Group #11797354 — subgroups

Subgroups — #11797354

Order 1152.

414 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 153 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	#10592388	C₂	1	1	yes	maximal, characteristic
2	C₃²×#346	C₂	1	1	yes	maximal, characteristic, complemented
3	#10589373	C₂	1	2	yes	maximal
4	#10589367	C₂	1	2	yes	maximal, complemented
5	C₄×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	#69074	—	3	4	no	maximal
7	C₂×#3743	C₄	1	1	yes	characteristic
8	C₂×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
9	#64081	C₄	1	1	yes	characteristic
10	C₂×C₃²×C₁₆	C₂²	1	2	yes	—
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	#63642	C₄	1	1	yes	characteristic
13	C₈×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
15	#63630	C₄	1	1	yes	characteristic
16	#4689	—	3	4	no	—
17	C₃×#346	D₃	1	4	yes	complemented
18	(C₂×D₃)⋊C₁₆	—	3	8	no	—
19	Dic₃⋊C₁₆	—	3	8	no	—
20	C₄×C₃⋊C₁₆	—	3	4	no	—
21	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
24	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
25	#3752	C₈	1	2	yes	—
26	#3750	C₈	1	2	yes	—
27	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
30	#3743	—	2	2	no	—
31	C₈×C₃⋊D₃	—	2	2	no	—
32	C₃²×C₁₆	—	2	2	no	—
33	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
34	#2164	—	9	1	no	Sylow
35	C₂×ΓL(1,𝔽₂₅)	—	3	4	no	—
36	C₂×D₃×C₈	—	3	4	no	—
37	#872	—	3	4	no	—
38	C₂×C₃×C₁₆	C₂×D₃	1	8	yes	—
39	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
40	#815	—	3	4	no	—
41	C₈×Dic₃	—	3	4	no	—
42	C₂×C₃⋊C₁₆	—	3	8	no	—
43	#803	—	3	4	no	—
44	C₂²×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
45	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
46	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₃⋊Dic₃	C₂×C₈	1	2	yes	—
48	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
49	C₄×C₃⋊D₃	—	2	1	no	—
50	C₄×C₃⋊D₃	—	2	1	no	—
51	C₃²×C₈	M₁₆	1	2	yes	—
52	C₃²×C₈	#41	1	2	yes	—
53	C₃⋊(C₃⋊C₈)	—	2	1	no	—
54	C₃⋊(C₃⋊C₈)	—	2	1	no	—
55	#405	—	9	1	no	—
56	C₄⋊C₁₆	—	9	2	no	—
57	C₂²⋊C₁₆	—	9	2	no	—
58	#346	C₃⋊D₃	1	1	yes	characteristic, complemented
59	C₄×C₁₆	—	9	1	no	—
60	C₂×C₄×D₃	—	3	4	no	—
61	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
62	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
63	C₃×C₄²	C₄×D₃	1	4	yes	—
64	(C₂×D₃)⋊C₄	—	3	8	no	—
65	Dic₃⋊C₄	—	3	8	no	—
66	C₄×Dic₃	—	3	4	no	—
67	C₂×C₃⋊C₈	—	3	4	no	—
68	C₂×C₃⋊C₈	—	3	4	no	—
69	ΓL(1,𝔽₂₅)	—	6	8	no	—
70	D₃×C₈	—	6	8	no	—
71	C₃×C₁₆	—	2	8	no	—
72	C₃⋊C₁₆	—	6	8	no	—
73	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₃⋊D₃	—	2	1	no	—
75	C₂×C₃⋊D₃	—	2	2	no	—
76	C₃²×C₄	—	2	2	no	—
77	C₃²×C₄	#117	1	1	yes	characteristic
78	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
79	C₃⋊Dic₃	—	2	1	no	—
80	C₃⋊Dic₃	—	2	2	no	—
81	C₂×M₁₆	—	9	1	no	—
82	C₂²×C₈	—	9	1	no	—
83	#117	—	9	1	no	—
84	C₂×C₁₆	—	9	2	no	—
85	C₂×C₁₆	C₂×C₃⋊D₃	1	2	yes	—
86	#97	—	9	1	no	—
87	#97	—	9	1	no	—
88	C₄×C₈	—	9	1	no	—
89	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
90	C₂²×D₃	—	3	4	no	—
91	C₂×C₃×C₄	D₃×C₈	1	8	yes	—
92	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
93	C₂×Dic₃	—	3	8	no	—
94	C₂×Dic₃	—	3	4	no	—
95	C₄×D₃	—	6	4	no	—
96	C₄×D₃	—	6	4	no	—
97	C₃×C₈	ΓL(1,𝔽₂₅)	1	8	yes	—
98	C₃×C₈	#235	1	8	yes	—
99	C₃⋊C₈	—	6	4	no	—
100	C₃⋊C₈	—	6	4	no	—
101	C₂×C₃²	#431	1	1	yes	characteristic
102	C₂×C₃²	#504	1	2	yes	—
103	C₃⋊D₃	—	4	1	no	—
104	C₂²×C₄	—	9	1	no	—
105	M₁₆	—	18	2	no	—
106	C₂×C₈	—	9	1	no	—
107	C₂×C₈	—	9	1	no	—
108	C₂×C₈	—	18	2	no	—
109	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
110	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
111	C₄⋊C₄	—	9	2	no	—
112	C₂²⋊C₄	—	9	2	no	—
113	C₄²	—	9	1	no	—
114	C₄²	C₄×C₃⋊D₃	1	1	yes	characteristic
115	C₁₆	—	2	2	no	—
116	C₁₆	—	18	2	no	—
117	C₂²×C₃	C₂×D₃×C₈	1	4	yes	—
118	C₂×D₃	—	6	8	no	—
119	C₂×D₃	—	6	4	no	—
120	C₃×C₄	—	2	8	no	—
121	C₃×C₄	#872	1	4	yes	—
122	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	4	yes	—
123	Dic₃	—	6	8	no	—
124	Dic₃	—	6	4	no	—
125	C₃²	#2164	1	1	yes	characteristic, complemented, Sylow
126	C₂³	—	9	1	no	—
127	C₂×C₄	—	9	2	no	—
128	C₂×C₄	—	9	1	no	—
129	C₂×C₄	—	18	1	no	—
130	C₂×C₄	—	18	1	no	—
131	C₂×C₄	C₈×C₃⋊D₃	1	2	yes	—
132	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
133	C₈	—	18	1	no	—
134	C₈	—	18	1	no	—
135	C₈	#3743	1	2	yes	—
136	C₈	#3828	1	2	yes	—
137	C₂×C₃	#4685	1	4	yes	—
138	C₂×C₃	#4907	1	8	yes	—
139	D₃	—	12	4	no	—
140	C₂²	—	18	2	no	—
141	C₂²	—	18	1	no	—
142	C₂²	C₂×C₈×C₃⋊D₃	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	18	2	no	—
145	C₄	—	18	1	no	—
146	C₄	#64081	1	1	yes	characteristic
147	C₄	C₂×#3743	1	1	yes	characteristic
148	C₃	#69074	1	4	yes	complemented
149	C₂	—	36	1	no	—
150	C₂	#10592384	1	1	yes	characteristic
151	C₂	#10592606	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10592388

