Group #11772286 — subgroups

Subgroups — #11772286

Order 1152.

230 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 53 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×#63767	C₂	1	1	yes	maximal, characteristic
2	#10590625	C₂	1	2	yes	maximal
3	C₂×C₄×#157	C₄	1	1	yes	characteristic
4	C₂×#3787	C₄	1	1	yes	characteristic
5	C₂×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
6	#63767	—	2	2	no	—
7	C₃×#812	—	2	2	no	—
8	#63544	—	2	2	no	—
9	#63542	C₄	1	4	yes	complemented
10	D₃×M₃₂	—	6	2	no	—
11	C₂²×#157	—	2	1	no	—
12	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
13	C₂×#622	C₈	1	2	yes	—
14	C₄×#157	—	2	1	no	—
15	C₄×#157	C₈	1	2	yes	—
16	#3787	—	2	1	no	—
17	#3787	—	2	1	no	—
18	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
19	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	1	yes	characteristic
20	C₈×C₃⋊D₃	C₂×C₄	1	2	yes	—
21	C₈×C₃⋊D₃	D₄	1	2	yes	—
22	C₃×C₃⋊C₁₆	—	2	4	no	—
23	#1375	—	9	1	no	maximal, Sylow
24	C₂×D₃×C₈	—	6	2	no	—
25	C₃×M₃₂	—	6	2	no	—
26	#812	—	2	2	no	—
27	#798	—	6	4	no	—
28	D₃×C₁₆	—	6	4	no	—
29	C₂²×C₃⋊D₃	M₁₆	1	1	yes	characteristic
30	C₂×#157	—	2	2	no	—
31	C₂×#157	—	4	2	no	—
32	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
34	C₄×C₃⋊D₃	C₄²	1	1	yes	characteristic
35	C₄×C₃⋊D₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₄×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
37	C₄×C₃⋊D₃	M₁₆	1	1	yes	characteristic
38	#622	—	2	2	no	—
39	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
40	C₃⋊(C₃⋊C₈)	C₄⋊C₄	1	2	yes	—
41	C₂×M₃₂	—	9	2	no	—
42	C₂×#97	—	9	1	no	—
43	C₂×C₄×D₃	—	6	2	no	—
44	C₂×C₃×C₈	—	2	2	no	—
45	C₂×C₃⋊C₈	—	6	2	no	—
46	D₃×C₈	—	6	2	no	—
47	D₃×C₈	—	6	2	no	—
48	D₃×C₈	—	6	2	no	—
49	D₃×C₈	—	6	2	no	—
50	C₃×C₁₆	—	6	4	no	—
51	C₃⋊C₁₆	—	2	4	no	—
52	C₂²×C₃²	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₃⋊D₃	—	2	2	no	—
54	C₂×C₃⋊D₃	#97	1	1	yes	characteristic
55	C₂×C₃⋊D₃	C₂²⋊C₈	1	1	yes	characteristic
56	#157	—	4	2	no	—
57	C₃²×C₄	#95	1	1	yes	characteristic
58	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
59	C₃⋊Dic₃	C₄×C₈	1	1	yes	characteristic
60	C₃⋊Dic₃	C₂²⋊C₈	1	1	yes	characteristic
61	C₂²×C₈	—	9	1	no	—
62	C₂²×C₈	—	9	1	no	—
63	C₂×C₄²	—	9	1	no	—
64	M₃₂	—	6	2	no	—
65	M₃₂	—	18	2	no	—
66	C₂×C₁₆	—	9	4	no	—
67	#97	—	18	2	no	—
68	C₂²×D₃	—	6	2	no	—
69	C₂×C₃×C₄	—	2	2	no	—
70	C₂×Dic₃	—	6	2	no	—
71	C₄×D₃	—	6	2	no	—
72	C₄×D₃	—	6	2	no	—
73	C₄×D₃	—	6	2	no	—
74	C₄×D₃	—	6	2	no	—
75	C₃×C₈	—	2	2	no	—
76	C₃×C₈	—	2	2	no	—
77	C₃⋊C₈	—	6	2	no	—
78	C₃⋊C₈	—	6	2	no	—
79	C₂×C₃²	—	2	1	no	—
80	C₂×C₃²	#336	1	1	yes	characteristic
81	C₃⋊D₃	—	2	1	no	—
82	C₃⋊D₃	#347	1	2	yes	—
83	C₂²×C₄	—	9	1	no	—
84	C₂²×C₄	—	18	1	no	—
85	C₂×C₈	—	9	1	no	—
86	C₂×C₈	—	9	2	no	—
87	C₂×C₈	—	9	2	no	—
88	C₂×C₈	—	9	2	no	—
89	C₂×C₈	—	18	1	no	—
90	C₂×C₈	—	18	1	no	—
91	C₂×C₈	GO⁺(4,𝔽₂)	1	1	yes	characteristic
92	C₄²	—	9	2	no	—
93	C₄²	—	18	1	no	—
94	C₁₆	—	6	4	no	—
95	C₂²×C₃	—	2	2	no	—
96	C₂×D₃	—	6	2	no	—
97	C₂×D₃	—	6	2	no	—
98	C₂×D₃	—	12	4	no	—
99	C₃×C₄	—	2	2	no	—
100	C₃×C₄	—	2	2	no	—
101	Dic₃	—	6	2	no	—
102	Dic₃	—	6	2	no	—
103	C₃²	#1375	1	1	yes	characteristic, complemented, Sylow
104	C₂³	—	9	1	no	—
105	C₂×C₄	—	9	1	no	—
106	C₂×C₄	—	9	1	no	—
107	C₂×C₄	—	9	1	no	—
108	C₂×C₄	—	9	1	no	—
109	C₂×C₄	—	9	1	no	—
110	C₂×C₄	—	18	2	no	—
111	C₂×C₄	—	36	2	no	—
112	C₂×C₄	#3773	1	1	yes	characteristic
113	C₈	—	9	2	no	—
114	C₈	—	18	2	no	—
115	C₈	D₃²⋊C₄	1	2	yes	—
116	C₂×C₃	—	2	2	no	—
117	C₂×C₃	—	4	2	no	—
118	D₃	—	6	4	no	—
119	D₃	—	12	2	no	—
120	C₂²	—	9	1	no	—
121	C₂²	—	9	1	no	—
122	C₂²	—	18	2	no	—
123	C₂²	#63732	1	1	yes	characteristic
124	C₄	—	9	1	no	—
125	C₄	—	9	1	no	—
126	C₄	—	36	2	no	—
127	C₄	#63732	1	1	yes	characteristic
128	C₄	#63737	1	1	yes	characteristic
129	C₃	—	2	2	no	—
130	C₂	—	2	1	no	—
131	C₂	—	9	2	no	—
132	C₂	—	18	1	no	—
133	C₂	#10590044	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#63767

