Group #22626329 — subgroups

Subgroups — #22626329

Order 9344.

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×#64408	C₂	1	2	yes	maximal, complemented
2	#17989514	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989499	C₂	1	2	yes	maximal, complemented
4	#17989474	C₂	1	1	yes	maximal, characteristic, complemented
5	D₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
6	D₄×#64408	—	2	2	no	—
7	#14556664	C₂²	1	1	yes	characteristic
8	#14556660	C₂²	1	2	yes	—
9	#14556653	—	2	2	no	—
10	C₈×#64408	C₂²	1	1	yes	characteristic
11	D₈×D₇₃	C₂²	1	2	yes	—
12	D₁₁₆₈	C₄	1	1	yes	characteristic, complemented
13	#14556532	C₄	1	2	yes	complemented
14	C₁₆×D₇₃	C₂²	1	1	yes	characteristic
15	D₁₆×C₇₃	C₄	1	1	yes	characteristic, complemented
16	C₂²×#64408	—	4	2	no	—
17	#11924036	D₄	1	1	yes	characteristic
18	C₄⋊#64408	—	2	2	no	—
19	C₂²⋊#64408	—	4	2	no	—
20	C₄×#64408	D₄	1	1	yes	characteristic
21	#11924031	—	2	1	no	—
22	D₄×D₇₃	—	2	2	no	—
23	D₅₈₄	C₂×C₄	1	2	yes	complemented
24	#11924013	—	2	2	no	—
25	C₈×D₇₃	C₂³	1	1	yes	characteristic
26	#11924003	C₂×C₄	1	1	yes	characteristic, complemented
27	D₈×C₇₃	C₂×C₄	1	2	yes	complemented
28	C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
29	C₂²×D₇₃	—	4	2	no	—
30	C₂×#64408	—	4	2	no	—
31	C₂×#64408	—	8	4	no	—
32	C₂×#64408	D₈	1	2	yes	—
33	D₄×C₇₃	—	2	2	no	—
34	C₂²⋊D₇₃	—	4	2	no	—
35	D₂₉₂	—	2	2	no	—
36	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
37	#10596833	—	2	1	no	—
38	C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
39	#10596831	#41	1	1	yes	characteristic
40	C₂²×C₇₃	—	4	2	no	—
41	C₂×D₇₃	—	4	2	no	—
42	C₂×D₇₃	—	8	2	no	—
43	C₂×D₇₃	—	8	2	no	—
44	C₂×D₇₃	C₂×D₈	1	1	yes	characteristic
45	#64408	—	2	2	no	—
46	#64408	—	8	2	no	—
47	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
48	Dic₇₃	#135	1	1	yes	characteristic
49	C₂×C₇₃	—	8	2	no	—
50	C₂×C₇₃	C₄×D₈	1	1	yes	characteristic
51	D₇₃	—	8	2	no	—
52	D₇₃	#509	1	2	yes	—
53	#2177	—	73	1	no	maximal, Sylow
54	C₇₃	#2177	1	1	yes	characteristic, complemented, Sylow
55	C₂×D₁₆	—	73	1	no	—
56	C₄×D₈	—	73	2	no	—
57	#367	—	73	1	no	—
58	#357	—	73	2	no	—
59	#346	—	73	1	no	—
60	C₂×D₈	—	73	2	no	—
61	C₄×D₄	—	146	2	no	—
62	D₁₆	—	73	2	no	—
63	D₁₆	—	73	1	no	—
64	D₁₆	#64408	1	1	yes	characteristic, complemented
65	C₂×C₁₆	—	73	1	no	—
66	C₂×C₁₆	—	73	1	no	—
67	#107	—	73	2	no	—
68	#102	—	146	2	no	—
69	C₄×C₈	—	73	1	no	—
70	C₂×D₄	—	146	2	no	—
71	C₂²×C₄	—	292	2	no	—
72	D₈	—	73	2	no	—
73	D₈	—	146	2	no	—
74	D₈	C₂×#64408	1	2	yes	complemented
75	C₂×C₈	—	73	1	no	—
76	C₂×C₈	—	73	1	no	—
77	C₄⋊C₄	—	146	2	no	—
78	C₂²⋊C₄	—	292	2	no	—
79	C₄²	—	73	1	no	—
80	C₁₆	—	73	1	no	—
81	C₁₆	—	146	1	no	—
82	C₁₆	C₂×#64408	1	1	yes	characteristic, complemented
83	C₂³	—	292	2	no	—
84	D₄	—	2	2	no	—
85	D₄	—	292	2	no	—
86	D₄	—	146	2	no	—
87	C₂×C₄	—	292	2	no	—
88	C₂×C₄	—	584	4	no	—
89	C₂×C₄	—	73	1	no	—
90	C₂×C₄	—	73	2	no	—
91	C₈	—	73	1	no	—
92	C₈	—	146	1	no	—
93	C₈	C₂²×#64408	1	1	yes	characteristic
94	C₂²	—	4	2	no	—
95	C₂²	—	292	2	no	—
96	C₂²	—	584	2	no	—
97	C₂²	—	584	2	no	—
98	C₂²	—	73	1	no	—
99	C₄	—	584	2	no	—
100	C₄	—	73	1	no	—
101	C₄	—	146	2	no	—
102	C₄	D₄×#64408	1	1	yes	characteristic
103	C₂	—	8	2	no	—
104	C₂	—	584	2	no	—
105	C₂	—	73	2	no	—
106	C₂	D₈×#64408	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₈×#64408

