Group #30605955 — subgroups

Subgroups — #30605955

Order 15616.

212 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 86 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	#21462233	C₂	1	2	yes	maximal
2	#21460841	C₂	1	1	yes	maximal, characteristic
3	#21460786	C₂	1	2	yes	maximal
4	#21460732	C₂	1	1	yes	maximal, characteristic
5	#21444625	C₂	1	1	yes	maximal, characteristic
6	#16840683	—	2	2	no	—
7	C₄⋊#11758187	C₂²	1	2	yes	—
8	#16840648	—	2	2	no	—
9	C₄×#11758187	C₂²	1	1	yes	characteristic
10	C₄×#11758187	C₂²	1	1	yes	characteristic
11	#16839886	C₂²	1	2	yes	—
12	C₄×Dic₂₄₄	C₄	1	1	yes	characteristic
13	#16839492	C₄	1	2	yes	—
14	C₈×#92418	C₂²	1	1	yes	characteristic
15	C₆₁×#335	C₄	1	1	yes	characteristic
16	C₂×#11758187	—	2	2	no	—
17	C₂×#11758187	—	2	1	no	—
18	C₂×#11758187	D₄	1	2	yes	—
19	C₂×Dic₂₄₄	C₈	1	1	yes	characteristic
20	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
21	#14519711	C₈	1	1	yes	characteristic
22	#14519706	—	2	2	no	—
23	#14519702	C₈	1	2	yes	—
24	C₄×#92418	C₂×C₄	1	1	yes	characteristic
25	C₄×#92418	C₂³	1	1	yes	characteristic
26	C₈×Dic₆₁	—	2	2	no	—
27	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₆₁	—	2	2	no	—
31	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
33	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
34	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
35	#11758196	—	2	2	no	—
36	C₄×Dic₆₁	—	2	2	no	—
37	C₂×#92418	—	2	1	no	—
38	C₂×#92418	C₂×D₄	1	1	yes	characteristic
39	C₂×#92418	#41	1	1	yes	characteristic
40	Dic₂₄₄	C₁₆	1	4	yes	complemented
41	#11758187	—	2	2	no	—
42	#11758187	—	4	1	no	—
43	#11758187	—	4	2	no	—
44	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
47	C₂×Dic₆₁	—	2	2	no	—
48	C₂×Dic₆₁	—	2	2	no	—
49	Dic₁₂₂	—	2	2	no	—
50	Dic₁₂₂	C₂×C₁₆	1	4	yes	—
51	C₈×C₆₁	—	2	1	no	—
52	C₈×C₆₁	—	4	2	no	—
53	C₈×C₆₁	C₂×C₁₆	1	2	yes	—
54	#92418	—	2	1	no	—
55	#92418	—	4	1	no	—
56	#92418	#137	1	2	yes	—
57	#12059	—	61	1	no	maximal, Sylow
58	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
59	C₄×C₆₁	—	2	1	no	—
60	C₄×C₆₁	#441	1	1	yes	characteristic
61	C₄×C₆₁	#444	1	1	yes	characteristic
62	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
63	C₄×C₆₁	#504	1	1	yes	characteristic
64	Dic₆₁	—	2	4	no	—
65	Dic₆₁	—	4	2	no	—
66	Q₈×C₁₆	—	61	2	no	—
67	C₈×Q₁₆	—	61	1	no	—
68	#1367	—	61	1	no	—
69	Q₈⋊C₁₆	—	61	2	no	—
70	#1312	—	61	1	no	—
71	C₂×C₆₁	#1579	1	1	yes	characteristic
72	C₂×C₆₁	D₄×C₁₆	1	1	yes	characteristic
73	C₂×C₆₁	#2171	1	1	yes	characteristic
74	C₈×Q₈	—	61	2	no	—
75	C₄×Q₁₆	—	61	1	no	—
76	C₄⋊C₁₆	—	61	2	no	—
77	C₄⋊C₁₆	—	122	2	no	—
78	C₄×C₁₆	—	61	1	no	—
79	C₄×C₁₆	—	61	1	no	—
80	C₄×C₁₆	—	122	2	no	—
81	#335	#6923	1	1	yes	characteristic
82	Q₈⋊C₈	—	61	2	no	—
83	C₈²	—	61	1	no	—
84	C₆₁	#12059	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	—	61	1	no	—
86	C₄×Q₈	—	61	2	no	—
87	C₂×C₁₆	—	61	2	no	—
88	C₂×C₁₆	—	122	1	no	—
89	C₂×C₁₆	—	122	2	no	—
90	#107	—	61	1	no	—
91	C₄⋊C₈	—	122	2	no	—
92	C₄⋊C₈	C₂×#6923	1	2	yes	—
93	#103	—	61	2	no	—
94	C₄×C₈	—	61	1	no	—
95	C₄×C₈	—	61	1	no	—
96	C₄×C₈	—	122	2	no	—
97	C₄×C₈	C₂×#6923	1	1	yes	characteristic
98	C₂×Q₈	—	61	2	no	—
99	Q₁₆	—	61	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	61	1	no	—
102	C₂×C₈	—	61	1	no	—
103	C₂×C₈	—	122	1	no	—
104	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
105	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
106	C₄⋊C₄	—	61	2	no	—
107	C₄⋊C₄	—	122	2	no	—
108	C₄²	—	122	2	no	—
109	C₄²	C₂²×#6923	1	1	yes	characteristic
110	C₁₆	—	122	2	no	—
111	C₁₆	—	244	2	no	—
112	C₁₆	—	244	1	no	—
113	Q₈	—	61	4	no	—
114	Q₈	—	122	2	no	—
115	C₂×C₄	—	122	2	no	—
116	C₂×C₄	—	122	2	no	—
117	C₂×C₄	D₄×#6923	1	1	yes	characteristic
118	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
119	C₂×C₄	#14519865	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	4	2	no	—
122	C₈	—	61	2	no	—
123	C₈	—	122	1	no	—
124	C₈	—	244	1	no	—
125	C₈	D₆₁⋊C₁₆	1	2	yes	—
126	C₂²	#16840790	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	122	4	no	—
129	C₄	—	244	2	no	—
130	C₄	#16840802	1	1	yes	characteristic
131	C₄	#16840809	1	1	yes	characteristic
132	C₄	C₂×D₆₁⋊C₁₆	1	1	yes	characteristic
133	C₄	#16840821	1	1	yes	characteristic
134	C₂	#21461414	1	1	yes	characteristic
135	C₂	#21462198	1	1	yes	characteristic
136	C₂	#21462211	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462233

