Group #11786041 — subgroups

Subgroups — #11786041

Order 1152.

302 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 91 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	#10590093	C₂	1	4	yes	maximal
4	C₂×C₄⋊#157	—	2	2	no	—
5	C₂×C₄⋊#157	—	2	2	no	—
6	C₂×C₄⋊#157	C₂²	1	4	yes	—
7	C₂×#3787	C₂²	1	2	yes	—
8	C₄²×C₃⋊D₃	C₂²	1	1	yes	characteristic
9	C₄×#622	C₄	1	2	yes	complemented
10	C₄×#622	C₄	1	2	yes	complemented
11	C₂²×#157	—	2	4	no	—
12	C₂×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
13	C₂×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
14	C₂×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
15	C₂×#622	C₂×C₄	1	4	yes	—
16	C₄⋊#157	—	2	8	no	—
17	C₄⋊#157	—	4	4	no	—
18	C₄⋊#157	—	4	4	no	—
19	#3787	—	2	4	no	—
20	C₃²×C₄²	Q₈	1	1	yes	characteristic
21	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₃⋊Dic₃	Q₈	1	1	yes	characteristic
24	#1834	—	9	1	no	maximal, Sylow
25	C₄²×D₃	—	12	1	no	—
26	C₂²×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
27	C₂×#157	—	4	4	no	—
28	C₂×#157	—	4	4	no	—
29	C₂×#157	—	4	4	no	—
30	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₃⋊D₃	D₈	1	2	yes	—
37	C₄×C₃⋊D₃	SD₁₆	1	4	yes	—
38	C₄×C₃⋊D₃	Q₁₆	1	2	yes	—
39	C₄×C₃⋊D₃	#41	1	4	yes	—
40	#622	—	2	4	no	—
41	C₂×C₄×C₈	—	9	1	no	—
42	#384	—	9	2	no	—
43	#340	—	9	4	no	—
44	C₂×C₄×D₃	—	12	1	no	—
45	C₂×C₄×D₃	—	12	1	no	—
46	C₂×C₄×D₃	—	12	1	no	—
47	C₃×C₄²	—	4	1	no	—
48	C₄×Dic₃	—	12	1	no	—
49	C₄×Dic₃	—	12	1	no	—
50	C₄×Dic₃	—	12	1	no	—
51	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₃⋊D₃	#124	1	2	yes	—
53	C₂×C₃⋊D₃	C₂×D₈	1	1	yes	characteristic
54	C₂×C₃⋊D₃	C₂×SD₁₆	1	2	yes	—
55	C₂×C₃⋊D₃	C₂×Q₁₆	1	1	yes	characteristic
56	#157	—	8	4	no	—
57	C₃²×C₄	#106	1	2	yes	—
58	C₃²×C₄	#107	1	2	yes	—
59	C₃²×C₄	#125	1	2	yes	—
60	C₃⋊Dic₃	#106	1	2	yes	—
61	C₃⋊Dic₃	#107	1	2	yes	—
62	C₃⋊Dic₃	#117	1	2	yes	—
63	C₂²×C₈	—	9	2	no	—
64	C₂×C₄⋊C₄	—	9	4	no	—
65	C₂×C₄⋊C₄	—	18	2	no	—
66	C₂×C₄⋊C₄	—	18	2	no	—
67	C₂×C₄²	—	9	1	no	—
68	C₄×C₈	—	9	2	no	—
69	C₄×C₈	—	9	2	no	—
