Subgroups — #36351516

Order 22144.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011273	C₂	1	2	yes	maximal
2	#424×C₁₇₃	C₂	1	1	yes	maximal, characteristic
3	#19149619	—	2	2	no	—
4	C₂×M₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
5	C₂²×C₈×C₁₇₃	C₄	1	1	yes	characteristic
6	#117×C₁₇₃	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₁₇₃	—	2	2	no	—
8	#14583801	—	4	2	no	—
9	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
10	M₁₆×C₁₇₃	—	4	1	no	—
11	C₂×C₈×C₁₇₃	—	2	1	no	—
12	C₂×C₈×C₁₇₃	—	4	1	no	—
13	C₂×C₈×C₁₇₃	D₄	1	2	yes	—
14	C₄⋊C₄×C₁₇₃	—	2	1	no	—
15	C₂²⋊C₄×C₁₇₃	C₈	1	2	yes	—
16	C₄²×C₁₇₃	—	2	1	no	—
17	C₂³×C₁₇₃	C₂×C₈	1	1	yes	characteristic
18	C₂×C₄×C₁₇₃	—	2	1	no	—
19	C₂×C₄×C₁₇₃	—	2	1	no	—
20	C₂×C₄×C₁₇₃	—	2	2	no	—
21	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₇₃	M₁₆	1	1	yes	characteristic
23	C₈×C₁₇₃	—	2	2	no	—
24	C₈×C₁₇₃	—	4	1	no	—
25	C₂²×C₁₇₃	—	2	1	no	—
26	C₂²×C₁₇₃	—	4	1	no	—
27	C₂²×C₁₇₃	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₁₇₃	—	2	1	no	—
29	C₄×C₁₇₃	—	4	2	no	—
30	C₄×C₁₇₃	#99	1	1	yes	characteristic
31	C₄×C₁₇₃	#101	1	1	yes	characteristic
32	C₂×C₁₇₃	—	2	1	no	—
33	C₂×C₁₇₃	—	4	1	no	—
34	C₂×C₁₇₃	#324	1	1	yes	characteristic
35	C₁₇₃	#1328	1	1	yes	characteristic, complemented, Sylow
36	#1328	—	173	1	no	maximal, Sylow
37	#424	D₁₇₃	1	1	yes	characteristic
38	#349	—	173	2	no	—
39	C₂×M₁₆	C₂×D₁₇₃	1	1	yes	characteristic
40	C₂²×C₈	Dic₁₇₃	1	1	yes	characteristic
41	#117	Dic₁₇₃	1	1	yes	characteristic
42	M₃₂	—	346	2	no	—
43	C₄⋊C₈	—	2	2	no	—
44	C₂²×C₄	C₂×Dic₁₇₃	1	1	yes	characteristic
45	M₁₆	—	4	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	C₂²⋊D₁₇₃	1	2	yes	—
49	C₄⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	#13063436	1	2	yes	—
51	C₄²	—	2	1	no	—
52	C₁₆	—	692	2	no	—
53	C₂³	C₂×#13063436	1	1	yes	characteristic
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	C₂²⋊Dic₁₇₃	1	1	yes	characteristic
58	C₂×C₄	#14583810	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	4	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	4	1	no	—
63	C₂²	C₂²⋊#13063436	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	4	2	no	—
66	C₄	#19149594	1	1	yes	characteristic
67	C₄	#19149598	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	#26011217	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19149594
commutator, prime radical	C₂³×C₁₇₃	C₂×C₈
Frattini	C₂×M₁₆	C₂×D₁₇₃
Fitting	#424×C₁₇₃	C₂
socle	C₂×C₁₇₃	#324
cosocle	C₂×M₁₆×C₁₇₃	C₂²
hypercenter	#424	D₁₇₃
hypocenter	C₁₇₃	#1328
LCS	G ▷ C₂³×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #424

Complements

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

N	H ≅ G/N	type
35. C₁₇₃	#1328	semidirect

Maximal subgroups

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

subgroup	index
#26011273	2
#424×C₁₇₃	2
#1328	173