Group #11777304 — subgroups

Subgroups — #11777304

Order 1152.

480 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 155 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	#10592529	C₂	1	2	yes	maximal
2	C₄×#3746	C₂	1	1	yes	maximal, characteristic
3	C₃²×#334	C₂	1	1	yes	maximal, characteristic
4	#10589350	C₂	1	2	yes	maximal
5	#10589321	C₂	1	1	yes	maximal, characteristic
6	#68517	—	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	2	yes	—
10	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
11	#63645	C₄	1	1	yes	characteristic
12	#63643	C₄	1	2	yes	—
13	#63641	—	2	2	no	—
14	C₈×C₃⋊Dic₃	—	2	2	no	—
15	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
17	#4830	—	3	8	no	—
18	C₄×Dic₁₂	—	3	4	no	—
19	C₃×#334	D₃	1	4	yes	—
20	#4484	—	3	8	no	—
21	#4455	—	3	4	no	—
22	C₂×#634	C₂×C₄	1	2	yes	—
23	C₂×C₃²×C₈	—	2	2	no	—
24	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₃²×C₄²	C₂³	1	1	yes	characteristic
27	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	2	yes	—
28	#3750	—	2	2	no	—
29	C₄×C₃⋊Dic₃	—	2	2	no	—
30	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
31	C₂×C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
32	#3746	C₈	1	4	yes	complemented
33	#1579	—	9	1	no	Sylow
34	C₂×Dic₁₂	—	3	4	no	—
35	C₄×Dic₆	—	3	8	no	—
36	C₃×C₄⋊C₈	C₂×D₃	1	8	yes	—
37	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
38	#818	—	3	4	no	—
39	#816	—	3	8	no	—
40	Dic₃⋊C₈	—	6	8	no	—
41	C₈×Dic₃	—	6	8	no	—
42	C₄×C₃⋊C₈	—	3	4	no	—
43	C₄×C₃⋊C₈	—	3	4	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	C₂×C₈	1	4	yes	—
51	C₃²×C₈	—	2	1	no	—
52	C₃²×C₈	—	4	2	no	—
53	C₃²×C₈	C₂×C₈	1	2	yes	—
54	C₃⋊(C₃⋊C₈)	—	2	2	no	—
55	C₃⋊(C₃⋊C₈)	—	4	1	no	—
56	C₈×Q₈	—	9	2	no	—
57	C₄×Q₁₆	—	9	1	no	—
58	#334	C₃⋊D₃	1	1	yes	characteristic
59	Q₈⋊C₈	—	9	2	no	—
60	#322	—	9	1	no	—
61	C₂×Dic₆	—	3	8	no	—
62	C₂×C₃×C₈	—	2	8	no	—
63	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
64	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
65	C₃×C₄²	C₂²×D₃	1	4	yes	—
66	C₄⋊Dic₃	—	3	8	no	—
67	Dic₃⋊C₄	—	6	8	no	—
68	C₄×Dic₃	—	6	8	no	—
69	C₂×C₃⋊C₈	—	3	8	no	—
70	C₂×C₃⋊C₈	—	6	4	no	—
71	Dic₁₂	—	3	16	no	—
72	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
73	C₃²×C₄	—	2	1	no	—
74	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
75	C₃²×C₄	#131	1	1	yes	characteristic
76	C₃²×C₄	#137	1	2	yes	—
77	C₃⋊Dic₃	—	2	4	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	—	18	2	no	—
83	C₄⋊C₈	C₂×C₃⋊D₃	1	2	yes	—
84	#103	—	9	2	no	—
85	C₄×C₈	—	9	1	no	—
86	C₄×C₈	—	9	1	no	—
87	C₄×C₈	—	18	2	no	—
88	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
90	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
91	C₂×C₃×C₄	#239	1	4	yes	—
92	C₂×Dic₃	—	6	8	no	—
93	C₂×Dic₃	—	6	8	no	—
94	Dic₆	—	3	16	no	—
95	Dic₆	—	6	8	no	—
96	C₃×C₈	—	2	4	no	—
97	C₃×C₈	—	4	8	no	—
98	C₃×C₈	D₃×C₈	1	8	yes	—
99	C₃⋊C₈	—	6	8	no	—
100	C₃⋊C₈	—	12	4	no	—
101	C₂×C₃²	C₈×D₄	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	—	9	4	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	9	2	no	—
108	C₂×C₈	—	18	1	no	—
109	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
110	C₂×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
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	#893	1	4	yes	—
116	C₃×C₄	—	2	4	no	—
117	C₃×C₄	C₂×D₃×C₈	1	4	yes	—
118	C₃×C₄	#907	1	4	yes	—
119	C₃×C₄	#915	1	8	yes	—
120	Dic₃	—	6	16	no	—
121	Dic₃	—	12	8	no	—
122	C₃²	#1579	1	1	yes	characteristic, complemented, Sylow
123	Q₈	—	9	4	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	D₄×C₃⋊D₃	1	1	yes	characteristic
129	C₂×C₄	#3832	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	4	2	no	—
132	C₈	—	18	2	no	—
133	C₈	—	36	1	no	—
134	C₈	C₈×C₃⋊D₃	1	2	yes	—
135	C₂×C₃	#4834	1	4	yes	—
136	C₂×C₃	#4853	1	4	yes	—
137	C₂×C₃	#4894	1	4	yes	—
138	C₂²	#64102	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	18	4	no	—
141	C₄	—	36	2	no	—
142	C₄	C₂×C₈×C₃⋊D₃	1	1	yes	characteristic
143	C₄	#64116	1	1	yes	characteristic
144	C₄	#64124	1	2	yes	—
145	C₃	#68517	1	4	yes	complemented
146	C₂	#10592533	1	1	yes	characteristic
147	C₂	#10592552	1	1	yes	characteristic
148	C₂	#10592593	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10592529

