Group #10590381 — subgroups

Subgroups — #10590381

Order 576.

236 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 78 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×C₄×Dic₆	C₂	1	2	yes	maximal
2	C₂×#3718	C₂	1	1	yes	maximal, characteristic
3	#63630	C₂	1	1	yes	maximal, characteristic
4	#63574	C₂	1	1	yes	maximal, characteristic
5	#63565	C₂	1	2	yes	maximal
6	#5030	—	3	2	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	1	yes	characteristic
10	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₃×Dic₃⋊C₄	—	2	2	no	—
13	C₃×C₄×Dic₃	—	2	2	no	—
14	#3718	C₄	1	4	yes	complemented
15	C₃×C₄×Q₈	—	3	2	no	—
16	C₂×#216	—	3	2	no	—
17	C₄×Dic₆	D₃	1	2	yes	—
18	#835	—	3	2	no	—
19	#810	—	3	2	no	—
20	#808	—	3	2	no	—
21	#803	—	3	2	no	—
22	#803	—	6	1	no	—
23	C₂×C₃²×C₄	D₄	1	2	yes	—
24	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
25	C₂×C₃×Dic₃	—	2	2	no	—
26	C₂×C₃×Dic₃	—	2	2	no	—
27	C₃×Dic₆	—	2	2	no	—
28	C₃×Dic₆	C₂×C₄	1	4	yes	—
29	C₃⋊(C₃⋊C₈)	—	2	1	no	—
30	C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
31	#441	—	9	1	no	Sylow
32	C₂×C₃×Q₈	—	3	2	no	—
33	C₂×Dic₆	C₂×D₃	1	2	yes	—
34	C₃×C₄⋊C₄	—	3	2	no	—
35	C₃×C₄⋊C₄	—	6	2	no	—
36	C₃×C₄²	—	2	1	no	—
37	C₃×C₄²	—	6	2	no	—
38	C₃×C₄²	C₂×D₃	1	2	yes	—
39	#216	—	3	8	no	—
40	C₄⋊Dic₃	C₂×D₃	1	2	yes	—
41	Dic₃⋊C₄	—	2	2	no	—
42	C₄×Dic₃	—	2	2	no	—
43	C₂×C₃⋊C₈	—	3	2	no	—
44	C₂×C₃⋊C₈	—	3	2	no	—
45	C₂×C₃⋊C₈	—	6	1	no	—
46	C₂×C₃⋊C₈	—	6	1	no	—
47	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
48	C₃²×C₄	—	2	2	no	—
49	C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
50	C₃²×C₄	#41	1	1	yes	characteristic
51	C₃×Dic₃	—	2	4	no	—
52	C₃×Dic₃	—	4	2	no	—
53	C₂×Q₁₆	—	9	1	no	—
54	C₄×Q₈	—	3	2	no	—
55	#106	—	9	1	no	—
56	#103	—	9	2	no	—
57	#97	—	9	1	no	—
58	C₃×Q₈	—	3	4	no	—
59	C₃×Q₈	—	6	2	no	—
60	C₂×C₃×C₄	—	2	2	no	—
61	C₂×C₃×C₄	—	2	1	no	—
62	C₂×C₃×C₄	—	6	2	no	—
63	C₂×C₃×C₄	—	6	2	no	—
64	C₂×C₃×C₄	#67	1	2	yes	—
65	C₂×C₃×C₄	#67	1	2	yes	—
66	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
67	C₂×Dic₃	—	2	2	no	—
68	C₂×Dic₃	—	2	2	no	—
69	Dic₆	—	2	2	no	—
70	Dic₆	C₄×D₃	1	4	yes	—
71	C₃⋊C₈	—	3	4	no	—
72	C₃⋊C₈	—	6	2	no	—
73	C₃⋊C₈	—	6	2	no	—
74	C₃⋊C₈	—	6	2	no	—
75	C₂×C₃²	C₄×D₄	1	1	yes	characteristic
76	C₂×C₃²	#137	1	2	yes	—
77	C₂×Q₈	—	3	2	no	—
78	Q₁₆	—	9	4	no	—
79	C₂×C₈	—	9	1	no	—
80	C₂×C₈	—	9	1	no	—
81	C₄⋊C₄	—	3	2	no	—
82	C₄⋊C₄	—	6	2	no	—
83	C₄²	—	6	2	no	—
84	C₄²	D₃²	1	1	yes	characteristic
85	C₂²×C₃	—	2	1	no	—
86	C₂²×C₃	C₂×#67	1	2	yes	—
87	C₃×C₄	—	2	2	no	—
88	C₃×C₄	—	2	2	no	—
89	C₃×C₄	—	2	1	no	—
90	C₃×C₄	—	2	1	no	—
91	C₃×C₄	—	4	2	no	—
92	C₃×C₄	—	6	4	no	—
93	C₃×C₄	—	12	2	no	—
94	C₃×C₄	C₂×C₄×D₃	1	2	yes	—
95	C₃×C₄	#235	1	2	yes	—
96	Dic₃	—	2	4	no	—
97	Dic₃	—	4	2	no	—
98	C₃²	#441	1	1	yes	characteristic, complemented, Sylow
99	Q₈	—	3	4	no	—
100	Q₈	—	6	2	no	—
101	C₂×C₄	—	6	2	no	—
102	C₂×C₄	—	6	2	no	—
103	C₂×C₄	#625	1	2	yes	—
104	C₂×C₄	C₂×D₃²	1	1	yes	characteristic
105	C₈	—	9	2	no	—
106	C₈	—	18	1	no	—
107	C₂×C₃	—	2	2	no	—
108	C₂×C₃	—	2	1	no	—
109	C₂×C₃	C₄×#67	1	2	yes	—
110	C₂×C₃	#942	1	2	yes	—
111	C₂×C₃	#951	1	2	yes	—
112	C₂²	C₂×#625	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	6	4	no	—
115	C₄	—	12	2	no	—
116	C₄	#3798	1	1	yes	characteristic
117	C₄	C₄×D₃²	1	1	yes	characteristic
118	C₃	—	2	1	no	—
119	C₃	#5030	1	2	yes	complemented
120	C₂	#63827	1	2	yes	—
121	C₂	C₄×#625	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₄×Dic₆

