Group #11797023 — subgroups

Subgroups — #11797023

Order 1152.

290 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 90 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×#63773	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3708	C₂	1	1	yes	maximal, characteristic
3	#10592696	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#3708	C₂	1	4	yes	maximal, complemented
5	C₃²×#117	C₄	1	1	yes	characteristic
6	C₂×#3748	C₄	1	1	yes	characteristic
7	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
8	#63773	—	2	2	no	—
9	C₂×#3708	—	2	2	no	—
10	C₂×#3708	C₂²	1	2	yes	—
11	C₂×#3708	C₂²	1	2	yes	—
12	#63630	C₄	1	2	yes	—
13	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	#4997	—	6	2	no	—
15	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
16	C₃²×C₄⋊C₄	C₈	1	2	yes	—
17	C₃²×C₂²⋊C₄	C₈	1	2	yes	—
18	C₃²×C₄²	C₂×C₄	1	2	yes	—
19	#3748	—	2	2	no	—
20	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
21	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
23	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	—
24	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	1	yes	characteristic
25	#3708	—	2	2	no	—
26	#3708	—	2	2	no	—
27	#2151	—	9	1	no	maximal, Sylow
28	C₃×#117	—	2	2	no	—
29	C₂×#208	—	6	2	no	—
30	C₂²×C₃⋊C₈	—	6	2	no	—
31	#803	—	6	4	no	—
32	C₄×C₃⋊C₈	—	6	4	no	—
33	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
34	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
35	C₂×C₃²×C₄	C₂×C₈	1	4	yes	—
36	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
37	C₂×C₃²×C₄	M₁₆	1	2	yes	—
38	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
39	C₃⋊(C₃⋊C₈)	—	2	2	no	—
40	C₃⋊(C₃⋊C₈)	C₄⋊C₄	1	2	yes	—
41	C₃⋊(C₃⋊C₈)	C₂×D₄	1	1	yes	characteristic
42	C₃⋊(C₃⋊C₈)	C₂×Q₈	1	1	yes	characteristic
43	C₂×M₃₂	—	9	1	no	—
44	C₂²×C₁₆	—	9	1	no	—
45	#405	—	9	1	no	—
46	C₄⋊C₁₆	—	9	4	no	—
47	C₂²×C₃×C₄	—	2	2	no	—
48	C₃×C₄⋊C₄	—	2	4	no	—
49	C₃×C₂²⋊C₄	—	2	4	no	—
50	C₃×C₄²	—	2	4	no	—
51	#208	—	6	8	no	—
52	C₂×C₃⋊C₈	—	6	2	no	—
53	C₂×C₃⋊C₈	—	6	4	no	—
54	C₂×C₃⋊C₈	—	6	2	no	—
55	C₂×C₃⋊C₈	—	6	4	no	—
56	C₂×C₃⋊C₈	—	6	4	no	—
57	C₂²×C₃²	—	2	2	no	—
58	C₂²×C₃²	C₄⋊C₈	1	2	yes	—
59	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
60	C₃²×C₄	—	2	4	no	—
61	C₃²×C₄	C₄⋊C₈	1	2	yes	—
62	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
64	C₂×M₁₆	—	9	1	no	—
65	C₂²×C₈	—	9	1	no	—
66	#117	#157	1	1	yes	characteristic
67	M₃₂	—	18	2	no	—
68	C₂×C₁₆	—	9	2	no	—
69	C₂×C₁₆	—	9	2	no	—
70	C₂×C₁₆	—	18	2	no	—
71	#97	—	9	2	no	—
72	C₄×C₈	—	9	2	no	—
73	C₂³×C₃	—	2	2	no	—
74	C₂×C₃×C₄	—	2	2	no	—
75	C₂×C₃×C₄	—	2	4	no	—
76	C₂×C₃×C₄	—	2	4	no	—
77	C₂×C₃×C₄	—	2	2	no	—
78	C₂×C₃×C₄	—	2	8	no	—
79	C₃⋊C₈	—	6	4	no	—
80	C₃⋊C₈	—	6	2	no	—
81	C₃⋊C₈	—	6	2	no	—
82	C₃⋊C₈	—	6	8	no	—
83	C₂×C₃²	—	2	2	no	—
84	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₂×C₃²	#504	1	2	yes	—
86	C₂²×C₄	C₂×#157	1	1	yes	characteristic
87	M₁₆	—	18	2	no	—
88	C₂×C₈	—	9	1	no	—
89	C₂×C₈	—	9	1	no	—
90	C₂×C₈	—	9	2	no	—
91	C₂×C₈	—	9	2	no	—
92	C₂×C₈	—	9	2	no	—
93	C₄⋊C₄	#622	1	2	yes	—
94	C₂²⋊C₄	#622	1	2	yes	—
95	C₄²	C₂×#157	1	2	yes	—
96	C₁₆	—	18	2	no	—
97	C₁₆	—	18	2	no	—
98	C₂²×C₃	—	2	4	no	—
99	C₂²×C₃	—	2	2	no	—
100	C₂²×C₃	—	4	4	no	—
101	C₃×C₄	—	2	2	no	—
102	C₃×C₄	—	2	2	no	—
103	C₃×C₄	—	2	4	no	—
104	C₃×C₄	—	4	8	no	—
105	C₃²	#2151	1	1	yes	characteristic, complemented, Sylow
106	C₂³	C₂×#622	1	1	yes	characteristic
107	C₂×C₄	#3788	1	2	yes	—
108	C₂×C₄	C₄⋊#157	1	2	yes	—
109	C₂×C₄	C₂×#622	1	4	yes	—
110	C₂×C₄	C₂×#622	1	1	yes	characteristic
111	C₂×C₄	C₂²×#157	1	1	yes	characteristic
112	C₈	—	9	1	no	—
113	C₈	—	9	1	no	—
114	C₈	—	9	2	no	—
115	C₈	—	18	2	no	—
116	C₂×C₃	—	2	2	no	—
117	C₂×C₃	—	2	4	no	—
118	C₂×C₃	—	4	4	no	—
119	C₂²	—	2	2	no	—
120	C₂²	C₄⋊#622	1	2	yes	—
121	C₂²	C₂²×#622	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	C₄⋊#622	1	2	yes	—
124	C₄	C₂×#3788	1	1	yes	characteristic
125	C₄	C₂×C₄⋊#157	1	1	yes	characteristic
126	C₃	—	2	2	no	—
127	C₂	—	2	2	no	—
128	C₂	C₂×C₄⋊#622	1	1	yes	characteristic
129	C₂	#10593838	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#63773

