Subgroups — #36353181

Order 22144.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011896	C₂	1	4	yes	maximal
2	#26011871	C₂	1	2	yes	maximal
3	C₈²×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₉₂	—	2	4	no	—
5	#19149679	C₂²	1	4	yes	—
6	#19149613	—	2	8	no	—
7	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₁₇₃	C₂²	1	2	yes	—
9	C₂×Dic₃₄₆	—	2	4	no	—
10	Dic₆₉₂	—	4	8	no	—
11	C₄⋊Dic₁₇₃	—	2	8	no	—
12	C₂×C₈×C₁₇₃	D₄	1	2	yes	—
13	C₂×C₈×C₁₇₃	D₄	1	4	yes	—
14	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
15	Dic₃₄₆	—	4	8	no	—
16	C₂×Dic₁₇₃	—	4	4	no	—
17	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
19	C₈×C₁₇₃	D₈	1	4	yes	—
20	C₈×C₁₇₃	Q₁₆	1	8	yes	complemented
21	C₂²×C₁₇₃	#127	1	1	yes	characteristic
22	C₄×C₁₇₃	C₂×D₈	1	2	yes	—
23	C₄×C₁₇₃	C₂×Q₁₆	1	4	yes	—
24	Dic₁₇₃	—	8	4	no	—
25	C₂×C₁₇₃	#493	1	1	yes	characteristic
26	C₂×C₁₇₃	#494	1	2	yes	—
27	C₁₇₃	#1713	1	1	yes	characteristic, complemented, Sylow
28	#1713	—	173	1	no	maximal, Sylow
29	#500	—	173	4	no	—
30	#494	—	173	2	no	—
31	C₈²	D₁₇₃	1	1	yes	characteristic
32	C₂×Q₁₆	—	346	4	no	—
33	C₄⋊Q₈	—	173	4	no	—
34	#107	—	346	8	no	—
35	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
36	C₄×C₈	C₂×D₁₇₃	1	2	yes	—
37	C₂×Q₈	—	346	4	no	—
38	Q₁₆	—	692	8	no	—
39	C₂×C₈	D₆₉₂	1	2	yes	—
40	C₂×C₈	D₆₉₂	1	4	yes	—
41	C₄⋊C₄	—	346	8	no	—
42	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
43	Q₈	—	692	8	no	—
44	C₂×C₄	—	692	4	no	—
45	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
46	C₂×C₄	C₂×D₆₉₂	1	2	yes	—
47	C₈	D₁₃₈₄	1	4	yes	—
48	C₈	Dic₆₉₂	1	8	yes	complemented
49	C₂²	#19149676	1	1	yes	characteristic
50	C₄	—	1384	4	no	—
51	C₄	C₂×D₁₃₈₄	1	2	yes	—
52	C₄	C₂×Dic₆₉₂	1	4	yes	—
53	C₂	#26011868	1	1	yes	characteristic
54	C₂	#26011871	1	2	yes	—

	subgroup	G/H
center	C₂²	#19149676
commutator, cosocle	C₄²×C₁₇₃	C₂³
Frattini	C₄²	C₂²×D₁₇₃
Fitting	C₈²×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	#127
hypercenter	C₈²	D₁₇₃
hypocenter	C₁₇₃	#1713
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
20. C₈×C₁₇₃	Q₁₆	semidirect
27. C₁₇₃	#1713	semidirect
48. C₈	Dic₆₉₂	semidirect

Maximal subgroups

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

subgroup	index
#26011896	2
#26011871	2
C₈²×C₁₇₃	2
#1713	173