Group #10590291 — subgroups

Subgroups — #10590291

Order 576.

200 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 64 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	C₃×#870	C₂	1	1	yes	maximal, characteristic
2	C₃×#869	C₂	1	1	yes	maximal, characteristic
3	#63630	C₂	1	1	yes	maximal, characteristic
4	#63565	C₂	1	4	yes	maximal
5	#5082	—	3	1	no	maximal
6	#5069	—	3	1	no	maximal
7	C₂×C₃×Dic₆	C₂²	1	2	yes	—
8	C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
10	C₃×C₄⋊Dic₃	—	2	2	no	—
11	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₃×Dic₃⋊C₄	—	2	2	no	—
13	C₃×C₄⋊Q₈	—	3	1	no	—
14	C₃×#125	—	3	1	no	—
15	#870	D₃	1	1	yes	characteristic
16	#869	D₃	1	1	yes	characteristic
17	#835	—	3	4	no	—
18	#810	—	3	4	no	—
19	#803	—	3	1	no	—
20	#803	—	3	1	no	—
21	#803	—	6	1	no	—
22	C₂×C₃²×C₄	D₄	1	2	yes	—
23	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
24	C₂×C₃×Dic₃	—	2	2	no	—
25	C₂×C₃×Dic₃	—	2	2	no	—
26	C₃×Dic₆	—	2	4	no	—
27	C₃⋊(C₃⋊C₈)	—	2	2	no	—
28	#491	—	9	1	no	Sylow
29	C₂×C₃×Q₈	—	3	2	no	—
30	C₂×Dic₆	C₂×D₃	1	2	yes	—
31	C₃×C₄⋊C₄	—	3	2	no	—
32	C₃×C₄⋊C₄	—	6	2	no	—
33	C₃×C₄⋊C₄	—	6	2	no	—
34	C₃×C₄²	—	2	1	no	—
35	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
36	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
37	C₄⋊Dic₃	—	2	2	no	—
38	C₄⋊Dic₃	C₂×D₃	1	2	yes	—
39	Dic₃⋊C₄	—	2	2	no	—
40	C₂×C₃⋊C₈	—	3	2	no	—
41	C₂×C₃⋊C₈	—	3	2	no	—
42	C₂×C₃⋊C₈	—	6	2	no	—
43	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
44	C₃²×C₄	—	2	2	no	—
45	C₃²×C₄	#41	1	2	yes	—
46	C₃×Dic₃	—	4	2	no	—
47	C₃×Dic₃	—	4	2	no	—
48	C₄⋊Q₈	—	3	1	no	—
49	#125	—	3	1	no	—
50	#103	—	9	4	no	—
51	#97	—	9	1	no	—
52	C₃×Q₈	—	6	4	no	—
53	C₂×C₃×C₄	—	2	2	no	—
54	C₂×C₃×C₄	—	2	1	no	—
55	C₂×C₃×C₄	—	6	2	no	—
56	C₂×C₃×C₄	—	6	2	no	—
57	C₂×C₃×C₄	#67	1	2	yes	—
58	C₂×C₃×C₄	#67	1	2	yes	—
59	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
60	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
61	C₂×Dic₃	—	2	2	no	—
62	C₂×Dic₃	—	2	2	no	—
63	Dic₆	—	2	4	no	—
64	C₃⋊C₈	—	6	4	no	—
65	C₃⋊C₈	—	6	2	no	—
66	C₃⋊C₈	—	6	2	no	—
67	C₂×C₃²	#124	1	1	yes	characteristic
68	C₂×C₃²	#137	1	2	yes	—
69	C₂×Q₈	—	3	2	no	—
70	C₂×C₈	—	9	2	no	—
71	C₄⋊C₄	—	3	2	no	—
72	C₄⋊C₄	—	6	2	no	—
73	C₄⋊C₄	—	6	2	no	—
74	C₄²	D₃²	1	1	yes	characteristic
75	C₂²×C₃	—	2	1	no	—
76	C₂²×C₃	C₂×#67	1	1	yes	characteristic
77	C₂²×C₃	C₂×#67	1	1	yes	characteristic
78	C₃×C₄	—	2	2	no	—
79	C₃×C₄	—	2	2	no	—
80	C₃×C₄	—	2	2	no	—
81	C₃×C₄	—	4	2	no	—
82	C₃×C₄	—	12	2	no	—
83	C₃×C₄	—	12	2	no	—
84	C₃×C₄	#237	1	2	yes	—
85	C₃×C₄	#239	1	2	yes	—
86	Dic₃	—	4	2	no	—
87	Dic₃	—	4	2	no	—
88	C₃²	#491	1	1	yes	characteristic, complemented, Sylow
89	Q₈	—	6	4	no	—
90	C₂×C₄	—	6	2	no	—
91	C₂×C₄	—	6	2	no	—
92	C₂×C₄	#625	1	2	yes	—
93	C₂×C₄	C₂×D₃²	1	1	yes	characteristic
94	C₈	—	18	2	no	—
95	C₂×C₃	—	2	2	no	—
96	C₂×C₃	—	2	1	no	—
97	C₂×C₃	#936	1	1	yes	characteristic
98	C₂×C₃	#942	1	2	yes	—
99	C₂×C₃	#947	1	1	yes	characteristic
100	C₂×C₃	#951	1	2	yes	—
101	C₂²	C₂×#625	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	12	2	no	—
104	C₄	—	12	2	no	—
105	C₄	#3796	1	2	yes	—
106	C₃	—	2	1	no	—
107	C₃	#5069	1	1	yes	characteristic, complemented
108	C₃	#5082	1	1	yes	characteristic, complemented
109	C₂	#63827	1	2	yes	—
110	C₂	#63863	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#870

