Group #11797378 — subgroups

Subgroups — #11797378

Order 1152.

192 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 68 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	#10588388	C₂	1	1	yes	maximal, characteristic
2	C₉×#347	C₂	1	1	yes	maximal, characteristic, complemented
3	#10588191	C₂	1	2	yes	maximal
4	#10588188	C₂	1	2	yes	maximal, complemented
5	#10588143	C₂	1	1	yes	maximal, characteristic, complemented
6	#69088	—	3	1	no	maximal
7	C₂×#3663	C₄	1	1	yes	characteristic
8	C₂×#3663	C₂²	1	1	yes	characteristic
9	C₄²×D₉	C₄	1	1	yes	characteristic
10	C₉×M₃₂	C₂²	1	2	yes	—
11	C₉×#97	C₂²	1	1	yes	characteristic
12	C₈×Dic₉	C₄	1	1	yes	characteristic
13	C₈×Dic₉	C₂²	1	1	yes	characteristic
14	#63371	C₂²	1	2	yes	—
15	#63362	C₄	1	1	yes	characteristic
16	#4707	—	3	1	no	—
17	C₃×#347	D₃	1	1	yes	characteristic
18	#4514	—	3	2	no	—
19	#4511	—	3	2	no	—
20	#4466	—	3	1	no	—
21	C₂×C₄×D₉	—	2	1	no	—
22	C₂×C₄×D₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
25	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₉	—	2	1	no	—
27	C₄×Dic₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₉⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₉⋊C₈	C₂×C₄	1	1	yes	characteristic
30	#3663	—	2	2	no	—
31	#3663	—	2	2	no	—
32	C₉×C₁₆	—	2	2	no	—
33	C₉⋊C₁₆	—	2	2	no	—
34	#2165	—	9	1	no	Sylow
35	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
36	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
37	C₄²×D₃	—	3	1	no	—
38	C₃×M₃₂	C₂×D₃	1	2	yes	—
39	C₃×#97	C₂×D₃	1	1	yes	characteristic
40	C₈×Dic₃	—	3	1	no	—
41	C₈×Dic₃	—	3	1	no	—
42	#812	—	3	2	no	—
43	#803	—	3	1	no	—
44	C₂²×D₉	M₁₆	1	1	yes	characteristic
45	C₂×C₄×C₉	—	2	1	no	—
46	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₉	—	2	1	no	—
48	C₂×Dic₉	M₁₆	1	1	yes	characteristic
49	C₄×D₉	—	2	1	no	—
50	C₄×D₉	—	2	1	no	—
51	C₄×D₉	—	4	2	no	—
52	C₈×C₉	—	2	1	no	—
53	C₈×C₉	#41	1	2	yes	—
54	C₉⋊C₈	—	2	1	no	—
55	C₉⋊C₈	M₁₆	1	2	yes	—
56	C₄×M₁₆	—	9	1	no	—
57	#364	—	9	2	no	—
58	#349	—	9	2	no	—
59	#347	—	9	1	no	—
60	#347	D₉	1	1	yes	characteristic, complemented
61	C₂×C₄×D₃	—	3	1	no	—
62	C₂×C₄×D₃	—	6	1	no	—
63	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
65	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
66	C₄×Dic₃	—	3	1	no	—
67	C₄×Dic₃	—	6	1	no	—
68	C₂×C₃⋊C₈	—	3	1	no	—
69	C₂×C₃⋊C₈	—	3	1	no	—
70	ΓL(1,𝔽₂₅)	—	6	2	no	—
71	ΓL(1,𝔽₂₅)	—	6	2	no	—
72	C₃×C₁₆	—	2	2	no	—
73	C₃⋊C₁₆	—	6	2	no	—
74	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
75	C₂×D₉	—	2	1	no	—
76	C₂×D₉	—	4	1	no	—
77	C₄×C₉	—	4	1	no	—
78	C₄×C₉	#117	1	1	yes	characteristic
79	C₄×C₉	C₂×M₁₆	1	1	yes	characteristic
80	Dic₉	—	2	2	no	—
81	Dic₉	—	2	1	no	—
82	C₂×M₁₆	—	9	1	no	—
83	C₂×M₁₆	—	9	1	no	—
84	C₂×C₄²	—	9	1	no	—
85	M₃₂	—	9	2	no	—
86	M₃₂	C₂×D₉	1	2	yes	—
87	#97	—	9	1	no	—
88	#97	C₂×D₉	1	1	yes	characteristic
89	C₄×C₈	—	9	1	no	—
90	C₄×C₈	—	9	1	no	—
91	C₂²×D₃	—	3	1	no	—
92	C₂×C₃×C₄	—	2	1	no	—
93	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
94	C₂×Dic₃	—	3	1	no	—
95	C₂×Dic₃	—	6	1	no	—
96	C₄×D₃	—	6	1	no	—
97	C₄×D₃	—	6	1	no	—
98	C₄×D₃	—	12	2	no	—
99	C₃×C₈	—	2	1	no	—
100	C₃×C₈	#235	1	2	yes	—
101	C₃⋊C₈	—	3	2	no	—
102	C₃⋊C₈	—	6	1	no	—
103	C₂×C₉	—	2	1	no	—
104	C₂×C₉	#432	1	1	yes	characteristic
105	D₉	—	4	1	no	—
106	C₂²×C₄	—	9	1	no	—
107	C₂²×C₄	—	18	1	no	—
108	M₁₆	—	18	2	no	—
109	M₁₆	—	18	2	no	—
110	C₂×C₈	—	9	1	no	—
111	C₂×C₈	—	9	1	no	—
112	C₂×C₈	C₄×D₉	1	1	yes	characteristic
113	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
114	C₄²	—	9	1	no	—
115	C₄²	—	18	1	no	—
116	C₄²	C₄×D₉	1	1	yes	characteristic
117	C₁₆	—	2	2	no	—
118	C₁₆	—	18	2	no	—
119	C₂²×C₃	D₃×M₁₆	1	1	yes	characteristic
120	C₂×D₃	—	6	1	no	—
121	C₂×D₃	—	12	1	no	—
122	C₃×C₄	—	4	1	no	—
123	C₃×C₄	#872	1	1	yes	characteristic
124	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
125	Dic₃	—	6	2	no	—
126	Dic₃	—	6	1	no	—
127	C₉	#2165	1	1	yes	characteristic, complemented, Sylow
128	C₂³	—	9	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	9	1	no	—
131	C₂×C₄	—	18	1	no	—
132	C₂×C₄	—	18	1	no	—
133	C₂×C₄	—	18	1	no	—
134	C₂×C₄	—	36	2	no	—
135	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	9	2	no	—
138	C₈	—	18	1	no	—
139	C₈	#3697	1	2	yes	—
140	C₂×C₃	—	2	1	no	—
141	C₂×C₃	#4705	1	1	yes	characteristic
142	D₃	—	12	1	no	—
143	C₂²	—	18	1	no	—
144	C₂²	—	36	1	no	—
145	C₂²	M₁₆×D₉	1	1	yes	characteristic
146	C₄	—	4	1	no	—
147	C₄	—	18	2	no	—
148	C₄	—	18	1	no	—
149	C₄	#63434	1	1	yes	characteristic
150	C₄	M₁₆×D₉	1	1	yes	characteristic
151	C₃	#69088	1	1	yes	characteristic
152	C₂	—	2	1	no	—
153	C₂	—	36	1	no	—
154	C₂	#10588386	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10588388

