Group #10592828 — subgroups

Subgroups — #10592828

Order 576.

474 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 151 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×#3828	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#63647	C₂	1	4	yes	maximal, complemented
5	#5129	—	3	4	no	maximal
6	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₂×#638	C₄	1	2	yes	—
8	#3828	—	2	2	no	—
9	#3828	—	2	2	no	—
10	C₂×C₄⋊(C₃⋊D₃)	C₄	1	1	yes	characteristic
11	C₂×C₄×C₃⋊D₃	C₂²	1	2	yes	—
12	C₂×#634	C₄	1	1	yes	characteristic
13	C₃²×M₁₆	—	2	2	no	—
14	C₂×C₃²×C₈	C₂²	1	2	yes	—
15	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
16	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	C₂×#235	—	3	4	no	—
18	C₂×C₃×M₁₆	D₃	1	4	yes	complemented
19	C₂²×C₃⋊C₈	—	3	4	no	—
20	(C₂×D₃)⋊C₈	—	3	16	no	—
21	C₂³×C₃²	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃⋊D₃	C₂×C₄	1	2	yes	—
23	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃²×C₄	D₄	1	2	yes	—
25	C₂×C₃²×C₄	D₄	1	2	yes	—
26	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
27	#638	—	2	4	no	—
28	C₂×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
29	C₄⋊(C₃⋊D₃)	—	2	2	no	—
30	C₄×C₃⋊D₃	—	2	2	no	—
31	C₄×C₃⋊D₃	—	2	2	no	—
32	#634	—	2	2	no	—
33	C₃²×C₈	—	2	2	no	—
34	C₃⋊(C₃⋊C₈)	—	2	2	no	—
35	#408	—	9	1	no	Sylow
36	C₂²×C₃×C₄	C₂×D₃	1	4	yes	—
37	C₂×#67	—	3	8	no	—
38	#235	—	6	8	no	—
39	#235	—	6	8	no	—
40	C₂×D₁₂	—	3	4	no	—
41	C₂×C₄×D₃	—	3	8	no	—
42	C₂×Dic₆	—	3	4	no	—
43	C₃×M₁₆	—	2	8	no	—
44	C₂×C₃×C₈	C₂×D₃	1	8	yes	—
45	C₂×C₃⋊C₈	—	3	8	no	—
46	C₂×C₃⋊C₈	—	6	8	no	—
47	C₂²×C₃²	—	2	2	no	—
48	C₂²×C₃²	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₃⋊D₃	—	2	2	no	—
51	C₂×C₃⋊D₃	—	2	4	no	—
52	C₃²×C₄	C₂²⋊C₄	1	2	yes	—
53	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
54	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
55	C₃⋊Dic₃	—	2	2	no	—
56	C₂×#41	—	9	1	no	—
57	C₂×M₁₆	C₃⋊D₃	1	1	yes	characteristic, complemented
58	C₂²×C₈	—	9	1	no	—
59	C₂²⋊C₈	—	9	4	no	—
60	C₂³×C₃	C₄×D₃	1	4	yes	—
61	C₂²×D₃	—	3	8	no	—
62	C₂×C₃×C₄	C₄×D₃	1	4	yes	—
63	C₂×C₃×C₄	D₁₂	1	8	yes	—
64	C₂×C₃×C₄	#67	1	8	yes	—
65	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
66	#67	—	6	16	no	—
67	C₂×Dic₃	—	3	8	no	—
68	D₁₂	—	6	8	no	—
69	C₄×D₃	—	6	8	no	—
70	C₄×D₃	—	6	8	no	—
71	Dic₆	—	6	8	no	—
72	C₃×C₈	—	2	8	no	—
73	C₃⋊C₈	—	6	8	no	—
74	C₂×C₃²	—	2	2	no	—
75	C₂×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃²	#131	1	2	yes	—
77	C₃⋊D₃	—	4	2	no	—
78	#41	—	18	2	no	—
79	#41	—	18	2	no	—
80	C₂×Q₈	—	9	1	no	—
81	C₂×D₄	—	9	2	no	—
82	C₂×D₄	—	9	1	no	—
83	C₂²×C₄	—	9	2	no	—
84	C₂²×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	C₂×C₈	—	9	2	no	—
87	C₂×C₈	—	18	2	no	—
88	C₂×C₈	C₂×C₃⋊D₃	1	2	yes	—
89	C₂²×C₃	—	2	8	no	—
90	C₂²×C₃	(C₂×D₃)⋊C₄	1	8	yes	—
91	C₂²×C₃	C₂×C₄×D₃	1	4	yes	—
92	C₂×D₃	—	6	16	no	—
93	C₂×D₃	—	6	8	no	—
94	C₃×C₄	(C₂×D₃)⋊C₄	1	8	yes	—
95	C₃×C₄	C₂×D₁₂	1	4	yes	—
96	C₃×C₄	C₂×#67	1	4	yes	—
97	Dic₃	—	6	8	no	—
98	C₃²	#408	1	1	yes	characteristic, complemented, Sylow
99	C₂³	—	9	2	no	—
100	C₂³	C₄×C₃⋊D₃	1	1	yes	characteristic
101	Q₈	—	18	2	no	—
102	D₄	—	18	4	no	—
103	D₄	—	18	2	no	—
104	C₂×C₄	—	9	2	no	—
105	C₂×C₄	—	18	2	no	—
106	C₂×C₄	—	18	2	no	—
107	C₂×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
108	C₂×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
109	C₂×C₄	#638	1	2	yes	—
110	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	18	2	no	—
113	C₂×C₃	—	2	8	no	—
114	C₂×C₃	#907	1	8	yes	—
115	C₂×C₃	C₂×(C₂×D₃)⋊C₄	1	4	yes	—
116	D₃	—	12	8	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	18	4	no	—
119	C₂²	—	18	2	no	—
120	C₂²	#3752	1	2	yes	—
121	C₂²	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
122	C₄	—	18	2	no	—
123	C₄	#3752	1	2	yes	—
124	C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
125	C₄	C₂×#638	1	1	yes	characteristic
126	C₃	#5129	1	4	yes	complemented
127	C₂	—	2	2	no	—
128	C₂	—	36	2	no	—
129	C₂	#64116	1	2	yes	—
130	C₂	C₂×#3752	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#3828

