Group #11772504 — subgroups

Subgroups — #11772504

Order 1152.

170 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 46 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	#10596481	C₂	1	1	yes	maximal, characteristic
2	C₂×#63746	C₂	1	1	yes	maximal, characteristic
3	#10593331	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#157	—	2	1	no	—
5	C₂×#3787	C₂²	1	1	yes	characteristic
6	#64218	C₄	1	2	yes	complemented
7	#64214	—	2	1	no	—
8	#64213	—	2	1	no	—
9	#63818	—	2	1	no	—
10	#63746	—	2	1	no	—
11	#63746	C₄	1	2	yes	—
12	#63733	—	2	1	no	—
13	#63733	C₄	1	2	yes	complemented
14	C₂²×#157	—	4	1	no	—
15	C₂×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
16	C₂×#622	C₂×C₄	1	1	yes	characteristic
17	C₂×#622	Q₈	1	1	yes	characteristic
18	C₄⋊#157	—	2	2	no	—
19	C₄⋊#157	—	4	1	no	—
20	#3787	C₂×C₄	1	1	yes	characteristic
21	#3787	C₂×C₄	1	1	yes	characteristic
22	#3787	D₄	1	1	yes	characteristic
23	#3787	D₄	1	1	yes	characteristic
24	#3771	—	2	2	no	—
25	C₃×#208	—	4	1	no	—
26	#3712	—	2	2	no	—
27	#3710	—	4	1	no	—
28	#1387	—	9	1	no	maximal, Sylow
29	D₃×M₁₆	—	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	SD₁₆	1	1	yes	characteristic
34	C₂×C₃⋊Dic₃	C₄⋊C₄	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	C₄²	1	1	yes	characteristic
40	#622	C₂²⋊C₄	1	1	yes	characteristic
41	#622	C₄⋊C₄	1	1	yes	characteristic
42	#622	C₄⋊C₄	1	1	yes	characteristic
43	C₃×C₃⋊C₈	—	4	2	no	—
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	C₃×M₁₆	—	12	1	no	—
49	#208	—	4	1	no	—
50	ΓL(1,𝔽₂₅)	—	12	2	no	—
51	D₃×C₈	—	12	2	no	—
52	C₂²×C₃²	#106	1	1	yes	characteristic
53	C₂×C₃⋊D₃	—	2	1	no	—
54	C₂×C₃⋊D₃	—	2	1	no	—
55	C₂×C₃⋊D₃	#102	1	1	yes	characteristic
56	C₂×C₃⋊D₃	#103	1	1	yes	characteristic
57	#157	—	8	2	no	—
58	C₃²×C₄	#102	1	1	yes	characteristic
59	C₃²×C₄	#103	1	1	yes	characteristic
60	C₃⋊Dic₃	#95	1	1	yes	characteristic
61	C₃⋊Dic₃	#107	1	1	yes	characteristic
62	C₂×M₁₆	—	18	1	no	—
63	C₂²×C₈	—	9	1	no	—
64	C₂×C₄⋊C₄	—	18	1	no	—
65	M₃₂	—	9	2	no	—
66	M₃₂	—	18	1	no	—
67	C₂×C₁₆	—	18	1	no	—
68	#108	—	9	2	no	—
69	#108	—	18	1	no	—
70	#107	—	9	2	no	—
71	#107	—	18	1	no	—
72	C₂²×D₃	—	12	1	no	—
73	C₂×C₃×C₄	—	4	1	no	—
74	C₂×Dic₃	—	12	1	no	—
75	C₄×D₃	—	12	1	no	—
76	C₄×D₃	—	12	1	no	—
77	C₄×D₃	—	12	1	no	—
78	C₄×D₃	—	12	1	no	—
79	C₃×C₈	—	12	2	no	—
80	C₃⋊C₈	—	4	2	no	—
81	C₂×C₃²	—	2	1	no	—
82	C₂×C₃²	#340	1	1	yes	characteristic
83	C₃⋊D₃	—	2	1	no	—
84	C₃⋊D₃	#361	1	1	yes	characteristic
85	C₃⋊D₃	#362	1	1	yes	characteristic
86	C₂²×C₄	—	9	1	no	—
87	C₂²×C₄	—	36	1	no	—
88	M₁₆	—	12	1	no	—
89	M₁₆	—	18	2	no	—
90	C₂×C₈	—	9	1	no	—
91	C₂×C₈	—	9	1	no	—
92	C₂×C₈	—	9	1	no	—
93	C₂×C₈	—	9	1	no	—
94	C₂×C₈	—	9	1	no	—
95	C₂×C₈	—	9	1	no	—
96	C₂×C₈	—	36	1	no	—
97	C₄⋊C₄	—	18	2	no	—
98	C₄⋊C₄	—	36	1	no	—
99	C₁₆	—	18	2	no	—
100	C₂²×C₃	—	4	1	no	—
101	C₂×D₃	—	12	1	no	—
102	C₂×D₃	—	12	1	no	—
103	C₂×D₃	—	24	1	no	—
104	C₂×D₃	—	24	1	no	—
105	C₃×C₄	—	4	1	no	—
106	C₃×C₄	—	4	1	no	—
107	Dic₃	—	12	1	no	—
108	Dic₃	—	12	1	no	—
109	C₃²	#1387	1	1	yes	characteristic, complemented, Sylow
110	C₂³	—	9	1	no	—
111	C₂×C₄	—	9	1	no	—
112	C₂×C₄	—	9	1	no	—
113	C₂×C₄	—	9	1	no	—
114	C₂×C₄	—	9	1	no	—
115	C₂×C₄	—	9	1	no	—
116	C₂×C₄	—	36	2	no	—
117	C₂×C₄	—	72	2	no	—
118	C₂×C₄	AΓL(1,𝔽₉)	1	1	yes	characteristic
119	C₈	—	9	1	no	—
120	C₈	—	9	1	no	—
121	C₈	—	9	1	no	—
122	C₈	—	9	1	no	—
123	C₈	—	12	2	no	—
124	C₂×C₃	—	4	1	no	—
125	C₂×C₃	—	8	1	no	—
126	D₃	—	12	1	no	—
127	D₃	—	12	1	no	—
128	D₃	—	24	1	no	—
129	C₂²	—	9	1	no	—
130	C₂²	—	9	1	no	—
131	C₂²	—	18	1	no	—
132	C₂²	—	18	1	no	—
133	C₂²	AGL(1,𝔽₉)⋊C₄	1	1	yes	characteristic
134	C₄	—	9	1	no	—
135	C₄	—	9	1	no	—
136	C₄	—	72	2	no	—
137	C₄	GO⁺(4,𝔽₂)⋊C₄	1	1	yes	characteristic
138	C₄	PSU(3,𝔽₂)⋊C₄	1	1	yes	characteristic
139	C₃	—	4	1	no	—
140	C₂	—	2	1	no	—
141	C₂	—	9	1	no	—
142	C₂	—	9	1	no	—
143	C₂	—	18	1	no	—
144	C₂	#10593166	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10596481