C₂

yes

maximal, complemented

#17989514

C₂

yes

maximal, characteristic, complemented

#17989499

C₂

yes

maximal, complemented

#17989474

C₂

yes

maximal, characteristic, complemented

D₁₆×D₇₃

C₂

yes

maximal, characteristic

D₄×#64408

—

#14556664

C₂²

yes

characteristic

#14556660

C₂²

yes

—

#14556653

—

C₈×#64408

C₂²

yes

characteristic

D₈×D₇₃

C₂²

yes

—

D₁₁₆₈

C₄

yes

characteristic, complemented

#14556532

C₄

yes

complemented

C₁₆×D₇₃

C₂²

yes

characteristic

D₁₆×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

#11924036

D₄

yes

—

—

D₄

yes

characteristic

#11924031

—

D₄×D₇₃

—

D₅₈₄

C₂×C₄

yes

complemented

#11924013

—

C₈×D₇₃

C₂³

yes

characteristic

#11924003

C₂×C₄

yes

characteristic, complemented

D₈×C₇₃

C₂×C₄

yes

complemented

C₁₆×C₇₃

C₂×C₄

yes

characteristic, complemented

C₂²×D₇₃

—

C₂×#64408

—

C₂×#64408

—

C₂×#64408

D₈

yes

—

D₄×C₇₃

—

C₂²⋊D₇₃

—

D₂₉₂

—

C₄×D₇₃

C₂×D₄

yes

characteristic

#10596833

—

C₈×C₇₃

C₂²×C₄

yes

yes

C₂²×C₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂×D₈

yes

characteristic

#64408

—

#64408

—

C₄×C₇₃

C₄×D₄

yes

characteristic

Dic₇₃

#135

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

C₄×D₈

yes

characteristic

D₇₃

—

D₇₃

#509

yes

—

#2177

—

maximal, Sylow

C₇₃

#2177

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₄×D₈

—

#367

—

#357

—

#346

—

C₂×D₈

—

C₄×D₄

—

146

—

D₁₆

—

D₁₆

—

D₁₆

#64408

yes

characteristic, complemented

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#102

—

146

—

C₄×C₈

—

C₂×D₄

—

146

—

C₂²×C₄

—

292

—

D₈

—

D₈

—

146

—

D₈

C₂×#64408

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

292

—

C₄²

—

C₁₆

—

C₁₆

—

146

—

C₁₆

C₂×#64408

yes

characteristic, complemented

C₂³

—

292

—

D₄

—

D₄

—

292

—

D₄

—

146

—

C₂×C₄

—

292

—

C₂×C₄

—

584

—

C₂×C₄

—

C₂×C₄

—

C₈

—

C₈

—

146

—

C₈

C₂²×#64408

yes

characteristic

C₂²

—

C₂²

—

292

—

C₂²

—

584

—

C₂²

—

584

—

C₂²

—

C₄

—

584

—

100

C₄

—

101

C₄

—

146

—

102

C₄

D₄×#64408

yes

characteristic

103

C₂

—

104

C₂

—

584

—

105

C₂

—

106

C₂

D₈×#64408

yes

characteristic

	subgroup	G/H
center	C₂	D₈×#64408
commutator	C₈×C₇₃	C₂²×C₄
Frattini	C₈	C₂²×#64408
Fitting	D₁₆×C₇₃	C₄
socle	C₂×C₇₃	C₄×D₈
hypercenter	D₁₆	#64408
hypocenter	C₇₃	#2177
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

subgroup

G/H

center

C₂

D₈×#64408

commutator

C₈×C₇₃

C₂²×C₄

Frattini

C₈

C₂²×#64408

Fitting

D₁₆×C₇₃

C₄

socle

C₂×C₇₃

C₄×D₈

D₁₆

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

N	H ≅ G/N	type
1. D₈×#64408	C₂	semidirect
2. #17989514	C₂	semidirect
3. #17989499	C₂	semidirect
4. #17989474	C₂	semidirect
12. D₁₁₆₈	C₄	semidirect
13. #14556532	C₄	semidirect
15. D₁₆×C₇₃	C₄	semidirect
23. D₅₈₄	C₂×C₄	semidirect
26. #11924003	C₂×C₄	semidirect
27. D₈×C₇₃	C₂×C₄	semidirect
28. C₁₆×C₇₃	C₂×C₄	semidirect
54. C₇₃	#2177	semidirect
64. D₁₆	#64408	semidirect
74. D₈	C₂×#64408	semidirect
82. C₁₆	C₂×#64408	semidirect

H ≅ G/N

type

1. D₈×#64408

C₂

semidirect

2. #17989514

C₂

semidirect

3. #17989499

C₂

semidirect

4. #17989474

C₂

semidirect

12. D₁₁₆₈

C₄

semidirect

13. #14556532

C₄

semidirect

15. D₁₆×C₇₃

C₄

semidirect

23. D₅₈₄

C₂×C₄

semidirect

26. #11924003

C₂×C₄

semidirect

27. D₈×C₇₃

C₂×C₄

semidirect

28. C₁₆×C₇₃

C₂×C₄

semidirect

54. C₇₃

#2177

semidirect

64. D₁₆

#64408

semidirect

74. D₈

C₂×#64408

semidirect

82. C₁₆

C₂×#64408

semidirect

Subgroups — #22626329

Conjugacy classes of subgroups

Complements