Subgroups — #36351732

Order 22144.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011290	C₂	1	2	yes	maximal
2	#494×C₁₇₃	C₂	1	1	yes	maximal, characteristic
3	#19149598	—	2	2	no	—
4	C₂×Q₁₆×C₁₇₃	—	2	2	no	—
5	C₄⋊Q₈×C₁₇₃	C₄	1	1	yes	characteristic
6	C₄⋊Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
7	C₄×C₈×C₁₇₃	C₄	1	1	yes	characteristic
8	#14583810	—	4	2	no	—
9	C₂×Q₈×C₁₇₃	—	2	1	no	—
10	C₂×Q₈×C₁₇₃	D₄	1	2	yes	—
11	Q₁₆×C₁₇₃	—	4	4	no	—
12	C₂×C₈×C₁₇₃	—	2	1	no	—
13	C₄⋊C₄×C₁₇₃	—	4	1	no	—
14	C₄⋊C₄×C₁₇₃	—	4	1	no	—
15	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
16	#13063436	—	8	2	no	—
17	Q₈×C₁₇₃	—	4	2	no	—
18	Q₈×C₁₇₃	—	4	2	no	—
19	C₂×C₄×C₁₇₃	—	2	1	no	—
20	C₂×C₄×C₁₇₃	—	2	2	no	—
21	C₂×C₄×C₁₇₃	—	4	1	no	—
22	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₇₃	—	2	2	no	—
24	C₂²×C₁₇₃	#99	1	1	yes	characteristic
25	C₄×C₁₇₃	—	2	2	no	—
26	C₄×C₁₇₃	—	2	1	no	—
27	C₄×C₁₇₃	—	4	2	no	—
28	C₄×C₁₇₃	—	8	1	no	—
29	C₂×C₁₇₃	—	2	1	no	—
30	C₂×C₁₇₃	#353	1	1	yes	characteristic
31	C₁₇₃	#1411	1	1	yes	characteristic, complemented, Sylow
32	#1411	—	173	1	no	maximal, Sylow
33	#494	D₁₇₃	1	1	yes	characteristic
34	#356	—	173	2	no	—
35	C₂×Q₁₆	—	2	2	no	—
36	C₄⋊Q₈	Dic₁₇₃	1	1	yes	characteristic
37	C₄⋊Q₈	C₂×D₁₇₃	1	1	yes	characteristic
38	#101	—	346	2	no	—
39	C₄×C₈	Dic₁₇₃	1	1	yes	characteristic
40	C₂×Q₈	—	2	1	no	—
41	C₂×Q₈	C₂²⋊D₁₇₃	1	2	yes	—
42	Q₁₆	—	4	4	no	—
43	M₁₆	—	692	2	no	—
44	C₂×C₈	—	2	1	no	—
45	C₄⋊C₄	—	4	1	no	—
46	C₄⋊C₄	—	4	1	no	—
47	C₄²	C₂×Dic₁₇₃	1	1	yes	characteristic
48	Q₈	—	4	2	no	—
49	Q₈	—	4	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	4	1	no	—
53	C₂×C₄	C₂²⋊Dic₁₇₃	1	1	yes	characteristic
54	C₈	—	2	2	no	—
55	C₈	—	1384	2	no	—
56	C₂²	#19149594	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	2	1	no	—
59	C₄	—	4	2	no	—
60	C₄	—	8	1	no	—
61	C₂	—	2	1	no	—
62	C₂	#26011284	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26011284
commutator	C₄²×C₁₇₃	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₁₇₃
Fitting	#494×C₁₇₃	C₂
socle	C₂×C₁₇₃	#353
prime radical	C₂²×C₁₇₃	#99
cosocle	C₄⋊Q₈×C₁₇₃	C₂²
hypercenter	#494	D₁₇₃
hypocenter	C₁₇₃	#1411
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂×C₄×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #494

Complements

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

N	H ≅ G/N	type
31. C₁₇₃	#1411	semidirect

Maximal subgroups

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

subgroup	index
#26011290	2
#494×C₁₇₃	2
#1411	173