Group #11790477 — subgroups

Subgroups — #11790477

Order 1152.

230 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 63 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	#10593775	C₂	1	1	yes	maximal, characteristic
2	#10593546	C₂	1	1	yes	maximal, characteristic
3	#10593544	C₂	1	2	yes	maximal
4	C₂×#63745	C₂	1	1	yes	maximal, characteristic
5	#10590188	C₂	1	2	yes	maximal
6	#64286	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#157	C₂²	1	2	yes	—
8	#64283	C₂²	1	2	yes	—
9	C₂×#3787	C₂²	1	1	yes	characteristic
10	M₁₆×C₃⋊D₃	C₂²	1	1	yes	characteristic
11	#63767	—	2	1	no	—
12	C₈×#157	—	2	1	no	—
13	#63748	C₄	1	4	yes	complemented
14	#63746	—	2	2	no	—
15	#63745	—	2	2	no	—
16	#63745	—	2	1	no	—
17	#63745	—	2	1	no	—
18	C₂²×#157	—	2	2	no	—
19	C₂×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
20	C₂²⋊#157	—	2	1	no	—
21	#3792	C₂×C₄	1	4	yes	—
22	C₂×#622	C₂×C₄	1	2	yes	—
23	C₄⋊#157	—	2	2	no	—
24	C₄⋊#157	—	2	2	no	—
25	C₄⋊#157	D₄	1	2	yes	—
26	C₄×#157	—	2	1	no	—
27	#3788	—	2	2	no	—
28	#3787	—	2	2	no	—
29	#3787	D₄	1	1	yes	characteristic
30	#3787	D₄	1	1	yes	characteristic
31	#3787	Q₈	1	2	yes	—
32	C₃²×M₁₆	Q₈	1	1	yes	characteristic
33	#3748	Q₈	1	1	yes	characteristic
34	#3743	—	2	1	no	—
35	C₈×C₃⋊D₃	—	2	1	no	—
36	#1951	—	9	1	no	maximal, Sylow
37	D₃×M₁₆	—	12	1	no	—
38	C₂²×C₃⋊D₃	#41	1	1	yes	characteristic
39	C₂×#157	—	2	2	no	—
40	C₂×#157	—	4	2	no	—
41	C₂×#157	—	8	2	no	—
42	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
44	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
45	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
46	C₄×C₃⋊D₃	C₂×Q₈	1	1	yes	characteristic
47	C₄×C₃⋊D₃	#41	1	1	yes	characteristic
48	#622	—	2	4	no	—
49	#622	C₄⋊C₄	1	2	yes	—
50	#622	C₄⋊C₄	1	2	yes	—
51	C₃²×C₈	—	2	1	no	—
52	C₃⋊(C₃⋊C₈)	—	2	1	no	—
53	C₂×#108	—	9	1	no	—
54	#428	—	9	2	no	—
55	C₂×#106	—	9	1	no	—
56	#405	—	9	1	no	—
57	#343	—	9	2	no	—
58	C₂×C₄×D₃	—	12	1	no	—
59	C₃×M₁₆	—	4	1	no	—
60	#208	—	12	1	no	—
61	ΓL(1,𝔽₂₅)	—	12	2	no	—
62	D₃×C₈	—	12	2	no	—
63	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
64	C₂×C₃⋊D₃	—	2	2	no	—
65	C₂×C₃⋊D₃	#121	1	1	yes	characteristic
66	C₂×C₃⋊D₃	#125	1	1	yes	characteristic
67	#157	—	4	2	no	—
68	#157	—	8	2	no	—
