Group #30597380 — subgroups

Subgroups — #30597380

Order 15616.

446 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 184 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	#21461533	C₂	1	2	yes	maximal
2	#21460641	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#92420	C₂²	1	4	yes	—
5	#16840628	C₄	1	8	yes	—
6	C₂×C₄×#92418	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
8	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
9	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
11	C₂²×#92420	C₂×C₄	1	8	yes	—
12	C₄×#92420	—	2	8	no	—
13	C₂²×#92418	C₂×C₄	1	2	yes	—
14	C₂×C₄×Dic₆₁	C₂×C₄	1	4	yes	—
15	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
17	C₄×#92418	C₈	1	4	yes	complemented
18	C₈×Dic₆₁	—	2	4	no	—
19	C₈×Dic₆₁	—	2	4	no	—
20	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₆₁	C₈	1	4	yes	complemented
23	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
24	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
25	C₂²×Dic₆₁	C₄²	1	2	yes	—
26	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
27	C₂×#92420	—	2	8	no	—
28	C₂×#92420	—	2	16	no	—
29	C₂×C₈×C₆₁	—	2	2	no	—
30	C₂×C₈×C₆₁	—	2	2	no	—
31	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
32	C₄²×C₆₁	C₂×C₈	1	2	yes	—
33	C₄²×C₆₁	C₂×C₈	1	2	yes	—
34	C₄×Dic₆₁	—	2	8	no	—
35	C₄×Dic₆₁	M₁₆	1	4	yes	—
36	C₄×Dic₆₁	#41	1	4	yes	—
37	C₂×#92418	—	2	2	no	—
38	C₂×#92418	—	2	2	no	—
39	C₂×#92418	C₂×C₈	1	4	yes	—
40	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
41	C₂×C₄×C₆₁	—	2	4	no	—
42	C₂×C₄×C₆₁	C₄×C₈	1	4	yes	—
43	C₂×C₄×C₆₁	#117	1	2	yes	—
44	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
47	C₂×Dic₆₁	—	2	4	no	—
48	C₂×Dic₆₁	#97	1	4	yes	—
49	C₂×Dic₆₁	#117	1	4	yes	—
50	C₂×Dic₆₁	#131	1	8	yes	—
51	#92420	—	4	8	no	—
52	C₈×C₆₁	—	2	4	no	—
53	#92418	—	2	4	no	—
54	#8133	—	61	1	no	maximal, Sylow
55	C₂²×C₆₁	#376	1	1	yes	characteristic
56	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
57	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
58	C₂²×C₆₁	#405	1	4	yes	—
59	C₄×C₆₁	—	2	4	no	—
60	C₄×C₆₁	#431	1	2	yes	—
61	C₄×C₆₁	#431	1	2	yes	—
62	Dic₆₁	—	2	4	no	—
63	Dic₆₁	#433	1	8	yes	—
64	#1744	—	61	2	no	—
65	C₄²×C₈	—	61	1	no	—
66	#1273	—	61	4	no	—
67	C₂×C₆₁	#1450	1	2	yes	—
68	C₂×C₆₁	#1452	1	2	yes	—
69	C₂×C₆₁	#1744	1	1	yes	characteristic
70	C₂×C₆₁	#1746	1	2	yes	—
71	C₂×C₄×C₈	—	61	1	no	—
72	C₂×C₄×C₈	—	61	4	no	—
73	C₂×C₄×C₈	—	61	2	no	—
74	C₂×C₄×C₈	—	61	2	no	—
75	C₂×C₄×C₈	#6923	1	1	yes	characteristic
76	C₄³	—	61	1	no	—
77	#336	—	61	8	no	—
78	C₆₁	#8133	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₈	—	61	2	no	—
80	C₂²×C₈	—	61	8	no	—
81	C₂²×C₈	C₂×#6923	1	2	yes	—
82	C₂×C₄²	—	61	1	no	—
83	C₂×C₄²	—	61	1	no	—
84	C₂×C₄²	—	61	4	no	—
85	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
86	C₄×C₈	—	61	4	no	—
87	C₄×C₈	—	122	4	no	—
88	C₄×C₈	—	122	4	no	—
89	C₄×C₈	—	122	8	no	—
90	C₄×C₈	#92420	1	4	yes	complemented
91	C₂²×C₄	—	61	2	no	—
92	C₂²×C₄	—	61	2	no	—
93	C₂²×C₄	C₄×#6923	1	2	yes	—
94	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	61	4	no	—
98	C₂×C₈	—	122	2	no	—
99	C₂×C₈	—	122	2	no	—
100	C₂×C₈	—	122	8	no	—
101	C₂×C₈	—	122	16	no	—
102	C₂×C₈	C₂×#92420	1	4	yes	—
103	C₄²	—	61	4	no	—
104	C₄²	—	61	4	no	—
105	C₄²	—	122	8	no	—
106	C₄²	D₆₁⋊C₈	1	2	yes	—
107	C₄²	C₂×#92420	1	2	yes	—
108	C₂³	C₂×C₄×#6923	1	1	yes	characteristic
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	61	8	no	—
111	C₂×C₄	—	61	4	no	—
112	C₂×C₄	—	61	4	no	—
113	C₂×C₄	—	122	4	no	—
114	C₂×C₄	C₄×#92420	1	4	yes	—
115	C₂×C₄	#14519856	1	2	yes	—
116	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
117	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
118	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
119	C₈	—	2	4	no	—
120	C₈	—	122	4	no	—
121	C₈	—	244	8	no	—
122	C₂²	#16840698	1	1	yes	characteristic
123	C₂²	C₂×C₄×#92420	1	1	yes	characteristic
124	C₂²	#16840721	1	1	yes	characteristic
125	C₂²	#16840725	1	4	yes	—
126	C₄	—	2	4	no	—
127	C₄	—	61	8	no	—
128	C₄	—	122	4	no	—
129	C₄	#16840778	1	2	yes	—
130	C₄	#16840780	1	2	yes	—
131	C₂	#21461022	1	2	yes	—
132	C₂	#21461033	1	2	yes	—
133	C₂	#21461533	1	1	yes	characteristic
134	C₂	#21461537	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21461533

