Group #21461605 — subgroups

Subgroups — #21461605

Order 7808.

292 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 130 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	2	yes	maximal
2	C₂×#14519832	C₂	1	2	yes	maximal
3	#16840628	C₂	1	2	yes	maximal
4	C₄²×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#92420	C₂²	1	4	yes	—
6	C₄⋊#92420	C₄	1	8	yes	complemented
7	#14519832	—	2	4	no	—
8	C₂×C₄×Dic₆₁	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₆₁	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₆₁	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₆₁	C₄	1	1	yes	characteristic
15	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
22	C₂×#92420	—	2	4	no	—
23	C₂×#92420	—	2	4	no	—
24	C₂×#92420	C₂×C₄	1	8	yes	—
25	C₄²×C₆₁	—	2	2	no	—
26	C₄×Dic₆₁	—	2	2	no	—
27	C₄×Dic₆₁	—	2	2	no	—
28	C₄×Dic₆₁	—	2	2	no	—
29	C₄×Dic₆₁	—	2	2	no	—
30	C₄×Dic₆₁	C₂×C₄	1	4	yes	—
31	C₄×Dic₆₁	D₄	1	2	yes	—
32	C₄×Dic₆₁	Q₈	1	2	yes	—
33	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	—	2	1	no	—
35	C₂×C₄×C₆₁	—	2	1	no	—
36	C₂×C₄×C₆₁	—	2	1	no	—
37	C₂×C₄×C₆₁	—	2	1	no	—
38	C₂×C₄×C₆₁	C₄²	1	2	yes	—
39	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₆₁	M₁₆	1	2	yes	—
41	C₂×C₄×C₆₁	M₁₆	1	2	yes	—
42	C₂×C₄×C₆₁	M₁₆	1	2	yes	—
43	C₂×Dic₆₁	—	2	1	no	—
44	C₂×Dic₆₁	—	2	1	no	—
45	C₂×Dic₆₁	—	2	1	no	—
46	C₂×Dic₆₁	—	2	1	no	—
47	C₂×Dic₆₁	—	2	1	no	—
48	C₂×Dic₆₁	—	2	1	no	—
49	C₂×Dic₆₁	C₄²	1	2	yes	—
50	C₂×Dic₆₁	C₄⋊C₄	1	2	yes	—
51	C₂×Dic₆₁	M₁₆	1	2	yes	—
52	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
53	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
55	C₂×Dic₆₁	#41	1	2	yes	—
56	#92420	—	2	8	no	—
57	C₂²×C₆₁	C₂×C₄²	1	1	yes	characteristic
58	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₆₁	#117	1	1	yes	characteristic
60	C₂²×C₆₁	C₂×M₁₆	1	1	yes	characteristic
61	C₂²×C₆₁	C₂×M₁₆	1	1	yes	characteristic
62	C₂²×C₆₁	C₂×M₁₆	1	1	yes	characteristic
63	C₂²×C₆₁	C₂×M₁₆	1	1	yes	characteristic
64	C₄×C₆₁	—	2	2	no	—
65	C₄×C₆₁	—	2	2	no	—
66	C₄×C₆₁	#97	1	4	yes	complemented
67	Dic₆₁	—	2	2	no	—
68	Dic₆₁	—	2	2	no	—
69	Dic₆₁	—	2	2	no	—
70	Dic₆₁	C₄×D₄	1	2	yes	—
71	Dic₆₁	C₄×Q₈	1	2	yes	—
72	#1767	—	61	1	no	maximal, Sylow
73	C₂×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₆₁	C₂×#97	1	1	yes	characteristic
75	C₂×C₆₁	C₄×M₁₆	1	1	yes	characteristic
76	C₂×C₆₁	#423	1	1	yes	characteristic
77	C₂×C₆₁	#423	1	1	yes	characteristic
