Group #21462217 — subgroups

Subgroups — #21462217

Order 7808.

172 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 48 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	D₈×#6923	C₂	1	2	yes	maximal, complemented
2	#16840689	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840674	C₂	1	2	yes	maximal, complemented
4	#16840649	C₂	1	1	yes	maximal, characteristic, complemented
5	D₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
6	D₄×#6923	—	2	2	no	—
7	D₆₁⋊C₁₆	C₂²	1	1	yes	characteristic
8	#14519846	C₂²	1	2	yes	—
9	#14519839	—	2	2	no	—
10	C₈×#6923	C₂²	1	1	yes	characteristic
11	D₈×D₆₁	C₂²	1	2	yes	—
12	D₉₇₆	C₄	1	1	yes	characteristic, complemented
13	#14519718	C₄	1	2	yes	complemented
14	C₁₆×D₆₁	C₂²	1	1	yes	characteristic
15	D₁₆×C₆₁	C₄	1	1	yes	characteristic, complemented
16	C₂²×#6923	—	4	2	no	—
17	D₄×D₆₁	—	2	2	no	—
18	C₂²⋊#6923	—	4	2	no	—
19	C₄⋊#6923	—	2	2	no	—
20	C₄×#6923	D₄	1	1	yes	characteristic
21	D₆₁⋊C₈	D₄	1	1	yes	characteristic
22	D₈×C₆₁	C₂×C₄	1	2	yes	complemented
23	#11758199	—	2	2	no	—
24	D₄₈₈	C₂×C₄	1	2	yes	complemented
25	C₈×D₆₁	C₂³	1	1	yes	characteristic
26	#11758187	—	2	1	no	—
27	C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
28	#11758185	C₂×C₄	1	1	yes	characteristic, complemented
29	C₂²×D₆₁	—	4	2	no	—
30	C₂×#6923	—	4	2	no	—
31	C₂×#6923	—	8	4	no	—
32	C₂×#6923	D₈	1	2	yes	—
33	D₄×C₆₁	—	2	2	no	—
34	C₂²⋊D₆₁	—	4	2	no	—
35	D₂₄₄	—	2	2	no	—
36	C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
37	#92420	—	2	1	no	—
38	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	#92418	#41	1	1	yes	characteristic
40	C₂²×C₆₁	—	4	2	no	—
41	C₂×D₆₁	—	4	2	no	—
42	C₂×D₆₁	—	8	2	no	—
43	C₂×D₆₁	—	8	2	no	—
44	C₂×D₆₁	C₂×D₈	1	1	yes	characteristic
45	#6923	—	2	2	no	—
46	#6923	—	8	2	no	—
47	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
48	Dic₆₁	#135	1	1	yes	characteristic
49	#2177	—	61	1	no	maximal, Sylow
50	C₂×C₆₁	—	8	2	no	—
51	C₂×C₆₁	C₄×D₈	1	1	yes	characteristic
52	D₆₁	—	8	2	no	—
53	D₆₁	#509	1	2	yes	—
54	C₂×D₁₆	—	61	1	no	—
55	C₄×D₈	—	61	2	no	—
56	#367	—	61	1	no	—
57	#357	—	61	2	no	—
58	#346	—	61	1	no	—
59	C₆₁	#2177	1	1	yes	characteristic, complemented, Sylow
60	C₂×D₈	—	61	2	no	—
61	C₄×D₄	—	122	2	no	—
62	D₁₆	—	61	2	no	—
63	D₁₆	—	61	1	no	—
64	D₁₆	#6923	1	1	yes	characteristic, complemented
65	C₂×C₁₆	—	61	1	no	—
66	C₂×C₁₆	—	61	1	no	—
67	#107	—	61	2	no	—
68	#102	—	122	2	no	—
69	C₄×C₈	—	61	1	no	—
70	C₂×D₄	—	122	2	no	—
71	C₂²×C₄	—	244	2	no	—
72	D₈	—	61	2	no	—
73	D₈	—	122	2	no	—
74	D₈	C₂×#6923	1	2	yes	complemented
75	C₂×C₈	—	61	1	no	—
76	C₂×C₈	—	61	1	no	—
77	C₄⋊C₄	—	122	2	no	—
78	C₂²⋊C₄	—	244	2	no	—
79	C₄²	—	61	1	no	—
80	C₁₆	—	61	1	no	—
81	C₁₆	—	122	1	no	—
82	C₁₆	C₂×#6923	1	1	yes	characteristic, complemented
83	C₂³	—	244	2	no	—
84	D₄	—	2	2	no	—
85	D₄	—	122	2	no	—
86	D₄	—	244	2	no	—
87	C₂×C₄	—	61	1	no	—
88	C₂×C₄	—	61	2	no	—
89	C₂×C₄	—	488	4	no	—
90	C₂×C₄	—	244	2	no	—
91	C₈	—	61	1	no	—
92	C₈	—	122	1	no	—
93	C₈	C₂²×#6923	1	1	yes	characteristic
94	C₂²	—	4	2	no	—
95	C₂²	—	61	1	no	—
96	C₂²	—	488	2	no	—
97	C₂²	—	488	2	no	—
98	C₂²	—	244	2	no	—
99	C₄	—	61	1	no	—
100	C₄	—	122	2	no	—
101	C₄	—	488	2	no	—
102	C₄	D₄×#6923	1	1	yes	characteristic
103	C₂	—	8	2	no	—
104	C₂	—	61	2	no	—
105	C₂	—	488	2	no	—
106	C₂	D₈×#6923	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₈×#6923

