Group #21449011 — subgroups

Subgroups — #21449011

Order 7808.

456 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 213 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	#16839661	C₂	1	3	yes	maximal, complemented
2	#16839655	C₂	1	3	yes	maximal
3	#16839648	C₂	1	3	yes	maximal, complemented
4	#16839637	C₂	1	3	yes	maximal
5	#16839632	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839630	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#126×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₆₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₆₁	C₂²	1	3	yes	—
10	C₂×C₄⋊Dic₆₁	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₆₁	C₂²	1	3	yes	—
12	C₂×C₂²⋊Dic₆₁	C₂²	1	3	yes	—
13	C₂×C₄×Dic₆₁	C₂²	1	3	yes	—
14	#14519684	C₂²	1	3	yes	—
15	#14519684	C₂²	1	3	yes	—
16	#14519684	C₂²	1	3	yes	—
17	#14519684	C₂²	1	3	yes	—
18	#126×C₆₁	C₄	1	8	yes	complemented
19	C₂×C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
20	C₂×C₂²⋊C₄×C₆₁	C₂²	1	3	yes	—
21	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
22	C₂⁴×C₆₁	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
24	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
25	C₂²×Dic₆₁	—	4	3	no	—
26	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
27	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
28	C₂²×Dic₆₁	C₂³	1	3	yes	—
29	C₂²×Dic₆₁	C₂³	1	3	yes	—
30	C₄⋊C₄×C₆₁	C₂×C₄	1	12	yes	complemented
31	C₂²⋊C₄×C₆₁	C₂×C₄	1	12	yes	—
32	C₄²×C₆₁	C₂×C₄	1	4	yes	complemented
33	C₂²⋊Dic₆₁	—	4	3	no	—
34	C₄⋊Dic₆₁	—	4	3	no	—
35	C₄⋊Dic₆₁	—	4	3	no	—
36	C₄⋊Dic₆₁	—	4	3	no	—
37	C₄×Dic₆₁	—	4	3	no	—
38	C₂³×C₆₁	—	2	3	no	—
39	C₂³×C₆₁	—	2	3	no	—
40	C₂³×C₆₁	C₂²×C₄	1	2	yes	—
41	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₆₁	—	2	3	no	—
43	C₂×C₄×C₆₁	—	2	3	no	—
44	C₂×C₄×C₆₁	—	2	3	no	—
45	C₂×C₄×C₆₁	—	2	3	no	—
46	C₂×C₄×C₆₁	C₂²×C₄	1	6	yes	—
47	C₂×C₄×C₆₁	C₂²×C₄	1	6	yes	—
48	C₂×Dic₆₁	—	2	3	no	—
49	C₂×Dic₆₁	—	2	3	no	—
50	C₂×Dic₆₁	—	2	3	no	—
51	C₂×Dic₆₁	—	2	3	no	—
52	C₂×Dic₆₁	—	2	3	no	—
53	C₂×Dic₆₁	—	2	3	no	—
54	C₂×Dic₆₁	—	2	6	no	—
55	C₂×Dic₆₁	—	4	4	no	—
56	C₂²×C₆₁	—	2	6	no	—
57	C₂²×C₆₁	—	4	1	no	—
58	C₂²×C₆₁	—	4	3	no	—
59	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
60	C₂²×C₆₁	H(5,𝔽₂)	1	3	yes	—
61	C₂²×C₆₁	2₋¹⁺⁴	1	3	yes	—
