Group #10592564 — subgroups

Subgroups — #10592564

Order 576.

390 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 97 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×#3745	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#3743	C₂	1	1	yes	maximal, characteristic, complemented
3	#64104	C₂	1	2	yes	maximal, complemented
4	C₃²×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	#63636	C₂	1	2	yes	maximal, complemented
6	#4865	—	3	4	no	maximal
7	C₂×C₄⋊(C₃⋊D₃)	—	2	2	no	—
8	C₂×C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
9	C₂×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
12	#3752	—	2	2	no	—
13	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
14	#3745	—	2	2	no	—
15	#3743	—	2	2	no	—
16	C₂×D₂₄	—	3	4	no	—
17	C₂×ΓL(1,𝔽₂₅)	—	3	4	no	—
18	#895	—	3	8	no	—
19	C₃×#106	D₃	1	4	yes	complemented
20	#809	—	3	8	no	—
21	C₂²×C₃⋊D₃	—	2	2	no	—
22	C₂²×C₃⋊D₃	D₄	1	1	yes	characteristic
23	C₂×C₃²×C₄	—	2	2	no	—
24	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
25	C₂×C₃⋊Dic₃	D₄	1	1	yes	characteristic
26	C₄⋊(C₃⋊D₃)	—	2	2	no	—
27	C₄⋊(C₃⋊D₃)	—	2	2	no	—
28	C₄⋊(C₃⋊D₃)	—	4	4	no	—
29	C₄×C₃⋊D₃	—	2	1	no	—
30	C₄×C₃⋊D₃	—	2	1	no	—
31	C₃²×C₈	D₄	1	2	yes	complemented
32	C₃⋊(C₃⋊C₈)	—	2	1	no	—
33	#469	—	9	1	no	Sylow
34	C₂×D₁₂	—	3	8	no	—
35	C₂×D₁₂	—	6	8	no	—
36	C₂×C₄×D₃	—	3	4	no	—
37	C₂×C₃×C₈	C₂×D₃	1	4	yes	—
38	C₃×C₄⋊C₄	C₂×D₃	1	8	yes	—
39	(C₂×D₃)⋊C₄	—	6	8	no	—
40	C₂×C₃⋊C₈	—	3	4	no	—
41	D₂₄	—	6	8	no	—
42	ΓL(1,𝔽₂₅)	—	6	8	no	—
43	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
44	C₂×C₃⋊D₃	—	2	1	no	—
45	C₂×C₃⋊D₃	—	2	2	no	—
46	C₂×C₃⋊D₃	—	4	2	no	—
47	C₂×C₃⋊D₃	—	4	2	no	—
48	C₂×C₃⋊D₃	—	4	2	no	—
49	C₃²×C₄	—	4	2	no	—
50	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
51	C₃²×C₄	#41	1	1	yes	characteristic
52	C₃⋊Dic₃	—	2	1	no	—
53	C₂×D₈	—	9	1	no	—
54	C₂×M₁₆	—	9	1	no	—
55	#121	—	9	2	no	—
56	#106	C₃⋊D₃	1	1	yes	characteristic, complemented
57	#102	—	9	2	no	—
58	C₂²×D₃	—	3	4	no	—
59	C₂²×D₃	—	6	8	no	—
60	C₂×C₃×C₄	—	2	8	no	—
61	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
62	C₂×Dic₃	—	3	4	no	—
63	D₁₂	—	6	8	no	—
64	D₁₂	—	6	8	no	—
65	D₁₂	—	12	16	no	—
66	C₄×D₃	—	6	4	no	—
67	C₄×D₃	—	6	4	no	—
68	C₃×C₈	D₁₂	1	8	yes	complemented
69	C₃⋊C₈	—	6	4	no	—
70	C₂×C₃²	#121	1	1	yes	characteristic
71	C₂×C₃²	AGL(1,ℤ/8)	1	2	yes	—
72	C₃⋊D₃	—	4	1	no	—
73	C₃⋊D₃	—	8	2	no	—
74	C₂×D₄	—	9	2	no	—
75	C₂×D₄	—	18	2	no	—
76	C₂²×C₄	—	9	1	no	—
77	D₈	—	18	2	no	—
78	M₁₆	—	18	2	no	—
79	C₂×C₈	—	9	1	no	—
80	C₂×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
81	C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
82	C₂²⋊C₄	—	18	2	no	—
83	C₂²×C₃	D₃×D₄	1	4	yes	—
84	C₂×D₃	—	6	4	no	—
85	C₂×D₃	—	6	8	no	—
86	C₂×D₃	—	12	8	no	—
87	C₂×D₃	—	12	8	no	—
88	C₂×D₃	—	12	8	no	—
89	C₃×C₄	—	4	8	no	—
90	C₃×C₄	C₂×D₁₂	1	4	yes	—
91	C₃×C₄	#239	1	4	yes	—
92	Dic₃	—	6	4	no	—
93	C₃²	#469	1	1	yes	characteristic, complemented, Sylow
94	C₂³	—	9	1	no	—
95	C₂³	—	18	2	no	—
96	D₄	—	18	2	no	—
97	D₄	—	18	2	no	—
98	D₄	—	36	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	9	1	no	—
101	C₂×C₄	—	18	1	no	—
102	C₂×C₄	—	18	1	no	—
103	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
104	C₈	—	18	1	no	—
105	C₈	C₄⋊(C₃⋊D₃)	1	2	yes	complemented
106	C₂×C₃	#895	1	4	yes	—
107	C₂×C₃	#914	1	8	yes	—
108	D₃	—	12	4	no	—
109	D₃	—	24	8	no	—
110	C₂²	—	18	1	no	—
111	C₂²	—	18	2	no	—
112	C₂²	—	36	2	no	—
113	C₂²	—	36	2	no	—
114	C₂²	—	36	2	no	—
115	C₂²	D₄×C₃⋊D₃	1	1	yes	characteristic
116	C₄	—	4	2	no	—
117	C₄	—	18	1	no	—
118	C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
119	C₄	#3832	1	1	yes	characteristic
120	C₃	#4865	1	4	yes	complemented
121	C₂	—	36	1	no	—
122	C₂	—	72	2	no	—
123	C₂	#64104	1	1	yes	characteristic
124	C₂	#64123	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#3745

