Group #11772027 — subgroups

Subgroups — #11772027

Order 1152.

153 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 153 conjugacy classes of subgroups — too many to draw. Only its 50 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	#10588375	C₂	1	1	yes	maximal, characteristic
2	C₉×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	#10588140	C₂	1	1	yes	maximal, characteristic, complemented
4	#67570	—	3	1	no	maximal
5	C₂×#3664	—	2	1	no	—
6	C₂×#3664	—	2	1	no	—
7	#63436	C₄	1	1	yes	characteristic
8	#63431	C₄	1	1	yes	characteristic
9	C₉×M₃₂	—	2	1	no	—
10	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
11	#63371	—	2	1	no	—
12	#4693	—	3	1	no	—
13	C₃×#364	D₃	1	1	yes	characteristic
14	#4463	—	3	1	no	—
15	C₂×D₃₆	—	2	1	no	—
16	C₂×D₃₆	—	4	1	no	—
17	C₂×Dic₁₈	—	2	1	no	—
18	C₂×C₈×C₉	D₄	1	1	yes	characteristic
19	C₂×C₈×C₉	D₄	1	1	yes	characteristic
20	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
21	C₄⋊Dic₉	—	4	1	no	—
22	#3664	—	4	1	no	—
23	#3664	—	4	1	no	—
24	#3664	—	4	1	no	—
25	#3664	—	4	1	no	—
26	C₉×C₁₆	—	4	1	no	—
27	C₉⋊C₁₆	—	4	1	no	—
28	#1359	—	9	1	no	Sylow
29	C₂×#204	—	6	1	no	—
30	C₂×#204	—	6	1	no	—
31	#874	—	3	1	no	—
32	#869	—	3	1	no	—
33	C₃×M₃₂	—	2	1	no	—
34	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
35	#812	—	6	1	no	—
36	C₂²×D₉	—	4	1	no	—
37	C₂×C₄×C₉	—	2	1	no	—
38	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×Dic₉	—	4	1	no	—
40	D₃₆	—	4	1	no	—
41	D₃₆	—	4	1	no	—
42	D₃₆	—	8	2	no	—
43	Dic₁₈	—	4	1	no	—
44	Dic₁₈	—	4	1	no	—
45	C₈×C₉	D₈	1	1	yes	characteristic
46	C₈×C₉	D₈	1	1	yes	characteristic
47	C₈×C₉	SD₁₆	1	1	yes	characteristic
48	C₈×C₉	SD₁₆	1	1	yes	characteristic
49	#492	—	9	1	no	—
50	#364	—	9	1	no	—
51	#364	D₉	1	1	yes	characteristic, complemented
52	C₂×D₁₂	—	6	1	no	—
53	C₂×D₁₂	—	12	1	no	—
54	C₂×Dic₆	—	6	1	no	—
55	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
56	C₂×C₃×C₈	#67	1	1	yes	characteristic
57	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
58	C₄⋊Dic₃	—	12	1	no	—
59	#204	—	12	1	no	—
60	#204	—	12	1	no	—
61	#204	—	12	1	no	—
62	#204	—	12	1	no	—
63	C₃×C₁₆	—	4	1	no	—
64	C₃⋊C₁₆	—	12	1	no	—
65	C₂²×C₉	#100	1	1	yes	characteristic
66	C₂×D₉	—	8	2	no	—