69	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
70	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
71	C₃⋊Dic₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₃⋊Dic₃	C₄×D₄	1	1	yes	characteristic
73	C₂×M₁₆	—	9	2	no	—
74	C₂×M₁₆	—	9	1	no	—
75	C₂²×C₈	—	9	1	no	—
76	#117	—	9	1	no	—
77	C₂×C₄⋊C₄	—	9	2	no	—
78	#108	—	9	4	no	—
79	#107	—	18	2	no	—
80	#106	—	18	2	no	—
81	#106	—	18	1	no	—
82	#106	—	18	1	no	—
83	#97	—	18	1	no	—
84	C₄×C₈	—	18	1	no	—
85	C₂²×D₃	—	12	1	no	—
86	C₂×C₃×C₄	—	4	1	no	—
87	C₂×Dic₃	—	12	1	no	—
88	C₄×D₃	—	12	1	no	—
89	C₄×D₃	—	12	1	no	—
90	C₄×D₃	—	12	1	no	—
91	C₄×D₃	—	12	1	no	—
92	C₃×C₈	—	4	2	no	—
93	C₃⋊C₈	—	12	2	no	—
94	C₂×C₃²	—	2	1	no	—
95	C₂×C₃²	#389	1	1	yes	characteristic
96	C₃⋊D₃	—	2	1	no	—
97	C₃⋊D₃	#471	1	2	yes	—
98	C₂²×C₄	—	9	1	no	—
99	C₂²×C₄	—	18	2	no	—
100	M₁₆	—	9	4	no	—
101	M₁₆	—	9	1	no	—
102	M₁₆	—	18	1	no	—
103	M₁₆	—	18	2	no	—
104	M₁₆	PSU(3,𝔽₂)	1	1	yes	characteristic
105	C₂×C₈	—	9	1	no	—
106	C₂×C₈	—	9	1	no	—
107	C₂×C₈	—	9	2	no	—
108	C₂×C₈	—	9	2	no	—
109	C₂×C₈	—	18	1	no	—
110	C₂×C₈	—	18	2	no	—
111	C₄⋊C₄	—	9	2	no	—
112	C₄⋊C₄	—	18	2	no	—
113	C₄⋊C₄	—	18	2	no	—
114	C₂²⋊C₄	—	18	1	no	—
115	C₄²	—	18	1	no	—
116	C₂²×C₃	—	4	1	no	—
117	C₂×D₃	—	12	1	no	—
118	C₂×D₃	—	12	1	no	—
119	C₂×D₃	—	24	2	no	—
120	C₃×C₄	—	4	1	no	—
121	C₃×C₄	—	4	1	no	—
122	Dic₃	—	12	1	no	—
123	Dic₃	—	12	1	no	—
124	C₃²	#1951	1	1	yes	characteristic, complemented, Sylow
125	C₂³	—	9	1	no	—
126	C₂×C₄	—	9	1	no	—
127	C₂×C₄	—	9	1	no	—
128	C₂×C₄	—	9	1	no	—
129	C₂×C₄	—	9	1	no	—
130	C₂×C₄	—	9	1	no	—
131	C₂×C₄	—	18	2	no	—
132	C₂×C₄	—	36	2	no	—
133	C₂×C₄	—	72	2	no	—
134	C₂×C₄	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	—	9	2	no	—
137	C₈	—	9	2	no	—
138	C₈	—	18	4	no	—
139	C₈	—	18	1	no	—
140	C₂×C₃	—	4	1	no	—
141	C₂×C₃	—	8	1	no	—
142	D₃	—	12	2	no	—
143	D₃	—	24	1	no	—
144	C₂²	—	9	1	no	—
145	C₂²	—	9	1	no	—
146	C₂²	—	18	2	no	—
147	C₂²	C₄×PSU(3,𝔽₂)	1	1	yes	characteristic
148	C₄	—	9	1	no	—
149	C₄	—	9	1	no	—
150	C₄	—	36	2	no	—
151	C₄	—	72	2	no	—
152	C₄	C₄⋊PSU(3,𝔽₂)	1	1	yes	characteristic
153	C₄	C₂²⋊PSU(3,𝔽₂)	1	1	yes	characteristic
154	C₃	—	4	1	no	—
155	C₂	—	2	1	no	—
156	C₂	—	9	2	no	—
157	C₂	—	18	1	no	—
158	C₂	#10593548	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593775

