Subgroups — #36353175

Order 22144.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011842	C₂	1	3	yes	maximal, complemented
2	#26011838	C₂	1	3	yes	maximal
3	C₈²×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19149679	C₂²	1	3	yes	—
5	#19149676	C₂²	1	1	yes	characteristic
6	#19149604	—	2	6	no	—
7	#19149601	—	2	6	no	—
8	C₄×C₈×C₁₇₃	C₂²	1	3	yes	—
9	C₂×Dic₃₄₆	—	2	3	no	—
10	C₂×D₆₉₂	—	2	3	no	—
11	C₄⋊Dic₁₇₃	—	2	3	no	—
12	C₄⋊Dic₁₇₃	—	2	3	no	—
13	C₂×C₈×C₁₇₃	D₄	1	6	yes	—
14	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
15	C₂²×D₁₇₃	—	4	1	no	—
16	Dic₃₄₆	—	4	6	no	—
17	D₆₉₂	—	4	6	no	—
18	C₂×Dic₁₇₃	—	4	3	no	—
19	C₂×C₄×C₁₇₃	C₂×D₄	1	3	yes	—
20	C₈×C₁₇₃	—	2	6	no	—
21	C₂²×C₁₇₃	#127	1	1	yes	characteristic
22	C₂×D₁₇₃	—	8	3	no	—
23	C₄×C₁₇₃	#135	1	6	yes	—
24	Dic₁₇₃	—	8	3	no	—
25	C₂×C₁₇₃	#495	1	3	yes	—
26	D₁₇₃	—	16	1	no	—
27	C₁₇₃	#1711	1	1	yes	characteristic, complemented, Sylow
28	#1711	—	173	1	no	maximal, Sylow
29	#487	—	173	3	no	—
30	#486	—	173	3	no	—
31	C₈²	D₁₇₃	1	1	yes	characteristic, complemented
32	C₄⋊Q₈	—	173	3	no	—
33	#127	—	173	1	no	—
34	#103	—	346	6	no	—
35	#102	—	346	6	no	—
36	C₄×C₈	C₂×D₁₇₃	1	3	yes	—
37	C₂×Q₈	—	346	3	no	—
38	C₂×D₄	—	346	3	no	—
39	C₂×C₈	D₆₉₂	1	6	yes	—
40	C₄⋊C₄	—	346	3	no	—
41	C₄⋊C₄	—	346	3	no	—
42	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
43	C₂³	—	692	1	no	—
44	Q₈	—	692	6	no	—
45	D₄	—	692	6	no	—
46	C₂×C₄	—	692	3	no	—
47	C₂×C₄	C₂×D₆₉₂	1	3	yes	—
48	C₈	—	2	6	no	—
49	C₂²	—	1384	3	no	—
50	C₂²	#19149676	1	1	yes	characteristic
51	C₄	—	1384	3	no	—
52	C₄	#19149701	1	6	yes	—
53	C₂	—	2768	1	no	—
54	C₂	#26011875	1	3	yes	—

	subgroup	G/H
center	C₂²	#19149676
commutator, cosocle	C₄²×C₁₇₃	C₂³
Frattini	C₄²	C₂²×D₁₇₃
Fitting	C₈²×C₁₇₃	C₂
socle	C₂²×C₁₇₃	#127
prime radical	#19149676	C₂²
hypercenter	C₈²	D₁₇₃
hypocenter	C₁₇₃	#1711
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #26011842	C₂	semidirect
3. C₈²×C₁₇₃	C₂	semidirect
27. C₁₇₃	#1711	semidirect
31. C₈²	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26011842	2
#26011838	2
C₈²×C₁₇₃	2
#1711	173