Group #22612245 — subgroups

Subgroups — #22612245

Order 9344.

112 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 35 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	#17989020	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988424	C₂	1	1	yes	maximal, characteristic, complemented
4	D₂₃₃₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988422	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988420	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₆₄×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14556614	—	2	1	no	—
9	D₈×D₇₃	—	2	1	no	—
10	#14556534	—	2	1	no	—
11	D₁₁₆₈	C₂²	1	1	yes	characteristic
12	D₁₁₆₈	C₂²	1	1	yes	characteristic, complemented
13	#14556532	—	2	1	no	—
14	C₁₆×D₇₃	C₂²	1	1	yes	characteristic
15	#14556497	C₂²	1	1	yes	characteristic, complemented
16	Q₃₂×C₇₃	C₂²	1	1	yes	characteristic, complemented
17	D₁₆×C₇₃	C₂²	1	1	yes	characteristic
18	C₃₂×C₇₃	C₂²	1	1	yes	characteristic, complemented
19	#11924029	—	4	1	no	—
20	D₄×D₇₃	—	4	1	no	—
21	#11924015	—	4	1	no	—
22	D₅₈₄	—	2	1	no	—
23	D₅₈₄	—	2	1	no	—
24	#11924013	—	4	1	no	—
25	C₈×D₇₃	D₄	1	1	yes	characteristic
26	#11924003	D₄	1	1	yes	characteristic
27	Q₁₆×C₇₃	—	2	1	no	—
28	D₈×C₇₃	—	2	1	no	—
29	C₁₆×C₇₃	C₂³	1	1	yes	characteristic
30	C₂²×D₇₃	—	8	1	no	—
31	Q₈×C₇₃	—	4	1	no	—
32	D₄×C₇₃	—	4	1	no	—
33	C₂²⋊D₇₃	—	8	1	no	—
34	D₂₉₂	—	4	1	no	—
35	D₂₉₂	—	4	1	no	—
36	D₂₉₂	—	8	1	no	—
37	C₄×D₇₃	—	8	1	no	—
38	C₄×D₇₃	D₈	1	1	yes	characteristic
39	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
40	#10596831	D₈	1	1	yes	characteristic
41	C₂²×C₇₃	—	8	1	no	—
42	C₂×D₇₃	—	8	1	no	—
43	C₂×D₇₃	—	8	1	no	—
44	C₂×D₇₃	—	16	2	no	—
45	C₂×D₇₃	D₁₆	1	1	yes	characteristic
46	C₄×C₇₃	—	8	1	no	—
47	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
48	Dic₇₃	D₁₆	1	1	yes	characteristic
49	C₂×C₇₃	—	16	1	no	—
50	C₂×C₇₃	C₂×D₁₆	1	1	yes	characteristic
51	D₇₃	—	2	1	no	—
52	D₇₃	—	16	1	no	—
53	D₇₃	—	16	1	no	—
54	#2263	—	73	1	no	maximal, Sylow
55	C₇₃	#2263	1	1	yes	characteristic, complemented, Sylow
56	#508	—	73	1	no	—
57	C₂×D₁₆	—	73	1	no	—
58	SD₆₄	—	73	1	no	—
59	SD₆₄	D₇₃	1	1	yes	characteristic, complemented
60	D₃₂	—	73	1	no	—
61	D₃₂	—	73	1	no	—
62	M₆₄	—	73	1	no	—
63	#135	—	146	1	no	—
64	C₂×D₈	—	146	1	no	—
65	Q₃₂	C₂×D₇₃	1	1	yes	characteristic, complemented
66	SD₃₂	—	146	1	no	—
67	D₁₆	—	73	1	no	—
68	D₁₆	—	73	1	no	—
69	D₁₆	—	146	1	no	—
70	D₁₆	C₂×D₇₃	1	1	yes	characteristic
71	C₂×C₁₆	—	73	1	no	—
72	C₃₂	—	73	1	no	—
73	C₃₂	C₂×D₇₃	1	1	yes	characteristic, complemented
74	#41	—	292	1	no	—
75	C₂×D₄	—	292	1	no	—
76	Q₁₆	—	2	1	no	—
77	SD₁₆	—	292	1	no	—
78	D₈	—	2	1	no	—
79	D₈	—	292	1	no	—
80	D₈	—	146	1	no	—
81	D₈	—	146	1	no	—
82	C₂×C₈	—	73	1	no	—
83	C₁₆	—	73	1	no	—
84	C₁₆	C₂²×D₇₃	1	1	yes	characteristic
85	C₂³	—	584	1	no	—
86	Q₈	—	4	1	no	—
87	D₄	—	4	1	no	—
88	D₄	—	292	1	no	—
89	D₄	—	292	1	no	—
90	D₄	—	584	1	no	—
91	D₄	—	584	1	no	—
92	C₂×C₄	—	584	1	no	—
93	C₂×C₄	—	73	1	no	—
94	C₈	—	73	1	no	—
95	C₈	D₄×D₇₃	1	1	yes	characteristic
96	C₂²	—	8	1	no	—
97	C₂²	—	584	1	no	—
98	C₂²	—	584	1	no	—
99	C₂²	—	73	1	no	—
100	C₂²	—	1168	2	no	—
101	C₄	—	8	1	no	—
102	C₄	—	73	1	no	—
103	C₄	D₈×D₇₃	1	1	yes	characteristic
104	C₂	—	16	1	no	—
105	C₂	—	1168	1	no	—
106	C₂	—	1168	1	no	—
107	C₂	—	146	1	no	—
108	C₂	D₁₆×D₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989020

