Group #11783508 — subgroups

Subgroups — #11783508

Order 1152.

720 conjugacy classes of subgroups; only the 321 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 321 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	C₃²×#376	C₂	1	1	yes	maximal, characteristic
2	C₂×#63630	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10589390	C₂	1	4	yes	maximal
5	#69435	—	3	4	no	maximal
6	C₂×C₃²×C₄²	C₂²	1	2	yes	—
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
9	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
10	C₃²×#95	C₄	1	4	yes	—
11	#63630	C₄	1	4	yes	complemented
12	C₄×C₃⋊(C₃⋊C₈)	C₄	1	4	yes	complemented
13	C₃×#376	D₃	1	4	yes	—
14	C₂×#803	—	3	4	no	—
15	C₂×C₄×C₃⋊C₈	—	3	4	no	—
16	#4524	—	3	16	no	—
17	C₂²×C₃²×C₄	C₂×C₄	1	4	yes	—
18	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
19	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
20	C₃²×C₄²	—	2	4	no	—
21	C₃²×C₄²	C₂×C₄	1	2	yes	—
22	C₃²×C₄²	C₂×C₄	1	2	yes	—
23	C₂×C₃⋊(C₃⋊C₈)	—	2	4	no	—
24	C₂×C₃⋊(C₃⋊C₈)	—	2	4	no	—
25	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
26	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
27	#1746	—	9	1	no	Sylow
28	C₂×C₃×C₄²	C₂×D₃	1	8	yes	—
29	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
30	C₂²×C₃⋊C₈	—	3	8	no	—
31	C₂²×C₃⋊C₈	—	3	8	no	—
32	C₃×#95	Dic₃	1	16	yes	—
33	#803	—	3	16	no	—
34	C₄×C₃⋊C₈	—	3	16	no	—
35	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₃²×C₄	—	2	4	no	—
37	C₂×C₃²×C₄	—	2	4	no	—
38	C₂×C₃²×C₄	—	2	8	no	—
39	C₂×C₃²×C₄	C₄²	1	4	yes	—
40	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₃²×C₄	#41	1	4	yes	—
43	C₃⋊(C₃⋊C₈)	—	2	4	no	—
44	C₃⋊(C₃⋊C₈)	—	2	4	no	—
45	C₂×#97	—	9	1	no	—
46	C₂×C₄×C₈	—	9	1	no	—
47	#376	C₃⋊D₃	1	1	yes	characteristic
48	#336	—	9	4	no	—
49	C₂²×C₃×C₄	C₂×Dic₃	1	16	yes	—
50	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
51	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
52	C₃×C₄²	—	2	16	no	—
53	C₃×C₄²	C₄×D₃	1	8	yes	—
54	C₃×C₄²	C₄×D₃	1	8	yes	—
55	C₂×C₃⋊C₈	—	3	16	no	—
56	C₂×C₃⋊C₈	—	3	16	no	—
57	C₂×C₃⋊C₈	—	6	16	no	—
58	C₂×C₃⋊C₈	—	6	16	no	—
