Group #11807951 — subgroups

Subgroups — #11807951

Order 1152.

636 conjugacy classes of subgroups; only the 294 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 294 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×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#10588909	C₂	1	4	yes	maximal
3	#10588660	C₂	1	8	yes	maximal
4	#10588617	C₂	1	2	yes	maximal
5	#83231	—	3	1	no	maximal
6	C₂²×Q₈×C₉	C₂²	1	2	yes	—
7	C₉×C₄⋊Q₈	C₄	1	8	yes	complemented
8	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₉	C₂²	1	4	yes	—
11	C₂×C₄×Dic₉	C₂²	1	4	yes	—
12	C₉⋊#95	C₂²	1	4	yes	—
13	C₉⋊#95	C₂²	1	8	yes	—
14	C₉⋊#95	C₂²	1	8	yes	—
15	C₂×C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
16	#5229	—	3	4	no	—
17	#4980	—	3	8	no	—
18	#4937	—	3	2	no	—
19	C₂×Q₈×C₉	—	4	2	no	—
20	C₂×Q₈×C₉	—	4	2	no	—
21	C₂×Q₈×C₉	C₂×C₄	1	8	yes	—
22	C₂²×C₄×C₉	C₂³	1	2	yes	—
23	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₉	C₂³	1	4	yes	—
25	C₂²×Dic₉	C₂³	1	8	yes	—
26	C₉×C₄⋊C₄	C₂×C₄	1	16	yes	—
27	C₄²×C₉	C₂×C₄	1	4	yes	—
28	C₄⋊Dic₉	—	4	4	no	—
29	C₄×Dic₉	—	4	4	no	—
30	#2381	—	9	1	no	Sylow
31	C₂²×C₃×Q₈	C₂×D₃	1	2	yes	—
32	C₃×C₄⋊Q₈	Dic₃	1	8	yes	—
33	C₂×C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
34	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
35	C₂×C₄⋊Dic₃	—	3	4	no	—
36	C₂×C₄×Dic₃	—	3	4	no	—
37	C₃⋊#95	—	3	4	no	—
38	C₃⋊#95	—	3	8	no	—
39	C₃⋊#95	—	3	8	no	—
40	C₂³×C₉	C₂⁴	1	1	yes	characteristic
41	Q₈×C₉	—	4	8	no	—
42	C₂×C₄×C₉	—	2	4	no	—
43	C₂×C₄×C₉	—	2	4	no	—
44	C₂×C₄×C₉	—	2	2	no	—
45	C₂×C₄×C₉	—	2	4	no	—
46	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₉	C₂²×C₄	1	8	yes	—
49	C₂×Dic₉	—	2	8	no	—
50	C₂×Dic₉	—	2	16	no	—
51	C₂×C₄⋊Q₈	D₉	1	1	yes	characteristic
52	#391	—	9	4	no	—
53	#387	—	9	8	no	—
54	#383	—	9	2	no	—
55	C₂×C₃×Q₈	—	4	2	no	—
56	C₂×C₃×Q₈	—	4	2	no	—
57	C₂×C₃×Q₈	C₂×Dic₃	1	8	yes	—
58	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
59	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
61	C₂²×Dic₃	—	3	8	no	—
62	C₃×C₄⋊C₄	C₂×Dic₃	1	16	yes	—
63	C₃×C₄²	C₂×Dic₃	1	4	yes	—
64	C₄⋊Dic₃	—	12	4	no	—
65	C₄×Dic₃	—	12	4	no	—