C₂

yes

maximal, characteristic

#10590625

C₂

yes

maximal

C₂×C₄×#157

C₄

yes

characteristic

C₂×#3787

C₄

yes

characteristic

C₂×C₈×C₃⋊D₃

C₂²

yes

—

—

—

C₄

yes

complemented

D₃×M₃₂

—

C₂²×#157

—

C₂×C₄×C₃⋊D₃

C₂×C₄

yes

characteristic

C₂×#622

C₈

yes

—

C₄×#157

—

C₄×#157

C₈

yes

—

#3787

—

#3787

—

C₂×C₃²×C₈

D₄

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

Q₈

yes

characteristic

C₈×C₃⋊D₃

C₂×C₄

yes

—

C₈×C₃⋊D₃

D₄

yes

—

C₃×C₃⋊C₁₆

—

#1375

—

maximal, Sylow

C₂×D₃×C₈

—

C₃×M₃₂

—

#812

—

#798

—

D₃×C₁₆

—

C₂²×C₃⋊D₃

M₁₆

yes

characteristic

C₂×#157

—

C₂×#157

—

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

C₂×C₈

yes

characteristic

C₄×C₃⋊D₃

C₄²

yes

characteristic

C₄×C₃⋊D₃

C₂²⋊C₄

yes

characteristic

C₄×C₃⋊D₃

C₂×C₈

yes

characteristic

C₄×C₃⋊D₃

M₁₆

yes

characteristic

#622

—

C₃²×C₈

C₂²⋊C₄

yes

—

C₃⋊(C₃⋊C₈)

C₄⋊C₄

yes

—

C₂×M₃₂

—

C₂×#97

—

C₂×C₄×D₃

—

C₂×C₃×C₈

—

C₂×C₃⋊C₈

—

D₃×C₈

—

D₃×C₈

—

D₃×C₈

—

D₃×C₈

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

C₄⋊C₈

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

#97

yes

characteristic

C₂×C₃⋊D₃

C₂²⋊C₈

yes

characteristic

#157

—

C₃²×C₄

#95

yes

characteristic

C₃²×C₄

C₄⋊C₈

yes

characteristic

C₃⋊Dic₃

C₄×C₈

yes

characteristic

C₃⋊Dic₃

C₂²⋊C₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#97

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#336

yes

characteristic

C₃⋊D₃

—

C₃⋊D₃

#347

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

GO⁺(4,𝔽₂)

yes

characteristic

C₄²

—

C₄²

—

C₁₆

—

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

100

C₃×C₄

—

101

Dic₃

—

102

Dic₃

—

103

C₃²

#1375

yes

characteristic, complemented, Sylow

104

C₂³

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#3773

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

D₃²⋊C₄

yes

—

116

C₂×C₃

—

117

C₂×C₃

—

118

D₃

—

119

D₃

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

#63732

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

#63732

yes

characteristic

128

C₄

#63737

yes

characteristic

129

C₃

—

130

C₂

—

131

C₂

—

132

C₂

—

133

C₂

#10590044

yes

characteristic

	subgroup	G/H
center	C₄	#63737
commutator	C₃⋊Dic₃	C₄×C₈
Frattini, hypercenter	C₂×C₈	GO⁺(4,𝔽₂)
Fitting	C₂×C₃²×C₈	D₄
socle	C₂×C₃²	#336
prime radical	C₂²×C₃⋊D₃	M₁₆
cosocle	C₂×C₈×C₃⋊D₃	C₂²
hypocenter	C₃²	#1375
LCS	G ▷ C₃⋊Dic₃ ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₈

subgroup

G/H

center

C₄

#63737

commutator

C₃⋊Dic₃

C₄×C₈

Frattini, hypercenter

C₂×C₈

GO⁺(4,𝔽₂)

Fitting

C₂×C₃²×C₈

D₄

socle

C₂×C₃²

#336

prime radical

C₂²×C₃⋊D₃

M₁₆

cosocle

C₂×C₈×C₃⋊D₃

C₂²

C₃²

G ▷ C₃⋊Dic₃ ▷ C₃²

C₁ ◁ C₄ ◁ C₂×C₈

N	H ≅ G/N	type
9. #63542	C₄	semidirect
103. C₃²	#1375	semidirect

H ≅ G/N

type

9. #63542

C₄

semidirect

103. C₃²

#1375

semidirect

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

subgroup	index
C₂×#63767	2
#10590625	2
#1375	9

subgroup

index

C₂×#63767

#10590625

#1375

Subgroups — #11772286

Conjugacy classes of subgroups

Complements

Maximal subgroups