Group #22612102 — subgroups

Subgroups — #22612102

Order 9344.

126 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 49 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	#17988985	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988411	C₂	1	2	yes	maximal, complemented
3	#17988408	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988397	C₂	1	2	yes	maximal
5	#443×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#14556591	—	2	1	no	—
7	#14556531	C₂²	1	2	yes	—
8	#14556523	—	2	1	no	—
9	#14556523	C₂²	1	2	yes	—
10	#135×C₇₃	C₂²	1	1	yes	characteristic
11	#131×C₇₃	—	2	1	no	—
12	#108×C₇₃	C₂²	1	1	yes	characteristic
13	#104×C₇₃	—	2	1	no	—
14	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₄₆	—	4	1	no	—
16	C₄⋊Dic₇₃	—	2	2	no	—
17	C₄⋊Dic₇₃	—	4	1	no	—
18	#11924003	—	2	2	no	—
19	#41×C₇₃	—	2	1	no	—
20	Q₁₆×C₇₃	Q₈	1	1	yes	characteristic, complemented
21	SD₁₆×C₇₃	—	2	1	no	—
22	D₈×C₇₃	Q₈	1	1	yes	characteristic, complemented
23	M₁₆×C₇₃	—	2	1	no	—
24	M₁₆×C₇₃	—	4	1	no	—
25	C₂×C₈×C₇₃	—	4	1	no	—
26	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
27	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
28	C₄²×C₇₃	D₄	1	1	yes	characteristic
29	Q₈×C₇₃	—	2	1	no	—
30	D₄×C₇₃	—	2	1	no	—
31	D₄×C₇₃	—	4	1	no	—
32	C₂×C₄×C₇₃	—	2	1	no	—
33	C₂×C₄×C₇₃	—	4	1	no	—
34	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₇₃	—	4	2	no	—
36	Dic₁₄₆	—	8	2	no	—
37	C₈×C₇₃	—	4	1	no	—
38	C₈×C₇₃	SD₁₆	1	1	yes	characteristic
39	C₈×C₇₃	SD₁₆	1	1	yes	characteristic
40	C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
41	C₈×C₇₃	#41	1	1	yes	characteristic
42	C₂²×C₇₃	—	4	1	no	—
43	C₂²×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
44	C₄×C₇₃	—	2	2	no	—
45	C₄×C₇₃	—	4	1	no	—
46	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
47	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
48	Dic₇₃	—	8	2	no	—
49	C₂×C₇₃	—	2	1	no	—
50	C₂×C₇₃	—	8	1	no	—
51	C₂×C₇₃	#476	1	1	yes	characteristic
52	#2230	—	73	1	no	maximal, Sylow
53	C₇₃	#2230	1	1	yes	characteristic, complemented, Sylow
54	#498	—	73	1	no	—
55	#443	D₇₃	1	1	yes	characteristic
56	#365	—	73	2	no	—
57	#364	—	73	1	no	—
58	#360	—	73	2	no	—
59	#135	C₂×D₇₃	1	1	yes	characteristic
60	#131	—	2	1	no	—
61	C₄⋊Q₈	—	146	1	no	—
62	M₃₂	—	73	2	no	—
63	#108	C₂×D₇₃	1	1	yes	characteristic
64	#106	—	73	2	no	—
65	#106	—	146	1	no	—
66	#104	—	2	1	no	—
67	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
68	#41	—	2	1	no	—
69	C₂×Q₈	—	292	1	no	—
70	Q₁₆	Dic₁₄₆	1	1	yes	characteristic, complemented
71	SD₁₆	—	2	1	no	—
72	D₈	Dic₁₄₆	1	1	yes	characteristic, complemented
73	M₁₆	—	2	1	no	—
74	M₁₆	—	4	1	no	—
75	C₂×C₈	—	4	1	no	—
76	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
77	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
78	C₄⋊C₄	—	292	1	no	—
79	C₄⋊C₄	—	146	2	no	—
80	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
81	C₁₆	—	146	2	no	—
82	Q₈	—	2	1	no	—
83	Q₈	—	584	2	no	—
84	D₄	—	2	1	no	—
85	D₄	—	4	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	—	292	2	no	—
89	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
90	C₈	—	4	1	no	—
91	C₈	#11924016	1	1	yes	characteristic
92	C₈	#11924016	1	1	yes	characteristic
93	C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
94	C₈	#11924028	1	1	yes	characteristic
95	C₂²	—	4	1	no	—
96	C₂²	#14556616	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	4	1	no	—
99	C₄	—	584	2	no	—
100	C₄	#14556570	1	1	yes	characteristic
101	C₄	C₂×#11924016	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	8	1	no	—
104	C₂	#17988877	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988985

