Group #11782666 — subgroups

Subgroups — #11782666

Order 1152.

330 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 90 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	C₉×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	#10588773	C₂	1	2	yes	maximal, complemented
3	#10588770	C₂	1	2	yes	maximal, complemented
4	#10588377	C₂	1	1	yes	maximal, characteristic
5	#10588132	C₂	1	1	yes	maximal, characteristic, complemented
6	#71336	—	3	1	no	maximal
7	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×#3675	—	2	4	no	—
9	C₂×D₇₂	—	2	2	no	—
10	#63436	C₂²	1	1	yes	characteristic
11	#63436	C₂²	1	1	yes	characteristic
12	C₉×#106	—	2	2	no	—
13	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
14	#63369	—	2	4	no	—
15	C₄×C₉⋊C₈	C₂²	1	2	yes	—
16	C₃×#498	D₃	1	1	yes	characteristic
17	#5088	—	3	2	no	—
18	#5085	—	3	2	no	—
19	#4695	—	3	1	no	—
20	#4455	—	3	1	no	—
21	C₂×Q₈×C₉	—	2	2	no	—
22	C₂×D₃₆	—	2	2	no	—
23	C₂×D₃₆	—	2	2	no	—
24	C₂×D₃₆	—	2	1	no	—
25	C₂×D₃₆	—	2	1	no	—
26	C₂×C₈×C₉	D₄	1	2	yes	—
27	C₉×C₄⋊C₄	—	2	2	no	—
28	C₉×C₄⋊C₄	—	2	2	no	—
29	C₄²×C₉	C₂³	1	1	yes	characteristic
30	#3675	—	4	8	no	—
31	C₂×C₉⋊C₈	D₄	1	4	yes	—
32	D₇₂	—	4	4	no	—
33	#1715	—	9	1	no	Sylow
34	C₃×C₄⋊Q₈	C₂×D₃	1	2	yes	—
35	C₂×#215	—	6	4	no	—
36	C₂×D₂₄	—	6	2	no	—
37	#874	—	3	1	no	—
38	#874	—	3	1	no	—
39	C₃×#106	—	2	2	no	—
40	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
41	#809	—	6	4	no	—
42	C₄×C₃⋊C₈	—	3	2	no	—
43	C₂²×D₉	—	4	1	no	—
44	C₂²×D₉	—	4	1	no	—
45	Q₈×C₉	—	4	4	no	—
46	C₂×C₄×C₉	—	4	2	no	—
47	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
49	D₃₆	—	4	2	no	—
50	D₃₆	—	4	2	no	—
51	D₃₆	—	4	4	no	—
52	D₃₆	—	4	4	no	—
53	C₈×C₉	SD₁₆	1	4	yes	complemented
54	C₉⋊C₈	—	2	4	no	—
55	#498	D₉	1	1	yes	characteristic, complemented
56	#493	—	9	1	no	—
57	#492	—	9	2	no	—
58	#486	—	9	2	no	—
59	#322	—	9	1	no	—
60	C₂×C₃×Q₈	—	2	2	no	—
61	C₂×D₁₂	—	6	2	no	—
62	C₂×D₁₂	—	6	2	no	—
63	C₂×D₁₂	—	6	1	no	—
64	C₂×D₁₂	—	6	1	no	—
65	C₂×C₃×C₈	#67	1	2	yes	—
66	C₃×C₄⋊C₄	—	2	2	no	—
67	C₃×C₄⋊C₄	—	2	2	no	—
68	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
69	#215	—	12	8	no	—
70	C₂×C₃⋊C₈	—	3	4	no	—
71	D₂₄	—	12	4	no	—
72	C₂²×C₉	#127	1	1	yes	characteristic
