Group #11797021 — subgroups

Subgroups — #11797021

Order 1152.

480 conjugacy classes of subgroups; only the 225 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 225 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×C₃²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#10592800	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#10589367	C₂	1	4	yes	maximal, complemented
5	#71421	—	3	4	no	maximal
6	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
7	#64135	C₄	1	1	yes	characteristic
8	C₂×#3748	C₄	1	1	yes	characteristic
9	C₃²×M₃₂	—	2	2	no	—
10	C₂×C₃²×C₁₆	C₂²	1	2	yes	—
11	#63642	C₄	1	2	yes	—
12	C₈×C₃⋊Dic₃	C₂²	1	2	yes	—
13	C₂×C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
14	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
15	C₂×C₃×M₃₂	D₃	1	4	yes	—
16	#5101	—	3	4	no	—
17	C₂²×C₃⋊C₁₆	—	3	4	no	—
18	Dic₃⋊C₁₆	—	3	16	no	—
19	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃²×C₈	D₄	1	2	yes	—
22	C₂×C₃²×C₈	Q₈	1	2	yes	—
23	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
24	#3757	C₈	1	2	yes	—
25	C₄⋊(C₃⋊Dic₃)	C₈	1	2	yes	—
26	C₄×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
27	#3748	—	2	2	no	—
28	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
29	C₃²×C₁₆	—	2	2	no	—
30	C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
31	#2151	—	9	1	no	Sylow
32	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
33	#926	—	3	4	no	—
34	C₂×#208	—	3	4	no	—
35	C₃×M₃₂	—	2	8	no	—
36	C₂×C₃×C₁₆	C₂×D₃	1	8	yes	—
37	#815	—	3	8	no	—
38	C₈×Dic₃	—	3	8	no	—
39	C₂×C₃⋊C₁₆	—	3	8	no	—
40	C₂×C₃⋊C₁₆	—	6	8	no	—
41	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
42	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
43	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
44	C₂×C₃²×C₄	M₁₆	1	2	yes	—
45	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₃⋊Dic₃	C₂×C₈	1	4	yes	—
47	C₃²×C₈	C₄⋊C₄	1	2	yes	—
48	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
49	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₃⋊(C₃⋊C₈)	—	2	2	no	—
51	C₂×M₃₂	C₃⋊D₃	1	1	yes	characteristic
52	C₂²×C₁₆	—	9	1	no	—
53	#405	—	9	1	no	—
54	C₄⋊C₁₆	—	9	4	no	—
55	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
56	C₂×C₃×C₈	Dic₆	1	8	yes	—
57	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
58	C₂×C₃×C₈	#67	1	8	yes	—
59	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
60	#217	—	3	8	no	—
61	C₄⋊Dic₃	—	3	8	no	—
62	C₄×Dic₃	—	3	8	no	—
63	#208	—	6	8	no	—
64	C₂×C₃⋊C₈	—	3	8	no	—
65	C₃×C₁₆	—	2	8	no	—
66	C₃⋊C₁₆	—	6	8	no	—
67	C₂²×C₃²	—	2	2	no	—
68	C₂²×C₃²	C₄⋊C₈	1	2	yes	—
69	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
70	C₃²×C₄	C₄⋊C₈	1	2	yes	—
71	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
73	C₃⋊Dic₃	—	2	4	no	—
74	C₂×M₁₆	—	9	1	no	—
75	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
76	#117	—	9	1	no	—
77	M₃₂	—	2	2	no	—
78	C₂×C₁₆	—	9	2	no	—
79	C₂×C₁₆	—	18	2	no	—
80	C₂×C₁₆	C₂×C₃⋊D₃	1	2	yes	—
81	#97	—	9	2	no	—
82	C₄×C₈	—	9	2	no	—
83	C₂³×C₃	D₃×C₈	1	4	yes	—
84	C₂×C₃×C₄	D₃×C₈	1	4	yes	—
85	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	8	yes	—
86	C₂×C₃×C₄	Dic₃⋊C₄	1	8	yes	—
87	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
88	C₂×Dic₃	—	3	16	no	—
89	C₃×C₈	Dic₃⋊C₄	1	8	yes	—
90	C₃×C₈	C₂×Dic₆	1	4	yes	—
91	C₃×C₈	C₂×#67	1	4	yes	—
92	C₃⋊C₈	—	6	8	no	—
93	C₂×C₃²	—	2	2	no	—
94	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₃²	#504	1	2	yes	—
96	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
97	M₁₆	—	18	2	no	—
98	C₂×C₈	—	9	2	no	—
99	C₂×C₈	#634	1	2	yes	—
100	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
101	C₂×C₈	#638	1	2	yes	—
102	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
103	C₄⋊C₄	—	9	2	no	—
104	C₂²⋊C₄	—	9	2	no	—
105	C₄²	—	9	2	no	—
106	C₁₆	—	2	2	no	—
107	C₁₆	—	18	2	no	—
108	C₂²×C₃	—	2	8	no	—
109	C₂²×C₃	Dic₃⋊C₈	1	8	yes	—
110	C₂²×C₃	C₂×D₃×C₈	1	4	yes	—
111	C₃×C₄	Dic₃⋊C₈	1	8	yes	—
112	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	4	yes	—
113	C₃×C₄	C₂×Dic₃⋊C₄	1	4	yes	—
114	Dic₃	—	6	16	no	—
115	C₃²	#2151	1	1	yes	characteristic, complemented, Sylow
116	C₂³	C₈×C₃⋊D₃	1	1	yes	characteristic
117	C₂×C₄	—	9	4	no	—
118	C₂×C₄	C₈×C₃⋊D₃	1	1	yes	characteristic
119	C₂×C₄	#3743	1	2	yes	—
120	C₂×C₄	#3750	1	2	yes	—
121	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
122	C₈	—	18	2	no	—
123	C₈	#3750	1	2	yes	—
124	C₈	C₂×#634	1	1	yes	characteristic
125	C₈	C₂×#638	1	1	yes	characteristic
126	C₂×C₃	—	2	8	no	—
127	C₂×C₃	#4907	1	8	yes	—
128	C₂×C₃	C₂×Dic₃⋊C₈	1	4	yes	—
129	C₂²	—	2	2	no	—
130	C₂²	#63641	1	2	yes	—
131	C₂²	C₂×C₈×C₃⋊D₃	1	1	yes	characteristic
132	C₄	—	18	4	no	—
133	C₄	#63641	1	2	yes	—
134	C₄	C₂×#3743	1	1	yes	characteristic
135	C₄	C₂×#3750	1	1	yes	characteristic
136	C₃	#71421	1	4	yes	complemented
137	C₂	—	2	2	no	—
138	C₂	#10592606	1	2	yes	—
139	C₂	C₂×#63641	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃²×M₃₂

