Group #21461623 — subgroups

Subgroups — #21461623

Order 7808.

232 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 90 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×#14519848	C₂	1	1	yes	maximal, characteristic
2	#16840727	C₂	1	2	yes	maximal
3	C₂×C₈×#6923	C₂	1	1	yes	maximal, characteristic
4	#16840632	C₂	1	2	yes	maximal
5	#16839799	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758213	C₂²	1	2	yes	—
7	C₂×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×#6923	C₂²	1	1	yes	characteristic
9	#14519848	C₄	1	4	yes	complemented
10	#14519832	—	2	2	no	—
11	C₄×#92420	—	2	2	no	—
12	C₈×#6923	C₄	1	4	yes	complemented
13	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
14	#14519735	C₂²	1	2	yes	—
15	#14519711	C₄	1	1	yes	characteristic, complemented
16	#14519708	C₄	1	2	yes	complemented
17	#107×C₆₁	C₄	1	1	yes	characteristic, complemented
18	C₂²×#6923	—	2	1	no	—
19	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
20	C₂×#92420	—	2	2	no	—
21	C₂×#92420	D₄	1	1	yes	characteristic
22	C₂×#92420	D₄	1	1	yes	characteristic
23	C₄×#6923	—	2	1	no	—
24	C₄×#6923	C₂×C₄	1	2	yes	—
25	#11758213	—	2	2	no	—
26	#11758213	C₂×C₄	1	4	yes	—
27	D₆₁⋊C₈	C₂×C₄	1	2	yes	—
28	D₆₁⋊C₈	Q₈	1	2	yes	—
29	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
31	#11758198	—	2	2	no	—
32	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
33	C₄×Dic₆₁	—	2	2	no	—
34	C₂×#92418	C₂×C₄	1	1	yes	characteristic
35	C₈×D₆₁	C₂×C₄	1	2	yes	—
36	C₈×D₆₁	C₂×C₄	1	2	yes	—
37	C₂²×D₆₁	#41	1	1	yes	characteristic
38	C₂×#6923	—	2	2	no	—
39	C₂×#6923	—	2	2	no	—
40	C₂×#6923	—	2	2	no	—
41	C₂×C₄×C₆₁	—	2	2	no	—
42	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₆₁	—	2	2	no	—
44	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
45	C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₄×D₆₁	C₂×Q₈	1	1	yes	characteristic
48	C₄×D₆₁	#41	1	1	yes	characteristic
49	#92420	—	2	4	no	—
50	#92420	C₄⋊C₄	1	2	yes	—
51	#92420	C₄⋊C₄	1	2	yes	—
52	C₈×C₆₁	C₄²	1	2	yes	—
53	#92418	C₄²	1	2	yes	—
54	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
55	C₂×D₆₁	—	2	1	no	—
56	C₂×D₆₁	—	2	1	no	—
57	C₂×D₆₁	#117	1	1	yes	characteristic
58	C₂×D₆₁	C₄×Q₈	1	1	yes	characteristic
59	#6923	—	2	4	no	—
60	C₄×C₆₁	—	4	2	no	—
