Group #10590295 — subgroups

Subgroups — #10590295

Order 576.

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

normal subgroups only

This group has 214 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	#64078	C₂	1	1	yes	maximal, characteristic
2	C₃×#870	C₂	1	1	yes	maximal, characteristic
3	C₃×#803	C₂	1	1	yes	maximal, characteristic
4	#63567	C₂	1	4	yes	maximal
5	#5069	—	3	1	no	maximal
6	#4703	—	3	1	no	maximal
7	C₂×#634	C₂²	1	2	yes	—
8	C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
10	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₃×Dic₃⋊C₄	—	2	2	no	—
12	C₂×C₃×C₃⋊C₈	C₂²	1	2	yes	—
13	C₃×#125	—	3	1	no	—
14	#870	D₃	1	1	yes	characteristic
15	#869	—	3	1	no	—
16	#869	—	3	1	no	—
17	#869	—	6	1	no	—
18	C₃×#97	—	3	1	no	—
19	#816	—	3	4	no	—
20	#810	—	3	4	no	—
21	#803	D₃	1	1	yes	characteristic
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	#634	—	2	4	no	—
26	C₂×C₃×Dic₃	—	2	2	no	—
27	C₃×C₃⋊C₈	—	2	2	no	—
28	#491	—	9	1	no	Sylow
29	C₂×Dic₆	—	3	2	no	—
30	C₂×Dic₆	—	3	2	no	—
31	C₂×Dic₆	—	6	2	no	—
32	C₂×C₃×C₈	—	3	2	no	—
33	C₃×C₄⋊C₄	—	3	2	no	—
34	C₃×C₄⋊C₄	—	6	2	no	—
35	C₃×C₄²	—	2	1	no	—
36	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
37	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
38	C₄⋊Dic₃	—	6	4	no	—
39	C₄⋊Dic₃	—	6	2	no	—
40	C₄⋊Dic₃	—	6	2	no	—
41	C₄⋊Dic₃	C₂×D₃	1	2	yes	—
42	Dic₃⋊C₄	—	2	2	no	—
43	C₂×C₃⋊C₈	C₂×D₃	1	2	yes	—
44	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
45	C₃²×C₄	—	2	2	no	—
46	C₃²×C₄	#41	1	2	yes	—
47	C₃⋊Dic₃	—	4	2	no	—
48	C₃×Dic₃	—	4	2	no	—
49	C₄⋊Q₈	—	9	1	no	—
50	#125	—	3	1	no	—
51	#103	—	9	4	no	—
52	#97	—	3	1	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	D₁₂	1	2	yes	—
57	C₂×C₃×C₄	#67	1	2	yes	—
58	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
59	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
60	C₂×Dic₃	—	2	2	no	—
61	C₂×Dic₃	—	6	4	no	—
62	C₂×Dic₃	—	6	2	no	—
63	C₂×Dic₃	—	6	2	no	—
64	Dic₆	—	6	4	no	—
65	Dic₆	—	6	4	no	—
66	Dic₆	—	12	4	no	—
67	C₃×C₈	—	6	2	no	—
68	C₃⋊C₈	—	2	2	no	—
69	C₂×C₃²	#124	1	1	yes	characteristic
70	C₂×C₃²	#137	1	2	yes	—
71	C₂×Q₈	—	9	2	no	—
72	C₂×C₈	—	3	2	no	—
73	C₄⋊C₄	—	3	2	no	—
74	C₄⋊C₄	—	6	2	no	—
75	C₄⋊C₄	—	18	2	no	—
76	C₄²	D₃²	1	1	yes	characteristic
77	C₂²×C₃	—	2	1	no	—
78	C₂²×C₃	C₂×D₁₂	1	1	yes	characteristic
79	C₂²×C₃	C₂×#67	1	1	yes	characteristic
80	C₃×C₄	—	2	4	no	—
81	C₃×C₄	—	2	2	no	—
82	C₃×C₄	—	2	2	no	—
83	C₃×C₄	—	2	2	no	—
84	C₃×C₄	—	12	2	no	—
85	C₃×C₄	#235	1	2	yes	—
86	C₃×C₄	#239	1	2	yes	—
87	Dic₃	—	4	2	no	—
88	Dic₃	—	12	4	no	—
89	Dic₃	—	12	2	no	—
90	Dic₃	—	12	2	no	—
91	C₃²	#491	1	1	yes	characteristic, complemented, Sylow
92	Q₈	—	18	4	no	—
93	C₂×C₄	—	6	2	no	—
94	C₂×C₄	—	18	2	no	—
95	C₂×C₄	#626	1	2	yes	—
96	C₂×C₄	C₂×D₃²	1	1	yes	characteristic
97	C₈	—	6	2	no	—
98	C₂×C₃	—	2	2	no	—
99	C₂×C₃	—	2	1	no	—
100	C₂×C₃	#875	1	1	yes	characteristic
101	C₂×C₃	#909	1	2	yes	—
102	C₂×C₃	#947	1	1	yes	characteristic
103	C₂×C₃	#951	1	2	yes	—
104	C₂²	C₂×#626	1	1	yes	characteristic
105	C₄	—	2	2	no	—
106	C₄	—	12	2	no	—
107	C₄	—	36	2	no	—
108	C₄	#3799	1	2	yes	—
109	C₃	—	2	1	no	—
110	C₃	#4703	1	1	yes	characteristic, complemented
111	C₃	#5069	1	1	yes	characteristic, complemented
112	C₂	#63831	1	2	yes	—
113	C₂	#63864	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#64078

