Subgroups — #36351751

Order 22144.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011923	C₂	1	1	yes	maximal, characteristic
2	#26011319	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19149701	—	2	1	no	—
5	Dic₁₃₈₄	C₄	1	1	yes	characteristic, complemented
6	#19149624	—	2	1	no	—
7	D₂₇₆₈	C₄	1	1	yes	characteristic, complemented
8	#19149613	—	2	1	no	—
9	C₂×C₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₇₃	—	2	1	no	—
11	#14583826	—	4	1	no	—
12	Dic₆₉₂	—	2	1	no	—
13	#14583815	—	4	1	no	—
14	D₁₃₈₄	—	2	1	no	—
15	C₄⋊Dic₁₇₃	—	4	1	no	—
16	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
17	C₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₇₃	D₄	1	1	yes	characteristic
19	Dic₃₄₆	—	4	1	no	—
20	D₆₉₂	—	4	1	no	—
21	C₂²⋊D₁₇₃	—	8	1	no	—
22	C₂×Dic₁₇₃	—	8	1	no	—
23	C₄×D₁₇₃	—	8	1	no	—
24	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
25	C₈×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₇₃	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₇₃	D₁₆	1	1	yes	characteristic
28	C₂×D₁₇₃	—	8	1	no	—
29	C₄×C₁₇₃	#102	1	1	yes	characteristic
30	C₄×C₁₇₃	SD₃₂	1	1	yes	characteristic
31	Dic₁₇₃	—	8	1	no	—
32	Dic₁₇₃	—	16	1	no	—
33	C₂×C₁₇₃	—	2	1	no	—
34	C₂×C₁₇₃	#357	1	1	yes	characteristic
35	D₁₇₃	—	16	1	no	—
36	C₁₇₃	#1418	1	1	yes	characteristic, complemented, Sylow
37	#1418	—	173	1	no	maximal, Sylow
38	#508	—	173	1	no	—
39	M₆₄	D₁₇₃	1	1	yes	characteristic, complemented
40	#367	—	173	1	no	—
41	#135	—	346	1	no	—
42	Q₃₂	—	173	1	no	—
43	SD₃₂	—	346	1	no	—
44	D₁₆	—	173	1	no	—
45	C₂×C₁₆	C₂×D₁₇₃	1	1	yes	characteristic
46	#107	—	346	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	692	1	no	—
49	Q₁₆	—	346	1	no	—
50	SD₁₆	—	692	1	no	—
51	D₈	—	346	1	no	—
52	C₂×C₈	D₆₉₂	1	1	yes	characteristic
53	C₄⋊C₄	—	692	1	no	—
54	C₁₆	C₄×D₁₇₃	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₁₇₃	1	1	yes	characteristic
56	Q₈	—	692	1	no	—
57	D₄	—	1384	1	no	—
58	D₄	—	692	1	no	—
59	C₂×C₄	—	1384	1	no	—
60	C₂×C₄	—	1384	1	no	—
61	C₂×C₄	D₁₃₈₄	1	1	yes	characteristic
62	C₈	#14583803	1	1	yes	characteristic
63	C₈	#14583815	1	1	yes	characteristic
64	C₂²	—	1384	1	no	—
65	C₂²	D₂₇₆₈	1	1	yes	characteristic
66	C₄	—	1384	1	no	—
67	C₄	—	2768	1	no	—
68	C₄	#19149601	1	1	yes	characteristic
69	C₄	#19149624	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	2768	1	no	—
72	C₂	#26011293	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26011293
commutator	C₁₆×C₁₇₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₇₃
Fitting	M₆₄×C₁₇₃	C₂
socle	C₂×C₁₇₃	#357
prime radical	#26011923	C₂
cosocle	C₂×C₁₆×C₁₇₃	C₂²
hypercenter	M₆₄	D₁₇₃
hypocenter	C₁₇₃	#1418
LCS	G ▷ C₁₆×C₁₇₃ ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #26011319	C₂	semidirect
3. M₆₄×C₁₇₃	C₂	semidirect
5. Dic₁₃₈₄	C₄	semidirect
7. D₂₇₆₈	C₄	semidirect
36. C₁₇₃	#1418	semidirect
39. M₆₄	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26011923	2
#26011319	2
M₆₄×C₁₇₃	2
#1418	173