Subgroups — #36368566

Order 22144.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012383	C₂	1	2	yes	maximal, complemented
2	#26011857	C₂	1	1	yes	maximal, characteristic
3	#19149753	—	2	2	no	—
4	#19149676	C₄	1	1	yes	characteristic
5	#19149599	—	2	2	no	—
6	#19149590	C₂²	1	1	yes	characteristic
7	#125×C₁₇₃	C₄	1	1	yes	characteristic
8	#14583829	—	2	4	no	—
9	C₂×D₆₉₂	—	2	1	no	—
10	C₂×D₆₉₂	—	4	1	no	—
11	C₂×#13063436	D₄	1	2	yes	—
12	C₄⋊C₄×C₁₇₃	—	2	1	no	—
13	C₄⋊C₄×C₁₇₃	—	4	1	no	—
14	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×D₁₇₃	—	4	1	no	—
16	D₆₉₂	—	4	2	no	—
17	D₆₉₂	—	8	2	no	—
18	#13063436	—	2	2	no	—
19	C₂×C₄×C₁₇₃	—	2	1	no	—
20	C₂×C₄×C₁₇₃	—	4	1	no	—
21	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂²×C₁₇₃	#100	1	1	yes	characteristic
23	C₂×D₁₇₃	—	8	2	no	—
24	C₂×D₁₇₃	—	8	1	no	—
25	C₄×C₁₇₃	—	4	1	no	—
26	C₄×C₁₇₃	—	8	1	no	—
27	C₄×C₁₇₃	#104	1	2	yes	—
28	C₂×C₁₇₃	—	2	1	no	—
29	C₂×C₁₇₃	#329	1	1	yes	characteristic
30	D₁₇₃	—	16	1	no	—
31	C₁₇₃	#1355	1	1	yes	characteristic, complemented, Sylow
32	#1355	—	173	1	no	maximal, Sylow
33	#490	—	173	1	no	—
34	#347	—	173	2	no	—
35	#127	—	173	1	no	—
36	#125	#10621924	1	1	yes	characteristic
37	M₃₂	—	346	2	no	—
38	#102	—	346	2	no	—
39	#97	—	173	1	no	—
40	C₂×D₄	—	346	1	no	—
41	C₂×D₄	—	692	1	no	—
42	C₂×C₈	—	173	2	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	—	4	1	no	—
45	C₄²	C₂×#10621924	1	1	yes	characteristic
46	C₁₆	—	346	4	no	—
47	C₂³	—	692	1	no	—
48	D₄	—	1384	2	no	—
49	D₄	—	692	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	4	1	no	—
52	C₂×C₄	C₂²⋊#10621924	1	1	yes	characteristic
53	C₈	—	346	2	no	—
54	C₂²	—	1384	2	no	—
55	C₂²	—	1384	1	no	—
56	C₂²	#19149739	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	8	1	no	—
59	C₄	#19149744	1	2	yes	—
60	C₂	—	2	1	no	—
61	C₂	—	2768	1	no	—
62	C₂	#26012347	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26012347
commutator	C₄²×C₁₇₃	C₂×C₄
Frattini	C₄²	C₂×#10621924
Fitting	#125×C₁₇₃	C₄
socle	C₂×C₁₇₃	#329
prime radical	#19149676	C₄
cosocle	#19149590	C₂²
hypercenter	#125	#10621924
hypocenter	C₁₇₃	#1355
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #26012383	C₂	semidirect
31. C₁₇₃	#1355	semidirect

Maximal subgroups

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

subgroup	index
#26012383	2
#26011857	2
#1355	173