Group #29706568 — subgroups

Subgroups — #29706568

Order 15616.

912 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 255 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	#21446117	C₂	1	12	yes	maximal, complemented
2	#21445962	C₂	1	2	yes	maximal, complemented
3	C₄×#127×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11758194	C₂²	1	4	yes	—
5	C₂×C₂²⋊#92418	C₂²	1	12	yes	—
6	#16839517	C₂²	1	12	yes	—
7	C₂×#127×C₆₁	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄×C₆₁	C₂²	1	6	yes	—
9	C₆₁×#390	C₄	1	6	yes	—
10	C₆₁×#384	C₄	1	1	yes	characteristic
11	C₄³×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×#11758194	—	4	12	no	—
13	C₂²×#92418	C₂³	1	8	yes	—
14	C₂²⋊#92418	—	4	12	no	—
15	C₂²×D₄×C₆₁	C₂×C₄	1	6	yes	—
16	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
17	#127×C₆₁	—	2	4	no	—
18	C₄×D₄×C₆₁	—	4	6	no	—
19	C₄×D₄×C₆₁	—	4	6	no	—
20	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	6	yes	—
21	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	12	yes	—
22	C₂×C₄²×C₆₁	C₂×C₄	1	3	yes	—
23	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₆₁	C₂³	1	3	yes	—
25	C₂⁴×C₆₁	C₂²×C₄	1	4	yes	—
26	C₂×D₄×C₆₁	—	2	12	no	—
27	C₂×D₄×C₆₁	—	4	6	no	—
28	C₂×D₄×C₆₁	—	4	6	no	—
29	C₂²×C₄×C₆₁	—	2	12	no	—
30	C₂²×C₄×C₆₁	—	2	12	no	—
31	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
32	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
33	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
34	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
35	C₄⋊C₄×C₆₁	—	4	6	no	—
36	C₂²⋊C₄×C₆₁	—	4	12	no	—
37	C₄²×C₆₁	—	2	6	no	—
38	C₄²×C₆₁	—	4	3	no	—
39	C₄²×C₆₁	M₁₆	1	4	yes	complemented
40	#11758194	—	4	16	no	—
41	C₂×#92418	—	2	24	no	—
42	C₂³×C₆₁	—	2	12	no	—
43	C₂³×C₆₁	—	2	4	no	—
44	C₂³×C₆₁	—	2	12	no	—
45	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
46	D₄×C₆₁	—	4	24	no	—
47	C₂×C₄×C₆₁	—	2	3	no	—
48	C₂×C₄×C₆₁	—	2	3	no	—
49	C₂×C₄×C₆₁	—	2	3	no	—
50	C₂×C₄×C₆₁	—	2	12	no	—
51	C₂×C₄×C₆₁	—	2	6	no	—
52	C₂×C₄×C₆₁	—	4	4	no	—
53	C₂×C₄×C₆₁	—	4	12	no	—
54	C₂×C₄×C₆₁	C₂×M₁₆	1	6	yes	—
55	C₂×C₄×C₆₁	H(5,𝔽₂)	1	6	yes	—
56	#92418	—	4	8	no	—
57	#23357	—	61	1	no	maximal, Sylow
58	C₂²×C₆₁	—	4	4	no	—
59	C₂²×C₆₁	—	4	12	no	—
60	C₂²×C₆₁	—	4	12	no	—
61	C₂²×C₆₁	#518	1	3	yes	—
62	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
63	C₂²×C₆₁	#568	1	3	yes	—
64	C₄×C₆₁	—	2	6	no	—
65	C₄×C₆₁	—	4	4	no	—
66	C₄×C₆₁	—	4	3	no	—
67	C₄×C₆₁	#560	1	4	yes	—
68	C₄×#127	D₆₁	1	1	yes	characteristic
69	#1870	—	61	12	no	—
70	#1839	—	61	2	no	—
71	C₂×C₆₁	—	8	4	no	—
72	C₂×C₆₁	#2380	1	1	yes	characteristic
73	C₂×C₆₁	#2975	1	3	yes	—
74	C₂×C₆₁	#2980	1	3	yes	—
75	C₂²×M₁₆	—	61	4	no	—
76	C₂×#127	Dic₆₁	1	1	yes	characteristic
77	C₂×C₄×D₄	C₂×D₆₁	1	6	yes	—
78	C₂×C₂²⋊C₈	—	61	12	no	—
79	#390	Dic₆₁	1	6	yes	—
80	#384	Dic₆₁	1	1	yes	characteristic
81	C₄³	C₂×D₆₁	1	1	yes	characteristic
82	#336	—	61	12	no	—
83	C₆₁	#23357	1	1	yes	characteristic, complemented, Sylow
84	C₂²×D₄	C₂×Dic₆₁	1	6	yes	—
85	C₂³×C₄	C₂²×D₆₁	1	4	yes	—
86	C₂×M₁₆	—	244	12	no	—
87	C₂²×C₈	—	61	8	no	—
88	#127	—	2	4	no	—
89	C₄×D₄	—	4	6	no	—
90	C₄×D₄	—	4	6	no	—
91	C₂×C₄⋊C₄	C₂×Dic₆₁	1	6	yes	—
92	C₂×C₂²⋊C₄	C₂×Dic₆₁	1	12	yes	—
93	C₂×C₄²	C₂×Dic₆₁	1	3	yes	—
94	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
95	C₂×C₄²	C₂²×D₆₁	1	3	yes	—
96	C₂²⋊C₈	—	244	12	no	—
97	C₂⁴	C₂²×Dic₆₁	1	4	yes	—
98	C₂×D₄	—	2	12	no	—
99	C₂×D₄	—	4	6	no	—
100	C₂×D₄	—	4	6	no	—
101	C₂²×C₄	—	2	12	no	—
102	C₂²×C₄	—	2	12	no	—
103	C₂²×C₄	C₂²×Dic₆₁	1	3	yes	—
104	C₂²×C₄	C₂²×Dic₆₁	1	4	yes	—
105	C₂²×C₄	C₂²×Dic₆₁	1	3	yes	—
106	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
107	M₁₆	—	244	16	no	—
108	C₂×C₈	—	122	24	no	—
109	C₄⋊C₄	—	4	6	no	—
110	C₂²⋊C₄	—	4	12	no	—
111	C₄²	—	2	6	no	—
112	C₄²	—	4	3	no	—
113	C₄²	#11758194	1	4	yes	complemented
114	C₂³	—	2	12	no	—
115	C₂³	—	2	4	no	—
116	C₂³	—	2	12	no	—
117	C₂³	C₂³×Dic₆₁	1	1	yes	characteristic
118	D₄	—	4	24	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	2	3	no	—
121	C₂×C₄	—	2	3	no	—
122	C₂×C₄	—	2	12	no	—
123	C₂×C₄	—	2	6	no	—
124	C₂×C₄	—	4	4	no	—
125	C₂×C₄	—	4	12	no	—
126	C₂×C₄	C₂×#11758194	1	6	yes	—
127	C₂×C₄	#14519822	1	6	yes	—
128	C₈	—	244	8	no	—
129	C₂²	—	4	12	no	—
130	C₂²	—	4	4	no	—
131	C₂²	—	4	12	no	—
132	C₂²	#16840237	1	3	yes	—
133	C₂²	C₂²×#11758194	1	1	yes	characteristic
134	C₂²	#16840529	1	3	yes	—
135	C₄	—	2	6	no	—
136	C₄	—	4	4	no	—
137	C₄	—	4	3	no	—
138	C₄	#16840499	1	4	yes	—
139	C₂	—	8	4	no	—
140	C₂	#21448996	1	1	yes	characteristic
141	C₂	#21454860	1	3	yes	—
142	C₂	#21454890	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21446117

