Group #10596159 — subgroups

Subgroups — #10596159

Order 576.

666 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 195 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	C₃²×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#64105	C₂	1	4	yes	maximal
3	#64101	C₂	1	2	yes	maximal
4	#64097	C₂	1	2	yes	maximal, complemented
5	#64096	C₂	1	4	yes	maximal, complemented
6	C₄²×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
7	#64079	C₂	1	1	yes	maximal, characteristic, complemented
8	#5688	—	3	4	no	maximal
9	C₂×C₄×C₃⋊D₃	C₂²	1	2	yes	—
10	C₂×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
11	C₂×#634	C₂²	1	4	yes	—
12	C₃²×C₄⋊C₄	C₂²	1	4	yes	—
13	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
14	C₃²×C₄²	C₂²	1	1	yes	characteristic
15	#3752	C₂²	1	4	yes	—
16	C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
17	#3750	C₂²	1	4	yes	—
18	#3750	C₂²	1	4	yes	—
19	C₄×C₃⋊Dic₃	C₂²	1	2	yes	—
20	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
21	C₄×C₃⋊Dic₃	C₂²	1	4	yes	—
22	C₃×#125	D₃	1	4	yes	complemented
23	#896	—	3	16	no	—
24	#892	—	3	8	no	—
25	#888	—	3	8	no	—
26	#887	—	3	16	no	—
27	C₄²×D₃	—	3	4	no	—
28	#870	—	3	4	no	—
29	C₂²×C₃⋊D₃	C₂³	1	1	yes	characteristic
30	C₂×C₃²×C₄	C₂³	1	2	yes	—
31	C₂×C₃²×C₄	C₂³	1	4	yes	—
32	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃⋊Dic₃	C₂³	1	4	yes	—
34	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
35	C₂×C₃⋊Dic₃	C₂³	1	2	yes	—
36	C₄×C₃⋊D₃	—	2	4	no	—
37	C₄×C₃⋊D₃	Q₈	1	4	yes	complemented
38	#634	—	2	8	no	—
39	#533	—	9	1	no	Sylow
40	C₂×C₄×D₃	—	3	4	no	—
41	C₂×C₄×D₃	—	3	8	no	—
42	C₂×Dic₆	—	3	16	no	—
43	C₃×C₄⋊C₄	C₂×D₃	1	16	yes	—
44	C₃×C₄⋊C₄	C₂×D₃	1	8	yes	—
45	C₃×C₄²	C₂×D₃	1	4	yes	—
46	(C₂×D₃)⋊C₄	—	3	16	no	—
47	C₄⋊Dic₃	—	3	8	no	—
48	Dic₃⋊C₄	—	3	16	no	—
49	Dic₃⋊C₄	—	3	16	no	—
50	C₄×Dic₃	—	3	8	no	—
51	C₄×Dic₃	—	3	4	no	—
52	C₄×Dic₃	—	3	16	no	—
53	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
54	C₂×C₃⋊D₃	—	2	2	no	—
55	C₂×C₃⋊D₃	C₂×Q₈	1	2	yes	—
56	C₃²×C₄	—	2	4	no	—
