Group #21461415 — subgroups

Subgroups — #21461415

Order 7808.

192 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 72 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	D₄×#92420	C₂	1	2	yes	maximal, complemented
2	#16840623	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#92420	C₂	1	2	yes	maximal, complemented
4	#16840603	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#92420	—	2	2	no	—
7	C₄⋊#92420	C₂²	1	2	yes	—
8	C₂²⋊#92420	—	2	2	no	—
9	C₄×#92420	C₂²	1	1	yes	characteristic
10	C₄×#92420	C₂²	1	1	yes	characteristic
11	D₄×Dic₆₁	C₂²	1	2	yes	—
12	#14519711	C₄	1	1	yes	characteristic
13	#14519698	C₄	1	2	yes	—
14	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₂×D₈×C₆₁	C₄	1	1	yes	characteristic
16	C₂×D₄×C₆₁	C₂×C₄	1	2	yes	—
17	C₂²×Dic₆₁	—	2	2	no	—
18	C₂×#92420	—	2	2	no	—
19	C₂×#92420	—	2	1	no	—
20	C₂×#92420	—	4	4	no	—
21	C₂×#92420	D₄	1	2	yes	—
22	D₈×C₆₁	C₈	1	4	yes	complemented
23	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²⋊Dic₆₁	—	2	2	no	—
25	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
26	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
27	C₂×#92418	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₆₁	—	2	2	no	—
29	D₄×C₆₁	—	2	2	no	—
30	D₄×C₆₁	C₂×C₈	1	4	yes	complemented
31	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₆₁	—	2	2	no	—
33	C₂×Dic₆₁	—	4	2	no	—
34	C₂×Dic₆₁	—	4	2	no	—
35	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₆₁	#41	1	1	yes	characteristic
37	#92420	—	2	2	no	—
38	#92420	—	4	1	no	—
39	#92420	—	4	2	no	—
40	C₈×C₆₁	C₂×C₈	1	2	yes	complemented
41	#92418	—	2	1	no	—
42	C₂²×C₆₁	—	2	4	no	—
43	C₂²×C₆₁	—	4	2	no	—
44	C₂²×C₆₁	—	4	2	no	—
45	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
46	C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
47	C₄×C₆₁	#131	1	1	yes	characteristic
48	Dic₆₁	—	2	1	no	—
49	Dic₆₁	—	4	2	no	—
50	Dic₆₁	AGL(1,ℤ/8)	1	2	yes	—
51	#1580	—	61	1	no	maximal, Sylow
52	C₂×C₆₁	—	4	4	no	—
53	C₂×C₆₁	C₈×D₄	1	1	yes	characteristic
54	C₂×C₆₁	#442	1	1	yes	characteristic
55	C₂×C₆₁	#444	1	1	yes	characteristic
56	C₄×D₈	—	61	1	no	—
57	C₈×D₄	—	61	2	no	—
58	#334	—	61	1	no	—
59	D₄⋊C₈	—	61	2	no	—
60	#322	—	61	1	no	—
61	C₆₁	#1580	1	1	yes	characteristic, complemented, Sylow
62	C₂×D₈	#6923	1	1	yes	characteristic
63	C₂²×C₈	—	122	2	no	—
64	C₄×D₄	—	61	2	no	—
65	#107	—	61	1	no	—
66	C₄⋊C₈	—	61	2	no	—
67	#102	—	61	2	no	—
68	C₂²⋊C₈	—	122	2	no	—
69	C₄×C₈	—	61	1	no	—
70	C₄×C₈	—	61	1	no	—
71	C₄×C₈	—	61	1	no	—
72	C₂×D₄	C₂×#6923	1	2	yes	—
73	C₂²×C₄	—	122	2	no	—
74	D₈	#92420	1	4	yes	complemented
75	C₂×C₈	—	61	2	no	—
76	C₂×C₈	—	61	1	no	—
77	C₂×C₈	—	122	1	no	—
78	C₂×C₈	—	122	2	no	—
79	C₂×C₈	—	244	4	no	—
80	C₂×C₈	C₂×#6923	1	1	yes	characteristic
81	C₄⋊C₄	—	61	2	no	—
82	C₂²⋊C₄	—	122	2	no	—
83	C₄²	—	61	1	no	—
84	C₂³	—	2	2	no	—
85	D₄	—	2	2	no	—
86	D₄	C₂×#92420	1	4	yes	complemented
87	C₂×C₄	—	61	1	no	—
88	C₂×C₄	—	61	1	no	—
89	C₂×C₄	—	122	2	no	—
90	C₂×C₄	—	244	2	no	—
91	C₂×C₄	—	244	2	no	—
92	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
93	C₈	—	122	2	no	—
94	C₈	—	122	1	no	—
95	C₈	—	244	2	no	—
96	C₈	—	244	1	no	—
97	C₈	C₂×#92420	1	2	yes	complemented
98	C₂²	—	2	4	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	2	no	—
101	C₂²	D₄×#6923	1	1	yes	characteristic
102	C₄	—	61	2	no	—
103	C₄	—	122	1	no	—
104	C₄	—	244	2	no	—
105	C₄	C₂²×#92420	1	1	yes	characteristic
106	C₄	#14519864	1	1	yes	characteristic
107	C₂	—	4	4	no	—
108	C₂	D₄×#92420	1	1	yes	characteristic
109	C₂	#16840803	1	1	yes	characteristic
110	C₂	#16840809	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×#92420

