Group #11786086 — subgroups

Subgroups — #11786086

Order 1152.

564 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 181 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	#10592817	C₂	1	2	yes	maximal
2	C₂×C₈×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₃²×#341	C₂	1	1	yes	maximal, characteristic
4	#10589424	C₂	1	2	yes	maximal
5	#10589418	C₂	1	1	yes	maximal, characteristic
6	#68928	—	3	4	no	maximal
7	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
10	C₂×#3748	C₂²	1	2	yes	—
11	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
12	#63641	—	2	4	no	—
13	C₈×C₃⋊Dic₃	C₄	1	2	yes	—
14	C₈×C₃⋊Dic₃	C₄	1	2	yes	—
15	#5118	—	3	8	no	—
16	C₂×C₈×Dic₃	—	3	4	no	—
17	C₃×#341	D₃	1	4	yes	—
18	#4558	—	3	8	no	—
19	#4552	—	3	4	no	—
20	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
21	C₂²×C₃⋊Dic₃	Q₈	1	2	yes	—
22	C₃²×M₁₆	—	2	2	no	—
23	C₃²×M₁₆	—	2	2	no	—
24	C₂×C₃²×C₈	—	2	2	no	—
25	C₂×C₃²×C₈	—	2	2	no	—
26	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
27	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₃⋊Dic₃	D₄	1	2	yes	—
30	#3748	—	2	2	no	—
31	#3748	—	2	2	no	—
32	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
33	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
34	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
35	#1837	—	9	1	no	Sylow
36	C₂×C₃×M₁₆	C₂×D₃	1	8	yes	—
37	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
38	C₂×C₄×Dic₃	—	3	4	no	—
39	C₂×#208	—	3	8	no	—
40	C₂²×C₃⋊C₈	—	3	4	no	—
41	Dic₃⋊C₈	—	6	16	no	—
42	C₈×Dic₃	—	3	8	no	—
43	C₈×Dic₃	—	3	8	no	—
44	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃²×C₄	#41	1	2	yes	—
48	C₂×C₃²×C₄	#41	1	2	yes	—
49	C₂×C₃⋊Dic₃	—	2	4	no	—
50	C₂×C₃⋊Dic₃	C₄⋊C₄	1	2	yes	—
51	C₂×C₃⋊Dic₃	C₄⋊C₄	1	2	yes	—
52	C₃²×C₈	—	2	2	no	—
53	C₃²×C₈	—	4	2	no	—
54	C₃⋊(C₃⋊C₈)	—	2	2	no	—
55	C₃⋊(C₃⋊C₈)	—	4	2	no	—
56	#423	—	9	2	no	—
57	C₂×C₄×C₈	—	9	1	no	—
58	#341	—	9	2	no	—
59	#341	—	9	1	no	—
60	#341	C₃⋊D₃	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
62	C₂²×Dic₃	—	3	8	no	—
63	C₃×M₁₆	—	2	8	no	—
64	C₃×M₁₆	—	2	8	no	—
65	C₂×C₃×C₈	—	2	8	no	—
66	C₂×C₃×C₈	—	2	8	no	—
67	C₂×C₃×C₈	C₄×D₃	1	8	yes	—
68	C₄×Dic₃	—	3	4	no	—
69	C₄×Dic₃	—	3	4	no	—
70	C₄×Dic₃	—	3	8	no	—
71	#208	—	6	8	no	—
72	#208	—	6	8	no	—
73	C₂×C₃⋊C₈	—	3	8	no	—
74	C₂×C₃⋊C₈	—	6	8	no	—
75	C₂×C₃⋊C₈	—	6	8	no	—
