Group #21456320 — subgroups

Subgroups — C₂×#16840161 #21456320

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×#14519769	C₂	1	1	yes	maximal, characteristic
2	#16840161	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14519710	C₂	1	2	yes	maximal
4	C₂×#14519708	C₂	1	2	yes	maximal
5	C₂×C₄×#92418	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×#92418	C₂²	1	1	yes	characteristic
8	C₂²×#92418	C₂²	1	1	yes	characteristic
9	#14519769	—	2	2	no	—
10	#14519769	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
12	C₂×#11758196	—	2	2	no	—
13	#14519710	C₂²	1	8	yes	—
14	#14519708	C₂²	1	8	yes	—
15	C₄×#92418	C₂²	1	4	yes	—
16	#125×C₆₁	—	2	2	no	—
17	#125×C₆₁	C₂²	1	4	yes	—
18	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
19	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
20	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	—	2	2	no	—
22	C₂²×C₄×C₆₁	D₄	1	2	yes	—
23	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
24	C₂²×Dic₆₁	—	2	2	no	—
25	C₄⋊C₄×C₆₁	—	2	8	no	—
26	C₄⋊C₄×C₆₁	—	2	2	no	—
27	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
28	C₄²×C₆₁	D₄	1	2	yes	—
29	C₄²×C₆₁	C₂³	1	2	yes	—
30	C₄⋊Dic₆₁	—	2	2	no	—
31	C₄⋊Dic₆₁	C₂³	1	4	yes	—
32	#11758196	—	2	8	no	—
33	C₂×#92418	Q₈	1	4	yes	—
34	C₂×#92418	Q₈	1	4	yes	—
35	C₂×#92418	C₂³	1	2	yes	—
36	C₂×#92418	C₂³	1	2	yes	—
37	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₆₁	—	2	4	no	—
39	C₂×C₄×C₆₁	—	2	4	no	—
40	C₂×C₄×C₆₁	—	4	4	no	—
41	C₂×C₄×C₆₁	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₆₁	—	2	4	no	—
47	C₂×Dic₆₁	—	4	4	no	—
48	#92418	C₂×Q₈	1	4	yes	—
49	#92418	C₂×Q₈	1	4	yes	—
50	C₂²×C₆₁	C₄⋊Q₈	1	2	yes	—
51	C₂²×C₆₁	#135	1	4	yes	—
52	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
53	C₄×C₆₁	—	2	4	no	—
54	C₄×C₆₁	—	4	4	no	—
55	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
56	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
57	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
58	Dic₆₁	—	4	4	no	—
59	C₂×#499	—	61	1	no	maximal, Sylow
60	C₂×C₆₁	#499	1	4	yes	complemented
61	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₆₁	C₂×#135	1	2	yes	—
63	C₂×#125	—	61	1	no	—
64	C₂×#125	D₆₁	1	1	yes	characteristic
65	#499	—	61	8	no	—
66	C₂×#107	—	61	2	no	—
67	C₂×#106	—	61	2	no	—
68	C₂×C₄×C₈	—	61	1	no	—
69	C₆₁	C₂×#499	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₈	—	61	1	no	—
71	C₂²×C₈	—	61	1	no	—
72	#125	—	2	2	no	—
73	#125	—	61	4	no	—
74	#125	—	122	2	no	—
75	#125	C₂×D₆₁	1	4	yes	—
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	—	61	2	no	—
78	C₂×C₄⋊C₄	—	122	2	no	—
79	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
80	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
81	#107	—	61	8	no	—
82	#106	—	61	8	no	—
83	C₄×C₈	—	61	4	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	122	2	no	—
86	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
87	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
88	C₂×C₈	—	61	4	no	—
89	C₂×C₈	—	61	4	no	—
90	C₂×C₈	—	61	2	no	—
91	C₂×C₈	—	61	2	no	—
92	C₄⋊C₄	—	2	8	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	61	4	no	—
95	C₄⋊C₄	—	122	2	no	—
96	C₄⋊C₄	—	122	8	no	—
97	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
98	C₄²	C₂²⋊D₆₁	1	2	yes	—
99	C₄²	C₂²×D₆₁	1	2	yes	—
100	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	—	122	4	no	—
105	C₂×C₄	—	244	4	no	—
106	C₂×C₄	Q₈×D₆₁	1	2	yes	—
107	C₂×C₄	Q₈×D₆₁	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₆₁	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
110	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
111	C₈	—	61	4	no	—
112	C₈	—	61	4	no	—
113	C₂²	#14519775	1	2	yes	—
114	C₂²	#14519797	1	4	yes	—
115	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	—	4	4	no	—
118	C₄	—	244	4	no	—
119	C₄	#14519775	1	2	yes	—
120	C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
121	C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
122	C₂	#16840161	1	4	yes	direct product factor
123	C₂	C₂×#14519775	1	1	yes	characteristic
124	C₂	C₂×#14519797	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519769

C₂

yes

maximal, characteristic

#16840161

C₂

yes

maximal, direct product factor

C₂×#14519710

C₂

yes

maximal

C₂×#14519708

C₂

yes

maximal

C₂×C₄×#92418

C₂

yes

maximal, characteristic

C₂×#125×C₆₁

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

characteristic

C₂²×#92418

C₂²

yes

characteristic

#14519769

—

#14519769

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

—

#14519710

C₂²

yes

—

#14519708

C₂²

yes

—

C₄×#92418

C₂²

yes

—

#125×C₆₁

—

#125×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₂×#92418

Q₈

yes

—

C₂×#92418

Q₈

yes

—

C₂×#92418

C₂³

yes

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

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

—

#92418

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×C₆₁

C₄⋊Q₈

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

C₂²×Q₈

yes

characteristic

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

C₂×#135

yes

—

C₂×#125

—

C₂×#125

D₆₁

yes

—

—

—

C₂×C₄×C₈

—

C₆₁

C₂×#499

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

#125

—

#125

—

#125

—

122

—

#125

C₂×D₆₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#107

—

#106

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²⋊D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

C₂²×D₆₁

yes

—

C₄²

C₂²⋊D₆₁

yes

—

C₄²

C₂²×D₆₁

yes

—

100

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

122

—

105

C₂×C₄

—

244

—

106

C₂×C₄

Q₈×D₆₁

yes

—

107

C₂×C₄

Q₈×D₆₁

yes

—

108

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

109

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

110

C₂×C₄

C₂³×D₆₁

yes

characteristic

111

C₈

—

112

C₈

—

113

C₂²

#14519775

yes

—

114

C₂²

#14519797

yes

—

115

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

244

—

119

C₄

#14519775

yes

—

120

C₄

C₂×Q₈×D₆₁

yes

characteristic

121

C₄

C₂×Q₈×D₆₁

yes

characteristic

122

C₂

#16840161

yes

direct product factor

123

C₂

C₂×#14519775

yes

characteristic

124

C₂

C₂×#14519797

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×#125×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×#125	D₆₁
hypocenter	C₆₁	C₂×#499
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×#125×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #16840161	C₂	direct (N × H)
60. C₂×C₆₁	#499	semidirect
69. C₆₁	C₂×#499	semidirect
122. C₂	#16840161	direct (N × H)

H ≅ G/N

type

2. #16840161

C₂

direct (N × H)

60. C₂×C₆₁

#499

semidirect

69. C₆₁

C₂×#499

semidirect

122. C₂

#16840161

direct (N × H)

Subgroups — C₂×#16840161 #21456320

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16840161 #21456320

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840161 #21456320