C₂

yes

maximal

#21460641

C₂

yes

maximal

C₄×C₈×Dic₆₁

C₂

yes

maximal, characteristic

C₂×C₄×#92420

C₂²

yes

—

#16840628

C₄

yes

—

C₂×C₄×#92418

C₄

yes

characteristic

C₂×C₈×Dic₆₁

C₄

yes

—

C₂×C₈×Dic₆₁

C₂²

yes

—

C₄²×Dic₆₁

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₁

C₄

yes

characteristic

C₂²×#92420

C₂×C₄

yes

—

C₄×#92420

—

C₂²×#92418

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₂³

yes

characteristic

C₄×#92418

C₈

yes

complemented

C₈×Dic₆₁

—

C₈×Dic₆₁

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₈

yes

complemented

C₂²×C₄×C₆₁

C₄²

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×Dic₆₁

C₄²

yes

—

C₂²×Dic₆₁

C₂²×C₄

yes

—

C₂×#92420

—

C₂×#92420

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₄²×C₆₁

C₂×C₈

yes

—

C₄²×C₆₁

C₂×C₈

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

M₁₆

yes

—

C₄×Dic₆₁

#41

yes

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

C₂×C₈

yes

—

C₂³×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄×C₈

yes

—

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

#97

yes

—

C₂×Dic₆₁

#117

yes

—

C₂×Dic₆₁

#131

yes

—

#92420

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

C₂×C₄×C₈

yes

characteristic

C₂²×C₆₁

C₂×#97

yes

characteristic

C₂²×C₆₁

#405

yes

—

C₄×C₆₁

—

C₄×C₆₁

#431

yes

—

C₄×C₆₁

#431

yes

—

Dic₆₁

—

Dic₆₁

#433

yes

—

#1744

—

C₄²×C₈

—

#1273

—

C₂×C₆₁

#1450

yes

—

C₂×C₆₁

#1452

yes

—

C₂×C₆₁

#1744

yes

characteristic

C₂×C₆₁

#1746

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#6923

yes

characteristic

C₄³

—

#336

—

C₆₁

#8133

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#6923

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×#6923

yes

characteristic

C₄×C₈

—

C₄×C₈

—

122

—

C₄×C₈

—

122

—

C₄×C₈

—

122

—

C₄×C₈

#92420

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×#6923

yes

—

C₂²×C₄

C₂²×#6923

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

100

C₂×C₈

—

122

—

101

C₂×C₈

—

122

—

102

C₂×C₈

C₂×#92420

yes

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

122

—

106

C₄²

D₆₁⋊C₈

yes

—

107

C₄²

C₂×#92420

yes

—

108

C₂³

C₂×C₄×#6923

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

122

—

114

C₂×C₄

C₄×#92420

yes

—

115

C₂×C₄

#14519856

yes

—

116

C₂×C₄

C₂²×#92420

yes

characteristic

117

C₂×C₄

C₂×D₆₁⋊C₈

yes

characteristic

118

C₂×C₄

D₆₁⋊M₁₆

yes

—

119

C₈

—

120

C₈

—

122

—

121

C₈

—

244

—

122

C₂²

#16840698

yes

characteristic

123

C₂²

C₂×C₄×#92420

yes

characteristic

124

C₂²

#16840721

yes

characteristic

125

C₂²

#16840725

yes

—

126

C₄

—

127

C₄

—

128

C₄

—

122

—

129

C₄

#16840778

yes

—

130

C₄

#16840780

yes

—

131

C₂

#21461022

yes

—

132

C₂

#21461033

yes

—

133

C₂

#21461533

yes

characteristic

134

C₂

#21461537

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator	C₂²×C₆₁	C₂×C₄×C₈
Fitting	C₂×C₄×C₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₄²
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×C₄×C₈	#6923
hypocenter	C₆₁	#8133
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×#6923

commutator

C₂²×C₆₁

C₂×C₄×C₈

Fitting

C₂×C₄×C₈×C₆₁

C₄

socle, prime radical

C₂³×C₆₁

C₂×C₄²

cosocle

C₂×C₄×Dic₆₁

C₂³

C₂×C₄×C₈

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
17. C₄×#92418	C₈	semidirect
22. C₄×C₈×C₆₁	C₈	semidirect
78. C₆₁	#8133	semidirect
90. C₄×C₈	#92420	semidirect

H ≅ G/N

type

17. C₄×#92418

C₈

semidirect

22. C₄×C₈×C₆₁

C₈

semidirect

78. C₆₁

#8133

semidirect

90. C₄×C₈

#92420

semidirect

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

subgroup	index
#21461533	2
#21460641	2
C₄×C₈×Dic₆₁	2
#8133	61

subgroup

index

#21461533

#21460641

C₄×C₈×Dic₆₁

#8133

Subgroups — #30597380

Conjugacy classes of subgroups

Complements

Maximal subgroups