Finite Group Databasebeta

Subgroups — #11739478

Order 896.

350 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 113 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	#90462	C₂	1	2	yes	maximal, complemented
2	C₇×#508	C₂	1	1	yes	maximal, characteristic, complemented
3	#89967	C₂	1	2	yes	maximal, complemented
4	D₇×Q₃₂	C₂	1	2	yes	maximal, complemented
5	#89687	C₂	1	4	yes	maximal, complemented
6	#89684	C₂	1	2	yes	maximal, complemented
7	C₂×Dic₅₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#89673	C₂	1	1	yes	maximal, characteristic, complemented
9	#6554	—	2	2	no	—
10	C₇×#135	C₂²	1	2	yes	—
11	#6514	—	2	2	no	—
12	D₇×Q₁₆	C₂²	1	4	yes	—
13	#6478	—	2	4	no	—
14	#6475	C₂²	1	4	yes	—
15	C₂×Dic₂₈	C₂²	1	2	yes	—
16	#6464	C₂²	1	1	yes	characteristic
17	C₇×Q₃₂	C₂²	1	1	yes	characteristic
18	C₇×SD₃₂	C₂²	1	2	yes	—
19	C₇×D₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₁₆	C₂²	1	1	yes	characteristic
21	#6403	C₂²	1	4	yes	—
22	#6401	C₂²	1	4	yes	—
23	#6386	C₂²	1	1	yes	characteristic
24	Dic₅₆	C₂²	1	2	yes	—
25	Dic₅₆	C₂²	1	2	yes	—
26	#6372	C₂²	1	2	yes	—
27	D₇×C₁₆	C₂²	1	2	yes	—
28	#3417	—	7	1	no	maximal, Sylow
29	C₇×#41	—	2	2	no	—
30	Q₈×D₇	—	2	4	no	—
31	#1159	—	2	4	no	—
32	#1159	—	4	4	no	—
33	#1157	—	4	2	no	—
34	#1157	D₄	1	1	yes	characteristic
35	C₂×Dic₁₄	—	2	2	no	—
36	C₂×Dic₁₄	—	4	2	no	—
37	C₇×Q₁₆	C₂³	1	2	yes	—
38	C₇×SD₁₆	—	2	2	no	—
39	C₇×D₈	C₂³	1	2	yes	—
40	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
41	#1144	—	2	4	no	—
42	#1142	—	2	4	no	—
43	#1136	D₄	1	1	yes	characteristic
44	Dic₂₈	—	2	2	no	—
45	Dic₂₈	C₂³	1	4	yes	—
46	#1131	D₄	1	2	yes	—
47	C₈×D₇	C₂³	1	2	yes	—
48	C₇×C₁₆	C₂³	1	1	yes	characteristic
49	C₇×C₁₆	C₂³	1	1	yes	characteristic
50	C₇⋊C₁₆	C₂³	1	2	yes	—
51	#578	—	7	2	no	—
52	#510	—	7	2	no	—
53	#510	—	7	4	no	—
54	#508	—	7	2	no	—
55	#508	D₇	1	1	yes	characteristic, complemented
56	C₂×Q₃₂	—	7	2	no	—
57	C₂×Q₃₂	—	7	1	no	—
58	#504	—	7	1	no	—
59	C₇×Q₈	—	2	2	no	—
60	C₇×D₄	—	2	2	no	—
61	C₇×D₄	—	4	2	no	—
62	C₂×C₄×C₇	—	4	2	no	—
63	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂²⋊D₇	—	4	4	no	—
65	C₂²⋊D₇	D₈	1	2	yes	—
66	C₂×Dic₇	—	4	2	no	—
67	C₂×Dic₇	—	4	4	no	—
68	D₂₈	D₈	1	1	yes	characteristic
69	C₄×D₇	—	4	4	no	—
70	C₄×D₇	C₂×D₄	1	2	yes	—
71	Dic₁₄	—	2	4	no	—
72	Dic₁₄	—	4	2	no	—
73	Dic₁₄	—	4	4	no	—
74	Dic₁₄	D₈	1	1	yes	characteristic
75	C₇×C₈	C₂×D₄	1	1	yes	characteristic
76	C₇×C₈	C₂⁴	1	1	yes	characteristic
77	C₇⋊C₈	C₂×D₄	1	2	yes	—
78	2₋¹⁺⁴	—	14	2	no	—
79	#137	—	14	4	no	—
80	#137	—	14	2	no	—
81	#135	—	7	4	no	—
82	#135	C₂×D₇	1	2	yes	—
83	C₂×Q₁₆	—	7	4	no	—
84	C₂×Q₁₆	—	7	2	no	—
85	#131	—	7	1	no	—