C₂

yes

maximal, characteristic

C₃²×#346

C₂

yes

maximal, characteristic, complemented

#10589373

C₂

yes

maximal

#10589367

C₂

yes

maximal, complemented

C₄×C₃⋊(C₃⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

#69074

—

maximal

C₂×#3743

C₄

yes

characteristic

C₂×C₈×C₃⋊D₃

C₂²

yes

characteristic

#64081

C₄

yes

characteristic

C₂×C₃²×C₁₆

C₂²

yes

—

C₃²×C₄×C₈

C₂²

yes

characteristic

#63642

C₄

yes

characteristic

C₈×C₃⋊Dic₃

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊C₁₆)

C₂²

yes

—

#63630

C₄

yes

characteristic

#4689

—

C₃×#346

D₃

yes

complemented

(C₂×D₃)⋊C₁₆

—

Dic₃⋊C₁₆

—

C₄×C₃⋊C₁₆

—

C₂×C₄×C₃⋊D₃

C₂×C₄

yes

characteristic

C₂×C₃²×C₈

C₂×C₄

yes

characteristic

C₂×C₃²×C₈

C₂³

yes

characteristic

C₃²×C₄²

C₂×C₄

yes

characteristic

#3752

C₈

yes

—

#3750

C₈

yes

—

C₄×C₃⋊Dic₃

C₂×C₄

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

characteristic

#3743

—

C₈×C₃⋊D₃

—

C₃²×C₁₆

—

C₃⋊(C₃⋊C₁₆)

—

#2164

—

Sylow

C₂×ΓL(1,𝔽₂₅)

