Group #29600495 — subgroups

Subgroups — #29600495

Order 15616.

218 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₂×#16839596	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#21444014	C₂	1	4	yes	maximal, complemented
4	C₂×C₆₁×M₆₄	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758185	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
7	#16839596	—	2	2	no	—
8	C₂×#14519683	C₂²	1	2	yes	—
9	C₁₆×Dic₆₁	C₄	1	2	yes	—
10	C₁₆×Dic₆₁	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×M₆₄	—	2	2	no	—
13	C₂×C₃₂×C₆₁	C₂²	1	2	yes	—
14	C₂²×#92418	C₈	1	1	yes	characteristic
15	C₂×C₄×Dic₆₁	C₈	1	1	yes	characteristic
16	C₂×#11758185	—	2	2	no	—
17	C₂×#11758185	C₂×C₄	1	2	yes	—
18	C₈×Dic₆₁	C₈	1	2	yes	—
19	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
20	#14519683	—	2	2	no	—
21	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
24	C₂×C₁₆×C₆₁	Q₈	1	2	yes	—
25	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
26	C₃₂×C₆₁	—	2	2	no	—
27	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
28	C₂²×Dic₆₁	C₁₆	1	2	yes	—
29	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
32	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₄×Dic₆₁	C₁₆	1	1	yes	characteristic
34	C₄×Dic₆₁	C₁₆	1	1	yes	characteristic
35	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
36	C₂×#92418	—	2	2	no	—
37	C₂×#92418	C₂×C₈	1	2	yes	—
38	C₁₆×C₆₁	C₄⋊C₄	1	2	yes	—
39	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	#11758185	—	2	2	no	—
42	C₂³×C₆₁	C₂×C₁₆	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
44	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₆₁	M₃₂	1	2	yes	—
46	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
47	C₂×Dic₆₁	—	2	4	no	—
48	C₂×Dic₆₁	C₂×C₁₆	1	2	yes	—
49	C₂×Dic₆₁	C₂×C₁₆	1	2	yes	—
50	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
51	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₈×C₆₁	C₂×M₁₆	1	1	yes	characteristic
53	#92418	—	2	2	no	—
54	#13627	—	61	1	no	maximal, Sylow
55	C₂²×C₆₁	—	2	2	no	—
56	C₂²×C₆₁	C₄⋊C₁₆	1	2	yes	—
57	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
58	C₄×C₆₁	C₄⋊C₁₆	1	2	yes	—
59	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
60	C₄×C₆₁	C₂×M₃₂	1	1	yes	characteristic
61	Dic₆₁	—	2	2	no	—
62	Dic₆₁	M₆₄	1	4	yes	complemented
