Group #11729543 — subgroups

Subgroups — #11729543

Order 896.

292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 105 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×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#6410	C₂	1	4	yes	maximal
3	#89696	C₂	1	1	yes	maximal, characteristic
4	C₄×#1136	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇×Q₈	C₂²	1	2	yes	—
6	C₇×C₄⋊Q₈	—	2	1	no	—
7	C₇×C₄⋊Q₈	—	2	1	no	—
8	C₇×C₄⋊Q₈	—	2	2	no	—
9	C₂×C₇×C₄⋊C₄	C₄	1	4	yes	—
10	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₂×#1136	C₂²	1	2	yes	—
12	C₂×#1136	C₂²	1	2	yes	—
13	#6410	—	2	8	no	—
14	C₄⋊(C₇⋊C₈)	—	2	2	no	—
15	#6377	—	2	1	no	—
16	C₄×C₇⋊C₈	—	2	1	no	—
17	#1966	—	7	1	no	maximal, Sylow
18	C₂×C₇×Q₈	—	2	4	no	—
19	C₂×C₇×Q₈	—	2	2	no	—
20	C₂×C₇×Q₈	—	2	2	no	—
21	C₂×C₇×Q₈	—	2	4	no	—
22	C₂²×C₄×C₇	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₇×C₄⋊C₄	—	2	4	no	—
26	C₇×C₄⋊C₄	—	2	4	no	—
27	C₄²×C₇	D₄	1	1	yes	characteristic
28	C₄²×C₇	D₄	1	1	yes	characteristic
29	C₄²×C₇	D₄	1	2	yes	—
30	#1136	—	2	2	no	—
31	#1136	—	2	4	no	—
32	#1136	D₄	1	4	yes	—
33	C₂×C₇⋊C₈	—	2	2	no	—
34	C₂×C₇⋊C₈	—	2	2	no	—
35	C₂×C₄⋊Q₈	D₇	1	1	yes	characteristic
36	#423	—	7	1	no	—
37	C₂×#101	—	7	4	no	—
38	C₄×M₁₆	—	7	1	no	—
39	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
40	C₇×Q₈	—	4	4	no	—
41	C₇×Q₈	—	4	2	no	—
42	C₇×Q₈	—	4	2	no	—
43	C₂×C₄×C₇	—	2	4	no	—
44	C₂×C₄×C₇	—	2	4	no	—
45	C₂×C₄×C₇	—	2	8	no	—
46	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇	#41	1	2	yes	—
52	C₇⋊C₈	—	2	4	no	—
53	C₇⋊C₈	—	4	2	no	—
54	C₂²×Q₈	C₂×D₇	1	2	yes	—
55	C₂×M₁₆	—	7	2	no	—
56	C₂×M₁₆	—	7	2	no	—
57	C₄⋊Q₈	—	2	1	no	—
58	C₄⋊Q₈	—	2	1	no	—
59	C₄⋊Q₈	—	2	2	no	—
60	C₂×C₄⋊C₄	Dic₇	1	4	yes	—
61	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
62	C₄⋊C₈	—	14	2	no	—
63	#101	—	14	8	no	—
64	#97	—	14	1	no	—
65	C₄×C₈	—	14	1	no	—
66	C₂²×C₇	—	2	2	no	—
67	C₂²×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₇	C₄×D₄	1	2	yes	—
69	C₄×C₇	—	2	2	no	—
70	C₄×C₇	—	4	4	no	—
71	C₄×C₇	#101	1	4	yes	complemented
72	C₄×C₇	#121	1	2	yes	—
73	C₄×C₇	#124	1	1	yes	characteristic
74	C₄×C₇	#127	1	1	yes	characteristic
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	—	2	2	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂²×C₄	C₂×Dic₇	1	4	yes	—
80	C₂²×C₄	C₂×Dic₇	1	2	yes	—
81	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
82	M₁₆	—	7	4	no	—
83	M₁₆	—	14	2	no	—
84	M₁₆	—	14	4	no	—
85	C₂×C₈	—	14	2	no	—
86	C₂×C₈	—	14	2	no	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	2	4	no	—
89	C₄²	C₂²⋊D₇	1	1	yes	characteristic
90	C₄²	C₂²⋊D₇	1	1	yes	characteristic
91	C₄²	C₂²⋊D₇	1	2	yes	—
92	C₂×C₇	—	2	2	no	—
93	C₂×C₇	#390	1	1	yes	characteristic
94	C₂×C₇	C₂×#101	1	2	yes	—
95	C₂³	C₂²×Dic₇	1	1	yes	characteristic
96	Q₈	—	4	2	no	—
97	Q₈	—	4	4	no	—
98	Q₈	—	4	2	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	8	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	C₂²⋊Dic₇	1	2	yes	—
103	C₂×C₄	C₂²⋊Dic₇	1	2	yes	—
104	C₂×C₄	D₄×D₇	1	2	yes	—
105	C₂×C₄	#1159	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
107	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
108	C₈	—	14	4	no	—
109	C₈	—	28	2	no	—
110	C₇	#1966	1	1	yes	characteristic, complemented, Sylow
111	C₂²	—	2	2	no	—
112	C₂²	D₄×Dic₇	1	2	yes	—
113	C₂²	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	#6410	1	4	yes	complemented
117	C₄	#6499	1	1	yes	characteristic
118	C₄	#6501	1	2	yes	—
119	C₄	#6503	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#89993	1	1	yes	characteristic
122	C₂	C₂×#6410	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#6410

