Group #36361832 — subgroups

Subgroups — #36361832

Order 22144.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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	#26011430	C₂	1	3	yes	maximal
2	#26011424	C₂	1	6	yes	maximal
3	#26011411	C₂	1	3	yes	maximal
4	#26011359	C₂	1	2	yes	maximal
5	#384×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₄₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₇₃	C₂²	1	3	yes	—
8	C₂×#14583805	C₂²	1	6	yes	—
9	#19149586	C₂²	1	6	yes	—
10	#19149586	C₂²	1	3	yes	—
11	#19149586	C₂²	1	3	yes	—
12	#19149586	C₂²	1	6	yes	—
13	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	3	yes	—
14	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	3	yes	—
15	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₄₆	—	4	3	no	—
17	C₂²×Dic₁₇₃	C₂³	1	6	yes	—
18	C₂²×Dic₁₇₃	C₂³	1	2	yes	—
19	C₄⋊Dic₁₇₃	—	4	3	no	—
20	#14583805	—	4	6	no	—
21	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₇₃	C₂³	1	3	yes	—
23	C₂²×C₄×C₁₇₃	C₂³	1	3	yes	—
24	C₄⋊C₄×C₁₇₃	—	4	3	no	—
25	C₄⋊C₄×C₁₇₃	—	4	3	no	—
26	C₄²×C₁₇₃	Q₈	1	4	yes	complemented
27	Dic₃₄₆	—	4	4	no	—
28	C₂×Dic₁₇₃	—	2	6	no	—
29	C₂×Dic₁₇₃	—	2	6	no	—
30	C₂×Dic₁₇₃	—	2	12	no	—
31	C₂³×C₁₇₃	C₂⁴	1	1	yes	characteristic
32	C₂×C₄×C₁₇₃	—	2	3	no	—
33	C₂×C₄×C₁₇₃	—	2	3	no	—
34	C₂×C₄×C₁₇₃	—	2	3	no	—
35	C₂×C₄×C₁₇₃	—	2	6	no	—
36	C₂×C₄×C₁₇₃	—	2	3	no	—
37	C₂×C₄×C₁₇₃	C₂×Q₈	1	6	yes	—
38	C₂²×C₁₇₃	C₂²×Q₈	1	1	yes	characteristic
39	C₂²×C₁₇₃	H(5,𝔽₂)	1	3	yes	—
40	C₂²×C₁₇₃	2₋¹⁺⁴	1	3	yes	—
41	C₄×C₁₇₃	—	2	6	no	—
42	C₄×C₁₇₃	—	4	3	no	—
43	C₄×C₁₇₃	—	4	1	no	—
44	Dic₁₇₃	—	4	2	no	—
45	Dic₁₇₃	—	4	6	no	—
46	C₂×C₁₇₃	#544	1	3	yes	—
47	C₂×C₁₇₃	#560	1	1	yes	characteristic
48	C₂×C₁₇₃	#563	1	3	yes	—
49	C₁₇₃	#2843	1	1	yes	characteristic, complemented, Sylow
50	#2843	—	173	1	no	maximal, Sylow
51	#400	—	173	3	no	—
52	#398	—	173	6	no	—
53	#395	—	173	3	no	—
54	#384	D₁₇₃	1	1	yes	characteristic
55	#383	—	173	2	no	—
56	C₂²×Q₈	—	173	1	no	—
57	C₂×C₄⋊C₄	—	173	6	no	—
58	C₂×C₄⋊C₄	—	173	3	no	—
59	C₂×C₄⋊C₄	C₂×D₁₇₃	1	3	yes	—
60	C₂×C₄⋊C₄	C₂×D₁₇₃	1	3	yes	—
61	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
62	#95	—	173	6	no	—
63	#95	—	173	6	no	—
64	#95	—	173	3	no	—
65	#95	—	173	3	no	—
66	C₂×Q₈	—	692	3	no	—
67	C₂²×C₄	—	173	6	no	—
68	C₂²×C₄	—	173	2	no	—
69	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₁₇₃	1	3	yes	—
71	C₂²×C₄	C₂²×D₁₇₃	1	3	yes	—
72	C₄⋊C₄	—	4	3	no	—
73	C₄⋊C₄	—	4	3	no	—
74	C₄⋊C₄	—	692	6	no	—
75	C₄⋊C₄	—	692	3	no	—
76	C₄²	Dic₃₄₆	1	4	yes	complemented
77	C₂³	C₂³×D₁₇₃	1	1	yes	characteristic
78	Q₈	—	692	4	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	2	3	no	—
84	C₂×C₄	—	346	6	no	—
85	C₂×C₄	—	346	6	no	—
86	C₂×C₄	—	346	12	no	—
87	C₂×C₄	C₂×Dic₃₄₆	1	6	yes	—
88	C₂²	C₂²×Dic₃₄₆	1	1	yes	characteristic
89	C₂²	#19149724	1	3	yes	—
90	C₂²	#19149727	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	4	1	no	—
93	C₄	—	4	3	no	—
94	C₄	—	692	2	no	—
95	C₄	—	692	6	no	—
96	C₂	#26012122	1	3	yes	—
97	C₂	#26012228	1	1	yes	characteristic
98	C₂	#26012240	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26011430

