Group #11782420 — subgroups

Subgroups — #11782420

Order 1152.

321 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 98 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	#10588770	C₂	1	2	yes	maximal
3	#10588760	C₂	1	2	yes	maximal, complemented
4	#10588375	C₂	1	1	yes	maximal, characteristic
5	C₈×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#71332	—	3	1	no	maximal
7	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×#3675	—	2	4	no	—
9	C₂×#3664	—	2	2	no	—
10	#63436	C₂²	1	1	yes	characteristic
11	#63431	C₂²	1	1	yes	characteristic
12	C₉×#106	—	2	2	no	—
13	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
14	#63367	—	2	4	no	—
15	C₄×C₉⋊C₈	C₂²	1	2	yes	—
16	C₃×#498	D₃	1	1	yes	characteristic
17	#5085	—	3	2	no	—
18	#5080	—	3	2	no	—
19	#4693	—	3	1	no	—
20	C₈×C₃⋊C₈	—	3	1	no	—
21	C₂×Q₈×C₉	—	2	2	no	—
22	C₂×D₃₆	—	2	2	no	—
23	C₂×D₃₆	—	2	1	no	—
24	C₂×Dic₁₈	—	2	1	no	—
25	C₂×C₈×C₉	D₄	1	2	yes	—
26	C₉×C₄⋊C₄	—	2	2	no	—
27	C₉×C₄⋊C₄	—	2	2	no	—
28	C₄²×C₉	C₂³	1	1	yes	characteristic
29	#3675	—	4	8	no	—
30	C₄⋊Dic₉	—	2	2	no	—
31	C₂×C₉⋊C₈	D₄	1	4	yes	—
32	#3664	—	4	4	no	—
33	#1705	—	9	1	no	Sylow
34	C₃×C₄⋊Q₈	C₂×D₃	1	2	yes	—
35	C₂×#215	—	6	4	no	—
36	C₂×#204	—	6	2	no	—
37	#874	—	3	1	no	—
38	#869	—	3	1	no	—
39	C₃×#106	—	2	2	no	—
40	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
41	#808	—	6	4	no	—
42	C₄×C₃⋊C₈	—	3	2	no	—
43	C₂²×D₉	—	4	1	no	—
44	Q₈×C₉	—	4	4	no	—
45	C₂×C₄×C₉	—	4	2	no	—
46	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₉	—	4	1	no	—
49	D₃₆	—	4	2	no	—
50	D₃₆	—	4	4	no	—
51	Dic₁₈	—	4	2	no	—
52	C₈×C₉	SD₁₆	1	4	yes	complemented
53	C₉⋊C₈	SD₁₆	1	8	yes	complemented
54	#498	—	9	2	no	—
55	#498	D₉	1	1	yes	characteristic, complemented
56	#492	—	9	2	no	—
57	#492	—	9	1	no	—
58	C₈²	—	9	1	no	—
59	C₂×C₃×Q₈	—	2	2	no	—
60	C₂×D₁₂	—	6	2	no	—
61	C₂×D₁₂	—	6	1	no	—
62	C₂×Dic₆	—	6	1	no	—
63	C₂×C₃×C₈	#67	1	2	yes	—
64	C₃×C₄⋊C₄	—	2	2	no	—
65	C₃×C₄⋊C₄	—	2	2	no	—
66	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
67	#215	—	12	8	no	—
68	C₄⋊Dic₃	—	6	2	no	—
69	C₂×C₃⋊C₈	—	3	4	no	—
