Group #11772304 — subgroups

Subgroups — #11772304

Order 1152.

186 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 55 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×C₈×#157	C₂	1	1	yes	maximal, characteristic
2	C₂×#63765	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	C₂×#3708	—	2	2	no	—
7	C₈×#157	—	2	2	no	—
8	#63765	—	2	2	no	—
9	#63765	C₄	1	4	yes	complemented
10	C₂²×#157	—	2	1	no	—
11	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
12	C₂×#622	C₈	1	2	yes	—
13	C₄×#157	—	2	1	no	—
14	C₄×#157	C₈	1	2	yes	—
15	#3787	—	2	1	no	—
16	#3787	—	2	1	no	—
17	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
18	C₂×C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
19	C₈×C₃⋊D₃	C₂×C₄	1	2	yes	—
20	C₈×C₃⋊D₃	D₄	1	2	yes	—
21	#3708	—	2	4	no	—
22	#1376	—	9	1	no	maximal, Sylow
23	C₂×D₃×C₈	—	12	1	no	—
24	C₂²×C₃⋊D₃	M₁₆	1	1	yes	characteristic
25	C₂×#157	—	2	4	no	—
26	C₂×#157	—	2	2	no	—
27	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
29	C₄×C₃⋊D₃	C₄²	1	1	yes	characteristic
30	C₄×C₃⋊D₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₄×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
32	C₄×C₃⋊D₃	M₁₆	1	1	yes	characteristic
33	#622	—	2	2	no	—
34	C₃²×C₈	C₄⋊C₄	1	2	yes	—
35	C₃⋊(C₃⋊C₈)	C₂²⋊C₄	1	2	yes	—
36	C₂×M₃₂	—	9	2	no	—
37	C₂×C₄×C₈	—	9	1	no	—
38	C₂×C₄×D₃	—	12	1	no	—
39	C₂×C₃×C₈	—	4	1	no	—
40	C₂×C₃⋊C₈	—	12	1	no	—
41	D₃×C₈	—	12	2	no	—
42	D₃×C₈	—	12	2	no	—
43	C₂²×C₃²	C₄⋊C₈	1	1	yes	characteristic
44	C₂×C₃⋊D₃	—	2	2	no	—
45	C₂×C₃⋊D₃	#97	1	1	yes	characteristic
46	C₂×C₃⋊D₃	C₂²⋊C₈	1	1	yes	characteristic
47	#157	—	2	4	no	—
48	C₃²×C₄	#95	1	1	yes	characteristic
49	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
50	C₃⋊Dic₃	C₄×C₈	1	1	yes	characteristic
51	C₃⋊Dic₃	C₂²⋊C₈	1	1	yes	characteristic
52	C₂²×C₈	—	9	1	no	—
53	C₂²×C₈	—	9	1	no	—
54	C₂×C₄²	—	9	1	no	—
55	M₃₂	—	9	4	no	—
56	M₃₂	—	18	2	no	—
57	C₂×C₁₆	—	18	2	no	—
58	C₄×C₈	—	18	2	no	—
59	C₂²×D₃	—	12	1	no	—
60	C₂×C₃×C₄	—	4	1	no	—
61	C₂×Dic₃	—	12	1	no	—
62	C₄×D₃	—	12	1	no	—
63	C₄×D₃	—	12	1	no	—
64	C₄×D₃	—	12	1	no	—
65	C₄×D₃	—	12	1	no	—
66	C₃×C₈	—	4	2	no	—
67	C₃⋊C₈	—	12	2	no	—
68	C₂×C₃²	#336	1	1	yes	characteristic
69	C₂×C₃²	#364	1	2	yes	—
70	C₃⋊D₃	—	2	1	no	—
71	C₃⋊D₃	—	2	1	no	—
72	C₂²×C₄	—	9	1	no	—
73	C₂²×C₄	—	18	1	no	—
74	C₂×C₈	—	9	1	no	—
75	C₂×C₈	—	9	2	no	—
76	C₂×C₈	—	9	2	no	—
77	C₂×C₈	—	9	2	no	—
78	C₂×C₈	—	18	1	no	—
79	C₂×C₈	—	18	1	no	—
80	C₂×C₈	PSU(3,𝔽₂)	1	1	yes	characteristic
81	C₄²	—	9	2	no	—
82	C₄²	—	18	1	no	—
83	C₁₆	—	18	4	no	—
84	C₂²×C₃	—	4	1	no	—
85	C₂×D₃	—	12	1	no	—
86	C₂×D₃	—	12	1	no	—
87	C₂×D₃	—	12	4	no	—
88	C₃×C₄	—	4	1	no	—
89	C₃×C₄	—	4	1	no	—
90	Dic₃	—	12	1	no	—
91	Dic₃	—	12	1	no	—
92	C₃²	#1376	1	1	yes	characteristic, complemented, Sylow
93	C₂³	—	9	1	no	—
94	C₂×C₄	—	9	1	no	—
95	C₂×C₄	—	9	1	no	—
96	C₂×C₄	—	9	1	no	—
97	C₂×C₄	—	9	1	no	—
98	C₂×C₄	—	9	1	no	—
99	C₂×C₄	—	18	2	no	—
100	C₂×C₄	—	18	4	no	—
101	C₂×C₄	#3777	1	1	yes	characteristic
102	C₈	—	9	2	no	—
103	C₈	—	18	2	no	—
104	C₈	#3777	1	2	yes	—
105	C₂×C₃	—	4	2	no	—
106	C₂×C₃	—	4	1	no	—
107	D₃	—	12	2	no	—
108	D₃	—	12	2	no	—
109	C₂²	—	9	1	no	—
110	C₂²	—	9	1	no	—
111	C₂²	—	18	2	no	—
112	C₂²	#63744	1	1	yes	characteristic
113	C₄	—	9	1	no	—
114	C₄	—	9	1	no	—
115	C₄	—	18	4	no	—
116	C₄	#63744	1	1	yes	characteristic
117	C₄	#63747	1	1	yes	characteristic
118	C₃	—	4	1	no	—
119	C₂	—	18	1	no	—
120	C₂	—	18	1	no	—
121	C₂	#10590083	1	2	yes	—
122	C₂	#10590090	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×#157