C₂

yes

maximal

C₄×#3746

C₂

yes

maximal, characteristic

C₃²×#334

C₂

yes

maximal, characteristic

#10589350

C₂

yes

maximal

#10589321

C₂

yes

maximal, characteristic

#68517

—

maximal

C₂×#3746

C₄

yes

characteristic

C₄×#634

C₂²

yes

—

C₃²×C₄⋊C₈

C₂²

yes

—

C₃²×C₄×C₈

C₂²

yes

characteristic

#63645

C₄

yes

characteristic

#63643

C₄

yes

—

#63641

—

C₈×C₃⋊Dic₃

—

C₄×C₃⋊(C₃⋊C₈)

C₂²

yes

characteristic

C₄×C₃⋊(C₃⋊C₈)

C₂²

yes

characteristic

#4830

—

C₄×Dic₁₂

—

C₃×#334

D₃

yes

—

#4484

—

#4455

—

C₂×#634

C₂×C₄

yes

—

C₂×C₃²×C₈

—

C₂×C₃²×C₈

C₂×C₄

yes

characteristic

C₂×C₃²×C₈

C₂×C₄

yes

characteristic

C₃²×C₄²

C₂³

yes

characteristic

C₄⋊(C₃⋊Dic₃)

C₂×C₄

yes

—

#3750

—

C₄×C₃⋊Dic₃

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

—

#3746

C₈

yes

complemented

#1579

—

Sylow

C₂×Dic₁₂

—

C₄×Dic₆

—

C₃×C₄⋊C₈

C₂×D₃

yes

—

C₃×C₄×C₈

C₂×D₃

yes

—

#818

—

#816

—

Dic₃⋊C₈

—

C₈×Dic₃

—

C₄×C₃⋊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

C₂×C₈

yes

—

C₃²×C₈

—

C₃²×C₈

—

C₃²×C₈

C₂×C₈

yes

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

C₈×Q₈

—

C₄×Q₁₆

—

#334

C₃⋊D₃

yes

characteristic

Q₈⋊C₈

—

#322

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄×D₃

yes

—

C₂×C₃×C₈

C₄×D₃

yes

—

C₃×C₄²

C₂²×D₃

yes

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

Dic₁₂

—

C₂²×C₃²

C₄×D₄

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₂²×C₈

yes

C₃²×C₄

yes

C₃²×C₄

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₃⋊D₃

yes

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃×C₄

C₂×C₄×D₃

yes

—

C₂×C₃×C₄

D₃×D₄

yes

—

C₂×C₃×C₄

#239

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

D₃×C₈

yes

—

C₃⋊C₈

—

100

C₃⋊C₈

—

101

C₂×C₃²

C₈×D₄

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

—

109

C₂×C₈

C₄×C₃⋊D₃

yes

characteristic

110

C₂×C₈

C₄×C₃⋊D₃

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

—

114

C₄²

C₂²×C₃⋊D₃

yes

characteristic

115

C₂²×C₃

#893

yes

—

116

C₃×C₄

—

117

C₃×C₄

C₂×D₃×C₈

yes

—

118

C₃×C₄

#907

yes

—

119

C₃×C₄

#915

yes

—

120

Dic₃

—

121

Dic₃

—

122

C₃²

#1579

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

D₄×C₃⋊D₃

yes

characteristic

129

C₂×C₄

#3832

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

C₈×C₃⋊D₃

yes

—

135

C₂×C₃

#4834

yes

—

136

C₂×C₃

#4853

yes

—

137

C₂×C₃

#4894

yes

—

138

C₂²

#64102

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

C₂×C₈×C₃⋊D₃

yes

characteristic

143

C₄

#64116

yes

characteristic

144

C₄

#64124

yes

—

145

C₃

#68517

yes

complemented

146

C₂

#10592533

yes

characteristic

147

C₂

#10592552

yes

characteristic

148

C₂

#10592593

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×C₃⋊D₃
commutator	C₃²×C₄	C₂²×C₈
Frattini	C₄²	C₂²×C₃⋊D₃
Fitting	C₃²×#334	C₂
socle, prime radical	C₂²×C₃²	C₄×D₄
cosocle	C₃²×C₄²	C₂³
hypercenter	#334	C₃⋊D₃
hypocenter	C₃²	#1579
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×C₃⋊D₃

commutator

C₃²×C₄

C₂²×C₈

Frattini

C₄²

C₂²×C₃⋊D₃

Fitting

C₃²×#334

C₂

socle, prime radical

C₂²×C₃²

C₄×D₄

cosocle

C₃²×C₄²

C₂³

C₃⋊D₃

C₃²

G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
32. #3746	C₈	semidirect
122. C₃²	#1579	semidirect
145. C₃	#68517	semidirect

H ≅ G/N

type

32. #3746

C₈

semidirect

122. C₃²

#1579

semidirect

145. C₃

#68517

semidirect

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

subgroup	index
#10592529	2
C₄×#3746	2
C₃²×#334	2
#10589350	2
#10589321	2
#68517	3

subgroup

index

C₃²×#334

Subgroups — #11777304

Conjugacy classes of subgroups

Complements

Maximal subgroups