Group #11772503 — subgroups

Subgroups — #11772503

Order 1152.

182 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 48 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×#64218	C₂	1	1	yes	maximal, characteristic
2	C₂×#63748	C₂	1	1	yes	maximal, characteristic
3	C₂×#63743	C₂	1	1	yes	maximal, characteristic
4	C₂×#3792	—	2	1	no	—
5	C₂²×#622	C₂²	1	1	yes	characteristic
6	#64218	—	2	1	no	—
7	#64218	C₄	1	2	yes	complemented
8	C₂×#3771	—	2	1	no	—
9	C₂×#3724	—	2	1	no	—
10	#63748	—	2	1	no	—
11	#63748	C₄	1	2	yes	complemented
12	#63743	—	2	1	no	—
13	#63743	C₄	1	2	yes	—
14	C₂²×C₃⋊Dic₃	D₄	1	1	yes	characteristic
15	C₂²×C₃×Dic₃	—	4	1	no	—
16	#3792	—	2	2	no	—
17	#3792	—	4	1	no	—
18	C₂×#622	—	4	1	no	—
19	C₂×#622	C₂×C₄	1	1	yes	characteristic
20	C₂×#622	C₂×C₄	1	1	yes	characteristic
21	C₂×#622	C₂×C₄	1	1	yes	characteristic
22	C₂×#622	D₄	1	1	yes	characteristic
23	C₂×#622	D₄	1	1	yes	characteristic
24	C₂×#622	Q₈	1	1	yes	characteristic
25	#3771	—	2	2	no	—
26	#3724	—	2	2	no	—
27	#3724	—	4	1	no	—
28	#1387	—	9	1	no	maximal, Sylow
29	C₂×Dic₃⋊C₄	—	12	1	no	—
30	C₂³×C₃²	SD₁₆	1	1	yes	characteristic
31	C₂×C₃⋊Dic₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₃⋊Dic₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₃⋊Dic₃	D₈	1	1	yes	characteristic
35	C₂×C₃⋊Dic₃	SD₁₆	1	1	yes	characteristic
36	C₂×C₃⋊Dic₃	Q₁₆	1	1	yes	characteristic
37	C₂×C₃×Dic₃	—	4	2	no	—
38	C₂×C₃×Dic₃	—	8	2	no	—
39	#622	—	4	2	no	—
40	#622	C₄²	1	1	yes	characteristic
41	#622	C₂²⋊C₄	1	1	yes	characteristic
42	#622	C₄⋊C₄	1	1	yes	characteristic
43	#622	C₄⋊C₄	1	1	yes	characteristic
44	C₂×M₃₂	—	9	1	no	—
45	C₂×#108	—	9	1	no	—
46	C₂×#107	—	9	1	no	—
47	C₂²×C₃×C₄	—	12	1	no	—
48	C₂²×Dic₃	—	4	1	no	—
49	C₂²×Dic₃	—	12	1	no	—
50	Dic₃⋊C₄	—	12	2	no	—
51	Dic₃⋊C₄	—	12	2	no	—
52	C₂²×C₃²	—	2	1	no	—
53	C₂²×C₃²	—	2	1	no	—
54	C₂²×C₃²	#102	1	1	yes	characteristic
55	C₂²×C₃²	#103	1	1	yes	characteristic
56	C₂²×C₃²	#106	1	1	yes	characteristic
57	C₃⋊Dic₃	#95	1	1	yes	characteristic
58	C₃⋊Dic₃	#102	1	1	yes	characteristic
59	C₃⋊Dic₃	#103	1	1	yes	characteristic
60	C₃⋊Dic₃	#107	1	1	yes	characteristic
61	C₃×Dic₃	—	8	2	no	—
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	C₂³×C₃	—	4	1	no	—
73	C₂×C₃×C₄	—	12	2	no	—
74	C₂×C₃×C₄	—	24	2	no	—
75	C₂×Dic₃	—	4	2	no	—
76	C₂×Dic₃	—	8	2	no	—
77	C₂×Dic₃	—	12	1	no	—
78	C₂×Dic₃	—	12	1	no	—
79	C₂×Dic₃	—	12	1	no	—
80	C₂×Dic₃	—	12	1	no	—
81	C₂×Dic₃	—	12	1	no	—
82	C₂×Dic₃	—	12	1	no	—
83	C₂×C₃²	—	2	1	no	—
84	C₂×C₃²	—	2	1	no	—
85	C₂×C₃²	#340	1	1	yes	characteristic
86	C₂×C₃²	#361	1	1	yes	characteristic
87	C₂×C₃²	#362	1	1	yes	characteristic
88	C₂²×C₄	—	9	1	no	—
89	C₂²×C₄	—	12	1	no	—
90	M₁₆	—	18	2	no	—
91	M₁₆	—	36	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	—	9	1	no	—
97	C₂×C₈	—	9	1	no	—
98	C₂×C₈	—	36	1	no	—
99	C₄⋊C₄	—	18	2	no	—
100	C₄⋊C₄	—	36	1	no	—
101	C₁₆	—	18	2	no	—
102	C₂²×C₃	—	4	2	no	—
103	C₂²×C₃	—	4	2	no	—
104	C₂²×C₃	—	4	1	no	—
105	C₂²×C₃	—	4	1	no	—
106	C₂²×C₃	—	4	1	no	—
107	C₃×C₄	—	24	2	no	—
108	Dic₃	—	8	2	no	—
109	Dic₃	—	12	1	no	—
110	Dic₃	—	12	1	no	—
111	Dic₃	—	12	1	no	—
112	Dic₃	—	12	1	no	—
113	C₃²	#1387	1	1	yes	characteristic, complemented, Sylow
114	C₂³	AΓL(1,𝔽₉)	1	1	yes	characteristic
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	—	9	1	no	—
121	C₂×C₄	—	12	2	no	—
122	C₂×C₄	—	24	2	no	—
123	C₈	—	9	1	no	—
124	C₈	—	9	1	no	—
125	C₈	—	9	1	no	—
126	C₈	—	9	1	no	—
127	C₈	—	36	2	no	—
128	C₂×C₃	—	4	2	no	—
129	C₂×C₃	—	4	2	no	—
130	C₂×C₃	—	4	1	no	—
131	C₂×C₃	—	4	1	no	—
132	C₂×C₃	—	4	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	GO⁺(4,𝔽₂)⋊C₄	1	1	yes	characteristic
136	C₂²	PSU(3,𝔽₂)⋊C₄	1	1	yes	characteristic
137	C₂²	AGL(1,𝔽₉)⋊C₄	1	1	yes	characteristic
138	C₄	—	9	1	no	—
139	C₄	—	9	1	no	—
140	C₄	—	9	1	no	—
141	C₄	—	9	1	no	—
142	C₄	—	24	2	no	—
143	C₃	—	4	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#10593166	1	1	yes	characteristic
147	C₂	#10593168	1	1	yes	characteristic
148	C₂	#10593169	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64218