C₂

yes

maximal, complemented

#16840623

C₂

yes

maximal, characteristic, complemented

D₄⋊#92420

C₂

yes

maximal, complemented

#16840603

C₂

yes

maximal, characteristic, complemented

D₈×Dic₆₁

C₂

yes

maximal, characteristic

C₂²×#92420

—

C₄⋊#92420

C₂²

yes

—

C₂²⋊#92420

—

C₄×#92420

C₂²

yes

characteristic

C₄×#92420

C₂²

yes

characteristic

D₄×Dic₆₁

C₂²

yes

—

#14519711

C₄

yes

characteristic

#14519698

C₄

yes

—

C₈×Dic₆₁

C₂²

yes

characteristic

C₂×D₈×C₆₁

C₄

yes

characteristic

C₂×D₄×C₆₁

C₂×C₄

yes

—

C₂²×Dic₆₁

—

—

—

—

D₄

yes

—

D₈×C₆₁

C₈

yes

complemented

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂×#92418

C₂×C₄

yes

characteristic

C₂³×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

C₂×C₈

yes

complemented

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

C₂×Dic₆₁

yes

—

—

—

C₈×C₆₁

C₂×C₈

yes

complemented

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×D₄

yes

characteristic

C₄×C₆₁

C₂²×C₈

yes

characteristic

C₄×C₆₁

#131

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

AGL(1,ℤ/8)

yes

—

#1580

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₈×D₄

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₄×D₈

—

C₈×D₄

—

#334

—

D₄⋊C₈

—

#322

—

C₆₁

#1580

yes

characteristic, complemented, Sylow

C₂×D₈

#6923

yes

characteristic

C₂²×C₈

—

122

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×D₄

C₂×#6923

yes

—

C₂²×C₄

—

122

—

D₈

#92420

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

C₂×#6923

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

122

—

C₄²

—

C₂³

—

D₄

—

D₄

C₂×#92420

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

244

—

C₂×C₄

—

244

—

C₂×C₄

C₂²×#6923

yes

characteristic

C₈

—

122

—

C₈

—

122

—

C₈

—

244

—

C₈

—

244

—

C₈

C₂×#92420

yes

complemented

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

D₄×#6923

yes

characteristic

102

C₄

—

103

C₄

—

122

—

104

C₄

—

244

—

105

C₄

C₂²×#92420

yes

characteristic

106

C₄

#14519864

yes

characteristic

107

C₂

—

108

C₂

D₄×#92420

yes

characteristic

109

C₂

#16840803

yes

characteristic

110

C₂

#16840809

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#6923
commutator	C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₂×D₈×C₆₁	C₄
socle	C₂²×C₆₁	C₄×D₄
hypercenter	C₂×D₈	#6923
hypocenter	C₆₁	#1580
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

subgroup

G/H

center

C₂²

D₄×#6923

commutator

C₄×C₆₁

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6923

Fitting

C₂×D₈×C₆₁

C₄

socle

C₂²×C₆₁

C₄×D₄

C₂×D₈

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

N	H ≅ G/N	type
1. D₄×#92420	C₂	semidirect
2. #16840623	C₂	semidirect
3. D₄⋊#92420	C₂	semidirect
4. #16840603	C₂	semidirect
22. D₈×C₆₁	C₈	semidirect
30. D₄×C₆₁	C₂×C₈	semidirect
40. C₈×C₆₁	C₂×C₈	semidirect
61. C₆₁	#1580	semidirect
74. D₈	#92420	semidirect
86. D₄	C₂×#92420	semidirect
97. C₈	C₂×#92420	semidirect

H ≅ G/N

type

1. D₄×#92420

C₂

semidirect

2. #16840623

C₂

semidirect

3. D₄⋊#92420

C₂

semidirect

4. #16840603

C₂

semidirect

22. D₈×C₆₁

C₈

semidirect

30. D₄×C₆₁

C₂×C₈

semidirect

40. C₈×C₆₁

C₂×C₈

semidirect

61. C₆₁

#1580

semidirect

74. D₈

#92420

semidirect

86. D₄

C₂×#92420

semidirect

97. C₈

C₂×#92420

semidirect

Subgroups — #21461415

Conjugacy classes of subgroups

Complements