C₂

yes

maximal, characteristic

C₂×C₃²×M₁₆

C₂

yes

maximal, characteristic, complemented

C₂²×C₃⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic, complemented

#63647

C₂

yes

maximal, complemented

#5129

—

maximal

C₂²×C₃²×C₄

C₂²

yes

characteristic

C₂×#638

C₄

yes

—

#3828

—

#3828

—

C₂×C₄⋊(C₃⋊D₃)

C₄

yes

characteristic

C₂×C₄×C₃⋊D₃

C₂²

yes

—

C₂×#634

C₄

yes

characteristic

C₃²×M₁₆

—

C₂×C₃²×C₈

C₂²

yes

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

C₂²

yes

—

C₂×#235

—

C₂×C₃×M₁₆

D₃

yes

complemented

C₂²×C₃⋊C₈

—

(C₂×D₃)⋊C₈

—

C₂³×C₃²

C₂×C₄

yes

characteristic

C₂²×C₃⋊D₃

C₂×C₄

yes

—

C₂×C₃²×C₄

C₂×C₄

yes

characteristic

C₂×C₃²×C₄

D₄

yes

—

C₂×C₃²×C₄

D₄

yes

—

C₂×C₃²×C₄

C₂³

yes

characteristic

#638

—

C₂×C₃⋊Dic₃

C₂×C₄

yes

—

C₄⋊(C₃⋊D₃)

