Subgroups — #36368715

Order 22144.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₇₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#26012419	C₂	1	1	yes	maximal, characteristic, complemented
3	#26011924	C₂	1	1	yes	maximal, characteristic
4	#19149750	—	2	1	no	—
5	#19149729	—	2	1	no	—
6	#19149702	—	2	1	no	—
7	#19149622	—	2	1	no	—
8	D₂₇₆₈	C₄	1	1	yes	characteristic
9	C₁₆×D₁₇₃	C₂²	1	1	yes	characteristic
10	Q₃₂×C₁₇₃	C₄	1	1	yes	characteristic
11	#14583836	—	4	1	no	—
12	#14583827	—	4	1	no	—
13	#14583813	—	4	1	no	—
14	D₁₃₈₄	—	2	1	no	—
15	C₈×D₁₇₃	D₄	1	1	yes	characteristic
16	#14583801	D₄	1	1	yes	characteristic
17	Q₁₆×C₁₇₃	—	2	1	no	—
18	C₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic
19	#13063443	—	8	1	no	—
20	D₆₉₂	—	4	1	no	—
21	D₆₉₂	—	8	1	no	—
22	C₄×D₁₇₃	—	8	1	no	—
23	C₄×D₁₇₃	D₈	1	1	yes	characteristic
24	#13063436	SD₁₆	1	1	yes	characteristic
25	Q₈×C₁₇₃	—	4	1	no	—
26	C₈×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₁₇₃	—	8	1	no	—
28	C₂×D₁₇₃	SD₃₂	1	1	yes	characteristic
29	C₄×C₁₇₃	—	8	1	no	—
30	C₄×C₁₇₃	#102	1	1	yes	characteristic
31	Dic₁₇₃	D₁₆	1	1	yes	characteristic
32	C₂×C₁₇₃	#357	1	1	yes	characteristic
33	D₁₇₃	—	2	1	no	—
34	D₁₇₃	—	16	1	no	—
35	C₁₇₃	#1417	1	1	yes	characteristic, complemented, Sylow
36	#1417	—	173	1	no	maximal, Sylow
37	#508	—	173	1	no	—
38	C₂×C₃₂	—	173	1	no	—
39	#368	—	173	1	no	—
40	#135	—	346	1	no	—
41	Q₃₂	#10621924	1	1	yes	characteristic
42	SD₃₂	—	346	1	no	—
43	D₁₆	—	173	1	no	—
44	C₂×C₁₆	—	173	1	no	—
45	#108	—	346	1	no	—
46	C₃₂	—	346	1	no	—
47	#41	—	692	1	no	—
48	Q₁₆	—	2	1	no	—
49	SD₁₆	—	692	1	no	—
50	D₈	—	346	1	no	—
51	M₁₆	—	692	1	no	—
52	C₂×C₈	—	173	1	no	—
53	C₁₆	—	173	1	no	—
54	C₁₆	C₂×#10621924	1	1	yes	characteristic
55	Q₈	—	4	1	no	—
56	D₄	—	1384	1	no	—
57	D₄	—	692	1	no	—
58	C₂×C₄	—	1384	1	no	—
59	C₂×C₄	—	173	1	no	—
60	C₈	—	1384	1	no	—
61	C₈	—	173	1	no	—
62	C₈	C₂²⋊#10621924	1	1	yes	characteristic
63	C₂²	—	1384	1	no	—
64	C₂²	—	173	1	no	—
65	C₄	—	8	1	no	—
66	C₄	—	173	1	no	—
67	C₄	#19149741	1	1	yes	characteristic
68	C₂	—	346	1	no	—
69	C₂	—	2768	1	no	—
70	C₂	#26012403	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26012403
commutator	C₁₆×C₁₇₃	C₂×C₄
Frattini	C₁₆	C₂×#10621924
Fitting	Q₃₂×C₁₇₃	C₄
socle	C₂×C₁₇₃	#357
prime radical	#26011924	C₂
cosocle	C₁₆×D₁₇₃	C₂²
hypercenter	Q₃₂	#10621924
hypocenter	C₁₇₃	#1417
LCS	G ▷ C₁₆×C₁₇₃ ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
1. D₁₇₃⋊C₃₂	C₂	semidirect
2. #26012419	C₂	semidirect
35. C₁₇₃	#1417	semidirect

Maximal subgroups

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

subgroup	index
D₁₇₃⋊C₃₂	2
#26012419	2
#26011924	2
#1417	173