Group #10589995 — subgroups

Subgroups — C₉×#563 #10589995

Order 576.

294 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 213 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×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
2	C₉×#126	C₂	1	4	yes	maximal, complemented
3	C₉×#125	C₂	1	2	yes	maximal, complemented
4	C₉×#124	C₂	1	1	yes	maximal, characteristic
5	C₉×#123	C₂	1	2	yes	maximal
6	C₉×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
7	C₃×#563	C₃	1	1	yes	maximal, characteristic
8	C₂×Q₈×C₉	C₂²	1	2	yes	—
9	C₂×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×D₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₉	C₂²	1	2	yes	—
12	C₉×C₄⋊C₄	C₂²	1	2	yes	—
13	C₉×C₄⋊C₄	C₂²	1	4	yes	—
14	C₉×C₄⋊C₄	C₂²	1	4	yes	—
15	C₉×C₄⋊C₄	C₂²	1	4	yes	—
16	C₉×C₄⋊C₄	C₂²	1	2	yes	—
17	C₉×C₂²⋊C₄	C₂²	1	4	yes	—
18	C₉×C₂²⋊C₄	C₂²	1	4	yes	—
19	C₉×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₄²×C₉	C₂²	1	1	yes	characteristic
21	C₄²×C₉	C₂²	1	2	yes	—
22	C₃×C₄⋊Q₈	C₂×C₃	1	2	yes	—
23	C₃×#126	C₂×C₃	1	4	yes	—
24	C₃×#125	C₂×C₃	1	2	yes	—
25	C₃×#124	C₂×C₃	1	1	yes	characteristic
26	C₃×#123	C₂×C₃	1	2	yes	—
27	C₃×C₂²⋊Q₈	C₂×C₃	1	4	yes	—
28	C₂³×C₉	C₂³	1	2	yes	—
29	Q₈×C₉	—	4	2	no	—
30	Q₈×C₉	—	4	1	no	—
31	D₄×C₉	—	4	1	no	—
32	C₂×C₄×C₉	—	4	2	no	—
33	C₂×C₄×C₉	C₂³	1	2	yes	—
34	C₂×C₄×C₉	C₂³	1	4	yes	—
35	C₂×C₄×C₉	C₂³	1	2	yes	—
36	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₉	C₂³	1	4	yes	—
38	#563	C₉	1	1	yes	characteristic, direct product factor, Sylow
39	C₂×C₃×Q₈	C₂²×C₃	1	2	yes	—
40	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂²×C₃	1	2	yes	—
43	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
44	C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
45	C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
46	C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
47	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
48	C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
49	C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
50	C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
51	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
52	C₃×C₄²	C₂²×C₃	1	2	yes	—
53	C₂²×C₉	—	2	2	no	—
54	C₂²×C₉	—	2	4	no	—
55	C₂²×C₉	C₂⁴	1	1	yes	characteristic
56	C₄×C₉	—	2	4	no	—
57	C₄×C₉	—	2	1	no	—
58	C₄×C₉	—	2	2	no	—
59	C₄×C₉	—	2	2	no	—
60	C₄×C₉	—	2	4	no	—
61	C₄⋊Q₈	C₂×C₉	1	2	yes	complemented
62	#126	C₂×C₉	1	4	yes	complemented
63	#125	C₂×C₉	1	2	yes	complemented
64	#124	C₂×C₉	1	1	yes	characteristic
65	#123	C₂×C₉	1	2	yes	—
66	C₂²⋊Q₈	C₂×C₉	1	4	yes	complemented
67	C₂³×C₃	C₂³×C₃	1	2	yes	—
68	C₃×Q₈	—	4	2	no	—
69	C₃×Q₈	—	4	1	no	—
70	C₃×D₄	—	4	1	no	—
71	C₂×C₃×C₄	—	4	2	no	—
72	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
73	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
74	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
75	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
76	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
77	C₂×C₉	—	4	2	no	—
78	C₂×C₉	H(5,𝔽₂)	1	1	yes	characteristic
79	C₂×C₉	2₋¹⁺⁴	1	2	yes	—
80	C₂×Q₈	C₂²×C₉	1	2	yes	—
81	C₂×Q₈	C₂²×C₉	1	1	yes	characteristic
82	C₂×D₄	C₂²×C₉	1	1	yes	characteristic
83	C₂²×C₄	C₂²×C₉	1	2	yes	—
84	C₄⋊C₄	C₂²×C₉	1	2	yes	—
85	C₄⋊C₄	C₂²×C₉	1	4	yes	—
86	C₄⋊C₄	C₂²×C₉	1	4	yes	—
87	C₄⋊C₄	C₂²×C₉	1	4	yes	—
88	C₄⋊C₄	C₂²×C₉	1	2	yes	—
89	C₂²⋊C₄	C₂²×C₉	1	4	yes	—
90	C₂²⋊C₄	C₂²×C₉	1	4	yes	—
91	C₂²⋊C₄	C₂²×C₉	1	2	yes	—
92	C₄²	C₂²×C₉	1	1	yes	characteristic
93	C₄²	C₂²×C₉	1	2	yes	—
94	C₂²×C₃	—	2	2	no	—
95	C₂²×C₃	—	2	4	no	—
96	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
97	C₃×C₄	—	2	4	no	—
98	C₃×C₄	—	2	1	no	—
99	C₃×C₄	—	2	2	no	—
100	C₃×C₄	—	2	2	no	—
101	C₃×C₄	—	2	4	no	—
102	C₉	#563	1	1	yes	characteristic, direct product factor, Sylow
103	C₂³	C₂³×C₉	1	2	yes	—
104	Q₈	—	4	2	no	—
105	Q₈	—	4	1	no	—
106	D₄	—	4	1	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	C₂³×C₉	1	2	yes	—
109	C₂×C₄	C₂³×C₉	1	4	yes	—
110	C₂×C₄	C₂³×C₉	1	2	yes	—
111	C₂×C₄	C₂³×C₉	1	1	yes	characteristic
112	C₂×C₄	C₂³×C₉	1	4	yes	—
113	C₂×C₃	—	4	2	no	—
114	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	4	no	—
118	C₂²	C₂⁴×C₉	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	2	1	no	—
121	C₄	—	2	2	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	4	no	—
124	C₃	C₃×#563	1	1	yes	characteristic
125	C₂	—	4	2	no	—
126	C₂	C₉×H(5,𝔽₂)	1	1	yes	characteristic
127	C₂	C₉×2₋¹⁺⁴	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×C₄⋊Q₈

