Group #11729824 — subgroups

Subgroups — #11729824

Order 896.

256 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 71 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	C₂²×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#6410	C₂	1	2	yes	maximal
3	#89997	C₂	1	2	yes	maximal
4	#89874	C₂	1	1	yes	maximal, characteristic
5	#89347	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₈	C₂²	1	2	yes	—
7	C₂×C₇×Q₁₆	—	2	4	no	—
8	C₂×C₇×Q₁₆	—	2	2	no	—
9	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
10	#6489	C₂²	1	1	yes	characteristic
11	C₂×#1136	C₂²	1	2	yes	—
12	C₂×#1136	C₂²	1	1	yes	characteristic
13	#6410	—	2	4	no	—
14	#6409	—	2	4	no	—
15	Dic₇⋊C₈	—	2	2	no	—
16	#2047	—	7	1	no	maximal, Sylow
17	C₂×C₇×Q₈	—	2	2	no	—
18	C₂×C₇×Q₈	—	2	2	no	—
19	C₂×C₇×Q₈	—	2	2	no	—
20	C₂×C₇×Q₈	—	2	2	no	—
21	C₂×C₇×Q₈	—	2	2	no	—
22	C₂×C₇×Q₈	D₄	1	4	yes	—
23	C₂²×C₄×C₇	—	2	2	no	—
24	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₇×Q₁₆	—	2	8	no	—
26	C₂×C₇×C₈	—	2	2	no	—
27	C₂×C₇×C₈	D₄	1	2	yes	—
28	C₂²⋊Dic₇	—	2	1	no	—
29	C₄⋊Dic₇	D₄	1	2	yes	—
30	C₄×Dic₇	—	2	1	no	—
31	#1136	—	2	2	no	—
32	#1136	—	2	2	no	—
33	#1136	—	2	2	no	—
34	C₂×C₇⋊C₈	—	2	2	no	—
35	C₂×C₇⋊C₈	—	2	1	no	—
36	C₂²×Q₁₆	D₇	1	1	yes	characteristic
37	#424	—	7	1	no	—
38	#419	—	7	2	no	—
39	C₂×#101	—	7	2	no	—
40	#341	—	7	1	no	—
41	C₂³×C₇	#41	1	1	yes	characteristic
42	C₇×Q₈	—	2	4	no	—
43	C₇×Q₈	—	2	4	no	—
44	C₇×Q₈	—	4	2	no	—
45	C₇×Q₈	—	4	2	no	—
46	C₂×C₄×C₇	—	2	4	no	—
47	C₂×C₄×C₇	—	4	2	no	—
48	C₂×C₄×C₇	—	4	2	no	—
49	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇	#41	1	2	yes	—
53	C₂×Dic₇	—	2	2	no	—
54	C₇×C₈	—	2	2	no	—
55	C₇⋊C₈	—	4	1	no	—
56	C₇⋊C₈	—	4	2	no	—
57	C₂²×Q₈	C₂×D₇	1	2	yes	—
58	C₂×Q₁₆	—	2	4	no	—
59	C₂×Q₁₆	—	2	2	no	—
60	C₂×M₁₆	—	7	2	no	—
61	C₂×M₁₆	—	7	1	no	—
62	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
63	#117	—	7	1	no	—
64	C₄⋊C₈	—	14	2	no	—
65	#103	—	14	4	no	—
66	#101	—	14	4	no	—
67	C₂²×C₇	—	2	2	no	—
68	C₂²×C₇	#121	1	1	yes	characteristic
69	C₂²×C₇	#123	1	2	yes	—
70	C₄×C₇	—	4	4	no	—
71	C₄×C₇	#120	1	1	yes	characteristic
72	C₄×C₇	#121	1	2	yes	—
73	C₄×C₇	#124	1	1	yes	characteristic
74	Dic₇	—	4	2	no	—
75	C₂×Q₈	—	2	2	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	2	2	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	—	2	2	no	—
80	C₂×Q₈	C₂²⋊D₇	1	4	yes	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
83	Q₁₆	—	2	8	no	—
84	M₁₆	—	14	2	no	—
85	M₁₆	—	14	2	no	—
86	M₁₆	—	14	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	14	2	no	—
89	C₂×C₈	—	14	1	no	—
90	C₂×C₈	C₂²⋊D₇	1	2	yes	—
91	C₄⋊C₄	—	7	2	no	—
92	C₂²⋊C₄	—	14	1	no	—
93	C₄²	—	14	1	no	—
94	C₂×C₇	—	2	2	no	—
95	C₂×C₇	#394	1	1	yes	characteristic
96	C₂×C₇	#473	1	2	yes	—
97	C₂³	#1159	1	1	yes	characteristic
98	Q₈	—	2	4	no	—
99	Q₈	—	2	4	no	—
100	Q₈	—	4	2	no	—
101	Q₈	—	4	2	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	14	2	no	—
106	C₂×C₄	D₄×D₇	1	1	yes	characteristic
107	C₂×C₄	#1159	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
109	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	28	2	no	—
112	C₈	—	28	1	no	—
113	C₇	#2047	1	1	yes	characteristic, complemented, Sylow
114	C₂²	—	2	2	no	—
115	C₂²	#6498	1	2	yes	—
116	C₂²	#6501	1	1	yes	characteristic
117	C₄	—	4	4	no	—
118	C₄	—	28	2	no	—
119	C₄	#6499	1	1	yes	characteristic
120	C₄	#6502	1	2	yes	—
121	C₄	#6516	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#89964	1	2	yes	—
124	C₂	#89994	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₇×Q₁₆