62	C₄×C₆₁	—	2	6	no	—
63	C₄×C₆₁	—	2	6	no	—
64	Dic₆₁	—	4	3	no	—
65	Dic₆₁	—	4	1	no	—
66	Dic₆₁	—	4	1	no	—
67	Dic₆₁	—	4	3	no	—
68	#2383	—	61	1	no	maximal, Sylow
69	C₂×C₆₁	—	4	2	no	—
70	C₂×C₆₁	#520	1	3	yes	—
71	C₂×C₆₁	#560	1	1	yes	characteristic
72	C₂×C₆₁	#563	1	3	yes	—
73	C₂×#126	D₆₁	1	1	yes	characteristic
74	#389	—	61	3	no	—
75	#388	—	61	3	no	—
76	#387	—	61	3	no	—
77	#385	—	61	3	no	—
78	#384	—	61	1	no	—
79	#383	—	61	1	no	—
80	C₆₁	#2383	1	1	yes	characteristic, complemented, Sylow
81	C₂³×C₄	—	61	1	no	—
82	#126	Dic₆₁	1	8	yes	complemented
83	C₂×C₄⋊C₄	—	61	3	no	—
84	C₂×C₄⋊C₄	—	61	3	no	—
85	C₂×C₄⋊C₄	—	61	3	no	—
86	C₂×C₄⋊C₄	C₂×D₆₁	1	3	yes	—
87	C₂×C₂²⋊C₄	—	61	3	no	—
88	C₂×C₂²⋊C₄	C₂×D₆₁	1	3	yes	—
89	C₂×C₄²	—	61	3	no	—
90	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
91	#95	—	61	3	no	—
92	#95	—	61	3	no	—
93	#95	—	61	3	no	—
94	#95	—	61	3	no	—
95	C₂⁴	C₂²×D₆₁	1	1	yes	characteristic
96	C₂²×C₄	—	61	3	no	—
97	C₂²×C₄	—	61	1	no	—
98	C₂²×C₄	—	61	1	no	—
99	C₂²×C₄	—	61	3	no	—
100	C₂²×C₄	—	244	3	no	—
101	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
102	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
103	C₄⋊C₄	—	244	3	no	—
104	C₄⋊C₄	—	244	3	no	—
105	C₄⋊C₄	—	244	3	no	—
106	C₄⋊C₄	C₂×Dic₆₁	1	12	yes	complemented
107	C₂²⋊C₄	—	244	3	no	—
108	C₂²⋊C₄	C₂×Dic₆₁	1	12	yes	—
109	C₄²	—	244	3	no	—
110	C₄²	C₂×Dic₆₁	1	4	yes	complemented
111	C₂³	—	2	3	no	—
112	C₂³	—	2	3	no	—
113	C₂³	C₂²×Dic₆₁	1	2	yes	—
114	C₂³	C₂³×D₆₁	1	1	yes	characteristic
115	C₂×C₄	—	2	3	no	—
116	C₂×C₄	—	2	3	no	—
117	C₂×C₄	—	2	3	no	—
118	C₂×C₄	—	2	3	no	—
119	C₂×C₄	—	122	3	no	—
120	C₂×C₄	—	122	3	no	—
121	C₂×C₄	—	122	3	no	—
122	C₂×C₄	—	122	3	no	—
123	C₂×C₄	—	122	3	no	—
124	C₂×C₄	—	122	3	no	—
125	C₂×C₄	—	122	6	no	—
126	C₂×C₄	—	244	4	no	—
127	C₂×C₄	C₂²×Dic₆₁	1	6	yes	—
128	C₂×C₄	C₂²×Dic₆₁	1	6	yes	—
129	C₂²	—	2	6	no	—
130	C₂²	—	4	1	no	—
131	C₂²	—	4	3	no	—
132	C₂²	C₂³×Dic₆₁	1	1	yes	characteristic
133	C₂²	#14519823	1	3	yes	—
134	C₂²	#14519824	1	3	yes	—
135	C₄	—	2	6	no	—
136	C₄	—	2	6	no	—
137	C₄	—	244	3	no	—
138	C₄	—	244	1	no	—
139	C₄	—	244	1	no	—
140	C₄	—	244	3	no	—
141	C₂	—	4	2	no	—
142	C₂	#16840245	1	3	yes	—
143	C₂	#16840503	1	1	yes	characteristic
144	C₂	#16840514	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16839661

