Group #22612165 — subgroups

Subgroups — #22612165

Order 9344.

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

normal subgroups only

This group has 138 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	#17988971	C₂	1	1	yes	maximal, characteristic
2	#17988390	C₂	1	2	yes	maximal, complemented
3	#17988387	C₂	1	2	yes	maximal
4	C₄×#11924003	C₂	1	1	yes	maximal, characteristic, complemented
5	#499×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₉₂	C₂²	1	1	yes	characteristic
7	C₂×#11924017	—	2	1	no	—
8	C₂×D₅₈₄	C₂²	1	1	yes	characteristic
9	#14556579	—	2	1	no	—
10	C₂×#11924003	C₂²	1	2	yes	—
11	#125×C₇₃	—	2	1	no	—
12	#107×C₇₃	C₂²	1	2	yes	—
13	#106×C₇₃	—	2	1	no	—
14	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₄₆	—	2	1	no	—
16	C₂×D₂₉₂	—	2	1	no	—
17	Dic₂₉₂	—	2	2	no	—
18	#11924017	—	4	2	no	—
19	D₅₈₄	—	2	2	no	—
20	#11924005	—	4	2	no	—
21	#11924003	—	2	2	no	—
22	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
23	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₇₃	—	2	2	no	—
25	C₄⋊C₄×C₇₃	—	4	2	no	—
26	C₄²×C₇₃	D₄	1	1	yes	characteristic
27	C₂²×D₇₃	—	4	1	no	—
28	C₂×C₄×C₇₃	—	4	2	no	—
29	C₂×C₄×C₇₃	D₈	1	2	yes	—
30	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₇₃	—	4	1	no	—
32	D₂₉₂	—	4	1	no	—
33	D₂₉₂	—	4	1	no	—
34	Dic₁₄₆	—	4	1	no	—
35	Dic₁₄₆	—	4	1	no	—
36	C₈×C₇₃	SD₁₆	1	2	yes	—
37	C₈×C₇₃	#41	1	2	yes	—
38	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
39	C₂×D₇₃	—	8	1	no	—
40	C₂×D₇₃	—	8	2	no	—
41	C₄×C₇₃	—	2	2	no	—
42	C₄×C₇₃	—	8	2	no	—
43	C₄×C₇₃	#124	1	1	yes	characteristic
44	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
45	Dic₇₃	—	8	1	no	—
46	C₂×C₇₃	#486	1	1	yes	characteristic
47	C₂×C₇₃	#508	1	2	yes	—
48	D₇₃	—	16	1	no	—
49	#2242	—	73	1	no	maximal, Sylow
50	C₇₃	#2242	1	1	yes	characteristic, complemented, Sylow
51	#499	D₇₃	1	1	yes	characteristic, complemented
52	#495	—	73	1	no	—
53	#358	—	73	2	no	—
54	#357	—	73	2	no	—
55	C₄×C₁₆	—	73	1	no	—
56	C₂×Q₁₆	—	73	1	no	—
57	C₂×SD₁₆	—	146	1	no	—
58	C₂×D₈	—	73	1	no	—
59	#125	—	2	1	no	—
60	#124	—	146	1	no	—
61	C₂×C₁₆	—	73	2	no	—
62	#107	C₂×D₇₃	1	2	yes	—
63	#106	—	2	1	no	—
64	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
65	C₂×Q₈	—	146	1	no	—
66	C₂×D₄	—	146	1	no	—
67	Q₁₆	—	146	2	no	—
68	SD₁₆	—	292	2	no	—
69	D₈	—	146	2	no	—
70	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
71	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	4	2	no	—
74	C₂²⋊C₄	—	292	2	no	—
75	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
76	C₁₆	—	146	2	no	—
77	C₂³	—	292	1	no	—
78	Q₈	—	292	1	no	—
79	Q₈	—	292	1	no	—
80	D₄	—	292	1	no	—
81	D₄	—	292	1	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	292	1	no	—
84	C₂×C₄	#11924013	1	2	yes	—
85	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
86	C₈	#11924015	1	2	yes	—
87	C₈	#11924029	1	2	yes	—
88	C₂²	—	584	1	no	—
89	C₂²	—	584	2	no	—
90	C₂²	C₂×#11924013	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	8	2	no	—
93	C₄	—	584	1	no	—
94	C₄	#14556580	1	1	yes	characteristic
95	C₄	C₂×#11924015	1	1	yes	characteristic
96	C₂	—	1168	1	no	—
97	C₂	#17988935	1	1	yes	characteristic
98	C₂	#17989017	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17988971

C₂

yes

maximal, characteristic

#17988390

C₂

yes

maximal, complemented

#17988387

C₂

yes

maximal

C₄×#11924003

C₂

yes

maximal, characteristic, complemented

#499×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₉₂

C₂²

yes

characteristic

C₂×#11924017

—

C₂×D₅₈₄

C₂²

yes

characteristic

#14556579

—

C₂×#11924003

C₂²

yes

—

#125×C₇₃

—

#107×C₇₃

C₂²

yes

—

#106×C₇₃

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

Dic₂₉₂

—

#11924017

—

D₅₈₄

—

#11924005

—

#11924003

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

D₄

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

SD₁₆

yes

—

C₈×C₇₃

#41

yes

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#124

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

Dic₇₃

—

C₂×C₇₃

#486

yes

characteristic

C₂×C₇₃

#508

yes

—

D₇₃

—

#2242

—

maximal, Sylow

C₇₃

#2242

yes

characteristic, complemented, Sylow

#499

D₇₃

yes

characteristic, complemented

#495

—

#358

—

#357

—

C₄×C₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

146

—

C₂×D₈

—

#125

—

#124

—

146

—

C₂×C₁₆

—

#107

C₂×D₇₃

yes

—

#106

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×Q₈

—

146

—

C₂×D₄

—

146

—

Q₁₆

—

146

—

SD₁₆

—

292

—

D₈

—

146

—

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

292

—

C₄²

C₂²⋊D₇₃

yes

characteristic

C₁₆

—

146

—

C₂³

—

292

—

Q₈

—

292

—

Q₈

—

292

—

D₄

—

292

—

D₄

—

292

—

C₂×C₄

—

C₂×C₄

—

292

—

C₂×C₄

#11924013

yes

—

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

C₈

#11924015

yes

—

C₈

#11924029

yes

—

C₂²

—

584

—

C₂²

—

584

—

C₂²

C₂×#11924013

yes

characteristic

C₄

—

C₄

—

C₄

—

584

—

C₄

yes

C₄

yes

C₂

—

1168

—

C₂

#17988935

yes

characteristic

C₂

#17989017

yes

—

	subgroup	G/H
center	C₂²	C₂×#11924013
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#499×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	#499	D₇₃
hypocenter	C₇₃	#2242
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

subgroup

G/H

center

C₂²

C₂×#11924013

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

#499×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

N	H ≅ G/N	type
2. #17988390	C₂	semidirect
4. C₄×#11924003	C₂	semidirect
5. #499×C₇₃	C₂	semidirect
50. C₇₃	#2242	semidirect
51. #499	D₇₃	semidirect

H ≅ G/N

type

2. #17988390

C₂

semidirect

4. C₄×#11924003

C₂

semidirect

5. #499×C₇₃

C₂

semidirect

50. C₇₃

#2242

semidirect

51. #499

D₇₃

semidirect

Subgroups — #22612165

Conjugacy classes of subgroups

Complements