C₂

yes

maximal, characteristic

C₃×#870

C₂

yes

maximal, characteristic

C₃×#803

C₂

yes

maximal, characteristic

#63567

C₂

yes

—

—

C₂²

yes

—

C₃²×C₄²

C₂²

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

C₃×C₄⋊Dic₃

C₂²

yes

—

C₃×Dic₃⋊C₄

—

C₂×C₃×C₃⋊C₈

C₂²

yes

—

C₃×#125

—

#870

D₃

yes

—

—

—

C₃×#97

—

#816

—

#810

—

#803

D₃

yes

characteristic

C₂×C₃²×C₄

D₄

yes

—

C₂×C₃²×C₄

C₂³

yes

characteristic

C₂×C₃⋊Dic₃

—

#634

—

C₂×C₃×Dic₃

—

C₃×C₃⋊C₈

—

#491

—

Sylow

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×D₃

yes

characteristic

C₃×C₄²

C₂×D₃

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂×D₃

yes

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

C₂×D₃

yes

—

C₂²×C₃²

C₂×D₄

yes

characteristic

C₃²×C₄

—

C₃²×C₄

#41

yes

—

C₃⋊Dic₃

—

C₃×Dic₃

—

C₄⋊Q₈

—

#125

—

#103

—

#97

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₁₂

yes

—

C₂×C₃×C₄

#67

yes

—

C₂×C₃×C₄

C₂²×D₃

yes

characteristic

C₂×C₃×C₄

C₂²×D₃

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

Dic₆

—

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×D₁₂

yes

characteristic

C₂²×C₃

C₂×#67

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#235

yes

—

C₃×C₄

#239

yes

—

Dic₃

—

Dic₃

—

Dic₃

—

Dic₃

—

C₃²

#491

yes

characteristic, complemented, Sylow

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#626

yes

—

C₂×C₄

C₂×D₃²

yes

characteristic

C₈

—

C₂×C₃

—

C₂×C₃

—

100

C₂×C₃

#875

yes

characteristic

101

C₂×C₃

#909

yes

—

102

C₂×C₃

#947

yes

characteristic

103

C₂×C₃

#951

yes

—

104

C₂²

C₂×#626

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

#3799

yes

—

109

C₃

—

110

C₃

#4703

yes

characteristic, complemented

111

C₃

#5069

yes

characteristic, complemented

112

C₂

#63831

yes

—

113

C₂

#63864

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#626
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×#626

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
91. C₃²	#491	semidirect
110. C₃	#4703	semidirect
111. C₃	#5069	semidirect

H ≅ G/N

type

91. C₃²

#491

semidirect

110. C₃

#4703

semidirect

111. C₃

#5069

semidirect

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

subgroup	index
#64078	2
C₃×#870	2
C₃×#803	2
#63567	2
#5069	3
#4703	3

subgroup

index

Subgroups — #10590295

Conjugacy classes of subgroups

Complements

Maximal subgroups