Group #21446678 — subgroups

Subgroups — #21446678

Order 7808.

312 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂²×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519700	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	#16839667	C₂	1	2	yes	maximal
5	C₂×#107×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₂	C₂²	1	2	yes	—
7	C₂×Dic₂₄₄	—	2	2	no	—
8	C₂×Dic₂₄₄	—	2	2	no	—
9	C₂×Dic₂₄₄	C₄	1	4	yes	complemented
10	C₂²×#92418	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
12	#14519700	—	2	4	no	—
13	C₈×Dic₆₁	—	2	2	no	—
14	#14519684	—	2	4	no	—
15	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
17	#107×C₆₁	—	2	2	no	—
18	C₂²×C₄×C₆₁	—	2	2	no	—
19	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
20	C₂²×Dic₆₁	—	2	2	no	—
21	C₂²×Dic₆₁	D₄	1	2	yes	—
22	C₂×Dic₁₂₂	—	2	2	no	—
23	C₂×Dic₁₂₂	—	2	2	no	—
24	C₂×Dic₁₂₂	—	2	2	no	—
25	C₂×Dic₁₂₂	—	2	2	no	—
26	C₂×Dic₁₂₂	—	2	2	no	—
27	C₂×Dic₁₂₂	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₆₁	D₄	1	2	yes	—
30	C₂×C₈×C₆₁	D₄	1	2	yes	—
31	C₄⋊C₄×C₆₁	—	2	2	no	—
32	C₄⋊C₄×C₆₁	—	2	2	no	—
33	C₄×Dic₆₁	—	2	1	no	—
34	C₄×Dic₆₁	—	2	1	no	—
35	C₂×#92418	—	2	1	no	—
36	C₂×#92418	—	2	1	no	—
37	C₂×#92418	D₄	1	2	yes	—
38	Dic₂₄₄	—	2	8	no	—
39	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	—	4	2	no	—
41	C₂×C₄×C₆₁	—	4	4	no	—
42	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
47	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
48	C₂×Dic₆₁	—	2	2	no	—
49	C₂×Dic₆₁	—	2	2	no	—
50	C₂×Dic₆₁	—	2	4	no	—
51	C₂×Dic₆₁	—	4	2	no	—
52	C₂×Dic₆₁	—	4	2	no	—
53	C₂×Dic₆₁	Q₁₆	1	4	yes	complemented
54	Dic₁₂₂	—	2	4	no	—
55	Dic₁₂₂	—	2	4	no	—
56	Dic₁₂₂	—	4	2	no	—
57	Dic₁₂₂	—	4	2	no	—
58	C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
59	#92418	—	2	2	no	—
60	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₆₁	#121	1	1	yes	characteristic
62	C₂²×C₆₁	#127	1	1	yes	characteristic
63	C₂²×C₆₁	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₆₁	#135	1	2	yes	—
65	C₄×C₆₁	—	8	2	no	—
66	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
68	C₄×C₆₁	#121	1	1	yes	characteristic
69	C₄×C₆₁	#124	1	1	yes	characteristic
70	Dic₆₁	—	2	4	no	—
71	Dic₆₁	—	4	4	no	—
72	#1969	—	61	1	no	maximal, Sylow
73	C₂×C₆₁	#390	1	1	yes	characteristic
74	C₂×C₆₁	C₄×Q₁₆	1	2	yes	—
75	C₂×C₆₁	#467	1	2	yes	—
76	C₂×C₆₁	#494	1	1	yes	characteristic
77	C₂×C₆₁	#495	1	1	yes	characteristic
78	C₂²×Q₁₆	—	61	1	no	—
79	C₂×#107	D₆₁	1	1	yes	characteristic
80	C₂×#103	—	61	2	no	—
81	C₂×C₄×C₈	—	61	1	no	—
82	#391	—	61	2	no	—
83	C₆₁	#1969	1	1	yes	characteristic, complemented, Sylow
84	C₂²×Q₈	—	61	2	no	—
85	C₂×Q₁₆	—	61	4	no	—
86	C₂×Q₁₆	—	122	2	no	—
87	C₂×Q₁₆	—	122	2	no	—
88	C₂²×C₈	—	61	1	no	—
89	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
91	C₂×C₄²	—	61	1	no	—
92	#107	—	2	2	no	—
93	#103	—	122	4	no	—
94	C₄×C₈	—	122	2	no	—
95	#95	—	122	4	no	—
96	C₂×Q₈	—	61	4	no	—
97	C₂×Q₈	—	122	2	no	—
98	C₂×Q₈	—	122	2	no	—
99	C₂×Q₈	—	122	2	no	—
100	C₂×Q₈	—	122	2	no	—
101	C₂×Q₈	—	122	2	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	61	2	no	—
104	C₂²×C₄	—	122	2	no	—
105	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
106	Q₁₆	—	122	8	no	—
107	C₂×C₈	—	61	2	no	—
108	C₂×C₈	—	122	1	no	—
109	C₂×C₈	—	122	1	no	—
110	C₂×C₈	C₄×D₆₁	1	2	yes	—
111	C₂×C₈	D₂₄₄	1	2	yes	—
112	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄²	—	122	1	no	—
116	C₄²	—	122	1	no	—
117	C₂³	D₄×D₆₁	1	1	yes	characteristic
118	Q₈	—	122	4	no	—
119	Q₈	—	122	4	no	—
120	Q₈	—	244	2	no	—
121	Q₈	—	244	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	4	no	—
124	C₂×C₄	—	61	4	no	—
125	C₂×C₄	—	122	2	no	—
126	C₂×C₄	—	122	2	no	—
127	C₂×C₄	—	122	4	no	—
128	C₂×C₄	—	244	2	no	—
129	C₂×C₄	—	244	2	no	—
130	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
131	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
132	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
133	C₂×C₄	#11758226	1	1	yes	characteristic
134	C₂×C₄	#11758226	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
136	C₈	—	122	2	no	—
137	C₈	#11758198	1	4	yes	—
138	C₂²	#14519739	1	1	yes	characteristic
139	C₂²	#14519752	1	1	yes	characteristic
140	C₂²	#14519773	1	1	yes	characteristic
141	C₂²	Q₁₆×D₆₁	1	2	yes	—
142	C₂²	#14519796	1	2	yes	—
143	C₄	—	8	2	no	—
144	C₄	—	122	4	no	—
145	C₄	—	244	4	no	—
146	C₄	C₂×#11758198	1	1	yes	characteristic
147	C₄	#14519739	1	1	yes	characteristic
148	C₄	#14519752	1	1	yes	characteristic
149	C₄	#14519766	1	1	yes	characteristic
150	C₂	#16839663	1	1	yes	characteristic
151	C₂	#16839856	1	2	yes	—
152	C₂	#16840004	1	2	yes	—
153	C₂	#16840140	1	1	yes	characteristic
154	C₂	#16840145	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₄₄

