Group #21446213 — subgroups

Subgroups — #21446213

Order 7808.

324 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16840240	C₂	1	1	yes	maximal, characteristic
2	#16839771	C₂	1	2	yes	maximal
3	C₂×#14519696	C₂	1	1	yes	maximal, characteristic
4	#16839522	C₂	1	2	yes	maximal
5	C₂×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758194	C₂²	1	2	yes	—
7	Q₈×Dic₆₁	—	2	4	no	—
8	#14519736	—	2	2	no	—
9	#14519736	C₂²	1	2	yes	—
10	#14519701	—	2	4	no	—
11	#14519696	—	2	2	no	—
12	C₂²×Q₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₆₁	—	2	2	no	—
15	C₄⋊Q₈×C₆₁	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
17	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₆₁	—	2	2	no	—
19	C₂×Q₈×C₆₁	—	2	1	no	—
20	C₂×Q₈×C₆₁	—	2	2	no	—
21	C₂×Q₈×C₆₁	—	2	2	no	—
22	C₂×Q₈×C₆₁	—	2	2	no	—
23	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
24	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₂×Q₈×C₆₁	D₄	1	2	yes	—
26	C₂²×C₄×C₆₁	—	2	1	no	—
27	C₂²×C₄×C₆₁	—	2	1	no	—
28	C₂²×C₄×C₆₁	D₄	1	2	yes	—
29	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₆₁	—	2	8	no	—
31	C₄²×C₆₁	—	2	1	no	—
32	C₄²×C₆₁	C₂×C₄	1	2	yes	—
33	C₂²⋊Dic₆₁	—	2	2	no	—
34	C₄⋊Dic₆₁	—	2	4	no	—
35	C₄⋊Dic₆₁	D₄	1	4	yes	—
36	C₄×Dic₆₁	—	2	4	no	—
37	C₄×Dic₆₁	—	2	2	no	—
38	#11758194	—	2	2	no	—
39	#11758194	—	2	2	no	—
40	C₂×#92418	—	2	2	no	—
41	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₆₁	—	2	4	no	—
43	Q₈×C₆₁	—	2	4	no	—
44	Q₈×C₆₁	—	2	4	no	—
45	Q₈×C₆₁	—	4	2	no	—
46	C₂×C₄×C₆₁	—	2	4	no	—
47	C₂×C₄×C₆₁	—	2	2	no	—
48	C₂×C₄×C₆₁	—	2	2	no	—
49	C₂×C₄×C₆₁	—	2	2	no	—
50	C₂×C₄×C₆₁	—	2	2	no	—
51	C₂×C₄×C₆₁	—	4	1	no	—
52	C₂×C₄×C₆₁	—	4	1	no	—
53	C₂×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
54	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#41	1	2	yes	—
58	C₂×Dic₆₁	—	2	4	no	—
59	#92418	—	4	2	no	—
60	C₂²×C₆₁	—	2	2	no	—
61	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₆₁	#120	1	1	yes	characteristic
63	C₂²×C₆₁	#123	1	1	yes	characteristic
64	C₄×C₆₁	—	2	4	no	—
65	C₄×C₆₁	—	2	4	no	—
66	C₄×C₆₁	—	4	2	no	—
67	C₄×C₆₁	C₄×D₄	1	2	yes	—
68	C₄×C₆₁	#121	1	2	yes	—
69	Dic₆₁	—	4	4	no	—
70	#1886	—	61	1	no	maximal, Sylow
71	C₂×C₆₁	—	2	1	no	—
72	C₂×C₆₁	—	2	1	no	—
73	C₂×C₆₁	#386	1	1	yes	characteristic
74	C₂×C₆₁	#456	1	2	yes	—
75	C₂×C₄⋊Q₈	D₆₁	1	1	yes	characteristic
76	#519	—	61	1	no	—
77	#419	—	61	2	no	—
78	C₂×#101	—	61	1	no	—
79	#339	—	61	2	no	—
80	C₆₁	#1886	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	C₂×D₆₁	1	1	yes	characteristic
82	C₂²×Q₈	C₂×D₆₁	1	1	yes	characteristic
83	C₂×M₁₆	—	61	2	no	—
84	C₄⋊Q₈	—	2	2	no	—
85	C₄⋊Q₈	Dic₆₁	1	4	yes	complemented
86	C₄×Q₈	—	122	4	no	—
87	#117	—	61	2	no	—
88	#117	—	122	2	no	—
89	C₂×C₄⋊C₄	—	2	2	no	—
90	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
91	#103	—	122	4	no	—
92	#101	—	122	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	C₂×Dic₆₁	1	2	yes	—
99	C₂×Q₈	C₂×Dic₆₁	1	2	yes	—
100	C₂×Q₈	C₂²⋊D₆₁	1	2	yes	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
104	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
105	M₁₆	—	122	2	no	—
106	M₁₆	—	122	2	no	—
107	C₂×C₈	—	122	2	no	—
108	C₄⋊C₄	—	2	8	no	—
109	C₄⋊C₄	—	61	4	no	—
110	C₄⋊C₄	—	122	4	no	—
111	C₂²⋊C₄	—	122	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	122	4	no	—
114	C₄²	—	122	2	no	—
115	C₄²	C₂×Dic₆₁	1	2	yes	—
116	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
117	Q₈	—	2	4	no	—
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	Q₈	—	4	2	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	122	4	no	—
129	C₂×C₄	C₂²⋊Dic₆₁	1	4	yes	—
130	C₂×C₄	D₄×D₆₁	1	2	yes	—
131	C₂×C₄	#11758224	1	2	yes	—
132	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
134	C₈	—	244	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
137	C₂²	#14519745	1	1	yes	characteristic
138	C₂²	#14519758	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	2	4	no	—
141	C₄	—	4	2	no	—
142	C₄	—	244	4	no	—
143	C₄	D₄×Dic₆₁	1	2	yes	—
144	C₄	#14519749	1	2	yes	—
145	C₂	—	2	1	no	—
146	C₂	—	2	1	no	—
147	C₂	#16839642	1	1	yes	characteristic
148	C₂	#16839944	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840240