C₂

yes

maximal

#21460841

C₂

yes

maximal, characteristic

#21460786

C₂

yes

maximal

#21460732

C₂

yes

maximal, characteristic

#21444625

C₂

yes

maximal, characteristic

#16840683

—

C₄⋊#11758187

C₂²

yes

—

#16840648

—

C₄×#11758187

C₂²

yes

characteristic

C₄×#11758187

C₂²

yes

characteristic

#16839886

C₂²

yes

—

C₄×Dic₂₄₄

C₄

yes

characteristic

#16839492

C₄

yes

—

C₈×#92418

C₂²

yes

characteristic

C₆₁×#335

C₄

yes

—

—

D₄

yes

—

C₂×Dic₂₄₄

C₈

yes

characteristic

C₄×Dic₁₂₂

C₂×C₄

yes

—

#14519711

C₈

yes

characteristic

#14519706

—

#14519702

C₈

yes

—

C₄×#92418

C₂×C₄

yes

characteristic

C₄×#92418

C₂³

yes

characteristic

C₈×Dic₆₁

—

C₄⋊C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×Dic₁₂₂

C₂×C₈

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄⋊Dic₆₁

C₂×C₈

yes

—

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

C₂×D₄

yes

yes

Dic₂₄₄

C₁₆

yes

—

—

—

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

#131

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

C₂×C₁₆

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂×C₁₆

yes

—

—

—

yes

—

#12059

—

maximal, Sylow

C₂²×C₆₁

C₈×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₂²×C₁₆

yes

characteristic

C₄×C₆₁

#504

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Q₈×C₁₆

—

C₈×Q₁₆

—

#1367

—

Q₈⋊C₁₆

—

#1312

—

C₂×C₆₁

#1579

yes

characteristic

C₂×C₆₁

D₄×C₁₆

yes

characteristic

C₂×C₆₁

#2171

yes

characteristic

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

122

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

122

—

#335

#6923

yes

characteristic

Q₈⋊C₈

—

C₈²

—

C₆₁

#12059

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

—

122

—

#107

—

C₄⋊C₈

—

122

—

C₄⋊C₈

C₂×#6923

yes

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

122

—

C₄×C₈

C₂×#6923

yes

characteristic

C₂×Q₈

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

122

—

104

C₂×C₈

D₆₁⋊C₈

yes

characteristic

105

C₂×C₈

D₆₁⋊C₈

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

122

—

108

C₄²

—

122

—

109

C₄²

C₂²×#6923

yes

characteristic

110

C₁₆

—

122

—

111

C₁₆

—

244

—

112

C₁₆

—

244

—

113

Q₈

—

114

Q₈

—

122

—

115

C₂×C₄

—

122

—

116

C₂×C₄

—

122

—

117

C₂×C₄

D₄×#6923

yes

characteristic

118

C₂×C₄

C₂×D₆₁⋊C₈

yes

characteristic

119

C₂×C₄

#14519865

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

122

—

124

C₈

—

244

—

125

C₈

D₆₁⋊C₁₆

yes

—

126

C₂²

#16840790

yes

characteristic

127

C₄

—

128

C₄

—

122

—

129

C₄

—

244

—

130

C₄

#16840802

yes

characteristic

131

C₄

#16840809

yes

characteristic

132

C₄

C₂×D₆₁⋊C₁₆

yes

characteristic

133

C₄

#16840821

yes

characteristic

134

C₂

#21461414

yes

characteristic

135

C₂

#21462198

yes

characteristic

136

C₂

#21462211

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6923
commutator	C₄×C₆₁	C₂²×C₁₆
Frattini	C₄²	C₂²×#6923
Fitting	C₆₁×#335	C₄
socle, prime radical	C₂²×C₆₁	C₈×D₄
cosocle	C₄×#92418	C₂³
hypercenter	#335	#6923
hypocenter	C₆₁	#12059
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×#6923

commutator

C₄×C₆₁

C₂²×C₁₆

Frattini

C₄²

C₂²×#6923

Fitting

C₆₁×#335

C₄

socle, prime radical

C₂²×C₆₁

C₈×D₄

C₂³

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
40. Dic₂₄₄	C₁₆	semidirect
84. C₆₁	#12059	semidirect

H ≅ G/N

type

40. Dic₂₄₄

C₁₆

semidirect

84. C₆₁

#12059

semidirect

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

subgroup	index
#21462233	2
#21460841	2
#21460786	2
#21460732	2
#21444625	2
#12059	61

subgroup

index

Subgroups — #30605955

Conjugacy classes of subgroups

Complements

Maximal subgroups