C₂

yes

maximal, characteristic

#10592800

C₂

yes

maximal, characteristic

C₂²×C₃⋊(C₃⋊C₁₆)

C₂

yes

maximal, characteristic

#10589367

C₂

yes

maximal, complemented

#71421

—

maximal

C₂²×C₃²×C₈

C₂²

yes

characteristic

#64135

C₄

yes

characteristic

C₂×#3748

C₄

yes

characteristic

C₃²×M₃₂

—

C₂×C₃²×C₁₆

C₂²

yes

—

#63642

C₄

yes

—

C₈×C₃⋊Dic₃

C₂²

yes

—

C₂×C₃⋊(C₃⋊C₁₆)

—

C₂×C₃⋊(C₃⋊C₁₆)

C₂²

yes

—

C₂×C₃×M₃₂

D₃

yes

—

#5101

—

C₂²×C₃⋊C₁₆

—

Dic₃⋊C₁₆

—

C₂²×C₃²×C₄

C₂×C₄

yes

characteristic

C₂×C₃²×C₈

C₂×C₄

yes

characteristic

C₂×C₃²×C₈

D₄

yes

—

C₂×C₃²×C₈

Q₈

yes

—

C₂×C₃²×C₈

C₂³

yes

characteristic

#3757

C₈

yes

—

C₄⋊(C₃⋊Dic₃)

C₈

yes

—

C₄×C₃⋊Dic₃

C₂×C₄

yes

—

#3748

—

C₂×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

—

C₃²×C₁₆

—

C₃⋊(C₃⋊C₁₆)

—

#2151

—

Sylow

C₂²×C₃×C₈

C₂×D₃

yes

—

#926

—

C₂×#208

—

C₃×M₃₂

—

C₂×C₃×C₁₆

C₂×D₃

yes

—

#815

—

C₈×Dic₃

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂³×C₃²

C₂×C₈

yes

characteristic

C₂×C₃²×C₄

C₄⋊C₄

yes

—

C₂×C₃²×C₄

C₂×C₈

yes

characteristic

C₂×C₃²×C₄

M₁₆

yes

—

C₂×C₃²×C₄

C₂²×C₄

yes

characteristic

C₂×C₃⋊Dic₃

C₂×C₈

yes

—

C₃²×C₈

C₄⋊C₄

yes

—

C₃²×C₈

C₂×D₄

yes

characteristic

C₃²×C₈

C₂×Q₈

yes

characteristic

C₃⋊(C₃⋊C₈)

