Group #11786267 — subgroups

Subgroups — #11786267

Order 1152.

286 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 87 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×#63768	C₂	1	2	yes	maximal
2	C₂×C₈×#157	C₂	1	1	yes	maximal, characteristic
3	#10593545	C₂	1	2	yes	maximal
4	C₂×#63746	C₂	1	2	yes	maximal
5	C₂×C₄⋊#157	C₂²	1	4	yes	—
6	C₂×C₄×#157	C₂²	1	1	yes	characteristic
7	C₂×#3787	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
9	#63768	—	2	4	no	—
10	C₈×#157	C₄	1	4	yes	complemented
11	#63747	—	2	4	no	—
12	#63746	—	2	4	no	—
13	C₂²×#157	—	2	4	no	—
14	C₂²×#157	D₄	1	2	yes	—
15	C₂×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
16	C₂×#622	Q₈	1	2	yes	—
17	C₄⋊#157	—	2	4	no	—
18	C₄⋊#157	—	2	4	no	—
19	C₄×#157	C₂×C₄	1	2	yes	—
20	C₄×#157	Q₈	1	2	yes	—
21	#3787	C₂×C₄	1	2	yes	—
22	#3787	D₄	1	2	yes	—
23	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
24	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	1	yes	characteristic
25	C₈×C₃⋊D₃	C₂×C₄	1	2	yes	—
26	C₈×C₃⋊D₃	D₄	1	2	yes	—
27	#1846	—	9	1	no	maximal, Sylow
28	C₂×D₃×C₈	—	12	1	no	—
29	C₂²×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
30	C₂×#157	—	2	4	no	—
31	C₂×#157	—	4	4	no	—
32	C₂×#157	—	4	8	no	—
33	C₂×#157	C₄⋊C₄	1	4	yes	—
34	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₃⋊Dic₃	C₂×Q₈	1	1	yes	characteristic
36	C₄×C₃⋊D₃	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
38	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
39	C₄×C₃⋊D₃	C₂×Q₈	1	1	yes	characteristic
40	#622	C₄⋊C₄	1	4	yes	—
41	C₃²×C₈	C₄⋊C₄	1	2	yes	—
42	C₃⋊(C₃⋊C₈)	C₄⋊C₄	1	2	yes	—
43	C₂×#107	—	9	2	no	—
44	C₂×#106	—	9	2	no	—
45	C₂×C₄×C₈	—	9	1	no	—
46	#382	—	9	2	no	—
47	C₂×C₄×D₃	—	12	1	no	—
48	C₂×C₃×C₈	—	4	1	no	—
49	C₂×C₃⋊C₈	—	12	1	no	—
50	D₃×C₈	—	12	2	no	—
51	D₃×C₈	—	12	2	no	—
52	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₃⋊D₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₃⋊D₃	#127	1	1	yes	characteristic
55	C₂×C₃⋊D₃	#135	1	4	yes	—
56	#157	—	2	4	no	—
57	#157	—	8	4	no	—