70	#204	—	12	4	no	—
71	C₂²×C₉	#127	1	1	yes	characteristic
72	C₂×D₉	—	8	2	no	—
73	C₂×D₉	—	8	1	no	—
74	C₄×C₉	—	8	2	no	—
75	C₄×C₉	C₂×SD₁₆	1	4	yes	—
76	C₄×C₉	C₂×SD₁₆	1	2	yes	—
77	Dic₉	—	8	1	no	—
78	C₂×SD₁₆	—	18	4	no	—
79	C₂×SD₁₆	—	18	2	no	—
80	C₄⋊Q₈	—	9	1	no	—
81	C₄⋊Q₈	C₂×D₉	1	2	yes	—
82	#127	—	9	1	no	—
83	#106	—	2	2	no	—
84	#106	—	18	4	no	—
85	C₄×C₈	—	9	2	no	—
86	C₄×C₈	C₂×D₉	1	1	yes	characteristic
87	C₂²×D₃	—	12	1	no	—
88	C₃×Q₈	—	4	4	no	—
89	C₂×C₃×C₄	—	4	2	no	—
90	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
91	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
92	C₂×Dic₃	—	12	1	no	—
93	D₁₂	—	12	4	no	—
94	D₁₂	—	12	2	no	—
95	Dic₆	—	12	2	no	—
96	C₃×C₈	#215	1	4	yes	—
97	C₃⋊C₈	—	3	8	no	—
98	C₂×C₉	#492	1	2	yes	—
99	C₂×C₉	#492	1	1	yes	characteristic
100	D₉	—	16	1	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	18	1	no	—
103	C₂×D₄	—	18	2	no	—
104	C₂×D₄	—	18	1	no	—
105	SD₁₆	—	36	8	no	—
106	SD₁₆	—	36	4	no	—
107	C₂×C₈	—	9	4	no	—
108	C₂×C₈	#611	1	2	yes	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	18	2	no	—
112	C₄²	C₂²×D₉	1	1	yes	characteristic
113	C₂²×C₃	#940	1	1	yes	characteristic
114	C₂×D₃	—	24	2	no	—
115	C₂×D₃	—	24	1	no	—
116	C₃×C₄	—	8	2	no	—
117	C₃×C₄	D₃×SD₁₆	1	4	yes	—
118	C₃×C₄	C₂×#215	1	2	yes	—
119	Dic₃	—	24	1	no	—
120	C₉	#1705	1	1	yes	characteristic, complemented, Sylow
121	C₂³	—	36	1	no	—
122	Q₈	—	4	4	no	—
123	Q₈	—	36	2	no	—
124	D₄	—	36	2	no	—
125	D₄	—	36	4	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	36	1	no	—
128	C₂×C₄	D₄×D₉	1	2	yes	—
129	C₂×C₄	C₂×#611	1	1	yes	characteristic
130	C₈	—	9	8	no	—
131	C₈	#3675	1	4	yes	complemented
132	C₂×C₃	#5085	1	1	yes	characteristic
133	C₂×C₃	#5175	1	2	yes	—
134	D₃	—	48	1	no	—
135	C₂²	—	72	1	no	—
136	C₂²	—	72	2	no	—
137	C₂²	#63502	1	1	yes	characteristic
138	C₄	—	8	2	no	—
139	C₄	—	72	1	no	—
140	C₄	SD₁₆×D₉	1	4	yes	—
141	C₄	C₂×#3675	1	2	yes	—
142	C₃	#71332	1	1	yes	characteristic
143	C₂	—	144	1	no	—
144	C₂	#10588770	1	1	yes	characteristic
145	C₂	#10588855	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#498

