Group #36352889 — subgroups

Subgroups — #36352889

Order 22144.

172 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 45 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	C₁₇₃⋊#564	C₂	1	1	yes	maximal, characteristic
2	#26011897	C₂	1	2	yes	maximal
3	#26011233	C₂	1	1	yes	maximal, characteristic
4	#26011232	C₂	1	2	yes	maximal
5	#491×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	#19149671	C₂²	1	2	yes	—
7	#19149670	—	2	2	no	—
8	#19149669	—	2	2	no	—
9	#19149669	—	2	2	no	—
10	#19149612	—	2	2	no	—
11	#19149610	—	2	2	no	—
12	#19149590	C₂²	1	2	yes	—
13	C₄⋊Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
14	#125×C₁₇₃	C₂²	1	1	yes	characteristic
15	#103×C₁₇₃	—	2	2	no	—
16	#97×C₁₇₃	C₂²	1	1	yes	characteristic
17	C₂×#13063436	—	2	2	no	—
18	C₄⋊Dic₁₇₃	—	2	2	no	—
19	C₄⋊Dic₁₇₃	D₄	1	2	yes	—
20	C₄⋊Dic₁₇₃	D₄	1	2	yes	—
21	#14583805	—	2	2	no	—
22	#14583805	—	2	2	no	—
23	#14583805	—	2	2	no	—
24	#14583805	—	2	2	no	—
25	C₄×Dic₁₇₃	—	2	2	no	—
26	C₂×Q₈×C₁₇₃	—	2	1	no	—
27	C₂×C₈×C₁₇₃	—	2	1	no	—
28	C₄⋊C₄×C₁₇₃	—	2	2	no	—
29	C₄⋊C₄×C₁₇₃	—	2	2	no	—
30	C₄⋊C₄×C₁₇₃	D₄	1	2	yes	—
31	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₇₃	—	2	2	no	—
33	C₂×Dic₁₇₃	—	2	2	no	—
34	#13063436	—	2	4	no	—
35	Q₈×C₁₇₃	—	4	2	no	—
36	C₂×C₄×C₁₇₃	—	2	2	no	—
37	C₂×C₄×C₁₇₃	—	4	1	no	—
38	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
40	C₈×C₁₇₃	—	4	1	no	—
41	C₂²×C₁₇₃	#120	1	1	yes	characteristic
42	C₄×C₁₇₃	—	2	1	no	—
43	C₄×C₁₇₃	—	2	2	no	—
44	C₄×C₁₇₃	—	4	2	no	—
45	C₄×C₁₇₃	—	8	1	no	—
46	Dic₁₇₃	—	4	2	no	—
47	Dic₁₇₃	—	4	2	no	—
48	C₂×C₁₇₃	#454	1	1	yes	characteristic
49	C₂×C₁₇₃	#456	1	2	yes	—
50	C₁₇₃	#1663	1	1	yes	characteristic, complemented, Sylow
51	#1663	—	173	1	no	maximal, Sylow
52	#564	—	173	1	no	—
53	#501	—	173	2	no	—
54	#491	D₁₇₃	1	1	yes	characteristic
55	#330	—	173	1	no	—
56	#330	—	173	2	no	—
57	C₄⋊Q₈	C₂×D₁₇₃	1	1	yes	characteristic
58	#125	—	346	2	no	—
59	#125	—	346	2	no	—
60	#125	—	346	2	no	—
61	#125	—	173	2	no	—
62	#125	C₂×D₁₇₃	1	1	yes	characteristic
63	#107	—	346	2	no	—
64	#106	—	346	2	no	—
65	#103	—	2	2	no	—
66	#97	—	173	2	no	—
67	#97	C₂×D₁₇₃	1	1	yes	characteristic
68	C₂×Q₈	—	2	1	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	346	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	346	2	no	—
74	C₄⋊C₄	—	346	2	no	—
75	C₄⋊C₄	—	346	2	no	—
76	C₄⋊C₄	—	346	2	no	—
77	C₄⋊C₄	—	346	2	no	—
78	C₄⋊C₄	—	173	2	no	—
79	C₄⋊C₄	—	173	2	no	—
80	C₄⋊C₄	C₂²⋊D₁₇₃	1	2	yes	—
81	C₄²	—	346	2	no	—
82	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
83	Q₈	—	4	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	4	1	no	—
86	C₂×C₄	—	346	2	no	—
87	C₂×C₄	—	346	2	no	—
88	C₂×C₄	D₄×D₁₇₃	1	2	yes	—
89	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
90	C₈	—	4	1	no	—
91	C₈	—	346	4	no	—
92	C₂²	#19149647	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	8	1	no	—
97	C₄	—	692	2	no	—
98	C₄	—	692	2	no	—
99	C₂	#26011663	1	1	yes	characteristic
100	C₂	#26011672	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇₃⋊#564