C₂

yes

maximal, characteristic

#10593546

C₂

yes

maximal, characteristic

#10593544

C₂

yes

maximal

C₂×#63745

C₂

yes

maximal, characteristic

#10590188

C₂

yes

maximal

#64286

C₂²

yes

characteristic

C₂×C₄⋊#157

C₂²

yes

—

#64283

C₂²

yes

—

C₂×#3787

C₂²

yes

characteristic

M₁₆×C₃⋊D₃

C₂²

yes

—

—

C₄

yes

—

—

—

—

C₂²×#157

—

C₂×C₄×C₃⋊D₃

C₂³

yes

characteristic

C₂²⋊#157

—

#3792

C₂×C₄

yes

—

C₂×#622

C₂×C₄

yes

—

C₄⋊#157

—

C₄⋊#157

—

C₄⋊#157

D₄

yes

—

—

—

—

D₄

yes

characteristic

#3787

D₄

yes

characteristic

#3787

Q₈

yes

—

C₃²×M₁₆

Q₈

yes

characteristic

#3748

Q₈

yes

characteristic

#3743

—

C₈×C₃⋊D₃

—

#1951

—

maximal, Sylow

D₃×M₁₆

—

C₂²×C₃⋊D₃

yes

—

—

—

C₂×C₃²×C₄

C₂×Q₈

yes

characteristic

C₂×C₃⋊Dic₃

C₂²×C₄

yes

characteristic

C₄×C₃⋊D₃

C₂×D₄

yes

characteristic

C₄×C₃⋊D₃

C₂×D₄

yes

characteristic

C₄×C₃⋊D₃

C₂×Q₈

yes

C₄×C₃⋊D₃

yes

—

C₄⋊C₄

yes

—

#622

C₄⋊C₄

yes

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

—

—

—

—

—

C₂×C₄×D₃

—

C₃×M₁₆

—

#208

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

C₂²×C₃²

C₄×Q₈

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

yes

C₂×C₃⋊D₃

yes

—

—

C₃²×C₄

C₂²⋊Q₈

yes

characteristic

C₃²×C₄

C₄⋊Q₈

yes

characteristic

C₃⋊Dic₃

C₂×C₄⋊C₄

yes

characteristic

C₃⋊Dic₃

C₄×D₄

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

—

—

—

—

—

—

—

C₄×C₈

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#389

yes

characteristic

C₃⋊D₃

—

C₃⋊D₃

#471

yes

—

C₂²×C₄

—

C₂²×C₄

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

PSU(3,𝔽₂)

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

—

116

C₂²×C₃

—

117

C₂×D₃

—

118

C₂×D₃

—

119

C₂×D₃

—

120

C₃×C₄

—

121

C₃×C₄

—

122

Dic₃

—

123

Dic₃

—

124

C₃²

#1951

yes

characteristic, complemented, Sylow

125

C₂³

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₂×PSU(3,𝔽₂)

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₂×C₃

—

141

C₂×C₃

—

142

D₃

—

143

D₃

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

C₄×PSU(3,𝔽₂)

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

C₄⋊PSU(3,𝔽₂)

yes

characteristic

153

C₄

C₂²⋊PSU(3,𝔽₂)

yes

characteristic

154

C₃

—

155

C₂

—

156

C₂

—

157

C₂

—

158

C₂

#10593548

yes

characteristic

	subgroup	G/H
center	C₂	#10593548
commutator	C₂×C₃⋊Dic₃	C₂²×C₄
Frattini	C₂×C₄	C₂×PSU(3,𝔽₂)
Fitting	C₃²×M₁₆	Q₈
socle	C₂×C₃²	#389
prime radical	C₂²×C₃⋊D₃	#41
cosocle	C₂×C₄×C₃⋊D₃	C₂³
hypercenter	M₁₆	PSU(3,𝔽₂)
hypocenter	C₃²	#1951
LCS	G ▷ C₂×C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

subgroup

G/H

center

C₂

#10593548

commutator

C₂×C₃⋊Dic₃

C₂²×C₄

Frattini

C₂×C₄

C₂×PSU(3,𝔽₂)

Fitting

C₃²×M₁₆

Q₈

C₂×C₃²

C₂²×C₃⋊D₃

C₂×C₄×C₃⋊D₃

C₂³

M₁₆

C₃²

G ▷ C₂×C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

N	H ≅ G/N	type
13. #63748	C₄	semidirect
124. C₃²	#1951	semidirect

H ≅ G/N

type

13. #63748

C₄

semidirect

124. C₃²

#1951

semidirect

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

subgroup	index
#10593775	2
#10593546	2
#10593544	2
C₂×#63745	2
#10590188	2
#1951	9

subgroup

index

Subgroups — #11790477

Conjugacy classes of subgroups

Complements

Maximal subgroups