Group #11777925 — subgroups

Subgroups — #11777925

Order 1152.

444 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 157 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	C₄×#3746	C₂	1	1	yes	maximal, characteristic
2	#10592378	C₂	1	2	yes	maximal
3	C₃²×#322	C₂	1	1	yes	maximal, characteristic
4	#10589323	C₂	1	1	yes	maximal, characteristic
5	#10589322	C₂	1	2	yes	maximal
6	#68089	—	3	4	no	maximal
7	C₂×#3746	C₄	1	1	yes	characteristic
8	C₄×#634	C₂²	1	2	yes	—
9	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	#63645	C₄	1	1	yes	characteristic
13	#63643	C₄	1	2	yes	—
14	#63642	—	2	2	no	—
15	#63641	—	2	2	no	—
16	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
17	C₄×Dic₁₂	—	3	4	no	—
18	#4679	—	3	8	no	—
19	C₃×#322	D₃	1	4	yes	—
20	#4457	—	3	4	no	—
21	#4456	—	3	8	no	—
22	C₂×#634	C₂×C₄	1	2	yes	—
23	C₂×C₃²×C₈	—	2	1	no	—
24	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	D₄	1	2	yes	—
27	C₃²×C₄²	C₂³	1	1	yes	characteristic
28	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	2	yes	—
29	#3750	—	2	2	no	—
30	C₄×C₃⋊Dic₃	—	2	2	no	—
31	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
32	#3746	—	2	2	no	—
33	#1593	—	9	1	no	Sylow
34	C₂×Dic₁₂	—	3	4	no	—
35	C₄×Dic₆	—	3	8	no	—
36	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
37	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
38	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
39	#818	—	3	4	no	—
40	#816	—	3	8	no	—
41	#815	—	6	8	no	—
42	Dic₃⋊C₈	—	6	8	no	—
43	C₄⋊(C₃⋊C₈)	—	3	8	no	—
44	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃²×C₄	#41	1	1	yes	characteristic
47	C₂×C₃⋊Dic₃	—	2	2	no	—
48	C₂×C₃⋊Dic₃	—	2	2	no	—
49	#634	—	2	2	no	—
50	#634	—	2	2	no	—
51	C₃²×C₈	—	2	1	no	—
52	C₃²×C₈	—	2	2	no	—
53	C₃²×C₈	—	4	1	no	—
54	C₃²×C₈	M₁₆	1	2	yes	—
55	C₃⋊(C₃⋊C₈)	—	4	2	no	—
56	#446	—	9	2	no	—
57	C₄×Q₁₆	—	9	1	no	—
58	#334	—	9	1	no	—
59	Q₈⋊C₈	—	9	2	no	—
60	#322	C₃⋊D₃	1	1	yes	characteristic
61	C₂×Dic₆	—	3	8	no	—
62	C₂×C₃×C₈	—	2	4	no	—
63	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
64	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
65	C₂×C₃×C₈	D₁₂	1	8	yes	—
66	C₃×C₄²	C₂²×D₃	1	4	yes	—
67	C₄⋊Dic₃	—	3	8	no	—
68	Dic₃⋊C₄	—	6	8	no	—
69	C₄×Dic₃	—	6	8	no	—
70	C₂×C₃⋊C₈	—	6	8	no	—
71	Dic₁₂	—	6	8	no	—
72	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
73	C₃²×C₄	—	2	1	no	—
74	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
75	C₃²×C₄	#131	1	1	yes	characteristic
76	C₃²×C₄	#137	1	2	yes	—
77	C₃⋊Dic₃	—	4	2	no	—
78	C₃⋊Dic₃	—	4	2	no	—
79	C₂×Q₁₆	—	9	1	no	—
80	C₄×Q₈	—	9	2	no	—
81	#107	—	9	1	no	—
82	C₄⋊C₈	—	9	2	no	—
83	C₄⋊C₈	—	18	2	no	—
84	#103	—	9	2	no	—
85	#97	—	18	2	no	—
86	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
87	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
88	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
90	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
91	C₂×C₃×C₄	#235	1	4	yes	—
92	C₂×Dic₃	—	6	8	no	—
93	C₂×Dic₃	—	6	8	no	—
94	Dic₆	—	6	8	no	—
95	Dic₆	—	6	8	no	—
96	C₃×C₈	—	2	4	no	—
97	C₃×C₈	—	2	8	no	—
98	C₃×C₈	—	4	4	no	—
99	C₃×C₈	ΓL(1,𝔽₂₅)	1	8	yes	—
100	C₃⋊C₈	—	12	8	no	—
101	C₂×C₃²	#436	1	1	yes	characteristic
102	C₂×C₃²	#441	1	1	yes	characteristic
103	C₂×C₃²	#444	1	1	yes	characteristic
104	C₂×Q₈	—	9	2	no	—
105	Q₁₆	—	18	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	18	2	no	—
108	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
109	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
110	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
111	C₄⋊C₄	—	9	2	no	—
112	C₄⋊C₄	—	18	2	no	—
113	C₄²	—	18	2	no	—
114	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
115	C₂²×C₃	C₄×D₁₂	1	4	yes	—
116	C₃×C₄	—	2	4	no	—
117	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	4	yes	—
118	C₃×C₄	#901	1	4	yes	—
119	C₃×C₄	#909	1	8	yes	—
120	Dic₃	—	12	8	no	—
121	Dic₃	—	12	8	no	—
122	C₃²	#1593	1	1	yes	characteristic, complemented, Sylow
123	Q₈	—	18	2	no	—
124	Q₈	—	18	2	no	—
125	C₂×C₄	—	18	2	no	—
126	C₂×C₄	—	18	2	no	—
127	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
128	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
129	C₂×C₄	#3828	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	2	2	no	—
132	C₈	—	4	1	no	—
133	C₈	—	36	2	no	—
134	C₈	#3743	1	2	yes	—
135	C₂×C₃	#4688	1	4	yes	—
136	C₂×C₃	#4716	1	4	yes	—
137	C₂×C₃	#4717	1	4	yes	—
138	C₂²	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	36	2	no	—
141	C₄	—	36	2	no	—
142	C₄	C₂×#3743	1	1	yes	characteristic
143	C₄	#64110	1	1	yes	characteristic
144	C₄	#64118	1	2	yes	—
145	C₃	#68089	1	4	yes	complemented
146	C₂	#10592387	1	1	yes	characteristic
147	C₂	#10592415	1	1	yes	characteristic
148	C₂	#10592416	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#3746