78	C₂×C₆₁	#432	1	1	yes	characteristic
79	C₂×C₆₁	#432	1	1	yes	characteristic
80	C₂×C₄⋊C₈	—	61	2	no	—
81	C₂×#97	—	61	2	no	—
82	C₄³	—	61	1	no	—
83	#336	—	61	2	no	—
84	C₆₁	#1767	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	—	61	4	no	—
86	C₂×C₄²	—	61	1	no	—
87	C₂×C₄²	—	61	1	no	—
88	C₂×C₄²	—	61	1	no	—
89	C₂×C₄²	—	61	1	no	—
90	C₂×C₄²	—	61	1	no	—
91	C₂×C₄²	—	61	1	no	—
92	C₂×C₄²	#6923	1	1	yes	characteristic
93	C₄⋊C₈	—	61	8	no	—
94	#97	—	122	4	no	—
95	C₂²×C₄	—	61	1	no	—
96	C₂²×C₄	—	61	1	no	—
97	C₂²×C₄	—	61	1	no	—
98	C₂²×C₄	—	61	1	no	—
99	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
100	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
101	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
102	C₂×C₈	—	61	8	no	—
103	C₂×C₈	—	122	4	no	—
104	C₂×C₈	—	122	4	no	—
105	C₄²	—	2	2	no	—
106	C₄²	—	61	4	no	—
107	C₄²	—	61	2	no	—
108	C₄²	—	61	2	no	—
109	C₄²	—	122	2	no	—
110	C₄²	—	122	2	no	—
111	C₄²	—	122	2	no	—
112	C₄²	—	122	2	no	—
113	C₂³	C₂²×#6923	1	1	yes	characteristic
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	61	2	no	—
119	C₂×C₄	—	61	1	no	—
120	C₂×C₄	—	61	1	no	—
121	C₂×C₄	—	61	2	no	—
122	C₂×C₄	—	61	2	no	—
123	C₂×C₄	—	61	2	no	—
124	C₂×C₄	—	61	2	no	—
125	C₂×C₄	—	122	1	no	—
126	C₂×C₄	—	122	1	no	—
127	C₂×C₄	—	122	1	no	—
128	C₂×C₄	—	122	1	no	—
129	C₂×C₄	—	122	1	no	—
130	C₂×C₄	—	122	1	no	—
131	C₂×C₄	#11758213	1	2	yes	—
132	C₂×C₄	#11758213	1	2	yes	—
133	C₂×C₄	C₄×#6923	1	2	yes	—
134	C₂×C₄	C₄⋊#6923	1	2	yes	—
135	C₂×C₄	#11758217	1	2	yes	—
136	C₈	—	122	8	no	—
137	C₂²	C₂×C₄×#6923	1	1	yes	characteristic
138	C₂²	C₂×C₄⋊#6923	1	1	yes	characteristic
139	C₂²	#14519856	1	1	yes	characteristic
140	C₂²	C₂×#11758217	1	1	yes	characteristic
141	C₂²	D₆₁⋊M₁₆	1	1	yes	characteristic
142	C₂²	C₂×#11758213	1	1	yes	characteristic
143	C₂²	C₂×#11758213	1	1	yes	characteristic
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	61	2	no	—
147	C₄	—	61	2	no	—
148	C₄	—	122	2	no	—
149	C₄	—	122	2	no	—
150	C₄	—	122	2	no	—
151	C₄	#14519832	1	4	yes	complemented
152	C₂	C₄×C₄⋊#6923	1	1	yes	characteristic
153	C₂	C₂×#14519832	1	1	yes	characteristic
154	C₂	C₄×#11758213	1	1	yes	characteristic
155	C₂	#16840765	1	1	yes	characteristic
156	C₂	#16840766	1	1	yes	characteristic
157	C₂	#16840783	1	1	yes	characteristic
158	C₂	#16840784	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#92420