C₂

yes

maximal, characteristic

C₉×#347

C₂

yes

maximal, characteristic, complemented

#10588191

C₂

yes

maximal

#10588188

C₂

yes

maximal, complemented

#10588143

C₂

yes

maximal, characteristic, complemented

#69088

—

maximal

C₂×#3663

C₄

yes

characteristic

C₂×#3663

C₂²

yes

characteristic

C₄²×D₉

C₄

yes

characteristic

C₉×M₃₂

C₂²

yes

—

C₉×#97

C₂²

yes

characteristic

C₈×Dic₉

C₄

yes

characteristic

C₈×Dic₉

C₂²

yes

characteristic

#63371

C₂²

yes

—

#63362

C₄

yes

characteristic

#4707

—

C₃×#347

D₃

yes

—

—

—

C₂×C₄×D₉

—

C₂×C₄×D₉

C₂×C₄

yes

characteristic

C₂×C₈×C₉

C₂×C₄

yes

characteristic

C₂×C₈×C₉

C₂³

yes

characteristic

C₄²×C₉

C₂×C₄

yes

characteristic

C₄×Dic₉

—

C₄×Dic₉

C₂×C₄

yes

characteristic

C₂×C₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₉⋊C₈

C₂×C₄

yes

characteristic

#3663

—

#3663

—

C₉×C₁₆

—

C₉⋊C₁₆

—

#2165

—

Sylow

C₂×ΓL(1,𝔽₂₅)

