Group #21461962 — subgroups

Subgroups — #21461962

Order 7808.

292 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 76 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	#16840765	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519838	C₂	1	4	yes	maximal
3	C₄×#11758217	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519776	C₂	1	1	yes	maximal, characteristic
5	C₂×#11758217	C₂²	1	2	yes	—
6	C₂×#11758217	C₂²	1	2	yes	—
7	#14519844	—	2	1	no	—
8	C₄⋊#92420	—	2	1	no	—
9	#14519838	—	2	8	no	—
10	#14519832	—	2	1	no	—
11	C₄×#92420	—	2	1	no	—
12	C₂²×Dic₁₂₂	C₂²	1	2	yes	—
13	#14519776	—	2	1	no	—
14	#14519776	—	2	1	no	—
15	#14519776	—	2	1	no	—
16	#14519776	—	2	1	no	—
17	C₂×C₄⋊Dic₆₁	C₄	1	1	yes	characteristic
18	C₂×#11758196	C₄	1	2	yes	—
19	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₆₁	C₂×C₄	1	2	yes	—
24	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₂₂	—	2	4	no	—
27	C₂×Dic₁₂₂	—	2	2	no	—
28	C₂×Dic₁₂₂	—	2	2	no	—
29	C₂×Dic₁₂₂	—	2	4	no	—
30	#11758217	—	2	2	no	—
31	#11758217	—	2	4	no	—
32	#11758217	D₄	1	4	yes	—
33	C₂×#92420	—	2	2	no	—
34	C₂×#92420	—	2	2	no	—
35	C₄⋊C₄×C₆₁	—	2	1	no	—
36	C₄⋊C₄×C₆₁	—	2	1	no	—
37	C₄⋊Dic₆₁	—	2	1	no	—
38	C₄⋊Dic₆₁	—	2	1	no	—
39	#11758196	—	2	2	no	—
40	#11758196	—	2	2	no	—
41	C₄×Dic₆₁	D₄	1	1	yes	characteristic
42	C₄×Dic₆₁	D₄	1	1	yes	characteristic
43	C₄×Dic₆₁	D₄	1	1	yes	characteristic
44	C₄×Dic₆₁	D₄	1	1	yes	characteristic
45	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	—	2	2	no	—
47	C₂×C₄×C₆₁	—	2	2	no	—
48	C₂×C₄×C₆₁	—	2	4	no	—
49	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₆₁	—	2	2	no	—
52	C₂×Dic₆₁	—	2	2	no	—
53	C₂×Dic₆₁	—	2	4	no	—
54	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
58	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂×Dic₆₁	#41	1	2	yes	—
60	Dic₁₂₂	—	4	4	no	—
61	Dic₁₂₂	—	4	2	no	—
62	Dic₁₂₂	—	4	2	no	—
63	#92420	—	2	4	no	—
64	#92420	—	4	2	no	—
65	C₂²×C₆₁	—	2	1	no	—
66	C₂²×C₆₁	—	2	1	no	—
67	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₆₁	C₄×D₄	1	2	yes	—
69	C₄×C₆₁	—	2	1	no	—
70	C₄×C₆₁	—	4	2	no	—
71	C₄×C₆₁	#101	1	2	yes	complemented
72	Dic₆₁	—	2	1	no	—
73	Dic₆₁	—	4	2	no	—
74	Dic₆₁	#101	1	2	yes	complemented
75	Dic₆₁	#121	1	2	yes	—
76	Dic₆₁	#124	1	1	yes	characteristic
77	Dic₆₁	#127	1	1	yes	characteristic
78	#1966	—	61	1	no	maximal, Sylow
79	C₂×C₆₁	—	2	2	no	—
80	C₂×C₆₁	#390	1	1	yes	characteristic
81	C₂×C₆₁	C₂×#101	1	1	yes	characteristic
82	C₂×C₆₁	C₂×#101	1	1	yes	characteristic
83	C₂×C₄⋊Q₈	—	61	1	no	—
84	#423	—	61	1	no	—
85	C₂×#101	—	61	4	no	—
86	C₄×M₁₆	—	61	1	no	—
87	C₆₁	#1966	1	1	yes	characteristic, complemented, Sylow
88	C₂²×Q₈	—	61	2	no	—
89	C₂×M₁₆	—	61	2	no	—
90	C₂×M₁₆	—	61	2	no	—
91	C₄⋊Q₈	—	122	1	no	—
92	C₄⋊Q₈	—	122	1	no	—
93	C₄⋊Q₈	—	122	1	no	—
94	C₄⋊Q₈	—	122	1	no	—
95	C₂×C₄⋊C₄	—	61	2	no	—
96	C₂×C₄⋊C₄	—	61	1	no	—
97	C₂×C₄⋊C₄	#6923	1	1	yes	characteristic
98	C₂×C₄²	—	61	1	no	—
99	C₄⋊C₈	—	122	1	no	—
100	C₄⋊C₈	—	122	1	no	—
101	#101	—	122	8	no	—
102	#97	—	122	1	no	—
103	C₄×C₈	—	122	1	no	—
104	C₂×Q₈	—	122	4	no	—
105	C₂×Q₈	—	122	2	no	—
106	C₂×Q₈	—	122	2	no	—
107	C₂×Q₈	—	122	4	no	—
108	C₂²×C₄	—	61	1	no	—
109	C₂²×C₄	—	61	1	no	—
110	C₂²×C₄	—	61	2	no	—
111	C₂²×C₄	C₂×#6923	1	2	yes	—
112	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
113	M₁₆	—	61	4	no	—
114	M₁₆	—	122	2	no	—
115	M₁₆	—	122	4	no	—
116	C₂×C₈	—	122	2	no	—
117	C₂×C₈	—	122	2	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	122	2	no	—
121	C₄⋊C₄	—	122	2	no	—
122	C₄⋊C₄	—	122	1	no	—
123	C₄⋊C₄	—	122	1	no	—
124	C₄²	—	61	1	no	—
125	C₄²	—	61	1	no	—
126	C₄²	—	61	1	no	—
127	C₄²	—	61	1	no	—
128	C₂³	C₂²×#6923	1	1	yes	characteristic
129	Q₈	—	244	4	no	—
130	Q₈	—	244	2	no	—
131	Q₈	—	244	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	61	1	no	—
136	C₂×C₄	—	61	1	no	—
137	C₂×C₄	—	61	1	no	—
138	C₂×C₄	—	61	2	no	—
139	C₂×C₄	—	61	2	no	—
140	C₂×C₄	—	61	1	no	—
141	C₂×C₄	—	122	2	no	—
142	C₂×C₄	—	122	2	no	—
143	C₂×C₄	—	122	4	no	—
144	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
145	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
146	C₈	—	122	4	no	—
147	C₈	—	244	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
151	C₂²	D₄×#6923	1	2	yes	—
152	C₄	—	2	1	no	—
153	C₄	—	4	2	no	—
154	C₄	—	61	2	no	—
155	C₄	—	61	1	no	—
156	C₄	—	61	1	no	—
157	C₄	—	61	2	no	—
158	C₄	—	122	1	no	—
159	C₄	—	244	2	no	—
160	C₄	#14519838	1	2	yes	complemented
161	C₂	—	2	2	no	—
162	C₂	#16840714	1	1	yes	characteristic
163	C₂	C₂×#14519838	1	1	yes	characteristic
164	C₂	D₆₁⋊#101	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840765