—

C₂×M₃₂

C₃⋊D₃

yes

characteristic

C₂²×C₁₆

—

#405

—

C₄⋊C₁₆

—

C₂²×C₃×C₄

C₄×D₃

yes

—

C₂×C₃×C₈

Dic₆

yes

—

C₂×C₃×C₈

C₄×D₃

yes

—

C₂×C₃×C₈

#67

yes

—

C₂×C₃×C₈

C₂²×D₃

yes

—

#217

—

C₄⋊Dic₃

—

C₄×Dic₃

—

#208

—

C₂×C₃⋊C₈

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

C₄⋊C₈

yes

—

C₂²×C₃²

C₂²×C₈

yes

characteristic

C₃²×C₄

C₄⋊C₈

yes

—

C₃²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₃²×C₄

C₂×M₁₆

yes

characteristic

C₃⋊Dic₃

—

C₂×M₁₆

—

C₂²×C₈

C₂×C₃⋊D₃

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₃⋊D₃

yes

—

#97

—

C₄×C₈

—

C₂³×C₃

D₃×C₈

yes

—

C₂×C₃×C₄

D₃×C₈

yes

—

C₂×C₃×C₄

ΓL(1,𝔽₂₅)

yes

—

C₂×C₃×C₄

Dic₃⋊C₄

yes

—

C₂×C₃×C₄

C₂×C₄×D₃

yes

—

C₂×Dic₃

—

C₃×C₈

Dic₃⋊C₄

yes

—

C₃×C₈

C₂×Dic₆

yes

—

C₃×C₈

C₂×#67

yes

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

C₂×C₄⋊C₈

yes

characteristic

C₂×C₃²

#504

yes

—

C₂²×C₄

C₄×C₃⋊D₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

#634

yes

—

100

C₂×C₈

C₄×C₃⋊D₃

yes

characteristic

101

C₂×C₈

#638

yes

—

102

C₂×C₈

C₂²×C₃⋊D₃

yes

characteristic

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₁₆

—

107

C₁₆

—

108

C₂²×C₃

—

109

C₂²×C₃

Dic₃⋊C₈

yes

—

110

C₂²×C₃

C₂×D₃×C₈

yes

—

111

C₃×C₄

Dic₃⋊C₈

yes

—

112

C₃×C₄

C₂×ΓL(1,𝔽₂₅)

yes

—

113

C₃×C₄

C₂×Dic₃⋊C₄

yes

—

114

Dic₃

—

115

C₃²

#2151

yes

characteristic, complemented, Sylow

116

C₂³

C₈×C₃⋊D₃

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

C₈×C₃⋊D₃

yes

characteristic

119

C₂×C₄

#3743

yes

—

120

C₂×C₄

#3750

yes

—

121

C₂×C₄

C₂×C₄×C₃⋊D₃

yes

characteristic

122

C₈

—

123

C₈

#3750

yes

—

124

C₈

C₂×#634

yes

characteristic

125

C₈

C₂×#638

yes

characteristic

126

C₂×C₃

—

127

C₂×C₃

#4907

yes

—

128

C₂×C₃

C₂×Dic₃⋊C₈

yes

—

129

C₂²

—

130

C₂²

#63641

yes

—

131

C₂²

C₂×C₈×C₃⋊D₃

yes

characteristic

132

C₄

—

133

C₄

#63641

yes

—

134

C₄

C₂×#3743

yes

characteristic

135

C₄

C₂×#3750

yes

characteristic

136

C₃

#71421

yes

complemented

137

C₂

—

138

C₂

#10592606

yes

—

139

C₂

C₂×#63641

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×C₃⋊D₃
commutator, socle	C₂²×C₃²	C₂²×C₈
Fitting	C₂×C₃²×M₃₂	C₂
prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂×M₃₂	C₃⋊D₃
hypocenter	C₃²	#2151
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×C₃⋊D₃

commutator, socle

C₂²×C₃²

C₂²×C₈

Fitting

C₂×C₃²×M₃₂

C₂

prime radical

C₂³×C₃²

C₂×C₈

cosocle

C₂×C₃²×C₈

C₂³

hypercenter

C₂×M₃₂

C₃⋊D₃

C₃²

G ▷ C₂²×C₃² ▷ C₃²

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
4. #10589367	C₂	semidirect
115. C₃²	#2151	semidirect
136. C₃	#71421	semidirect

H ≅ G/N

type

4. #10589367

C₂

semidirect

115. C₃²

#2151

semidirect

136. C₃

#71421

semidirect

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

subgroup	index
C₂×C₃²×M₃₂	2
#10592800	2
C₂²×C₃⋊(C₃⋊C₁₆)	2
#10589367	2
#71421	3

subgroup

index

C₂×C₃²×M₃₂

#10592800

C₂²×C₃⋊(C₃⋊C₁₆)

#10589367

#71421

Subgroups — #11797021

Conjugacy classes of subgroups

Complements

Maximal subgroups