C₂

yes

maximal, characteristic

#16839771

C₂

yes

maximal

C₂×#14519696

C₂

yes

maximal, characteristic

#16839522

C₂

yes

maximal

C₂×C₄⋊Q₈×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

Q₈×Dic₆₁

—

#14519736

—

#14519736

C₂²

yes

—

#14519701

—

#14519696

—

C₂²×Q₈×C₆₁

C₂²

yes

characteristic

C₂²×Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

C₂×C₄

yes

—

C₂×Q₈×C₆₁

C₂×C₄

yes

—

C₂×Q₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

#11758194

—

#11758194

—

C₂×#92418

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×Dic₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

—

C₄×C₆₁

#121

yes

—

Dic₆₁

—

#1886

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#386

yes

characteristic

C₂×C₆₁

#456

yes

—

C₂×C₄⋊Q₈

D₆₁

yes

—

—

—

—

C₆₁

#1886

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₆₁

yes

characteristic

C₂²×Q₈

C₂×D₆₁

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₆₁

yes

complemented

C₄×Q₈

—

122

—

#117

—

#117

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#103

—

122

—

#101

—

122

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₆₁

yes

—

C₂×Q₈

C₂×Dic₆₁

yes

—

100

C₂×Q₈

C₂²⋊D₆₁

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²⋊D₆₁

yes

—

104

C₂²×C₄

C₂²×D₆₁

yes

characteristic

105

M₁₆

—

122

—

106

M₁₆

—

122

—

107

C₂×C₈

—

122

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

122

—

111

C₂²⋊C₄

—

122

—

112

C₄²

—

113

C₄²

—

122

—

114

C₄²

—

122

—

115

C₄²

C₂×Dic₆₁

yes

—

116

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

122

—

129

C₂×C₄

C₂²⋊Dic₆₁

yes

—

130

C₂×C₄

D₄×D₆₁

yes

—

131

C₂×C₄

#11758224

yes

—

132

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

134

C₈

—

244

—

135

C₂²

—

136

C₂²

C₂×C₂²⋊Dic₆₁

yes

characteristic

137

C₂²

#14519745

yes

characteristic

138

C₂²

#14519758

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

244

—

143

C₄

D₄×Dic₆₁

yes

—

144

C₄

#14519749

yes

—

145

C₂

—

146

C₂

—

147

C₂

#16839642

yes

characteristic

148

C₂

#16839944

yes

—

	subgroup	G/H
center	C₂²	#14519745
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×C₄⋊Q₈×C₆₁	C₂
socle	C₂²×C₆₁	#120
hypercenter	C₂×C₄⋊Q₈	D₆₁
hypocenter	C₆₁	#1886
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

#14519745

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×C₄⋊Q₈×C₆₁

C₂

socle

C₂²×C₆₁

#120

hypercenter

C₂×C₄⋊Q₈

D₆₁

hypocenter

C₆₁

#1886

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
15. C₄⋊Q₈×C₆₁	C₄	semidirect
80. C₆₁	#1886	semidirect
85. C₄⋊Q₈	Dic₆₁	semidirect

H ≅ G/N

type

15. C₄⋊Q₈×C₆₁

C₄

semidirect

80. C₆₁

#1886

semidirect

85. C₄⋊Q₈

Dic₆₁

semidirect

Subgroups — #21446213

Conjugacy classes of subgroups

Complements