Group #11790512 — subgroups

Subgroups — #11790512

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×#63746	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	#63746	—	2	1	no	—
16	#63746	—	2	1	no	—
17	#63745	—	2	2	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	1	yes	characteristic
26	C₄⋊#157	D₄	1	1	yes	characteristic
27	C₄×#157	—	2	1	no	—
28	#3788	—	2	2	no	—
29	#3787	—	2	2	no	—
30	#3787	D₄	1	1	yes	characteristic
31	#3787	D₄	1	1	yes	characteristic
32	#3787	Q₈	1	2	yes	—
33	C₃²×M₁₆	Q₈	1	1	yes	characteristic
34	#3748	Q₈	1	1	yes	characteristic
35	#3743	—	2	1	no	—
36	C₈×C₃⋊D₃	—	2	1	no	—
37	#1952	—	9	1	no	maximal, Sylow
38	D₃×M₁₆	—	12	1	no	—
39	C₂²×C₃⋊D₃	#41	1	1	yes	characteristic
40	C₂×#157	—	2	1	no	—
41	C₂×#157	—	2	1	no	—
42	C₂×#157	—	4	2	no	—
43	C₂×#157	—	8	2	no	—
44	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
46	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
47	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
48	C₄×C₃⋊D₃	C₂×Q₈	1	1	yes	characteristic
49	C₄×C₃⋊D₃	#41	1	1	yes	characteristic
50	#622	—	2	4	no	—
51	#622	C₄⋊C₄	1	2	yes	—
52	#622	C₄⋊C₄	1	2	yes	—
53	C₃²×C₈	—	2	1	no	—
54	C₃⋊(C₃⋊C₈)	—	2	1	no	—
55	C₂×#108	—	9	1	no	—
56	#428	—	9	2	no	—
57	C₂×#107	—	9	1	no	—
58	#405	—	9	1	no	—
59	#343	—	9	2	no	—
60	C₂×C₄×D₃	—	12	1	no	—
61	C₃×M₁₆	—	4	1	no	—
62	#208	—	12	1	no	—
63	ΓL(1,𝔽₂₅)	—	12	2	no	—
64	D₃×C₈	—	12	2	no	—
65	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
66	C₂×C₃⋊D₃	—	2	1	no	—
67	C₂×C₃⋊D₃	—	2	1	no	—
68	C₂×C₃⋊D₃	#121	1	1	yes	characteristic
69	C₂×C₃⋊D₃	#125	1	1	yes	characteristic
70	#157	—	4	1	no	—
71	#157	—	4	1	no	—
72	#157	—	8	2	no	—
73	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
74	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
75	C₃⋊Dic₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₃⋊Dic₃	C₄×D₄	1	1	yes	characteristic
77	C₂×M₁₆	—	9	2	no	—
78	C₂×M₁₆	—	9	1	no	—
79	C₂²×C₈	—	9	1	no	—
80	#117	—	9	1	no	—
81	C₂×C₄⋊C₄	—	9	2	no	—
82	#108	—	9	4	no	—
83	#107	—	18	2	no	—
84	#107	—	18	1	no	—
85	#107	—	18	1	no	—
86	#106	—	18	2	no	—
87	#97	—	18	1	no	—
88	C₄×C₈	—	18	1	no	—
89	C₂²×D₃	—	12	1	no	—
90	C₂×C₃×C₄	—	4	1	no	—
91	C₂×Dic₃	—	12	1	no	—
92	C₄×D₃	—	12	1	no	—
93	C₄×D₃	—	12	1	no	—
94	C₄×D₃	—	12	1	no	—
95	C₄×D₃	—	12	1	no	—
96	C₃×C₈	—	4	2	no	—
97	C₃⋊C₈	—	12	2	no	—
98	C₂×C₃²	—	2	1	no	—
99	C₂×C₃²	#389	1	1	yes	characteristic
100	C₃⋊D₃	—	2	1	no	—
101	C₃⋊D₃	#472	1	1	yes	characteristic
102	C₃⋊D₃	#473	1	1	yes	characteristic
103	C₂²×C₄	—	9	1	no	—
104	C₂²×C₄	—	18	2	no	—
105	M₁₆	—	9	4	no	—
106	M₁₆	—	9	1	no	—
107	M₁₆	—	18	1	no	—
108	M₁₆	—	18	2	no	—
109	M₁₆	PSU(3,𝔽₂)	1	1	yes	characteristic
110	C₂×C₈	—	9	1	no	—
111	C₂×C₈	—	9	1	no	—
112	C₂×C₈	—	9	2	no	—
113	C₂×C₈	—	9	2	no	—
114	C₂×C₈	—	18	1	no	—
115	C₂×C₈	—	18	2	no	—
116	C₄⋊C₄	—	9	1	no	—
117	C₄⋊C₄	—	9	1	no	—
118	C₄⋊C₄	—	18	2	no	—
119	C₄⋊C₄	—	18	2	no	—
120	C₂²⋊C₄	—	18	1	no	—
121	C₄²	—	18	1	no	—
122	C₂²×C₃	—	4	1	no	—
123	C₂×D₃	—	12	1	no	—
124	C₂×D₃	—	12	1	no	—
125	C₂×D₃	—	24	1	no	—
126	C₂×D₃	—	24	1	no	—
127	C₃×C₄	—	4	1	no	—
128	C₃×C₄	—	4	1	no	—
129	Dic₃	—	12	1	no	—
130	Dic₃	—	12	1	no	—
131	C₃²	#1952	1	1	yes	characteristic, complemented, Sylow
132	C₂³	—	9	1	no	—
133	C₂×C₄	—	9	1	no	—
134	C₂×C₄	—	9	1	no	—
135	C₂×C₄	—	9	1	no	—
136	C₂×C₄	—	9	1	no	—
137	C₂×C₄	—	9	1	no	—
138	C₂×C₄	—	18	1	no	—
139	C₂×C₄	—	18	1	no	—
140	C₂×C₄	—	36	2	no	—
141	C₂×C₄	—	72	2	no	—
142	C₂×C₄	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	9	2	no	—
145	C₈	—	9	2	no	—
146	C₈	—	18	4	no	—
147	C₈	—	18	1	no	—
148	C₂×C₃	—	4	1	no	—
149	C₂×C₃	—	8	1	no	—
150	D₃	—	12	1	no	—
151	D₃	—	12	1	no	—
152	D₃	—	24	1	no	—
153	C₂²	—	9	1	no	—
154	C₂²	—	9	1	no	—
155	C₂²	—	18	1	no	—
156	C₂²	—	18	1	no	—
157	C₂²	C₄×PSU(3,𝔽₂)	1	1	yes	characteristic
158	C₄	—	9	1	no	—
159	C₄	—	9	1	no	—
160	C₄	—	36	1	no	—
161	C₄	—	36	1	no	—
162	C₄	—	72	2	no	—
163	C₄	C₄⋊PSU(3,𝔽₂)	1	1	yes	characteristic
164	C₄	C₂²⋊PSU(3,𝔽₂)	1	1	yes	characteristic
165	C₃	—	4	1	no	—
166	C₂	—	2	1	no	—
167	C₂	—	9	1	no	—
168	C₂	—	9	1	no	—
169	C₂	—	18	1	no	—
170	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×#63746

