Group #11731665 — subgroups

Subgroups — #11731665

Order 896.

472 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.

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

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6453	C₂	1	2	yes	maximal
2	C₇×#382	C₂	1	1	yes	maximal, characteristic
3	#89748	C₂	1	2	yes	maximal
4	#89708	C₂	1	1	yes	maximal, characteristic
5	C₄²×Dic₇	C₂	1	1	yes	maximal, characteristic
6	#89419	C₂	1	8	yes	maximal
7	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
11	C₂×Dic₇⋊C₄	C₂²	1	4	yes	—
12	C₂×Dic₇⋊C₄	C₂²	1	8	yes	—
13	C₂×C₄×Dic₇	C₂²	1	4	yes	—
14	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
16	#6453	—	2	8	no	—
17	C₇×#95	C₂²	1	4	yes	—
18	C₇⋊#95	C₂²	1	4	yes	—
19	#2859	—	7	1	no	maximal, Sylow
20	C₂²×C₄×C₇	C₂³	1	4	yes	—
21	C₂²×C₄×C₇	C₂³	1	2	yes	—
22	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₇	C₂³	1	4	yes	—
24	C₂²×Dic₇	C₂³	1	2	yes	—
25	C₂²×Dic₇	C₂³	1	2	yes	—
26	C₇×C₄⋊C₄	—	2	4	no	—
27	C₄²×C₇	Q₈	1	4	yes	—
28	C₄⋊Dic₇	—	2	4	no	—
29	C₄⋊Dic₇	—	2	8	no	—
30	Dic₇⋊C₄	—	2	8	no	—
31	Dic₇⋊C₄	—	2	16	no	—
32	C₄×Dic₇	—	2	8	no	—
33	C₄×Dic₇	D₄	1	4	yes	—
34	C₄×Dic₇	Q₈	1	4	yes	—
35	C₂×#125	—	7	2	no	—
36	#389	—	7	8	no	—
37	#384	—	7	2	no	—
38	#382	—	7	1	no	—
39	#382	D₇	1	1	yes	characteristic
40	C₄³	—	7	1	no	—
41	C₂³×C₇	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₇	—	2	4	no	—
43	C₂×C₄×C₇	—	2	4	no	—
44	C₂×C₄×C₇	—	2	8	no	—
45	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
47	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
49	C₂×Dic₇	—	2	4	no	—
50	C₂×Dic₇	—	2	4	no	—
51	C₂×Dic₇	—	2	8	no	—
52	C₂×Dic₇	C₂×D₄	1	4	yes	—
53	C₂×Dic₇	C₂×Q₈	1	4	yes	—
54	C₂×Dic₇	#41	1	8	yes	—
55	#125	—	14	8	no	—
56	C₂×C₄⋊C₄	—	7	4	no	—
57	C₂×C₄⋊C₄	—	7	2	no	—
58	C₂×C₄⋊C₄	—	7	4	no	—
59	C₂×C₄⋊C₄	—	7	8	no	—
60	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
61	C₂×C₄²	—	7	1	no	—
62	C₂×C₄²	—	7	1	no	—
63	C₂×C₄²	—	7	4	no	—
64	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
65	#95	—	7	4	no	—
66	#95	C₂×D₇	1	4	yes	—
67	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
68	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
69	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
70	C₂²×C₇	C₂×#41	1	4	yes	—
71	C₄×C₇	—	2	4	no	—
72	C₄×C₇	—	4	4	no	—
73	C₄×C₇	C₄⋊Q₈	1	4	yes	—
74	Dic₇	—	2	4	no	—
75	Dic₇	—	4	4	no	—
76	Dic₇	#125	1	8	yes	complemented
77	C₂²×C₄	—	7	2	no	—
78	C₂²×C₄	—	7	2	no	—
79	C₂²×C₄	—	7	4	no	—
80	C₂²×C₄	C₂²×D₇	1	4	yes	—
81	C₂²×C₄	C₂²×D₇	1	2	yes	—
82	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
83	C₄⋊C₄	—	2	4	no	—
84	C₄⋊C₄	—	14	16	no	—
85	C₄⋊C₄	—	14	8	no	—
86	C₄⋊C₄	—	14	8	no	—
87	C₄⋊C₄	—	14	4	no	—
88	C₄²	—	7	4	no	—
89	C₄²	—	7	4	no	—
90	C₄²	—	14	8	no	—
91	C₄²	Dic₁₄	1	4	yes	—
92	C₂×C₇	C₂×#125	1	2	yes	—
93	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₇	#532	1	2	yes	—
95	C₂×C₇	#533	1	2	yes	—
96	C₂³	C₂³×D₇	1	1	yes	characteristic
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	8	no	—
100	C₂×C₄	—	7	4	no	—
101	C₂×C₄	—	7	8	no	—
102	C₂×C₄	—	7	4	no	—
103	C₂×C₄	—	14	4	no	—
104	C₂×C₄	—	14	4	no	—
105	C₂×C₄	—	14	8	no	—
106	C₂×C₄	C₂×Dic₁₄	1	4	yes	—
107	C₂×C₄	C₂×Dic₁₄	1	2	yes	—
108	C₂×C₄	D₄×D₇	1	2	yes	—
109	C₂×C₄	Q₈×D₇	1	2	yes	—
110	C₇	#2859	1	1	yes	characteristic, complemented, Sylow
111	C₂²	C₂²×Dic₁₄	1	1	yes	characteristic
112	C₂²	C₂×D₄×D₇	1	1	yes	characteristic
113	C₂²	C₂×Q₈×D₇	1	1	yes	characteristic
114	C₂²	D₇×#41	1	4	yes	—
115	C₄	—	2	4	no	—
116	C₄	—	4	4	no	—
117	C₄	—	7	8	no	—
118	C₄	—	14	4	no	—
119	C₄	—	28	4	no	—
120	C₄	#6451	1	4	yes	—
121	C₂	C₂×#6451	1	1	yes	characteristic
122	C₂	#90208	1	2	yes	—
123	C₂	#90359	1	2	yes	—
124	C₂	D₇×#125	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#6453

