Group #11772502 — subgroups

Subgroups — #11772502

Order 1152.

174 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 49 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	#10593771	C₂	1	1	yes	maximal, characteristic
2	C₂×#63769	C₂	1	1	yes	maximal, characteristic
3	#10593762	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#157	—	2	1	no	—
5	#64283	—	2	1	no	—
6	C₂×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	#63773	C₄	1	2	yes	complemented
8	#63771	C₄	1	2	yes	complemented
9	#63770	—	2	1	no	—
10	#63769	—	2	1	no	—
11	#63769	C₄	1	2	yes	—
12	#63765	—	2	1	no	—
13	#63764	—	2	1	no	—
14	C₂²×#157	—	4	1	no	—
15	C₂×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
16	#3792	—	4	1	no	—
17	C₄⋊#157	—	2	2	no	—
18	C₄⋊#157	—	4	1	no	—
19	#3788	—	2	2	no	—
20	#3787	—	4	1	no	—
21	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
22	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
23	C₈×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
24	C₈×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
25	C₈×C₃⋊D₃	D₄	1	1	yes	characteristic
26	C₈×C₃⋊D₃	D₄	1	1	yes	characteristic
27	#3708	—	2	2	no	—
28	#1387	—	9	1	no	maximal, Sylow
29	C₂×D₃×C₈	—	12	1	no	—
30	C₂²×C₃⋊D₃	SD₁₆	1	1	yes	characteristic
31	C₂×#157	—	4	2	no	—
32	C₂×#157	—	8	2	no	—
33	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₃⋊Dic₃	SD₁₆	1	1	yes	characteristic
35	C₄×C₃⋊D₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₄×C₃⋊D₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₄×C₃⋊D₃	D₈	1	1	yes	characteristic
38	C₄×C₃⋊D₃	Q₁₆	1	1	yes	characteristic
39	#622	—	4	2	no	—
40	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₃⋊(C₃⋊C₈)	C₄²	1	1	yes	characteristic
43	C₃⋊(C₃⋊C₈)	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×M₃₂	—	9	1	no	—
45	C₂×#108	—	9	1	no	—
46	C₂×#107	—	9	1	no	—
47	C₂×C₄×D₃	—	12	1	no	—
48	C₂×C₃×C₈	—	4	1	no	—
49	C₂×C₃⋊C₈	—	12	1	no	—
50	D₃×C₈	—	12	1	no	—
51	D₃×C₈	—	12	1	no	—
52	D₃×C₈	—	12	1	no	—
53	D₃×C₈	—	12	1	no	—
54	C₂²×C₃²	#106	1	1	yes	characteristic
55	C₂×C₃⋊D₃	—	2	1	no	—
56	C₂×C₃⋊D₃	—	2	1	no	—
57	C₂×C₃⋊D₃	#102	1	1	yes	characteristic
58	C₂×C₃⋊D₃	#103	1	1	yes	characteristic
59	#157	—	8	2	no	—
60	C₃²×C₄	#95	1	1	yes	characteristic
