Group #21456153 — subgroups

Subgroups — C₂×#16840126 #21456153

Order 7808.

488 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14519769	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840126	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14519698	C₂	1	4	yes	maximal, complemented
4	C₂×#14519688	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#92418	C₂²	1	2	yes	—
7	#14519769	—	2	2	no	—
8	#14519769	C₂²	1	4	yes	complemented
9	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
10	C₂×#11758196	—	2	2	no	—
11	#14519698	C₂²	1	16	yes	complemented
12	#14519688	C₂²	1	4	yes	complemented
13	C₂²×D₄×C₆₁	—	2	2	no	—
14	C₂²×D₄×C₆₁	C₂²	1	2	yes	—
15	#127×C₆₁	—	2	2	no	—
16	#127×C₆₁	C₂²	1	4	yes	—
17	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
18	C₂⁴×C₆₁	—	2	2	no	—
19	C₂×D₄×C₆₁	—	2	2	no	—
20	C₂×D₄×C₆₁	—	2	8	no	—
21	C₂×D₄×C₆₁	—	2	8	no	—
22	C₂×D₄×C₆₁	—	4	8	no	—
23	C₂×D₄×C₆₁	C₂³	1	4	yes	—
24	C₂²×C₄×C₆₁	D₄	1	2	yes	—
25	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₆₁	—	2	2	no	—
27	C₄²×C₆₁	D₄	1	2	yes	—
28	C₄²×C₆₁	C₂³	1	2	yes	—
29	C₄⋊Dic₆₁	—	2	2	no	—
30	C₄⋊Dic₆₁	C₂³	1	4	yes	—
31	#11758196	—	2	8	no	—
32	C₂×#92418	—	2	4	no	—
33	C₂×#92418	C₂³	1	4	yes	—
34	C₂³×C₆₁	—	2	4	no	—
35	C₂³×C₆₁	—	4	4	no	—
36	C₂³×C₆₁	—	4	8	no	—
37	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
38	D₄×C₆₁	—	2	8	no	—
39	D₄×C₆₁	—	4	16	no	—
40	D₄×C₆₁	—	4	4	no	—
41	C₂×C₄×C₆₁	—	2	4	no	—
42	C₂×C₄×C₆₁	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	#41	1	4	yes	—
45	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₆₁	—	2	4	no	—
47	C₂×Dic₆₁	—	4	4	no	—
48	#92418	—	2	4	no	—
49	C₂²×C₆₁	—	4	4	no	—
50	C₂²×C₆₁	—	4	8	no	—
51	C₂²×C₆₁	—	8	8	no	—
52	C₂²×C₆₁	#124	1	2	yes	—
53	C₂²×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
54	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
55	C₄×C₆₁	—	2	4	no	—
56	C₄×C₆₁	#124	1	2	yes	—
57	C₄×C₆₁	C₂×#41	1	2	yes	—
58	Dic₆₁	—	4	4	no	—
59	C₂×#490	—	61	1	no	maximal, Sylow
60	C₂×C₆₁	—	8	4	no	—
61	C₂×C₆₁	#490	1	4	yes	complemented
62	C₂×C₆₁	C₂×#124	1	1	yes	characteristic
63	C₂×C₆₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
64	C₂×#127	D₆₁	1	1	yes	characteristic
65	C₂×#125	—	61	1	no	—
66	#490	—	61	8	no	—
67	C₂×#102	—	61	4	no	—
68	C₂×#97	—	61	1	no	—
69	C₆₁	C₂×#490	1	1	yes	characteristic, complemented, Sylow
70	C₂²×D₄	—	2	2	no	—
71	C₂²×D₄	C₂×D₆₁	1	2	yes	—
72	C₂²×C₈	—	61	2	no	—
73	#127	—	2	2	no	—
74	#127	C₂×D₆₁	1	4	yes	—
75	#125	—	61	4	no	—
76	#125	—	122	2	no	—
77	C₂×C₄⋊C₄	—	61	2	no	—
78	C₂×C₄⋊C₄	—	122	2	no	—
79	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
80	#102	—	61	16	no	—
81	#97	—	61	4	no	—
82	C₂⁴	—	2	2	no	—
83	C₂×D₄	—	2	8	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	8	no	—
86	C₂×D₄	—	4	8	no	—
87	C₂×D₄	C₂²×D₆₁	1	4	yes	—
88	C₂²×C₄	—	122	2	no	—
89	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
90	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
91	C₂×C₈	—	61	4	no	—
92	C₂×C₈	—	122	4	no	—
93	C₄⋊C₄	—	61	4	no	—
94	C₄⋊C₄	—	122	2	no	—
95	C₄⋊C₄	—	122	8	no	—
96	C₄²	C₂²⋊D₆₁	1	2	yes	—
97	C₄²	C₂²×D₆₁	1	2	yes	—
98	C₂³	—	2	4	no	—
99	C₂³	—	4	4	no	—
100	C₂³	—	4	8	no	—
101	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
102	D₄	—	2	8	no	—
103	D₄	—	4	16	no	—
104	D₄	—	4	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	122	4	no	—
107	C₂×C₄	—	244	4	no	—
108	C₂×C₄	#11758224	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₆₁	1	4	yes	—
110	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
111	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
112	C₈	—	122	4	no	—
113	C₂²	—	4	4	no	—
114	C₂²	—	4	8	no	—
115	C₂²	—	8	8	no	—
116	C₂²	#14519764	1	2	yes	—
117	C₂²	#14519803	1	4	yes	—
118	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	244	4	no	—
121	C₄	#14519764	1	2	yes	—
122	C₄	C₂×#11758224	1	2	yes	—
123	C₂	—	8	4	no	—
124	C₂	#16840126	1	4	yes	direct product factor
125	C₂	C₂×#14519764	1	1	yes	characteristic
126	C₂	C₂×#14519803	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519769