73	C₂×D₉	—	8	1	no	—
74	C₂×D₉	—	8	1	no	—
75	C₂×D₉	—	8	2	no	—
76	C₂×D₉	—	8	2	no	—
77	C₄×C₉	—	8	2	no	—
78	C₄×C₉	C₂×SD₁₆	1	2	yes	—
79	C₄×C₉	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×SD₁₆	—	18	4	no	—
81	C₂×D₈	—	18	2	no	—
82	C₄⋊Q₈	C₂×D₉	1	2	yes	—
83	#127	—	9	1	no	—
84	#127	—	9	1	no	—
85	#106	—	2	2	no	—
86	#102	—	18	4	no	—
87	C₄×C₈	—	9	2	no	—
88	C₄×C₈	C₂×D₉	1	1	yes	characteristic
89	C₂²×D₃	—	12	1	no	—
90	C₂²×D₃	—	12	1	no	—
91	C₃×Q₈	—	4	4	no	—
92	C₂×C₃×C₄	—	4	2	no	—
93	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
94	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
95	D₁₂	—	12	4	no	—
96	D₁₂	—	12	4	no	—
97	D₁₂	—	12	2	no	—
98	D₁₂	—	12	2	no	—
99	C₃×C₈	#215	1	4	yes	—
100	C₃⋊C₈	—	6	4	no	—
101	C₂×C₉	#492	1	1	yes	characteristic
102	C₂×C₉	#496	1	2	yes	—
103	D₉	—	16	1	no	—
104	D₉	—	16	1	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×D₄	—	18	2	no	—
107	C₂×D₄	—	18	2	no	—
108	C₂×D₄	—	18	1	no	—
109	C₂×D₄	—	18	1	no	—
110	SD₁₆	—	36	8	no	—
111	D₈	—	36	4	no	—
112	C₂×C₈	—	9	4	no	—
113	C₂×C₈	#611	1	2	yes	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄²	C₂²×D₉	1	1	yes	characteristic
117	C₂²×C₃	#940	1	1	yes	characteristic
118	C₂×D₃	—	24	2	no	—
119	C₂×D₃	—	24	2	no	—
120	C₂×D₃	—	24	1	no	—
121	C₂×D₃	—	24	1	no	—
122	C₃×C₄	—	8	2	no	—
123	C₃×C₄	#914	1	4	yes	—
124	C₃×C₄	C₂×#215	1	2	yes	—
125	C₉	#1715	1	1	yes	characteristic, complemented, Sylow
126	C₂³	—	36	1	no	—
127	C₂³	—	36	1	no	—
128	Q₈	—	4	4	no	—
129	D₄	—	36	2	no	—
130	D₄	—	36	2	no	—
131	D₄	—	36	4	no	—
132	D₄	—	36	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	D₄×D₉	1	2	yes	—
135	C₂×C₄	C₂×#611	1	1	yes	characteristic
136	C₈	—	18	4	no	—
137	C₈	#3675	1	4	yes	complemented
138	C₂×C₃	#5085	1	1	yes	characteristic
139	C₂×C₃	#5176	1	2	yes	—
140	D₃	—	48	1	no	—
141	D₃	—	48	1	no	—
142	C₂²	—	72	1	no	—
143	C₂²	—	72	1	no	—
144	C₂²	—	72	2	no	—
145	C₂²	—	72	2	no	—
146	C₂²	#63502	1	1	yes	characteristic
147	C₄	—	8	2	no	—
148	C₄	#63476	1	4	yes	—
149	C₄	C₂×#3675	1	2	yes	—
150	C₃	#71336	1	1	yes	characteristic
151	C₂	—	144	1	no	—
152	C₂	—	144	1	no	—
153	C₂	#10588770	1	1	yes	characteristic
154	C₂	#10588856	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#498

