Group #30608513 — subgroups

Subgroups — #30608513

Order 15616.

198 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 96 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	D₆₁⋊M₆₄	C₂	1	2	yes	maximal
2	#21460979	C₂	1	2	yes	maximal, complemented
3	#21460978	C₂	1	2	yes	maximal, complemented
4	#21447417	C₂	1	1	yes	maximal, characteristic
5	#16840695	—	2	2	no	—
6	#16840694	C₂²	1	4	yes	—
7	D₆₁⋊C₃₂	—	2	2	no	—
8	C₂×#14519716	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
10	#16839716	C₄	1	1	yes	characteristic
11	#16839537	C₄	1	1	yes	characteristic
12	C₄×#11758185	C₂²	1	1	yes	characteristic
13	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#346	C₄	1	1	yes	characteristic
15	#14519827	—	2	4	no	—
16	C₂×#11758189	C₈	1	1	yes	characteristic
17	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
18	#14519734	C₈	1	1	yes	characteristic
19	#14519716	—	2	2	no	—
20	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
21	C₂×#11758185	C₂³	1	1	yes	characteristic
22	C₁₆×D₆₁	D₄	1	2	yes	—
23	C₁₆×D₆₁	Q₈	1	2	yes	—
24	#14519688	C₈	1	1	yes	characteristic
25	#14519687	C₈	1	1	yes	characteristic
26	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
34	#11758198	—	2	1	no	—
35	#11758196	—	2	1	no	—
36	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
37	C₂×#92418	C₂×C₈	1	1	yes	characteristic
38	C₂×#92418	C₂×C₈	1	1	yes	characteristic
39	#11758189	C₁₆	1	4	yes	—
40	C₈×D₆₁	C₄⋊C₄	1	2	yes	—
41	C₈×D₆₁	M₁₆	1	2	yes	—
42	C₁₆×C₆₁	—	2	1	no	—
43	C₁₆×C₆₁	C₄⋊C₄	1	2	yes	—
44	#11758185	—	2	1	no	—
45	#11758185	C₂×D₄	1	1	yes	characteristic
46	#11758185	C₂×Q₈	1	1	yes	characteristic
47	C₂²×D₆₁	M₃₂	1	1	yes	characteristic
48	C₂×C₄×C₆₁	—	2	1	no	—
49	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
50	C₂×Dic₆₁	—	2	1	no	—
51	C₂×Dic₆₁	M₃₂	1	1	yes	characteristic
52	C₄×D₆₁	C₄⋊C₈	1	2	yes	—
53	C₄×D₆₁	C₂×C₁₆	1	2	yes	—
54	C₈×C₆₁	C₂×C₁₆	1	2	yes	—
55	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₈×C₆₁	C₂×M₁₆	1	1	yes	characteristic
57	#92418	C₄⋊C₈	1	2	yes	—
58	#92418	C₂×C₁₆	1	2	yes	—
59	#13630	—	61	1	no	maximal, Sylow
60	C₂²×C₆₁	C₂×M₃₂	1	1	yes	characteristic
61	C₂×D₆₁	—	4	1	no	—
62	C₂×D₆₁	C₄⋊C₁₆	1	2	yes	—
63	C₄×C₆₁	—	4	1	no	—