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

characteristic

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

—

#1952

—

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

—

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

100

C₃⋊D₃

—

101

C₃⋊D₃

#472

yes

characteristic

102

C₃⋊D₃

#473

yes

characteristic

103

C₂²×C₄

—

104

C₂²×C₄

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

PSU(3,𝔽₂)

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₂²×C₃

—

123

C₂×D₃

—

124

C₂×D₃

—

125

C₂×D₃

—

126

C₂×D₃

—

127

C₃×C₄

—

128

C₃×C₄

—

129

Dic₃

—

130

Dic₃

—

131

C₃²

#1952

yes

characteristic, complemented, Sylow

132

C₂³

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

C₂×PSU(3,𝔽₂)

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₂×C₃

—

149

C₂×C₃

—

150

D₃

—

151

D₃

—

152

D₃

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₄×PSU(3,𝔽₂)

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

C₄⋊PSU(3,𝔽₂)

yes

characteristic

164

C₄

C₂²⋊PSU(3,𝔽₂)

yes

characteristic

165

C₃

—

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

170

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	#1952
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
131. C₃²	#1952	semidirect

H ≅ G/N

type

13. #63748

C₄

semidirect

131. C₃²

#1952

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×#63746	2
#10590188	2
#1952	9

subgroup

index

Subgroups — #11790512

Conjugacy classes of subgroups

Complements

Maximal subgroups