C₂

yes

maximal, characteristic, complemented

D₁₆×D₇₃

C₂

yes

maximal, characteristic, complemented

#17988424

C₂

yes

maximal, characteristic, complemented

D₂₃₃₆

C₂

yes

maximal, characteristic, complemented

#17988422

C₂

yes

maximal, characteristic, complemented

#17988420

C₂

yes

maximal, characteristic, complemented

SD₆₄×C₇₃

C₂

yes

maximal, characteristic, complemented

#14556614

—

D₈×D₇₃

—

#14556534

—

D₁₁₆₈

C₂²

yes

characteristic

D₁₁₆₈

C₂²

yes

characteristic, complemented

#14556532

—

C₁₆×D₇₃

C₂²

yes

characteristic

#14556497

C₂²

yes

characteristic, complemented

Q₃₂×C₇₃

C₂²

yes

characteristic, complemented

D₁₆×C₇₃

C₂²

yes

characteristic

C₃₂×C₇₃

C₂²

yes

characteristic, complemented

#11924029

—

D₄×D₇₃

—

#11924015

—

D₅₈₄

—

D₅₈₄

—

#11924013

—

C₈×D₇₃

D₄

yes

characteristic

#11924003

D₄

yes

characteristic

Q₁₆×C₇₃

—

D₈×C₇₃

—

C₁₆×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

C₂²⋊D₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

D₈

yes

characteristic

C₈×C₇₃

C₂×D₄

yes

characteristic

#10596831

D₈

yes

characteristic

C₂²×C₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₂×D₈

yes

characteristic

Dic₇₃

D₁₆

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

C₂×D₁₆

yes

characteristic

D₇₃

—

D₇₃

—

D₇₃

—

#2263

—

maximal, Sylow

C₇₃

#2263

yes

characteristic, complemented, Sylow

#508

—

C₂×D₁₆

—

SD₆₄

—

SD₆₄

D₇₃

yes

characteristic, complemented

D₃₂

—

D₃₂

—

M₆₄

—

#135

—

146

—

C₂×D₈

—

146

—

Q₃₂

C₂×D₇₃

yes

characteristic, complemented

SD₃₂

—

146

—

D₁₆

—

D₁₆

—

D₁₆

—

146

—

D₁₆

C₂×D₇₃

yes

characteristic

C₂×C₁₆

—

C₃₂

—

C₃₂

C₂×D₇₃

yes

characteristic, complemented

#41

—

292

—

C₂×D₄

—

292

—

Q₁₆

—

SD₁₆

—

292

—

D₈

—

D₈

—

292

—

D₈

—

146

—

D₈

—

146

—

C₂×C₈

—

C₁₆

—

C₁₆

C₂²×D₇₃

yes

characteristic

C₂³

—

584

—

Q₈

—

D₄

—

D₄

—

292

—

D₄

—

292

—

D₄

—

584

—

D₄

—

584

—

C₂×C₄

—

584

—

C₂×C₄

—

C₈

—

C₈

D₄×D₇₃

yes

characteristic

C₂²

—

C₂²

—

584

—

C₂²

—

584

—

C₂²

—

100

C₂²

—

1168

—

101

C₄

—

102

C₄

—

103

C₄

D₈×D₇₃

yes

characteristic

104

C₂

—

105

C₂

—

1168

—

106

C₂

—

1168

—

107

C₂

—

146

—

108

C₂

D₁₆×D₇₃

yes

characteristic

	subgroup	G/H
center	C₂	D₁₆×D₇₃
commutator	C₁₆×C₇₃	C₂³
Frattini	C₁₆	C₂²×D₇₃
Fitting	SD₆₄×C₇₃	C₂
socle	C₂×C₇₃	C₂×D₁₆
hypercenter	SD₆₄	D₇₃
hypocenter	C₇₃	#2263
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ SD₆₄

subgroup

G/H

center

C₂

D₁₆×D₇₃

commutator

C₁₆×C₇₃

C₂³

Frattini

C₁₆

C₂²×D₇₃

Fitting

SD₆₄×C₇₃

C₂

socle

C₂×C₇₃

C₂×D₁₆

hypercenter

SD₆₄

D₇₃

hypocenter

C₇₃

#2263

LCS

G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ SD₆₄

N	H ≅ G/N	type
1. #17989020	C₂	semidirect
2. D₁₆×D₇₃	C₂	semidirect
3. #17988424	C₂	semidirect
4. D₂₃₃₆	C₂	semidirect
5. #17988422	C₂	semidirect
6. #17988420	C₂	semidirect
7. SD₆₄×C₇₃	C₂	semidirect
12. D₁₁₆₈	C₂²	semidirect
15. #14556497	C₂²	semidirect
16. Q₃₂×C₇₃	C₂²	semidirect
18. C₃₂×C₇₃	C₂²	semidirect
55. C₇₃	#2263	semidirect
59. SD₆₄	D₇₃	semidirect
65. Q₃₂	C₂×D₇₃	semidirect
73. C₃₂	C₂×D₇₃	semidirect

H ≅ G/N

type

1. #17989020

C₂

semidirect

2. D₁₆×D₇₃

C₂

semidirect

3. #17988424

C₂

semidirect

4. D₂₃₃₆

C₂

semidirect

5. #17988422

C₂

semidirect

6. #17988420

C₂

semidirect

7. SD₆₄×C₇₃

C₂

semidirect

12. D₁₁₆₈

C₂²

semidirect

15. #14556497

C₂²

semidirect

16. Q₃₂×C₇₃

C₂²

semidirect

18. C₃₂×C₇₃

C₂²

semidirect

55. C₇₃

#2263

semidirect

59. SD₆₄

D₇₃

semidirect

65. Q₃₂

C₂×D₇₃

semidirect

73. C₃₂

C₂×D₇₃

semidirect

Subgroups — #22612245

Conjugacy classes of subgroups

Complements