Group #10592454 — subgroups

Subgroups — #10592454

Order 576.

300 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 75 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×#3756	C₂	1	1	yes	maximal, characteristic, complemented
2	#64151	C₂	1	1	yes	maximal, characteristic
3	#64118	C₂	1	1	yes	maximal, characteristic
4	#64116	C₂	1	1	yes	maximal, characteristic
5	C₃²×#101	C₂	1	1	yes	maximal, characteristic, complemented
6	#63651	C₂	1	1	yes	maximal, characteristic, complemented
7	#63649	C₂	1	1	yes	maximal, characteristic, complemented
8	#4755	—	3	4	no	maximal
9	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
10	#3828	C₂²	1	1	yes	characteristic
11	C₂×#634	C₂²	1	1	yes	characteristic
12	C₃²×M₁₆	C₂²	1	1	yes	characteristic
13	C₃²×M₁₆	C₂²	1	1	yes	characteristic
14	#3756	—	2	1	no	—
15	#3756	—	2	1	no	—
16	#3756	—	2	1	no	—
17	#3755	—	2	1	no	—
18	#3748	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
20	#3746	—	2	1	no	—
21	#3744	—	2	1	no	—
22	#3743	—	2	1	no	—
23	C₈×C₃⋊D₃	—	2	1	no	—
24	C₂×#216	—	3	4	no	—
25	#942	—	3	4	no	—
26	#909	—	3	4	no	—
27	#907	—	3	4	no	—
28	C₃×#101	D₃	1	4	yes	complemented
29	#824	—	3	4	no	—
30	#822	—	3	4	no	—
31	C₃²×Q₈	—	2	1	no	—
32	C₃²×Q₈	—	2	1	no	—
33	C₂×C₃²×C₄	—	2	1	no	—
34	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
35	#638	—	2	1	no	—
36	C₂×C₃⋊Dic₃	—	2	1	no	—
37	C₄⋊(C₃⋊D₃)	D₄	1	1	yes	characteristic
38	C₄×C₃⋊D₃	—	2	1	no	—
39	#634	—	2	1	no	—
40	#634	—	2	1	no	—
41	#634	D₄	1	1	yes	characteristic
42	C₃²×C₈	—	2	1	no	—
43	C₃²×C₈	—	2	1	no	—
44	C₃⋊(C₃⋊C₈)	—	2	1	no	—
45	C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
46	C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
47	#473	—	9	1	no	Sylow
48	C₂×C₃×Q₈	C₂×D₃	1	4	yes	—
49	#235	—	3	4	no	—
50	C₂×Dic₆	—	3	4	no	—
51	C₃×M₁₆	C₂×D₃	1	4	yes	—
52	C₃×M₁₆	C₂×D₃	1	4	yes	—
53	#216	—	6	4	no	—
54	#216	—	6	4	no	—
55	#216	—	6	4	no	—
56	#215	—	6	4	no	—
57	#208	—	3	4	no	—
58	C₂×C₃⋊C₈	—	3	4	no	—
59	Dic₁₂	—	6	4	no	—
60	#204	—	6	4	no	—
61	ΓL(1,𝔽₂₅)	—	6	4	no	—
62	D₃×C₈	—	6	4	no	—
63	C₂²×C₃²	#41	1	1	yes	characteristic
64	C₂×C₃⋊D₃	—	2	1	no	—
65	C₃²×C₄	—	4	1	no	—
66	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