67	C₂×D₉	—	8	1	no	—
68	C₄×C₉	—	2	2	no	—
69	C₄×C₉	#102	1	1	yes	characteristic
70	C₄×C₉	#102	1	1	yes	characteristic
71	Dic₉	—	8	1	no	—
72	C₂×SD₁₆	—	18	1	no	—
73	C₂×SD₁₆	—	18	1	no	—
74	C₄⋊Q₈	—	9	1	no	—
75	#127	—	9	1	no	—
76	M₃₂	—	2	1	no	—
77	M₃₂	—	18	1	no	—
78	C₄×C₈	C₂×D₉	1	1	yes	characteristic
79	C₂²×D₃	—	12	1	no	—
80	C₂×C₃×C₄	—	2	1	no	—
81	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
82	C₂×Dic₃	—	12	1	no	—
83	D₁₂	—	12	1	no	—
84	D₁₂	—	12	1	no	—
85	D₁₂	—	24	2	no	—
86	Dic₆	—	12	1	no	—
87	Dic₆	—	12	1	no	—
88	C₃×C₈	#204	1	1	yes	characteristic
89	C₃×C₈	D₂₄	1	1	yes	characteristic
90	C₃×C₈	#213	1	1	yes	characteristic
91	C₃×C₈	#215	1	1	yes	characteristic
92	C₂×C₉	—	2	1	no	—
93	C₂×C₉	#331	1	1	yes	characteristic
94	D₉	—	16	1	no	—
95	C₂×Q₈	—	18	1	no	—
96	C₂×D₄	—	18	1	no	—
97	C₂×D₄	—	36	1	no	—
98	SD₁₆	—	36	1	no	—
99	SD₁₆	—	36	1	no	—
100	SD₁₆	—	36	1	no	—
101	SD₁₆	—	36	1	no	—
102	C₂×C₈	D₃₆	1	1	yes	characteristic
103	C₂×C₈	#611	1	1	yes	characteristic
104	C₄⋊C₄	—	36	1	no	—
105	C₄²	C₄×D₉	1	1	yes	characteristic
106	C₁₆	—	4	1	no	—
107	C₁₆	—	36	1	no	—
108	C₂²×C₃	#823	1	1	yes	characteristic
109	C₂×D₃	—	24	2	no	—
110	C₂×D₃	—	24	1	no	—
111	C₃×C₄	—	2	2	no	—
112	C₃×C₄	#809	1	1	yes	characteristic
113	C₃×C₄	#821	1	1	yes	characteristic
114	Dic₃	—	24	1	no	—
115	C₉	#1359	1	1	yes	characteristic, complemented, Sylow
116	C₂³	—	36	1	no	—
117	Q₈	—	36	1	no	—
118	Q₈	—	36	1	no	—
119	D₄	—	36	1	no	—
120	D₄	—	36	1	no	—
121	D₄	—	72	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	36	1	no	—
124	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
125	C₈	#3664	1	1	yes	characteristic
126	C₈	D₇₂	1	1	yes	characteristic
127	C₈	#3673	1	1	yes	characteristic
128	C₈	#3675	1	1	yes	characteristic
129	C₂×C₃	—	2	1	no	—
130	C₂×C₃	#4485	1	1	yes	characteristic
131	D₃	—	48	1	no	—
132	C₂²	—	72	2	no	—
133	C₂²	—	72	1	no	—
134	C₂²	#63383	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	72	1	no	—
137	C₄	#63369	1	1	yes	characteristic
138	C₄	#63380	1	1	yes	characteristic
139	C₃	#67570	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	144	1	no	—
142	C₂	#10588163	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10588375