—

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

—

#634

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

#408

—

Sylow

C₂²×C₃×C₄

C₂×D₃

yes

—

C₂×#67

—

#235

—

#235

—

C₂×D₁₂

—

C₂×C₄×D₃

—

C₂×Dic₆

—

C₃×M₁₆

—

C₂×C₃×C₈

C₂×D₃

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

C₂²⋊C₄

yes

—

C₂²×C₃²

C₂²×C₄

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

C₂²⋊C₄

yes

—

C₃²×C₄

C₂×D₄

yes

characteristic

C₃²×C₄

C₂×D₄

yes

characteristic

C₃⋊Dic₃

—

C₂×#41

—

C₂×M₁₆

C₃⋊D₃

yes

characteristic, complemented

C₂²×C₈

—

C₂²⋊C₈

—

C₂³×C₃

C₄×D₃

yes

—

C₂²×D₃

—

C₂×C₃×C₄

C₄×D₃

yes

—

C₂×C₃×C₄

D₁₂

yes

—

C₂×C₃×C₄

#67

yes

—

C₂×C₃×C₄

C₂²×D₃

yes

—

#67

—

C₂×Dic₃

—

D₁₂

—

C₄×D₃

—

C₄×D₃

—

Dic₆

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₃²

#131

yes

—

C₃⋊D₃

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₃⋊D₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₃⋊D₃

yes

—

C₂²×C₃

—

C₂²×C₃

(C₂×D₃)⋊C₄

yes

—

C₂²×C₃

C₂×C₄×D₃

yes

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

(C₂×D₃)⋊C₄

yes

—

C₃×C₄

C₂×D₁₂

yes

—

C₃×C₄

C₂×#67

yes

—

Dic₃

—

C₃²

#408

yes

characteristic, complemented, Sylow

C₂³

—

100

C₂³

C₄×C₃⋊D₃

yes

characteristic

101

Q₈

—

102

D₄

—

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄×C₃⋊D₃

yes

characteristic

108

C₂×C₄

C₄⋊(C₃⋊D₃)

yes

—

109

C₂×C₄

#638

yes

—

110

C₂×C₄

C₂²×C₃⋊D₃

yes

characteristic

111

C₈

—

112

C₈

—

113

C₂×C₃

—

114

C₂×C₃

#907

yes

—

115

C₂×C₃

C₂×(C₂×D₃)⋊C₄

yes

—

116

D₃

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

#3752

yes

—

121

C₂²

C₂×C₄×C₃⋊D₃

yes

characteristic

122

C₄

—

123

C₄

#3752

yes

—

124

C₄

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

125

C₄

C₂×#638

yes

characteristic

126

C₃

#5129

yes

complemented

127

C₂

—

128

C₂

—

129

C₂

#64116

yes

—

130

C₂

C₂×#3752

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×C₃⋊D₃
commutator, socle	C₂²×C₃²	C₂²×C₄
Fitting	C₂×C₃²×M₁₆	C₂
prime radical	C₂×#3828	C₂
cosocle	C₂×C₃²×C₄	C₂³
hypercenter	C₂×M₁₆	C₃⋊D₃
hypocenter	C₃²	#408
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×C₃⋊D₃

commutator, socle

C₂²×C₃²

C₂²×C₄

Fitting

C₂×C₃²×M₁₆

C₂

prime radical

C₂×#3828

C₂

cosocle

C₂×C₃²×C₄

C₂³

hypercenter

C₂×M₁₆

C₃⋊D₃

C₃²

G ▷ C₂²×C₃² ▷ C₃²

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
2. C₂×C₃²×M₁₆	C₂	semidirect
3. C₂²×C₃⋊(C₃⋊C₈)	C₂	semidirect
4. #63647	C₂	semidirect
18. C₂×C₃×M₁₆	D₃	semidirect
57. C₂×M₁₆	C₃⋊D₃	semidirect
98. C₃²	#408	semidirect
126. C₃	#5129	semidirect

H ≅ G/N

type

2. C₂×C₃²×M₁₆

C₂

semidirect

3. C₂²×C₃⋊(C₃⋊C₈)

C₂

semidirect

4. #63647

C₂

semidirect

18. C₂×C₃×M₁₆

D₃

semidirect

57. C₂×M₁₆

C₃⋊D₃

semidirect

98. C₃²

#408

semidirect

126. C₃

#5129

semidirect

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

subgroup	index
C₂×#3828	2
C₂×C₃²×M₁₆	2
C₂²×C₃⋊(C₃⋊C₈)	2
#63647	2
#5129	3

subgroup

index

C₂×#3828

C₂×C₃²×M₁₆

C₂²×C₃⋊(C₃⋊C₈)

#63647

#5129

Subgroups — #10592828

Conjugacy classes of subgroups

Complements

Maximal subgroups