Group #11786108 — subgroups

Subgroups — #11786108

Order 1152.

540 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 217 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×#423	C₂	1	1	yes	maximal, characteristic
2	#10592623	C₂	1	1	yes	maximal, characteristic
3	C₂×#63630	C₂	1	1	yes	maximal, characteristic
4	#10589424	C₂	1	4	yes	maximal
5	#70517	—	3	4	no	maximal
6	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂×#3748	C₂²	1	2	yes	—
9	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
10	C₃²×C₄⋊C₈	—	2	2	no	—
11	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
12	#63630	C₄	1	4	yes	—
13	C₃×#423	D₃	1	4	yes	—
14	#4924	—	3	4	no	—
15	C₂×#803	—	3	4	no	—
16	#4558	—	3	16	no	—
17	C₂²×C₃²×C₄	Q₈	1	2	yes	—
18	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
19	C₃²×M₁₆	—	2	4	no	—
20	C₂×C₃²×C₈	—	2	2	no	—
21	C₃²×C₄²	C₂×C₄	1	2	yes	—
22	C₃²×C₄²	D₄	1	2	yes	—
23	#3748	—	2	4	no	—
24	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
25	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
26	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
27	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
28	#1838	—	9	1	no	Sylow
29	C₂×C₃×M₁₆	C₂×D₃	1	8	yes	—
30	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
31	C₂×#208	—	3	8	no	—
32	C₂²×C₃⋊C₈	—	3	8	no	—
33	C₃×C₄⋊C₈	—	2	8	no	—
34	C₄⋊(C₃⋊C₈)	—	6	8	no	—
35	#803	—	3	16	no	—
36	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₃²×C₄	—	4	2	no	—
38	C₂×C₃²×C₄	C₄⋊C₄	1	4	yes	—
39	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
41	C₂×C₃²×C₄	#41	1	2	yes	—
42	C₂×C₃²×C₄	#41	1	2	yes	—
43	C₃²×C₈	—	4	2	no	—
44	C₃⋊(C₃⋊C₈)	—	2	4	no	—
45	C₃⋊(C₃⋊C₈)	—	4	2	no	—
46	#423	—	9	1	no	—
47	#423	C₃⋊D₃	1	1	yes	characteristic
48	C₂×#97	—	9	1	no	—
49	#341	—	9	4	no	—
50	C₂²×C₃×C₄	Dic₆	1	8	yes	—
51	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
52	C₃×M₁₆	—	2	16	no	—
53	C₂×C₃×C₈	—	2	8	no	—
54	C₃×C₄²	C₄×D₃	1	8	yes	—
55	C₃×C₄²	#67	1	8	yes	—
56	#208	—	6	16	no	—
57	C₂×C₃⋊C₈	—	3	16	no	—
58	C₂×C₃⋊C₈	—	6	8	no	—
59	C₂×C₃⋊C₈	—	6	8	no	—
60	C₂×C₃⋊C₈	—	6	8	no	—
61	C₂²×C₃²	—	2	2	no	—
62	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₃²	#125	1	2	yes	—
64	C₃²×C₄	—	2	4	no	—
65	C₃²×C₄	#117	1	2	yes	—
66	C₃²×C₄	#124	1	1	yes	characteristic
67	C₃²×C₄	#124	1	1	yes	characteristic
68	C₂×M₁₆	—	9	2	no	—
69	C₂×M₁₆	C₂×C₃⋊D₃	1	2	yes	—
70	C₂²×C₈	—	9	2	no	—
71	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
72	C₄⋊C₈	—	2	2	no	—
73	C₄⋊C₈	—	18	2	no	—
74	#97	—	9	4	no	—
75	C₂³×C₃	C₂×Dic₆	1	4	yes	—
76	C₂×C₃×C₄	—	4	8	no	—
77	C₂×C₃×C₄	Dic₃⋊C₄	1	16	yes	—
78	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
79	C₂×C₃×C₄	#237	1	8	yes	—
80	C₂×C₃×C₄	#239	1	8	yes	—
81	C₂×C₃×C₄	C₂×#67	1	4	yes	—
82	C₃×C₈	—	4	8	no	—
83	C₃⋊C₈	—	6	16	no	—
84	C₃⋊C₈	—	12	8	no	—
85	C₂×C₃²	—	2	2	no	—
86	C₂×C₃²	#382	1	1	yes	characteristic
87	C₂×C₃²	#430	1	2	yes	—
88	C₂²×C₄	#634	1	2	yes	—
89	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
90	M₁₆	—	2	4	no	—
91	M₁₆	—	18	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	9	4	no	—
94	C₂×C₈	—	18	2	no	—
95	C₂×C₈	—	18	2	no	—
96	C₂×C₈	—	18	2	no	—
97	C₄²	C₄×C₃⋊D₃	1	2	yes	—
98	C₄²	#638	1	2	yes	—
99	C₂²×C₃	—	2	8	no	—
100	C₂²×C₃	#890	1	8	yes	—
101	C₂²×C₃	C₂×Dic₃⋊C₄	1	4	yes	—
102	C₃×C₄	—	2	16	no	—
103	C₃×C₄	#892	1	8	yes	—
104	C₃×C₄	#936	1	4	yes	—
105	C₃×C₄	#947	1	4	yes	—
106	C₃²	#1838	1	1	yes	characteristic, complemented, Sylow
107	C₂³	C₂×#634	1	1	yes	characteristic
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	#3750	1	4	yes	—
110	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
111	C₂×C₄	#3830	1	2	yes	—
112	C₂×C₄	#3832	1	2	yes	—
113	C₂×C₄	C₂×#638	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₈	—	18	4	no	—
116	C₈	—	36	2	no	—
117	C₂×C₃	—	2	8	no	—
118	C₂×C₃	#4977	1	4	yes	—
119	C₂×C₃	#5124	1	8	yes	—
120	C₂²	—	2	2	no	—
121	C₂²	#64099	1	2	yes	—
122	C₂²	C₂×#3750	1	1	yes	characteristic
123	C₄	—	2	4	no	—
124	C₄	#64101	1	2	yes	—
125	C₄	#64145	1	1	yes	characteristic
126	C₄	#64156	1	1	yes	characteristic
127	C₃	#70517	1	4	yes	complemented
128	C₂	—	2	2	no	—
129	C₂	#10592676	1	1	yes	characteristic
130	C₂	#10592823	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#423