67	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
68	C₃⋊Dic₃	—	2	1	no	—
69	C₃⋊Dic₃	—	4	1	no	—
70	#137	—	9	1	no	—
71	#137	—	9	1	no	—
72	C₂×Q₁₆	—	9	1	no	—
73	#131	—	9	1	no	—
74	#108	—	9	1	no	—
75	#101	—	9	1	no	—
76	#101	C₃⋊D₃	1	1	yes	characteristic, complemented
77	C₃×Q₈	—	2	4	no	—
78	C₃×Q₈	—	2	4	no	—
79	C₂×C₃×C₄	—	2	4	no	—
80	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
81	#67	—	6	4	no	—
82	C₂×Dic₃	—	6	4	no	—
83	D₁₂	—	3	4	no	—
84	C₄×D₃	—	6	4	no	—
85	Dic₆	—	3	4	no	—
86	Dic₆	—	6	4	no	—
87	Dic₆	—	6	4	no	—
88	C₃×C₈	—	2	4	no	—
89	C₃×C₈	—	2	4	no	—
90	C₃⋊C₈	—	3	4	no	—
91	C₃⋊C₈	—	3	4	no	—
92	C₃⋊C₈	—	6	4	no	—
93	C₂×C₃²	—	2	1	no	—
94	C₂×C₃²	#121	1	1	yes	characteristic
95	C₃⋊D₃	—	4	1	no	—
96	#41	—	9	1	no	—
97	C₂×Q₈	—	9	1	no	—
98	C₂×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
99	Q₁₆	—	18	1	no	—
100	Q₁₆	—	18	1	no	—
101	Q₁₆	—	18	1	no	—
102	Q₁₆	—	18	1	no	—
103	SD₁₆	—	18	1	no	—
104	SD₁₆	—	18	1	no	—
105	M₁₆	—	9	1	no	—
106	M₁₆	—	18	1	no	—
107	M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
108	M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
109	C₂×C₈	—	9	1	no	—
110	C₂×C₈	—	18	1	no	—
111	C₂²×C₃	#235	1	4	yes	—
112	C₂×D₃	—	6	4	no	—
113	C₃×C₄	—	4	4	no	—
114	C₃×C₄	D₃×D₄	1	4	yes	—
115	C₃×C₄	D₃×D₄	1	4	yes	—
116	Dic₃	—	6	4	no	—
117	Dic₃	—	12	4	no	—
118	C₃²	#473	1	1	yes	characteristic, complemented, Sylow
119	Q₈	—	2	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	—	9	1	no	—
122	Q₈	—	18	1	no	—
123	Q₈	—	18	1	no	—
124	D₄	—	9	1	no	—
125	D₄	—	18	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	18	1	no	—
128	C₂×C₄	—	18	1	no	—
129	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	9	1	no	—
133	C₈	—	9	1	no	—
134	C₈	—	18	1	no	—
135	C₂×C₃	—	2	4	no	—
136	C₂×C₃	#884	1	4	yes	—
137	D₃	—	12	4	no	—
138	C₂²	—	18	1	no	—
139	C₂²	#3828	1	1	yes	characteristic
140	C₄	—	4	1	no	—
141	C₄	—	18	1	no	—
142	C₄	—	36	1	no	—
143	C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
144	C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
145	C₃	#4755	1	4	yes	complemented
146	C₂	—	2	1	no	—
147	C₂	—	36	1	no	—
148	C₂	#64093	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#3756

