Group #10589669 — subgroups

Subgroups — #10589669

Order 576.

300 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 131 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	C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	#63506	C₂	1	2	yes	maximal
3	#63451	C₂	1	4	yes	maximal
4	#63449	C₂	1	2	yes	maximal
5	#63432	C₂	1	2	yes	maximal
6	#63431	C₂	1	1	yes	maximal, characteristic
7	C₄×Dic₁₈	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₁₈	C₂	1	2	yes	maximal
9	#5567	—	3	1	no	maximal
10	C₂×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₈	C₂²	1	2	yes	—
12	C₂×Dic₁₈	C₂²	1	1	yes	characteristic
13	C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₉×C₄⋊C₄	C₂²	1	2	yes	—
15	C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₄²×C₉	C₂²	1	2	yes	—
17	C₄⋊Dic₉	C₂²	1	2	yes	—
18	C₄⋊Dic₉	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₉	C₂²	1	2	yes	—
20	Dic₉⋊C₄	C₂²	1	4	yes	—
21	Dic₉⋊C₄	C₂²	1	4	yes	—
22	Dic₉⋊C₄	C₂²	1	2	yes	—
23	Dic₉⋊C₄	C₂²	1	4	yes	—
24	C₄×Dic₉	C₂²	1	2	yes	—
25	C₄×Dic₉	C₂²	1	4	yes	—
26	C₃×C₄×Q₈	D₃	1	1	yes	characteristic
27	#944	—	3	2	no	—
28	#889	—	3	4	no	—
29	#887	—	3	2	no	—
30	#870	—	3	2	no	—
31	#869	—	3	1	no	—
32	C₄×Dic₆	—	3	2	no	—
33	C₄×Dic₆	—	3	1	no	—
34	Q₈×C₉	—	2	2	no	—
35	C₂×C₄×C₉	C₂³	1	2	yes	—
36	C₂×C₄×C₉	C₂³	1	2	yes	—
37	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂×Dic₉	C₂³	1	2	yes	—
41	C₂×Dic₉	C₂³	1	4	yes	—
42	C₂×Dic₉	C₂³	1	2	yes	—
43	Dic₁₈	—	2	4	no	—
44	Dic₁₈	Q₈	1	4	yes	complemented
45	#557	—	9	1	no	Sylow
46	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
47	C₂×Dic₆	—	3	1	no	—
48	C₂×Dic₆	—	3	2	no	—
49	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
50	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
51	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
52	C₃×C₄²	C₂×D₃	1	2	yes	—
53	C₄⋊Dic₃	—	3	2	no	—
54	C₄⋊Dic₃	—	3	1	no	—
55	C₄⋊Dic₃	—	3	2	no	—
56	Dic₃⋊C₄	—	3	2	no	—
57	Dic₃⋊C₄	—	3	4	no	—
58	Dic₃⋊C₄	—	3	4	no	—
59	Dic₃⋊C₄	—	3	4	no	—
60	C₄×Dic₃	—	3	4	no	—
61	C₄×Dic₃	—	3	2	no	—
62	C₂²×C₉	C₂⁴	1	1	yes	characteristic
63	C₄×C₉	—	2	2	no	—
64	C₄×C₉	—	2	2	no	—
65	C₄×C₉	—	2	1	no	—
66	C₄×C₉	C₂×Q₈	1	2	yes	—
67	C₄×C₉	#41	1	2	yes	—
68	Dic₉	—	2	4	no	—
69	Dic₉	—	2	2	no	—
70	Dic₉	C₂×Q₈	1	4	yes	—