57	C₃²×C₄	—	2	2	no	—
58	C₃²×C₄	C₂×Q₈	1	2	yes	—
59	C₃⋊Dic₃	—	2	4	no	—
60	C₃⋊Dic₃	C₂×Q₈	1	2	yes	—
61	C₃⋊Dic₃	#41	1	4	yes	—
62	C₄⋊Q₈	—	9	2	no	—
63	#125	—	9	1	no	—
64	#125	C₃⋊D₃	1	1	yes	characteristic, complemented
65	C₂²⋊Q₈	—	9	4	no	—
66	C₄×Q₈	—	9	4	no	—
67	#117	—	9	2	no	—
68	C₂×C₄²	—	9	1	no	—
69	C₂²×D₃	—	3	4	no	—
70	C₂×C₃×C₄	C₂²×D₃	1	8	yes	—
71	C₂×C₃×C₄	C₂²×D₃	1	16	yes	—
72	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
73	C₂×Dic₃	—	3	4	no	—
74	C₂×Dic₃	—	3	8	no	—
75	C₂×Dic₃	—	3	16	no	—
76	C₄×D₃	—	3	16	no	—
77	C₄×D₃	—	6	16	no	—
78	Dic₆	—	6	32	no	—
79	C₂×C₃²	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₃²	C₂×#41	1	2	yes	—
81	C₃⋊D₃	—	2	2	no	—
82	C₂×Q₈	—	9	4	no	—
83	C₂²×C₄	—	9	1	no	—
84	C₂²×C₄	—	9	2	no	—
85	C₄⋊C₄	—	9	2	no	—
86	C₄⋊C₄	—	9	4	no	—
87	C₄⋊C₄	—	9	4	no	—
88	C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
89	C₄⋊C₄	C₂×C₃⋊D₃	1	4	yes	—
90	C₂²⋊C₄	—	9	4	no	—
91	C₄²	—	9	2	no	—
92	C₄²	—	9	1	no	—
93	C₄²	—	9	4	no	—
94	C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
95	C₂²×C₃	C₂³×D₃	1	4	yes	—
96	C₂×D₃	—	3	8	no	—
97	C₂×D₃	—	6	8	no	—
98	C₃×C₄	—	2	16	no	—
99	C₃×C₄	—	2	8	no	—
100	C₃×C₄	D₃×Q₈	1	8	yes	—
101	Dic₃	—	3	8	no	—
102	Dic₃	—	3	16	no	—
103	Dic₃	—	6	16	no	—
104	C₃²	#533	1	1	yes	characteristic, complemented, Sylow
105	C₂³	—	9	1	no	—
106	Q₈	—	18	8	no	—
107	C₂×C₄	—	9	1	no	—
108	C₂×C₄	—	9	2	no	—
109	C₂×C₄	—	9	4	no	—
110	C₂×C₄	—	9	4	no	—
111	C₂×C₄	—	18	4	no	—
112	C₂×C₄	C₂²×C₃⋊D₃	1	2	yes	—
113	C₂×C₄	C₂²×C₃⋊D₃	1	4	yes	—
114	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
115	C₂×C₃	C₂×D₃×Q₈	1	4	yes	—
116	C₂×C₃	D₃×#41	1	8	yes	—
117	D₃	—	6	8	no	—
118	C₂²	—	9	2	no	—
119	C₂²	—	18	2	no	—
120	C₂²	C₂³×C₃⋊D₃	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	4	no	—
123	C₄	—	9	2	no	—
124	C₄	—	9	4	no	—
125	C₄	—	18	4	no	—
126	C₄	Q₈×C₃⋊D₃	1	2	yes	—
127	C₃	#5688	1	4	yes	complemented
128	C₂	—	18	2	no	—
129	C₂	C₂×Q₈×C₃⋊D₃	1	1	yes	characteristic
130	C₂	#41×C₃⋊D₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#125

