Group #21456321 — subgroups

Subgroups — C₂×#16840162 #21456321

Order 7808.

360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14519768	C₂	1	2	yes	maximal
2	#16840162	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14519710	C₂	1	2	yes	maximal
4	C₂×#14519709	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×#92418	C₂²	1	1	yes	characteristic
8	#14519768	—	2	4	no	—
9	#14519768	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₆₁	—	2	2	no	—
11	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
12	C₂×#11758196	—	2	2	no	—
13	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
14	#14519710	C₂²	1	8	yes	—
15	#14519709	C₂²	1	4	yes	—
16	C₈×Dic₆₁	C₂²	1	4	yes	—
17	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
19	#106×C₆₁	C₂²	1	4	yes	—
20	C₂²×C₄×C₆₁	—	2	2	no	—
21	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
22	C₂²×Dic₆₁	—	2	2	no	—
23	C₂²×Dic₆₁	D₄	1	2	yes	—
24	C₂×C₈×C₆₁	Q₈	1	4	yes	—
25	C₂×C₈×C₆₁	C₂³	1	2	yes	—
26	C₄⋊C₄×C₆₁	—	2	2	no	—
27	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
28	C₄⋊Dic₆₁	—	2	2	no	—
29	C₄⋊Dic₆₁	—	2	8	no	—
30	C₄⋊Dic₆₁	C₂³	1	4	yes	—
31	#11758196	—	2	8	no	—
32	C₄×Dic₆₁	D₄	1	2	yes	—
33	C₄×Dic₆₁	C₂³	1	2	yes	—
34	C₂×#92418	Q₈	1	4	yes	—
35	C₂×#92418	C₂³	1	2	yes	—
36	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₆₁	—	2	4	no	—
38	C₂×C₄×C₆₁	—	4	4	no	—
39	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
43	C₂×Dic₆₁	—	2	4	no	—
44	C₂×Dic₆₁	—	2	4	no	—
45	C₂×Dic₆₁	—	4	4	no	—
46	C₂×Dic₆₁	C₂×D₄	1	4	yes	—
47	C₈×C₆₁	C₂×Q₈	1	4	yes	—
48	#92418	C₂×Q₈	1	4	yes	—
49	C₂²×C₆₁	C₄⋊Q₈	1	2	yes	—
50	C₂²×C₆₁	#135	1	4	yes	—
51	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
52	C₄×C₆₁	—	4	4	no	—
53	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
54	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
55	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
56	Dic₆₁	—	2	4	no	—
57	Dic₆₁	—	4	4	no	—
58	C₂×#499	—	61	1	no	maximal, Sylow
59	C₂×C₆₁	#499	1	4	yes	complemented
60	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₆₁	C₂×#135	1	2	yes	—
62	C₂×#125	—	61	2	no	—
63	#499	—	61	8	no	—
64	C₂×#107	—	61	2	no	—
65	C₂×#106	—	61	1	no	—
66	C₂×#106	D₆₁	1	1	yes	characteristic
67	C₂×C₄×C₈	—	61	1	no	—
68	C₆₁	C₂×#499	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	—	61	1	no	—
70	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
71	#125	—	61	8	no	—
72	#125	—	122	4	no	—
73	C₂×C₄⋊C₄	—	61	2	no	—
74	C₂×C₄⋊C₄	—	122	2	no	—
75	C₂×C₄⋊C₄	—	122	2	no	—
76	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
77	C₂×C₄²	—	61	1	no	—
78	#107	—	61	8	no	—
79	#106	—	61	4	no	—
80	#106	C₂×D₆₁	1	4	yes	—
81	C₄×C₈	—	61	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	61	2	no	—
84	C₂²×C₄	—	122	2	no	—
85	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
86	C₂×C₈	—	61	4	no	—
87	C₂×C₈	—	61	2	no	—
88	C₂×C₈	Dic₁₂₂	1	4	yes	—
89	C₂×C₈	C₂²×D₆₁	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	61	4	no	—
92	C₄⋊C₄	—	122	8	no	—
93	C₄⋊C₄	—	122	8	no	—
94	C₄⋊C₄	—	122	2	no	—
95	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
96	C₄²	—	61	2	no	—
97	C₄²	—	61	2	no	—
98	C₂³	D₄×D₆₁	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	—	61	4	no	—
102	C₂×C₄	—	122	4	no	—
103	C₂×C₄	—	122	4	no	—
104	C₂×C₄	—	244	4	no	—
105	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
106	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₆₁	1	2	yes	—
108	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
109	C₈	—	61	4	no	—
110	C₈	C₂×Dic₁₂₂	1	4	yes	—
111	C₂²	#14519776	1	2	yes	—
112	C₂²	#14519798	1	4	yes	—
113	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
114	C₄	—	4	4	no	—
115	C₄	—	122	4	no	—
116	C₄	—	244	4	no	—
117	C₄	#14519776	1	2	yes	—
118	C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
119	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
120	C₂	#16840162	1	4	yes	direct product factor
121	C₂	C₂×#14519776	1	1	yes	characteristic
122	C₂	C₂×#14519798	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519768