71	C₄⋊Q₈	—	9	2	no	—
72	C₄⋊Q₈	—	9	1	no	—
73	#125	—	9	4	no	—
74	#125	—	9	2	no	—
75	C₄×Q₈	—	9	2	no	—
76	C₄×Q₈	—	9	2	no	—
77	C₄×Q₈	—	9	1	no	—
78	C₄×Q₈	D₉	1	1	yes	characteristic
79	C₃×Q₈	—	2	2	no	—
80	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
81	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
82	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
83	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂×Dic₃	—	3	2	no	—
86	C₂×Dic₃	—	3	2	no	—
87	C₂×Dic₃	—	3	4	no	—
88	Dic₆	—	3	4	no	—
89	Dic₆	—	6	4	no	—
90	C₂×C₉	C₂²×Q₈	1	1	yes	characteristic
91	C₂×C₉	C₂×#41	1	1	yes	characteristic
92	C₂×C₉	2₋¹⁺⁴	1	1	yes	characteristic
93	C₂×Q₈	—	9	1	no	—
94	C₂×Q₈	—	9	2	no	—
95	C₂×Q₈	C₂×D₉	1	1	yes	characteristic
96	C₄⋊C₄	—	9	2	no	—
97	C₄⋊C₄	—	9	4	no	—
98	C₄⋊C₄	—	9	1	no	—
99	C₄⋊C₄	—	9	4	no	—
100	C₄⋊C₄	—	9	2	no	—
101	C₄⋊C₄	—	9	4	no	—
102	C₄⋊C₄	—	9	2	no	—
103	C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
104	C₄⋊C₄	C₂×D₉	1	2	yes	—
105	C₄²	—	9	4	no	—
106	C₄²	—	9	2	no	—
107	C₄²	C₂×D₉	1	2	yes	—
108	C₄²	C₂×D₉	1	1	yes	characteristic
109	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
110	C₃×C₄	—	2	2	no	—
111	C₃×C₄	—	2	2	no	—
112	C₃×C₄	—	2	1	no	—
113	C₃×C₄	#235	1	2	yes	—
114	C₃×C₄	D₃×Q₈	1	2	yes	—
115	Dic₃	—	3	4	no	—
116	Dic₃	—	6	4	no	—
117	Dic₃	—	6	2	no	—
118	C₉	#557	1	1	yes	characteristic, complemented, Sylow
119	Q₈	—	2	2	no	—
120	Q₈	—	9	4	no	—
121	Q₈	—	18	4	no	—
122	C₂×C₄	—	9	2	no	—
123	C₂×C₄	—	9	2	no	—
124	C₂×C₄	—	9	4	no	—
125	C₂×C₄	C₂²×D₉	1	2	yes	—
126	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
127	C₂×C₄	C₂²×D₉	1	2	yes	—
128	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
129	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
130	C₂×C₃	C₂×#235	1	1	yes	characteristic
131	C₂×C₃	C₂×D₃×Q₈	1	1	yes	characteristic
132	C₂×C₃	#1010	1	1	yes	characteristic
133	C₂²	C₂³×D₉	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	2	2	no	—
136	C₄	—	2	1	no	—
137	C₄	—	9	4	no	—
138	C₄	—	18	4	no	—
139	C₄	—	18	2	no	—
140	C₄	#3697	1	2	yes	—
141	C₄	Q₈×D₉	1	2	yes	—
142	C₃	#5567	1	1	yes	characteristic
143	C₂	C₂×#3697	1	1	yes	characteristic
144	C₂	C₂×Q₈×D₉	1	1	yes	characteristic
145	C₂	#63716	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Q₈×C₉

