Group #22612156 — subgroups

Subgroups — #22612156

Order 9344.

160 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 61 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	#17988964	C₂	1	1	yes	maximal, characteristic
2	#17988387	C₂	1	4	yes	maximal, complemented
3	C₄×#11924003	C₂	1	1	yes	maximal, characteristic, complemented
4	#500×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₅₈₄	—	2	1	no	—
6	C₂×D₅₈₄	C₂²	1	2	yes	—
7	#14556588	—	2	1	no	—
8	C₂×#11924003	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₇₃	—	2	1	no	—
10	#107×C₇₃	—	2	1	no	—
11	#107×C₇₃	C₂²	1	2	yes	—
12	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×D₂₉₂	—	2	2	no	—
14	C₂×D₂₉₂	—	4	1	no	—
15	C₂×D₂₉₂	—	4	1	no	—
16	D₅₈₄	—	2	4	no	—
17	D₅₈₄	—	4	2	no	—
18	#11924003	—	2	2	no	—
19	C₂×Q₈×C₇₃	—	4	1	no	—
20	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₇₃	—	2	2	no	—
23	C₄⋊C₄×C₇₃	—	4	1	no	—
24	C₄²×C₇₃	D₄	1	1	yes	characteristic
25	C₂²×D₇₃	—	4	2	no	—
26	Q₈×C₇₃	—	8	2	no	—
27	C₂×C₄×C₇₃	—	4	2	no	—
28	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
30	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
31	D₂₉₂	—	4	2	no	—
32	D₂₉₂	—	4	2	no	—
33	D₂₉₂	—	8	2	no	—
34	D₂₉₂	—	8	2	no	—
35	C₈×C₇₃	SD₁₆	1	2	yes	—
36	C₈×C₇₃	#41	1	2	yes	—
37	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
38	C₂×D₇₃	—	8	2	no	—
39	C₂×D₇₃	—	8	2	no	—
40	C₂×D₇₃	—	8	2	no	—
41	C₄×C₇₃	—	8	2	no	—
42	C₄×C₇₃	D₁₆	1	2	yes	—
43	C₄×C₇₃	SD₃₂	1	2	yes	—
44	C₄×C₇₃	#124	1	1	yes	characteristic
45	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
46	C₂×C₇₃	#486	1	1	yes	characteristic
47	C₂×C₇₃	C₂×D₁₆	1	1	yes	characteristic
48	C₂×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
49	D₇₃	—	16	2	no	—
50	#2240	—	73	1	no	maximal, Sylow
51	C₇₃	#2240	1	1	yes	characteristic, complemented, Sylow
52	#500	D₇₃	1	1	yes	characteristic, complemented
53	#493	—	73	1	no	—
54	#357	—	73	4	no	—
55	C₄×C₁₆	—	73	1	no	—
56	C₂×D₈	—	73	2	no	—
57	C₂×D₈	—	146	1	no	—
58	C₄⋊Q₈	—	2	1	no	—
59	#127	—	146	1	no	—
60	C₂×C₁₆	—	73	2	no	—
61	#107	—	2	1	no	—
62	#107	C₂×D₇₃	1	2	yes	—
63	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
64	C₂×Q₈	—	4	1	no	—
65	C₂×D₄	—	292	1	no	—
66	C₂×D₄	—	292	1	no	—
67	C₂×D₄	—	146	2	no	—
68	D₈	—	292	2	no	—
69	D₈	—	146	4	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	1	no	—
74	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
75	C₁₆	—	146	2	no	—
76	C₂³	—	292	2	no	—
77	Q₈	—	8	2	no	—
78	D₄	—	292	2	no	—
79	D₄	—	292	2	no	—
80	D₄	—	584	2	no	—
81	D₄	—	584	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	#11924013	1	1	yes	characteristic
84	C₂×C₄	#11924013	1	1	yes	characteristic
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	2	no	—
89	C₂²	—	584	2	no	—
90	C₂²	—	584	2	no	—
91	C₂²	C₂×#11924013	1	1	yes	characteristic
92	C₄	—	8	2	no	—
93	C₄	#14556532	1	2	yes	—
94	C₄	#14556534	1	2	yes	—
95	C₄	#14556580	1	1	yes	characteristic
96	C₄	C₂×#11924015	1	1	yes	characteristic
97	C₂	—	1168	2	no	—
98	C₂	#17988935	1	1	yes	characteristic
99	C₂	C₂×#14556532	1	1	yes	characteristic
100	C₂	C₂×#14556534	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988964

C₂

yes

maximal, characteristic

#17988387

C₂

yes

maximal, complemented

C₄×#11924003

C₂

yes

maximal, characteristic, complemented

#500×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×D₅₈₄

—

C₂×D₅₈₄

C₂²

yes

—

#14556588

—

C₂×#11924003

C₂²

yes

—

C₄⋊Q₈×C₇₃

—

#107×C₇₃

—

#107×C₇₃

C₂²

yes

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

D₅₈₄

—

#11924003

—

C₂×Q₈×C₇₃

—

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

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

SD₁₆

yes

—

C₈×C₇₃

#41

yes

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

D₁₆

yes

—

C₄×C₇₃

SD₃₂

yes

—

C₄×C₇₃

#124

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₇₃

#486

yes

characteristic

C₂×C₇₃

C₂×D₁₆

yes

characteristic

C₂×C₇₃

C₂×SD₃₂

yes

characteristic

D₇₃

—

#2240

—

maximal, Sylow

C₇₃

#2240

yes

characteristic, complemented, Sylow

#500

D₇₃

yes

characteristic, complemented

#493

—

#357

—

C₄×C₁₆

—

C₂×D₈

—

C₂×D₈

—

146

—

C₄⋊Q₈

—

#127

—

146

—

C₂×C₁₆

—

#107

—

#107

C₂×D₇₃

yes

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

C₂×D₄

—

146

—

D₈

—

292

—

D₈

—

146

—

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₇₃

yes

characteristic

C₁₆

—

146

—

C₂³

—

292

—

Q₈

—

D₄

—

292

—

D₄

—

292

—

D₄

—

584

—

D₄

—

584

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

C₈

#11924015

yes

—

C₈

#11924029

yes

—

C₂²

—

584

—

C₂²

—

584

—

C₂²

—

584

—

C₂²

C₂×#11924013

yes

characteristic

C₄

—

C₄

#14556532

yes

—

C₄

#14556534

yes

—

C₄

yes

C₄

yes

C₂

—

1168

—

C₂

yes

C₂

yes

100

C₂

C₂×#14556534

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11924013
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#500×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	#500	D₇₃
hypocenter	C₇₃	#2240
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

subgroup

G/H

center

C₂²

C₂×#11924013

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

#500×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

N	H ≅ G/N	type
2. #17988387	C₂	semidirect
3. C₄×#11924003	C₂	semidirect
4. #500×C₇₃	C₂	semidirect
51. C₇₃	#2240	semidirect
52. #500	D₇₃	semidirect

H ≅ G/N

type

2. #17988387

C₂

semidirect

3. C₄×#11924003

C₂

semidirect

4. #500×C₇₃

C₂

semidirect

51. C₇₃

#2240

semidirect

52. #500

D₇₃

semidirect

Subgroups — #22612156

Conjugacy classes of subgroups

Complements