C₂

yes

maximal

C₂×#3718

C₂

yes

maximal, characteristic

#63630

C₂

yes

maximal, characteristic

#63574

C₂

yes

maximal, characteristic

#63565

C₂

yes

maximal

#5030

—

maximal

C₂×C₃×Dic₆

C₂²

yes

—

C₃²×C₄²

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₂²

yes

characteristic

C₃×C₄⋊Dic₃

C₂²

yes

—

C₃×Dic₃⋊C₄

—

C₃×C₄×Dic₃

—

#3718

C₄

yes

complemented

C₃×C₄×Q₈

—

C₂×#216

—

C₄×Dic₆

D₃

yes

—

—

—

—

—

—

C₂×C₃²×C₄

D₄

yes

—

C₂×C₃²×C₄

C₂³

yes

characteristic

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×Dic₆

C₂×C₄

yes

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

C₂×C₄

yes

—

#441

—

Sylow

C₂×C₃×Q₈

—

C₂×Dic₆

C₂×D₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×D₃

yes

—

#216

—

C₄⋊Dic₃

C₂×D₃

yes

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

C₂×D₄

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₂²×C₄

yes

characteristic

C₃²×C₄

#41

yes

characteristic

C₃×Dic₃

—

C₃×Dic₃

—

C₂×Q₁₆

—

C₄×Q₈

—

#106

—

#103

—

#97

—

C₃×Q₈

—

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

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

C₄×D₃

yes

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

C₄×D₄

yes

characteristic

C₂×C₃²

#137

yes

—

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

D₃²

yes

characteristic

C₂²×C₃

—

C₂²×C₃

C₂×#67

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×C₄×D₃

yes

—

C₃×C₄

#235

yes

—

Dic₃

—

Dic₃

—

C₃²

#441

yes

characteristic, complemented, Sylow

Q₈

—

100

Q₈

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#625

yes

—

104

C₂×C₄

C₂×D₃²

yes

characteristic

105

C₈

—

106

C₈

—

107

C₂×C₃

—

108

C₂×C₃

—

109

C₂×C₃

C₄×#67

yes

—

110

C₂×C₃

#942

yes

—

111

C₂×C₃

#951

yes

—

112

C₂²

C₂×#625

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#3798

yes

characteristic

117

C₄

C₄×D₃²

yes

characteristic

118

C₃

—

119

C₃

#5030

yes

complemented

120

C₂

#63827

yes

—

121

C₂

C₄×#625

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#625
commutator	C₃²×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂×D₃²
Fitting	C₃²×C₄²	C₂²
socle, prime radical	C₂²×C₃²	C₂×D₄
cosocle	C₂×C₃²×C₄	C₂³
hypercenter	C₄²	D₃²
hypocenter	C₃²	#441
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

C₂×#625

commutator

C₃²×C₄

C₂²×C₄

Frattini

C₂×C₄

C₂×D₃²

Fitting

C₃²×C₄²

C₂²

socle, prime radical

C₂²×C₃²

C₂×D₄

cosocle

C₂×C₃²×C₄

C₂³

hypercenter

C₄²

D₃²

hypocenter

C₃²

#441

LCS

G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
14. #3718	C₄	semidirect
98. C₃²	#441	semidirect
119. C₃	#5030	semidirect

H ≅ G/N

type

14. #3718

C₄

semidirect

98. C₃²

#441

semidirect

119. C₃

#5030

semidirect

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

subgroup	index
C₃×C₄×Dic₆	2
C₂×#3718	2
#63630	2
#63574	2
#63565	2
#5030	3

subgroup

index

C₃×C₄×Dic₆

Subgroups — #10590381

Conjugacy classes of subgroups

Complements

Maximal subgroups