58	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
60	C₃⋊Dic₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₃⋊Dic₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₈	—	9	1	no	—
63	C₂²×C₈	—	9	1	no	—
64	C₂×C₄⋊C₄	—	9	4	no	—
65	C₂×C₄²	—	9	1	no	—
66	#107	—	18	4	no	—
67	#106	—	18	4	no	—
68	C₄×C₈	—	9	4	no	—
69	#95	—	18	4	no	—
70	C₂²×D₃	—	12	1	no	—
71	C₂×C₃×C₄	—	4	1	no	—
72	C₂×Dic₃	—	12	1	no	—
73	C₄×D₃	—	12	1	no	—
74	C₄×D₃	—	12	1	no	—
75	C₄×D₃	—	12	1	no	—
76	C₄×D₃	—	12	1	no	—
77	C₃×C₈	—	4	2	no	—
78	C₃⋊C₈	—	12	2	no	—
79	C₂×C₃²	#384	1	1	yes	characteristic
80	C₂×C₃²	#499	1	2	yes	—
81	C₃⋊D₃	#427	1	2	yes	—
82	C₃⋊D₃	#495	1	2	yes	—
83	C₂²×C₄	—	9	1	no	—
84	C₂²×C₄	—	9	2	no	—
85	C₂²×C₄	—	18	4	no	—
86	C₂×C₈	—	9	2	no	—
87	C₂×C₈	—	9	2	no	—
88	C₂×C₈	—	9	2	no	—
89	C₂×C₈	—	9	1	no	—
90	C₂×C₈	—	9	2	no	—
91	C₂×C₈	—	9	2	no	—
92	C₂×C₈	PSU(3,𝔽₂)	1	1	yes	characteristic
93	C₄⋊C₄	—	18	4	no	—
94	C₄⋊C₄	—	18	4	no	—
95	C₄²	—	9	2	no	—
96	C₄²	—	9	2	no	—
97	C₂²×C₃	—	4	1	no	—
98	C₂×D₃	—	12	1	no	—
99	C₂×D₃	—	12	1	no	—
100	C₂×D₃	—	12	4	no	—
101	C₃×C₄	—	4	1	no	—
102	C₃×C₄	—	4	1	no	—
103	Dic₃	—	12	1	no	—
104	Dic₃	—	12	1	no	—
105	C₃²	#1846	1	1	yes	characteristic, complemented, Sylow
106	C₂³	—	9	1	no	—
107	C₂×C₄	—	9	1	no	—
108	C₂×C₄	—	9	1	no	—
109	C₂×C₄	—	9	1	no	—
110	C₂×C₄	—	9	1	no	—
111	C₂×C₄	—	9	1	no	—
112	C₂×C₄	—	9	4	no	—
113	C₂×C₄	—	18	4	no	—
114	C₂×C₄	—	36	4	no	—
115	C₂×C₄	—	36	8	no	—
116	C₂×C₄	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
117	C₈	—	9	4	no	—
118	C₈	—	9	2	no	—
119	C₈	#3777	1	2	yes	—
120	C₂×C₃	—	4	1	no	—
121	C₂×C₃	—	4	2	no	—
122	D₃	—	12	2	no	—
123	D₃	—	12	2	no	—
124	C₂²	—	9	1	no	—
125	C₂²	—	9	1	no	—
126	C₂²	—	9	4	no	—
127	C₂²	C₄⋊PSU(3,𝔽₂)	1	1	yes	characteristic
128	C₄	—	9	1	no	—
129	C₄	—	9	1	no	—
130	C₄	—	18	4	no	—
131	C₄	—	72	4	no	—
132	C₄	C₄⋊PSU(3,𝔽₂)	1	1	yes	characteristic
133	C₄	C₂×#3777	1	1	yes	characteristic
134	C₃	—	4	1	no	—
135	C₂	—	9	2	no	—
136	C₂	—	9	2	no	—
137	C₂	#10593533	1	2	yes	—
138	C₂	#10593553	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#63768