66	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
67	C₂²×C₉	H(5,𝔽₂)	1	2	yes	—
68	C₂²×C₉	2₋¹⁺⁴	1	4	yes	—
69	C₄×C₉	—	2	8	no	—
70	C₄×C₉	—	2	2	no	—
71	C₄×C₉	—	2	4	no	—
72	Dic₉	—	4	8	no	—
73	C₂²×Q₈	C₂×D₉	1	2	yes	—
74	C₄⋊Q₈	Dic₉	1	8	yes	complemented
75	C₂×C₄⋊C₄	—	9	4	no	—
76	C₂×C₄⋊C₄	C₂×D₉	1	4	yes	—
77	C₂×C₄²	—	9	4	no	—
78	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
79	#95	—	9	8	no	—
80	#95	—	9	4	no	—
81	#95	—	9	8	no	—
82	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
83	C₃×Q₈	—	4	8	no	—
84	C₂×C₃×C₄	—	2	4	no	—
85	C₂×C₃×C₄	—	2	4	no	—
86	C₂×C₃×C₄	—	2	2	no	—
87	C₂×C₃×C₄	—	2	4	no	—
88	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
89	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
90	C₂×C₃×C₄	C₂²×Dic₃	1	8	yes	—
91	C₂×Dic₃	—	6	8	no	—
92	C₂×Dic₃	—	6	16	no	—
93	C₂×C₉	#518	1	1	yes	characteristic
94	C₂×C₉	#519	1	2	yes	—
95	C₂×C₉	#563	1	4	yes	—
96	C₂×Q₈	—	4	2	no	—
97	C₂×Q₈	—	4	2	no	—
98	C₂×Q₈	C₂×Dic₉	1	8	yes	—
99	C₂²×C₄	—	9	8	no	—
100	C₂²×C₄	C₂²×D₉	1	2	yes	—
101	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
102	C₂²×C₄	C₂²×D₉	1	4	yes	—
103	C₄⋊C₄	—	36	4	no	—
104	C₄⋊C₄	C₂×Dic₉	1	16	yes	—
105	C₄²	—	36	4	no	—
106	C₄²	C₂×Dic₉	1	4	yes	—
107	C₂²×C₃	#1004	1	2	yes	—
108	C₂²×C₃	#1007	1	4	yes	—
109	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
110	C₃×C₄	—	2	2	no	—
111	C₃×C₄	—	2	8	no	—
112	C₃×C₄	—	2	4	no	—
113	Dic₃	—	12	8	no	—
114	C₉	#2381	1	1	yes	characteristic, complemented, Sylow
115	C₂³	C₂³×D₉	1	1	yes	characteristic
116	Q₈	—	4	8	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	18	8	no	—
122	C₂×C₄	—	18	16	no	—
123	C₂×C₄	C₂²×Dic₉	1	4	yes	—
124	C₂×C₄	C₂²×Dic₉	1	2	yes	—
125	C₂×C₄	C₂²×Dic₉	1	8	yes	—
126	C₂×C₃	#5735	1	4	yes	—
127	C₂×C₃	#5798	1	1	yes	characteristic
128	C₂×C₃	#5812	1	2	yes	—
129	C₂²	#63710	1	2	yes	—
130	C₂²	#63713	1	4	yes	—
131	C₂²	C₂³×Dic₉	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	8	no	—
134	C₄	—	2	4	no	—
135	C₄	—	36	8	no	—
136	C₃	#83231	1	1	yes	characteristic
137	C₂	#10589837	1	4	yes	—
138	C₂	#10589900	1	1	yes	characteristic
139	C₂	#10589914	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₉×C₄⋊Q₈