C₂

yes

maximal, complemented

#21445962

C₂

yes

maximal, complemented

C₄×#127×C₆₁

C₂

yes

maximal, characteristic

C₂²×#11758194

C₂²

yes

—

C₂×C₂²⋊#92418

C₂²

yes

—

#16839517

C₂²

yes

—

C₂×#127×C₆₁

C₄

yes

characteristic

C₂×C₄×D₄×C₆₁

C₂²

yes

—

C₆₁×#390

C₄

yes

—

C₆₁×#384

C₄

yes

characteristic

C₄³×C₆₁

C₂²

yes

characteristic

C₂×#11758194

—

C₂²×#92418

C₂³

yes

—

C₂²⋊#92418

—

C₂²×D₄×C₆₁

C₂×C₄

yes

—

C₂³×C₄×C₆₁

C₂³

yes

—

#127×C₆₁

—

C₄×D₄×C₆₁

—

C₄×D₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₁

C₂³

yes

—

C₂⁴×C₆₁

C₂²×C₄

yes

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

—

C₂²×C₄×C₆₁

C₂⁴

yes

characteristic

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

M₁₆

yes

complemented

#11758194

—

C₂×#92418

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂³×C₄

yes

characteristic

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×C₄×C₆₁

H(5,𝔽₂)

