Group #11729186 — subgroups

Subgroups — #11729186

Order 896.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 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×#6464	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₇)⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂×#1157	C₄	1	1	yes	characteristic
6	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂×#1136	C₄	1	1	yes	characteristic
8	#6464	—	2	2	no	—
9	#6464	—	2	2	no	—
10	C₂×#1131	C₄	1	2	yes	—
11	C₂×C₈×D₇	C₂²	1	2	yes	—
12	C₇×M₃₂	—	2	2	no	—
13	C₂×C₇×C₁₆	C₂²	1	2	yes	—
14	C₂×C₇⋊C₁₆	—	2	2	no	—
15	C₂×C₇⋊C₁₆	C₂²	1	2	yes	—
16	#2113	—	7	1	no	maximal, Sylow
17	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₂²⋊D₇	C₈	1	2	yes	—
19	#1157	—	2	2	no	—
20	#1157	—	2	2	no	—
21	C₂×D₂₈	C₈	1	1	yes	characteristic
22	C₂×C₄×D₇	C₂×C₄	1	2	yes	—
23	C₂×Dic₁₄	C₈	1	1	yes	characteristic
24	C₂×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₇×C₈	D₄	1	2	yes	—
26	C₂×C₇×C₈	D₄	1	2	yes	—
27	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
28	#1136	—	2	2	no	—
29	C₂×C₇⋊C₈	C₂×C₄	1	2	yes	—
30	#1131	—	2	4	no	—
31	C₈×D₇	—	2	2	no	—
32	C₈×D₇	—	2	2	no	—
33	C₇×C₁₆	—	2	2	no	—
34	C₇⋊C₁₆	—	2	2	no	—
35	C₂×#131	—	7	1	no	—
36	C₂×M₃₂	D₇	1	1	yes	characteristic, complemented
37	C₂²×C₁₆	—	7	1	no	—
38	C₂²⋊C₁₆	—	7	4	no	—
39	C₂³×C₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×D₇	C₂×C₈	1	2	yes	—
41	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₇	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₇	M₁₆	1	2	yes	—
44	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²⋊D₇	—	2	4	no	—
46	C₂×Dic₇	C₂×C₈	1	2	yes	—
47	D₂₈	—	2	2	no	—
48	C₄×D₇	—	2	2	no	—
49	C₄×D₇	—	2	2	no	—
50	Dic₁₄	—	2	2	no	—
51	C₇×C₈	C₂²⋊C₄	1	2	yes	—
52	C₇×C₈	C₂×D₄	1	1	yes	characteristic
53	C₇×C₈	C₂×D₄	1	1	yes	characteristic
54	C₇⋊C₈	—	2	2	no	—
55	C₂×#41	—	7	1	no	—
56	#131	—	14	2	no	—
57	#131	—	14	2	no	—
58	C₂×M₁₆	—	7	1	no	—
