Group #21461123 — subgroups

Subgroups — #21461123

Order 7808.

248 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 84 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	#16840778	C₂	1	1	yes	maximal, characteristic
2	#16840778	C₂	1	1	yes	maximal, characteristic
3	#16840612	C₂	1	4	yes	maximal, complemented
4	C₂×#14519742	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#92420	—	2	1	no	—
6	C₄⋊#92420	C₂²	1	2	yes	—
7	C₂²⋊#92420	—	2	1	no	—
8	C₂²⋊#92420	—	2	1	no	—
9	C₄×#92420	—	2	1	no	—
10	C₄×#92420	C₂²	1	2	yes	—
11	C₂²×Dic₁₂₂	C₄	1	1	yes	characteristic
12	#14519742	—	2	1	no	—
13	#14519742	—	2	1	no	—
14	#14519742	C₄	1	2	yes	—
15	#14519742	C₂²	1	2	yes	—
16	C₂×#11758196	—	2	1	no	—
17	C₂×C₄×Dic₆₁	—	2	1	no	—
18	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
20	C₂²×C₄×C₆₁	—	2	1	no	—
21	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₆₁	—	2	1	no	—
23	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
24	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
25	C₂×Dic₁₂₂	—	2	1	no	—
26	C₂×Dic₁₂₂	—	2	2	no	—
27	C₂×Dic₁₂₂	C₈	1	4	yes	complemented
28	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
29	C₂×#92420	—	2	2	no	—
30	C₂×#92420	—	2	2	no	—
31	C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
33	#11758196	—	2	2	no	—
34	#11758196	—	2	2	no	—
35	C₄×Dic₆₁	—	2	2	no	—
36	C₄×Dic₆₁	—	2	2	no	—
37	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₆₁	D₄	1	1	yes	characteristic
39	C₄×Dic₆₁	D₄	1	1	yes	characteristic
40	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
41	C₂³×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₆₁	—	2	2	no	—
43	C₂×C₄×C₆₁	—	4	2	no	—
44	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂×C₈	1	2	yes	—
46	C₂×C₄×C₆₁	M₁₆	1	2	yes	—
47	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₆₁	—	2	4	no	—
49	C₂×Dic₆₁	—	2	2	no	—
50	C₂×Dic₆₁	—	2	2	no	—
51	C₂×Dic₆₁	—	4	1	no	—
52	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
56	Dic₁₂₂	—	2	4	no	—
57	Dic₁₂₂	—	2	2	no	—
58	Dic₁₂₂	C₂×C₈	1	4	yes	—
59	#92420	—	4	2	no	—
60	#92420	—	4	2	no	—
61	C₂²×C₆₁	—	2	1	no	—
62	C₂²×C₆₁	—	2	1	no	—
63	C₂²×C₆₁	C₂²⋊C₈	1	2	yes	—
64	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₆₁	—	4	2	no	—
66	C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
67	C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
68	C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
69	Dic₆₁	—	2	4	no	—
70	Dic₆₁	—	2	1	no	—
71	Dic₆₁	—	2	1	no	—
72	Dic₆₁	—	2	1	no	—
73	Dic₆₁	#137	1	2	yes	—
74	#1479	—	61	1	no	maximal, Sylow
75	C₂×C₆₁	—	2	2	no	—
76	C₂×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	C₂×C₆₁	#419	1	1	yes	characteristic
78	C₂×C₆₁	#421	1	1	yes	characteristic
79	C₂×C₄×Q₈	—	61	1	no	—
80	#431	—	61	1	no	—
81	#431	—	61	1	no	—
82	Q₈⋊C₈	—	61	4	no	—
83	C₆₁	#1479	1	1	yes	characteristic, complemented, Sylow
84	C₂²×Q₈	—	61	1	no	—
85	C₄×Q₈	—	61	2	no	—
86	C₄×Q₈	—	61	2	no	—
87	C₄×Q₈	—	122	1	no	—
88	C₄×Q₈	—	122	1	no	—
89	C₂×C₄⋊C₄	—	122	1	no	—
90	C₂×C₄⋊C₄	#6923	1	1	yes	characteristic
91	C₂×C₄²	—	61	1	no	—
92	C₂×C₄²	—	122	1	no	—
93	C₄⋊C₈	—	61	2	no	—
94	C₄⋊C₈	—	122	1	no	—
95	C₂²⋊C₈	—	122	1	no	—
96	C₂²⋊C₈	—	122	1	no	—
97	C₄×C₈	—	61	2	no	—
98	C₄×C₈	—	122	1	no	—
99	C₂×Q₈	—	61	4	no	—
100	C₂×Q₈	—	61	2	no	—
101	C₂×Q₈	—	122	2	no	—
102	C₂×Q₈	—	122	1	no	—
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	—	61	1	no	—
105	C₂²×C₄	—	61	1	no	—
106	C₂²×C₄	—	122	1	no	—
107	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
108	C₂×C₈	—	122	2	no	—
109	C₂×C₈	—	122	2	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	122	2	no	—
112	C₄⋊C₄	—	122	2	no	—
113	C₄⋊C₄	C₂×#6923	1	2	yes	—
114	C₄²	—	61	1	no	—
115	C₄²	—	61	1	no	—
116	C₄²	—	61	1	no	—
117	C₄²	—	61	1	no	—
118	C₄²	—	122	2	no	—
119	C₄²	—	122	2	no	—
120	C₂³	C₂²⋊#6923	1	1	yes	characteristic
121	Q₈	—	61	4	no	—
122	Q₈	—	122	4	no	—
123	Q₈	—	122	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	61	1	no	—
127	C₂×C₄	—	61	1	no	—
128	C₂×C₄	—	61	1	no	—
129	C₂×C₄	—	61	1	no	—
130	C₂×C₄	—	122	2	no	—
131	C₂×C₄	—	122	4	no	—
132	C₂×C₄	—	122	2	no	—
133	C₂×C₄	—	244	1	no	—
134	C₂×C₄	D₆₁⋊C₈	1	2	yes	—
135	C₂×C₄	#11758213	1	2	yes	—
136	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
137	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
138	C₈	—	244	2	no	—
139	C₈	—	244	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	#14519834	1	2	yes	—
143	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
144	C₄	—	4	2	no	—
145	C₄	—	61	2	no	—
146	C₄	—	122	4	no	—
147	C₄	—	122	1	no	—
148	C₄	—	122	1	no	—
149	C₄	—	122	1	no	—
150	C₄	#14519834	1	2	yes	—
151	C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
152	C₄	C₂×#11758213	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	C₂×#14519834	1	1	yes	characteristic
155	C₂	#16840755	1	1	yes	characteristic
156	C₂	#16840760	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840778

