Group #21454537 — subgroups

Subgroups — #21454537

Order 7808.

484 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 221 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×#14519784	C₂	1	2	yes	maximal, complemented
2	#16839834	C₂	1	8	yes	maximal, complemented
3	#16839786	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839729	C₂	1	2	yes	maximal, complemented
5	C₂×#14519688	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#532	C₂	1	1	yes	maximal, characteristic
7	#14519784	—	2	8	no	—
8	C₂×#11758194	C₂²	1	2	yes	—
9	C₂×#11758194	C₂²	1	4	yes	—
10	C₂²×#92418	C₂²	1	1	yes	characteristic
11	C₂²×#92418	C₂²	1	4	yes	—
12	C₂²×#92418	C₂²	1	1	yes	characteristic
13	C₄⋊#92418	C₂²	1	4	yes	—
14	C₂²⋊#92418	C₂²	1	8	yes	—
15	#14519688	C₂²	1	4	yes	—
16	C₂×#41×C₆₁	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
18	#127×C₆₁	C₄	1	1	yes	characteristic
19	#124×C₆₁	C₄	1	2	yes	—
20	#121×C₆₁	C₄	1	4	yes	—
21	C₄×D₄×C₆₁	C₂²	1	4	yes	—
22	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
23	#41×C₆₁	—	2	4	no	—
24	#41×C₆₁	—	2	4	no	—
25	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
26	C₂×D₄×C₆₁	C₂×C₄	1	4	yes	—
27	C₂×D₄×C₆₁	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₆₁	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₆₁	C₂³	1	4	yes	—
31	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₆₁	C₂×C₄	1	8	yes	—
34	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₆₁	C₂³	1	2	yes	—
37	#11758194	—	2	4	no	—
38	#11758194	—	2	8	no	—
39	C₂×#92418	—	2	2	no	—
40	C₂×#92418	—	2	8	no	—
41	C₂×#92418	D₄	1	4	yes	complemented
42	C₂×#92418	C₂³	1	2	yes	—
43	C₂×#92418	C₂³	1	4	yes	—
44	C₂×#92418	C₂³	1	2	yes	—
45	C₂³×C₆₁	C₂²×C₄	1	4	yes	—
46	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₆₁	—	2	4	no	—
48	D₄×C₆₁	—	2	8	no	—
49	D₄×C₆₁	—	2	4	no	—
50	D₄×C₆₁	—	4	4	no	—
51	C₂×C₄×C₆₁	—	2	4	no	—
52	C₂×C₄×C₆₁	—	2	4	no	—
53	C₂×C₄×C₆₁	—	4	2	no	—
54	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₆₁	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₆₁	#41	1	2	yes	—
60	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
61	#92418	—	2	4	no	—
62	#92418	—	2	2	no	—
63	#92418	C₂×D₄	1	4	yes	—
64	C₂²×C₆₁	—	2	2	no	—
65	C₂²×C₆₁	—	2	4	no	—
66	C₂²×C₆₁	—	2	8	no	—
67	C₂²×C₆₁	C₄×D₄	1	2	yes	—
68	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₆₁	—	2	4	no	—
70	C₄×C₆₁	—	2	2	no	—
71	C₄×C₆₁	—	2	2	no	—
72	C₄×C₆₁	C₄×D₄	1	2	yes	—
73	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
