Group #11790185 — subgroups

Subgroups — #11790185

Order 1152.

278 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 75 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×#63769	C₂	1	1	yes	maximal, characteristic
2	C₂×#63768	C₂	1	1	yes	maximal, characteristic
3	C₂×#63767	C₂	1	1	yes	maximal, characteristic
4	#10593554	C₂	1	2	yes	maximal
5	#10590093	C₂	1	2	yes	maximal
6	C₂×C₄⋊#157	C₂²	1	2	yes	—
7	C₂×C₄⋊#157	C₂²	1	2	yes	—
8	C₂×C₄×#157	C₂²	1	1	yes	characteristic
9	C₂×#3787	C₂²	1	1	yes	characteristic
10	C₂×#3777	—	2	2	no	—
11	C₂×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
12	#63769	C₄	1	4	yes	complemented
13	#63768	—	2	2	no	—
14	#63767	—	2	2	no	—
15	#63747	—	2	2	no	—
16	C₂²×#157	—	2	2	no	—
17	C₂²×#157	—	2	2	no	—
18	C₂²×#157	D₄	1	2	yes	—
19	C₂×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
20	C₂×#622	Q₈	1	2	yes	—
21	C₄⋊#157	—	2	2	no	—
22	C₄⋊#157	—	2	2	no	—
23	C₄⋊#157	—	2	2	no	—
24	C₄⋊#157	C₂×C₄	1	4	yes	—
25	C₄×#157	—	2	1	no	—
26	C₄×#157	—	2	1	no	—
27	#3787	—	2	1	no	—
28	#3787	—	2	1	no	—
29	#3777	—	4	4	no	—
30	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
31	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	1	yes	characteristic
32	C₈×C₃⋊D₃	C₂×C₄	1	2	yes	—
33	C₈×C₃⋊D₃	D₄	1	2	yes	—
34	#1943	—	9	1	no	maximal, Sylow
35	C₂×D₃×C₈	—	12	1	no	—
36	C₂²×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
37	C₂×#157	—	2	4	no	—
38	C₂×#157	—	2	2	no	—
39	C₂×#157	—	2	2	no	—
40	C₂×#157	—	2	2	no	—
41	C₂×#157	—	4	2	no	—
42	C₂×#157	—	4	2	no	—
43	C₂×#157	—	4	2	no	—
44	C₂×#157	—	4	2	no	—
45	C₂×#157	—	4	2	no	—
46	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₃⋊Dic₃	C₂×Q₈	1	1	yes	characteristic
48	C₄×C₃⋊D₃	C₂²×C₄	1	1	yes	characteristic
49	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
50	C₄×C₃⋊D₃	#41	1	1	yes	characteristic
51	C₄×C₃⋊D₃	#41	1	1	yes	characteristic
52	#622	—	2	2	no	—
53	C₃²×C₈	C₄⋊C₄	1	2	yes	complemented
54	C₃⋊(C₃⋊C₈)	C₄⋊C₄	1	2	yes	complemented
55	C₂×#107	—	9	1	no	—
56	C₂×#106	—	9	1	no	—
57	C₂×#97	—	9	1	no	—
58	#389	—	9	2	no	—
59	#340	—	9	2	no	—
60	C₂×C₄×D₃	—	12	1	no	—
61	C₂×C₃×C₈	—	4	1	no	—
62	C₂×C₃⋊C₈	—	12	1	no	—
63	D₃×C₈	—	12	2	no	—
64	D₃×C₈	—	12	2	no	—
65	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₃⋊D₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₃⋊D₃	#121	1	1	yes	characteristic
68	C₂×C₃⋊D₃	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×C₃⋊D₃	#137	1	2	yes	—
70	#157	—	4	4	no	—
71	#157	—	4	2	no	—
