Group #11772459 — subgroups

Subgroups — #11772459

Order 1152.

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

normal subgroups only

This group has 230 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×#63773	C₂	1	1	yes	maximal, characteristic
2	C₂×#63578	C₂	1	2	yes	maximal
3	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
4	C₂×C₃×#208	—	2	2	no	—
5	#63773	—	2	2	no	—
6	C₂×#3708	C₄	1	2	yes	—
7	#63578	—	2	2	no	—
8	#63578	C₄	1	4	yes	—
9	C₂×#822	—	6	2	no	—
10	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
11	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
12	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
13	C₂×C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
14	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	1	yes	characteristic
15	C₃×#208	—	2	4	no	—
16	C₃×#208	—	4	2	no	—
17	C₂×C₃×C₃⋊C₈	—	4	2	no	—
18	#3708	—	2	2	no	—
19	#1385	—	9	1	no	maximal, Sylow
20	C₂×C₃×M₁₆	—	6	2	no	—
21	C₂×#208	—	2	2	no	—
22	C₂²×C₃⋊C₈	—	6	2	no	—
23	#822	—	6	4	no	—
24	#822	—	6	4	no	—
25	C₂³×C₃²	Q₁₆	1	1	yes	characteristic
26	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
27	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
29	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
30	C₃⋊(C₃⋊C₈)	C₄²	1	1	yes	characteristic
31	C₃⋊(C₃⋊C₈)	C₂²⋊C₄	1	1	yes	characteristic
32	C₃⋊(C₃⋊C₈)	C₄⋊C₄	1	2	yes	—
33	C₃×C₃⋊C₈	—	4	4	no	—
34	C₂×M₃₂	—	9	1	no	—
35	C₂×#108	—	9	2	no	—
36	C₂²×C₃×C₄	—	2	2	no	—
37	C₃×M₁₆	—	6	4	no	—
38	C₃×M₁₆	—	12	2	no	—
39	C₂×C₃×C₈	—	12	2	no	—
40	#208	—	2	4	no	—
41	#208	—	4	2	no	—
42	C₂×C₃⋊C₈	—	4	2	no	—
43	C₂×C₃⋊C₈	—	6	2	no	—
44	C₂×C₃⋊C₈	—	6	2	no	—
45	C₂×C₃⋊C₈	—	6	2	no	—
46	C₂×C₃⋊C₈	—	6	2	no	—
47	C₂×C₃⋊C₈	—	6	2	no	—
48	C₂×C₃⋊C₈	—	6	2	no	—
49	C₂²×C₃²	—	4	1	no	—
50	C₂²×C₃²	#103	1	2	yes	—
51	C₂²×C₃²	#107	1	1	yes	characteristic
52	C₃²×C₄	#95	1	1	yes	characteristic
53	C₃²×C₄	#102	1	2	yes	—
54	C₃²×C₄	#106	1	1	yes	characteristic
55	C₂×M₁₆	—	6	2	no	—
56	C₂²×C₈	—	9	1	no	—
57	M₃₂	—	18	2	no	—
58	C₂×C₁₆	—	9	2	no	—
59	#108	—	9	4	no	—
60	#108	—	18	2	no	—
61	C₂³×C₃	—	2	2	no	—
62	C₂×C₃×C₄	—	2	2	no	—
63	C₂×C₃×C₄	—	2	2	no	—
64	C₂×C₃×C₄	—	2	2	no	—
65	C₂×C₃×C₄	—	2	2	no	—
66	C₂×C₃×C₄	—	2	2	no	—
67	C₂×C₃×C₄	—	2	2	no	—
68	C₃×C₈	—	12	4	no	—
69	C₃⋊C₈	—	4	4	no	—
70	C₃⋊C₈	—	6	2	no	—
71	C₃⋊C₈	—	6	2	no	—
72	C₃⋊C₈	—	6	2	no	—
73	C₃⋊C₈	—	6	2	no	—
74	C₂×C₃²	—	2	2	no	—
75	C₂×C₃²	—	2	1	no	—
76	C₂×C₃²	#340	1	1	yes	characteristic
77	C₂²×C₄	GO⁺(4,𝔽₂)	1	1	yes	characteristic
78	M₁₆	—	6	4	no	—
79	M₁₆	—	12	2	no	—
80	C₂×C₈	—	9	2	no	—
81	C₂×C₈	—	9	2	no	—
82	C₂×C₈	—	9	1	no	—
83	C₂×C₈	—	9	1	no	—
84	C₂×C₈	—	12	2	no	—
85	C₁₆	—	18	2	no	—
86	C₂²×C₃	—	2	2	no	—
87	C₂²×C₃	—	2	2	no	—
88	C₂²×C₃	—	2	2	no	—
89	C₂²×C₃	—	4	4	no	—
90	C₃×C₄	—	2	2	no	—
91	C₃×C₄	—	2	2	no	—
92	C₃×C₄	—	2	2	no	—
93	C₃×C₄	—	2	2	no	—
94	C₃²	#1385	1	1	yes	characteristic, complemented, Sylow
95	C₂³	#3776	1	1	yes	characteristic
96	C₂×C₄	D₃²⋊C₄	1	2	yes	—
97	C₂×C₄	#3773	1	1	yes	characteristic
98	C₂×C₄	#3774	1	1	yes	characteristic
99	C₂×C₄	#3775	1	2	yes	—
100	C₈	—	9	1	no	—
101	C₈	—	9	1	no	—
102	C₈	—	9	2	no	—
103	C₈	—	12	4	no	—
104	C₂×C₃	—	2	4	no	—
105	C₂×C₃	—	2	2	no	—
106	C₂×C₃	—	4	2	no	—
107	C₂×C₃	—	4	2	no	—
108	C₂²	—	4	1	no	—
109	C₂²	#63740	1	2	yes	—
110	C₂²	#63743	1	1	yes	characteristic
111	C₄	#63737	1	1	yes	characteristic
112	C₄	#63739	1	2	yes	—
113	C₄	#63742	1	1	yes	characteristic
114	C₃	—	2	2	no	—
115	C₂	—	2	2	no	—
116	C₂	—	2	1	no	—
117	C₂	#10590075	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#63773

