Group #10588862 — subgroups

Subgroups — #10588862

Order 576.

162 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 57 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×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₉	C₂	1	1	yes	maximal, characteristic
3	#63506	C₂	1	1	yes	maximal, characteristic
4	C₂×#3674	C₂	1	1	yes	maximal, characteristic
5	#63395	C₂	1	1	yes	maximal, characteristic
6	#63376	C₂	1	1	yes	maximal, characteristic
7	Dic₉⋊C₈	C₂	1	1	yes	maximal, characteristic
8	#5182	—	3	1	no	maximal
9	C₂×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₈	C₂²	1	1	yes	characteristic
12	C₉×Q₁₆	—	2	2	no	—
13	C₂×C₈×C₉	C₂²	1	1	yes	characteristic
14	#3674	—	2	2	no	—
15	C₄⋊Dic₉	—	2	1	no	—
16	C₄⋊Dic₉	C₂²	1	1	yes	characteristic
17	Dic₉⋊C₄	—	2	1	no	—
18	Dic₉⋊C₄	—	2	1	no	—
19	C₄×Dic₉	—	2	1	no	—
20	C₄×Dic₉	C₂²	1	1	yes	characteristic
21	C₂×C₉⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃×Q₁₆	D₃	1	1	yes	characteristic
23	Q₈×Dic₃	—	3	1	no	—
24	#944	—	3	1	no	—
25	C₂×#216	—	3	1	no	—
26	#835	—	3	1	no	—
27	#816	—	3	1	no	—
28	Dic₃⋊C₈	—	3	1	no	—
29	Q₈×C₉	—	2	1	no	—
30	Q₈×C₉	—	2	1	no	—
31	Q₈×C₉	—	2	1	no	—
32	Q₈×C₉	D₄	1	2	yes	—
33	C₂×C₄×C₉	—	2	1	no	—
34	C₂×C₄×C₉	—	2	1	no	—
35	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₂×Dic₉	—	2	1	no	—
37	C₂×Dic₉	—	2	1	no	—
38	C₂×Dic₉	D₄	1	1	yes	characteristic
39	C₂×Dic₉	D₄	1	1	yes	characteristic
40	Dic₁₈	—	2	1	no	—
41	Dic₁₈	—	2	1	no	—
42	C₈×C₉	—	2	1	no	—
43	C₉⋊C₈	—	2	1	no	—
44	#462	—	9	1	no	Sylow
45	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
46	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
47	C₂×Dic₆	—	3	1	no	—
48	C₃×Q₁₆	—	2	2	no	—
49	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
50	#216	—	6	2	no	—
51	C₄⋊Dic₃	—	3	1	no	—
52	C₄⋊Dic₃	—	6	1	no	—
53	Dic₃⋊C₄	—	6	1	no	—
54	Dic₃⋊C₄	—	6	1	no	—
55	C₄×Dic₃	—	3	1	no	—
56	C₄×Dic₃	—	6	1	no	—
57	C₂×C₃⋊C₈	—	3	1	no	—
58	C₂²×C₉	C₂×D₄	1	1	yes	characteristic
59	C₄×C₉	—	2	2	no	—
60	C₄×C₉	—	4	1	no	—
61	C₄×C₉	C₂×D₄	1	1	yes	characteristic
62	C₄×C₉	#41	1	1	yes	characteristic
63	Dic₉	—	2	1	no	—
64	Dic₉	—	4	1	no	—
65	Dic₉	—	4	1	no	—
66	Dic₉	Q₁₆	1	2	yes	complemented