72	#157	—	8	2	no	—
73	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₃²×C₄	#125	1	1	yes	characteristic
75	C₃⋊Dic₃	C₄×Q₈	1	1	yes	characteristic
76	C₃⋊Dic₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂²×C₈	—	9	1	no	—
78	C₂²×C₈	—	9	1	no	—
79	C₂×C₄⋊C₄	—	9	2	no	—
80	C₂×C₄⋊C₄	—	9	2	no	—
81	C₂×C₄⋊C₄	—	18	2	no	—
82	C₂×C₄²	—	9	1	no	—
83	#107	—	9	4	no	—
84	#106	—	18	2	no	—
85	#97	—	18	2	no	—
86	#95	—	18	2	no	—
87	C₂²×D₃	—	12	1	no	—
88	C₂×C₃×C₄	—	4	1	no	—
89	C₂×Dic₃	—	12	1	no	—
90	C₄×D₃	—	12	1	no	—
91	C₄×D₃	—	12	1	no	—
92	C₄×D₃	—	12	1	no	—
93	C₄×D₃	—	12	1	no	—
94	C₃×C₈	—	4	2	no	—
95	C₃⋊C₈	—	12	2	no	—
96	C₂×C₃²	#389	1	1	yes	characteristic
97	C₂×C₃²	#501	1	2	yes	—
98	C₃⋊D₃	#441	1	1	yes	characteristic
99	C₃⋊D₃	#442	1	1	yes	characteristic
100	C₃⋊D₃	#468	1	2	yes	—
101	C₂²×C₄	—	9	1	no	—
102	C₂²×C₄	—	9	2	no	—
103	C₂²×C₄	—	18	2	no	—
104	C₂²×C₄	—	18	2	no	—
105	C₂×C₈	—	9	1	no	—
106	C₂×C₈	—	9	2	no	—
107	C₂×C₈	—	9	2	no	—
108	C₂×C₈	—	9	2	no	—
109	C₂×C₈	—	18	1	no	—
110	C₂×C₈	—	18	1	no	—
111	C₂×C₈	PSU(3,𝔽₂)	1	1	yes	characteristic
112	C₄⋊C₄	—	9	4	no	—
113	C₄⋊C₄	—	18	2	no	—
114	C₄⋊C₄	—	18	2	no	—
115	C₄⋊C₄	—	18	2	no	—
116	C₄⋊C₄	—	36	4	no	—
117	C₄²	—	18	1	no	—
118	C₄²	—	18	1	no	—
119	C₂²×C₃	—	4	1	no	—
120	C₂×D₃	—	12	1	no	—
121	C₂×D₃	—	12	1	no	—
122	C₂×D₃	—	12	2	no	—
123	C₂×D₃	—	12	2	no	—
124	C₃×C₄	—	4	1	no	—
125	C₃×C₄	—	4	1	no	—
126	Dic₃	—	12	1	no	—
127	Dic₃	—	12	1	no	—
128	C₃²	#1943	1	1	yes	characteristic, complemented, Sylow
129	C₂³	—	9	1	no	—
130	C₂×C₄	—	9	1	no	—
131	C₂×C₄	—	9	1	no	—
132	C₂×C₄	—	9	1	no	—
133	C₂×C₄	—	9	1	no	—
134	C₂×C₄	—	9	1	no	—
135	C₂×C₄	—	18	2	no	—
136	C₂×C₄	—	18	4	no	—
137	C₂×C₄	—	18	2	no	—
138	C₂×C₄	—	18	2	no	—
139	C₂×C₄	—	36	2	no	—
140	C₂×C₄	—	36	2	no	—
141	C₂×C₄	—	36	2	no	—
142	C₂×C₄	—	36	2	no	—
143	C₂×C₄	—	36	2	no	—
144	C₂×C₄	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
145	C₈	—	9	2	no	—
146	C₈	—	18	2	no	—
147	C₈	#3777	1	2	yes	complemented
148	C₂×C₃	—	4	2	no	—
149	C₂×C₃	—	4	1	no	—
150	D₃	—	12	1	no	—
151	D₃	—	12	1	no	—
152	D₃	—	12	2	no	—
153	C₂²	—	9	2	no	—
154	C₂²	—	9	2	no	—
155	C₂²	—	9	1	no	—
156	C₂²	—	9	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	2	no	—
161	C₄	—	36	4	no	—
162	C₄	—	72	2	no	—
163	C₄	#64243	1	1	yes	characteristic
164	C₄	C₂×#3777	1	1	yes	characteristic
165	C₃	—	4	1	no	—
166	C₂	—	9	1	no	—
167	C₂	—	9	2	no	—
168	C₂	—	9	1	no	—
169	C₂	#10593537	1	2	yes	—
170	C₂	#10593554	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#63769