C₂

yes

maximal, characteristic

C₂×#63578

C₂

yes

maximal

C₂²×C₃⋊(C₃⋊C₈)

C₂²

yes

characteristic

C₂×C₃×#208

—

#63773

—

C₂×#3708

C₄

yes

—

#63578

—

#63578

C₄

yes

—

C₂×#822

—

C₂²×C₃²×C₄

D₄

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

—

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

—

C₂×C₃⋊(C₃⋊C₈)

Q₈

yes

characteristic

C₃×#208

—

C₃×#208

—

C₂×C₃×C₃⋊C₈

—

#3708

—

#1385

—

maximal, Sylow

C₂×C₃×M₁₆

—

C₂×#208

—

C₂²×C₃⋊C₈

—

#822

—

#822

—

C₂³×C₃²

Q₁₆

yes

characteristic

C₂×C₃²×C₄

C₂²⋊C₄

yes

—

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃²×C₄

D₈

yes

characteristic

C₂×C₃²×C₄

SD₁₆

yes

—

C₃⋊(C₃⋊C₈)

C₄²

yes

characteristic

C₃⋊(C₃⋊C₈)

C₂²⋊C₄

yes

characteristic

C₃⋊(C₃⋊C₈)

C₄⋊C₄

yes

—

C₃×C₃⋊C₈

—

C₂×M₃₂

—

C₂×#108

—

C₂²×C₃×C₄

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

#103

yes

—

C₂²×C₃²

yes

C₃²×C₄

yes

C₃²×C₄

yes

—

C₃²×C₄

#106

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

M₃₂

—

C₂×C₁₆

—

#108

—

#108

—

C₂³×C₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

#340

yes

characteristic

C₂²×C₄

GO⁺(4,𝔽₂)

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₁₆

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃²

#1385

yes

characteristic, complemented, Sylow

C₂³

#3776

yes

characteristic

C₂×C₄

D₃²⋊C₄

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₂×C₃

—

105

C₂×C₃

—

106

C₂×C₃

—

107

C₂×C₃

—

108

C₂²

—

109

C₂²

#63740

yes

—

110

C₂²

#63743

yes

characteristic

111

C₄

#63737

yes

characteristic

112

C₄

#63739

yes

—

113

C₄

#63742

yes

characteristic

114

C₃

—

115

C₂

—

116

C₂

—

117

C₂

#10590075

yes

characteristic

	subgroup	G/H
center	C₄	#63742
commutator	C₃⋊(C₃⋊C₈)	C₄²
Frattini, hypercenter	C₂²×C₄	GO⁺(4,𝔽₂)
Fitting	C₂²×C₃²×C₄	D₄
socle	C₂×C₃²	#340
prime radical	C₂³×C₃²	Q₁₆
cosocle	C₂²×C₃⋊(C₃⋊C₈)	C₂²
hypocenter	C₃²	#1385
LCS	G ▷ C₃⋊(C₃⋊C₈) ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄

subgroup

G/H

center

C₄

#63742

commutator

C₃⋊(C₃⋊C₈)

C₄²

Frattini, hypercenter

C₂²×C₄

GO⁺(4,𝔽₂)

Fitting

C₂²×C₃²×C₄

D₄

socle

C₂×C₃²

#340

prime radical

C₂³×C₃²

Q₁₆

cosocle

C₂²×C₃⋊(C₃⋊C₈)

C₂²

hypocenter

C₃²

#1385

LCS

G ▷ C₃⋊(C₃⋊C₈) ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₄ ◁ C₂²×C₄

N	H ≅ G/N	type
94. C₃²	#1385	semidirect

H ≅ G/N

type

94. C₃²

#1385

semidirect

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

subgroup	index
C₂×#63773	2
C₂×#63578	2
#1385	9

subgroup

index

C₂×#63773

C₂×#63578

#1385

Subgroups — #11772459

Conjugacy classes of subgroups

Complements

Maximal subgroups