C₂

yes

maximal, complemented

C₉×#126

C₂

yes

maximal, complemented

C₉×#125

C₂

yes

maximal, complemented

C₉×#124

C₂

yes

maximal, characteristic

C₉×#123

C₂

yes

maximal

C₉×C₂²⋊Q₈

C₂

yes

maximal, complemented

C₃×#563

C₃

yes

maximal, characteristic

C₂×Q₈×C₉

C₂²

yes

—

C₂×Q₈×C₉

C₂²

yes

characteristic

C₂×D₄×C₉

C₂²

yes

characteristic

C₂²×C₄×C₉

C₂²

yes

—

C₉×C₄⋊C₄

C₂²

yes

—

C₉×C₄⋊C₄

C₂²

yes

—

C₉×C₄⋊C₄

C₂²

yes

—

C₉×C₄⋊C₄

C₂²

yes

—

C₉×C₄⋊C₄

C₂²

yes

—

C₉×C₂²⋊C₄

C₂²

yes

—

C₉×C₂²⋊C₄

C₂²

yes

—

C₉×C₂²⋊C₄

C₂²

yes

—

C₄²×C₉

C₂²

yes

characteristic

C₄²×C₉

C₂²

yes

—

C₃×C₄⋊Q₈

C₂×C₃

yes

—

C₃×#126

C₂×C₃

yes

—

C₃×#125

C₂×C₃

yes

—

C₃×#124

C₂×C₃

yes

characteristic

C₃×#123

C₂×C₃

yes

—

C₃×C₂²⋊Q₈

C₂×C₃

yes

—

C₂³×C₉

C₂³

yes

—

Q₈×C₉

—

Q₈×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂³

yes

—

C₂×C₄×C₉

C₂³

yes

—

C₂×C₄×C₉

C₂³

yes

—

C₂×C₄×C₉

C₂³

yes

characteristic

C₂×C₄×C₉

C₂³

yes

—

#563

C₉

yes

characteristic, direct product factor, Sylow

C₂×C₃×Q₈

C₂²×C₃

yes

—

C₂×C₃×Q₈

C₂²×C₃

yes

characteristic

C₂×C₃×D₄

C₂²×C₃

yes

characteristic

C₂²×C₃×C₄

C₂²×C₃

yes

—

C₃×C₄⋊C₄

C₂²×C₃

yes

—

C₃×C₄⋊C₄

C₂²×C₃

yes

—

C₃×C₄⋊C₄

C₂²×C₃

yes

—

C₃×C₄⋊C₄

C₂²×C₃

yes

—

C₃×C₄⋊C₄

C₂²×C₃

yes

—

C₃×C₂²⋊C₄

C₂²×C₃

yes

—

C₃×C₂²⋊C₄

C₂²×C₃

yes

—

C₃×C₂²⋊C₄

C₂²×C₃

yes

—

C₃×C₄²

C₂²×C₃

yes

characteristic

C₃×C₄²

C₂²×C₃

yes

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂⁴

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄⋊Q₈

C₂×C₉

yes

complemented

#126

C₂×C₉

yes

complemented

#125

C₂×C₉

yes

complemented

#124

C₂×C₉

yes

characteristic

#123

C₂×C₉

yes

—

C₂²⋊Q₈

C₂×C₉

yes

complemented

C₂³×C₃

yes

—

C₃×Q₈

—

C₃×Q₈

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂³×C₃

yes

—

C₂×C₃×C₄

C₂³×C₃

yes

—

C₂×C₃×C₄

C₂³×C₃

yes

—

C₂×C₃×C₄

C₂³×C₃

yes

characteristic

C₂×C₃×C₄

C₂³×C₃

yes

—

C₂×C₉

—

C₂×C₉

H(5,𝔽₂)

yes

characteristic

C₂×C₉

2₋¹⁺⁴

yes

—

C₂×Q₈

C₂²×C₉

yes

—

C₂×Q₈

C₂²×C₉

yes

characteristic

C₂×D₄

C₂²×C₉

yes

characteristic

C₂²×C₄

C₂²×C₉

yes

—

C₄⋊C₄

C₂²×C₉

yes

—

C₄⋊C₄

C₂²×C₉

yes

—

C₄⋊C₄

C₂²×C₉