C₂

yes

maximal, complemented

#16839655

C₂

yes

maximal

#16839648

C₂

yes

maximal, complemented

#16839637

C₂

yes

maximal

#16839632

C₂

yes

maximal, characteristic, complemented

#16839630

C₂

yes

maximal, characteristic, complemented

C₂×#126×C₆₁

C₂

yes

maximal, characteristic

C₂³×Dic₆₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₂²⋊Dic₆₁

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

#14519684

C₂²

yes

—

#14519684

C₂²

yes

—

#14519684

C₂²

yes

—

#14519684

C₂²

yes

—

#126×C₆₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂⁴×C₆₁

C₂³

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

complemented

C₂²⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

complemented

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂²×C₄

yes

—

C₂³×C₆₁

C₂⁴

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₂²×C₆₁

H(5,𝔽₂)

yes

—

C₂²×C₆₁

2₋¹⁺⁴

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#2383

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#520

yes

—

C₂×C₆₁

#560

yes

characteristic

C₂×C₆₁

#563

yes

—

C₂×#126

D₆₁

yes

—

—

—

—

—

—

C₆₁

#2383

yes

characteristic, complemented, Sylow

C₂³×C₄

—

#126

Dic₆₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₁

yes

—

—

—

—

C₂⁴

C₂²×D₆₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

244

—

101

C₂²×C₄

C₂²×D₆₁

yes

—

102

C₂²×C₄

C₂²×D₆₁

yes

—

103

C₄⋊C₄

—

244

—

104

C₄⋊C₄

—

244

—

105

C₄⋊C₄

—

244

—

106

C₄⋊C₄

C₂×Dic₆₁

yes

complemented

107

C₂²⋊C₄

—

244

—

108

C₂²⋊C₄

C₂×Dic₆₁

yes

—

109

C₄²

—

244

—

110

C₄²

C₂×Dic₆₁

yes

complemented

111

C₂³

—

112

C₂³

—

113

C₂³

C₂²×Dic₆₁

yes

—

114

C₂³

C₂³×D₆₁

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

122

—

120

C₂×C₄

—

122

—

121

C₂×C₄

—

122

—

122

C₂×C₄

—

122

—

123

C₂×C₄

—

122

—

124

C₂×C₄

—

122

—

125

C₂×C₄

—

122

—

126

C₂×C₄

—

244

—

127

C₂×C₄

C₂²×Dic₆₁

yes

—

128

C₂×C₄

C₂²×Dic₆₁

yes

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×Dic₆₁

yes

characteristic

133

C₂²

#14519823

yes

—

134

C₂²

#14519824

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

244

—

138

C₄

—

244

—

139

C₄

—

244

—

140

C₄

—

244

—

141

C₂

—

142

C₂

#16840245

yes

—

143

C₂

#16840503

yes

characteristic

144

C₂

#16840514

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator	C₂²×C₆₁	C₂³×C₄
Fitting	C₂×#126×C₆₁	C₂
socle	C₂³×C₆₁	C₂⁴
hypercenter	C₂×#126	D₆₁
hypocenter	C₆₁	#2383
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×#126

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₁

commutator

C₂²×C₆₁

C₂³×C₄

Fitting

C₂×#126×C₆₁

C₂

socle

C₂³×C₆₁

C₂⁴

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ C₂×#126

N	H ≅ G/N	type
1. #16839661	C₂	semidirect
3. #16839648	C₂	semidirect
5. #16839632	C₂	semidirect
6. #16839630	C₂	semidirect
18. #126×C₆₁	C₄	semidirect
30. C₄⋊C₄×C₆₁	C₂×C₄	semidirect
32. C₄²×C₆₁	C₂×C₄	semidirect
80. C₆₁	#2383	semidirect
82. #126	Dic₆₁	semidirect
106. C₄⋊C₄	C₂×Dic₆₁	semidirect
110. C₄²	C₂×Dic₆₁	semidirect

H ≅ G/N

type

1. #16839661

C₂

semidirect

3. #16839648

C₂

semidirect

5. #16839632

C₂

semidirect

6. #16839630

C₂

semidirect

18. #126×C₆₁

C₄

semidirect

30. C₄⋊C₄×C₆₁

C₂×C₄

semidirect

32. C₄²×C₆₁

C₂×C₄

semidirect

80. C₆₁

#2383

semidirect

82. #126

Dic₆₁

semidirect

106. C₄⋊C₄

C₂×Dic₆₁

semidirect

110. C₄²

C₂×Dic₆₁

semidirect

Subgroups — #21449011

Conjugacy classes of subgroups

Complements