59	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
60	C₂²×C₃²	#117	1	2	yes	—
61	C₂²×C₃²	#131	1	4	yes	—
62	C₃²×C₄	—	2	4	no	—
63	C₃²×C₄	—	4	4	no	—
64	C₃²×C₄	#117	1	2	yes	—
65	C₃²×C₄	#117	1	2	yes	—
66	C₂²×C₈	—	9	2	no	—
67	C₂²×C₈	—	9	2	no	—
68	C₂×C₄²	C₂×C₃⋊D₃	1	2	yes	—
69	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
70	#97	—	9	4	no	—
71	C₄×C₈	—	9	4	no	—
72	#95	C₃⋊Dic₃	1	4	yes	—
73	C₂³×C₃	C₂²×Dic₃	1	4	yes	—
74	C₂×C₃×C₄	—	2	16	no	—
75	C₂×C₃×C₄	—	2	16	no	—
76	C₂×C₃×C₄	—	2	32	no	—
77	C₂×C₃×C₄	C₄×Dic₃	1	16	yes	—
78	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
79	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
80	C₂×C₃×C₄	#235	1	16	yes	—
81	C₃⋊C₈	—	6	16	no	—
82	C₃⋊C₈	—	6	16	no	—
83	C₂×C₃²	#376	1	1	yes	characteristic
84	C₂×C₃²	#405	1	2	yes	—
85	C₂×C₃²	#433	1	4	yes	—
86	C₂²×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
87	C₂²×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
88	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
89	C₂×C₈	—	9	4	no	—
90	C₂×C₈	—	9	4	no	—
91	C₂×C₈	—	18	4	no	—
92	C₂×C₈	—	18	4	no	—
93	C₄²	—	2	4	no	—
94	C₄²	C₄×C₃⋊D₃	1	2	yes	—
95	C₄²	C₄×C₃⋊D₃	1	2	yes	—
96	C₂²×C₃	C₂×C₄×Dic₃	1	4	yes	—
97	C₂²×C₃	#926	1	8	yes	—
98	C₂²×C₃	#948	1	16	yes	—
99	C₃×C₄	—	2	16	no	—
100	C₃×C₄	—	4	16	no	—
101	C₃×C₄	#872	1	8	yes	—
102	C₃×C₄	#872	1	8	yes	—
103	C₃²	#1746	1	1	yes	characteristic, complemented, Sylow
104	C₂³	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	8	no	—
108	C₂×C₄	C₄×C₃⋊Dic₃	1	4	yes	—
109	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
111	C₂×C₄	#3828	1	4	yes	—
112	C₈	—	18	4	no	—
113	C₈	—	18	4	no	—
114	C₂×C₃	#4932	1	4	yes	—
115	C₂×C₃	#4997	1	8	yes	—
116	C₂×C₃	#5000	1	16	yes	—
117	C₂²	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
118	C₂²	#64135	1	2	yes	—
119	C₂²	#64157	1	4	yes	—
120	C₄	—	2	4	no	—
121	C₄	—	4	4	no	—
122	C₄	#64081	1	2	yes	—
123	C₄	#64081	1	2	yes	—
124	C₃	#69435	1	4	yes	complemented
125	C₂	#10592631	1	1	yes	characteristic
126	C₂	#10592696	1	2	yes	—
127	C₂	#10592699	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#376