61	C₃²×C₄	#107	1	1	yes	characteristic
62	C₃⋊Dic₃	#102	1	1	yes	characteristic
63	C₃⋊Dic₃	#103	1	1	yes	characteristic
64	C₂×M₁₆	—	18	1	no	—
65	C₂²×C₈	—	9	1	no	—
66	C₂×C₄⋊C₄	—	18	1	no	—
67	M₃₂	—	9	2	no	—
68	M₃₂	—	18	1	no	—
69	C₂×C₁₆	—	18	1	no	—
70	#108	—	9	2	no	—
71	#108	—	18	1	no	—
72	#107	—	9	2	no	—
73	#107	—	18	1	no	—
74	C₂²×D₃	—	12	1	no	—
75	C₂×C₃×C₄	—	4	1	no	—
76	C₂×Dic₃	—	12	1	no	—
77	C₄×D₃	—	12	1	no	—
78	C₄×D₃	—	12	1	no	—
79	C₄×D₃	—	12	1	no	—
80	C₄×D₃	—	12	1	no	—
81	C₃×C₈	—	4	1	no	—
82	C₃×C₈	—	4	1	no	—
83	C₃⋊C₈	—	12	1	no	—
84	C₃⋊C₈	—	12	1	no	—
85	C₂×C₃²	—	2	1	no	—
86	C₂×C₃²	#340	1	1	yes	characteristic
87	C₃⋊D₃	—	2	1	no	—
88	C₃⋊D₃	#361	1	1	yes	characteristic
89	C₃⋊D₃	#362	1	1	yes	characteristic
90	C₂²×C₄	—	9	1	no	—
91	C₂²×C₄	—	36	1	no	—
92	M₁₆	—	18	2	no	—
93	M₁₆	—	36	1	no	—
94	C₂×C₈	—	9	1	no	—
95	C₂×C₈	—	9	1	no	—
96	C₂×C₈	—	9	1	no	—
97	C₂×C₈	—	9	1	no	—
98	C₂×C₈	—	9	1	no	—
99	C₂×C₈	—	36	1	no	—
100	C₂×C₈	PSU(3,𝔽₂)	1	1	yes	characteristic
101	C₄⋊C₄	—	18	2	no	—
102	C₄⋊C₄	—	36	1	no	—
103	C₁₆	—	18	2	no	—
104	C₂²×C₃	—	4	1	no	—
105	C₂×D₃	—	12	1	no	—
106	C₂×D₃	—	12	1	no	—
107	C₂×D₃	—	12	2	no	—
108	C₂×D₃	—	12	2	no	—
109	C₃×C₄	—	4	1	no	—
110	C₃×C₄	—	4	1	no	—
111	Dic₃	—	12	1	no	—
112	Dic₃	—	12	1	no	—
113	C₃²	#1387	1	1	yes	characteristic, complemented, Sylow
114	C₂³	—	9	1	no	—
115	C₂×C₄	—	9	1	no	—
116	C₂×C₄	—	9	1	no	—
117	C₂×C₄	—	9	1	no	—
118	C₂×C₄	—	9	1	no	—
119	C₂×C₄	—	9	1	no	—
120	C₂×C₄	—	36	2	no	—
121	C₂×C₄	—	72	2	no	—
122	C₂×C₄	#3777	1	1	yes	characteristic
123	C₈	—	9	1	no	—
124	C₈	—	9	1	no	—
125	C₈	—	36	2	no	—
126	C₈	#3777	1	1	yes	characteristic
127	C₈	#3777	1	1	yes	characteristic
128	C₂×C₃	—	4	2	no	—
129	C₂×C₃	—	4	1	no	—
130	D₃	—	12	1	no	—
131	D₃	—	12	1	no	—
132	D₃	—	12	2	no	—
133	C₂²	—	9	1	no	—
134	C₂²	—	9	1	no	—
135	C₂²	—	18	1	no	—
136	C₂²	—	18	1	no	—
137	C₂²	#63745	1	1	yes	characteristic
138	C₄	—	9	1	no	—
139	C₄	—	9	1	no	—
140	C₄	—	72	2	no	—
141	C₄	#63746	1	1	yes	characteristic
142	C₄	#63747	1	1	yes	characteristic
143	C₃	—	4	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	9	1	no	—
146	C₂	—	9	1	no	—
147	C₂	—	18	1	no	—
148	C₂	#10590093	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593771