C₂

yes

maximal, characteristic, complemented

C₂×#3743

C₂

yes

maximal, characteristic, complemented

#64104

C₂

yes

maximal, complemented

C₃²×#106

C₂

yes

maximal, characteristic, complemented

#63636

C₂

yes

maximal, complemented

#4865

—

maximal

C₂×C₄⋊(C₃⋊D₃)

—

C₂×C₄⋊(C₃⋊D₃)

C₂²

yes

—

C₂×C₄×C₃⋊D₃

C₂²

yes

characteristic

C₂×C₃²×C₈

C₂²

yes

characteristic

C₃²×C₄⋊C₄

C₂²

yes

—

#3752

—

C₂×C₃⋊(C₃⋊C₈)

C₂²

yes

characteristic

#3745

—

#3743

—

C₂×D₂₄

—

C₂×ΓL(1,𝔽₂₅)

—

#895

—

C₃×#106

D₃

yes

complemented

#809

—

C₂²×C₃⋊D₃

—

C₂²×C₃⋊D₃

D₄

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂³

yes

characteristic

C₂×C₃⋊Dic₃

D₄

yes

characteristic

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

—

C₃²×C₈

D₄

yes

complemented

C₃⋊(C₃⋊C₈)

—

#469

—

Sylow

C₂×D₁₂

—

C₂×D₁₂

—

C₂×C₄×D₃

—

C₂×C₃×C₈

C₂×D₃

yes

—

C₃×C₄⋊C₄

C₂×D₃

yes

—

(C₂×D₃)⋊C₄

—

C₂×C₃⋊C₈

—

D₂₄

—

ΓL(1,𝔽₂₅)

—

C₂²×C₃²

C₂×D₄

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

—

C₃²×C₄

C₂×D₄

yes

characteristic

C₃²×C₄

#41

yes

characteristic

C₃⋊Dic₃

—

C₂×D₈

—

C₂×M₁₆

—

#121

—

#106

C₃⋊D₃

yes

characteristic, complemented

#102

—

C₂²×D₃

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×D₃

yes

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

D₁₂

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

D₁₂

yes

complemented

C₃⋊C₈

—

C₂×C₃²

#121

yes

characteristic

C₂×C₃²

AGL(1,ℤ/8)

yes

—

C₃⋊D₃

—

C₃⋊D₃

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂×C₃⋊D₃

yes

characteristic

C₄⋊C₄

C₂×C₃⋊D₃

yes

—

C₂²⋊C₄

—

C₂²×C₃

D₃×D₄

yes

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

C₂×D₁₂

yes

—

C₃×C₄

#239

yes

—

Dic₃

—

C₃²

#469

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²×C₃⋊D₃

yes

characteristic

104

C₈

—

105

C₈

C₄⋊(C₃⋊D₃)

yes

complemented

106

C₂×C₃

#895

yes

—

107

C₂×C₃

#914

yes

—

108

D₃

—

109

D₃

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

D₄×C₃⋊D₃

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

119

C₄

#3832

yes

characteristic

120

C₃

#4865

yes

complemented

121

C₂

—

122

C₂

—

123

C₂

#64104

yes

characteristic

124

C₂

#64123

yes

—

	subgroup	G/H
center	C₂²	D₄×C₃⋊D₃
commutator, cosocle	C₂×C₃²×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#106	C₂
socle	C₂²×C₃²	C₂×D₄
hypercenter	#106	C₃⋊D₃
hypocenter	C₃²	#469
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

subgroup

G/H

center

C₂²

D₄×C₃⋊D₃

commutator, cosocle

C₂×C₃²×C₄

C₂³

Frattini

C₂×C₄

C₂²×C₃⋊D₃

Fitting

C₃²×#106

C₂

socle

C₂²×C₃²

C₂×D₄

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

N	H ≅ G/N	type
1. C₂×#3745	C₂	semidirect
2. C₂×#3743	C₂	semidirect
3. #64104	C₂	semidirect
4. C₃²×#106	C₂	semidirect
5. #63636	C₂	semidirect
19. C₃×#106	D₃	semidirect
31. C₃²×C₈	D₄	semidirect
56. #106	C₃⋊D₃	semidirect
68. C₃×C₈	D₁₂	semidirect
93. C₃²	#469	semidirect
105. C₈	C₄⋊(C₃⋊D₃)	semidirect
120. C₃	#4865	semidirect

H ≅ G/N

type

1. C₂×#3745

C₂

semidirect

2. C₂×#3743

C₂

semidirect

3. #64104

C₂

semidirect

4. C₃²×#106

C₂

semidirect

5. #63636

C₂

semidirect

19. C₃×#106

D₃

semidirect

31. C₃²×C₈

D₄

semidirect

56. #106

C₃⋊D₃

semidirect

68. C₃×C₈

D₁₂

semidirect

93. C₃²

#469

semidirect

105. C₈

C₄⋊(C₃⋊D₃)

semidirect

120. C₃

#4865

semidirect

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

subgroup	index
C₂×#3745	2
C₂×#3743	2
#64104	2
C₃²×#106	2
#63636	2
#4865	3

subgroup

index

C₃²×#106

Subgroups — #10592564

Conjugacy classes of subgroups

Complements

Maximal subgroups