C₂

yes

maximal, characteristic, complemented

#10588770

C₂

yes

maximal

#10588760

C₂

yes

maximal, complemented

#10588375

C₂

yes

maximal, characteristic

C₈×C₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#71332

—

maximal

C₉×C₄⋊Q₈

C₂²

yes

—

C₂×#3675

—

C₂×#3664

—

#63436

C₂²

yes

characteristic

#63431

C₂²

yes

characteristic

C₉×#106

—

C₄×C₈×C₉

C₂²

yes

characteristic

#63367

—

C₄×C₉⋊C₈

C₂²

yes

—

C₃×#498

D₃

yes

—

—

—

C₈×C₃⋊C₈

—

C₂×Q₈×C₉

—

C₂×D₃₆

—

C₂×D₃₆

—

C₂×Dic₁₈

—

C₂×C₈×C₉

D₄

yes

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₄²×C₉

C₂³

yes

characteristic

#3675

—

C₄⋊Dic₉

—

C₂×C₉⋊C₈

D₄

yes

—

#3664

—

#1705

—

Sylow

C₃×C₄⋊Q₈

C₂×D₃

yes

—

—

—

—

—

—

C₃×C₄×C₈

C₂×D₃

yes

characteristic

#808

—

C₄×C₃⋊C₈

—

C₂²×D₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×Dic₉

—

D₃₆

—

D₃₆

—

Dic₁₈

—

C₈×C₉

SD₁₆

yes

complemented

C₉⋊C₈

SD₁₆

yes

complemented

#498

—

#498

D₉

yes

characteristic, complemented

#492

—

#492

—

C₈²

—

C₂×C₃×Q₈

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×D₃

yes

characteristic

#215

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

#204

—

C₂²×C₉

#127

yes

characteristic

C₂×D₉

—

C₂×D₉

—

C₄×C₉

—

C₄×C₉

C₂×SD₁₆

yes

—

C₄×C₉

C₂×SD₁₆

yes

—

Dic₉

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₉

yes

—

#127

—

#106

—

#106

—

C₄×C₈

—

C₄×C₈

C₂×D₉

yes

characteristic

C₂²×D₃

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₃×D₄

yes

—

C₂×C₃×C₄

C₂×#67

yes

characteristic

C₂×Dic₃

—

D₁₂

—

D₁₂

—

Dic₆

—

C₃×C₈

#215

yes

—

C₃⋊C₈

—

C₂×C₉

#492

yes

—

C₂×C₉

#492

yes

characteristic

100

D₉

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

105

SD₁₆

—

106

SD₁₆

—

107

C₂×C₈

—

108

C₂×C₈

#611

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

C₂²×D₉

yes

characteristic

113

C₂²×C₃

#940

yes

characteristic

114

C₂×D₃

—

115

C₂×D₃

—

116

C₃×C₄

—

117

C₃×C₄

D₃×SD₁₆

yes

—

118

C₃×C₄

C₂×#215

yes

—

119

Dic₃

—

120

C₉

#1705

yes

characteristic, complemented, Sylow

121

C₂³

—

122

Q₈

—

123

Q₈

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

D₄×D₉

yes

—

129

C₂×C₄

C₂×#611

yes

characteristic

130

C₈

—

131

C₈

#3675

yes

complemented

132

C₂×C₃

#5085

yes

characteristic

133

C₂×C₃

#5175

yes

—

134

D₃

—

135

C₂²

—

136

C₂²

—

137

C₂²

#63502

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

SD₁₆×D₉

yes

—

141

C₄

C₂×#3675

yes

—

142

C₃

#71332

yes

characteristic

143

C₂

—

144

—

144

C₂

#10588770

yes

characteristic

145

C₂

#10588855

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	#63436	C₂²
hypercenter	#498	D₉
hypocenter	C₉	#1705
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

C₂²

D₉

C₉

G ▷ C₄²×C₉ ▷ C₂²×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
1. C₉×#498	C₂	semidirect
3. #10588760	C₂	semidirect
5. C₈×C₉⋊C₈	C₂	semidirect
52. C₈×C₉	SD₁₆	semidirect
53. C₉⋊C₈	SD₁₆	semidirect
55. #498	D₉	semidirect
120. C₉	#1705	semidirect
131. C₈	#3675	semidirect

H ≅ G/N

type

1. C₉×#498

C₂

semidirect

3. #10588760

C₂

semidirect

5. C₈×C₉⋊C₈

C₂

semidirect

52. C₈×C₉

SD₁₆

semidirect

53. C₉⋊C₈

SD₁₆

semidirect

55. #498

D₉

semidirect

120. C₉

#1705

semidirect

131. 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
#10588770	2
#10588760	2
#10588375	2
C₈×C₉⋊C₈	2
#71332	3

subgroup

index

C₈×C₉⋊C₈

Subgroups — #11782420

Conjugacy classes of subgroups

Complements

Maximal subgroups