Subgroups — #36351754

Order 22144.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₇₆₈	C₂	1	1	yes	maximal, characteristic
2	#26011321	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₃₈₄	—	2	1	no	—
5	D₂₇₆₈	—	2	1	no	—
6	D₂₇₆₈	C₄	1	2	yes	—
7	#19149615	—	2	1	no	—
8	C₂×C₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
9	C₃₂×C₁₇₃	—	2	1	no	—
10	C₂×D₆₉₂	—	4	1	no	—
11	D₁₃₈₄	—	2	2	no	—
12	D₁₃₈₄	—	4	1	no	—
13	#14583810	—	4	1	no	—
14	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
15	C₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₇₃	D₄	1	1	yes	characteristic
17	C₂²×D₁₇₃	—	8	1	no	—
18	D₆₉₂	—	4	2	no	—
19	D₆₉₂	—	8	1	no	—
20	#13063436	—	8	1	no	—
21	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
22	C₈×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₇₃	SD₁₆	1	1	yes	characteristic
24	C₂²×C₁₇₃	SD₃₂	1	1	yes	characteristic
25	C₂×D₁₇₃	—	8	2	no	—
26	C₂×D₁₇₃	—	16	2	no	—
27	C₄×C₁₇₃	#102	1	1	yes	characteristic
28	C₄×C₁₇₃	D₁₆	1	1	yes	characteristic
29	C₂×C₁₇₃	—	2	1	no	—
30	C₂×C₁₇₃	#357	1	1	yes	characteristic
31	D₁₇₃	—	16	2	no	—
32	C₁₇₃	#1419	1	1	yes	characteristic, complemented, Sylow
33	#1419	—	173	1	no	maximal, Sylow
34	C₂×D₁₆	—	173	1	no	—
35	M₆₄	D₁₇₃	1	1	yes	characteristic, complemented
36	#368	—	173	1	no	—
37	C₂×D₈	—	346	1	no	—
38	D₁₆	—	346	1	no	—
39	D₁₆	—	173	2	no	—
40	C₂×C₁₆	C₂×D₁₇₃	1	1	yes	characteristic
41	#108	—	346	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×D₄	—	692	1	no	—
44	D₈	—	346	2	no	—
45	D₈	—	692	1	no	—
46	M₁₆	—	692	1	no	—
47	C₂×C₈	D₆₉₂	1	1	yes	characteristic
48	C₁₆	C₄×D₁₇₃	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₁₇₃	1	1	yes	characteristic
50	C₂³	—	1384	1	no	—
51	D₄	—	1384	1	no	—
52	D₄	—	692	2	no	—
53	C₂×C₄	D₁₃₈₄	1	1	yes	characteristic
54	C₈	—	1384	1	no	—
55	C₈	#14583803	1	1	yes	characteristic
56	C₈	#14583815	1	1	yes	characteristic
57	C₂²	—	1384	2	no	—
58	C₂²	—	2768	2	no	—
59	C₂²	#19149624	1	1	yes	characteristic
60	C₄	#19149601	1	1	yes	characteristic
61	C₄	D₂₇₆₈	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	2768	2	no	—
64	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	C₂×D₂₇₆₈	C₂
cosocle	C₂×C₁₆×C₁₇₃	C₂²
hypercenter	M₆₄	D₁₇₃
hypocenter	C₁₇₃	#1419
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. #26011321	C₂	semidirect
3. M₆₄×C₁₇₃	C₂	semidirect
32. C₁₇₃	#1419	semidirect
35. M₆₄	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₇₆₈	2
#26011321	2
M₆₄×C₁₇₃	2
#1419	173