67	C₂×Q₁₆	—	9	1	no	—
68	C₂×Q₁₆	D₉	1	1	yes	characteristic
69	C₄⋊Q₈	—	9	1	no	—
70	C₄×Q₈	—	9	1	no	—
71	C₄⋊C₈	—	9	1	no	—
72	#103	—	9	1	no	—
73	#103	—	9	1	no	—
74	C₃×Q₈	—	2	1	no	—
75	C₃×Q₈	—	2	1	no	—
76	C₃×Q₈	—	2	1	no	—
77	C₃×Q₈	#67	1	2	yes	—
78	C₂×C₃×C₄	—	2	1	no	—
79	C₂×C₃×C₄	—	2	1	no	—
80	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
81	C₂×Dic₃	—	3	1	no	—
82	C₂×Dic₃	—	3	1	no	—
83	C₂×Dic₃	—	6	1	no	—
84	C₂×Dic₃	—	6	1	no	—
85	Dic₆	—	6	1	no	—
86	Dic₆	—	6	1	no	—
87	C₃×C₈	—	2	1	no	—
88	C₃⋊C₈	—	6	1	no	—
89	C₂×C₉	#121	1	1	yes	characteristic
90	C₂×C₉	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₉	#137	1	1	yes	characteristic
92	C₂×Q₈	—	9	1	no	—
93	C₂×Q₈	C₂×D₉	1	1	yes	characteristic
94	C₂×Q₈	C₂×D₉	1	1	yes	characteristic
95	Q₁₆	—	2	2	no	—
96	Q₁₆	—	18	2	no	—
97	C₂×C₈	—	9	1	no	—
98	C₂×C₈	C₂×D₉	1	1	yes	characteristic
99	C₄⋊C₄	—	9	1	no	—
100	C₄⋊C₄	—	18	1	no	—
101	C₄⋊C₄	—	18	1	no	—
102	C₄⋊C₄	—	18	1	no	—
103	C₄²	—	9	1	no	—
104	C₄²	—	18	1	no	—
105	C₂²×C₃	D₃×D₄	1	1	yes	characteristic
106	C₃×C₄	—	2	2	no	—
107	C₃×C₄	—	4	1	no	—
108	C₃×C₄	#237	1	1	yes	characteristic
109	C₃×C₄	C₂×#67	1	1	yes	characteristic
110	Dic₃	—	3	2	no	—
111	Dic₃	—	6	1	no	—
112	Dic₃	—	12	1	no	—
113	Dic₃	—	12	1	no	—
114	C₉	#462	1	1	yes	characteristic, complemented, Sylow
115	Q₈	—	2	1	no	—
116	Q₈	—	2	1	no	—
117	Q₈	—	2	1	no	—
118	Q₈	—	18	1	no	—
119	Q₈	—	18	1	no	—
120	Q₈	#611	1	2	yes	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	9	1	no	—
124	C₂×C₄	—	9	1	no	—
125	C₂×C₄	—	18	1	no	—
126	C₂×C₄	—	18	1	no	—
127	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	18	1	no	—
130	C₂×C₃	D₃×Q₁₆	1	1	yes	characteristic
131	C₂×C₃	#918	1	1	yes	characteristic
132	C₂×C₃	#939	1	1	yes	characteristic
133	C₂²	D₄×D₉	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	4	1	no	—
136	C₄	—	9	2	no	—
137	C₄	—	18	1	no	—
138	C₄	—	36	1	no	—
139	C₄	—	36	1	no	—
140	C₄	#3699	1	1	yes	characteristic
141	C₄	C₂×#611	1	1	yes	characteristic
142	C₃	#5182	1	1	yes	characteristic
143	C₂	Q₁₆×D₉	1	1	yes	characteristic
144	C₂	#63480	1	1	yes	characteristic
145	C₂	#63501	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₉×Q₁₆