—

C₂×ΓL(1,𝔽₂₅)

—

C₄²×D₃

—

C₃×M₃₂

C₂×D₃

yes

—

C₃×#97

C₂×D₃

yes

characteristic

C₈×Dic₃

—

C₈×Dic₃

—

#812

—

#803

—

C₂²×D₉

M₁₆

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂²×C₄

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

M₁₆

yes

characteristic

C₄×D₉

—

C₄×D₉

—

C₄×D₉

—

C₈×C₉

—

C₈×C₉

#41

yes

—

C₉⋊C₈

—

C₉⋊C₈

M₁₆

yes

—

C₄×M₁₆

—

—

—

—

D₉

yes

characteristic, complemented

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₂×C₃×C₈

C₄×D₃

yes

characteristic

C₂×C₃×C₈

C₂²×D₃

yes

characteristic

C₃×C₄²

C₄×D₃

yes

characteristic

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

ΓL(1,𝔽₂₅)

—

ΓL(1,𝔽₂₅)

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₉

C₂×M₁₆

yes

characteristic

C₂×D₉

—

C₂×D₉

—

C₄×C₉

—

C₄×C₉

#117

yes

characteristic

C₄×C₉

C₂×M₁₆

yes

characteristic

Dic₉

—

Dic₉

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×C₄²

—

M₃₂

—

M₃₂

C₂×D₉

yes

—

#97

—

#97

C₂×D₉

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₄×D₃

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

100

C₃×C₈

#235

yes

—

101

C₃⋊C₈

—

102

C₃⋊C₈

—

103

C₂×C₉

—

104

C₂×C₉

#432

yes

characteristic

105

D₉

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₄×D₉

yes

characteristic

113

C₂×C₈

C₂²×D₉

yes

characteristic

114

C₄²

—

115

C₄²

—

116

C₄²

C₄×D₉

yes

characteristic

117

C₁₆

—

118

C₁₆

—

119

C₂²×C₃

D₃×M₁₆

yes

characteristic

120

C₂×D₃

—

121

C₂×D₃

—

122

C₃×C₄

—

123

C₃×C₄

#872

yes

characteristic

124

C₃×C₄

D₃×M₁₆

yes

characteristic

125

Dic₃

—

126

Dic₃

—

127

C₉

#2165

yes

characteristic, complemented, Sylow

128

C₂³

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄×D₉

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

#3697

yes

—

140

C₂×C₃

—

141

C₂×C₃

#4705

yes

characteristic

142

D₃

—

143

C₂²

—

144

C₂²

—

145

C₂²

M₁₆×D₉

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#63434

yes

characteristic

150

C₄

M₁₆×D₉

yes

characteristic

151

C₃

#69088

yes

characteristic

152

C₂

—

153

C₂

—

154

C₂

#10588386

yes

characteristic

	subgroup	G/H
center	C₄	#63434
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₈	C₂²×D₃
Fitting	C₉×#347	C₂
socle	C₂×C₃	#4705
prime radical	C₂²×D₉	M₁₆
cosocle	C₂×C₈×C₉	C₂³
hypercenter	#347	D₉
hypocenter	C₉	#2165
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ #97 ◁ #347

subgroup

G/H

center

C₄

#63434

commutator

C₂×C₄×C₉

C₂²×C₄

Frattini

C₂×C₃×C₈

C₂²×D₃

C₂

C₂×C₃

C₂²×D₉

M₁₆

cosocle

C₂×C₈×C₉

C₂³

D₉

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₄ ◁ #97 ◁ #347

N	H ≅ G/N	type
2. C₉×#347	C₂	semidirect
4. #10588188	C₂	semidirect
5. #10588143	C₂	semidirect
60. #347	D₉	semidirect
127. C₉	#2165	semidirect

H ≅ G/N

type

2. C₉×#347

C₂

semidirect

4. #10588188

C₂

semidirect

5. #10588143

C₂

semidirect

60. #347

D₉

semidirect

127. C₉

#2165

semidirect

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

subgroup	index
#10588388	2
C₉×#347	2
#10588191	2
#10588188	2
#10588143	2
#69088	3

subgroup

index

Subgroups — #11797378

Conjugacy classes of subgroups

Complements

Maximal subgroups