Subgroups — #36353312

Order 22144.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19149590	C₂	1	2	yes	maximal
2	#26011246	C₂	1	4	yes	maximal
3	C₄³×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13063436	C₂²	1	4	yes	—
5	#19149590	C₄	1	8	yes	complemented
6	C₂×C₄²×C₁₇₃	C₄	1	4	yes	—
7	C₂×C₄²×C₁₇₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
9	C₂×#13063436	—	2	8	no	—
10	C₂×#13063436	C₂×C₄	1	8	yes	—
11	C₂²×C₄×C₁₇₃	C₂×C₄	1	2	yes	—
12	C₂²×C₄×C₁₇₃	C₂×C₄	1	4	yes	—
13	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
14	C₄²×C₁₇₃	—	2	8	no	—
15	C₄²×C₁₇₃	—	2	4	no	—
16	C₄²×C₁₇₃	C₂×C₄	1	4	yes	—
17	#13063436	—	2	8	no	—
18	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₇₃	—	2	4	no	—
20	C₂×C₄×C₁₇₃	—	2	4	no	—
21	C₂×C₄×C₁₇₃	—	2	4	no	—
22	C₂×C₄×C₁₇₃	C₄²	1	4	yes	—
23	C₂×C₄×C₁₇₃	M₁₆	1	8	yes	—
24	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
25	C₂×C₄×C₁₇₃	#41	1	4	yes	—
26	C₂²×C₁₇₃	C₂×C₄²	1	1	yes	characteristic
27	C₂²×C₁₇₃	#117	1	2	yes	—
28	C₂²×C₁₇₃	C₂×M₁₆	1	4	yes	—
29	C₄×C₁₇₃	—	2	4	no	—
30	C₄×C₁₇₃	—	2	8	no	—
31	C₄×C₁₇₃	#117	1	4	yes	—
32	C₂×C₁₇₃	#376	1	1	yes	characteristic
33	C₂×C₁₇₃	C₄×M₁₆	1	2	yes	—
34	C₂×C₁₇₃	#432	1	4	yes	—
35	C₁₇₃	#1745	1	1	yes	characteristic, complemented, Sylow
36	#1745	—	173	1	no	maximal, Sylow
37	C₂×#97	—	173	2	no	—
38	C₄³	D₁₇₃	1	1	yes	characteristic
39	#336	—	173	4	no	—
40	C₂²×C₈	—	173	4	no	—
41	C₂×C₄²	Dic₁₇₃	1	4	yes	—
42	C₂×C₄²	C₂×D₁₇₃	1	2	yes	—
43	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
44	#97	—	173	8	no	—
45	C₂²×C₄	C₂×Dic₁₇₃	1	4	yes	—
46	C₂²×C₄	C₂×Dic₁₇₃	1	2	yes	—
47	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
48	C₂×C₈	—	346	8	no	—
49	C₂×C₈	—	173	8	no	—
50	C₄²	—	2	8	no	—
51	C₄²	—	2	4	no	—
52	C₄²	C₄×D₁₇₃	1	4	yes	—
53	C₂³	C₂²×Dic₁₇₃	1	1	yes	characteristic
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	C₄×Dic₁₇₃	1	4	yes	—
58	C₂×C₄	#14583810	1	8	yes	—
59	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
60	C₂×C₄	#14583826	1	4	yes	—
61	C₈	—	346	8	no	—
62	C₂²	C₂×C₄×Dic₁₇₃	1	1	yes	characteristic
63	C₂²	#19149638	1	2	yes	—
64	C₂²	C₂×#14583810	1	4	yes	—
65	C₄	—	2	4	no	—
66	C₄	—	2	8	no	—
67	C₄	#19149636	1	4	yes	—
68	C₂	#26011337	1	1	yes	characteristic
69	C₂	C₄×#14583810	1	2	yes	—
70	C₂	#26011548	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₇₃
commutator	C₂²×C₁₇₃	C₂×C₄²
Fitting	C₄³×C₁₇₃	C₂
socle, prime radical	C₂³×C₁₇₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	C₄³	D₁₇₃
hypocenter	C₁₇₃	#1745
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
5. #19149590	C₄	semidirect
35. C₁₇₃	#1745	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19149590	2
#26011246	2
C₄³×C₁₇₃	2
#1745	173