C₂

yes

maximal, characteristic, complemented

#64151

C₂

yes

maximal, characteristic

#64118

C₂

yes

maximal, characteristic

#64116

C₂

yes

maximal, characteristic

C₃²×#101

C₂

yes

maximal, characteristic, complemented

#63651

C₂

yes

maximal, characteristic, complemented

#63649

C₂

yes

maximal, characteristic, complemented

#4755

—

maximal

C₂×C₃²×Q₈

C₂²

yes

characteristic

#3828

C₂²

yes

characteristic

C₂×#634

C₂²

yes

characteristic

C₃²×M₁₆

C₂²

yes

characteristic

C₃²×M₁₆

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₂²

yes

—

—

—

C₈×C₃⋊D₃

—

—

—

—

—

D₃

yes

complemented

#824

—

#822

—

C₃²×Q₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂³

yes

characteristic

#638

—

C₂×C₃⋊Dic₃

—

C₄⋊(C₃⋊D₃)

D₄

yes

characteristic

C₄×C₃⋊D₃

—

#634

—

#634

—

#634

D₄

yes

characteristic

C₃²×C₈

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

D₄

yes

characteristic

C₃⋊(C₃⋊C₈)

D₄

yes

characteristic

#473

—

Sylow

C₂×C₃×Q₈

C₂×D₃

yes

—

#235

—

C₂×Dic₆

—

C₃×M₁₆

C₂×D₃

yes

—

C₃×M₁₆

C₂×D₃

yes

—

—

—

—

—

—

C₂×C₃⋊C₈

—

Dic₁₂

—

#204

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

C₂²×C₃²

#41

yes

characteristic

C₂×C₃⋊D₃

—

C₃²×C₄

—

C₃²×C₄

C₂×D₄

yes

characteristic

C₃²×C₄

C₂×D₄

yes

characteristic

C₃⋊Dic₃

—

C₃⋊Dic₃

—

#137

—

#137

—

C₂×Q₁₆

—

—

—

—

C₃⋊D₃

yes

characteristic, complemented

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×D₃

yes

—

#67

—

C₂×Dic₃

—

D₁₂

—

C₄×D₃

—

Dic₆

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#121

yes

characteristic

C₃⋊D₃

—

#41

—

C₂×Q₈

—

C₂×Q₈

C₂×C₃⋊D₃

yes

characteristic

Q₁₆

—

100

Q₁₆

—

101

Q₁₆

—

102

Q₁₆

—

103

SD₁₆

—

104

SD₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

C₂×C₃⋊D₃

yes

characteristic

108

M₁₆

C₂×C₃⋊D₃

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂²×C₃

#235

yes

—

112

C₂×D₃

—

113

C₃×C₄

—

114

C₃×C₄

D₃×D₄

yes

—

115

C₃×C₄

D₃×D₄

yes

—

116

Dic₃

—

117

Dic₃

—

118

C₃²

#473

yes

characteristic, complemented, Sylow

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂²×C₃⋊D₃

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₂×C₃

—

136

C₂×C₃

#884

yes

—

137

D₃

—

138

C₂²

—

139

C₂²

#3828

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

D₄×C₃⋊D₃

yes

characteristic

144

C₄

D₄×C₃⋊D₃

yes

characteristic

145

C₃

#4755

yes

complemented

146

C₂

—

147

C₂

—

148

C₂

#64093

yes

characteristic

	subgroup	G/H
center	C₂	#64093
commutator, cosocle	C₂×C₃²×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#101	C₂
socle	C₂×C₃²	#121
prime radical	#3828	C₂²
hypercenter	#101	C₃⋊D₃
hypocenter	C₃²	#473
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

subgroup

G/H

center

C₂

#64093

commutator, cosocle

C₂×C₃²×C₄

C₂³

Frattini

C₂×C₄

C₂²×C₃⋊D₃

Fitting

C₃²×#101

C₂

C₂×C₃²

C₂²

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

N	H ≅ G/N	type
1. C₂×#3756	C₂	semidirect
5. C₃²×#101	C₂	semidirect
6. #63651	C₂	semidirect
7. #63649	C₂	semidirect
28. C₃×#101	D₃	semidirect
76. #101	C₃⋊D₃	semidirect
118. C₃²	#473	semidirect
145. C₃	#4755	semidirect

H ≅ G/N

type

1. C₂×#3756

C₂

semidirect

5. C₃²×#101

C₂

semidirect

6. #63651

C₂

semidirect

7. #63649

C₂

semidirect

28. C₃×#101

D₃

semidirect

76. #101

C₃⋊D₃

semidirect

118. C₃²

#473

semidirect

145. C₃

#4755

semidirect

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

subgroup	index
C₂×#3756	2
#64151	2
#64118	2
#64116	2
C₃²×#101	2
#63651	2
#63649	2
#4755	3

subgroup

index

C₃²×#101

Subgroups — #10592454

Conjugacy classes of subgroups

Complements

Maximal subgroups