Group #21461992 — subgroups

Subgroups — #21461992

Order 7808.

280 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 92 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×C₄⋊#92420	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519832	C₂	1	1	yes	maximal, characteristic
3	#16840628	C₂	1	2	yes	maximal
4	#16840628	C₂	1	2	yes	maximal
5	C₂×#14519742	C₂	1	1	yes	maximal, characteristic
6	C₂²×#92420	C₂²	1	2	yes	—
7	C₂²×#92420	C₂²	1	2	yes	—
8	C₄⋊#92420	—	2	2	no	—
9	#14519832	—	2	2	no	—
10	C₂²×Dic₁₂₂	C₄	1	1	yes	characteristic
11	#14519742	—	2	2	no	—
12	#14519742	—	2	2	no	—
13	C₂×#11758196	C₄	1	1	yes	characteristic
14	C₂×#11758196	C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₂₂	—	2	2	no	—
27	C₂×Dic₁₂₂	—	2	2	no	—
28	C₂×Dic₁₂₂	—	2	2	no	—
29	C₂×#92420	—	2	2	no	—
30	C₂×#92420	—	2	2	no	—
31	C₂×#92420	—	2	2	no	—
32	C₂×#92420	—	2	4	no	—
33	C₂×#92420	Q₈	1	4	yes	complemented
34	C₄⋊C₄×C₆₁	—	2	2	no	—
35	#11758196	—	2	2	no	—
36	#11758196	—	2	2	no	—
37	C₄×Dic₆₁	—	2	1	no	—
38	C₄×Dic₆₁	—	2	1	no	—
39	C₄×Dic₆₁	—	2	1	no	—
40	C₄×Dic₆₁	—	2	1	no	—
41	C₄×Dic₆₁	D₄	1	2	yes	—
42	C₄×Dic₆₁	D₄	1	2	yes	—
43	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	—	2	1	no	—
45	C₂×C₄×C₆₁	—	2	1	no	—
46	C₂×C₄×C₆₁	—	2	2	no	—
47	C₂×C₄×C₆₁	—	2	1	no	—
48	C₂×C₄×C₆₁	—	2	2	no	—
49	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₆₁	M₁₆	1	2	yes	—
51	C₂×Dic₆₁	—	2	1	no	—
52	C₂×Dic₆₁	—	2	1	no	—
53	C₂×Dic₆₁	—	2	1	no	—
54	C₂×Dic₆₁	—	2	1	no	—
55	C₂×Dic₆₁	—	2	1	no	—
56	C₂×Dic₆₁	—	2	2	no	—
57	C₂×Dic₆₁	C₂²⋊C₄	1	2	yes	—
58	C₂×Dic₆₁	M₁₆	1	2	yes	—
59	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₆₁	C₂×Q₈	1	2	yes	—
62	C₂×Dic₆₁	#41	1	2	yes	—
63	Dic₁₂₂	—	2	8	no	—
64	#92420	—	2	4	no	—
65	#92420	—	4	2	no	—
66	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₆₁	C₄×Q₈	1	2	yes	—
68	C₂²×C₆₁	C₂×M₁₆	1	1	yes	characteristic
69	C₂²×C₆₁	C₂×M₁₆	1	1	yes	characteristic
70	C₂²×C₆₁	#131	1	2	yes	—
71	C₄×C₆₁	—	2	2	no	—
72	C₄×C₆₁	—	2	2	no	—
73	C₄×C₆₁	—	4	1	no	—
74	Dic₆₁	—	2	2	no	—
75	Dic₆₁	—	2	2	no	—
76	Dic₆₁	—	4	1	no	—
77	Dic₆₁	C₂²⋊Q₈	1	2	yes	—
78	Dic₆₁	#124	1	1	yes	characteristic
79	Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
80	#1985	—	61	1	no	maximal, Sylow
81	C₂×C₆₁	#391	1	1	yes	characteristic
82	C₂×C₆₁	#407	1	1	yes	characteristic
83	C₂×C₆₁	#408	1	1	yes	characteristic
84	C₂×C₆₁	#446	1	2	yes	—
85	C₂×C₆₁	#446	1	2	yes	—
86	C₂×C₄×Q₈	—	61	1	no	—
87	C₂×C₄⋊C₈	—	61	1	no	—
88	C₂×#97	—	61	1	no	—
89	#336	—	61	2	no	—
90	#336	—	61	2	no	—
91	C₆₁	#1985	1	1	yes	characteristic, complemented, Sylow
92	C₂²×Q₈	—	61	1	no	—
93	C₂²×C₈	—	61	2	no	—
94	C₂²×C₈	—	61	2	no	—
95	C₄×Q₈	—	122	2	no	—
96	C₄×Q₈	—	122	2	no	—
97	C₂×C₄⋊C₄	—	61	1	no	—
98	C₂×C₄⋊C₄	—	61	1	no	—
99	C₂×C₄⋊C₄	#6923	1	1	yes	characteristic
100	C₂×C₄²	—	61	1	no	—
101	C₂×C₄²	—	61	1	no	—
102	C₂×C₄²	—	61	1	no	—
103	C₄⋊C₈	—	122	2	no	—
104	#97	—	122	2	no	—
105	C₂×Q₈	—	122	2	no	—
106	C₂×Q₈	—	122	2	no	—
107	C₂×Q₈	—	122	2	no	—
108	C₂²×C₄	—	61	1	no	—
109	C₂²×C₄	—	61	1	no	—
110	C₂²×C₄	—	61	1	no	—
111	C₂²×C₄	—	61	1	no	—
112	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
113	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
114	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
115	C₂×C₈	—	61	4	no	—
116	C₂×C₈	—	122	2	no	—
117	C₂×C₈	—	122	2	no	—
118	C₂×C₈	—	122	2	no	—
119	C₂×C₈	—	122	4	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	122	2	no	—
122	C₄⋊C₄	—	122	2	no	—
123	C₄²	—	61	2	no	—
124	C₄²	—	61	2	no	—
125	C₄²	—	122	1	no	—
126	C₄²	—	122	1	no	—
127	C₄²	—	122	1	no	—
128	C₄²	—	122	1	no	—
129	C₂³	C₂²×#6923	1	1	yes	characteristic
130	Q₈	—	122	8	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	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	2	no	—
141	C₂×C₄	—	61	2	no	—
142	C₂×C₄	—	122	1	no	—
143	C₂×C₄	—	122	1	no	—
144	C₂×C₄	—	122	1	no	—
145	C₂×C₄	—	122	1	no	—
146	C₂×C₄	—	122	1	no	—
147	C₂×C₄	—	122	2	no	—
148	C₂×C₄	#11758213	1	2	yes	—
149	C₂×C₄	C₂²⋊#6923	1	2	yes	—
150	C₈	—	122	4	no	—
151	C₈	—	244	2	no	—
152	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
153	C₂²	Q₈×#6923	1	2	yes	—
154	C₂²	D₆₁⋊M₁₆	1	1	yes	characteristic
155	C₂²	C₂×#11758213	1	1	yes	characteristic
156	C₂²	#14519864	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	1	no	—
160	C₄	—	61	2	no	—
161	C₄	—	61	1	no	—
162	C₄	—	61	1	no	—
163	C₄	—	122	2	no	—
164	C₄	—	122	2	no	—
165	C₄	—	244	1	no	—
166	C₂	(C₂×#6923)⋊Q₈	1	1	yes	characteristic
167	C₂	#16840733	1	1	yes	characteristic
168	C₂	#16840735	1	1	yes	characteristic
169	C₂	#16840812	1	2	yes	—
170	C₂	#16840814	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#92420