86	Q₃₂	—	7	4	no	—
87	Q₃₂	—	7	2	no	—
88	Q₃₂	—	7	2	no	—
89	Q₃₂	C₂×D₇	1	1	yes	characteristic
90	SD₃₂	—	7	4	no	—
91	SD₃₂	C₂×D₇	1	2	yes	—
92	D₁₆	C₂×D₇	1	1	yes	characteristic
93	M₃₂	—	7	1	no	—
94	M₃₂	—	7	2	no	—
95	C₂×C₁₆	—	7	2	no	—
96	C₂×C₁₆	C₂×D₇	1	1	yes	characteristic
97	C₂²×C₇	—	4	2	no	—
98	C₂²×C₇	C₂×D₈	1	1	yes	characteristic
99	C₂×D₇	C₂×D₈	1	2	yes	—
100	C₄×C₇	—	4	2	no	—
101	C₄×C₇	C₂×D₈	1	1	yes	characteristic
102	C₄×C₇	C₂²×D₄	1	1	yes	characteristic
103	Dic₇	—	4	4	no	—
104	Dic₇	C₂×D₈	1	2	yes	—
105	#41	—	2	2	no	—
106	#41	—	7	1	no	—
107	#41	—	14	4	no	—
108	#41	—	28	2	no	—
109	#41	—	28	4	no	—
110	C₂×Q₈	—	14	4	no	—
111	C₂×Q₈	—	14	2	no	—
112	C₂×Q₈	—	28	2	no	—
113	Q₁₆	—	7	4	no	—
114	Q₁₆	—	14	4	no	—
115	Q₁₆	—	14	2	no	—
116	Q₁₆	C₂²×D₇	1	2	yes	—
117	SD₁₆	—	2	2	no	—
118	SD₁₆	—	14	4	no	—
119	D₈	C₂²×D₇	1	2	yes	—
120	M₁₆	—	7	1	no	—
121	M₁₆	—	7	2	no	—
122	C₂×C₈	—	7	2	no	—
123	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
124	C₁₆	—	7	2	no	—
125	C₁₆	C₂²×D₇	1	1	yes	characteristic
126	C₁₆	C₂²×D₇	1	1	yes	characteristic
127	C₂×C₇	—	2	1	no	—
128	C₂×C₇	—	8	2	no	—
129	C₂×C₇	C₂²×D₈	1	1	yes	characteristic
130	D₇	—	2	2	no	—
131	Q₈	—	2	2	no	—
132	Q₈	—	7	1	no	—
133	Q₈	—	14	4	no	—
134	Q₈	—	28	4	no	—
135	Q₈	—	28	2	no	—
136	D₄	—	2	2	no	—
137	D₄	—	4	2	no	—
138	D₄	—	7	2	no	—
139	D₄	—	7	1	no	—
140	D₄	—	28	4	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	7	2	no	—
143	C₂×C₄	—	28	4	no	—
144	C₂×C₄	—	28	4	no	—
145	C₂×C₄	—	28	2	no	—
146	C₂×C₄	D₄×D₇	1	1	yes	characteristic
147	C₈	—	7	2	no	—
148	C₈	D₄×D₇	1	1	yes	characteristic
149	C₈	C₂³×D₇	1	1	yes	characteristic
150	C₇	#3417	1	1	yes	characteristic, complemented, Sylow
151	C₂²	—	4	2	no	—
152	C₂²	—	7	2	no	—
153	C₂²	D₇×D₈	1	1	yes	characteristic
154	C₄	—	4	2	no	—
155	C₄	—	7	2	no	—
156	C₄	—	28	4	no	—
157	C₄	D₇×D₈	1	1	yes	characteristic
158	C₄	C₂×D₄×D₇	1	1	yes	characteristic
159	C₂	—	2	1	no	—
160	C₂	—	8	2	no	—
161	C₂	—	14	2	no	—
162	C₂	C₂×D₇×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₇×D₈
commutator, cosocle	C₇×C₈	C₂⁴
Frattini	C₈	C₂³×D₇
Fitting	C₇×#508	C₂
socle	C₂×C₇	C₂²×D₈
hypercenter	#508	D₇
hypocenter	C₇	#3417
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #90462	C₂	semidirect
2. C₇×#508	C₂	semidirect
3. #89967	C₂	semidirect
4. D₇×Q₃₂	C₂	semidirect
5. #89687	C₂	semidirect
6. #89684	C₂	semidirect
7. C₂×Dic₅₆	C₂	semidirect
8. #89673	C₂	semidirect
55. #508	D₇	semidirect
150. C₇	#3417	semidirect

Maximal subgroups

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

subgroup	index
#90462	2
C₇×#508	2
#89967	2
D₇×Q₃₂	2
#89687	2
#89684	2
C₂×Dic₅₆	2
#89673	2
#3417	7