63	C₂×M₆₄	—	61	1	no	—
64	C₂×M₆₄	D₆₁	1	1	yes	characteristic
65	C₂×C₄×C₁₆	—	61	1	no	—
66	C₄⋊C₃₂	—	61	4	no	—
67	C₂×C₆₁	—	2	2	no	—
68	C₂×C₆₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
69	C₂×C₆₁	C₂×M₆₄	1	2	yes	—
70	C₂²×C₁₆	—	61	1	no	—
71	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
72	C₂×C₄×C₈	—	61	1	no	—
73	M₆₄	—	2	2	no	—
74	M₆₄	—	122	2	no	—
75	C₂×C₃₂	—	61	2	no	—
76	C₂×C₃₂	C₂×D₆₁	1	2	yes	—
77	C₄×C₁₆	—	61	2	no	—
78	C₄×C₁₆	—	61	2	no	—
79	C₆₁	#13627	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	—	61	1	no	—
81	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
82	C₂×C₄²	—	61	1	no	—
83	C₂×C₁₆	—	61	2	no	—
84	C₂×C₁₆	—	122	2	no	—
85	C₂×C₁₆	Dic₁₂₂	1	2	yes	—
86	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
87	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
88	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
89	C₄×C₈	—	61	2	no	—
90	C₄×C₈	—	61	2	no	—
91	C₃₂	—	2	2	no	—
92	C₃₂	—	122	2	no	—
93	C₂²×C₄	—	61	2	no	—
94	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
95	C₂×C₈	—	61	2	no	—
96	C₂×C₈	—	122	2	no	—
97	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
98	C₂×C₈	#11758189	1	2	yes	—
99	C₂×C₈	#11758196	1	2	yes	—
100	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
101	C₄²	—	61	1	no	—
102	C₄²	—	61	1	no	—
103	C₄²	—	61	2	no	—
104	C₁₆	—	122	2	no	—
105	C₁₆	#11758196	1	2	yes	—
106	C₁₆	C₂×Dic₁₂₂	1	1	yes	characteristic
107	C₁₆	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
108	C₂³	C₁₆×D₆₁	1	1	yes	characteristic
109	C₂×C₄	—	61	2	no	—
110	C₂×C₄	—	61	2	no	—
111	C₂×C₄	—	122	4	no	—
112	C₂×C₄	#14519706	1	2	yes	—
113	C₂×C₄	C₁₆×D₆₁	1	1	yes	characteristic
114	C₂×C₄	#14519716	1	2	yes	—
115	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
116	C₈	—	122	2	no	—
117	C₈	#14519706	1	2	yes	—
118	C₈	C₂×#11758196	1	1	yes	characteristic
119	C₈	C₂×#11758189	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	#16839580	1	2	yes	—
122	C₂²	C₂×C₁₆×D₆₁	1	1	yes	characteristic
123	C₄	—	61	4	no	—
124	C₄	—	122	2	no	—
125	C₄	#16839580	1	2	yes	—
126	C₄	C₂×#14519706	1	1	yes	characteristic
127	C₄	C₂×#14519716	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	C₂×#16839580	1	1	yes	characteristic
130	C₂	M₆₄×D₆₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16839596