C₂

yes

maximal, characteristic

C₂×#14519832

C₂

yes

maximal, characteristic

#16840628

C₂

yes

maximal

#16840628

C₂

yes

maximal

C₂×#14519742

C₂

yes

maximal, characteristic

C₂²×#92420

C₂²

yes

—

C₂²×#92420

C₂²

yes

—

C₄⋊#92420

—

#14519832

—

C₂²×Dic₁₂₂

C₄

yes

—

—

C₄

yes

characteristic

C₂×#11758196

C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

—

—

—

—

Q₈

yes

complemented

C₄⋊C₄×C₆₁

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

D₄

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

M₁₆

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

—

C₂×Dic₆₁

M₁₆

yes

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×Q₈

yes

—

C₂×Dic₆₁

#41

yes

—

Dic₁₂₂

—

#92420

—

#92420

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

C₄×Q₈

yes

—

C₂²×C₆₁

C₂×M₁₆

yes

characteristic

C₂²×C₆₁

C₂×M₁₆

yes

characteristic

C₂²×C₆₁

#131

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₂²⋊Q₈

yes

—

Dic₆₁

#124

yes

characteristic

Dic₆₁

C₄⋊Q₈

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#446

yes

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

#336

—

#336

—

C₆₁

#1985

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

122

—

C₄×Q₈

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#6923

yes

characteristic

100

C₂×C₄²

—

101

C₂×C₄²

—

102

C₂×C₄²

—

103

C₄⋊C₈

—

122

—

104

#97

—

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

—

112

C₂²×C₄

C₂×#6923

yes

characteristic

113

C₂²×C₄

C₂×#6923

yes

characteristic

114

C₂²×C₄

C₂×#6923

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

122

—

117

C₂×C₈

—

122

—

118

C₂×C₈

—

122

—

119

C₂×C₈

—

122

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

—

122

C₄⋊C₄

—

122

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

122

—

126

C₄²

—

122

—

127

C₄²

—

122

—

128

C₄²

—

122

—

129

C₂³

C₂²×#6923

yes

characteristic

130

Q₈

—

122

—

131

C₂×C₄

—

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

—

142

C₂×C₄

—

122

—

143

C₂×C₄

—

122

—

144

C₂×C₄

—

122

—

145

C₂×C₄

—

122

—

146

C₂×C₄

—

122

—

147

C₂×C₄

—

122

—

148

C₂×C₄

#11758213

yes

—

149

C₂×C₄

C₂²⋊#6923

yes

—

150

C₈

—

122

—

151

C₈

—

244

—

152

C₂²

C₂×C₂²⋊#6923

yes

characteristic

153

C₂²

Q₈×#6923

yes

—

154

C₂²

D₆₁⋊M₁₆

yes

characteristic

155

C₂²

C₂×#11758213

yes

characteristic

156

C₂²

#14519864

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

122

—

164

C₄

—

122

—

165

C₄

—

244

—

166

C₂

(C₂×#6923)⋊Q₈

yes

characteristic

167

C₂

#16840733

yes

characteristic

168

C₂

#16840735

yes

characteristic

169

C₂

#16840812

yes

—

170

C₂

#16840814

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#6923
commutator, socle	C₂³×C₆₁	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₆₁	C₄
hypercenter	C₂×C₄⋊C₄	#6923
hypocenter	C₆₁	#1985
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×#6923

commutator, socle

C₂³×C₆₁

C₂²×C₄

Fitting

C₂×C₄⋊C₄×C₆₁

C₄

C₂×C₄⋊C₄

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
33. C₂×#92420	Q₈	semidirect
91. C₆₁	#1985	semidirect

H ≅ G/N

type

33. C₂×#92420

Q₈

semidirect

91. C₆₁

#1985

semidirect

Subgroups — #21461992

Conjugacy classes of subgroups

Complements