Subgroups — #36351865

Order 22144.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14583809	C₂	1	3	yes	maximal
2	#26011212	C₂	1	3	yes	maximal, complemented
3	C₈²×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14583809	C₄	1	6	yes	—
5	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
6	#19149589	C₄	1	6	yes	—
7	C₄×#13063436	C₂²	1	3	yes	—
8	C₄×C₈×C₁₇₃	C₂²	1	3	yes	—
9	C₂×C₄×D₁₇₃	C₂×C₄	1	3	yes	—
10	#14583809	—	2	12	no	—
11	C₂×#13063436	C₂×C₄	1	6	yes	—
12	C₄×Dic₁₇₃	C₂×C₄	1	3	yes	—
13	C₂×C₈×C₁₇₃	C₂×C₄	1	6	yes	—
14	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
15	C₂²×D₁₇₃	C₄²	1	1	yes	characteristic
16	C₂×Dic₁₇₃	C₄²	1	3	yes	—
17	C₄×D₁₇₃	—	2	6	no	—
18	#13063436	—	2	6	no	—
19	C₂×C₄×C₁₇₃	C₂²×C₄	1	3	yes	—
20	C₈×C₁₇₃	M₁₆	1	12	yes	complemented
21	C₂²×C₁₇₃	C₂×C₄²	1	1	yes	characteristic
22	C₂×D₁₇₃	—	2	3	no	—
23	C₄×C₁₇₃	C₂×M₁₆	1	6	yes	—
24	Dic₁₇₃	—	2	3	no	—
25	C₂×C₁₇₃	C₄×M₁₆	1	3	yes	—
26	D₁₇₃	—	4	1	no	—
27	C₁₇₃	#1455	1	1	yes	characteristic, complemented, Sylow
28	#1455	—	173	1	no	maximal, Sylow
29	C₄×M₁₆	—	173	3	no	—
30	#322	—	173	3	no	—
31	C₈²	D₁₇₃	1	1	yes	characteristic, complemented
32	C₂×M₁₆	—	173	6	no	—
33	C₂×C₄²	—	173	1	no	—
34	#97	—	173	6	no	—
35	C₄×C₈	—	173	3	no	—
36	C₄×C₈	C₂×D₁₇₃	1	3	yes	—
37	C₂²×C₄	—	173	3	no	—
38	M₁₆	—	346	12	no	—
39	C₂×C₈	—	173	6	no	—
40	C₂×C₈	C₄×D₁₇₃	1	6	yes	—
41	C₄²	—	173	3	no	—
42	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
43	C₂³	—	173	1	no	—
44	C₂×C₄	—	346	6	no	—
45	C₂×C₄	—	173	3	no	—
46	C₂×C₄	C₂×C₄×D₁₇₃	1	3	yes	—
47	C₈	—	346	6	no	—
48	C₈	#14583809	1	12	yes	complemented
49	C₂²	—	346	3	no	—
50	C₂²	C₄²×D₁₇₃	1	1	yes	characteristic
51	C₄	—	346	3	no	—
52	C₄	C₂×#14583809	1	6	yes	—
53	C₂	—	692	1	no	—
54	C₂	C₄×#14583809	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₇₃
commutator, socle	C₂²×C₁₇₃	C₂×C₄²
Fitting	C₈²×C₁₇₃	C₂
prime radical	C₂²×D₁₇₃	C₄²
cosocle	C₄²×C₁₇₃	C₂³
hypercenter	C₈²	D₁₇₃
hypocenter	C₁₇₃	#1455
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #26011212	C₂	semidirect
3. C₈²×C₁₇₃	C₂	semidirect
20. C₈×C₁₇₃	M₁₆	semidirect
27. C₁₇₃	#1455	semidirect
31. C₈²	D₁₇₃	semidirect
48. C₈	#14583809	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14583809	2
#26011212	2
C₈²×C₁₇₃	2
#1455	173