yes

—

#92418

—

#23357

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#518

yes

—

C₂²×C₆₁

C₂²×M₁₆

yes

characteristic

C₂²×C₆₁

#568

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#560

yes

—

C₄×#127

D₆₁

yes

characteristic

#1870

—

#1839

—

C₂×C₆₁

—

C₂×C₆₁

#2380

yes

characteristic

C₂×C₆₁

#2975

yes

—

C₂×C₆₁

#2980

yes

—

C₂²×M₁₆

—

C₂×#127

Dic₆₁

yes

characteristic

C₂×C₄×D₄

C₂×D₆₁

yes

—

C₂×C₂²⋊C₈

—

#390

Dic₆₁

yes

—

#384

Dic₆₁

yes

characteristic

C₄³

C₂×D₆₁

yes

characteristic

#336

—

C₆₁

#23357

yes

characteristic, complemented, Sylow

C₂²×D₄

C₂×Dic₆₁

yes

—

C₂³×C₄

C₂²×D₆₁

yes

—

C₂×M₁₆

—

244

—

C₂²×C₈

—

#127

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

C₂×Dic₆₁

yes

—

C₂×C₂²⋊C₄

C₂×Dic₆₁

yes

—

C₂×C₄²

C₂×Dic₆₁

yes

—

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

C₂×C₄²

C₂²×D₆₁

yes

—

C₂²⋊C₈

—

244

—

C₂⁴

C₂²×Dic₆₁

yes

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²×Dic₆₁

yes

—

104

C₂²×C₄

C₂²×Dic₆₁

yes

—

105

C₂²×C₄

C₂²×Dic₆₁

yes

—

106

C₂²×C₄

C₂³×D₆₁

yes

characteristic

107

M₁₆

—

244

—

108

C₂×C₈

—

122

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₄²

#11758194

yes

complemented

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

C₂³×Dic₆₁

yes

characteristic

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂×#11758194

yes

—

127

C₂×C₄

#14519822

yes

—

128

C₈

—

244

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#16840237

yes

—

133

C₂²

C₂²×#11758194

yes

characteristic

134

C₂²

#16840529

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#16840499

yes

—

139

C₂

—

140

C₂

#21448996

yes

characteristic

141

C₂

#21454860

yes

—

142

C₂

#21454890

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₁
commutator, socle	C₂³×C₆₁	C₂³×C₄
Fitting	C₄×#127×C₆₁	C₂
prime radical	C₂×#127×C₆₁	C₄
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₄×#127	D₆₁
hypocenter	C₆₁	#23357
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#127

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₆₁

commutator, socle

C₂³×C₆₁

C₂³×C₄

Fitting

C₄×#127×C₆₁

C₂

prime radical

C₂×#127×C₆₁

C₄

cosocle

C₂²×C₄×C₆₁

C₂⁴

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂²×C₄ ◁ C₄×#127

N	H ≅ G/N	type
1. #21446117	C₂	semidirect
2. #21445962	C₂	semidirect
39. C₄²×C₆₁	M₁₆	semidirect
83. C₆₁	#23357	semidirect
113. C₄²	#11758194	semidirect

H ≅ G/N

type

1. #21446117

C₂

semidirect

2. #21445962

C₂

semidirect

39. C₄²×C₆₁

M₁₆

semidirect

83. C₆₁

#23357

semidirect

113. C₄²

#11758194

semidirect

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

subgroup	index
#21446117	2
#21445962	2
C₄×#127×C₆₁	2
#23357	61

subgroup

index

#21446117

#21445962

C₄×#127×C₆₁

#23357

Subgroups — #29706568

Conjugacy classes of subgroups

Complements

Maximal subgroups