C₂

yes

maximal, characteristic

C₂×#14519838

C₂

yes

maximal

C₄×#11758217

C₂

yes

maximal, characteristic

C₂×#14519776

C₂

yes

maximal, characteristic

C₂×#11758217

C₂²

yes

—

C₂×#11758217

C₂²

yes

—

—

—

—

—

—

C₂²×Dic₁₂₂

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Dic₆₁

C₄

yes

characteristic

C₂×#11758196

C₄

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₄

yes

—

C₂²×Dic₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

#11758217

—

#11758217

—

#11758217

D₄

yes

—

C₂×#92420

—

C₂×#92420

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

#11758196

—

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

#41

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

#92420

—

#92420

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

C₄×D₄

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#101

yes

complemented

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#101

yes

complemented

Dic₆₁

#121

yes

—

Dic₆₁

yes

Dic₆₁

yes

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄⋊Q₈

—

#423

—

C₂×#101

—

C₄×M₁₆

—

C₆₁

#1966

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#6923

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

122

—

100

C₄⋊C₈

—

122

—

101

#101

—

122

—

102

#97

—

122

—

103

C₄×C₈

—

122

—

104

C₂×Q₈

—

122

—

105

C₂×Q₈

—

122

—

106

C₂×Q₈

—

122

—

107

C₂×Q₈

—

122

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂×#6923

yes

—

112

C₂²×C₄

C₂×#6923

yes

characteristic

113

M₁₆

—

114

M₁₆

—

122

—

115

M₁₆

—

122

—

116

C₂×C₈

—

122

—

117

C₂×C₈

—

122

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

122

—

121

C₄⋊C₄

—

122

—

122

C₄⋊C₄

—

122

—

123

C₄⋊C₄

—

122

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₂³

C₂²×#6923

yes

characteristic

129

Q₈

—

244

—

130

Q₈

—

244

—

131

Q₈

—

244

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

122

—

142

C₂×C₄

—

122

—

143

C₂×C₄

—

122

—

144

C₂×C₄

C₂²⋊#6923

yes

characteristic

145

C₂×C₄

C₂²⋊#6923

yes

characteristic

146

C₈

—

122

—

147

C₈

—

244

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂×C₂²⋊#6923

yes

characteristic

151

C₂²

D₄×#6923

yes

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

122

—

159

C₄

—

244

—

160

C₄

#14519838

yes

complemented

161

C₂

—

162

C₂

#16840714

yes

characteristic

163

C₂

C₂×#14519838

yes

characteristic

164

C₂

D₆₁⋊#101

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6923
commutator	C₂³×C₆₁	C₂²×C₄
Frattini	C₂³	C₂²×#6923
Fitting	C₂×C₄⋊C₄×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	C₂×C₄⋊C₄	#6923
hypocenter	C₆₁	#1966
LCS	G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6923

commutator

C₂³×C₆₁

C₂²×C₄

Frattini

C₂³

C₂²×#6923

Fitting

C₂×C₄⋊C₄×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

C₂×C₄⋊C₄

C₆₁

G ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
71. C₄×C₆₁	#101	semidirect
74. Dic₆₁	#101	semidirect
87. C₆₁	#1966	semidirect
160. C₄	#14519838	semidirect

H ≅ G/N

type

71. C₄×C₆₁

#101

semidirect

74. Dic₆₁

#101

semidirect

87. C₆₁

#1966

semidirect

160. C₄

#14519838

semidirect

Subgroups — #21461962

Conjugacy classes of subgroups

Complements