64	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
66	Dic₆₁	—	4	1	no	—
67	Dic₆₁	C₄⋊C₁₆	1	2	yes	—
68	C₂×M₆₄	—	61	2	no	—
69	#2108	—	61	1	no	—
70	#1422	—	61	2	no	—
71	#1422	—	61	2	no	—
72	C₂×C₆₁	—	2	1	no	—
73	C₂×C₆₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
74	D₆₁	—	2	2	no	—
75	C₂×M₃₂	—	61	1	no	—
76	C₂²×C₁₆	—	61	1	no	—
77	#405	—	61	1	no	—
78	M₆₄	—	61	4	no	—
79	M₆₄	—	122	2	no	—
80	C₂×C₃₂	—	122	2	no	—
81	#346	—	61	1	no	—
82	#346	#6923	1	1	yes	characteristic
83	C₄×C₁₆	—	61	1	no	—
84	C₄×C₁₆	—	61	1	no	—
85	C₆₁	#13630	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₁₆	—	61	1	no	—
87	C₂²×C₈	—	61	1	no	—
88	#117	—	61	1	no	—
89	M₃₂	—	122	2	no	—
90	C₂×C₁₆	—	61	1	no	—
91	C₂×C₁₆	—	61	2	no	—
92	C₂×C₁₆	—	61	2	no	—
93	C₂×C₁₆	—	61	1	no	—
94	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
95	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
96	#97	—	61	1	no	—
97	#97	—	61	1	no	—
98	C₄×C₈	—	61	1	no	—
99	C₄×C₈	C₂×#6923	1	1	yes	characteristic
100	C₃₂	—	122	4	no	—
101	C₂²×C₄	—	61	1	no	—
102	M₁₆	—	61	4	no	—
103	C₂×C₈	—	61	1	no	—
104	C₂×C₈	—	61	1	no	—
105	C₂×C₈	—	61	2	no	—
106	C₂×C₈	—	61	2	no	—
107	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
108	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
109	C₄⋊C₄	—	122	1	no	—
110	C₂²⋊C₄	—	122	1	no	—
111	C₄²	—	61	1	no	—
112	C₄²	D₆₁⋊C₈	1	1	yes	characteristic
113	C₁₆	—	2	1	no	—
114	C₁₆	—	61	1	no	—
115	C₁₆	—	61	1	no	—
116	C₁₆	—	122	1	no	—
117	C₁₆	C₄⋊#6923	1	2	yes	—
118	C₂³	—	61	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	61	1	no	—
121	C₂×C₄	—	61	2	no	—
122	C₂×C₄	—	61	2	no	—
123	C₂×C₄	—	122	1	no	—
124	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
125	C₈	—	61	2	no	—
126	C₈	—	61	2	no	—
127	C₈	D₆₁⋊C₁₆	1	2	yes	—
128	C₈	C₂×C₄⋊#6923	1	1	yes	characteristic
129	C₈	D₆₁⋊M₁₆	1	1	yes	characteristic
130	C₂²	—	61	2	no	—
131	C₂²	—	244	1	no	—
132	C₂²	D₆₁⋊M₃₂	1	1	yes	characteristic
133	C₄	—	4	1	no	—
134	C₄	—	61	2	no	—
135	C₄	—	244	1	no	—
136	C₄	D₆₁⋊(C₄⋊C₈)	1	1	yes	characteristic
137	C₄	C₂×D₆₁⋊C₁₆	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	122	2	no	—
140	C₂	D₆₁⋊(C₄⋊C₁₆)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₆₁⋊M₆₄