C₂

yes

maximal, characteristic, complemented

#16840126

C₂

yes

maximal, direct product factor

C₂×#14519698

C₂

yes

maximal, complemented

C₂×#14519688

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₆₁

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

—

#14519769

—

#14519769

C₂²

yes

complemented

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

—

#14519698

C₂²

yes

complemented

#14519688

C₂²

yes

complemented

C₂²×D₄×C₆₁

—

C₂²×D₄×C₆₁

C₂²

yes

—

#127×C₆₁

—

#127×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂⁴×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₂×#92418

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#124

yes

—

C₂²×C₆₁

AGL(1,ℤ/8)

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

#124

yes

—

C₄×C₆₁

C₂×#41

yes

—

Dic₆₁

—

C₂×#490

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×#127

D₆₁

yes

—

—

—

C₂×#97

—

C₆₁

C₂×#490

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

C₂×D₆₁

yes

—

C₂²×C₈

—

#127

—

#127

C₂×D₆₁

yes

—

#125

—

#125

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#102

—

#97

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×D₆₁

yes

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²⋊D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

122

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄²

C₂²⋊D₆₁

yes

—

C₄²

C₂²×D₆₁

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

122

—

107

C₂×C₄

—

244

—

108

C₂×C₄

#11758224

yes

—

109

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

110

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

111

C₂×C₄

C₂³×D₆₁

yes

characteristic

112

C₈

—

122

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#14519764

yes

—

117

C₂²

#14519803

yes

—

118

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

119

C₄

—

120

C₄

—

244

—

121

C₄

#14519764

yes

—

122

C₄

C₂×#11758224

yes

—

123

C₂

—

124

C₂

#16840126

yes

direct product factor

125

C₂

C₂×#14519764

yes

characteristic

126

C₂

C₂×#14519803

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×#127×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×#127	D₆₁
hypocenter	C₆₁	C₂×#490
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×#127×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#14519769	C₂	semidirect
2. #16840126	C₂	direct (N × H)
3. C₂×#14519698	C₂	semidirect
4. C₂×#14519688	C₂	semidirect
8. #14519769	C₂²	semidirect
11. #14519698	C₂²	semidirect
12. #14519688	C₂²	semidirect
61. C₂×C₆₁	#490	semidirect
69. C₆₁	C₂×#490	semidirect
124. C₂	#16840126	direct (N × H)

H ≅ G/N

type

1. C₂×#14519769

C₂

semidirect

2. #16840126

C₂

direct (N × H)

3. C₂×#14519698

C₂

semidirect

4. C₂×#14519688

C₂

semidirect

8. #14519769

C₂²

semidirect

11. #14519698

C₂²

semidirect

12. #14519688

C₂²

semidirect

61. C₂×C₆₁

#490

semidirect

69. C₆₁

C₂×#490

semidirect

124. C₂

#16840126

direct (N × H)

Subgroups — C₂×#16840126 #21456153

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16840126 #21456153

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840126 #21456153