C₂

yes

maximal, characteristic

C₂×#63746

C₂

yes

maximal, characteristic

#10593331

C₂

yes

maximal, characteristic

C₂×C₄⋊#157

—

C₂×#3787

C₂²

yes

characteristic

#64218

C₄

yes

—

—

—

—

C₄

yes

—

#63733

—

#63733

C₄

yes

complemented

C₂²×#157

—

C₂×C₄×C₃⋊D₃

D₄

yes

characteristic

C₂×#622

C₂×C₄

yes

characteristic

C₂×#622

Q₈

yes

—

—

C₂×C₄

yes

characteristic

#3787

C₂×C₄

yes

characteristic

#3787

D₄

yes

characteristic

#3787

D₄

yes

—

—

—

—

—

D₃×M₁₆

—

C₂²×C₃⋊D₃

SD₁₆

yes

characteristic

C₂×#157

—

C₂×#157

—

C₂×C₃²×C₄

SD₁₆

yes

characteristic

C₂×C₃⋊Dic₃

C₄⋊C₄

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

C₄²

yes

characteristic

#622

C₂²⋊C₄

yes

characteristic

#622

C₄⋊C₄

yes

characteristic

#622

C₄⋊C₄

yes

characteristic

C₃×C₃⋊C₈

—

C₂×M₃₂

—

C₂×#108

—

C₂×#107

—

C₂×C₄×D₃

—

C₃×M₁₆

—

#208

—

ΓL(1,𝔽₂₅)

—

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

—

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

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₁₆

—

100

C₂²×C₃

—

101

C₂×D₃

—

102

C₂×D₃

—

103

C₂×D₃

—

104

C₂×D₃

—

105

C₃×C₄

—

106

C₃×C₄

—

107

Dic₃

—

108

Dic₃

—

109

C₃²

#1387

yes

characteristic, complemented, Sylow

110

C₂³

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

AΓL(1,𝔽₉)

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₂×C₃

—

125

C₂×C₃

—

126

D₃

—

127

D₃

—

128

D₃

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

AGL(1,𝔽₉)⋊C₄

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

GO⁺(4,𝔽₂)⋊C₄

yes

characteristic

138

C₄

PSU(3,𝔽₂)⋊C₄

yes

characteristic

139

C₃

—

140

C₂

—

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

#10593166

yes

characteristic

	subgroup	G/H
center	C₂	#10593166
commutator	#622	C₄²
Frattini, hypercenter	C₂×C₄	AΓL(1,𝔽₉)
Fitting	C₂×C₃²×C₄	SD₁₆
socle	C₂×C₃²	#340
prime radical	C₂²×C₃⋊D₃	SD₁₆
cosocle	C₂×#3787	C₂²
hypocenter	C₃²	#1387
LCS	G ▷ #622 ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄

subgroup

G/H

C₂

C₄²

Frattini, hypercenter

C₂×C₄

AΓL(1,𝔽₉)

Fitting

C₂×C₃²×C₄

SD₁₆

socle

C₂×C₃²

#340

prime radical

C₂²×C₃⋊D₃

SD₁₆

C₂²

C₃²

G ▷ #622 ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂ ◁ C₂×C₄

N	H ≅ G/N	type
6. #64218	C₄	semidirect
13. #63733	C₄	semidirect
109. C₃²	#1387	semidirect

H ≅ G/N

type

6. #64218

C₄

semidirect

13. #63733

C₄

semidirect

109. C₃²

#1387

semidirect

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

subgroup	index
#10596481	2
C₂×#63746	2
#10593331	2
#1387	9

subgroup

index

Subgroups — #11772504

Conjugacy classes of subgroups

Complements

Maximal subgroups