76	C₂²×C₃²	—	2	2	no	—
77	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₃²	#125	1	2	yes	—
79	C₃²×C₄	#117	1	1	yes	characteristic
80	C₃²×C₄	#117	1	1	yes	characteristic
81	C₃²×C₄	#124	1	2	yes	—
82	C₃⋊Dic₃	—	2	2	no	—
83	C₃⋊Dic₃	#108	1	4	yes	—
84	C₂×M₁₆	—	9	2	no	—
85	C₂×M₁₆	C₂×C₃⋊D₃	1	2	yes	—
86	C₂²×C₈	—	9	1	no	—
87	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
88	C₂×C₄²	—	9	1	no	—
89	C₄⋊C₈	—	18	4	no	—
90	C₄×C₈	—	9	2	no	—
91	C₄×C₈	—	9	2	no	—
92	C₂³×C₃	D₃×Q₈	1	4	yes	—
93	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
94	C₂×C₃×C₄	#235	1	8	yes	—
95	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
96	C₂×C₃×C₄	#237	1	8	yes	—
97	C₂×Dic₃	—	3	8	no	—
98	C₂×Dic₃	—	3	8	no	—
99	C₂×Dic₃	—	6	16	no	—
100	C₃×C₈	—	2	8	no	—
101	C₃×C₈	—	4	8	no	—
102	C₃⋊C₈	—	6	8	no	—
103	C₃⋊C₈	—	12	8	no	—
104	C₂×C₃²	—	2	2	no	—
105	C₂×C₃²	#382	1	1	yes	characteristic
106	C₂×C₃²	C₂×#108	1	2	yes	—
107	C₂²×C₄	—	9	2	no	—
108	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	M₁₆	—	18	2	no	—
112	M₁₆	—	18	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	9	2	no	—
116	C₂×C₈	—	18	2	no	—
117	C₂×C₈	—	18	2	no	—
118	C₂×C₈	C₄×C₃⋊D₃	1	2	yes	—
119	C₄²	—	9	1	no	—
120	C₄²	—	9	1	no	—
121	C₄²	—	9	2	no	—
122	C₂²×C₃	—	2	8	no	—
123	C₂²×C₃	#889	1	8	yes	—
124	C₂²×C₃	D₃×C₄⋊C₄	1	4	yes	—
125	C₃×C₄	#872	1	4	yes	—
126	C₃×C₄	#877	1	4	yes	—
127	C₃×C₄	#885	1	8	yes	—
128	Dic₃	—	3	16	no	—
129	Dic₃	—	6	8	no	—
130	C₃²	#1837	1	1	yes	characteristic, complemented, Sylow
131	C₂³	Q₈×C₃⋊D₃	1	1	yes	characteristic
132	C₂×C₄	—	9	2	no	—
133	C₂×C₄	—	9	2	no	—
134	C₂×C₄	—	18	4	no	—
135	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
136	C₂×C₄	#3828	1	2	yes	—
137	C₂×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
138	C₂×C₄	#3830	1	2	yes	—
139	C₈	—	2	2	no	—
140	C₈	—	4	2	no	—
141	C₈	—	18	2	no	—
142	C₈	—	36	2	no	—
143	C₂×C₃	—	2	8	no	—
144	C₂×C₃	#4650	1	4	yes	—
145	C₂×C₃	D₃×#108	1	8	yes	—
146	C₂²	—	2	2	no	—
147	C₂²	#64098	1	2	yes	—
148	C₂²	C₄⋊C₄×C₃⋊D₃	1	1	yes	characteristic
149	C₄	—	9	4	no	—
150	C₄	—	18	2	no	—
151	C₄	#64081	1	1	yes	characteristic
152	C₄	#64086	1	1	yes	characteristic
153	C₄	#64094	1	2	yes	—
154	C₃	#68928	1	4	yes	complemented
155	C₂	—	2	2	no	—
156	C₂	#10592349	1	1	yes	characteristic
157	C₂	C₃⋊D₃×#108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10592817