C₂

yes

maximal

C₇×#382

C₂

yes

maximal, characteristic

#89748

C₂

yes

maximal

#89708

C₂

yes

maximal, characteristic

C₄²×Dic₇

C₂

yes

maximal, characteristic

#89419

C₂

yes

maximal

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×Dic₇⋊C₄

C₂²

yes

—

C₂×Dic₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

characteristic

C₂×C₄×Dic₇

C₂²

yes

characteristic

#6453

—

C₇×#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

#2859

—

maximal, Sylow

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₇×C₄⋊C₄

—

C₄²×C₇

Q₈

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

Dic₇⋊C₄

—

C₄×Dic₇

—

C₄×Dic₇

D₄

yes

—

C₄×Dic₇

Q₈

yes

—

—

—

—

—

D₇

yes

characteristic

C₄³

—

C₂³×C₇

C₂⁴

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂×D₄

yes

—

C₂×Dic₇

C₂×Q₈

yes

—

C₂×Dic₇

#41

yes

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇

yes

characteristic

#95

—

#95

C₂×D₇

yes

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₂²×C₇

C₂²×Q₈

yes

characteristic

C₂²×C₇

C₂²×Q₈

yes

characteristic

C₂²×C₇

C₂×#41

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₄⋊Q₈

yes

—

Dic₇

—

Dic₇

—

Dic₇

#125

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₁₄

yes

—

C₂×C₇

C₂×#125

yes

—

C₂×C₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₇

#532

yes

—

C₂×C₇

#533

yes

—

C₂³

C₂³×D₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×Dic₁₄

yes

—

107

C₂×C₄

C₂×Dic₁₄

yes

—

108

C₂×C₄

D₄×D₇

yes

—

109

C₂×C₄

Q₈×D₇

yes

—

110

C₇

#2859

yes

characteristic, complemented, Sylow

111

C₂²

C₂²×Dic₁₄

yes

characteristic

112

C₂²

C₂×D₄×D₇

yes

characteristic

113

C₂²

C₂×Q₈×D₇

yes

characteristic

114

C₂²

D₇×#41

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#6451

yes

—

121

C₂

C₂×#6451

yes

characteristic

122

C₂

#90208

yes

—

123

C₂

#90359

yes

—

124

C₂

D₇×#125

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂³×C₇	C₂⁴
Fitting	C₇×#382	C₂
hypercenter	#382	D₇
hypocenter	C₇	#2859
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇

commutator, socle, prime radical, cosocle

C₂³×C₇

C₂⁴

C₂

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
76. Dic₇	#125	semidirect
110. C₇	#2859	semidirect

H ≅ G/N

type

76. Dic₇

#125

semidirect

110. C₇

#2859

semidirect

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

subgroup	index
C₂×#6453	2
C₇×#382	2
#89748	2
#89708	2
C₄²×Dic₇	2
#89419	2
#2859	7

subgroup

index

C₄²×Dic₇

Subgroups — #11731665

Conjugacy classes of subgroups

Complements

Maximal subgroups