C₂

yes

maximal, characteristic, complemented

#64105

C₂

yes

maximal

#64101

C₂

yes

maximal

#64097

C₂

yes

maximal, complemented

#64096

C₂

yes

maximal, complemented

C₄²×C₃⋊D₃

C₂

yes

maximal, characteristic

#64079

C₂

yes

maximal, characteristic, complemented

#5688

—

maximal

C₂×C₄×C₃⋊D₃

C₂²

yes

—

C₂×C₄×C₃⋊D₃

C₂²

yes

characteristic

C₂×#634

C₂²

yes

—

C₃²×C₄⋊C₄

C₂²

yes

—

C₃²×C₄⋊C₄

C₂²

yes

—

C₃²×C₄²

C₂²

yes

characteristic

#3752

C₂²

yes

—

C₄⋊(C₃⋊Dic₃)

C₂²

yes

—

#3750

C₂²

yes

—

#3750

C₂²

yes

—

C₄×C₃⋊Dic₃

C₂²

yes

—

C₄×C₃⋊Dic₃

C₂²

yes

characteristic

C₄×C₃⋊Dic₃

C₂²

yes

—

C₃×#125

D₃

yes

—

—

—

—

C₄²×D₃

—

#870

—

C₂²×C₃⋊D₃

C₂³

yes

characteristic

C₂×C₃²×C₄

C₂³

yes

—

C₂×C₃²×C₄

C₂³

yes

—

C₂×C₃²×C₄

C₂³

yes

characteristic

C₂×C₃⋊Dic₃

C₂³

yes

—

C₂×C₃⋊Dic₃

C₂³

yes

characteristic

C₂×C₃⋊Dic₃

C₂³

yes

—

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

Q₈

yes

—

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₂×Dic₆

—

C₃×C₄⋊C₄

C₂×D₃

yes

—

C₃×C₄⋊C₄

C₂×D₃

yes

—

C₃×C₄²

C₂×D₃

yes

—

(C₂×D₃)⋊C₄

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

C₂⁴

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

C₂×Q₈

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×Q₈

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

C₂×Q₈

yes

—

C₃⋊Dic₃

#41

yes

—

C₄⋊Q₈

—

#125

—

#125

C₃⋊D₃

yes

characteristic, complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂²×D₃

—

C₂×C₃×C₄

C₂²×D₃

yes

—

C₂×C₃×C₄

C₂²×D₃

yes

—

C₂×C₃×C₄

C₂²×D₃

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

Dic₆

—

C₂×C₃²

C₂²×Q₈

yes

characteristic

C₂×C₃²

C₂×#41

yes

—

C₃⋊D₃

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₃⋊D₃

yes

—

C₄⋊C₄

C₂×C₃⋊D₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₃⋊D₃

yes

characteristic

C₂²×C₃

C₂³×D₃

yes

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

100

C₃×C₄

D₃×Q₈

yes

—

101

Dic₃

—

102

Dic₃

—

103

Dic₃

—

104

C₃²

#533

yes

characteristic, complemented, Sylow

105

C₂³

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×C₃⋊D₃

yes

—

113

C₂×C₄

C₂²×C₃⋊D₃

yes

—

114

C₂×C₄

C₂²×C₃⋊D₃

yes

characteristic

115

C₂×C₃

C₂×D₃×Q₈

yes

—

116

C₂×C₃

D₃×#41

yes

—

117

D₃

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂³×C₃⋊D₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

Q₈×C₃⋊D₃

yes

—

127

C₃

#5688

yes

complemented

128

C₂

—

129

C₂

C₂×Q₈×C₃⋊D₃

yes

characteristic

130

C₂

#41×C₃⋊D₃

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊D₃
commutator, socle, cosocle	C₂²×C₃²	C₂⁴
Fitting	C₃²×#125	C₂
prime radical	C₂²×C₃⋊D₃	C₂³
hypercenter	#125	C₃⋊D₃
hypocenter	C₃²	#533
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×C₃⋊D₃

commutator, socle, cosocle

C₂²×C₃²

C₂⁴

Fitting

C₃²×#125

C₂

prime radical

C₂²×C₃⋊D₃

C₂³

C₃⋊D₃

C₃²

G ▷ C₂²×C₃² ▷ C₃²

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. C₃²×#125	C₂	semidirect
4. #64097	C₂	semidirect
5. #64096	C₂	semidirect
7. #64079	C₂	semidirect
22. C₃×#125	D₃	semidirect
37. C₄×C₃⋊D₃	Q₈	semidirect
64. #125	C₃⋊D₃	semidirect
104. C₃²	#533	semidirect
127. C₃	#5688	semidirect

H ≅ G/N

type

1. C₃²×#125

C₂

semidirect

4. #64097

C₂

semidirect

5. #64096

C₂

semidirect

7. #64079

C₂

semidirect

22. C₃×#125

D₃

semidirect

37. C₄×C₃⋊D₃

Q₈

semidirect

64. #125

C₃⋊D₃

semidirect

104. C₃²

#533

semidirect

127. C₃

#5688

semidirect

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

subgroup	index
C₃²×#125	2
#64105	2
#64101	2
#64097	2
#64096	2
C₄²×C₃⋊D₃	2
#64079	2
#5688	3

subgroup

index

C₃²×#125

C₄²×C₃⋊D₃

Subgroups — #10596159

Conjugacy classes of subgroups

Complements

Maximal subgroups