C₂

yes

maximal, characteristic

C₂×#14519700

C₂

yes

maximal

C₂×C₈×Dic₆₁

C₂

yes

maximal, characteristic

#16839667

C₂

yes

maximal

C₂×#107×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

C₄

yes

complemented

C₂²×#92418

C₂²

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519700

—

C₈×Dic₆₁

—

#14519684

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

#107×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

D₄

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

D₄

yes

—

Dic₂₄₄

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

Q₁₆

yes

complemented

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

C₂²⋊C₄

yes

—

#92418

—

C₂²×C₆₁

C₄×D₄

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

C₂×Q₁₆

yes

—

C₂²×C₆₁

#135

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

C₄×D₄

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₄×Q₁₆

yes

—

C₂×C₆₁

#467

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂²×Q₁₆

—

C₂×#107

D₆₁

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

—

#391

—

C₆₁

#1969

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

122

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

#107

—

#103

—

122

—

C₄×C₈

—

122

—

#95

—

122

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

100

C₂×Q₈

—

122

—

101

C₂×Q₈

—

122

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

122

—

105

C₂²×C₄

C₂²×D₆₁

yes

characteristic

106

Q₁₆

—

122

—

107

C₂×C₈

—

108

C₂×C₈

—

122

—

109

C₂×C₈

—

122

—

110

C₂×C₈

C₄×D₆₁

yes

—

111

C₂×C₈

D₂₄₄

yes

—

112

C₂×C₈

C₂²⋊D₆₁

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

122

—

116

C₄²

—

122

—

117

C₂³

D₄×D₆₁

yes

characteristic

118

Q₈

—

122

—

119

Q₈

—

122

—

120

Q₈

—

244

—

121

Q₈

—

244

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

122

—

126

C₂×C₄

—

122

—

127

C₂×C₄

—

122

—

128

C₂×C₄

—

244

—

129

C₂×C₄

—

244

—

130

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

131

C₂×C₄

C₂×D₂₄₄

yes

characteristic

132

C₂×C₄

D₄×D₆₁

yes

characteristic

133

C₂×C₄

#11758226

yes

characteristic

134

C₂×C₄

#11758226

yes

characteristic

135

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

136

C₈

—

122

—

137

C₈

#11758198

yes

—

138

C₂²

#14519739

yes

characteristic

139

C₂²

#14519752

yes

characteristic

140

C₂²

#14519773

yes

characteristic

141

C₂²

Q₁₆×D₆₁

yes

—

142

C₂²

#14519796

yes

—

143

C₄

—

144

C₄

—

122

—

145

C₄

—

244

—

146

C₄

C₂×#11758198

yes

characteristic

147

C₄

#14519739

yes

characteristic

148

C₄

#14519752

yes

characteristic

149

C₄

#14519766

yes

characteristic

150

C₂

#16839663

yes

characteristic

151

C₂

#16839856

yes

—

152

C₂

#16840004

yes

—

153

C₂

#16840140

yes

characteristic

154

C₂

#16840145

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×#107×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×#107	D₆₁
hypocenter	C₆₁	#1969
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×#107×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
9. C₂×Dic₂₄₄	C₄	semidirect
53. C₂×Dic₆₁	Q₁₆	semidirect
83. C₆₁	#1969	semidirect

H ≅ G/N

type

9. C₂×Dic₂₄₄

C₄

semidirect

53. C₂×Dic₆₁

Q₁₆

semidirect

83. C₆₁

#1969

semidirect

Subgroups — #21446678

Conjugacy classes of subgroups

Complements