C₂

yes

maximal, characteristic

#16840778

C₂

yes

maximal, characteristic

#16840612

C₂

yes

maximal, complemented

C₂×#14519742

C₂

yes

maximal, characteristic

C₄⋊#92420

—

C₄⋊#92420

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂²×Dic₁₂₂

C₄

yes

—

—

C₄

yes

—

#14519742

C₂²

yes

—

C₂×#11758196

—

C₂×C₄×Dic₆₁

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

D₄

yes

characteristic

C₂²×Dic₆₁

D₄

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₈

yes

complemented

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×#92420

—

C₂×#92420

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

C₂³

yes

characteristic

C₂³×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₈

yes

—

C₂×C₄×C₆₁

M₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

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

characteristic

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

C₂×C₈

yes

—

#92420

—

#92420

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂²⋊C₈

yes

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₂²⋊C₈

yes

—

C₄×C₆₁

C₂²×C₈

yes

characteristic

C₄×C₆₁

C₂×M₁₆

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#137

yes

—

#1479

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×C₂²⋊C₈

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄×Q₈

—

#431

—

#431

—

Q₈⋊C₈

—

C₆₁

#1479

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

122

—

C₄×Q₈

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

#6923

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

122

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

122

—

C₂²⋊C₈

—

122

—

C₄×C₈

—

C₄×C₈

—

122

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

122

—

102

C₂×Q₈

—

122

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

122

—

107

C₂²×C₄

C₂×#6923

yes

characteristic

108

C₂×C₈

—

122

—

109

C₂×C₈

—

122

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

122

—

112

C₄⋊C₄

—

122

—

113

C₄⋊C₄

C₂×#6923

yes

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

122

—

119

C₄²

—

122

—

120

C₂³

C₂²⋊#6923

yes

characteristic

121

Q₈

—

122

Q₈

—

122

—

123

Q₈

—

122

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

122

—

131

C₂×C₄

—

122

—

132

C₂×C₄

—

122

—

133

C₂×C₄

—

244

—

134

C₂×C₄

D₆₁⋊C₈

yes

—

135

C₂×C₄

#11758213

yes

—

136

C₂×C₄

C₂²⋊#6923

yes

characteristic

137

C₂×C₄

C₂²×#6923

yes

characteristic

138

C₈

—

244

—

139

C₈

—

244

—

140

C₂²

—

141

C₂²

—

142

C₂²

#14519834

yes

—

143

C₂²

C₂×C₂²⋊#6923

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

122

—

147

C₄

—

122

—

148

C₄

—

122

—

149

C₄

—

122

—

150

C₄

#14519834

yes

—

151

C₄

C₂×D₆₁⋊C₈

yes

characteristic

152

C₄

C₂×#11758213

yes

characteristic

153

C₂

—

154

C₂

C₂×#14519834

yes

characteristic

155

C₂

#16840755

yes

characteristic

156

C₂

#16840760

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6923
commutator	C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₂×C₄⋊C₄×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	C₂×C₄⋊C₄	#6923
hypocenter	C₆₁	#1479
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6923

commutator

C₄×C₆₁

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6923

Fitting

C₂×C₄⋊C₄×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

C₂×C₄⋊C₄

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
3. #16840612	C₂	semidirect
27. C₂×Dic₁₂₂	C₈	semidirect
83. C₆₁	#1479	semidirect

H ≅ G/N

type

3. #16840612

C₂

semidirect

27. C₂×Dic₁₂₂

C₈

semidirect

83. C₆₁

#1479

semidirect

Subgroups — #21461123

Conjugacy classes of subgroups

Complements