C₂

yes

maximal, characteristic

#10592623

C₂

yes

maximal, characteristic

C₂×#63630

C₂

yes

maximal, characteristic

#10589424

C₂

yes

maximal

#70517

—

maximal

C₂×C₃²×M₁₆

C₂²

yes

—

C₂×C₃²×C₄²

C₂²

yes

characteristic

C₂×#3748

C₂²

yes

—

C₂²×C₃⋊(C₃⋊C₈)

C₂²

yes

—

C₃²×C₄⋊C₈

—

C₄⋊(C₃⋊(C₃⋊C₈))

—

#63630

C₄

yes

—

C₃×#423

D₃

yes

—

#4924

—

C₂×#803

—

#4558

—

C₂²×C₃²×C₄

Q₈

yes

—

C₂²×C₃²×C₄

C₂³

yes

characteristic

C₃²×M₁₆

—

C₂×C₃²×C₈

—

C₃²×C₄²

C₂×C₄

yes

—

C₃²×C₄²

D₄

yes

—

#3748

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

—

#1838

—

Sylow

C₂×C₃×M₁₆

C₂×D₃

yes

—

C₂×C₃×C₄²

C₂×D₃

yes

—

C₂×#208

—

C₂²×C₃⋊C₈

—

C₃×C₄⋊C₈

—

C₄⋊(C₃⋊C₈)

—

#803

—

C₂³×C₃²

C₂×Q₈

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₄⋊C₄

yes

—

C₂×C₃²×C₄

C₂²×C₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

#41

yes

—

C₂×C₃²×C₄

#41

yes

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

#423

—

#423

C₃⋊D₃

yes

characteristic

C₂×#97

—

#341

—

C₂²×C₃×C₄

Dic₆

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₃×C₄²

C₄×D₃

yes

—

C₃×C₄²

#67

yes

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃²

#125

yes

—

C₃²×C₄

—

C₃²×C₄

#117

yes

—

C₃²×C₄

yes

C₃²×C₄

yes

C₂×M₁₆

—

C₂×M₁₆

C₂×C₃⋊D₃

yes

—

C₂²×C₈

—

C₂×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₂³×C₃

C₂×Dic₆

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

Dic₃⋊C₄

yes

—

C₂×C₃×C₄

C₂×C₄×D₃

yes

—

C₂×C₃×C₄

#237

yes

—

C₂×C₃×C₄

#239

yes

—

C₂×C₃×C₄

C₂×#67

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#382

yes

characteristic

C₂×C₃²

#430

yes

—

C₂²×C₄

#634

yes

—

C₂²×C₄

C₂²×C₃⋊D₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₄×C₃⋊D₃

yes

—

C₄²

#638

yes

—

C₂²×C₃

—

100

C₂²×C₃

#890

yes

—

101

C₂²×C₃

C₂×Dic₃⋊C₄

yes

—

102

C₃×C₄

—

103

C₃×C₄

#892

yes

—

104

C₃×C₄

#936

yes

—

105

C₃×C₄

#947

yes

—

106

C₃²

#1838

yes

characteristic, complemented, Sylow

107

C₂³

C₂×#634

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

#3750

yes

—

110

C₂×C₄

C₂×C₄×C₃⋊D₃

yes

characteristic

111

C₂×C₄

#3830

yes

—

112

C₂×C₄

#3832

yes

—

113

C₂×C₄

C₂×#638

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

117

C₂×C₃

—

118

C₂×C₃

#4977

yes

—

119

C₂×C₃

#5124

yes

—

120

C₂²

—

121

C₂²

#64099

yes

—

122

C₂²

C₂×#3750

yes

characteristic

123

C₄

—

124

C₄

#64101

yes

—

125

C₄

#64145

yes

characteristic

126

C₄

#64156

yes

characteristic

127

C₃

#70517

yes

complemented

128

C₂

—

129

C₂

#10592676

yes

characteristic

130

C₂

#10592823

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#638
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#423	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₂×Q₈
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	#423	C₃⋊D₃
hypocenter	C₃²	#1838
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×#638

commutator

C₂×C₃²×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×C₃⋊D₃

Fitting

C₃²×#423

C₂

socle

C₂²×C₃²

C₂×C₄⋊C₄

prime radical

C₂³×C₃²

C₂×Q₈

cosocle

C₂²×C₃²×C₄

C₂³

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
106. C₃²	#1838	semidirect
127. C₃	#70517	semidirect

H ≅ G/N

type

106. C₃²

#1838

semidirect

127. C₃

#70517

semidirect

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

subgroup	index
C₃²×#423	2
#10592623	2
C₂×#63630	2
#10589424	2
#70517	3

subgroup

index

C₃²×#423

Subgroups — #11786108

Conjugacy classes of subgroups

Complements

Maximal subgroups