C₂

yes

maximal, characteristic

#10588909

C₂

yes

maximal

#10588660

C₂

yes

maximal

#10588617

C₂

yes

maximal

#83231

—

maximal

C₂²×Q₈×C₉

C₂²

yes

—

C₉×C₄⋊Q₈

C₄

yes

complemented

C₂×C₉×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₉

C₂²

yes

—

C₂×C₄×Dic₉

C₂²

yes

—

C₉⋊#95

C₂²

yes

—

C₉⋊#95

C₂²

yes

—

C₉⋊#95

C₂²

yes

—

C₂×C₃×C₄⋊Q₈

D₃

yes

—

—

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

C₂×C₄

yes

—

C₂²×C₄×C₉

C₂³

yes

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₂²×C₄×C₉

C₂³

yes

—

C₂²×Dic₉

C₂³

yes

—

C₉×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₉

C₂×C₄

yes

—

C₄⋊Dic₉

—

C₄×Dic₉

—

#2381

—

Sylow

C₂²×C₃×Q₈

C₂×D₃

yes

—

C₃×C₄⋊Q₈

Dic₃

yes

—

C₂×C₃×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₃×C₄²

C₂×D₃

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄×Dic₃

—

C₃⋊#95

—

C₃⋊#95

—

C₃⋊#95

—

C₂³×C₉

C₂⁴

yes

characteristic

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂²×C₄

yes

—

C₂×C₄×C₉

C₂²×C₄

yes

—

C₂×C₄×C₉

C₂²×C₄

yes

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×C₄⋊Q₈

D₉

yes

—

—

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₂×Dic₃

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

characteristic

C₂²×C₃×C₄

C₂²×D₃

yes

—

C₂²×Dic₃

—

C₃×C₄⋊C₄

C₂×Dic₃

yes

—

C₃×C₄²

C₂×Dic₃

yes

—

C₄⋊Dic₃

—

C₄×Dic₃

—

C₂²×C₉

C₂³×C₄

yes

characteristic

C₂²×C₉

H(5,𝔽₂)

yes

—

C₂²×C₉

2₋¹⁺⁴

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

Dic₉

—

C₂²×Q₈

C₂×D₉

yes

—

C₄⋊Q₈

Dic₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉

yes

—

—

—

C₂³×C₃

C₂³×D₃

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×Dic₃

yes

—

C₂×C₃×C₄

C₂²×Dic₃

yes

—

C₂×C₃×C₄

C₂²×Dic₃

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×C₉

#518

yes

characteristic

C₂×C₉

#519

yes

—

C₂×C₉

#563

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₉

yes

—

C₂²×C₄

—

100

C₂²×C₄

C₂²×D₉

yes

—

101

C₂²×C₄

C₂²×D₉

yes

characteristic

102

C₂²×C₄

C₂²×D₉

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

C₂×Dic₉

yes

—

105

C₄²

—

106

C₄²

C₂×Dic₉

yes

—

107

C₂²×C₃

#1004

yes

—

108

C₂²×C₃

#1007

yes

—

109

C₂²×C₃

C₂³×Dic₃

yes

characteristic

110

C₃×C₄

—

111

C₃×C₄

—

112

C₃×C₄

—

113

Dic₃

—

114

C₉

#2381

yes

characteristic, complemented, Sylow

115

C₂³

C₂³×D₉

yes

characteristic

116

Q₈

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂²×Dic₉

yes

—

124

C₂×C₄

C₂²×Dic₉

yes

—

125

C₂×C₄

C₂²×Dic₉

yes

—

126

C₂×C₃

#5735

yes

—

127

C₂×C₃

#5798

yes

characteristic

128

C₂×C₃

#5812

yes

—

129

C₂²

#63710

yes

—

130

C₂²

#63713

yes

—

131

C₂²

C₂³×Dic₉

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₃

#83231

yes

characteristic

137

C₂

#10589837

yes

—

138

C₂

#10589900

yes

characteristic

139

C₂

#10589914

yes

—

	subgroup	G/H
center	C₂³	C₂³×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini, socle, prime radical	C₂³×C₃	C₂³×D₃
Fitting	C₂×C₉×C₄⋊Q₈	C₂
cosocle	C₂³×C₉	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₉
hypocenter	C₉	#2381
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂³×D₉

commutator

C₂²×C₉

C₂³×C₄

Frattini, socle, prime radical

C₂³×C₃

C₂³×D₃

Fitting

C₂×C₉×C₄⋊Q₈

C₂

cosocle

C₂³×C₉

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₉

C₉

G ▷ C₂²×C₉ ▷ C₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
7. C₉×C₄⋊Q₈	C₄	semidirect
74. C₄⋊Q₈	Dic₉	semidirect
114. C₉	#2381	semidirect

H ≅ G/N

type

7. C₉×C₄⋊Q₈

C₄

semidirect

74. C₄⋊Q₈

Dic₉

semidirect

114. C₉

#2381

semidirect

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

subgroup	index
C₂×C₉×C₄⋊Q₈	2
#10588909	2
#10588660	2
#10588617	2
#83231	3

subgroup

index

C₂×C₉×C₄⋊Q₈

Subgroups — #11807951

Conjugacy classes of subgroups

Complements

Maximal subgroups