Group #11786239 — subgroups

Subgroups — #11786239

Order 1152.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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	#10593735	C₂	1	2	yes	maximal
2	C₄×#3787	C₂	1	1	yes	maximal, characteristic
3	C₂×#63746	C₂	1	4	yes	maximal
4	C₂×C₄⋊#157	—	2	4	no	—
5	C₂×C₄⋊#157	C₂²	1	4	yes	—
6	C₂×#3787	C₂²	1	2	yes	—
7	C₄²×C₃⋊D₃	C₂²	1	1	yes	characteristic
8	C₄×#622	C₄	1	4	yes	complemented
9	#63746	—	2	8	no	—
10	C₂²×#157	—	2	4	no	—
11	C₂×C₄×C₃⋊D₃	D₄	1	2	yes	—
12	C₂×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
13	C₂×#622	C₂×C₄	1	4	yes	—
14	C₄⋊#157	—	2	8	no	—
15	C₄⋊#157	—	4	8	no	—
16	#3787	D₄	1	4	yes	—
17	#3787	Q₈	1	4	yes	—
18	C₃²×C₄²	Q₈	1	1	yes	characteristic
19	C₄×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
20	C₄×C₃⋊Dic₃	Q₈	1	1	yes	characteristic
21	#1845	—	9	1	no	maximal, Sylow
22	C₄²×D₃	—	12	1	no	—
23	C₂²×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
24	C₂×#157	—	4	4	no	—
25	C₂×#157	—	4	8	no	—
26	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
27	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₃⋊Dic₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₃⋊D₃	D₈	1	4	yes	—
31	C₄×C₃⋊D₃	Q₁₆	1	4	yes	—
32	C₄×C₃⋊D₃	C₂×D₄	1	2	yes	—
33	C₄×C₃⋊D₃	C₂×Q₈	1	2	yes	—
34	#622	C₄⋊C₄	1	8	yes	complemented
35	C₂×#107	—	9	4	no	—
36	C₂×C₄×C₈	—	9	1	no	—
37	#384	—	9	2	no	—
38	C₂×C₄×D₃	—	12	2	no	—
39	C₂×C₄×D₃	—	12	1	no	—
40	C₃×C₄²	—	4	1	no	—
41	C₄×Dic₃	—	12	2	no	—
42	C₄×Dic₃	—	12	1	no	—
43	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₃⋊D₃	#127	1	1	yes	characteristic
45	C₂×C₃⋊D₃	C₄⋊Q₈	1	1	yes	characteristic
46	C₂×C₃⋊D₃	C₂×D₈	1	2	yes	—
47	C₂×C₃⋊D₃	C₂×Q₁₆	1	2	yes	—
48	#157	—	8	4	no	—
49	C₃²×C₄	#107	1	4	yes	complemented
50	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
51	C₃⋊Dic₃	#107	1	4	yes	complemented