C₂

yes

maximal, characteristic

C₂×#6410

C₂

yes

maximal

#89997

C₂

yes

maximal

#89874

C₂

yes

maximal, characteristic

#89347

C₂

yes

maximal, characteristic

C₂²×C₇×Q₈

C₂²

yes

—

C₂×C₇×Q₁₆

—

C₂×C₇×Q₁₆

—

C₂²×C₇×C₈

C₂²

yes

characteristic

#6489

C₂²

yes

characteristic

C₂×#1136

C₂²

yes

—

C₂×#1136

C₂²

yes

characteristic

#6410

—

#6409

—

Dic₇⋊C₈

—

#2047

—

maximal, Sylow

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

D₄

yes

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₇×Q₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

D₄

yes

—

C₂²⋊Dic₇

—

C₄⋊Dic₇

D₄

yes

—

C₄×Dic₇

—

#1136

—

#1136

—

#1136

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂²×Q₁₆

D₇

yes

—

—

—

—

C₂³×C₇

#41

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

#41

yes

—

C₂×Dic₇

—

C₇×C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₂²×Q₈

C₂×D₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₇

yes

characteristic

#117

—

C₄⋊C₈

—

#103

—

#101

—

C₂²×C₇

—

C₂²×C₇

#121

yes

characteristic

C₂²×C₇

#123

yes

—

C₄×C₇

—

C₄×C₇

#120

yes

characteristic

C₄×C₇

#121

yes

—

C₄×C₇

#124

yes

characteristic

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

characteristic

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂×C₇

—

C₂×C₇

#394

yes

characteristic

C₂×C₇

#473

yes

—

C₂³

#1159

yes

characteristic

Q₈

—

Q₈

—

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

D₄×D₇

yes

characteristic

107

C₂×C₄

#1159

yes

—

108

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

109

C₂×C₄

C₂×C₂²⋊D₇

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₇

#2047

yes

characteristic, complemented, Sylow

114

C₂²

—

115

C₂²

#6498

yes

—

116

C₂²

#6501

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

#6499

yes

characteristic

120

C₄

#6502

yes

—

121

C₄

#6516

yes

characteristic

122

C₂

—

123

C₂

#89964

yes

—

124

C₂

#89994

yes

characteristic

	subgroup	G/H
center	C₂²	#6501
commutator, cosocle	C₂²×C₄×C₇	C₂³
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₂²×C₇×Q₁₆	C₂
socle	C₂²×C₇	#121
prime radical	C₂³×C₇	#41
hypercenter	C₂²×Q₁₆	D₇
hypocenter	C₇	#2047
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#6501

commutator, cosocle

C₂²×C₄×C₇

C₂³

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₂²×C₇×Q₁₆

C₂

C₂²×C₇

C₂³×C₇

C₂²×Q₁₆

D₇

hypocenter

C₇

#2047

LCS

G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
113. C₇	#2047	semidirect

H ≅ G/N

type

113. C₇

#2047

semidirect

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

subgroup	index
C₂²×C₇×Q₁₆	2
C₂×#6410	2
#89997	2
#89874	2
#89347	2
#2047	7

subgroup

index

C₂²×C₇×Q₁₆

Subgroups — #11729824

Conjugacy classes of subgroups

Complements

Maximal subgroups