C₂

yes

maximal, characteristic

#63506

C₂

yes

maximal

#63451

C₂

yes

maximal

#63449

C₂

yes

maximal

#63432

C₂

yes

maximal

#63431

C₂

yes

maximal, characteristic

C₄×Dic₁₈

C₂

yes

maximal, characteristic

C₄×Dic₁₈

C₂

yes

maximal

#5567

—

maximal

C₂×Q₈×C₉

C₂²

yes

characteristic

C₂×Dic₁₈

C₂²

yes

—

C₂×Dic₁₈

C₂²

yes

characteristic

C₉×C₄⋊C₄

C₂²

yes

characteristic

C₉×C₄⋊C₄

C₂²

yes

—

C₄²×C₉

C₂²

yes

characteristic

C₄²×C₉

C₂²

yes

—

C₄⋊Dic₉

C₂²

yes

—

C₄⋊Dic₉

C₂²

yes

characteristic

C₄⋊Dic₉

C₂²

yes

—

Dic₉⋊C₄

C₂²

yes

—

Dic₉⋊C₄

C₂²

yes

—

Dic₉⋊C₄

C₂²

yes

—

Dic₉⋊C₄

C₂²

yes

—

C₄×Dic₉

C₂²

yes

—

C₄×Dic₉

C₂²

yes

—

C₃×C₄×Q₈

D₃

yes

—

—

—

—

—

C₄×Dic₆

—

C₄×Dic₆

—

Q₈×C₉

—

C₂×C₄×C₉

C₂³

yes

—

C₂×C₄×C₉

C₂³

yes

—

C₂×C₄×C₉

C₂³

yes

characteristic

C₂×C₄×C₉

C₂³

yes

characteristic

C₂×C₄×C₉

C₂³

yes

characteristic

C₂×Dic₉

C₂³

yes

—

C₂×Dic₉

C₂³

yes

—

C₂×Dic₉

C₂³

yes

—

Dic₁₈

—

Dic₁₈

Q₈

yes

complemented

#557

—

Sylow

C₂×C₃×Q₈

C₂×D₃

yes

characteristic

C₂×Dic₆

—

C₂×Dic₆

—

C₃×C₄⋊C₄

C₂×D₃

yes

characteristic

C₃×C₄⋊C₄

C₂×D₃

yes

—

C₃×C₄²

C₂×D₃

yes

characteristic

C₃×C₄²

C₂×D₃

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₉

C₂⁴

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×Q₈

yes

—

C₄×C₉

#41

yes

—

Dic₉

—

Dic₉

—

Dic₉

C₂×Q₈

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₉

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

C₂²×D₃

yes

—

C₂×C₃×C₄

C₂²×D₃

yes

—

C₂×C₃×C₄

C₂²×D₃

yes

characteristic

C₂×C₃×C₄

C₂²×D₃

yes

characteristic

C₂×C₃×C₄

C₂²×D₃

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₂×C₉

C₂²×Q₈

yes

characteristic

C₂×C₉

C₂×#41

yes

characteristic

C₂×C₉

2₋¹⁺⁴

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

C₂×D₉

yes

characteristic

104

C₄⋊C₄

C₂×D₉

yes

—

105

C₄²

—

106

C₄²

—

107

C₄²

C₂×D₉

yes

—

108

C₄²

C₂×D₉

yes

characteristic

109

C₂²×C₃

C₂³×D₃

yes

characteristic

110

C₃×C₄

—

111

C₃×C₄

—

112

C₃×C₄

—

113

C₃×C₄

#235

yes

—

114

C₃×C₄

D₃×Q₈

yes

—

115

Dic₃

—

116

Dic₃

—

117

Dic₃

—

118

C₉

#557

yes

characteristic, complemented, Sylow

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂²×D₉

yes

—

126

C₂×C₄

C₂²×D₉

yes

characteristic

127

C₂×C₄

C₂²×D₉

yes

—

128

C₂×C₄

C₂²×D₉

yes

characteristic

129

C₂×C₄

C₂²×D₉

yes

characteristic

130

C₂×C₃

C₂×#235

yes

characteristic

131

C₂×C₃

C₂×D₃×Q₈

yes

characteristic

132

C₂×C₃

#1010

yes

characteristic

133

C₂²

C₂³×D₉

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#3697

yes

—

141

C₄

Q₈×D₉

yes

—

142

C₃

#5567

yes

characteristic

143

C₂

C₂×#3697

yes

characteristic

144

C₂

C₂×Q₈×D₉

yes

characteristic

145

C₂

#63716

yes

characteristic

	subgroup	G/H
center	C₂²	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴
Frattini, socle, prime radical	C₂²×C₃	C₂³×D₃
Fitting	C₄×Q₈×C₉	C₂
hypercenter	C₄×Q₈	D₉
hypocenter	C₉	#557
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center

C₂²

C₂³×D₉

commutator, cosocle

C₂²×C₉

C₂⁴

Frattini, socle, prime radical

C₂²×C₃

C₂³×D₃

Fitting

C₄×Q₈×C₉

C₂

hypercenter

C₄×Q₈

D₉

C₉

G ▷ C₂²×C₉ ▷ C₉

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
44. Dic₁₈	Q₈	semidirect
118. C₉	#557	semidirect

H ≅ G/N

type

44. Dic₁₈

Q₈

semidirect

118. C₉

#557

semidirect

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

subgroup	index
C₄×Q₈×C₉	2
#63506	2
#63451	2
#63449	2
#63432	2
#63431	2
C₄×Dic₁₈	2
#5567	3

subgroup

index

C₄×Q₈×C₉

C₄×Dic₁₈

Subgroups — #10589669

Conjugacy classes of subgroups

Complements

Maximal subgroups