C₂

yes

maximal

C₂×C₈×#157

C₂

yes

maximal, characteristic

#10593545

C₂

yes

maximal

C₂×#63746

C₂

yes

maximal

C₂×C₄⋊#157

C₂²

yes

—

C₂×C₄×#157

C₂²

yes

characteristic

C₂×#3787

C₂²

yes

characteristic

C₂×C₈×C₃⋊D₃

C₂²

yes

characteristic

#63768

—

C₈×#157

C₄

yes

complemented

#63747

—

#63746

—

C₂²×#157

—

C₂²×#157

D₄

yes

—

C₂×C₄×C₃⋊D₃

C₂³

yes

characteristic

C₂×#622

Q₈

yes

—

C₄⋊#157

—

C₄⋊#157

—

C₄×#157

C₂×C₄

yes

—

C₄×#157

Q₈

yes

—

#3787

C₂×C₄

yes

—

#3787

D₄

yes

—

C₂×C₃²×C₈

Q₈

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

Q₈

yes

characteristic

C₈×C₃⋊D₃

C₂×C₄

yes

—

C₈×C₃⋊D₃

D₄

yes

—

#1846

—

maximal, Sylow

C₂×D₃×C₈

—

C₂²×C₃⋊D₃

C₂×D₄

yes

—

—

—

C₄⋊C₄

yes

—

C₂×C₃²×C₄

C₂×Q₈

yes

characteristic

C₂×C₃⋊Dic₃

C₂×Q₈

yes

characteristic

C₄×C₃⋊D₃

C₂²×C₄

yes

characteristic

C₄×C₃⋊D₃

C₂×D₄

yes

characteristic

C₄×C₃⋊D₃

C₂×D₄

yes

characteristic

C₄×C₃⋊D₃

C₂×Q₈

yes

characteristic

#622

C₄⋊C₄

yes

—

C₃²×C₈

C₄⋊C₄

yes

—

C₃⋊(C₃⋊C₈)

C₄⋊C₄

yes

—

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

—

#382

—

C₂×C₄×D₃

—

C₂×C₃×C₈

—

C₂×C₃⋊C₈

—

D₃×C₈

—

D₃×C₈

—

C₂²×C₃²

C₄⋊Q₈

yes

characteristic

C₂×C₃⋊D₃

C₂×C₄⋊C₄

yes

C₂×C₃⋊D₃

yes

C₂×C₃⋊D₃

yes

—

#157

—

#157

—

C₃²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₃²×C₄

C₄⋊Q₈

yes

characteristic

C₃⋊Dic₃

C₂×C₄⋊C₄

yes

characteristic

C₃⋊Dic₃

C₄⋊Q₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#107

—

#106

—

C₄×C₈

—

#95

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

#384

yes

characteristic

C₂×C₃²

#499

yes

—

C₃⋊D₃

#427

yes

—

C₃⋊D₃

#495

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

PSU(3,𝔽₂)

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

100

C₂×D₃

—

101

C₃×C₄

—

102

C₃×C₄

—

103

Dic₃

—

104

Dic₃

—

105

C₃²

#1846

yes

characteristic, complemented, Sylow

106

C₂³

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×PSU(3,𝔽₂)

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

#3777

yes

—

120

C₂×C₃

—

121

C₂×C₃

—

122

D₃

—

123

D₃

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₄⋊PSU(3,𝔽₂)

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₄⋊PSU(3,𝔽₂)

yes

characteristic

133

C₄

C₂×#3777

yes

characteristic

134

C₃

—

135

C₂

—

136

C₂

—

137

C₂

#10593533

yes

—

138

C₂

#10593553

yes

characteristic

	subgroup	G/H
center	C₂²	C₄⋊PSU(3,𝔽₂)
commutator	C₄×C₃⋊D₃	C₂²×C₄
Frattini	C₂×C₄	C₂×PSU(3,𝔽₂)
Fitting	C₂×C₃²×C₈	Q₈
socle	C₂²×C₃²	C₄⋊Q₈
prime radical	C₂²×C₃⋊D₃	C₂×D₄
cosocle	C₂×C₄×C₃⋊D₃	C₂³
hypercenter	C₂×C₈	PSU(3,𝔽₂)
hypocenter	C₃²	#1846
LCS	G ▷ C₄×C₃⋊D₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center

C₂²

C₄⋊PSU(3,𝔽₂)

commutator

C₄×C₃⋊D₃

C₂²×C₄

Frattini

C₂×C₄

C₂×PSU(3,𝔽₂)

Fitting

C₂×C₃²×C₈

Q₈

socle

C₂²×C₃²

C₄⋊Q₈

prime radical

C₂²×C₃⋊D₃

C₂×D₄

cosocle

C₂×C₄×C₃⋊D₃

C₂³

C₂×C₈

C₃²

G ▷ C₄×C₃⋊D₃ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
10. C₈×#157	C₄	semidirect
105. C₃²	#1846	semidirect

H ≅ G/N

type

10. C₈×#157

C₄

semidirect

105. C₃²

#1846

semidirect

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

subgroup	index
C₂×#63768	2
C₂×C₈×#157	2
#10593545	2
C₂×#63746	2
#1846	9

subgroup

index

C₂×C₈×#157

Subgroups — #11786267

Conjugacy classes of subgroups

Complements

Maximal subgroups