59	C₂×M₁₆	—	7	2	no	—
60	C₂²×C₈	—	7	2	no	—
61	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
62	M₃₂	—	2	2	no	—
63	C₂×C₁₆	—	7	2	no	—
64	C₂×C₁₆	—	14	2	no	—
65	C₂×C₁₆	C₂×D₇	1	2	yes	—
66	C₂²×C₇	—	2	2	no	—
67	C₂²×C₇	C₂²⋊C₈	1	2	yes	—
68	C₂²×C₇	C₂²×C₈	1	1	yes	characteristic
69	C₂×D₇	—	2	2	no	—
70	C₂×D₇	—	2	4	no	—
71	C₄×C₇	C₂²⋊C₈	1	2	yes	—
72	C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
74	Dic₇	—	2	2	no	—
75	#41	—	14	2	no	—
76	#41	—	14	2	no	—
77	C₂×Q₈	—	7	1	no	—
78	C₂×D₄	—	7	2	no	—
79	C₂×D₄	—	7	1	no	—
80	C₂²×C₄	—	7	2	no	—
81	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
82	M₁₆	—	14	2	no	—
83	M₁₆	—	14	4	no	—
84	C₂×C₈	—	7	2	no	—
85	C₂×C₈	—	14	2	no	—
86	C₂×C₈	—	14	2	no	—
87	C₂×C₈	C₄×D₇	1	1	yes	characteristic
88	C₂×C₈	D₂₈	1	2	yes	—
89	C₂×C₈	C₂²⋊D₇	1	2	yes	—
90	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
91	C₁₆	—	2	2	no	—
92	C₁₆	—	14	2	no	—
93	C₂×C₇	—	2	2	no	—
94	C₂×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₇	#504	1	2	yes	—
96	D₇	—	4	2	no	—
97	C₂³	—	7	2	no	—
98	C₂³	C₈×D₇	1	1	yes	characteristic
99	Q₈	—	14	2	no	—
100	D₄	—	14	4	no	—
101	D₄	—	14	2	no	—
102	C₂×C₄	—	7	2	no	—
103	C₂×C₄	—	14	2	no	—
104	C₂×C₄	—	14	2	no	—
105	C₂×C₄	C₈×D₇	1	1	yes	characteristic
106	C₂×C₄	#1131	1	2	yes	—
107	C₂×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
109	C₈	—	14	2	no	—
110	C₈	(C₂×D₇)⋊C₄	1	2	yes	—
111	C₈	C₂×D₂₈	1	1	yes	characteristic
112	C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
113	C₇	#2113	1	1	yes	characteristic, complemented, Sylow
114	C₂²	—	2	2	no	—
115	C₂²	—	14	2	no	—
116	C₂²	—	14	4	no	—
117	C₂²	(C₂×D₇)⋊C₈	1	2	yes	—
118	C₂²	C₂×C₈×D₇	1	1	yes	characteristic
119	C₄	—	14	2	no	—
120	C₄	(C₂×D₇)⋊C₈	1	2	yes	—
121	C₄	C₂×#1131	1	1	yes	characteristic
122	C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	28	2	no	—
125	C₂	#89679	1	2	yes	—
126	C₂	C₂×(C₂×D₇)⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#6464

