Subgroups — #34176157

Order 20224.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765709	C₂	1	1	yes	maximal, characteristic
2	#23761602	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₉×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098995	—	2	1	no	—
5	Dic₁₂₆₄	C₄	1	1	yes	characteristic, complemented
6	#19098402	—	2	1	no	—
7	D₂₅₂₈	C₄	1	1	yes	characteristic, complemented
8	#19098389	—	2	1	no	—
9	C₂×C₃₂×C₇₉	C₂²	1	1	yes	characteristic
10	C₆₄×C₇₉	—	2	1	no	—
11	#14579183	—	4	1	no	—
12	Dic₆₃₂	—	2	1	no	—
13	#14579106	—	4	1	no	—
14	D₁₂₆₄	—	2	1	no	—
15	#14579095	—	4	1	no	—
16	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
17	C₃₂×C₇₉	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₇₉	D₄	1	1	yes	characteristic
19	#11929624	—	8	1	no	—
20	Dic₃₁₆	—	4	1	no	—
21	#11929613	—	8	1	no	—
22	D₆₃₂	—	4	1	no	—
23	C₄⋊Dic₇₉	—	8	1	no	—
24	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
25	C₁₆×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₇₉	SD₁₆	1	1	yes	characteristic
27	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
28	C₂²⋊D₇₉	—	16	1	no	—
29	C₂×Dic₇₉	—	16	1	no	—
30	D₃₁₆	—	8	1	no	—
31	C₄×D₇₉	—	16	1	no	—
32	Dic₁₅₈	—	8	1	no	—
33	C₈×C₇₉	#102	1	1	yes	characteristic
34	C₈×C₇₉	SD₃₂	1	1	yes	characteristic
35	C₂²×C₇₉	D₃₂	1	1	yes	characteristic
36	C₂×D₇₉	—	16	1	no	—
37	C₄×C₇₉	#357	1	1	yes	characteristic
38	C₄×C₇₉	SD₆₄	1	1	yes	characteristic
39	Dic₇₉	—	16	1	no	—
40	Dic₇₉	—	32	1	no	—
41	#7540	—	79	1	no	maximal, Sylow
42	C₂×C₇₉	—	2	1	no	—
43	C₂×C₇₉	#1415	1	1	yes	characteristic
44	D₇₉	—	32	1	no	—
45	#2262	—	79	1	no	—
46	M₁₂₈	D₇₉	1	1	yes	characteristic, complemented
47	#1424	—	79	1	no	—
48	C₇₉	#7540	1	1	yes	characteristic, complemented, Sylow
49	#508	—	158	1	no	—
50	Q₆₄	—	79	1	no	—
51	SD₆₄	—	158	1	no	—
52	D₃₂	—	79	1	no	—
53	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
54	#366	—	158	1	no	—
55	C₆₄	—	2	1	no	—
56	#135	—	316	1	no	—
57	Q₃₂	—	158	1	no	—
58	SD₃₂	—	316	1	no	—
59	D₁₆	—	158	1	no	—
60	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
61	#107	—	316	1	no	—
62	C₃₂	C₄×D₇₉	1	1	yes	characteristic
63	C₃₂	C₂²⋊D₇₉	1	1	yes	characteristic
64	#41	—	632	1	no	—
65	Q₁₆	—	316	1	no	—
66	SD₁₆	—	632	1	no	—
67	D₈	—	316	1	no	—
68	C₂×C₈	D₆₃₂	1	1	yes	characteristic
69	C₄⋊C₄	—	632	1	no	—
70	C₁₆	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
71	C₁₆	#11929613	1	1	yes	characteristic
72	Q₈	—	632	1	no	—
73	D₄	—	632	1	no	—
74	D₄	—	1264	1	no	—
75	C₂×C₄	—	1264	1	no	—
76	C₂×C₄	—	1264	1	no	—
77	C₂×C₄	D₁₂₆₄	1	1	yes	characteristic
78	C₈	#14579083	1	1	yes	characteristic
79	C₈	#14579106	1	1	yes	characteristic
80	C₂²	—	1264	1	no	—
81	C₂²	D₂₅₂₈	1	1	yes	characteristic
82	C₄	—	2528	1	no	—
83	C₄	—	1264	1	no	—
84	C₄	#19098365	1	1	yes	characteristic
85	C₄	#19098402	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2528	1	no	—
88	C₂	#23761579	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23761579
commutator	C₃₂×C₇₉	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₇₉
Fitting	C₇₉×M₁₂₈	C₂
socle	C₂×C₇₉	#1415
prime radical	#23765709	C₂
cosocle	C₂×C₃₂×C₇₉	C₂²
hypercenter	M₁₂₈	D₇₉
hypocenter	C₇₉	#7540
LCS	G ▷ C₃₂×C₇₉ ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. #23761602	C₂	semidirect
3. C₇₉×M₁₂₈	C₂	semidirect
5. Dic₁₂₆₄	C₄	semidirect
7. D₂₅₂₈	C₄	semidirect
46. M₁₂₈	D₇₉	semidirect
48. C₇₉	#7540	semidirect

Maximal subgroups

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

subgroup	index
#23765709	2
#23761602	2
C₇₉×M₁₂₈	2
#7540	79