Group #21460652 — subgroups

Subgroups — #21460652

Order 7808.

172 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 70 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	C₂×C₄×#92420	C₂	1	1	yes	maximal, characteristic
3	M₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#92420	—	2	1	no	—
5	C₄⋊#92420	C₄	1	2	yes	—
6	C₂²⋊#92420	—	2	1	no	—
7	C₄×#92420	—	2	1	no	—
8	C₄×#92420	—	2	1	no	—
9	C₄×#92420	C₄	1	2	yes	—
10	C₂×#11758194	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
12	#14519687	—	2	1	no	—
13	C₈×Dic₆₁	—	2	1	no	—
14	C₂×M₁₆×C₆₁	C₄	1	1	yes	characteristic
15	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
17	C₂²×Dic₆₁	Q₈	1	1	yes	characteristic
18	C₂×#92420	—	2	4	no	—
19	C₂×#92420	—	2	2	no	—
20	C₂×#92420	—	2	2	no	—
21	M₁₆×C₆₁	—	2	1	no	—
22	M₁₆×C₆₁	C₈	1	2	yes	complemented
23	C₂×C₈×C₆₁	—	2	1	no	—
24	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₆₁	D₄	1	1	yes	characteristic
27	C₄×Dic₆₁	D₄	1	1	yes	characteristic
28	#11758194	—	2	1	no	—
29	#11758194	C₈	1	2	yes	complemented
30	C₂×#92418	—	2	1	no	—
31	C₂³×C₆₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₆₁	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₆₁	M₁₆	1	1	yes	characteristic
37	C₂×C₄×C₆₁	M₁₆	1	1	yes	characteristic
38	C₂×Dic₆₁	—	2	2	no	—
39	C₂×Dic₆₁	—	2	2	no	—
40	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
44	#92420	—	2	4	no	—
45	#92420	—	4	2	no	—
46	C₈×C₆₁	—	2	2	no	—
47	#92418	—	2	2	no	—
48	C₂²×C₆₁	—	2	1	no	—
49	C₂²×C₆₁	—	2	1	no	—
50	C₂²×C₆₁	#95	1	1	yes	characteristic
51	C₂²×C₆₁	C₄⋊C₈	1	1	yes	characteristic
52	C₂²×C₆₁	C₄⋊C₈	1	1	yes	characteristic
53	C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
54	C₄×C₆₁	#97	1	1	yes	characteristic
55	C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
56	C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
57	Dic₆₁	—	2	1	no	—
58	Dic₆₁	—	2	1	no	—
59	Dic₆₁	#104	1	2	yes	—
60	Dic₆₁	#108	1	2	yes	—
61	#1278	—	61	1	no	maximal, Sylow
62	C₂×C₆₁	—	2	1	no	—
63	C₂×C₆₁	—	2	1	no	—
64	C₂×C₆₁	#336	1	1	yes	characteristic
65	C₂×C₆₁	#339	1	1	yes	characteristic
66	C₂×C₆₁	#341	1	1	yes	characteristic
67	#431	—	61	1	no	—
68	C₄×M₁₆	—	61	1	no	—
69	C₂×C₄×C₈	—	61	1	no	—
70	C₆₁	#1278	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	—	61	1	no	—
72	C₂×M₁₆	#6923	1	1	yes	characteristic
73	C₂²×C₈	—	122	1	no	—
74	C₂×C₄²	—	61	1	no	—
75	C₄⋊C₈	—	61	2	no	—
76	C₂²⋊C₈	—	122	1	no	—
77	#97	—	122	1	no	—
78	C₄×C₈	—	61	2	no	—
79	C₄×C₈	—	122	1	no	—
80	C₄×C₈	—	122	1	no	—
81	C₄×C₈	—	122	1	no	—
82	C₂²×C₄	—	61	1	no	—
83	C₂²×C₄	—	61	1	no	—
84	C₂²×C₄	C₂×#6923	1	1	yes	characteristic
85	M₁₆	—	2	1	no	—
86	M₁₆	—	61	2	no	—
87	M₁₆	—	122	1	no	—
88	M₁₆	#92420	1	2	yes	complemented
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	122	2	no	—
91	C₂×C₈	—	122	4	no	—
92	C₂×C₈	—	122	1	no	—
93	C₂×C₈	—	122	2	no	—
94	C₄²	—	61	1	no	—
95	C₄²	—	61	1	no	—
96	C₄²	—	61	1	no	—
97	C₄²	—	61	1	no	—
98	C₂³	C₄⋊#6923	1	1	yes	characteristic
99	C₂×C₄	—	61	1	no	—
100	C₂×C₄	—	61	1	no	—
101	C₂×C₄	—	61	1	no	—
102	C₂×C₄	—	61	1	no	—
103	C₂×C₄	—	122	2	no	—
104	C₂×C₄	—	122	2	no	—
105	C₂×C₄	D₆₁⋊C₈	1	1	yes	characteristic
106	C₂×C₄	#11758213	1	1	yes	characteristic
107	C₂×C₄	C₄×#6923	1	1	yes	characteristic
108	C₂×C₄	C₂×#92420	1	1	yes	characteristic
109	C₂×C₄	#11758217	1	1	yes	characteristic
110	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	122	4	no	—
113	C₈	—	122	2	no	—
114	C₈	—	244	2	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	(C₂×#6923)⋊C₄	1	1	yes	characteristic
118	C₂²	C₄⋊#92420	1	1	yes	characteristic
119	C₂²	#14519844	1	1	yes	characteristic
120	C₄	—	61	2	no	—
121	C₄	—	61	2	no	—
122	C₄	—	122	1	no	—
123	C₄	—	122	1	no	—
124	C₄	C₄×#92420	1	1	yes	characteristic
125	C₄	#14519832	1	1	yes	characteristic
126	C₄	C₂²⋊#92420	1	1	yes	characteristic
127	C₄	#14519834	1	1	yes	characteristic
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#16840628	1	1	yes	characteristic
131	C₂	#16840634	1	1	yes	characteristic
132	C₂	#16840637	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840778