70	C₂²×D₃	—	12	1	no	—
71	C₂×C₃×C₄	—	4	1	no	—
72	C₂×C₃×C₄	—	4	1	no	—
73	C₂×C₃×C₄	—	4	1	no	—
74	C₂×Dic₃	—	12	1	no	—
75	C₂×Dic₃	—	12	1	no	—
76	C₂×Dic₃	—	12	1	no	—
77	C₄×D₃	—	12	2	no	—
78	C₄×D₃	—	12	4	no	—
79	C₄×D₃	—	12	2	no	—
80	C₄×D₃	—	12	4	no	—
81	C₂×C₃²	#382	1	1	yes	characteristic
82	C₂×C₃²	C₂×#106	1	1	yes	characteristic
83	C₂×C₃²	C₂×#107	1	1	yes	characteristic
84	C₃⋊D₃	#486	1	2	yes	—
85	C₃⋊D₃	#487	1	2	yes	—
86	C₂²×C₄	—	9	1	no	—
87	C₂²×C₄	—	9	1	no	—
88	C₂²×C₄	—	9	1	no	—
89	C₂²×C₄	—	18	4	no	—
90	C₂×C₈	—	9	4	no	—
91	C₂×C₈	—	18	4	no	—
92	C₄⋊C₄	—	18	8	no	—
93	C₄⋊C₄	—	36	4	no	—
94	C₄⋊C₄	—	36	4	no	—
95	C₄²	—	9	1	no	—
96	C₄²	—	9	1	no	—
97	C₄²	—	9	1	no	—
98	C₄²	PSU(3,𝔽₂)	1	1	yes	characteristic
99	C₂²×C₃	—	4	1	no	—
100	C₂×D₃	—	12	2	no	—
101	C₂×D₃	—	12	1	no	—
102	C₂×D₃	—	12	2	no	—
103	C₂×D₃	—	12	1	no	—
104	C₃×C₄	—	4	2	no	—
105	C₃×C₄	—	4	2	no	—
106	C₃×C₄	—	4	2	no	—
107	Dic₃	—	12	2	no	—
108	Dic₃	—	12	2	no	—
109	Dic₃	—	12	2	no	—
110	C₃²	#1834	1	1	yes	characteristic, complemented, Sylow
111	C₂³	—	9	1	no	—
112	C₂×C₄	—	9	1	no	—
113	C₂×C₄	—	9	1	no	—
114	C₂×C₄	—	9	1	no	—
115	C₂×C₄	—	9	4	no	—
116	C₂×C₄	—	9	2	no	—
117	C₂×C₄	—	9	2	no	—
118	C₂×C₄	—	9	4	no	—
119	C₂×C₄	—	36	4	no	—
120	C₂×C₄	—	36	4	no	—
121	C₂×C₄	—	36	4	no	—
122	C₂×C₄	#3777	1	1	yes	characteristic
123	C₂×C₄	#3777	1	1	yes	characteristic
124	C₂×C₄	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
125	C₈	—	18	4	no	—
126	C₂×C₃	—	4	1	no	—
127	C₂×C₃	—	4	1	no	—
128	C₂×C₃	—	4	1	no	—
129	D₃	—	12	2	no	—
130	D₃	—	12	2	no	—
131	C₂²	—	9	2	no	—
132	C₂²	—	9	1	no	—
133	C₂²	—	9	1	no	—
134	C₂²	—	9	2	no	—
135	C₂²	C₂×#3777	1	1	yes	characteristic
136	C₄	—	9	2	no	—
137	C₄	—	9	2	no	—
138	C₄	—	9	2	no	—
139	C₄	—	72	4	no	—
140	C₄	#63745	1	2	yes	—
141	C₄	#63746	1	2	yes	—
142	C₄	#64243	1	2	yes	—
143	C₃	—	4	1	no	—
144	C₂	—	9	2	no	—
145	C₂	—	9	2	no	—
146	C₂	C₂×#63745	1	1	yes	characteristic
147	C₂	C₂×#63746	1	1	yes	characteristic
148	C₂	#10593545	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593735