C₂

yes

maximal, characteristic

C₂²×#3708

C₂

yes

maximal, characteristic

#10592696

C₂

yes

maximal, characteristic

C₄⋊#3708

C₂

yes

maximal, complemented

C₃²×#117

C₄

yes

characteristic

C₂×#3748

C₄

yes

characteristic

C₂²×C₃⋊(C₃⋊C₈)

C₂²

yes

—

—

C₂²

yes

—

C₂×#3708

C₂²

yes

—

#63630

C₄

yes

—

C₄×C₃⋊(C₃⋊C₈)

C₂²

yes

—

#4997

—

C₂²×C₃²×C₄

C₂×C₄

yes

characteristic

C₃²×C₄⋊C₄

C₈

yes

—

C₃²×C₂²⋊C₄

C₈

yes

—

C₃²×C₄²

C₂×C₄

yes

—

#3748

—

C₂×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

—

C₂×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

—

C₂×C₃⋊(C₃⋊C₈)

Q₈

yes

—

C₂×C₃⋊(C₃⋊C₈)

C₂³

yes

—

—

—

—

—

C₂²×C₃⋊C₈

—

#803

—

C₄×C₃⋊C₈

—

C₂³×C₃²

C₂×C₈

yes

characteristic

C₂×C₃²×C₄

C₄⋊C₄

yes

—

C₂×C₃²×C₄

C₂×C₈

yes

—

C₂×C₃²×C₄

C₂×C₈

yes

characteristic

C₂×C₃²×C₄

M₁₆

yes

—

C₂×C₃²×C₄

C₂²×C₄

yes

characteristic

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

C₄⋊C₄

yes

—

C₃⋊(C₃⋊C₈)

C₂×D₄

yes

characteristic

C₃⋊(C₃⋊C₈)

C₂×Q₈

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

C₄⋊C₁₆

—

C₂²×C₃×C₄

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

C₄⋊C₈

yes

—

C₂²×C₃²

C₂²×C₈

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₄⋊C₈

yes

—

C₃²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₃²×C₄

C₂×M₁₆

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

#117

#157

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₂³×C₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

C₂×C₄⋊C₈

yes

characteristic

C₂×C₃²

#504

yes

—

C₂²×C₄

C₂×#157

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

#622

yes

—

C₂²⋊C₄

#622

yes

—

C₄²

C₂×#157

yes

—

C₁₆

—

C₁₆

—

C₂²×C₃

—

C₂²×C₃

—

100

C₂²×C₃

—

101

C₃×C₄

—

102

C₃×C₄

—

103

C₃×C₄

—

104

C₃×C₄

—

105

C₃²

#2151

yes

characteristic, complemented, Sylow

106

C₂³

C₂×#622

yes

characteristic

107

C₂×C₄

#3788

yes

—

108

C₂×C₄

C₄⋊#157

yes

—

109

C₂×C₄

C₂×#622

yes

—

110

C₂×C₄

C₂×#622

yes

characteristic

111

C₂×C₄

C₂²×#157

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₂×C₃

—

117

C₂×C₃

—

118

C₂×C₃

—

119

C₂²

—

120

C₂²

C₄⋊#622

yes

—

121

C₂²

C₂²×#622

yes

characteristic

122

C₄

—

123

C₄

C₄⋊#622

yes

—

124

C₄

C₂×#3788

yes

characteristic

125

C₄

C₂×C₄⋊#157

yes

characteristic

126

C₃

—

127

C₂

—

128

C₂

C₂×C₄⋊#622

yes

characteristic

129

C₂

#10593838

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#157
commutator, socle	C₂²×C₃²	C₂²×C₈
Fitting	C₃²×#117	C₄
prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂×C₃⋊(C₃⋊C₈)	C₂³
hypercenter	#117	#157
hypocenter	C₃²	#2151
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ #117

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#157

commutator, socle

C₂²×C₃²

C₂²×C₈

Fitting

C₃²×#117

C₄

prime radical

C₂³×C₃²

C₂×C₈

cosocle

C₂×C₃⋊(C₃⋊C₈)

C₂³

C₃²

G ▷ C₂²×C₃² ▷ C₃²

C₁ ◁ C₂×C₄ ◁ #117

N	H ≅ G/N	type
4. C₄⋊#3708	C₂	semidirect
105. C₃²	#2151	semidirect

H ≅ G/N

type

4. C₄⋊#3708

C₂

semidirect

105. C₃²

#2151

semidirect

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

subgroup	index
C₂×#63773	2
C₂²×#3708	2
#10592696	2
C₄⋊#3708	2
#2151	9

subgroup

index

C₂²×#3708

Subgroups — #11797023

Conjugacy classes of subgroups

Complements

Maximal subgroups