C₂

yes

maximal, characteristic

C₂×#63768

C₂

yes

maximal, characteristic

C₂×#63767

C₂

yes

maximal, characteristic

#10593554

C₂

yes

maximal

#10590093

C₂

yes

maximal

C₂×C₄⋊#157

C₂²

yes

—

C₂×C₄⋊#157

C₂²

yes

—

C₂×C₄×#157

C₂²

yes

characteristic

C₂×#3787

C₂²

yes

characteristic

C₂×#3777

—

C₂×C₈×C₃⋊D₃

C₂²

yes

characteristic

#63769

C₄

yes

—

—

—

C₂²×#157

—

C₂²×#157

—

C₂²×#157

D₄

yes

—

C₂×C₄×C₃⋊D₃

C₂³

yes

characteristic

C₂×#622

Q₈

yes

—

—

—

—

C₂×C₄

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

—

#1943

—

maximal, Sylow

C₂×D₃×C₈

—

C₂²×C₃⋊D₃

C₂×D₄

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

C₄×C₃⋊D₃

yes

C₄×C₃⋊D₃

yes

—

C₃²×C₈

C₄⋊C₄

yes

complemented

C₃⋊(C₃⋊C₈)

C₄⋊C₄

yes

complemented

C₂×#107

—

C₂×#106

—

C₂×#97

—

#389

—

#340

—

C₂×C₄×D₃

—

C₂×C₃×C₈

—

C₂×C₃⋊C₈

—

D₃×C₈

—

D₃×C₈

—

C₂²×C₃²

C₄⋊Q₈

yes

characteristic

C₂×C₃⋊D₃

C₄×D₄

yes

C₂×C₃⋊D₃

yes

C₂×C₃⋊D₃

yes

—

C₂×C₃⋊D₃

#137

yes

—

#157

—

#157

—

#157

—

C₃²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₃²×C₄

#125

yes

characteristic

C₃⋊Dic₃

C₄×Q₈

yes

characteristic

C₃⋊Dic₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

—

—

—

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

#389

yes

characteristic

C₂×C₃²

#501

yes

—

C₃⋊D₃

yes

C₃⋊D₃

yes

100

C₃⋊D₃

#468

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

PSU(3,𝔽₂)

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₂²×C₃

—

120

C₂×D₃

—

121

C₂×D₃

—

122

C₂×D₃

—

123

C₂×D₃

—

124

C₃×C₄

—

125

C₃×C₄

—

126

Dic₃

—

127

Dic₃

—

128

C₃²

#1943

yes

characteristic, complemented, Sylow

129

C₂³

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

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

—

143

C₂×C₄

—

144

C₂×C₄

C₂×PSU(3,𝔽₂)

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

#3777

yes

complemented

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

#64243

yes

characteristic

164

C₄

C₂×#3777

yes

characteristic

165

C₃

—

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

#10593537

yes

—

170

C₂

#10593554

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	#1943
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
12. #63769	C₄	semidirect
53. C₃²×C₈	C₄⋊C₄	semidirect
54. C₃⋊(C₃⋊C₈)	C₄⋊C₄	semidirect
128. C₃²	#1943	semidirect
147. C₈	#3777	semidirect

H ≅ G/N

type

12. #63769

C₄

semidirect

53. C₃²×C₈

C₄⋊C₄

semidirect

54. C₃⋊(C₃⋊C₈)

C₄⋊C₄

semidirect

128. C₃²

#1943

semidirect

147. C₈

#3777

semidirect

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

subgroup	index
C₂×#63769	2
C₂×#63768	2
C₂×#63767	2
#10593554	2
#10590093	2
#1943	9

subgroup

index

Subgroups — #11790185

Conjugacy classes of subgroups

Complements

Maximal subgroups