C₂

yes

maximal, characteristic

C₂×#63769

C₂

yes

maximal, characteristic

#10593762

C₂

yes

maximal, characteristic

C₂×C₄⋊#157

—

#64283

—

C₂×C₈×C₃⋊D₃

C₂²

yes

characteristic

#63773

C₄

yes

complemented

#63771

C₄

yes

—

—

C₄

yes

—

#63765

—

#63764

—

C₂²×#157

—

C₂×C₄×C₃⋊D₃

D₄

yes

—

—

—

—

—

C₂×C₃²×C₈

Q₈

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

characteristic

C₈×C₃⋊D₃

C₂×C₄

yes

characteristic

C₈×C₃⋊D₃

C₂×C₄

yes

characteristic

C₈×C₃⋊D₃

D₄

yes

characteristic

C₈×C₃⋊D₃

D₄

yes

—

—

C₂×D₃×C₈

—

C₂²×C₃⋊D₃

SD₁₆

yes

characteristic

C₂×#157

—

C₂×#157

—

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

SD₁₆

yes

characteristic

C₄×C₃⋊D₃

C₂²⋊C₄

yes

characteristic

C₄×C₃⋊D₃

C₂²⋊C₄

yes

characteristic

C₄×C₃⋊D₃

D₈

yes

characteristic

C₄×C₃⋊D₃

Q₁₆

yes

characteristic

#622

—

C₃²×C₈

C₄⋊C₄

yes

characteristic

C₃²×C₈

C₄⋊C₄

yes

characteristic

C₃⋊(C₃⋊C₈)

C₄²

yes

characteristic

C₃⋊(C₃⋊C₈)

C₂²⋊C₄

yes

characteristic

C₂×M₃₂

—

C₂×#108

—

C₂×#107

—

C₂×C₄×D₃

—

C₂×C₃×C₈

—

C₂×C₃⋊C₈

—

D₃×C₈

—

D₃×C₈

—

D₃×C₈

—

D₃×C₈

—

C₂²×C₃²

#106

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

yes

C₂×C₃⋊D₃

yes

—

C₃²×C₄

yes

C₃²×C₄

yes

C₃⋊Dic₃

yes

C₃⋊Dic₃

yes

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#340

yes

characteristic

C₃⋊D₃

—

C₃⋊D₃

yes

C₃⋊D₃

yes

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

PSU(3,𝔽₂)

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₁₆

—

104

C₂²×C₃

—

105

C₂×D₃

—

106

C₂×D₃

—

107

C₂×D₃

—

108

C₂×D₃

—

109

C₃×C₄

—

110

C₃×C₄

—

111

Dic₃

—

112

Dic₃

—

113

C₃²

#1387

yes

characteristic, complemented, Sylow

114

C₂³

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#3777

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

#3777

yes

characteristic

127

C₈

#3777

yes

characteristic

128

C₂×C₃

—

129

C₂×C₃

—

130

D₃

—

131

D₃

—

132

D₃

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

#63745

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#63746

yes

characteristic

142

C₄

#63747

yes

characteristic

143

C₃

—

144

C₂

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#10590093

yes

characteristic

	subgroup	G/H
center	C₂	#10590093
commutator	C₃⋊(C₃⋊C₈)	C₄²
Frattini, hypercenter	C₂×C₈	PSU(3,𝔽₂)
Fitting	C₂×C₃²×C₈	Q₈
socle	C₂×C₃²	#340
prime radical	C₂²×C₃⋊D₃	SD₁₆
cosocle	C₂×C₈×C₃⋊D₃	C₂²
hypocenter	C₃²	#1387
LCS	G ▷ C₃⋊(C₃⋊C₈) ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center

C₂

#10590093

commutator

C₃⋊(C₃⋊C₈)

C₄²

Frattini, hypercenter

C₂×C₈

PSU(3,𝔽₂)

Fitting

C₂×C₃²×C₈

Q₈

socle

C₂×C₃²

#340

prime radical

C₂²×C₃⋊D₃

SD₁₆

cosocle

C₂×C₈×C₃⋊D₃

C₂²

hypocenter

C₃²

#1387

LCS

G ▷ C₃⋊(C₃⋊C₈) ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
7. #63773	C₄	semidirect
8. #63771	C₄	semidirect
113. C₃²	#1387	semidirect

H ≅ G/N

type

7. #63773

C₄

semidirect

8. #63771

C₄

semidirect

113. C₃²

#1387

semidirect

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

subgroup	index
#10593771	2
C₂×#63769	2
#10593762	2
#1387	9

subgroup

index

Subgroups — #11772502

Conjugacy classes of subgroups

Complements

Maximal subgroups