Subgroups — #36361840

Order 22144.

3422 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19149648	C₂	1	7	yes	maximal, complemented
2	#26011347	C₂	1	7	yes	maximal, complemented
3	C₂⁶×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂²⋊D₁₇₃	C₂²	1	7	yes	complemented
5	#19149648	—	2	28	no	—
6	C₂×C₂²⋊Dic₁₇₃	C₂²	1	21	yes	complemented
7	C₂⁵×C₁₇₃	—	2	28	no	—
8	C₂⁵×C₁₇₃	C₂²	1	7	yes	—
9	C₂³×D₁₇₃	C₂³	1	1	yes	characteristic, complemented
10	C₂×C₂²⋊D₁₇₃	—	2	42	no	—
11	C₂²×Dic₁₇₃	C₂³	1	7	yes	complemented
12	C₂²⋊Dic₁₇₃	—	2	42	no	—
13	C₂⁴×C₁₇₃	—	2	224	no	—
14	C₂⁴×C₁₇₃	—	2	84	no	—
15	C₂⁴×C₁₇₃	D₄	1	28	yes	complemented
16	C₂⁴×C₁₇₃	C₂³	1	7	yes	—
17	C₂²×D₁₇₃	—	2	7	no	—
18	C₂²⋊D₁₇₃	—	4	28	no	—
19	C₂×Dic₁₇₃	—	2	21	no	—
20	C₂³×C₁₇₃	—	2	256	no	—
21	C₂³×C₁₇₃	—	2	224	no	—
22	C₂³×C₁₇₃	—	2	168	no	—
23	C₂³×C₁₇₃	—	2	28	no	—
24	C₂³×C₁₇₃	C₂×D₄	1	42	yes	—
25	C₂³×C₁₇₃	C₂⁴	1	1	yes	characteristic
26	C₂²×C₁₇₃	—	2	224	no	—
27	C₂²×C₁₇₃	—	2	84	no	—
28	C₂²×C₁₇₃	#120	1	28	yes	complemented
29	C₂²×C₁₇₃	C₂²×D₄	1	7	yes	—
30	C₂×D₁₇₃	—	4	7	no	—
31	Dic₁₇₃	—	4	7	no	—
32	C₂×C₁₇₃	—	2	28	no	—
33	C₂×C₁₇₃	C₂×#120	1	7	yes	—
34	D₁₇₃	—	8	1	no	—
35	C₁₇₃	#2846	1	1	yes	characteristic, complemented, Sylow
36	#2846	—	173	1	no	maximal, Sylow
37	C₂⁶	D₁₇₃	1	1	yes	characteristic, complemented
38	C₂×#120	—	173	7	no	—
39	#379	—	173	7	no	—
40	C₂⁵	—	2	28	no	—
41	C₂⁵	C₂×D₁₇₃	1	7	yes	—
42	C₂²×D₄	—	173	7	no	—
43	#120	—	346	28	no	—
44	C₂×C₂²⋊C₄	—	173	21	no	—
45	C₂⁴	—	2	224	no	—
46	C₂⁴	—	2	84	no	—
47	C₂⁴	—	173	1	no	—
48	C₂⁴	C₂²⋊D₁₇₃	1	28	yes	complemented
49	C₂⁴	C₂²×D₁₇₃	1	7	yes	—
50	C₂×D₄	—	346	42	no	—
51	C₂²×C₄	—	173	7	no	—
52	C₂²⋊C₄	—	346	42	no	—
53	C₂³	—	2	256	no	—
54	C₂³	—	2	224	no	—
55	C₂³	—	2	168	no	—
56	C₂³	—	2	28	no	—
57	C₂³	—	346	7	no	—
58	C₂³	C₂×C₂²⋊D₁₇₃	1	42	yes	—
59	C₂³	C₂³×D₁₇₃	1	1	yes	characteristic
60	D₄	—	692	28	no	—
61	C₂×C₄	—	346	21	no	—
62	C₂²	—	2	224	no	—
63	C₂²	—	2	84	no	—
64	C₂²	—	692	7	no	—
65	C₂²	#19149648	1	28	yes	complemented
66	C₂²	C₂²×C₂²⋊D₁₇₃	1	7	yes	—
67	C₄	—	692	7	no	—
68	C₂	—	2	28	no	—
69	C₂	—	1384	1	no	—
70	C₂	C₂×#19149648	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇₃
commutator, socle, cosocle	C₂³×C₁₇₃	C₂⁴
Fitting	C₂⁶×C₁₇₃	C₂
hypercenter	C₂⁶	D₁₇₃
hypocenter	C₁₇₃	#2846
LCS	G ▷ C₂³×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₂⁶

Complements

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

N	H ≅ G/N	type
1. C₂×#19149648	C₂	semidirect
2. #26011347	C₂	semidirect
3. C₂⁶×C₁₇₃	C₂	semidirect
4. C₂²×C₂²⋊D₁₇₃	C₂²	semidirect
6. C₂×C₂²⋊Dic₁₇₃	C₂²	semidirect
9. C₂³×D₁₇₃	C₂³	semidirect
11. C₂²×Dic₁₇₃	C₂³	semidirect
15. C₂⁴×C₁₇₃	D₄	semidirect
28. C₂²×C₁₇₃	#120	semidirect
35. C₁₇₃	#2846	semidirect
37. C₂⁶	D₁₇₃	semidirect
48. C₂⁴	C₂²⋊D₁₇₃	semidirect
65. C₂²	#19149648	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19149648	2
#26011347	2
C₂⁶×C₁₇₃	2
#2846	173