C₂

yes

maximal, characteristic

C₉×#364

C₂

yes

maximal, characteristic, complemented

#10588140

C₂

yes

maximal, characteristic, complemented

—

—

—

C₄

yes

characteristic

#63431

C₄

yes

characteristic

C₉×M₃₂

—

C₄×C₈×C₉

C₂²

yes

—

—

D₃

yes

characteristic

#4463

—

C₂×D₃₆

—

C₂×D₃₆

—

C₂×Dic₁₈

—

C₂×C₈×C₉

D₄

yes

characteristic

C₂×C₈×C₉

D₄

yes

characteristic

C₄²×C₉

C₂×C₄

yes

characteristic

C₄⋊Dic₉

—

—

—

—

—

C₉×C₁₆

—

C₉⋊C₁₆

—

—

—

—

—

—

C₃×M₃₂

—

C₃×C₄×C₈

C₂×D₃

yes

characteristic

#812

—

C₂²×D₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₉

—

D₃₆

—

D₃₆

—

D₃₆

—

Dic₁₈

—

Dic₁₈

—

C₈×C₉

D₈

yes

characteristic

C₈×C₉

D₈

yes

characteristic

C₈×C₉

SD₁₆

yes

characteristic

C₈×C₉

SD₁₆

yes

—

—

D₉

yes

characteristic, complemented

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×C₃×C₈

D₁₂

yes

characteristic

C₂×C₃×C₈

#67

yes

characteristic

C₃×C₄²

C₄×D₃

yes

characteristic

C₄⋊Dic₃

—

—

—

—

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₉

#100

yes

characteristic

C₂×D₉

—

C₂×D₉

—

C₄×C₉

—

C₄×C₉

yes

C₄×C₉

yes

Dic₉

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

#127

—

M₃₂

—

M₃₂

—

C₄×C₈

C₂×D₉

yes

characteristic

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

characteristic

C₂×Dic₃

—

D₁₂

—

D₁₂

—

D₁₂

—

Dic₆

—

Dic₆

—

C₃×C₈

#204

yes

characteristic

C₃×C₈

D₂₄

yes

C₃×C₈

yes

C₃×C₈

yes

C₂×C₉

—

C₂×C₉

#331

yes

characteristic

D₉

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

100

SD₁₆

—

101

SD₁₆

—

102

C₂×C₈

D₃₆

yes

characteristic

103

C₂×C₈

#611

yes

characteristic

104

C₄⋊C₄

—

105

C₄²

C₄×D₉

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₂²×C₃

#823

yes

characteristic

109

C₂×D₃

—

110

C₂×D₃

—

111

C₃×C₄

—

112

C₃×C₄

#809

yes

characteristic

113

C₃×C₄

#821

yes

characteristic

114

Dic₃

—

115

C₉

#1359

yes

characteristic, complemented, Sylow

116

C₂³

—

117

Q₈

—

118

Q₈

—

119

D₄

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

(C₂×D₉)⋊C₄

yes

characteristic

125

C₈

#3664

yes

characteristic

126

C₈

D₇₂

yes

characteristic

127

C₈

#3673

yes

characteristic

128

C₈

#3675

yes

characteristic

129

C₂×C₃

—

130

C₂×C₃

#4485

yes

characteristic

131

D₃

—

132

C₂²

—

133

C₂²

—

134

C₂²

#63383

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

#63369

yes

characteristic

138

C₄

#63380

yes

characteristic

139

C₃

#67570

yes

characteristic

140

C₂

—

141

C₂

—

144

—

142

C₂

#10588163

yes

characteristic

	subgroup	G/H
center	C₂	#10588163
commutator	C₄²×C₉	C₂×C₄
Frattini	C₃×C₄×C₈	C₂×D₃
Fitting	C₉×#364	C₂
socle	C₂×C₃	#4485
prime radical	#63436	C₄
cosocle	C₄×C₈×C₉	C₂²
hypercenter	#364	D₉
hypocenter	C₉	#1359
LCS	G ▷ C₄²×C₉ ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#10588163

commutator

C₄²×C₉

C₂×C₄

Frattini

C₃×C₄×C₈

C₂×D₃

C₂

C₂×C₃

C₄

C₄×C₈×C₉

C₂²

D₉

C₉

G ▷ C₄²×C₉ ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
2. C₉×#364	C₂	semidirect
3. #10588140	C₂	semidirect
51. #364	D₉	semidirect
115. C₉	#1359	semidirect

H ≅ G/N

type

2. C₉×#364

C₂

semidirect

3. #10588140

C₂

semidirect

51. #364

D₉

semidirect

115. C₉

#1359

semidirect

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

subgroup	index
#10588375	2
C₉×#364	2
#10588140	2
#67570	3

subgroup

index

Subgroups — #11772027

Conjugacy classes of subgroups

Complements

Maximal subgroups