Group #36367711 — subgroups

Subgroups — #36367711

Order 22144.

930 conjugacy classes of subgroups; only the 451 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 451 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	#26012245	C₂	1	10	yes	maximal, complemented
2	#26012239	C₂	1	5	yes	maximal, complemented
3	#26012047	C₂	1	5	yes	maximal, complemented
4	#26012026	C₂	1	10	yes	maximal, complemented
5	#564×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19149678	C₂²	1	10	yes	—
7	#19149674	C₂²	1	20	yes	—
8	#19149671	C₂²	1	5	yes	—
9	#19149667	C₂²	1	10	yes	—
10	#19149664	C₂²	1	20	yes	—
11	#19149659	C₂²	1	10	yes	—
12	#19149657	C₂²	1	20	yes	—
13	#19149654	C₂²	1	10	yes	—
14	#19149644	C₂²	1	10	yes	—
15	#19149641	C₂²	1	10	yes	—
16	#19149637	C₂²	1	10	yes	—
17	C₄²×D₁₇₃	C₂²	1	5	yes	—
18	#125×C₁₇₃	C₂²	1	5	yes	—
19	#125×C₁₇₃	C₂²	1	10	yes	—
20	C₂×Dic₃₄₆	C₂³	1	10	yes	—
21	C₂×D₆₉₂	C₂³	1	10	yes	—
22	C₂×C₄×D₁₇₃	C₂³	1	5	yes	—
23	C₂×C₄×D₁₇₃	C₂³	1	10	yes	—
24	C₄⋊Dic₁₇₃	C₂³	1	10	yes	—
25	#14583805	C₂³	1	20	yes	—
26	#14583803	C₂³	1	20	yes	—
27	#14583803	C₂³	1	20	yes	—
28	C₄×Dic₁₇₃	C₂³	1	10	yes	—
29	C₄×Dic₁₇₃	C₂³	1	5	yes	—
30	C₄⋊C₄×C₁₇₃	C₂³	1	10	yes	—
31	C₄⋊C₄×C₁₇₃	C₂³	1	10	yes	—
32	C₄⋊C₄×C₁₇₃	C₂³	1	10	yes	—
33	C₄²×C₁₇₃	C₂³	1	5	yes	—
34	C₂²×D₁₇₃	C₂⁴	1	5	yes	—
35	Dic₃₄₆	—	4	10	no	—
36	D₆₉₂	—	4	10	no	—
37	C₂×Dic₁₇₃	C₂⁴	1	10	yes	—
38	C₂×Dic₁₇₃	C₂⁴	1	1	yes	characteristic
39	C₄×D₁₇₃	—	2	10	no	—
40	C₄×D₁₇₃	—	4	10	no	—
41	C₂×C₄×C₁₇₃	C₂⁴	1	10	yes	—
42	C₂×C₄×C₁₇₃	C₂⁴	1	5	yes	—
43	C₂²×C₁₇₃	C₂⁵	1	1	yes	characteristic
44	C₂×D₁₇₃	—	2	5	no	—
45	C₂×D₁₇₃	—	2	10	no	—
46	C₄×C₁₇₃	—	2	5	no	—