C₂

yes

maximal, characteristic

C₂×C₇×M₃₂

C₂

yes

maximal, characteristic, complemented

C₂²×C₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

(C₂×D₇)⋊C₁₆

C₂

yes

maximal, complemented

C₂×#1157

C₄

yes

characteristic

C₂²×C₇×C₈

C₂²

yes

characteristic

C₂×#1136

C₄

yes

—

—

C₄

yes

—

C₂×C₈×D₇

C₂²

yes

—

C₇×M₃₂

—

C₂×C₇×C₁₆

C₂²

yes

—

C₂×C₇⋊C₁₆

—

C₂×C₇⋊C₁₆

C₂²

yes

—

#2113

—

maximal, Sylow

C₂²×C₄×C₇

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₇

C₈

yes

—

#1157

—

#1157

—

C₂×D₂₈

C₈

yes

characteristic

C₂×C₄×D₇

C₂×C₄

yes

—

C₂×Dic₁₄

C₈

yes

characteristic

C₂×C₇×C₈

C₂×C₄

yes

characteristic

C₂×C₇×C₈

D₄

yes

—

C₂×C₇×C₈

D₄

yes

—

C₂×C₇×C₈

C₂³

yes

characteristic

#1136

—

C₂×C₇⋊C₈

C₂×C₄

yes

—

#1131

—

C₈×D₇

—

C₈×D₇

—

C₇×C₁₆

—

C₇⋊C₁₆

—

C₂×#131

—

C₂×M₃₂

D₇

yes

characteristic, complemented

C₂²×C₁₆

—

C₂²⋊C₁₆

—

C₂³×C₇

C₂×C₈

yes

characteristic

C₂²×D₇

C₂×C₈

yes

—

C₂×C₄×C₇

C₂²⋊C₄

yes

—

C₂×C₄×C₇

C₂×C₈

yes

characteristic

C₂×C₄×C₇

M₁₆

yes

—

C₂×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²⋊D₇

—

C₂×Dic₇

C₂×C₈

yes

—

D₂₈

—

C₄×D₇

—

C₄×D₇

—

Dic₁₄

—

C₇×C₈

C₂²⋊C₄

yes

—

C₇×C₈

C₂×D₄

yes

characteristic

C₇×C₈

C₂×D₄

yes

characteristic

C₇⋊C₈

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇

yes

—

C₂²×C₇

—

C₂²×C₇

C₂²⋊C₈

yes

—

C₂²×C₇

C₂²×C₈

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₄×C₇

C₂²⋊C₈

yes

—

C₄×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇

C₂×M₁₆

yes

characteristic

Dic₇

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₇

yes

characteristic

C₂×C₈

D₂₈

yes

—

C₂×C₈

C₂²⋊D₇

yes

—

C₂×C₈

C₂²×D₇

yes

characteristic

C₁₆

—

C₁₆

—

C₂×C₇

—

C₂×C₇

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₇

#504

yes

—

D₇

—

C₂³

—

C₂³

C₈×D₇

yes

characteristic

Q₈

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₈×D₇

yes

characteristic

106

C₂×C₄

#1131

yes

—

107

C₂×C₄

(C₂×D₇)⋊C₄

yes

—

108

C₂×C₄

C₂×C₄×D₇

yes

characteristic

109

C₈

—

110

C₈

(C₂×D₇)⋊C₄

yes

—

111

C₈

C₂×D₂₈

yes

characteristic

112

C₈

C₂×C₂²⋊D₇

yes

characteristic

113

C₇

#2113

yes

characteristic, complemented, Sylow

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

(C₂×D₇)⋊C₈

yes

—

118

C₂²

C₂×C₈×D₇

yes

characteristic

119

C₄

—

120

C₄

(C₂×D₇)⋊C₈

yes

—

121

C₄

C₂×#1131

yes

characteristic

122

C₄

C₂×(C₂×D₇)⋊C₄

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#89679

yes

—

126

C₂

C₂×(C₂×D₇)⋊C₈

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₈
Fitting	C₂×C₇×M₃₂	C₂
prime radical	C₂×#1157	C₄
cosocle	C₂×C₇×C₈	C₂³
hypercenter	C₂×M₃₂	D₇
hypocenter	C₇	#2113
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₇

commutator, socle

C₂²×C₇

C₂²×C₈

Fitting

C₂×C₇×M₃₂

C₂

prime radical

C₂×#1157

C₄

cosocle

C₂×C₇×C₈

C₂³

hypercenter

C₂×M₃₂

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
2. C₂×C₇×M₃₂	C₂	semidirect
3. C₂²×C₇⋊C₁₆	C₂	semidirect
4. (C₂×D₇)⋊C₁₆	C₂	semidirect
36. C₂×M₃₂	D₇	semidirect
113. C₇	#2113	semidirect

H ≅ G/N

type

2. C₂×C₇×M₃₂

C₂

semidirect

3. C₂²×C₇⋊C₁₆

C₂

semidirect

4. (C₂×D₇)⋊C₁₆

C₂

semidirect

36. C₂×M₃₂

D₇

semidirect

113. C₇

#2113

semidirect

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

subgroup	index
C₂×#6464	2
C₂×C₇×M₃₂	2
C₂²×C₇⋊C₁₆	2
(C₂×D₇)⋊C₁₆	2
#2113	7

subgroup

index

C₂×#6464

C₂×C₇×M₃₂

C₂²×C₇⋊C₁₆

(C₂×D₇)⋊C₁₆

#2113

Subgroups — #11729186

Conjugacy classes of subgroups

Complements

Maximal subgroups