C₂

yes

maximal, characteristic

#10592378

C₂

yes

maximal

C₃²×#322

C₂

yes

maximal, characteristic

#10589323

C₂

yes

maximal, characteristic

#10589322

C₂

yes

—

C₄

yes

characteristic

C₄×#634

C₂²

yes

—

C₃²×C₄×C₈

C₂²

yes

characteristic

C₃²×C₄×C₈

C₂²

yes

characteristic

C₃²×C₄×C₈

C₂²

yes

characteristic

#63645

C₄

yes

characteristic

#63643

C₄

yes

—

#63642

—

#63641

—

C₄⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

—

C₄×Dic₁₂

—

#4679

—

C₃×#322

D₃

yes

—

#4457

—

#4456

—

C₂×#634

C₂×C₄

yes

—

C₂×C₃²×C₈

—

C₂×C₃²×C₈

C₂×C₄

yes

characteristic

C₂×C₃²×C₈

C₂×C₄

yes

characteristic

C₂×C₃²×C₈

D₄

yes

—

C₃²×C₄²

C₂³

yes

characteristic

C₄⋊(C₃⋊Dic₃)

C₂×C₄

yes

—

#3750

—

C₄×C₃⋊Dic₃

—

C₂×C₃⋊(C₃⋊C₈)