47	C₄×C₁₇₃	—	2	10	no	—
48	Dic₁₇₃	—	2	10	no	—
49	Dic₁₇₃	2₋¹⁺⁴	1	2	yes	—
50	C₂×C₁₇₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
51	C₂×C₁₇₃	#585	1	2	yes	—
52	D₁₇₃	—	4	5	no	—
53	C₁₇₃	#3531	1	1	yes	characteristic, complemented, Sylow
54	#3531	—	173	1	no	maximal, Sylow
55	#564	D₁₇₃	1	1	yes	characteristic, complemented
56	#563	—	173	10	no	—
57	#562	—	173	5	no	—
58	#533	—	173	5	no	—
59	#529	—	173	10	no	—
60	C₄⋊Q₈	—	173	10	no	—
61	#126	—	173	20	no	—
62	#125	—	173	5	no	—
63	#125	C₂×D₁₇₃	1	10	yes	—
64	#125	C₂×D₁₇₃	1	5	yes	—
65	#124	—	173	10	no	—
66	#123	—	173	20	no	—
67	C₂²⋊Q₈	—	173	10	no	—
68	C₂²⋊Q₈	—	173	20	no	—
69	#121	—	173	10	no	—
70	C₄×Q₈	—	173	10	no	—
71	C₄×D₄	—	173	10	no	—
72	#117	—	173	10	no	—
73	C₂×C₄²	—	173	5	no	—
74	C₂×Q₈	—	173	10	no	—
75	C₂×D₄	—	173	10	no	—
76	C₂²×C₄	—	173	5	no	—
77	C₂²×C₄	—	173	10	no	—
78	C₄⋊C₄	—	173	10	no	—
79	C₄⋊C₄	—	173	20	no	—
80	C₄⋊C₄	C₂²×D₁₇₃	1	10	yes	—
81	C₄⋊C₄	C₂²×D₁₇₃	1	10	yes	—
82	C₄⋊C₄	C₂²×D₁₇₃	1	10	yes	—
83	C₂²⋊C₄	—	173	20	no	—
84	C₂²⋊C₄	—	173	20	no	—
85	C₄²	—	173	10	no	—
86	C₄²	—	173	5	no	—
87	C₄²	C₂²×D₁₇₃	1	5	yes	—
88	C₂³	—	173	5	no	—
89	Q₈	—	692	10	no	—
90	D₄	—	692	10	no	—
91	C₂×C₄	—	346	10	no	—
92	C₂×C₄	—	173	10	no	—
93	C₂×C₄	—	173	1	no	—
94	C₂×C₄	—	692	10	no	—
95	C₂×C₄	C₂³×D₁₇₃	1	5	yes	—
96	C₂×C₄	C₂³×D₁₇₃	1	10	yes	—
97	C₂²	—	346	5	no	—
98	C₂²	—	346	10	no	—
99	C₂²	C₂⁴×D₁₇₃	1	1	yes	characteristic
100	C₄	—	2	5	no	—
101	C₄	—	2	10	no	—
102	C₄	—	346	10	no	—
103	C₄	—	173	2	no	—
104	C₂	—	692	5	no	—
105	C₂	2₋¹⁺⁴×D₁₇₃	1	1	yes	characteristic
106	C₂	#26012328	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26012245

