5. C₇×#343	C₂	semidirect
6. #89353	C₂	semidirect
7. #89349	C₂	semidirect
54. #343	D₇	semidirect
167. C₇	#2079	semidirect

Maximal subgroups

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

subgroup	index
#89905	2
#89901	2
#89738	2
C₂×#6382	2
C₇×#343	2
#89353	2
#89349	2
#2079	7