yes

—

C₄⋊C₄

C₂²×C₉

yes

—

C₄⋊C₄

C₂²×C₉

yes

—

C₂²⋊C₄

C₂²×C₉

yes

—

C₂²⋊C₄

C₂²×C₉

yes

—

C₂²⋊C₄

C₂²×C₉

yes

—

C₄²

C₂²×C₉

yes

characteristic

C₄²

C₂²×C₉

yes

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂⁴×C₃

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

—

102

C₉

#563

yes

characteristic, direct product factor, Sylow

103

C₂³

C₂³×C₉

yes

—

104

Q₈

—

105

Q₈

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂³×C₉

yes

—

109

C₂×C₄

C₂³×C₉

yes

—

110

C₂×C₄

C₂³×C₉

yes

—

111

C₂×C₄

C₂³×C₉

yes

characteristic

112

C₂×C₄

C₂³×C₉

yes

—

113

C₂×C₃

—

114

C₂×C₃

C₃×H(5,𝔽₂)

yes

characteristic

115

C₂×C₃

C₃×2₋¹⁺⁴

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂⁴×C₉

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₃

C₃×#563

yes

characteristic

125

C₂

—

126

C₂

C₉×H(5,𝔽₂)

yes

characteristic

127

C₂

C₉×2₋¹⁺⁴

yes

—

	subgroup	G/H
center	C₂²×C₉	C₂⁴
commutator	C₂²	C₂⁴×C₉
Frattini, socle, cosocle	C₂²×C₃	C₂⁴×C₃
prime radical	C₂×C₃×D₄	C₂²×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ G

subgroup

G/H

center

C₂²×C₉

C₂⁴

commutator

C₂²

C₂⁴×C₉

Frattini, socle, cosocle

C₂²×C₃

C₂⁴×C₃

prime radical

C₂×C₃×D₄

C₂²×C₃

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₉ ◁ G

N	H ≅ G/N	type
1. C₉×C₄⋊Q₈	C₂	semidirect
2. C₉×#126	C₂	semidirect
3. C₉×#125	C₂	semidirect
6. C₉×C₂²⋊Q₈	C₂	semidirect
38. #563	C₉	direct (N × H)
61. C₄⋊Q₈	C₂×C₉	semidirect
62. #126	C₂×C₉	semidirect
63. #125	C₂×C₉	semidirect
66. C₂²⋊Q₈	C₂×C₉	semidirect
102. C₉	#563	direct (N × H)

H ≅ G/N

type

1. C₉×C₄⋊Q₈

C₂

semidirect

2. C₉×#126

C₂

semidirect

3. C₉×#125

C₂

semidirect

6. C₉×C₂²⋊Q₈

C₂

semidirect

38. #563

C₉

direct (N × H)

61. C₄⋊Q₈

C₂×C₉

semidirect

62. #126

C₂×C₉

semidirect

63. #125

C₂×C₉

semidirect

66. C₂²⋊Q₈

C₂×C₉

semidirect

102. C₉

#563

direct (N × H)

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

subgroup	index
C₉×C₄⋊Q₈	2
C₉×#126	2
C₉×#125	2
C₉×#124	2
C₉×#123	2
C₉×C₂²⋊Q₈	2
C₃×#563	3

subgroup

index

C₉×C₄⋊Q₈

C₉×C₂²⋊Q₈

Subgroups — C₉×#563 #10589995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#563 #10589995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#563 #10589995