74	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
75	#2932	—	61	1	no	maximal, Sylow
76	C₂×C₆₁	—	2	2	no	—
77	C₂×C₆₁	—	4	4	no	—
78	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
79	C₂×C₆₁	#568	1	2	yes	—
80	C₂×#131	—	61	2	no	—
81	#532	D₆₁	1	1	yes	characteristic
82	#435	—	61	8	no	—
83	#423	—	61	1	no	—
84	#408	—	61	2	no	—
85	C₂×#97	—	61	1	no	—
86	C₆₁	#2932	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	C₂×D₆₁	1	2	yes	—
88	#131	—	122	8	no	—
89	C₂×M₁₆	—	61	2	no	—
90	C₂×M₁₆	—	61	4	no	—
91	C₂²×C₈	—	61	1	no	—
92	C₂²×C₈	—	61	4	no	—
93	C₂²×C₈	—	61	1	no	—
94	C₄⋊Q₈	Dic₆₁	1	1	yes	characteristic
95	#127	Dic₆₁	1	1	yes	characteristic
96	#124	Dic₆₁	1	2	yes	—
97	#121	Dic₆₁	1	4	yes	—
98	C₄×D₄	C₂×D₆₁	1	4	yes	—
99	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
100	C₄⋊C₈	—	61	4	no	—
101	C₂²⋊C₈	—	61	8	no	—
102	#97	—	61	4	no	—
103	#41	—	2	4	no	—
104	#41	—	2	4	no	—
105	C₂×Q₈	C₂×Dic₆₁	1	2	yes	—
106	C₂×D₄	C₂×Dic₆₁	1	4	yes	—
107	C₂×D₄	C₂×Dic₆₁	1	2	yes	—
108	C₂×D₄	C₂×Dic₆₁	1	4	yes	—
109	C₂²×C₄	C₂×Dic₆₁	1	2	yes	—
110	C₂²×C₄	C₂²×D₆₁	1	4	yes	—
111	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
112	M₁₆	—	122	8	no	—
113	M₁₆	—	122	4	no	—
114	C₂×C₈	—	61	4	no	—
115	C₂×C₈	—	61	2	no	—
116	C₂×C₈	—	61	2	no	—
117	C₂×C₈	—	61	4	no	—
118	C₂×C₈	—	122	8	no	—
119	C₂×C₈	—	122	2	no	—
120	C₄⋊C₄	C₂×Dic₆₁	1	4	yes	—
121	C₂²⋊C₄	C₂×Dic₆₁	1	8	yes	—
122	C₄²	C₂×Dic₆₁	1	1	yes	characteristic
123	C₄²	C₂×Dic₆₁	1	1	yes	characteristic
124	C₄²	C₂²×D₆₁	1	2	yes	—
125	C₂³	C₂²×Dic₆₁	1	4	yes	—
126	C₂³	C₂²×Dic₆₁	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	D₄	—	2	8	no	—
129	D₄	—	2	4	no	—
130	D₄	—	4	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	D₄×D₆₁	1	2	yes	—
135	C₂×C₄	#11758224	1	2	yes	—
136	C₂×C₄	C₂²×Dic₆₁	1	2	yes	—
137	C₂×C₄	C₂²×Dic₆₁	1	2	yes	—
138	C₂×C₄	C₂²×Dic₆₁	1	4	yes	—
139	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
140	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
141	C₈	—	61	4	no	—
142	C₈	—	122	4	no	—
143	C₈	—	122	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	4	no	—
146	C₂²	—	2	8	no	—
147	C₂²	D₄×Dic₆₁	1	2	yes	—
148	C₂²	C₂³×Dic₆₁	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	D₄×Dic₆₁	1	2	yes	—
153	C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
154	C₄	C₂×#11758224	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	4	4	no	—
157	C₂	C₂×D₄×Dic₆₁	1	1	yes	characteristic
158	C₂	#16840529	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519784