C₂

yes

maximal

C₄×#3787

C₂

yes

maximal, characteristic

#10590093

C₂

yes

maximal

C₂×C₄⋊#157

—

C₂×C₄⋊#157

—

C₂×C₄⋊#157

C₂²

yes

—

C₂×#3787

C₂²

yes

—

C₄²×C₃⋊D₃

C₂²

yes

characteristic

C₄×#622

C₄

yes

complemented

C₄×#622

C₄

yes

complemented

C₂²×#157

—

C₂×C₄×C₃⋊D₃

D₄

yes

characteristic

C₂×C₄×C₃⋊D₃

D₄

yes

characteristic

C₂×C₄×C₃⋊D₃

C₂³

yes

characteristic

C₂×#622

C₂×C₄

yes

—

—

—

—

—

C₃²×C₄²

Q₈

yes

characteristic

C₄×C₃⋊Dic₃

C₂×C₄

yes

characteristic

C₄×C₃⋊Dic₃

C₂×C₄

yes

characteristic

C₄×C₃⋊Dic₃

Q₈

yes

characteristic

#1834

—

maximal, Sylow

C₄²×D₃

—

C₂²×C₃⋊D₃

C₂×D₄

yes

—

—

—

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃²×C₄

C₂×Q₈

yes

characteristic

C₂×C₃⋊Dic₃

C₄⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

C₄⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

C₂²×C₄

yes

characteristic

C₄×C₃⋊D₃

D₈

yes

—

C₄×C₃⋊D₃

SD₁₆

yes

—

C₄×C₃⋊D₃

Q₁₆

yes

—

C₄×C₃⋊D₃

#41

yes

—

#622

—

C₂×C₄×C₈

—

#384

—

#340

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₃×C₄²

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

C₂×C₄⋊C₄

yes

characteristic

C₂×C₃⋊D₃

#124

yes

—

C₂×C₃⋊D₃

C₂×D₈

yes

characteristic

C₂×C₃⋊D₃

C₂×SD₁₆

yes

—

C₂×C₃⋊D₃

C₂×Q₁₆

yes

characteristic

#157

—

C₃²×C₄

#106

yes

—

C₃²×C₄

#107

yes

—

C₃²×C₄

#125

yes

—

C₃⋊Dic₃

#106

yes

—

C₃⋊Dic₃

#107

yes

—

C₃⋊Dic₃

#117

yes

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄×C₈

—

C₄×C₈

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₂×C₃²

yes

C₂×C₃²

yes

C₂×C₃²

yes

C₃⋊D₃

yes

—

C₃⋊D₃

#487

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

PSU(3,𝔽₂)

yes

characteristic

C₂²×C₃

—

100

C₂×D₃

—

101

C₂×D₃

—

102

C₂×D₃

—

103

C₂×D₃

—

104

C₃×C₄

—

105

C₃×C₄

—

106

C₃×C₄

—

107

Dic₃

—

108

Dic₃

—

109

Dic₃

—

110

C₃²

#1834

yes

characteristic, complemented, Sylow

111

C₂³

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#3777

yes

characteristic

123

C₂×C₄

#3777

yes

characteristic

124

C₂×C₄

C₂×PSU(3,𝔽₂)

yes

characteristic

125

C₈

—

126

C₂×C₃

—

127

C₂×C₃

—

128

C₂×C₃

—

129

D₃

—

130

D₃

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

C₂×#3777

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#63745

yes

—

141

C₄

#63746

yes

—

142

C₄

#64243

yes

—

143

C₃

—

144

C₂

—

145

C₂

—

146

C₂

C₂×#63745

yes

characteristic

147

C₂

C₂×#63746

yes

characteristic

148

C₂

#10593545

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	#1834
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
9. C₄×#622	C₄	semidirect
10. C₄×#622	C₄	semidirect
110. C₃²	#1834	semidirect

H ≅ G/N

type

9. C₄×#622

C₄

semidirect

10. C₄×#622

C₄

semidirect

110. C₃²

#1834

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
#10590093	2
#1834	9

subgroup

index

Subgroups — #11786041

Conjugacy classes of subgroups

Complements

Maximal subgroups