C₂

yes

maximal, characteristic

#26011897

C₂

yes

maximal

#26011233

C₂

yes

maximal, characteristic

#26011232

C₂

yes

maximal

#491×C₁₇₃

C₂

yes

maximal, characteristic

#19149671

C₂²

yes

—

—

—

—

—

—

C₂²

yes

—

C₄⋊Q₈×C₁₇₃

C₂²

yes

characteristic

#125×C₁₇₃

C₂²

yes

characteristic

#103×C₁₇₃

—

#97×C₁₇₃

C₂²

yes

characteristic

C₂×#13063436

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

D₄

yes

—

C₄⋊Dic₁₇₃

D₄

yes

—

—

—

—

—

C₄×Dic₁₇₃

—

C₂×Q₈×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

D₄

yes

—

C₄²×C₁₇₃

C₂³

yes

characteristic

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

#13063436

—

Q₈×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₈×C₁₇₃

—

C₂²×C₁₇₃

#120

yes

characteristic

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

#454

yes

characteristic

C₂×C₁₇₃

#456

yes

—

C₁₇₃

#1663

yes

characteristic, complemented, Sylow

#1663

—

173

maximal, Sylow

#564

—

173

—

#501

—

173

—

#491

D₁₇₃

yes

characteristic

#330

—

173

—

#330

—

173

—

C₄⋊Q₈

C₂×D₁₇₃

yes

characteristic

#125

—

346

—

#125

—

346

—

#125

—

346

—

#125

—

173

—

#125

C₂×D₁₇₃

yes

characteristic

#107

—

346

—

#106

—

346

—

#103

—

#97

—

173

—

#97

C₂×D₁₇₃

yes

characteristic

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

346

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

173

—

C₄⋊C₄

—

173

—

C₄⋊C₄

C₂²⋊D₁₇₃

yes

—

C₄²

—

346

—

C₄²

C₂²×D₁₇₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

346

—

C₂×C₄

—

346

—

C₂×C₄

D₄×D₁₇₃

yes

—

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

characteristic

C₈

—

C₈

—

346

—

C₂²

#19149647

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

692

—

C₄

—

692

—

C₂

#26011663

yes

characteristic

100

C₂

#26011672

yes

—

	subgroup	G/H
center	C₂²	#19149647
commutator, cosocle	C₄²×C₁₇₃	C₂³
Frattini	C₄²	C₂²×D₁₇₃
Fitting	#491×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	#120
hypercenter	#491	D₁₇₃
hypocenter	C₁₇₃	#1663
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #491

subgroup

G/H

center

C₂²

#19149647

commutator, cosocle

C₄²×C₁₇₃

C₂³

Frattini

C₄²

C₂²×D₁₇₃

Fitting

#491×C₁₇₃

C₂

C₂²×C₁₇₃

D₁₇₃

C₁₇₃

G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #491

N	H ≅ G/N	type
50. C₁₇₃	#1663	semidirect

H ≅ G/N

type

50. C₁₇₃

#1663

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₇₃⋊#564	2
#26011897	2
#26011233	2
#26011232	2
#491×C₁₇₃	2
#1663	173

subgroup

index

C₁₇₃⋊#564

#491×C₁₇₃

173

Subgroups — #36352889

Conjugacy classes of subgroups

Complements

Maximal subgroups