C₂

yes

maximal, characteristic, complemented

#10588773

C₂

yes

maximal, complemented

#10588770

C₂

yes

maximal, complemented

#10588377

C₂

yes

maximal, characteristic

#10588132

C₂

yes

maximal, characteristic, complemented

#71336

—

maximal

C₉×C₄⋊Q₈

C₂²

yes

—

C₂×#3675

—

C₂×D₇₂

—

#63436

C₂²

yes

characteristic

#63436

C₂²

yes

characteristic

C₉×#106

—

C₄×C₈×C₉

C₂²

yes

characteristic

#63369

—

C₄×C₉⋊C₈

C₂²

yes

—

C₃×#498

D₃

yes

—

—

—

—

C₂×Q₈×C₉

—

C₂×D₃₆

—

C₂×D₃₆

—

C₂×D₃₆

—

C₂×D₃₆

—

C₂×C₈×C₉

D₄

yes

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₄²×C₉

C₂³

yes

characteristic

#3675

—

C₂×C₉⋊C₈

D₄

yes

—

D₇₂

—

#1715

—

Sylow

C₃×C₄⋊Q₈

C₂×D₃

yes

—

C₂×#215

—

C₂×D₂₄

—

#874

—

#874

—

C₃×#106

—

C₃×C₄×C₈

C₂×D₃

yes

characteristic

#809

—

C₄×C₃⋊C₈

—

C₂²×D₉

—

C₂²×D₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

characteristic

D₃₆

—

D₃₆

—

D₃₆

—

D₃₆

—

C₈×C₉

SD₁₆

yes

complemented

C₉⋊C₈

—

#498

D₉

yes

characteristic, complemented

—

—

—

—

C₂×C₃×Q₈

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×D₃

yes

characteristic

#215

—

C₂×C₃⋊C₈

—

D₂₄

—

C₂²×C₉

#127

yes

characteristic

C₂×D₉

—

C₂×D₉

—

C₂×D₉

—

C₂×D₉

—

C₄×C₉

—

C₄×C₉

C₂×SD₁₆

yes

—

C₄×C₉

AGL(1,ℤ/8)

yes

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂×D₉

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×D₉

yes

characteristic

C₂²×D₃

—

C₂²×D₃

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₃×D₄

yes

—

C₂×C₃×C₄

C₂×#67

yes

characteristic

D₁₂

—

D₁₂

—

D₁₂

—

D₁₂

—

C₃×C₈

#215

yes

—

100

C₃⋊C₈

—

101

C₂×C₉

#492

yes

characteristic

102

C₂×C₉

#496

yes

—

103

D₉

—

104

D₉

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

SD₁₆

—

111

D₈

—

112

C₂×C₈

—

113

C₂×C₈

#611

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄²

C₂²×D₉

yes

characteristic

117

C₂²×C₃

#940

yes

characteristic

118

C₂×D₃

—

119

C₂×D₃

—

120

C₂×D₃

—

121

C₂×D₃

—

122

C₃×C₄

—

123

C₃×C₄

#914

yes

—

124

C₃×C₄

C₂×#215

yes

—

125

C₉

#1715

yes

characteristic, complemented, Sylow

126

C₂³

—

127

C₂³

—

128

Q₈

—

129

D₄

—

130

D₄

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

D₄×D₉

yes

—

135

C₂×C₄

C₂×#611

yes

characteristic

136

C₈

—

137

C₈

#3675

yes

complemented

138

C₂×C₃

#5085

yes

characteristic

139

C₂×C₃

#5176

yes

—

140

D₃

—

141

D₃

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#63502

yes

characteristic

147

C₄

—

148

C₄

#63476

yes

—

149

C₄

C₂×#3675

yes

—

150

C₃

#71336

yes

characteristic

151

C₂

—

144

—

152

C₂

—

144

—

153

C₂

#10588770

yes

characteristic

154

C₂

#10588856

yes

—

	subgroup	G/H
center	C₂²	#63502
commutator, cosocle	C₄²×C₉	C₂³
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₉×#498	C₂
socle	C₂²×C₃	#940
prime radical	#10588377	C₂
hypercenter	#498	D₉
hypocenter	C₉	#1715
LCS	G ▷ C₄²×C₉ ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#63502

commutator, cosocle

C₄²×C₉

C₂³

Frattini

C₃×C₄²

C₂²×D₃

C₂

C₂²×C₃

C₂

D₉

C₉

G ▷ C₄²×C₉ ▷ C₂²×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
1. C₉×#498	C₂	semidirect
2. #10588773	C₂	semidirect
3. #10588770	C₂	semidirect
5. #10588132	C₂	semidirect
53. C₈×C₉	SD₁₆	semidirect
55. #498	D₉	semidirect
125. C₉	#1715	semidirect
137. C₈	#3675	semidirect

H ≅ G/N

type

1. C₉×#498

C₂

semidirect

2. #10588773

C₂

semidirect

3. #10588770

C₂

semidirect

5. #10588132

C₂

semidirect

53. C₈×C₉

SD₁₆

semidirect

55. #498

D₉

semidirect

125. C₉

#1715

semidirect

137. C₈

#3675

semidirect

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

subgroup	index
C₉×#498	2
#10588773	2
#10588770	2
#10588377	2
#10588132	2
#71336	3

subgroup

index

Subgroups — #11782666

Conjugacy classes of subgroups

Complements

Maximal subgroups