C₂

yes

maximal

C₂×#14519832

C₂

yes

maximal

#16840628

C₂

yes

maximal

C₄²×Dic₆₁

C₂

yes

maximal, characteristic

C₂²×#92420

C₂²

yes

—

C₄⋊#92420

C₄

yes

complemented

#14519832

—

C₂×C₄×Dic₆₁

C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

C₂×C₄²×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

—

—

C₂×C₄

yes

—

C₄²×C₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

Q₈

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄²

yes

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

M₁₆

yes

—

C₂×C₄×C₆₁

M₁₆

yes

—

C₂×C₄×C₆₁

M₁₆

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₄²

yes

—

C₂×Dic₆₁

C₄⋊C₄

yes

—

C₂×Dic₆₁

M₁₆

yes

—

C₂×Dic₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×Q₈

yes

characteristic

C₂×Dic₆₁

#41

yes

—

#92420

—

C₂²×C₆₁

C₂×C₄²

yes

characteristic

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₁

#117

yes

characteristic

C₂²×C₆₁

C₂×M₁₆

yes

characteristic

C₂²×C₆₁

C₂×M₁₆

yes

characteristic

C₂²×C₆₁

C₂×M₁₆

yes

characteristic

C₂²×C₆₁

C₂×M₁₆

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#97

yes

complemented

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₄×D₄

yes

—

Dic₆₁

C₄×Q₈

yes

—

#1767

—

maximal, Sylow

C₂×C₆₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₆₁

C₂×#97

yes

characteristic

C₂×C₆₁

C₄×M₁₆

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄⋊C₈

—

C₂×#97

—

C₄³

—

#336

—

C₆₁

#1767

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

#6923

yes

characteristic

C₄⋊C₈

—

#97

—

122

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6923

yes

characteristic

100

C₂²×C₄

C₂×#6923

yes

characteristic

101

C₂²×C₄

C₂×#6923

yes

characteristic

102

C₂×C₈

—

103

C₂×C₈

—

122

—

104

C₂×C₈

—

122

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

122

—

110

C₄²

—

122

—

111

C₄²

—

122

—

112

C₄²

—

122

—

113

C₂³

C₂²×#6923

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

122

—

126

C₂×C₄

—

122

—

127

C₂×C₄

—

122

—

128

C₂×C₄

—

122

—

129

C₂×C₄

—

122

—

130

C₂×C₄

—

122

—

131

C₂×C₄

#11758213

yes

—

132

C₂×C₄

#11758213

yes

—

133

C₂×C₄

C₄×#6923

yes

—

134

C₂×C₄

C₄⋊#6923

yes

—

135

C₂×C₄

#11758217

yes

—

136

C₈

—

122

—

137

C₂²

C₂×C₄×#6923

yes

characteristic

138

C₂²

C₂×C₄⋊#6923

yes

characteristic

139

C₂²

#14519856

yes

characteristic

140

C₂²

C₂×#11758217

yes

characteristic

141

C₂²

D₆₁⋊M₁₆

yes

characteristic

142

C₂²

C₂×#11758213

yes

characteristic

143

C₂²

C₂×#11758213

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

122

—

149

C₄

—

122

—

150

C₄

—

122

—

151

C₄

#14519832

yes

complemented

152

C₂

C₄×C₄⋊#6923

yes

characteristic

153

C₂

C₂×#14519832

yes

characteristic

154

C₂

C₄×#11758213

yes

characteristic

155

C₂

#16840765

yes

characteristic

156

C₂

#16840766

yes

characteristic

157

C₂

#16840783

yes

characteristic

158

C₂

#16840784

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#6923
commutator	C₂²×C₆₁	C₂×C₄²
Fitting	C₂×C₄²×C₆₁	C₄
socle	C₂³×C₆₁	C₂²×C₄
hypercenter	C₂×C₄²	#6923
hypocenter	C₆₁	#1767
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×#6923

commutator

C₂²×C₆₁

C₂×C₄²

Fitting

C₂×C₄²×C₆₁

C₄

socle

C₂³×C₆₁

C₂²×C₄

C₂×C₄²

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
6. C₄⋊#92420	C₄	semidirect
66. C₄×C₆₁	#97	semidirect
84. C₆₁	#1767	semidirect
151. C₄	#14519832	semidirect

H ≅ G/N

type

6. C₄⋊#92420

C₄

semidirect

66. C₄×C₆₁

#97

semidirect

84. C₆₁

#1767

semidirect

151. C₄

#14519832

semidirect

Subgroups — #21461605

Conjugacy classes of subgroups

Complements