C₂

yes

maximal, characteristic

C₂×#63630

C₂

yes

maximal, characteristic

C₂×C₄×C₃⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic

#10589390

C₂

yes

maximal

#69435

—

maximal

C₂×C₃²×C₄²

C₂²

yes

—

C₂×C₃²×C₄²

C₂²

yes

characteristic

C₂²×C₃⋊(C₃⋊C₈)

C₂²

yes

—

C₂²×C₃⋊(C₃⋊C₈)

C₂²

yes

—

C₃²×#95

C₄

yes

—

#63630

C₄

yes

complemented

C₄×C₃⋊(C₃⋊C₈)

C₄

yes

complemented

C₃×#376

D₃

yes

—

C₂×#803

—

C₂×C₄×C₃⋊C₈

—

#4524

—

C₂²×C₃²×C₄

C₂×C₄

yes

—

C₂²×C₃²×C₄

C₂×C₄

yes

—

C₂²×C₃²×C₄

C₂³

yes

characteristic

C₃²×C₄²

—

C₃²×C₄²

C₂×C₄

yes

—

C₃²×C₄²

C₂×C₄

yes

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

—

C₂×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

—

#1746

—

Sylow

C₂×C₃×C₄²

C₂×D₃

yes

—

C₂×C₃×C₄²

C₂×D₃

yes

—

C₂²×C₃⋊C₈

—

C₂²×C₃⋊C₈

—

C₃×#95

Dic₃

yes

—

#803

—

C₄×C₃⋊C₈

—

C₂³×C₃²

C₂²×C₄

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₄²

yes

—

C₂×C₃²×C₄

C₂²×C₄

yes

characteristic

C₂×C₃²×C₄

C₂²×C₄

yes

characteristic

C₂×C₃²×C₄

#41

yes

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

C₂×#97

—

C₂×C₄×C₈

—

#376

C₃⋊D₃

yes

characteristic

#336

—

C₂²×C₃×C₄

C₂×Dic₃

yes

—

C₂²×C₃×C₄

C₂×Dic₃

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

—

C₃×C₄²

—

C₃×C₄²

C₄×D₃

yes

—

C₃×C₄²

C₄×D₃

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

C₂×C₄²

yes

characteristic

C₂²×C₃²

#117

yes

—

C₂²×C₃²

#131

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#117

yes

—

C₃²×C₄

#117

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×C₃⋊D₃

yes

—

C₂×C₄²

C₂×C₃⋊D₃

yes

characteristic

#97

—

C₄×C₈

—

#95

C₃⋊Dic₃

yes

—

C₂³×C₃

C₂²×Dic₃

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄×Dic₃

yes

—

C₂×C₃×C₄

C₂×C₄×D₃

yes

—

C₂×C₃×C₄

C₂×C₄×D₃

yes

—

C₂×C₃×C₄

#235

yes

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

#376

yes

characteristic

C₂×C₃²

#405

yes

—

C₂×C₃²

#433

yes

—

C₂²×C₄

C₂×C₃⋊Dic₃

yes

—

C₂²×C₄

C₂×C₃⋊Dic₃

yes

—

C₂²×C₄

C₂²×C₃⋊D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

C₄×C₃⋊D₃

yes

—

C₄²

C₄×C₃⋊D₃

yes

—

C₂²×C₃

C₂×C₄×Dic₃

yes

—

C₂²×C₃

#926

yes

—

C₂²×C₃

#948

yes

—

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

#872

yes

—

102

C₃×C₄

#872

yes

—

103

C₃²

#1746

yes

characteristic, complemented, Sylow

104

C₂³

C₂²×C₃⋊Dic₃

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₄×C₃⋊Dic₃

yes

—

109

C₂×C₄

C₂×C₄×C₃⋊D₃

yes

characteristic

110

C₂×C₄

C₂×C₄×C₃⋊D₃

yes

characteristic

111

C₂×C₄

#3828

yes

—

112

C₈

—

113

C₈

—

114

C₂×C₃

#4932

yes

—

115

C₂×C₃

#4997

yes

—

116

C₂×C₃

#5000

yes

—

117

C₂²

C₂×C₄×C₃⋊Dic₃

yes

characteristic

118

C₂²

#64135

yes

—

119

C₂²

#64157

yes

—

120

C₄

—

121

C₄

—

122

C₄

#64081

yes

—

123

C₄

#64081

yes

—

124

C₃

#69435

yes

complemented

125

C₂

#10592631

yes

characteristic

126

C₂

#10592696

yes

—

127

C₂

#10592699

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×C₃⋊D₃
commutator	C₂²×C₃²	C₂×C₄²
Fitting	C₃²×#376	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	#376	C₃⋊D₃
hypocenter	C₃²	#1746
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ #376

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×C₃⋊D₃

commutator

C₂²×C₃²

C₂×C₄²

Fitting

C₃²×#376

C₂

socle, prime radical

C₂³×C₃²

C₂²×C₄

cosocle

C₂²×C₃²×C₄

C₂³

C₃⋊D₃

C₃²

G ▷ C₂²×C₃² ▷ C₃²

C₁ ◁ C₂²×C₄ ◁ #376

N	H ≅ G/N	type
11. #63630	C₄	semidirect
12. C₄×C₃⋊(C₃⋊C₈)	C₄	semidirect
103. C₃²	#1746	semidirect
124. C₃	#69435	semidirect

H ≅ G/N

type

11. #63630

C₄

semidirect

12. C₄×C₃⋊(C₃⋊C₈)

C₄

semidirect

103. C₃²

#1746

semidirect

124. C₃

#69435

semidirect

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

subgroup	index
C₃²×#376	2
C₂×#63630	2
C₂×C₄×C₃⋊(C₃⋊C₈)	2
#10589390	2
#69435	3

subgroup

index

C₃²×#376

C₂×#63630

C₂×C₄×C₃⋊(C₃⋊C₈)

#10589390

#69435

Subgroups — #11783508

Conjugacy classes of subgroups

Complements

Maximal subgroups