Subgroups — #36353177

Order 22144.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011868	C₂	1	1	yes	maximal, characteristic, complemented
2	#26011867	C₂	1	4	yes	maximal, complemented
3	C₈×#13063436	C₂	1	1	yes	maximal, characteristic, complemented
4	#493×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₃₈₄	—	2	2	no	—
6	C₂×#14583811	—	2	8	no	—
7	#19149676	C₂²	1	2	yes	—
8	C₄×#13063436	C₂²	1	2	yes	—
9	C₂×D₈×C₁₇₃	—	2	2	no	—
10	#127×C₁₇₃	C₂²	1	2	yes	—
11	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
12	C₂×D₆₉₂	—	2	4	no	—
13	C₂×D₆₉₂	—	2	2	no	—
14	D₁₃₈₄	—	4	4	no	—
15	#14583811	—	4	16	no	—
16	C₂×#13063436	D₄	1	4	yes	—
17	C₂×D₄×C₁₇₃	—	2	2	no	—
18	C₂×D₄×C₁₇₃	—	2	4	no	—
19	D₈×C₁₇₃	—	4	4	no	—
20	C₂×C₈×C₁₇₃	D₄	1	2	yes	—
21	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
22	C₂²×D₁₇₃	—	4	2	no	—
23	D₆₉₂	—	4	8	no	—
24	D₆₉₂	—	4	4	no	—
25	#13063436	D₈	1	8	yes	complemented
26	C₂³×C₁₇₃	—	4	2	no	—
27	D₄×C₁₇₃	—	4	8	no	—
28	D₄×C₁₇₃	—	4	4	no	—
29	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
30	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
31	C₈×C₁₇₃	D₈	1	4	yes	complemented
32	C₂²×C₁₇₃	—	8	4	no	—
33	C₂²×C₁₇₃	—	8	2	no	—
34	C₂²×C₁₇₃	#127	1	1	yes	characteristic
35	C₂×D₁₇₃	—	8	4	no	—
36	C₂×D₁₇₃	—	8	2	no	—
37	C₄×C₁₇₃	C₂×D₈	1	4	yes	—
38	C₄×C₁₇₃	C₂×D₈	1	2	yes	—
39	C₂×C₁₇₃	—	16	2	no	—
40	C₂×C₁₇₃	#493	1	2	yes	—
41	C₂×C₁₇₃	#493	1	1	yes	characteristic
42	D₁₇₃	—	16	2	no	—
43	C₁₇₃	#1712	1	1	yes	characteristic, complemented, Sylow
44	#1712	—	173	1	no	maximal, Sylow
45	#493	—	173	4	no	—
46	#493	—	173	1	no	—
47	#493	D₁₇₃	1	1	yes	characteristic, complemented
48	C₈²	—	173	1	no	—
49	C₂×D₈	—	2	2	no	—
50	C₂×D₈	—	346	8	no	—
51	C₂×D₈	—	346	2	no	—
52	#127	—	173	2	no	—
53	#127	C₂×D₁₇₃	1	2	yes	—
54	C₄×C₈	—	173	2	no	—
55	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
56	C₂×D₄	—	2	4	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	—	346	4	no	—
59	C₂×D₄	—	346	2	no	—
60	D₈	—	4	4	no	—
61	D₈	—	692	16	no	—
62	D₈	—	692	4	no	—
63	C₂×C₈	—	173	4	no	—
64	C₂×C₈	C₂²⋊D₁₇₃	1	2	yes	—
65	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
66	C₂³	—	4	2	no	—
67	C₂³	—	692	2	no	—
68	D₄	—	4	8	no	—
69	D₄	—	4	4	no	—
70	D₄	—	692	8	no	—
71	D₄	—	692	4	no	—
72	C₂×C₄	D₄×D₁₇₃	1	2	yes	—
73	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
74	C₈	—	173	8	no	—
75	C₈	#14583811	1	4	yes	complemented
76	C₂²	—	8	4	no	—
77	C₂²	—	8	2	no	—
78	C₂²	—	1384	4	no	—
79	C₂²	—	1384	2	no	—
80	C₂²	#19149675	1	1	yes	characteristic
81	C₄	D₈×D₁₇₃	1	4	yes	—
82	C₄	C₂×#14583811	1	2	yes	—
83	C₂	—	16	2	no	—
84	C₂	—	2768	2	no	—
85	C₂	#26011866	1	2	yes	—
86	C₂	#26011867	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19149675
commutator, cosocle	C₄²×C₁₇₃	C₂³
Frattini	C₄²	C₂²×D₁₇₃
Fitting	#493×C₁₇₃	C₂
socle	C₂²×C₁₇₃	#127
hypercenter	#493	D₁₇₃
hypocenter	C₁₇₃	#1712
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #26011868	C₂	semidirect
2. #26011867	C₂	semidirect
3. C₈×#13063436	C₂	semidirect
4. #493×C₁₇₃	C₂	semidirect
25. #13063436	D₈	semidirect
31. C₈×C₁₇₃	D₈	semidirect
43. C₁₇₃	#1712	semidirect
47. #493	D₁₇₃	semidirect
75. C₈	#14583811	semidirect

Maximal subgroups

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

subgroup	index
#26011868	2
#26011867	2
C₈×#13063436	2
#493×C₁₇₃	2
#1712	173