C₂

yes

maximal

#21460979

C₂

yes

maximal, complemented

#21460978

C₂

yes

maximal, complemented

#21447417

C₂

yes

maximal, characteristic

#16840695

—

#16840694

C₂²

yes

—

D₆₁⋊C₃₂

—

C₂×#14519716

C₄

yes

characteristic

C₂×C₁₆×D₆₁

C₂²

yes

characteristic

#16839716

C₄

yes

characteristic

#16839537

C₄

yes

characteristic

C₄×#11758185

C₂²

yes

characteristic

C₁₆×Dic₆₁

C₂²

yes

characteristic

C₆₁×#346

C₄

yes

characteristic

#14519827

—

C₂×#11758189

C₈

yes

characteristic

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

#14519734

C₈

yes

characteristic

#14519716

—

C₂×#11758185

C₂×C₄

yes

characteristic

C₂×#11758185

C₂³

yes

characteristic

C₁₆×D₆₁

D₄

yes

—

C₁₆×D₆₁

Q₈

yes

—

#14519688

C₈

yes

characteristic

#14519687

C₈

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₈

yes

characteristic

#11758198

—

#11758196

—

C₄×Dic₆₁

C₂×C₈

yes

characteristic

C₂×#92418

C₂×C₈

yes

characteristic

C₂×#92418

C₂×C₈

yes

characteristic

#11758189

C₁₆

yes

—

C₈×D₆₁

C₄⋊C₄

yes

—

C₈×D₆₁

M₁₆

yes

—

C₁₆×C₆₁

—

C₁₆×C₆₁

C₄⋊C₄

yes

—

#11758185

—

#11758185

C₂×D₄

yes

characteristic

#11758185

C₂×Q₈

yes

characteristic

C₂²×D₆₁

M₃₂

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

M₃₂

yes

characteristic

C₄×D₆₁

C₄⋊C₈

yes

—

C₄×D₆₁

C₂×C₁₆

yes

—

C₈×C₆₁

C₂×C₁₆

yes

—

C₈×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₆₁

C₂×M₁₆

yes

characteristic

#92418

C₄⋊C₈

yes

—

#92418

C₂×C₁₆

yes

—

#13630

—

maximal, Sylow

C₂²×C₆₁

C₂×M₃₂

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

C₄⋊C₁₆

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₄×C₆₁

C₂²×C₁₆

yes

characteristic

Dic₆₁

—

Dic₆₁

C₄⋊C₁₆

yes

—

C₂×M₆₄

—

#2108

—

#1422

—

#1422

—

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄⋊C₁₆

yes

characteristic

D₆₁

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

M₆₄

—

M₆₄

—

122

—

C₂×C₃₂

—

122

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₆₁

#13630

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

122

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

C₄×C₈

—

C₄×C₈

C₂×#6923

yes

characteristic

100

C₃₂

—

122

—

101

C₂²×C₄

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

D₆₁⋊C₈

yes

characteristic

108

C₂×C₈

C₂²×#6923

yes

characteristic

109

C₄⋊C₄

—

122

—

110

C₂²⋊C₄

—

122

—

111

C₄²

—

112

C₄²

D₆₁⋊C₈

yes

characteristic

113

C₁₆

—

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

122

—

117

C₁₆

C₄⋊#6923

yes

—

118

C₂³

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

122

—

124

C₂×C₄

C₂×D₆₁⋊C₈

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

D₆₁⋊C₁₆

yes

—

128

C₈

C₂×C₄⋊#6923

yes

characteristic

129

C₈

D₆₁⋊M₁₆

yes

characteristic

130

C₂²

—

131

C₂²

—

244

—

132

C₂²

D₆₁⋊M₃₂

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

244

—

136

C₄

D₆₁⋊(C₄⋊C₈)

yes

characteristic

137

C₄

C₂×D₆₁⋊C₁₆

yes

characteristic

138

C₂

—

139

C₂

—

122

—

140

C₂

D₆₁⋊(C₄⋊C₁₆)

yes

characteristic

	subgroup	G/H
center	C₈	C₂×C₄⋊#6923
commutator	C₄×C₆₁	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₆₁×#346	C₄
socle	C₂×C₆₁	C₂×C₄⋊C₁₆
prime radical	C₂²×D₆₁	M₃₂
cosocle	C₂×#11758185	C₂³
hypercenter	#346	#6923
hypocenter	C₆₁	#13630
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #346

subgroup

G/H

center

C₈

C₂×C₄⋊#6923

commutator

C₄×C₆₁

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×#6923

Fitting

C₆₁×#346

C₄

socle

C₂×C₆₁

C₂×C₄⋊C₁₆

prime radical

C₂²×D₆₁

M₃₂

C₂³

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #346

N	H ≅ G/N	type
2. #21460979	C₂	semidirect
3. #21460978	C₂	semidirect
85. C₆₁	#13630	semidirect

H ≅ G/N

type

2. #21460979

C₂

semidirect

3. #21460978

C₂

semidirect

85. C₆₁

#13630

semidirect

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

subgroup	index
D₆₁⋊M₆₄	2
#21460979	2
#21460978	2
#21447417	2
#13630	61

subgroup

index

D₆₁⋊M₆₄

Subgroups — #30608513

Conjugacy classes of subgroups

Complements

Maximal subgroups