—

#3746

—

#1593

—

Sylow

C₂×Dic₁₂

—

C₄×Dic₆

—

C₃×C₄×C₈

C₂×D₃

yes

—

C₃×C₄×C₈

C₂×D₃

yes

—

C₃×C₄×C₈

C₂×D₃

yes

—

#818

—

#816

—

#815

—

Dic₃⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₂×C₃²×C₄

C₂²×C₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

#41

yes

characteristic

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#634

—

#634

—

C₃²×C₈

—

C₃²×C₈

—

C₃²×C₈

—

C₃²×C₈

M₁₆

yes

—

C₃⋊(C₃⋊C₈)

—

#446

—

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

#322

C₃⋊D₃

yes

characteristic

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄×D₃

yes

—

C₂×C₃×C₈

C₄×D₃

yes

—

C₂×C₃×C₈

D₁₂

yes

—

C₃×C₄²

C₂²×D₃

yes

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

Dic₁₂

—

C₂²×C₃²

C₄×D₄

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₂×M₁₆

yes

C₃²×C₄

yes

C₃²×C₄

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃×C₄

C₂×C₄×D₃

yes

—

C₂×C₃×C₄

C₂×D₁₂

yes

—

C₂×C₃×C₄

#235

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

ΓL(1,𝔽₂₅)

yes

—

100

C₃⋊C₈

—

101

C₂×C₃²

#436

yes

characteristic

102

C₂×C₃²

#441

yes

characteristic

103

C₂×C₃²

#444

yes

characteristic

104

C₂×Q₈

—

105

Q₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₄×C₃⋊D₃

yes

characteristic

109

C₂×C₈

C₄×C₃⋊D₃

yes

characteristic

110

C₂×C₈

C₄⋊(C₃⋊D₃)

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

—

114

C₄²

C₂²×C₃⋊D₃

yes

characteristic

115

C₂²×C₃

C₄×D₁₂

yes

—

116

C₃×C₄

—

117

C₃×C₄

C₂×ΓL(1,𝔽₂₅)

yes

—

118

C₃×C₄

#901

yes

—

119

C₃×C₄

#909

yes

—

120

Dic₃

—

121

Dic₃

—

122

C₃²

#1593

yes

characteristic, complemented, Sylow

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₂×C₄×C₃⋊D₃

yes

characteristic

128

C₂×C₄

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

129

C₂×C₄

#3828

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

#3743

yes

—

135

C₂×C₃

#4688

yes

—

136

C₂×C₃

#4716

yes

—

137

C₂×C₃

#4717

yes

—

138

C₂²

C₄×C₄⋊(C₃⋊D₃)

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

C₂×#3743

yes

characteristic

143

C₄

#64110

yes

characteristic

144

C₄

#64118

yes

—

145

C₃

#68089

yes

complemented

146

C₂

#10592387

yes

characteristic

147

C₂

#10592415

yes

characteristic

148

C₂

#10592416

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×#322	C₂
socle, prime radical	C₂²×C₃²	C₄×D₄
cosocle	C₃²×C₄²	C₂³
hypercenter	#322	C₃⋊D₃
hypocenter	C₃²	#1593
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊(C₃⋊D₃)

commutator

C₂×C₃²×C₄

C₂²×C₄

Frattini

C₄²

C₂²×C₃⋊D₃

Fitting

C₃²×#322

C₂

socle, prime radical

C₂²×C₃²

C₄×D₄

cosocle

C₃²×C₄²

C₂³

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
122. C₃²	#1593	semidirect
145. C₃	#68089	semidirect

H ≅ G/N

type

122. C₃²

#1593

semidirect

145. C₃

#68089

semidirect

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

subgroup	index
C₄×#3746	2
#10592378	2
C₃²×#322	2
#10589323	2
#10589322	2
#68089	3

subgroup

index

C₃²×#322

Subgroups — #11777925

Conjugacy classes of subgroups

Complements

Maximal subgroups