C₂

yes

maximal, characteristic

C₂×#63765

C₂

yes

maximal

C₂×C₄×#157

C₄

yes

characteristic

C₂×#3787

C₄

yes

characteristic

C₂×C₈×C₃⋊D₃

C₂²

yes

—

—

—

C₄

yes

complemented

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

Q₈

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

characteristic

C₈×C₃⋊D₃

C₂×C₄

yes

—

C₈×C₃⋊D₃

D₄

yes

—

#3708

—

#1376

—

maximal, Sylow

C₂×D₃×C₈

—

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

C₄⋊C₄

yes

—

C₃⋊(C₃⋊C₈)

C₂²⋊C₄

yes

—

C₂×M₃₂

—

C₂×C₄×C₈

—

C₂×C₄×D₃

—

C₂×C₃×C₈

—

C₂×C₃⋊C₈

—

D₃×C₈

—

D₃×C₈

—

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

—

C₄×C₈

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

#336

yes

characteristic

C₂×C₃²

#364

yes

—

C₃⋊D₃

—

C₃⋊D₃

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

PSU(3,𝔽₂)

yes

characteristic

C₄²

—

C₄²

—

C₁₆

—

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

C₃²

#1376

yes

characteristic, complemented, Sylow

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#3777

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

#3777

yes

—

105

C₂×C₃

—

106

C₂×C₃

—

107

D₃

—

108

D₃

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

#63744

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#63744

yes

characteristic

117

C₄

#63747

yes

characteristic

118

C₃

—

119

C₂

—

120

C₂

—

121

C₂

#10590083

yes

—

122

C₂

#10590090

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#3777
commutator	C₃⋊Dic₃	C₄×C₈
Frattini, hypercenter	C₂×C₈	PSU(3,𝔽₂)
Fitting	C₂×C₃²×C₈	Q₈
socle	C₂²×C₃²	C₄⋊C₈
prime radical	C₂²×C₃⋊D₃	M₁₆
cosocle	C₂×C₈×C₃⋊D₃	C₂²
hypocenter	C₃²	#1376
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center

C₂×C₄

#3777

commutator

C₃⋊Dic₃

C₄×C₈

Frattini, hypercenter

C₂×C₈

PSU(3,𝔽₂)

Fitting

C₂×C₃²×C₈

Q₈

socle

C₂²×C₃²

C₄⋊C₈

prime radical

C₂²×C₃⋊D₃

M₁₆

cosocle

C₂×C₈×C₃⋊D₃

C₂²

hypocenter

C₃²

#1376

LCS

G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
9. #63765	C₄	semidirect
92. C₃²	#1376	semidirect

H ≅ G/N

type

9. #63765

C₄

semidirect

92. C₃²

#1376

semidirect

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

subgroup	index
C₂×C₈×#157	2
C₂×#63765	2
#1376	9

subgroup

index

C₂×C₈×#157

C₂×#63765

#1376

Subgroups — #11772304

Conjugacy classes of subgroups

Complements

Maximal subgroups