Subgroups — #36353179

Order 22144.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011894	C₂	1	4	yes	maximal
2	#26011871	C₂	1	1	yes	maximal, characteristic
3	C₈×#13063436	C₂	1	1	yes	maximal, characteristic
4	#494×C₁₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₉₂	—	2	2	no	—
6	#19149679	C₂²	1	2	yes	—
7	#19149612	—	2	8	no	—
8	C₄×#13063436	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₁₇₃	—	2	2	no	—
10	C₄⋊Q₈×C₁₇₃	C₂²	1	2	yes	—
11	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₃₄₆	—	2	2	no	—
13	Dic₆₉₂	—	4	4	no	—
14	C₂×#13063436	D₄	1	4	yes	—
15	C₄⋊Dic₁₇₃	—	2	4	no	—
16	C₂×Q₈×C₁₇₃	—	2	2	no	—
17	Q₁₆×C₁₇₃	—	4	4	no	—
18	C₂×C₈×C₁₇₃	D₄	1	2	yes	—
19	C₄⋊C₄×C₁₇₃	—	2	4	no	—
20	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
21	Dic₃₄₆	—	4	4	no	—
22	C₂×Dic₁₇₃	—	4	2	no	—
23	#13063436	Q₁₆	1	8	yes	complemented
24	Q₈×C₁₇₃	—	4	4	no	—
25	C₂×C₄×C₁₇₃	—	4	2	no	—
26	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
27	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
28	C₈×C₁₇₃	D₈	1	4	yes	—
29	C₂²×C₁₇₃	#127	1	1	yes	characteristic
30	C₄×C₁₇₃	—	8	2	no	—
31	C₄×C₁₇₃	C₂×D₈	1	2	yes	—
32	C₄×C₁₇₃	C₂×Q₁₆	1	4	yes	—
33	Dic₁₇₃	—	8	2	no	—
34	C₂×C₁₇₃	#493	1	1	yes	characteristic
35	C₂×C₁₇₃	#494	1	2	yes	—
36	C₁₇₃	#1713	1	1	yes	characteristic, complemented, Sylow
37	#1713	—	173	1	no	maximal, Sylow
38	#500	—	173	4	no	—
39	#494	—	173	1	no	—
40	#494	D₁₇₃	1	1	yes	characteristic
41	C₈²	—	173	1	no	—
42	C₂×Q₁₆	—	2	2	no	—
43	C₂×Q₁₆	—	346	2	no	—
44	C₄⋊Q₈	—	173	2	no	—
45	C₄⋊Q₈	C₂×D₁₇₃	1	2	yes	—
46	#107	—	346	8	no	—
47	C₄×C₈	—	173	2	no	—
48	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
49	C₂×Q₈	—	2	2	no	—
50	C₂×Q₈	—	346	2	no	—
51	Q₁₆	—	4	4	no	—
52	Q₁₆	—	692	4	no	—
53	C₂×C₈	—	173	4	no	—
54	C₂×C₈	C₂²⋊D₁₇₃	1	2	yes	—
55	C₄⋊C₄	—	2	4	no	—
56	C₄⋊C₄	—	346	4	no	—
57	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
58	Q₈	—	4	4	no	—
59	Q₈	—	692	4	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	692	2	no	—
62	C₂×C₄	D₄×D₁₇₃	1	2	yes	—
63	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
64	C₈	—	173	8	no	—
65	C₈	#14583811	1	4	yes	—
66	C₂²	#19149675	1	1	yes	characteristic
67	C₄	—	8	2	no	—
68	C₄	—	1384	2	no	—
69	C₄	C₂×#14583811	1	2	yes	—
70	C₄	Q₁₆×D₁₇₃	1	4	yes	—
71	C₂	#26011867	1	1	yes	characteristic
72	C₂	#26011869	1	2	yes	—

	subgroup	G/H
center	C₂²	#19149675
commutator, cosocle	C₄²×C₁₇₃	C₂³
Frattini	C₄²	C₂²×D₁₇₃
Fitting	#494×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	#127
hypercenter	#494	D₁₇₃
hypocenter	C₁₇₃	#1713
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
23. #13063436	Q₁₆	semidirect
36. C₁₇₃	#1713	semidirect

Maximal subgroups

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

subgroup	index
#26011894	2
#26011871	2
C₈×#13063436	2
#494×C₁₇₃	2
#1713	173