Group #22611811 — subgroups

Subgroups — #22611811

Order 9344.

146 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 53 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	#17989019	C₂	1	1	yes	maximal, characteristic
2	#17988703	C₂	1	2	yes	maximal, complemented
3	#17988393	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#368×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14556613	C₂²	1	2	yes	—
7	#14556599	—	2	2	no	—
8	Dic₅₈₄	C₄	1	1	yes	characteristic, complemented
9	#14556536	C₄	1	2	yes	complemented
10	D₁₁₆₈	C₄	1	1	yes	characteristic, complemented
11	C₂×#11924003	C₂²	1	1	yes	characteristic
12	#14556517	—	2	2	no	—
13	C₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	#108×C₇₃	C₂²	1	2	yes	—
16	#11924028	—	2	2	no	—
17	Dic₂₉₂	C₂×C₄	1	2	yes	—
18	#11924017	—	2	2	no	—
19	D₅₈₄	C₂×C₄	1	2	yes	—
20	#11924011	—	4	2	no	—
21	C₂×#10596831	D₄	1	1	yes	characteristic
22	C₈×D₇₃	—	4	2	no	—
23	C₄×Dic₇₃	D₄	1	1	yes	characteristic
24	#11924003	—	2	1	no	—
25	M₁₆×C₇₃	—	2	2	no	—
26	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
27	C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
30	C₂²⋊D₇₃	—	4	2	no	—
31	C₂×Dic₇₃	—	2	1	no	—
32	D₂₉₂	—	2	2	no	—
33	C₄×D₇₃	—	4	2	no	—
34	Dic₁₄₆	—	2	2	no	—
35	C₈×C₇₃	—	4	2	no	—
36	C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₈×C₇₃	#41	1	1	yes	characteristic
38	#10596831	D₈	1	1	yes	characteristic
39	#10596831	D₈	1	1	yes	characteristic
40	C₂²×C₇₃	#135	1	1	yes	characteristic
41	C₂×D₇₃	—	4	2	no	—
42	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
43	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
44	Dic₇₃	—	2	2	no	—
45	Dic₇₃	—	4	2	no	—
46	C₂×C₇₃	—	2	1	no	—
47	C₂×C₇₃	C₄×D₈	1	1	yes	characteristic
48	D₇₃	—	8	2	no	—
49	#2178	—	73	1	no	maximal, Sylow
50	C₇₃	#2178	1	1	yes	characteristic, complemented, Sylow
51	#508	—	73	1	no	—
52	#443	—	73	2	no	—
53	#368	D₇₃	1	1	yes	characteristic, complemented
54	#359	—	73	2	no	—
55	C₄×C₁₆	—	73	1	no	—
56	#135	—	73	2	no	—
57	#131	—	146	2	no	—
58	Q₃₂	—	73	1	no	—
59	SD₃₂	—	73	2	no	—
60	D₁₆	—	73	1	no	—
61	C₂×C₁₆	—	73	1	no	—
62	C₂×C₁₆	C₂×D₇₃	1	1	yes	characteristic
63	#108	C₂×D₇₃	1	2	yes	—
64	#104	—	146	2	no	—
65	C₄×C₈	—	73	1	no	—
66	#41	—	146	2	no	—
67	Q₁₆	—	73	2	no	—
68	SD₁₆	—	146	2	no	—
69	D₈	—	73	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	292	2	no	—
72	C₂×C₈	—	292	2	no	—
73	C₂×C₈	—	73	1	no	—
74	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
75	C₄²	—	73	1	no	—
76	C₁₆	—	146	1	no	—
77	C₁₆	C₄×D₇₃	1	1	yes	characteristic
78	C₁₆	C₄×D₇₃	1	1	yes	characteristic
79	Q₈	—	146	2	no	—
80	D₄	—	292	2	no	—
81	D₄	—	146	2	no	—
82	C₂×C₄	—	292	2	no	—
83	C₂×C₄	—	146	1	no	—
84	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
85	C₈	—	4	2	no	—
86	C₈	—	73	1	no	—
87	C₈	—	73	1	no	—
88	C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
89	C₈	#11924029	1	1	yes	characteristic
90	C₂²	—	292	2	no	—
91	C₂²	#14556614	1	1	yes	characteristic
92	C₄	—	292	2	no	—
93	C₄	—	146	2	no	—
94	C₄	#14556553	1	1	yes	characteristic
95	C₄	D₈×D₇₃	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	584	2	no	—
98	C₂	#17988669	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989019

