Group #11786103 — subgroups

Subgroups — #11786103

Order 1152.

242 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 69 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	#10593805	C₂	1	2	yes	maximal
2	C₂×C₄×#622	C₂	1	1	yes	maximal, characteristic
3	#10590097	C₂	1	4	yes	maximal
4	C₂×#3792	C₂²	1	4	yes	—
5	C₂²×#622	C₂²	1	2	yes	—
6	C₂×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
7	#63779	—	2	2	no	—
8	C₄⋊#622	—	2	2	no	—
9	C₄×#622	C₄	1	2	yes	—
10	C₄×#622	C₄	1	2	yes	—
11	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
12	C₂²×C₃⋊Dic₃	Q₈	1	1	yes	characteristic
13	C₂²×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
14	#3792	—	2	8	no	—
15	C₂×#622	—	2	4	no	—
16	C₂×#622	—	2	4	no	—
17	C₂×#622	C₂×C₄	1	4	yes	—
18	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
19	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₃⋊Dic₃	D₄	1	1	yes	characteristic
21	C₄×C₃⋊Dic₃	D₄	1	1	yes	characteristic
22	#1837	—	9	1	no	maximal, Sylow
23	C₂×C₄×Dic₃	—	12	1	no	—
24	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₃²×C₄	—	2	2	no	—
26	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₃⋊Dic₃	—	2	2	no	—
29	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₃⋊Dic₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₃⋊Dic₃	#41	1	4	yes	—
34	#622	—	2	4	no	—
35	#622	—	4	4	no	—
36	#423	—	9	2	no	—
37	C₂×C₄×C₈	—	9	1	no	—
38	#341	—	9	4	no	—
39	C₂²×C₃×C₄	—	4	1	no	—
40	C₂²×Dic₃	—	12	1	no	—
41	C₂²×Dic₃	—	12	1	no	—
42	C₄×Dic₃	—	12	1	no	—
43	C₄×Dic₃	—	12	1	no	—
44	C₄×Dic₃	—	12	1	no	—
45	C₄×Dic₃	—	12	1	no	—
46	C₂²×C₃²	—	2	1	no	—
47	C₂²×C₃²	—	2	1	no	—
48	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₃²	#125	1	2	yes	—
50	C₃²×C₄	—	2	1	no	—
51	C₃²×C₄	#108	1	2	yes	—
52	C₃⋊Dic₃	—	2	1	no	—
53	C₃⋊Dic₃	#108	1	2	yes	—
54	C₃⋊Dic₃	#117	1	2	yes	—
55	C₃⋊Dic₃	#124	1	2	yes	—
56	C₂×M₁₆	—	9	4	no	—
57	C₂²×C₈	—	9	2	no	—
58	C₂×C₄²	—	9	1	no	—
59	C₄⋊C₈	—	18	2	no	—
60	C₄⋊C₈	—	18	2	no	—
61	C₄×C₈	—	9	2	no	—
62	C₄×C₈	—	9	2	no	—
63	C₂³×C₃	—	4	1	no	—
64	C₂×C₃×C₄	—	4	4	no	—
65	C₂×C₃×C₄	—	4	1	no	—
66	C₂×C₃×C₄	—	4	1	no	—
67	C₂×Dic₃	—	12	1	no	—
68	C₂×Dic₃	—	12	1	no	—
69	C₂×Dic₃	—	12	1	no	—
70	C₂×Dic₃	—	12	1	no	—
71	C₂×Dic₃	—	12	4	no	—
72	C₂×Dic₃	—	12	4	no	—
73	C₂×C₃²	—	2	2	no	—
74	C₂×C₃²	#382	1	1	yes	characteristic
75	C₂×C₃²	C₂×#108	1	1	yes	characteristic
76	C₂×C₃²	C₂×#108	1	1	yes	characteristic
77	C₂²×C₄	—	9	1	no	—
78	C₂²×C₄	—	9	1	no	—
79	C₂²×C₄	PSU(3,𝔽₂)	1	1	yes	characteristic
80	M₁₆	—	18	8	no	—
81	C₂×C₈	—	9	4	no	—
82	C₂×C₈	—	18	4	no	—
83	C₂×C₈	—	18	4	no	—
84	C₄²	—	9	1	no	—
85	C₄²	—	9	1	no	—
86	C₄²	—	9	1	no	—
87	C₄²	—	9	1	no	—
88	C₂²×C₃	—	4	2	no	—
89	C₂²×C₃	—	4	2	no	—
90	C₂²×C₃	—	4	2	no	—
91	C₂²×C₃	—	4	1	no	—
92	C₃×C₄	—	4	2	no	—
93	C₃×C₄	—	4	2	no	—
94	Dic₃	—	12	2	no	—
95	Dic₃	—	12	2	no	—
96	Dic₃	—	12	2	no	—
97	Dic₃	—	12	2	no	—
98	C₃²	#1837	1	1	yes	characteristic, complemented, Sylow
99	C₂³	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	9	4	no	—
102	C₂×C₄	—	9	1	no	—
103	C₂×C₄	—	9	1	no	—
104	C₂×C₄	—	9	1	no	—
105	C₂×C₄	—	9	1	no	—
106	C₂×C₄	—	18	2	no	—
107	C₂×C₄	#3777	1	1	yes	characteristic
108	C₂×C₄	#3777	1	1	yes	characteristic
109	C₈	—	18	4	no	—
110	C₈	—	36	4	no	—
111	C₂×C₃	—	4	4	no	—
112	C₂×C₃	—	4	1	no	—
113	C₂×C₃	—	4	1	no	—
114	C₂×C₃	—	4	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	#64243	1	2	yes	—
118	C₂²	C₂×#3777	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	9	2	no	—
121	C₄	—	9	2	no	—
122	C₄	—	9	2	no	—
123	C₄	—	18	1	no	—
124	C₄	#63748	1	2	yes	—
125	C₃	—	4	1	no	—
126	C₂	—	2	2	no	—
127	C₂	#10593545	1	1	yes	characteristic
128	C₂	#10593546	1	1	yes	characteristic
129	C₂	C₂×#63748	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593805