—

C₂×D₃×C₈

—

#872

—

C₂×C₃×C₁₆

C₂×D₃

yes

—

C₃×C₄×C₈

C₂×D₃

yes

—

#815

—

C₈×Dic₃

—

C₂×C₃⋊C₁₆

—

#803

—

C₂²×C₃⋊D₃

C₂×C₈

yes

characteristic

C₂×C₃²×C₄

C₂×C₈

yes

—

C₂×C₃²×C₄

C₂²×C₄

yes

characteristic

C₂×C₃⋊Dic₃

C₂×C₈

yes

—

C₂×C₃⋊Dic₃

C₂×C₈

yes

characteristic

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

—

C₃²×C₈

M₁₆

yes

—

C₃²×C₈

#41

yes

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

#405

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

C₃⋊D₃

yes

characteristic, complemented

C₄×C₁₆

—

C₂×C₄×D₃

—

C₂×C₃×C₈

C₄×D₃

yes

—

C₂×C₃×C₈

C₂²×D₃

yes

—

C₃×C₄²

C₄×D₃

yes

—

(C₂×D₃)⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

C₂²×C₈

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

—

C₃²×C₄

#117

yes

characteristic

C₃²×C₄

C₂×M₁₆

yes

characteristic

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₃⋊D₃

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₂²×D₃

—

C₂×C₃×C₄

D₃×C₈

yes

—

C₂×C₃×C₄

C₂×C₄×D₃

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

ΓL(1,𝔽₂₅)

yes

—

C₃×C₈

#235

yes

—

C₃⋊C₈

—

100

C₃⋊C₈

—

101

C₂×C₃²

#431

yes

characteristic

102

C₂×C₃²

#504

yes

—

103

C₃⋊D₃

—

104

C₂²×C₄

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₄×C₃⋊D₃

yes

characteristic

110

C₂×C₈

C₂²×C₃⋊D₃

yes

characteristic

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₄×C₃⋊D₃

yes

characteristic

115

C₁₆

—

116

C₁₆

—

117

C₂²×C₃

C₂×D₃×C₈

yes

—

118

C₂×D₃

—

119

C₂×D₃

—

120

C₃×C₄

—

121

C₃×C₄

#872

yes

—

122

C₃×C₄

C₂×ΓL(1,𝔽₂₅)

yes

—

123

Dic₃

—

124

Dic₃

—

125

C₃²

#2164

yes

characteristic, complemented, Sylow

126

C₂³

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₈×C₃⋊D₃

yes

—

132

C₂×C₄

C₂×C₄×C₃⋊D₃

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

#3743

yes

—

136

C₈

#3828

yes

—

137

C₂×C₃

#4685

yes

—

138

C₂×C₃

#4907

yes

—

139

D₃

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂×C₈×C₃⋊D₃

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#64081

yes

characteristic

147

C₄

C₂×#3743

yes

characteristic

148

C₃

#69074

yes

complemented

149

C₂

—

150

C₂

#10592384

yes

characteristic

151

C₂

#10592606

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×C₃⋊D₃
commutator, socle	C₂²×C₃²	C₂²×C₈
Fitting	C₃²×#346	C₂
prime radical	C₂²×C₃⋊D₃	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	#346	C₃⋊D₃
hypocenter	C₃²	#2164
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×C₃⋊D₃

commutator, socle

C₂²×C₃²

C₂²×C₈

Fitting

C₃²×#346

C₂

prime radical

C₂²×C₃⋊D₃

C₂×C₈

cosocle

C₂×C₃²×C₈

C₂³

C₃⋊D₃

C₃²

G ▷ C₂²×C₃² ▷ C₃²

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
2. C₃²×#346	C₂	semidirect
4. #10589367	C₂	semidirect
5. C₄×C₃⋊(C₃⋊C₁₆)	C₂	semidirect
17. C₃×#346	D₃	semidirect
58. #346	C₃⋊D₃	semidirect
125. C₃²	#2164	semidirect
148. C₃	#69074	semidirect

H ≅ G/N

type

2. C₃²×#346

C₂

semidirect

4. #10589367

C₂

semidirect

5. C₄×C₃⋊(C₃⋊C₁₆)

C₂

semidirect

17. C₃×#346

D₃

semidirect

58. #346

C₃⋊D₃

semidirect

125. C₃²

#2164

semidirect

148. C₃

#69074

semidirect

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

subgroup	index
#10592388	2
C₃²×#346	2
#10589373	2
#10589367	2
C₄×C₃⋊(C₃⋊C₁₆)	2
#69074	3

subgroup

index

C₃²×#346

C₄×C₃⋊(C₃⋊C₁₆)

Subgroups — #11797354

Conjugacy classes of subgroups

Complements

Maximal subgroups