C₂

yes

maximal

C₂×C₈×C₃⋊Dic₃

C₂

yes

maximal, characteristic

C₃²×#341

C₂

yes

maximal, characteristic

#10589424

C₂

yes

maximal

#10589418

C₂

yes

maximal, characteristic

#68928

—

maximal

C₂×C₃²×M₁₆

C₂²

yes

—

C₂²×C₃²×C₈

C₂²

yes

characteristic

C₂×C₄×C₃⋊Dic₃

C₂²

yes

characteristic

C₂×#3748

C₂²

yes

—

C₂²×C₃⋊(C₃⋊C₈)

C₂²

yes

characteristic

#63641

—

C₈×C₃⋊Dic₃

C₄

yes

—

C₈×C₃⋊Dic₃

C₄

yes

—

#5118

—

C₂×C₈×Dic₃

—

C₃×#341

D₃

yes

—

#4558

—

#4552

—

C₂²×C₃²×C₄

C₂³

yes

characteristic

C₂²×C₃⋊Dic₃

Q₈

yes

—

C₃²×M₁₆

—

C₃²×M₁₆

—

C₂×C₃²×C₈

—

C₂×C₃²×C₈

—

C₂×C₃²×C₈

C₂×C₄

yes

—

C₄×C₃⋊Dic₃

C₂×C₄

yes

characteristic

C₄×C₃⋊Dic₃

C₂×C₄

yes

characteristic

C₄×C₃⋊Dic₃

D₄

yes

—

#3748

—

#3748

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

—

#1837

—

Sylow

C₂×C₃×M₁₆

C₂×D₃

yes

—

C₂²×C₃×C₈

C₂×D₃

yes

—

C₂×C₄×Dic₃

—

C₂×#208

—

C₂²×C₃⋊C₈

—

Dic₃⋊C₈

—

C₈×Dic₃

—

C₈×Dic₃

—

C₂³×C₃²

C₂×Q₈

yes

characteristic

C₂×C₃²×C₄

C₂²×C₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

#41

yes

—

C₂×C₃²×C₄

#41

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

C₄⋊C₄

yes

—

C₂×C₃⋊Dic₃

C₄⋊C₄

yes

—

C₃²×C₈

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

#423

—

C₂×C₄×C₈

—

#341

—

#341

—

#341

C₃⋊D₃

yes

characteristic

C₂²×C₃×C₄

C₂²×D₃

yes

—

C₂²×Dic₃

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄×D₃

yes

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃²

#125

yes

—

C₃²×C₄

yes

C₃²×C₄

yes

C₃²×C₄

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

#108

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₃⋊D₃

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₃⋊D₃

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂³×C₃

D₃×Q₈

yes

—

C₂×C₃×C₄

C₂×C₄×D₃

yes

—

C₂×C₃×C₄

#235

yes

—

C₂×C₃×C₄

D₃×D₄

yes

—

C₂×C₃×C₄

#237

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

100

C₃×C₈

—

101

C₃×C₈

—

102

C₃⋊C₈

—

103

C₃⋊C₈

—

104

C₂×C₃²

—

105

C₂×C₃²

#382

yes

characteristic

106

C₂×C₃²

C₂×#108

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²×C₃⋊D₃

yes

characteristic

109

M₁₆

—

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₄×C₃⋊D₃

yes

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₂²×C₃

—

123

C₂²×C₃

#889

yes

—

124

C₂²×C₃

D₃×C₄⋊C₄

yes

—

125

C₃×C₄

#872

yes

—

126

C₃×C₄

#877

yes

—

127

C₃×C₄

#885

yes

—

128

Dic₃

—

129

Dic₃

—

130

C₃²

#1837

yes

characteristic, complemented, Sylow

131

C₂³

Q₈×C₃⋊D₃

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄×C₃⋊D₃

yes

characteristic

136

C₂×C₄

#3828

yes

—

137

C₂×C₄

D₄×C₃⋊D₃

yes

characteristic

138

C₂×C₄

#3830

yes

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₂×C₃

—

144

C₂×C₃

#4650

yes

—

145

C₂×C₃

D₃×#108

yes

—

146

C₂²

—

147

C₂²

#64098

yes

—

148

C₂²

C₄⋊C₄×C₃⋊D₃

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

#64081

yes

characteristic

152

C₄

#64086

yes

characteristic

153

C₄

#64094

yes

—

154

C₃

#68928

yes

complemented

155

C₂

—

156

C₂

#10592349

yes

characteristic

157

C₂

C₃⋊D₃×#108

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×C₃⋊D₃
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#341	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₂×Q₈
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	#341	C₃⋊D₃
hypocenter	C₃²	#1837
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×C₃⋊D₃

commutator

C₂×C₃²×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×C₃⋊D₃

Fitting

C₃²×#341

C₂

socle

C₂²×C₃²

C₂×C₄⋊C₄

prime radical

C₂³×C₃²

C₂×Q₈

cosocle

C₂²×C₃²×C₄

C₂³

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
130. C₃²	#1837	semidirect
154. C₃	#68928	semidirect

H ≅ G/N

type

130. C₃²

#1837

semidirect

154. C₃

#68928

semidirect

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

subgroup	index
#10592817	2
C₂×C₈×C₃⋊Dic₃	2
C₃²×#341	2
#10589424	2
#10589418	2
#68928	3

subgroup

index

#10592817

C₂×C₈×C₃⋊Dic₃

C₃²×#341

#10589424

#10589418

#68928

Subgroups — #11786086

Conjugacy classes of subgroups

Complements

Maximal subgroups