61	C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
62	C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
63	Dic₆₁	—	4	2	no	—
64	Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
65	Dic₆₁	C₄×D₄	1	1	yes	characteristic
66	#1772	—	61	1	no	maximal, Sylow
67	C₂×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₆₁	#443	1	1	yes	characteristic
69	C₂×C₆₁	#443	1	1	yes	characteristic
70	D₆₁	—	2	1	no	—
71	D₆₁	—	2	1	no	—
72	C₂×#108	—	61	1	no	—
73	#427	—	61	1	no	—
74	#405	—	61	2	no	—
75	C₂×C₄×C₈	—	61	1	no	—
76	#339	—	61	2	no	—
77	C₆₁	#1772	1	1	yes	characteristic, complemented, Sylow
78	C₂×M₁₆	—	61	2	no	—
79	C₂²×C₈	—	61	1	no	—
80	C₂²×C₈	—	61	1	no	—
81	#117	—	61	2	no	—
82	C₂×C₄²	—	61	1	no	—
83	#108	—	61	4	no	—
84	#107	—	61	1	no	—
85	#107	#6923	1	1	yes	characteristic, complemented
86	#106	—	61	2	no	—
87	#97	—	122	2	no	—
88	C₄×C₈	—	61	4	no	—
89	C₄×C₈	—	122	2	no	—
90	C₂²×C₄	—	61	1	no	—
91	C₂²×C₄	—	122	1	no	—
92	M₁₆	—	61	4	no	—
93	M₁₆	—	122	2	no	—
94	C₂×C₈	—	61	2	no	—
95	C₂×C₈	—	61	1	no	—
96	C₂×C₈	—	61	1	no	—
97	C₂×C₈	—	61	2	no	—
98	C₂×C₈	—	61	1	no	—
99	C₂×C₈	—	61	2	no	—
100	C₂×C₈	—	61	2	no	—
101	C₂×C₈	—	122	2	no	—
102	C₂×C₈	C₂×#6923	1	1	yes	characteristic
103	C₄⋊C₄	—	61	2	no	—
104	C₄⋊C₄	C₂×#6923	1	2	yes	—
105	C₂²⋊C₄	—	122	2	no	—
106	C₄²	—	61	2	no	—
107	C₄²	—	122	2	no	—
108	C₄²	—	122	1	no	—
109	C₂³	—	61	1	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	61	1	no	—
112	C₂×C₄	—	61	1	no	—
113	C₂×C₄	—	61	1	no	—
114	C₂×C₄	—	61	1	no	—
115	C₂×C₄	—	61	1	no	—
116	C₂×C₄	—	122	2	no	—
117	C₂×C₄	—	122	2	no	—
118	C₂×C₄	—	122	2	no	—
119	C₂×C₄	—	122	2	no	—
120	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
121	C₈	—	61	2	no	—
122	C₈	—	61	2	no	—
123	C₈	—	61	2	no	—
124	C₈	—	122	4	no	—
125	C₈	C₄×#6923	1	2	yes	—
126	C₂²	—	61	1	no	—
127	C₂²	—	61	1	no	—
128	C₂²	—	122	1	no	—
129	C₂²	—	122	1	no	—
130	C₂²	D₄×#6923	1	1	yes	characteristic
131	C₄	—	4	2	no	—
132	C₄	—	61	1	no	—
133	C₄	—	61	1	no	—
134	C₄	—	122	4	no	—
135	C₄	—	244	2	no	—
136	C₄	C₂×C₄×#6923	1	1	yes	characteristic
137	C₄	Q₈×#6923	1	1	yes	characteristic
138	C₂	—	122	1	no	—
139	C₂	—	122	1	no	—
140	C₂	C₄⋊C₄×#6923	1	1	yes	characteristic
141	C₂	#16840804	1	1	yes	characteristic
142	C₂	#16840806	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519848