C₂

yes

maximal, characteristic

C₃×#869

C₂

yes

maximal, characteristic

#63630

C₂

yes

maximal, characteristic

#63565

C₂

yes

—

—

C₂×C₃×Dic₆

C₂²

yes

—

C₃²×C₄²

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₂²

yes

—

C₃×C₄⋊Dic₃

—

C₃×C₄⋊Dic₃

C₂²

yes

—

C₃×Dic₃⋊C₄

—

C₃×C₄⋊Q₈

—

C₃×#125

—

#870

D₃

yes

characteristic

#869

D₃

yes

—

—

—

—

—

C₂×C₃²×C₄

D₄

yes

—

C₂×C₃²×C₄

C₂³

yes

characteristic

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃⋊(C₃⋊C₈)

—

#491

—

Sylow

C₂×C₃×Q₈

—

C₂×Dic₆

C₂×D₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×D₃

yes

characteristic

C₃×C₄²

C₂×D₃

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂×D₃

yes

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

C₂×D₄

yes

characteristic

C₃²×C₄

—

C₃²×C₄

#41

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

C₄⋊Q₈

—

#125

—

#103

—

#97

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#67

yes

—

C₂×C₃×C₄

#67

yes

—

C₂×C₃×C₄

C₂²×D₃

yes

characteristic

C₂×C₃×C₄

C₂²×D₃

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

#124

yes

characteristic

C₂×C₃²

#137

yes

—

C₂×Q₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₃²

yes

characteristic

C₂²×C₃

—

C₂²×C₃

C₂×#67

yes

characteristic

C₂²×C₃

C₂×#67

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#237

yes

—

C₃×C₄

#239

yes

—

Dic₃

—

Dic₃

—

C₃²

#491

yes

characteristic, complemented, Sylow

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#625

yes

—

C₂×C₄

C₂×D₃²

yes

characteristic

C₈

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

#936

yes

characteristic

C₂×C₃

#942

yes

—

C₂×C₃

#947

yes

characteristic

100

C₂×C₃

#951

yes

—

101

C₂²

C₂×#625

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#3796

yes

—

106

C₃

—

107

C₃

#5069

yes

characteristic, complemented

108

C₃

#5082

yes

characteristic, complemented

109

C₂

#63827

yes

—

110

C₂

#63863

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#625
commutator, cosocle	C₂×C₃²×C₄	C₂³
Frattini	C₂×C₄	C₂×D₃²
Fitting	C₃²×C₄²	C₂²
socle, prime radical	C₂²×C₃²	C₂×D₄
hypercenter	C₄²	D₃²
hypocenter	C₃²	#491
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

C₂×#625

commutator, cosocle

C₂×C₃²×C₄

C₂³

Frattini

C₂×C₄

C₂×D₃²

Fitting

C₃²×C₄²

C₂²

socle, prime radical

C₂²×C₃²

C₂×D₄

hypercenter

C₄²

D₃²

hypocenter

C₃²

#491

LCS

G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
88. C₃²	#491	semidirect
107. C₃	#5069	semidirect
108. C₃	#5082	semidirect

H ≅ G/N

type

88. C₃²

#491

semidirect

107. C₃

#5069

semidirect

108. C₃

#5082

semidirect

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

subgroup	index
C₃×#870	2
C₃×#869	2
#63630	2
#63565	2
#5082	3
#5069	3

subgroup

index

Subgroups — #10590291

Conjugacy classes of subgroups

Complements

Maximal subgroups