C₂

yes

maximal, complemented

#16839834

C₂

yes

maximal, complemented

#16839786

C₂

yes

maximal, characteristic, complemented

#16839729

C₂

yes

maximal, complemented

C₂×#14519688

C₂

yes

maximal, characteristic, complemented

C₆₁×#532

C₂

yes

maximal, characteristic

#14519784

—

C₂×#11758194

C₂²

yes

—

C₂×#11758194

C₂²

yes

—

C₂²×#92418

C₂²

yes

characteristic

C₂²×#92418

C₂²

yes

—

C₂²×#92418

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

—

C₂²⋊#92418

C₂²

yes

—

#14519688

C₂²

yes

—

C₂×#41×C₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₄

yes

characteristic

#127×C₆₁

C₄

yes

characteristic

#124×C₆₁

C₄

yes

—

#121×C₆₁

C₄

yes

—

C₄×D₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

C₂×C₄

yes

—

C₂×D₄×C₆₁

C₂×C₄

yes

—

C₂×D₄×C₆₁

C₂×C₄

yes

—

C₂×D₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂²⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

—

—

—

—

—

D₄

yes

complemented

C₂×#92418

C₂³

yes

—

C₂×#92418

C₂³

yes

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂²×C₄

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

—

—

C₂×D₄

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×D₄

yes

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

—

C₄×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

C₂×#41

yes

characteristic

#2932

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄×D₄

yes

characteristic

C₂×C₆₁

#568

yes

—

C₂×#131

—

#532

D₆₁

yes

—

—

—

C₂×#97

—

C₆₁

#2932

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₆₁

yes

—

#131

—

122

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₆₁

yes

characteristic

#127

Dic₆₁

yes

characteristic

#124

Dic₆₁

yes

—

#121

Dic₆₁

yes

—

C₄×D₄

C₂×D₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

#97

—

103

#41

—

104

#41

—

105

C₂×Q₈

C₂×Dic₆₁

yes

—

106

C₂×D₄

C₂×Dic₆₁

yes

—

107

C₂×D₄

C₂×Dic₆₁

yes

—

108

C₂×D₄

C₂×Dic₆₁

yes

—

109

C₂²×C₄

C₂×Dic₆₁

yes

—

110

C₂²×C₄

C₂²×D₆₁

yes

—

111

C₂²×C₄

C₂²×D₆₁

yes

characteristic

112

M₁₆

—

122

—

113

M₁₆

—

122

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

122

—

119

C₂×C₈

—

122

—

120

C₄⋊C₄

C₂×Dic₆₁

yes

—

121

C₂²⋊C₄

C₂×Dic₆₁

yes

—

122

C₄²

C₂×Dic₆₁

yes

characteristic

123

C₄²

C₂×Dic₆₁

yes

characteristic

124

C₄²

C₂²×D₆₁

yes

—

125

C₂³

C₂²×Dic₆₁

yes

—

126

C₂³

C₂²×Dic₆₁

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

D₄×D₆₁

yes

—

135

C₂×C₄

#11758224

yes

—

136

C₂×C₄

C₂²×Dic₆₁

yes

—

137

C₂×C₄

C₂²×Dic₆₁

yes

—

138

C₂×C₄

C₂²×Dic₆₁

yes

—

139

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

140

C₂×C₄

C₂³×D₆₁

yes

characteristic

141

C₈

—

142

C₈

—

122

—

143

C₈

—

122

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

D₄×Dic₆₁

yes

—

148

C₂²

C₂³×Dic₆₁

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

D₄×Dic₆₁

yes

—

153

C₄

C₂×D₄×D₆₁

yes

characteristic

154

C₄

C₂×#11758224

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

C₂×D₄×Dic₆₁

yes

characteristic

158

C₂

#16840529

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₆₁×#532	C₂
hypercenter	#532	D₆₁
hypocenter	C₆₁	#2932
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₁

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₆₁×#532

C₂

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. C₂×#14519784	C₂	semidirect
2. #16839834	C₂	semidirect
3. #16839786	C₂	semidirect
4. #16839729	C₂	semidirect
5. C₂×#14519688	C₂	semidirect
41. C₂×#92418	D₄	semidirect
86. C₆₁	#2932	semidirect

H ≅ G/N

type

1. C₂×#14519784

C₂

semidirect

2. #16839834

C₂

semidirect

3. #16839786

C₂

semidirect

4. #16839729

C₂

semidirect

5. C₂×#14519688

C₂

semidirect

41. C₂×#92418

D₄

semidirect

86. C₆₁

#2932

semidirect

Subgroups — #21454537

Conjugacy classes of subgroups

Complements