C₂

yes

maximal, characteristic

Q₈×Dic₉

C₂

yes

maximal, characteristic

#63506

C₂

yes

maximal, characteristic

C₂×#3674

C₂

yes

maximal, characteristic

#63395

C₂

yes

maximal, characteristic

#63376

C₂

yes

maximal, characteristic

Dic₉⋊C₈

C₂

yes

maximal, characteristic

#5182

—

maximal

C₂×Q₈×C₉

C₂²

yes

characteristic

C₂×Q₈×C₉

C₂²

yes

characteristic

C₂×Dic₁₈

C₂²

yes

characteristic

C₉×Q₁₆

—

C₂×C₈×C₉

C₂²

yes

characteristic

#3674

—

C₄⋊Dic₉

—

C₄⋊Dic₉

C₂²

yes

characteristic

Dic₉⋊C₄

—

Dic₉⋊C₄

—

C₄×Dic₉

—

C₄×Dic₉

C₂²

yes

characteristic

C₂×C₉⋊C₈

C₂²

yes

characteristic

C₂×C₃×Q₁₆

D₃

yes

characteristic

Q₈×Dic₃

—

—

—

—

—

Dic₃⋊C₈

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

D₄

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂³

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

D₄

yes

characteristic

C₂×Dic₉

D₄

yes

characteristic

Dic₁₈

—

Dic₁₈

—

C₈×C₉

—

C₉⋊C₈

—

#462

—

Sylow

C₂×C₃×Q₈

C₂×D₃

yes

characteristic

C₂×C₃×Q₈

C₂×D₃

yes

characteristic

C₂×Dic₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

C₂×D₃

yes

characteristic

#216

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂²×C₉

C₂×D₄

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×D₄

yes

characteristic

C₄×C₉

#41

yes

characteristic

Dic₉

—

Dic₉

—

Dic₉

—

Dic₉

Q₁₆

yes

complemented

C₂×Q₁₆

—

C₂×Q₁₆

D₉

yes

characteristic

C₄⋊Q₈

—

C₄×Q₈

—

C₄⋊C₈

—

#103

—

#103

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

#67

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×D₃

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₉

#121

yes

characteristic

C₂×C₉

C₂×Q₁₆

yes

characteristic

C₂×C₉

#137

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂×D₉

yes

characteristic

C₂×Q₈

C₂×D₉

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

C₂×D₉

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₂²×C₃

D₃×D₄

yes

characteristic

106

C₃×C₄

—

107

C₃×C₄

—

108

C₃×C₄

#237

yes

characteristic

109

C₃×C₄

C₂×#67

yes

characteristic

110

Dic₃

—

111

Dic₃

—

112

Dic₃

—

113

Dic₃

—

114

C₉

#462

yes

characteristic, complemented, Sylow

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

Q₈

#611

yes

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₂²×D₉

yes

characteristic

128

C₈

—

129

C₈

—

130

C₂×C₃

D₃×Q₁₆

yes

characteristic

131

C₂×C₃

#918

yes

characteristic

132

C₂×C₃

#939

yes

characteristic

133

C₂²

D₄×D₉

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#3699

yes

characteristic

141

C₄

C₂×#611

yes

characteristic

142

C₃

#5182

yes

characteristic

143

C₂

Q₁₆×D₉

yes

characteristic

144

C₂

#63480

yes

characteristic

145

C₂

#63501

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₉
commutator, cosocle	C₂×C₄×C₉	C₂³
Frattini	C₂×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×Q₁₆	C₂
socle, prime radical	C₂²×C₃	D₃×D₄
hypercenter	C₂×Q₁₆	D₉
hypocenter	C₉	#462
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×D₉

commutator, cosocle

C₂×C₄×C₉

C₂³

Frattini

C₂×C₃×C₄

C₂²×D₃

Fitting

C₂×C₉×Q₁₆

C₂

socle, prime radical

C₂²×C₃

D₃×D₄

hypercenter

C₂×Q₁₆

D₉

hypocenter

C₉

#462

LCS

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
66. Dic₉	Q₁₆	semidirect
114. C₉	#462	semidirect

H ≅ G/N

type

66. Dic₉

Q₁₆

semidirect

114. C₉

#462

semidirect

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

subgroup	index
C₂×C₉×Q₁₆	2
Q₈×Dic₉	2
#63506	2
C₂×#3674	2
#63395	2
#63376	2
Dic₉⋊C₈	2
#5182	3

subgroup

index

C₂×C₉×Q₁₆

Q₈×Dic₉

Dic₉⋊C₈

Subgroups — #10588862

Conjugacy classes of subgroups

Complements

Maximal subgroups