C₂

yes

maximal, complemented

#16840689

C₂

yes

maximal, characteristic, complemented

#16840674

C₂

yes

maximal, complemented

#16840649

C₂

yes

maximal, characteristic, complemented

D₁₆×D₆₁

C₂

yes

maximal, characteristic

D₄×#6923

—

D₆₁⋊C₁₆

C₂²

yes

characteristic

#14519846

C₂²

yes

—

#14519839

—

C₈×#6923

C₂²

yes

characteristic

D₈×D₆₁

C₂²

yes

—

D₉₇₆

C₄

yes

characteristic, complemented

#14519718

C₄

yes

complemented

C₁₆×D₆₁

C₂²

yes

characteristic

D₁₆×C₆₁

C₄

yes

characteristic, complemented

C₂²×#6923

—

D₄×D₆₁

—

C₂²⋊#6923

—

C₄⋊#6923

—

C₄×#6923

D₄

yes

characteristic

D₆₁⋊C₈

D₄

yes

characteristic

D₈×C₆₁

C₂×C₄

yes

complemented

#11758199

—

D₄₈₈

C₂×C₄

yes

complemented

C₈×D₆₁

C₂³

yes

characteristic

#11758187

—

C₁₆×C₆₁

C₂×C₄

yes

characteristic, complemented

#11758185

C₂×C₄

yes

characteristic, complemented

C₂²×D₆₁

—

C₂×#6923

—

C₂×#6923

—

C₂×#6923

D₈

yes

—

D₄×C₆₁

—

C₂²⋊D₆₁

—

D₂₄₄

—

C₄×D₆₁

C₂×D₄

yes

characteristic

#92420

—

C₈×C₆₁

C₂²×C₄

yes

yes

C₂²×C₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

C₂×D₈

yes

characteristic

#6923

—

#6923

—

C₄×C₆₁

C₄×D₄

yes

Dic₆₁

yes

—

C₂×C₆₁

—

C₂×C₆₁

C₄×D₈

yes

characteristic

D₆₁

—

D₆₁

#509

yes

—

C₂×D₁₆

—

C₄×D₈

—

#367

—

#357

—

#346

—

C₆₁

#2177

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₄×D₄

—

122

—

D₁₆

—

D₁₆

—

D₁₆

#6923

yes

characteristic, complemented

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#102

—

122

—

C₄×C₈

—

C₂×D₄

—

122

—

C₂²×C₄

—

244

—

D₈

—

D₈

—

122

—

D₈

C₂×#6923

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

122

—

C₂²⋊C₄

—

244

—

C₄²

—

C₁₆

—

C₁₆

—

122

—

C₁₆

C₂×#6923

yes

characteristic, complemented

C₂³

—

244

—

D₄

—

D₄

—

122

—

D₄

—

244

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

488

—

C₂×C₄

—

244

—

C₈

—

C₈

—

122

—

C₈

C₂²×#6923

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

488

—

C₂²

—

488

—

C₂²

—

244

—

C₄

—

100

C₄

—

122

—

101

C₄

—

488

—

102

C₄

D₄×#6923

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

488

—

106

C₂

D₈×#6923

yes

characteristic

	subgroup	G/H
center	C₂	D₈×#6923
commutator	C₈×C₆₁	C₂²×C₄
Frattini	C₈	C₂²×#6923
Fitting	D₁₆×C₆₁	C₄
socle	C₂×C₆₁	C₄×D₈
hypercenter	D₁₆	#6923
hypocenter	C₆₁	#2177
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

subgroup

G/H

center

C₂

D₈×#6923

commutator

C₈×C₆₁

C₂²×C₄

Frattini

C₈

C₂²×#6923

Fitting

D₁₆×C₆₁

C₄

socle

C₂×C₆₁

C₄×D₈

D₁₆

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

N	H ≅ G/N	type
1. D₈×#6923	C₂	semidirect
2. #16840689	C₂	semidirect
3. #16840674	C₂	semidirect
4. #16840649	C₂	semidirect
12. D₉₇₆	C₄	semidirect
13. #14519718	C₄	semidirect
15. D₁₆×C₆₁	C₄	semidirect
22. D₈×C₆₁	C₂×C₄	semidirect
24. D₄₈₈	C₂×C₄	semidirect
27. C₁₆×C₆₁	C₂×C₄	semidirect
28. #11758185	C₂×C₄	semidirect
59. C₆₁	#2177	semidirect
64. D₁₆	#6923	semidirect
74. D₈	C₂×#6923	semidirect
82. C₁₆	C₂×#6923	semidirect

H ≅ G/N

type

1. D₈×#6923

C₂

semidirect

2. #16840689

C₂

semidirect

3. #16840674

C₂

semidirect

4. #16840649

C₂

semidirect

12. D₉₇₆

C₄

semidirect

13. #14519718

C₄

semidirect

15. D₁₆×C₆₁

C₄

semidirect

22. D₈×C₆₁

C₂×C₄

semidirect

24. D₄₈₈

C₂×C₄

semidirect

27. C₁₆×C₆₁

C₂×C₄

semidirect

28. #11758185

C₂×C₄

semidirect

59. C₆₁

#2177

semidirect

64. D₁₆

#6923

semidirect

74. D₈

C₂×#6923

semidirect

82. C₁₆

C₂×#6923

semidirect

Subgroups — #21462217

Conjugacy classes of subgroups

Complements