Subgroups — #23761591

Order 10112.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₂₆₄	C₂	1	1	yes	maximal, characteristic
2	#19098393	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₆₃₂	—	2	1	no	—
5	D₁₂₆₄	—	2	1	no	—
6	D₁₂₆₄	C₄	1	2	yes	—
7	#14579097	—	2	1	no	—
8	C₂×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
9	C₃₂×C₇₉	—	2	1	no	—
10	C₂×D₃₁₆	—	4	1	no	—
11	D₆₃₂	—	2	2	no	—
12	D₆₃₂	—	4	1	no	—
13	#11929608	—	4	1	no	—
14	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
15	C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₇₉	D₄	1	1	yes	characteristic
17	C₂²×D₇₉	—	8	1	no	—
18	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
19	D₃₁₆	—	4	2	no	—
20	D₃₁₆	—	8	1	no	—
21	C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₇₉	SD₁₆	1	1	yes	characteristic
23	C₇₉⋊C₈	—	8	1	no	—
24	C₂²×C₇₉	SD₃₂	1	1	yes	characteristic
25	C₂×D₇₉	—	8	2	no	—
26	C₂×D₇₉	—	16	2	no	—
27	C₄×C₇₉	#102	1	1	yes	characteristic
28	C₄×C₇₉	D₁₆	1	1	yes	characteristic
29	C₂×C₇₉	—	2	1	no	—
30	C₂×C₇₉	#357	1	1	yes	characteristic
31	D₇₉	—	16	2	no	—
32	#1419	—	79	1	no	maximal, Sylow
33	C₇₉	#1419	1	1	yes	characteristic, complemented, Sylow
34	C₂×D₁₆	—	79	1	no	—
35	M₆₄	D₇₉	1	1	yes	characteristic, complemented
36	#368	—	79	1	no	—
37	C₂×D₈	—	158	1	no	—
38	D₁₆	—	79	2	no	—
39	D₁₆	—	158	1	no	—
40	C₂×C₁₆	C₂×D₇₉	1	1	yes	characteristic
41	#108	—	158	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×D₄	—	316	1	no	—
44	D₈	—	316	1	no	—
45	D₈	—	158	2	no	—
46	M₁₆	—	316	1	no	—
47	C₂×C₈	D₃₁₆	1	1	yes	characteristic
48	C₁₆	C₄×D₇₉	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
50	C₂³	—	632	1	no	—
51	D₄	—	316	2	no	—
52	D₄	—	632	1	no	—
53	C₂×C₄	D₆₃₂	1	1	yes	characteristic
54	C₈	—	632	1	no	—
55	C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
56	C₈	#11929613	1	1	yes	characteristic
57	C₂²	—	632	2	no	—
58	C₂²	—	1264	2	no	—
59	C₂²	#14579106	1	1	yes	characteristic
60	C₄	#14579083	1	1	yes	characteristic
61	C₄	D₁₂₆₄	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	1264	2	no	—
64	C₂	#19098365	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19098365
commutator	C₁₆×C₇₉	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₇₉
Fitting	M₆₄×C₇₉	C₂
socle	C₂×C₇₉	#357
prime radical	C₂×D₁₂₆₄	C₂
cosocle	C₂×C₁₆×C₇₉	C₂²
hypercenter	M₆₄	D₇₉
hypocenter	C₇₉	#1419
LCS	G ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
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. #19098393	C₂	semidirect
3. M₆₄×C₇₉	C₂	semidirect
33. C₇₉	#1419	semidirect
35. M₆₄	D₇₉	semidirect