C₂

yes

maximal, complemented

#26012239

C₂

yes

maximal, complemented

#26012047

C₂

yes

maximal, complemented

#26012026

C₂

yes

maximal, complemented

#564×C₁₇₃

C₂

yes

maximal, characteristic, complemented

#19149678

C₂²

yes

—

#19149674

C₂²

yes

—

#19149671

C₂²

yes

—

#19149667

C₂²

yes

—

#19149664

C₂²

yes

—

#19149659

C₂²

yes

—

#19149657

C₂²

yes

—

#19149654

C₂²

yes

—

#19149644

C₂²

yes

—

#19149641

C₂²

yes

—

#19149637

C₂²

yes

—

C₄²×D₁₇₃

C₂²

yes

—

#125×C₁₇₃

C₂²

yes

—

#125×C₁₇₃

C₂²

yes

—

C₂×Dic₃₄₆

C₂³

yes

—

C₂×D₆₉₂

C₂³

yes

—

C₂×C₄×D₁₇₃

C₂³

yes

—

C₂×C₄×D₁₇₃

C₂³

yes

—

C₄⋊Dic₁₇₃

C₂³

yes

—

#14583805

C₂³

yes

—

#14583803

C₂³

yes

—

#14583803

C₂³

yes

—

C₄×Dic₁₇₃

C₂³

yes

—

C₄×Dic₁₇₃

C₂³

yes

—

C₄⋊C₄×C₁₇₃

C₂³

yes

—

C₄⋊C₄×C₁₇₃

C₂³

yes

—

C₄⋊C₄×C₁₇₃

C₂³

yes

—

C₄²×C₁₇₃

C₂³

yes

—

C₂²×D₁₇₃

C₂⁴

yes

—

Dic₃₄₆

—

D₆₉₂

—

C₂×Dic₁₇₃

C₂⁴

yes

—

C₂×Dic₁₇₃

C₂⁴

yes

characteristic

C₄×D₁₇₃

—

C₄×D₁₇₃

—

C₂×C₄×C₁₇₃

C₂⁴

yes

—

C₂×C₄×C₁₇₃

C₂⁴

yes

—

C₂²×C₁₇₃

C₂⁵

yes

characteristic

C₂×D₁₇₃

—

C₂×D₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

Dic₁₇₃

—

Dic₁₇₃

2₋¹⁺⁴

yes

—

C₂×C₁₇₃

C₂×2₋¹⁺⁴

yes

characteristic

C₂×C₁₇₃

#585

yes

—

D₁₇₃

—

C₁₇₃

#3531

yes

characteristic, complemented, Sylow

#3531

—

173

maximal, Sylow

#564

D₁₇₃

yes

characteristic, complemented

#563

—

173

—

#562

—

173

—

#533

—

173

—

#529

—

173

—

C₄⋊Q₈

—

173

—

#126

—

173

—

#125

—

173

—

#125

C₂×D₁₇₃

yes

—

#125

C₂×D₁₇₃

yes

—

#124

—

173

—

#123

—

173

—

C₂²⋊Q₈

—

173

—

C₂²⋊Q₈

—

173

—

#121

—

173

—

C₄×Q₈

—

173

—

C₄×D₄

—

173

—

#117

—

173

—

C₂×C₄²

—

173

—

C₂×Q₈

—

173

—

C₂×D₄

—

173

—

C₂²×C₄

—

173

—

C₂²×C₄

—

173

—

C₄⋊C₄

—

173

—

C₄⋊C₄

—

173

—

C₄⋊C₄

C₂²×D₁₇₃

yes

—

C₄⋊C₄

C₂²×D₁₇₃

yes

—

C₄⋊C₄

C₂²×D₁₇₃

yes

—

C₂²⋊C₄

—

173

—

C₂²⋊C₄

—

173

—

C₄²

—

173

—

C₄²

—

173

—

C₄²

C₂²×D₁₇₃

yes

—

C₂³

—

173

—

Q₈

—

692

—

D₄

—

692

—

C₂×C₄

—

346

—

C₂×C₄

—

173

—

C₂×C₄

—

173

—

C₂×C₄

—

692

—

C₂×C₄

C₂³×D₁₇₃

yes

—

C₂×C₄

C₂³×D₁₇₃

yes

—

C₂²

—

346

—

C₂²

—

346

—

C₂²

C₂⁴×D₁₇₃

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

346

—

103

C₄

—

173

—

104

C₂

—

692

—

105

C₂

2₋¹⁺⁴×D₁₇₃

yes

characteristic

106

C₂

#26012328

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₇₃
commutator, socle, cosocle	C₂²×C₁₇₃	C₂⁵
Fitting	#564×C₁₇₃	C₂
hypercenter	#564	D₁₇₃
hypocenter	C₁₇₃	#3531
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₁₇₃

commutator, socle, cosocle

C₂²×C₁₇₃

C₂⁵

Fitting

#564×C₁₇₃

C₂

D₁₇₃

C₁₇₃

G ▷ C₂²×C₁₇₃ ▷ C₁₇₃

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
1. #26012245	C₂	semidirect
2. #26012239	C₂	semidirect
3. #26012047	C₂	semidirect
4. #26012026	C₂	semidirect
5. #564×C₁₇₃	C₂	semidirect
53. C₁₇₃	#3531	semidirect
55. #564	D₁₇₃	semidirect

H ≅ G/N

type

1. #26012245

C₂

semidirect

2. #26012239

C₂

semidirect

3. #26012047

C₂

semidirect

4. #26012026

C₂

semidirect

5. #564×C₁₇₃

C₂

semidirect

53. C₁₇₃

#3531

semidirect

55. #564

D₁₇₃

semidirect

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

subgroup	index
#26012245	2
#26012239	2
#26012047	2
#26012026	2
#564×C₁₇₃	2
#3531	173

subgroup

index

#564×C₁₇₃

173

Subgroups — #36367711

Conjugacy classes of subgroups

Complements

Maximal subgroups