C₂

yes

maximal, characteristic, complemented

#17988411

C₂

yes

maximal, complemented

#17988408

C₂

yes

maximal, characteristic, complemented

#17988397

C₂

yes

maximal

#443×C₇₃

C₂

yes

maximal, characteristic

#14556591

—

#14556531

C₂²

yes

—

#14556523

—

#14556523

C₂²

yes

—

#135×C₇₃

C₂²

yes

characteristic

#131×C₇₃

—

#108×C₇₃

C₂²

yes

characteristic

#104×C₇₃

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

—

#41×C₇₃

—

Q₁₆×C₇₃

Q₈

yes

characteristic, complemented

SD₁₆×C₇₃

—

D₈×C₇₃

Q₈

yes

characteristic, complemented

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

SD₁₆

yes

characteristic

C₈×C₇₃

SD₁₆

yes

characteristic

C₈×C₇₃

C₂×Q₈

yes

characteristic

C₈×C₇₃

#41

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

AGL(1,ℤ/8)

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#498

—

#443

D₇₃

yes

—

—

—

C₂×D₇₃

yes

characteristic

#131

—

C₄⋊Q₈

—

146

—

M₃₂

—

#108

C₂×D₇₃

yes

characteristic

#106

—

#106

—

146

—

#104

—

C₄×C₈

C₂×D₇₃

yes

characteristic

#41

—

C₂×Q₈

—

292

—

Q₁₆

Dic₁₄₆

yes

characteristic, complemented

SD₁₆

—

D₈

Dic₁₄₆

yes

characteristic, complemented

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₄²

C₂²⋊D₇₃

yes

characteristic

C₁₆

—

146

—

Q₈

—

Q₈

—

584

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

292

—

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

C₈

—

C₈

yes

C₈

yes

C₈

C₂×Dic₁₄₆

yes

characteristic

C₈

#11924028

yes

characteristic

C₂²

—

C₂²

#14556616

yes

characteristic

C₄

—

C₄

—

C₄

—

584

—

100

C₄

#14556570

yes

characteristic

101

C₄

C₂×#11924016

yes

characteristic

102

C₂

—

103

C₂

—

104

C₂

#17988877

yes

characteristic

	subgroup	G/H
center	C₂	#17988877
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#443×C₇₃	C₂
socle	C₂×C₇₃	#476
hypercenter	#443	D₇₃
hypocenter	C₇₃	#2230
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

subgroup

G/H

center

C₂

#17988877

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

#443×C₇₃

C₂

C₂×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

N	H ≅ G/N	type
1. #17988985	C₂	semidirect
2. #17988411	C₂	semidirect
3. #17988408	C₂	semidirect
20. Q₁₆×C₇₃	Q₈	semidirect
22. D₈×C₇₃	Q₈	semidirect
53. C₇₃	#2230	semidirect
70. Q₁₆	Dic₁₄₆	semidirect
72. D₈	Dic₁₄₆	semidirect

H ≅ G/N

type

1. #17988985

C₂

semidirect

2. #17988411

C₂

semidirect

3. #17988408

C₂

semidirect

20. Q₁₆×C₇₃

Q₈

semidirect

22. D₈×C₇₃

Q₈

semidirect

53. C₇₃

#2230

semidirect

70. Q₁₆

Dic₁₄₆

semidirect

72. D₈

Dic₁₄₆

semidirect

Subgroups — #22612102

Conjugacy classes of subgroups

Complements