C₂

yes

maximal

C₂×C₄×#622

C₂

yes

maximal, characteristic

#10590097

C₂

yes

maximal

C₂×#3792

C₂²

yes

—

C₂²×#622

C₂²

yes

—

C₂×C₄×C₃⋊Dic₃

C₂²

yes

—

—

C₄

yes

—

C₄×#622

C₄

yes

—

C₂²×C₃²×C₄

Q₈

yes

characteristic

C₂²×C₃⋊Dic₃

Q₈

yes

characteristic

C₂²×C₃⋊Dic₃

C₂³

yes

—

—

—

C₂×C₄

yes

—

C₄×C₃⋊Dic₃

C₂×C₄

yes

characteristic

C₄×C₃⋊Dic₃

C₂×C₄

yes

characteristic

C₄×C₃⋊Dic₃

D₄

yes

characteristic

C₄×C₃⋊Dic₃

D₄

yes

characteristic

#1837

—

maximal, Sylow

C₂×C₄×Dic₃

—

C₂³×C₃²

C₂×Q₈

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

C₄⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

C₄⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

C₂²×C₄

yes

characteristic

C₂×C₃⋊Dic₃

C₂×D₄

yes

characteristic

C₂×C₃⋊Dic₃

#41

yes

—

#622

—

#622

—

#423

—

C₂×C₄×C₈

—

#341

—

C₂²×C₃×C₄

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃²

#125

yes

—

C₃²×C₄

—

C₃²×C₄

#108

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

#108

yes

—

C₃⋊Dic₃

#117

yes

—

C₃⋊Dic₃

#124

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂³×C₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×C₃²

—

C₂×C₃²

yes

C₂×C₃²

yes

C₂×C₃²

yes

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

PSU(3,𝔽₂)

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

Dic₃

—

Dic₃

—

C₃²

#1837

yes

characteristic, complemented, Sylow

C₂³

C₂×PSU(3,𝔽₂)

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#3777

yes

characteristic

108

C₂×C₄

#3777

yes

characteristic

109

C₈

—

110

C₈

—

111

C₂×C₃

—

112

C₂×C₃

—

113

C₂×C₃

—

114

C₂×C₃

—

115

C₂²

—

116

C₂²

—

117

C₂²

#64243

yes

—

118

C₂²

C₂×#3777

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#63748

yes

—

125

C₃

—

126

C₂

—

127

C₂

#10593545

yes

characteristic

128

C₂

#10593546

yes

characteristic

129

C₂

C₂×#63748

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#3777
commutator	C₂×C₃⋊Dic₃	C₂²×C₄
Frattini	C₂³	C₂×PSU(3,𝔽₂)
Fitting	C₂²×C₃²×C₄	Q₈
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₂×Q₈
cosocle	C₂²×C₃⋊Dic₃	C₂³
hypercenter	C₂²×C₄	PSU(3,𝔽₂)
hypocenter	C₃²	#1837
LCS	G ▷ C₂×C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

subgroup

G/H

center

C₂²

C₂×#3777

commutator

C₂×C₃⋊Dic₃

C₂²×C₄

Frattini

C₂³

C₂×PSU(3,𝔽₂)

Fitting

C₂²×C₃²×C₄

Q₈

socle

C₂²×C₃²

C₂×C₄⋊C₄

prime radical

C₂³×C₃²

C₂×Q₈

cosocle

C₂²×C₃⋊Dic₃

C₂³

C₂²×C₄

C₃²

G ▷ C₂×C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₂²×C₄

N	H ≅ G/N	type
98. C₃²	#1837	semidirect

H ≅ G/N

type

98. C₃²

#1837

semidirect

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

subgroup	index
#10593805	2
C₂×C₄×#622	2
#10590097	2
#1837	9

subgroup

index

#10593805

C₂×C₄×#622

#10590097

#1837

Subgroups — #11786103

Conjugacy classes of subgroups

Complements

Maximal subgroups