C₂

yes

maximal, characteristic

#17988703

C₂

yes

maximal, complemented

#17988393

C₂

yes

maximal, complemented

C₁₆×Dic₇₃

C₂

yes

maximal, characteristic, complemented

#368×C₇₃

C₂

yes

maximal, characteristic, complemented

#14556613

C₂²

yes

—

#14556599

—

Dic₅₈₄

C₄

yes

characteristic, complemented

#14556536

C₄

yes

complemented

D₁₁₆₈

C₄

yes

characteristic, complemented

C₂×#11924003

C₂²

yes

characteristic

#14556517

—

C₈×Dic₇₃

C₂²

yes

characteristic

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

#108×C₇₃

C₂²

yes

—

#11924028

—

Dic₂₉₂

C₂×C₄

yes

—

#11924017

—

D₅₈₄

C₂×C₄

yes

—

#11924011

—

C₂×#10596831

D₄

yes

characteristic

C₈×D₇₃

—

C₄×Dic₇₃

D₄

yes

characteristic

#11924003

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂²×C₄

yes

C₈×C₇₃

yes

D₈

yes

characteristic

#10596831

D₈

yes

characteristic

C₂²×C₇₃

#135

yes

characteristic

C₂×D₇₃

—

C₄×C₇₃

C₄×D₄

yes

characteristic

C₄×C₇₃

C₂×D₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₄×D₈

yes

characteristic

D₇₃

—

#2178

—

maximal, Sylow

C₇₃

#2178

yes

characteristic, complemented, Sylow

#508

—

#443

—

#368

D₇₃

yes

characteristic, complemented

#359

—

C₄×C₁₆

—

#135

—

#131

—

146

—

Q₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₃

yes

characteristic

#108

C₂×D₇₃

yes

—

#104

—

146

—

C₄×C₈

—

#41

—

146

—

Q₁₆

—

SD₁₆

—

146

—

D₈

—

M₁₆

—

M₁₆

—

292

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄²

—

C₁₆

—

146

—

C₁₆

C₄×D₇₃

yes

characteristic

C₁₆

C₄×D₇₃

yes

characteristic

Q₈

—

146

—

D₄

—

292

—

D₄

—

146

—

C₂×C₄

—

292

—

C₂×C₄

—

146

—

C₂×C₄

D₄×D₇₃

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

C₂×C₄×D₇₃

yes

characteristic

C₈

#11924029

yes

characteristic

C₂²

—

292

—

C₂²

#14556614

yes

characteristic

C₄

—

292

—

C₄

—

146

—

C₄

#14556553

yes

characteristic

C₄

D₈×D₇₃

yes

characteristic

C₂

—

C₂

—

584

—

C₂

#17988669

yes

characteristic

	subgroup	G/H
center	C₄	D₈×D₇₃
commutator	C₈×C₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#368×C₇₃	C₂
socle	C₂×C₇₃	C₄×D₈
hypercenter	#368	D₇₃
hypocenter	C₇₃	#2178
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

subgroup

G/H

center

C₄

D₈×D₇₃

commutator

C₈×C₇₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

#368×C₇₃

C₂

socle

C₂×C₇₃

C₄×D₈

D₇₃

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

N	H ≅ G/N	type
2. #17988703	C₂	semidirect
3. #17988393	C₂	semidirect
4. C₁₆×Dic₇₃	C₂	semidirect
5. #368×C₇₃	C₂	semidirect
8. Dic₅₈₄	C₄	semidirect
9. #14556536	C₄	semidirect
10. D₁₁₆₈	C₄	semidirect
50. C₇₃	#2178	semidirect
53. #368	D₇₃	semidirect

H ≅ G/N

type

2. #17988703

C₂

semidirect

3. #17988393

C₂

semidirect

4. C₁₆×Dic₇₃

C₂

semidirect

5. #368×C₇₃

C₂

semidirect

8. Dic₅₈₄

C₄

semidirect

9. #14556536

C₄

semidirect

10. D₁₁₆₈

C₄

semidirect

50. C₇₃

#2178

semidirect

53. #368

D₇₃

semidirect

Subgroups — #22611811

Conjugacy classes of subgroups

Complements