C₂

yes

maximal

#16840162

C₂

yes

maximal, direct product factor

C₂×#14519710

C₂

yes

maximal

C₂×#14519709

C₂

yes

maximal, characteristic

C₂×C₈×Dic₆₁

C₂

yes

maximal, characteristic

C₂×#106×C₆₁

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

characteristic

#14519768

—

#14519768

C₂²

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519710

C₂²

yes

—

#14519709

C₂²

yes

—

C₈×Dic₆₁

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

#106×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

D₄

yes

—

C₂×C₈×C₆₁

Q₈

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂×#92418

Q₈

yes

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₈×C₆₁

C₂×Q₈

yes

—

#92418

C₂×Q₈

yes

—

C₂²×C₆₁

C₄⋊Q₈

yes

—

C₂²×C₆₁

#135

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄⋊Q₈

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

C₂²×Q₈

yes

characteristic

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

C₂×#135

yes

—

—

—

—

—

D₆₁

yes

characteristic

C₂×C₄×C₈

—

C₆₁

C₂×#499

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

#125

—

#125

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

#107

—

#106

—

#106

C₂×D₆₁

yes

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₂₂

yes

—

C₂×C₈

C₂²×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

C₂²×D₆₁

yes

—

C₄²

—

C₄²

—

C₂³

D₄×D₆₁

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

122

—

103

C₂×C₄

—

122

—

104

C₂×C₄

—

244

—

105

C₂×C₄

C₂×Dic₁₂₂

yes

—

106

C₂×C₄

D₄×D₆₁

yes

characteristic

107

C₂×C₄

Q₈×D₆₁

yes

—

108

C₂×C₄

C₂³×D₆₁

yes

characteristic

109

C₈

—

110

C₈

C₂×Dic₁₂₂

yes

—

111

C₂²

#14519776

yes

—

112

C₂²

#14519798

yes

—

113

C₂²

C₂×D₄×D₆₁

yes

characteristic

114

C₄

—

115

C₄

—

122

—

116

C₄

—

244

—

117

C₄

#14519776

yes

—

118

C₄

C₂×Q₈×D₆₁

yes

characteristic

119

C₄

C₂²×Dic₁₂₂

yes

characteristic

120

C₂

#16840162

yes

direct product factor

121

C₂

C₂×#14519776

yes

characteristic

122

C₂

C₂×#14519798

yes

—

	subgroup	G/H
center	C₂³	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×#106×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×#106	D₆₁
hypocenter	C₆₁	C₂×#499
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×#106×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. #16840162	C₂	direct (N × H)
59. C₂×C₆₁	#499	semidirect
68. C₆₁	C₂×#499	semidirect
120. C₂	#16840162	direct (N × H)

H ≅ G/N

type

2. #16840162

C₂

direct (N × H)

59. C₂×C₆₁

#499

semidirect

68. C₆₁

C₂×#499

semidirect

120. C₂

#16840162

direct (N × H)

Subgroups — C₂×#16840162 #21456321

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16840162 #21456321

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840162 #21456321