C₂

yes

maximal, characteristic

C₂×#63748

C₂

yes

maximal, characteristic

C₂×#63743

C₂

yes

maximal, characteristic

C₂×#3792

—

C₂²×#622

C₂²

yes

characteristic

#64218

—

#64218

C₄

yes

—

—

—

C₄

yes

complemented

#63743

—

#63743

C₄

yes

—

C₂²×C₃⋊Dic₃

D₄

yes

characteristic

C₂²×C₃×Dic₃

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#622

C₂×C₄

yes

characteristic

C₂×#622

C₂×C₄

yes

characteristic

C₂×#622

D₄

yes

characteristic

C₂×#622

D₄

yes

characteristic

C₂×#622

Q₈

yes

—

—

—

—

C₂×Dic₃⋊C₄

—

C₂³×C₃²

SD₁₆

yes

characteristic

C₂×C₃⋊Dic₃

C₂²⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

C₂²⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

C₄⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

D₈

yes

characteristic

C₂×C₃⋊Dic₃

SD₁₆

yes

characteristic

C₂×C₃⋊Dic₃

Q₁₆

yes

characteristic

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

#622

—

#622

C₄²

yes

characteristic

#622

C₂²⋊C₄

yes

characteristic

#622

C₄⋊C₄

yes

characteristic

#622

C₄⋊C₄

yes

characteristic

C₂×M₃₂

—

C₂×#108

—

C₂×#107

—

C₂²×C₃×C₄

—

C₂²×Dic₃

—

C₂²×Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

yes

C₂²×C₃²

yes

C₂²×C₃²

yes

C₃⋊Dic₃

yes

C₃⋊Dic₃

yes

C₃⋊Dic₃

yes

C₃⋊Dic₃

yes

C₃×Dic₃

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

—

C₂³×C₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

yes

C₂×C₃²

yes

C₂×C₃²

yes

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₁₆

—

102

C₂²×C₃

—

103

C₂²×C₃

—

104

C₂²×C₃

—

105

C₂²×C₃

—

106

C₂²×C₃

—

107

C₃×C₄

—

108

Dic₃

—

109

Dic₃

—

110

Dic₃

—

111

Dic₃

—

112

Dic₃

—

113

C₃²

#1387

yes

characteristic, complemented, Sylow

114

C₂³

AΓL(1,𝔽₉)

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₂×C₃

—

129

C₂×C₃

—

130

C₂×C₃

—

131

C₂×C₃

—

132

C₂×C₃

—

133

C₂²

—

134

C₂²

—

135

C₂²

GO⁺(4,𝔽₂)⋊C₄

yes

characteristic

136

C₂²

PSU(3,𝔽₂)⋊C₄

yes

characteristic

137

C₂²

AGL(1,𝔽₉)⋊C₄

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₃

—

144

C₂

—

145

C₂

—

146

C₂

#10593166

yes

characteristic

147

C₂

#10593168

yes

characteristic

148

C₂

#10593169

yes

characteristic

	subgroup	G/H
center	C₂²	GO⁺(4,𝔽₂)⋊C₄
commutator	#622	C₄²
Frattini, hypercenter	C₂³	AΓL(1,𝔽₉)
Fitting, prime radical	C₂³×C₃²	SD₁₆
socle	C₂²×C₃²	#102
cosocle	C₂²×#622	C₂²
hypocenter	C₃²	#1387
LCS	G ▷ #622 ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂³

subgroup

G/H

center

C₂²

GO⁺(4,𝔽₂)⋊C₄

commutator

#622

C₄²

Frattini, hypercenter

C₂³

AΓL(1,𝔽₉)

Fitting, prime radical

C₂³×C₃²

SD₁₆

socle

C₂²×C₃²

#102

cosocle

C₂²×#622

C₂²

hypocenter

C₃²

#1387

LCS

G ▷ #622 ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₂³

N	H ≅ G/N	type
7. #64218	C₄	semidirect
11. #63748	C₄	semidirect
113. C₃²	#1387	semidirect

H ≅ G/N

type

7. #64218

C₄

semidirect

11. #63748

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
C₂×#64218	2
C₂×#63748	2
C₂×#63743	2
#1387	9

subgroup

index

Subgroups — #11772503

Conjugacy classes of subgroups

Complements

Maximal subgroups