C₂

yes

maximal

#26011424

C₂

yes

maximal

#26011411

C₂

yes

maximal

#26011359

C₂

yes

maximal

#384×C₁₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₃₄₆

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

C₂×#14583805

C₂²

yes

—

#19149586

C₂²

yes

—

#19149586

C₂²

yes

—

#19149586

C₂²

yes

—

#19149586

C₂²

yes

—

C₂×C₄⋊C₄×C₁₇₃

C₂²

yes

—

C₂×C₄⋊C₄×C₁₇₃

C₂²

yes

—

C₂×C₄²×C₁₇₃

C₂²

yes

characteristic

C₂×Dic₃₄₆

—

C₂²×Dic₁₇₃

C₂³

yes

—

C₂²×Dic₁₇₃

C₂³

yes

—

C₄⋊Dic₁₇₃

—

#14583805

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

C₂²×C₄×C₁₇₃

C₂³

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄²×C₁₇₃

Q₈

yes

complemented

Dic₃₄₆

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂³×C₁₇₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×Q₈

yes

—

C₂²×C₁₇₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₇₃

H(5,𝔽₂)

yes

—

C₂²×C₁₇₃

2₋¹⁺⁴

yes

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

#544

yes

—

C₂×C₁₇₃

#560

yes

characteristic

C₂×C₁₇₃

#563

yes

—

C₁₇₃

#2843

yes

characteristic, complemented, Sylow

#2843

—

173

maximal, Sylow

#400

—

173

—

#398

—

173

—

#395

—

173

—

#384

D₁₇₃

yes

characteristic

#383

—

173

—

C₂²×Q₈

—

173

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄⋊C₄

C₂×D₁₇₃

yes

—

C₂×C₄⋊C₄

C₂×D₁₇₃

yes

—

C₂×C₄²

C₂×D₁₇₃

yes

characteristic

#95

—

173

—

#95

—

173

—

#95

—

173

—

#95

—

173

—

C₂×Q₈

—

692

—

C₂²×C₄

—

173

—

C₂²×C₄

—

173

—

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

C₂²×C₄

C₂²×D₁₇₃

yes

—

C₂²×C₄

C₂²×D₁₇₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

692

—

C₄⋊C₄

—

692

—

C₄²

Dic₃₄₆

yes

complemented

C₂³

C₂³×D₁₇₃

yes

characteristic

Q₈

—

692

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

346

—

C₂×C₄

—

346

—

C₂×C₄

—

346

—

C₂×C₄

C₂×Dic₃₄₆

yes

—

C₂²

C₂²×Dic₃₄₆

yes

characteristic

C₂²

#19149724

yes

—

C₂²

#19149727

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

692

—

C₄

—

692

—

C₂

#26012122

yes

—

C₂

#26012228

yes

characteristic

C₂

#26012240

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇₃
commutator, socle, prime radical, cosocle	C₂³×C₁₇₃	C₂⁴
Fitting	#384×C₁₇₃	C₂
hypercenter	#384	D₁₇₃
hypocenter	C₁₇₃	#2843
LCS	G ▷ C₂³×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₇₃

commutator, socle, prime radical, cosocle

C₂³×C₁₇₃

C₂⁴

Fitting

#384×C₁₇₃

C₂

D₁₇₃

C₁₇₃

G ▷ C₂³×C₁₇₃ ▷ C₁₇₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
26. C₄²×C₁₇₃	Q₈	semidirect
49. C₁₇₃	#2843	semidirect
76. C₄²	Dic₃₄₆	semidirect

H ≅ G/N

type

26. C₄²×C₁₇₃

Q₈

semidirect

49. C₁₇₃

#2843

semidirect

76. C₄²

Dic₃₄₆

semidirect

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

subgroup	index
#26011430	2
#26011424	2
#26011411	2
#26011359	2
#384×C₁₇₃	2
#2843	173

subgroup

index

#384×C₁₇₃

173

Subgroups — #36361832

Conjugacy classes of subgroups

Complements

Maximal subgroups