C₂

yes

maximal, characteristic

C₂×C₄×#92420

C₂

yes

maximal, characteristic

M₁₆×Dic₆₁

C₂

yes

maximal, characteristic

C₂²×#92420

—

C₄⋊#92420

C₄

yes

—

—

—

—

C₄

yes

—

C₂×#11758194

C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519687

—

C₈×Dic₆₁

—

C₂×M₁₆×C₆₁

C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

D₄

yes

characteristic

C₂²×Dic₆₁

Q₈

yes

—

—

—

M₁₆×C₆₁

—

M₁₆×C₆₁

C₈

yes

complemented

C₂×C₈×C₆₁

—

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

#11758194

—

#11758194

C₈

yes

complemented

C₂×#92418

—

C₂³×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂×C₄×C₆₁

M₁₆

yes

characteristic

C₂×C₄×C₆₁

M₁₆

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

C₄⋊C₄

yes

characteristic

C₂×Dic₆₁

C₄⋊C₄

yes

characteristic

#92420

—

#92420

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#95

yes

characteristic

C₂²×C₆₁

C₄⋊C₈

yes

characteristic

C₂²×C₆₁

C₄⋊C₈

yes

characteristic

C₄×C₆₁

C₄×C₈

yes

characteristic

C₄×C₆₁

#97

yes

characteristic

C₄×C₆₁

C₂²⋊C₈

yes

characteristic

C₄×C₆₁

C₂²⋊C₈

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#104

yes

—

Dic₆₁

#108

yes

—

#1278

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

—

C₆₁

#1278

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

#6923

yes

characteristic

C₂²×C₈

—

122

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

122

—

#97

—

122

—

C₄×C₈

—

C₄×C₈

—

122

—

C₄×C₈

—

122

—

C₄×C₈

—

122

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6923

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

122

—

M₁₆

#92420

yes

complemented

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₄⋊#6923

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

122

—

104

C₂×C₄

—

122

—

105

C₂×C₄

D₆₁⋊C₈

yes

characteristic

106

C₂×C₄

#11758213

yes

characteristic

107

C₂×C₄

C₄×#6923

yes

characteristic

108

C₂×C₄

C₂×#92420

yes

characteristic

109

C₂×C₄

#11758217

yes

characteristic

110

C₂×C₄

C₂²⋊#6923

yes

characteristic

111

C₈

—

112

C₈

—

122

—

113

C₈

—

122

—

114

C₈

—

244

—

115

C₂²

—

116

C₂²

—

117

C₂²

(C₂×#6923)⋊C₄

yes

characteristic

118

C₂²

C₄⋊#92420

yes

characteristic

119

C₂²

#14519844

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

122

—

123

C₄

—

122

—

124

C₄

C₄×#92420

yes

characteristic

125

C₄

#14519832

yes

characteristic

126

C₄

C₂²⋊#92420

yes

characteristic

127

C₄

#14519834

yes

characteristic

128

C₂

—

129

C₂

—

130

C₂

#16840628

yes

characteristic

131

C₂

#16840634

yes

characteristic

132

C₂

#16840637

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#6923
commutator	C₄×C₆₁	C₄×C₈
Frattini	C₂²×C₄	C₂×#6923
Fitting	C₂×M₁₆×C₆₁	C₄
socle	C₂²×C₆₁	#95
hypercenter	C₂×M₁₆	#6923
hypocenter	C₆₁	#1278
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₂×C₄

C₂²⋊#6923

commutator

C₄×C₆₁

C₄×C₈

Frattini

C₂²×C₄

C₂×#6923

Fitting

C₂×M₁₆×C₆₁

C₄

C₂²×C₆₁

C₂×M₁₆

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
22. M₁₆×C₆₁	C₈	semidirect
29. #11758194	C₈	semidirect
70. C₆₁	#1278	semidirect
88. M₁₆	#92420	semidirect

H ≅ G/N

type

22. M₁₆×C₆₁

C₈

semidirect

29. #11758194

C₈

semidirect

70. C₆₁

#1278

semidirect

88. M₁₆

#92420

semidirect

Subgroups — #21460652

Conjugacy classes of subgroups

Complements