C₂

yes

maximal

#89696

C₂

yes

maximal, characteristic

C₄×#1136

C₂

yes

maximal, characteristic

C₂²×C₇×Q₈

C₂²

yes

—

C₇×C₄⋊Q₈

—

C₇×C₄⋊Q₈

—

C₇×C₄⋊Q₈

—

C₂×C₇×C₄⋊C₄

C₄

yes

—

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×#1136

C₂²

yes

—

C₂×#1136

C₂²

yes

—

#6410

—

C₄⋊(C₇⋊C₈)

—

#6377

—

C₄×C₇⋊C₈

—

#1966

—

maximal, Sylow

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

C₂×C₄

yes

—

C₂²×C₄×C₇

C₂×C₄

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

D₄

yes

characteristic

C₄²×C₇

D₄

yes

characteristic

C₄²×C₇

D₄

yes

—

#1136

—

#1136

—

#1136

D₄

yes

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₄⋊Q₈

D₇

yes

characteristic

#423

—

C₂×#101

—

C₄×M₁₆

—

C₂³×C₇

C₂²×C₄

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂²⋊C₄

yes

—

C₂×C₄×C₇

C₂²⋊C₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

#41

yes

—

C₇⋊C₈

—

C₇⋊C₈

—

C₂²×Q₈

C₂×D₇

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

Dic₇

yes

—

C₂×C₄²

C₂×D₇

yes

characteristic

C₄⋊C₈

—

#101

—

#97

—

C₄×C₈

—

C₂²×C₇

—

C₂²×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇

C₄×D₄

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#101

yes

complemented

C₄×C₇

#121

yes

—

C₄×C₇

yes

C₄×C₇

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₇

yes

—

C₂²×C₄

C₂×Dic₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₇

yes

characteristic

C₄²

C₂²⋊D₇

yes

characteristic

C₄²

C₂²⋊D₇

yes

—

C₂×C₇

—

C₂×C₇

#390

yes

characteristic

C₂×C₇

C₂×#101

yes

—

C₂³

C₂²×Dic₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂²⋊Dic₇

yes

—

103

C₂×C₄

C₂²⋊Dic₇

yes

—

104

C₂×C₄

D₄×D₇

yes

—

105

C₂×C₄

#1159

yes

—

106

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

107

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

108

C₈

—

109

C₈

—

110

C₇

#1966

yes

characteristic, complemented, Sylow

111

C₂²

—

112

C₂²

D₄×Dic₇

yes

—

113

C₂²

C₂×C₂²⋊Dic₇

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

#6410

yes

complemented

117

C₄

#6499

yes

characteristic

118

C₄

#6501

yes

—

119

C₄

#6503

yes

characteristic

120

C₂

—

121

C₂

#89993

yes

characteristic

122

C₂

C₂×#6410

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₇
commutator, prime radical	C₂³×C₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₂×C₇×C₄⋊Q₈	C₂
socle	C₂²×C₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₇
hypocenter	C₇	#1966
LCS	G ▷ C₂³×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₇

commutator, prime radical

C₂³×C₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₂×C₇×C₄⋊Q₈

C₂

socle

C₂²×C₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₇

C₂³

hypercenter

C₂×C₄⋊Q₈

D₇

hypocenter

C₇

#1966

LCS

G ▷ C₂³×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
71. C₄×C₇	#101	semidirect
110. C₇	#1966	semidirect
116. C₄	#6410	semidirect

H ≅ G/N

type

71. C₄×C₇

#101

semidirect

110. C₇

#1966

semidirect

116. C₄

#6410

semidirect

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

subgroup	index
C₂×C₇×C₄⋊Q₈	2
C₂×#6410	2
#89696	2
C₄×#1136	2
#1966	7

subgroup

index

C₂×C₇×C₄⋊Q₈

Subgroups — #11729543

Conjugacy classes of subgroups

Complements

Maximal subgroups