52	C₃⋊Dic₃	C₂×C₄⋊C₄	1	2	yes	—
53	C₂²×C₈	—	9	2	no	—
54	C₂×C₄⋊C₄	—	9	4	no	—
55	C₂×C₄⋊C₄	—	18	4	no	—
56	C₂×C₄²	—	9	1	no	—
57	#107	—	18	8	no	—
58	C₄×C₈	—	9	4	no	—
59	C₂²×D₃	—	12	1	no	—
60	C₂×C₃×C₄	—	4	1	no	—
61	C₂×C₃×C₄	—	4	2	no	—
62	C₂×Dic₃	—	12	2	no	—
63	C₂×Dic₃	—	12	1	no	—
64	C₄×D₃	—	12	4	no	—
65	C₄×D₃	—	12	4	no	—
66	C₄×D₃	—	12	2	no	—
67	C₄×D₃	—	12	2	no	—
68	C₂×C₃²	#384	1	1	yes	characteristic
69	C₂×C₃²	C₂×#107	1	2	yes	—
70	C₃⋊D₃	#493	1	1	yes	characteristic
71	C₃⋊D₃	#494	1	1	yes	characteristic
72	C₃⋊D₃	#500	1	2	yes	—
73	C₂²×C₄	—	9	1	no	—
74	C₂²×C₄	—	9	2	no	—
75	C₂²×C₄	—	18	4	no	—
76	C₂×C₈	—	9	4	no	—
77	C₂×C₈	—	9	4	no	—
78	C₂×C₈	—	9	4	no	—
79	C₄⋊C₄	—	18	8	no	—
80	C₄⋊C₄	—	36	8	no	—
81	C₄²	—	9	1	no	—
82	C₄²	—	9	2	no	—
83	C₄²	PSU(3,𝔽₂)	1	1	yes	characteristic
84	C₂²×C₃	—	4	1	no	—
85	C₂×D₃	—	12	1	no	—
86	C₂×D₃	—	12	1	no	—
87	C₂×D₃	—	12	2	no	—
88	C₂×D₃	—	12	2	no	—
89	C₃×C₄	—	4	4	no	—
90	C₃×C₄	—	4	2	no	—
91	Dic₃	—	12	4	no	—
92	Dic₃	—	12	2	no	—
93	C₃²	#1845	1	1	yes	characteristic, complemented, Sylow
94	C₂³	—	9	1	no	—
95	C₂×C₄	—	9	1	no	—
96	C₂×C₄	—	9	2	no	—
97	C₂×C₄	—	9	4	no	—
98	C₂×C₄	—	9	4	no	—
99	C₂×C₄	—	9	2	no	—
100	C₂×C₄	—	9	2	no	—
101	C₂×C₄	—	36	4	no	—
102	C₂×C₄	—	36	8	no	—
103	C₂×C₄	#3777	1	2	yes	—
104	C₂×C₄	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
105	C₈	—	9	8	no	—
106	C₂×C₃	—	4	2	no	—
107	C₂×C₃	—	4	1	no	—
108	D₃	—	12	1	no	—
109	D₃	—	12	1	no	—
110	D₃	—	12	2	no	—
111	C₂²	—	9	2	no	—
112	C₂²	—	9	2	no	—
113	C₂²	—	9	1	no	—
114	C₂²	—	9	1	no	—
115	C₂²	C₂×#3777	1	1	yes	characteristic
116	C₄	—	9	4	no	—
117	C₄	—	9	2	no	—
118	C₄	—	72	4	no	—
119	C₄	#63746	1	4	yes	complemented
120	C₄	C₄⋊PSU(3,𝔽₂)	1	2	yes	—
121	C₃	—	4	1	no	—
122	C₂	—	9	1	no	—
123	C₂	—	9	2	no	—
124	C₂	—	9	1	no	—
125	C₂	C₂×#63746	1	2	yes	—
126	C₂	#10593553	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593735