C₂

yes

maximal, characteristic

#16840727

C₂

yes

maximal

C₂×C₈×#6923

C₂

yes

maximal, characteristic

#16840632

C₂

yes

maximal

#16839799

C₂

yes

maximal, characteristic

C₂×#11758213

C₂²

yes

—

C₂×D₆₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×#6923

C₂²

yes

characteristic

#14519848

C₄

yes

—

—

C₄

yes

complemented

C₂×C₈×D₆₁

C₂²

yes

characteristic

#14519735

C₂²

yes

—

#14519711

C₄

yes

characteristic, complemented

#14519708

C₄

yes

complemented

#107×C₆₁

C₄

yes

characteristic, complemented

C₂²×#6923

—

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×#92420

—

C₂×#92420

D₄

yes

characteristic

C₂×#92420

D₄

yes

characteristic

C₄×#6923

—

C₄×#6923

C₂×C₄

yes

—

#11758213

—

#11758213

C₂×C₄

yes

—

D₆₁⋊C₈

C₂×C₄

yes

—

D₆₁⋊C₈

Q₈

yes

—

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

#11758198

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

—

C₂×#92418

C₂×C₄

yes

characteristic

C₈×D₆₁

C₂×C₄

yes

—

C₈×D₆₁

C₂×C₄

yes

—

C₂²×D₆₁

yes

—

—

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₄×D₆₁

C₂²×C₄

yes

characteristic

C₄×D₆₁

C₂²×C₄

yes

characteristic

C₄×D₆₁

C₂×Q₈

yes

C₄×D₆₁

yes

—

C₄⋊C₄

yes

—

#92420

C₄⋊C₄

yes

—

C₈×C₆₁

C₄²

yes

—

#92418

C₄²

yes

—

C₂²×C₆₁

C₄×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

#117

yes

characteristic

C₂×D₆₁

C₄×Q₈

yes

characteristic

#6923

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄²

yes

characteristic

C₄×C₆₁

C₄×Q₈

yes

characteristic

Dic₆₁

—

Dic₆₁

C₂×C₄⋊C₄

yes

characteristic

Dic₆₁

C₄×D₄

yes

characteristic

#1772

—

maximal, Sylow

C₂×C₆₁

C₄×C₄⋊C₄

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

D₆₁

—

C₂×#108

—

#427

—

#405

—

C₂×C₄×C₈

—

#339

—

C₆₁

#1772

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

C₂×C₄²

—

—

—

yes

characteristic, complemented

#106

—

#97

—

122

—

C₄×C₈

—

C₄×C₈

—

122

—

C₂²×C₄

—

C₂²×C₄

—

122

—

M₁₆

—

M₁₆

—

122

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

122

—

102

C₂×C₈

C₂×#6923

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

C₂×#6923

yes

—

105

C₂²⋊C₄

—

122

—

106

C₄²

—

107

C₄²

—

122

—

108

C₄²

—

122

—

109

C₂³

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

122

—

117

C₂×C₄

—

122

—

118

C₂×C₄

—

122

—

119

C₂×C₄

—

122

—

120

C₂×C₄

C₂²×#6923

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

122

—

125

C₈

C₄×#6923

yes

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

122

—

129

C₂²

—

122

—

130

C₂²

D₄×#6923

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

122

—

135

C₄

—

244

—

136

C₄

C₂×C₄×#6923

yes

characteristic

137

C₄

Q₈×#6923

yes

characteristic

138

C₂

—

122

—

139

C₂

—

122

—

140

C₂

C₄⋊C₄×#6923

yes

characteristic

141

C₂

#16840804

yes

characteristic

142

C₂

#16840806

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#6923
commutator	C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄	C₂²×#6923
Fitting	#107×C₆₁	C₄
socle	C₂²×C₆₁	C₄×D₄
hypercenter	#107	#6923
hypocenter	C₆₁	#1772
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×#6923

commutator

C₄×C₆₁

C₂×C₄²

Frattini

C₂×C₄

C₂²×#6923

Fitting

#107×C₆₁

C₄

socle

C₂²×C₆₁

C₄×D₄

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
9. #14519848	C₄	semidirect
12. C₈×#6923	C₄	semidirect
15. #14519711	C₄	semidirect
16. #14519708	C₄	semidirect
17. #107×C₆₁	C₄	semidirect
77. C₆₁	#1772	semidirect
85. #107	#6923	semidirect

H ≅ G/N

type

9. #14519848

C₄

semidirect

12. C₈×#6923

C₄

semidirect

15. #14519711

C₄

semidirect

16. #14519708

C₄

semidirect

17. #107×C₆₁

C₄

semidirect

77. C₆₁

#1772

semidirect

85. #107

#6923

semidirect

Subgroups — #21461623

Conjugacy classes of subgroups

Complements