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₆₁

C₂

yes

maximal, characteristic

#21444014

C₂

yes

maximal, complemented

C₂×C₆₁×M₆₄

C₂

yes

maximal, characteristic

C₂²×#11758185

C₄

yes

characteristic

C₂×C₈×Dic₆₁

C₄

yes

characteristic

#16839596

—

C₂×#14519683

C₂²

yes

—

C₁₆×Dic₆₁

C₄

yes

—

C₁₆×Dic₆₁

C₂²

yes

—

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₆₁×M₆₄

—

C₂×C₃₂×C₆₁

C₂²

yes

—

C₂²×#92418

C₈

yes

characteristic

C₂×C₄×Dic₆₁

C₈

yes

characteristic

C₂×#11758185

—

C₂×#11758185

C₂×C₄

yes

—

C₈×Dic₆₁

C₈

yes

—

C₈×Dic₆₁

C₂×C₄

yes

—

#14519683

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

D₄

yes

—

C₂×C₁₆×C₆₁

Q₈

yes

—

C₂×C₁₆×C₆₁

C₂³

yes

characteristic

C₃₂×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂²×Dic₆₁

C₁₆

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

M₁₆

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₁₆

yes

characteristic

C₄×Dic₆₁

C₁₆

yes

characteristic

C₄×Dic₆₁

C₂×C₈

yes

—

C₂×#92418

—

C₂×#92418

C₂×C₈

yes

—

C₁₆×C₆₁

C₄⋊C₄

yes

—

C₁₆×C₆₁

C₂×D₄

yes

characteristic

C₁₆×C₆₁

C₂×Q₈

yes

characteristic

#11758185

—

C₂³×C₆₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₈

yes

—

C₂×C₄×C₆₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₁

M₃₂

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×C₁₆

yes

—

C₂×Dic₆₁

C₂×C₁₆

yes

—

C₈×C₆₁

C₄⋊C₈

yes

—

C₈×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₆₁

C₂×M₁₆

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄⋊C₁₆

yes

—

C₂²×C₆₁

C₂²×C₁₆

yes

characteristic

C₄×C₆₁

C₄⋊C₁₆

yes

—

C₄×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₄×C₆₁

C₂×M₃₂

yes

characteristic

Dic₆₁

—

Dic₆₁

M₆₄

yes

complemented

C₂×M₆₄

—

C₂×M₆₄

D₆₁

yes

characteristic

C₂×C₄×C₁₆

—

C₄⋊C₃₂

—

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₆₁

C₂×M₆₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

C₂×C₄×C₈

—

M₆₄

—

M₆₄

—

122

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₆₁

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₆₁

#13627

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₆₁

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

Dic₁₂₂

yes

—

C₂×C₁₆

C₄×D₆₁

yes

characteristic

C₂×C₁₆

C₂²⋊D₆₁

yes

—

C₂×C₁₆

C₂²×D₆₁

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₃₂

—

122

—

C₂²×C₄

—

C₂²×C₄

C₈×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

C₈×D₆₁

yes

characteristic

C₂×C₈

#11758189

yes

—

C₂×C₈

#11758196

yes

—

100

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₁₆

—

122

—

105

C₁₆

#11758196

yes

—

106

C₁₆

C₂×Dic₁₂₂

yes

characteristic

107

C₁₆

C₂×C₂²⋊D₆₁

yes

characteristic

108

C₂³

C₁₆×D₆₁

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

122

—

112

C₂×C₄

#14519706

yes

—

113

C₂×C₄

C₁₆×D₆₁

yes

characteristic

114

C₂×C₄

#14519716

yes

—

115

C₂×C₄

C₂×C₈×D₆₁

yes

characteristic

116

C₈

—

122

—

117

C₈

#14519706

yes

—

118

C₈

C₂×#11758196

yes

characteristic

119

C₈

C₂×#11758189

yes

characteristic

120

C₂²

—

121

C₂²

#16839580

yes

—

122

C₂²

C₂×C₁₆×D₆₁

yes

characteristic

123

C₄

—

124

C₄

—

122

—

125

C₄

#16839580

yes

—

126

C₄

C₂×#14519706

yes

characteristic

127

C₄

C₂×#14519716

yes

characteristic

128

C₂

—

129

C₂

C₂×#16839580

yes

characteristic

130

C₂

M₆₄×D₆₁

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₆₁
commutator, socle	C₂²×C₆₁	C₂²×C₁₆
Fitting	C₂×C₆₁×M₆₄	C₂
prime radical	C₂³×C₆₁	C₂×C₁₆
cosocle	C₂×C₁₆×C₆₁	C₂³
hypercenter	C₂×M₆₄	D₆₁
hypocenter	C₆₁	#13627
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₆₁

commutator, socle

C₂²×C₆₁

C₂²×C₁₆

Fitting

C₂×C₆₁×M₆₄

C₂

prime radical

C₂³×C₆₁

C₂×C₁₆

cosocle

C₂×C₁₆×C₆₁

C₂³

hypercenter

C₂×M₆₄

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

N	H ≅ G/N	type
3. #21444014	C₂	semidirect
62. Dic₆₁	M₆₄	semidirect
79. C₆₁	#13627	semidirect

H ≅ G/N

type

3. #21444014

C₂

semidirect

62. Dic₆₁

M₆₄

semidirect

79. C₆₁

#13627

semidirect

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

subgroup	index
C₂×#16839596	2
C₂×C₁₆×Dic₆₁	2
#21444014	2
C₂×C₆₁×M₆₄	2
#13627	61

subgroup

index

C₂×#16839596

C₂×C₁₆×Dic₆₁

#21444014

C₂×C₆₁×M₆₄

#13627

Subgroups — #29600495

Conjugacy classes of subgroups

Complements

Maximal subgroups