C₂

yes

maximal

C₄×#3787

C₂

yes

maximal, characteristic

C₂×#63746

C₂

yes

maximal

C₂×C₄⋊#157

—

C₂×C₄⋊#157

C₂²

yes

—

C₂×#3787

C₂²

yes

—

C₄²×C₃⋊D₃

C₂²

yes

characteristic

C₄×#622

C₄

yes

complemented

#63746

—

C₂²×#157

—

C₂×C₄×C₃⋊D₃

D₄

yes

—

C₂×C₄×C₃⋊D₃

C₂³

yes

characteristic

C₂×#622

C₂×C₄

yes

—

C₄⋊#157

—

C₄⋊#157

—

#3787

D₄

yes

—

#3787

Q₈

yes

—

C₃²×C₄²

Q₈

yes

characteristic

C₄×C₃⋊Dic₃

C₂×C₄

yes

—

C₄×C₃⋊Dic₃

Q₈

yes

characteristic

#1845

—

maximal, Sylow

C₄²×D₃

—

C₂²×C₃⋊D₃

C₂×D₄

yes

characteristic

C₂×#157

—

C₂×#157

—

C₂×C₃²×C₄

C₄⋊C₄

yes

—

C₂×C₃²×C₄

C₂×Q₈

yes

characteristic

C₂×C₃⋊Dic₃

C₄⋊C₄

yes

—

C₂×C₃⋊Dic₃

C₂²×C₄

yes

characteristic

C₄×C₃⋊D₃

D₈

yes

—

C₄×C₃⋊D₃

Q₁₆

yes

—

C₄×C₃⋊D₃

C₂×D₄

yes

—

C₄×C₃⋊D₃

C₂×Q₈

yes

—

#622

C₄⋊C₄

yes

complemented

C₂×#107

—

C₂×C₄×C₈

—

#384

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₃×C₄²

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

C₂×C₄⋊C₄

yes

characteristic

C₂×C₃⋊D₃

#127

yes

characteristic

C₂×C₃⋊D₃

C₄⋊Q₈

yes

characteristic

C₂×C₃⋊D₃

C₂×D₈

yes

—

C₂×C₃⋊D₃

C₂×Q₁₆

yes

—

#157

—

C₃²×C₄

#107

yes

complemented

C₃²×C₄

C₄⋊Q₈

yes

—

C₃⋊Dic₃

#107

yes

complemented

C₃⋊Dic₃

C₂×C₄⋊C₄

yes

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#107

—

C₄×C₈

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₂×C₃²

#384

yes

characteristic

C₂×C₃²

C₂×#107

yes

—

C₃⋊D₃

yes

C₃⋊D₃

yes

C₃⋊D₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

PSU(3,𝔽₂)

yes

characteristic

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

C₃²

#1845

yes

characteristic, complemented, Sylow

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#3777

yes

—

104

C₂×C₄

C₂×PSU(3,𝔽₂)

yes

characteristic

105

C₈

—

106

C₂×C₃

—

107

C₂×C₃

—

108

D₃

—

109

D₃

—

110

D₃

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×#3777

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#63746

yes

complemented

120

C₄

C₄⋊PSU(3,𝔽₂)

yes

—

121

C₃

—

122

C₂

—

123

C₂

—

124

C₂

—

125

C₂

C₂×#63746

yes

—

126

C₂

#10593553

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#3777
commutator	C₂×C₃⋊Dic₃	C₂²×C₄
Frattini	C₂×C₄	C₂×PSU(3,𝔽₂)
Fitting	C₃²×C₄²	Q₈
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂²×C₃⋊D₃	C₂×D₄
cosocle	C₂×C₄×C₃⋊D₃	C₂³
hypercenter	C₄²	PSU(3,𝔽₂)
hypocenter	C₃²	#1845
LCS	G ▷ C₂×C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

C₂×#3777

commutator

C₂×C₃⋊Dic₃

C₂²×C₄

Frattini

C₂×C₄

C₂×PSU(3,𝔽₂)

Fitting

C₃²×C₄²

Q₈

socle

C₂²×C₃²

C₂×C₄⋊C₄

prime radical

C₂²×C₃⋊D₃

C₂×D₄

cosocle

C₂×C₄×C₃⋊D₃

C₂³

C₄²

C₃²

G ▷ C₂×C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
8. C₄×#622	C₄	semidirect
34. #622	C₄⋊C₄	semidirect
49. C₃²×C₄	#107	semidirect
51. C₃⋊Dic₃	#107	semidirect
93. C₃²	#1845	semidirect
119. C₄	#63746	semidirect

H ≅ G/N

type

8. C₄×#622

C₄

semidirect

34. #622

C₄⋊C₄

semidirect

49. C₃²×C₄

#107

semidirect

51. C₃⋊Dic₃

#107

semidirect

93. C₃²

#1845

semidirect

119. C₄

#63746

semidirect

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

subgroup	index
#10593735	2
C₄×#3787	2
C₂×#63746	2
#1845	9

subgroup

index

Subgroups — #11786239

Conjugacy classes of subgroups

Complements

Maximal subgroups