Subgroups — #34209323

Order 20224.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765709	C₂	1	2	yes	maximal, complemented
2	Dic₂₅₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761616	C₂	1	2	yes	maximal, complemented
4	D₅₀₅₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₆₄×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098995	—	2	2	no	—
7	Dic₁₂₆₄	C₂²	1	2	yes	—
8	#19098402	—	2	2	no	—
9	D₂₅₂₈	C₂²	1	2	yes	—
10	C₂×C₃₂×C₇₉	C₂²	1	1	yes	characteristic
11	C₆₄×C₇₉	C₂²	1	1	yes	characteristic
12	C₆₄×C₇₉	C₂²	1	1	yes	characteristic
13	#14579183	—	4	2	no	—
14	Dic₆₃₂	—	2	2	no	—
15	#14579106	—	4	2	no	—
16	D₁₂₆₄	—	2	2	no	—
17	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
18	C₃₂×C₇₉	D₄	1	1	yes	characteristic
19	C₃₂×C₇₉	C₂³	1	1	yes	characteristic
20	#11929624	—	8	2	no	—
21	Dic₃₁₆	—	4	2	no	—
22	#11929613	—	8	2	no	—
23	D₆₃₂	—	4	2	no	—
24	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
25	C₁₆×C₇₉	D₈	1	1	yes	characteristic
26	C₁₆×C₇₉	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
28	C₂²⋊D₇₉	—	16	2	no	—
29	D₃₁₆	—	8	2	no	—
30	C₄×D₇₉	—	16	2	no	—
31	Dic₁₅₈	—	8	2	no	—
32	C₈×C₇₉	D₁₆	1	1	yes	characteristic
33	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
34	C₂²×C₇₉	D₃₂	1	1	yes	characteristic
35	C₂×D₇₉	—	16	2	no	—
36	C₄×C₇₉	D₃₂	1	1	yes	characteristic
37	C₄×C₇₉	C₂×D₁₆	1	1	yes	characteristic
38	Dic₇₉	—	16	2	no	—
39	#13741	—	79	1	no	maximal, Sylow
40	C₂×C₇₉	—	2	1	no	—
41	C₂×C₇₉	C₂×D₃₂	1	1	yes	characteristic
42	D₇₉	—	32	2	no	—
43	#2262	—	79	2	no	—
44	Q₁₂₈	—	79	1	no	—
45	SD₁₂₈	—	79	2	no	—
46	D₆₄	—	79	1	no	—
47	C₂×C₆₄	D₇₉	1	1	yes	characteristic, complemented
48	C₇₉	#13741	1	1	yes	characteristic, complemented, Sylow
49	#508	—	158	2	no	—
50	Q₆₄	—	79	2	no	—
51	SD₆₄	—	158	2	no	—
52	D₃₂	—	79	2	no	—
53	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
54	C₆₄	C₂×D₇₉	1	1	yes	characteristic
55	C₆₄	C₂×D₇₉	1	1	yes	characteristic
56	#135	—	316	2	no	—
57	Q₃₂	—	158	2	no	—
58	SD₃₂	—	316	2	no	—
59	D₁₆	—	158	2	no	—
60	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
61	C₃₂	D₃₁₆	1	1	yes	characteristic
62	C₃₂	C₂²×D₇₉	1	1	yes	characteristic
63	#41	—	632	2	no	—
64	Q₁₆	—	316	2	no	—
65	SD₁₆	—	632	2	no	—
66	D₈	—	316	2	no	—
67	C₂×C₈	D₆₃₂	1	1	yes	characteristic
68	C₁₆	D₆₃₂	1	1	yes	characteristic
69	C₁₆	C₂×D₃₁₆	1	1	yes	characteristic
70	Q₈	—	632	2	no	—
71	D₄	—	632	2	no	—
72	D₄	—	1264	2	no	—
73	C₂×C₄	—	1264	2	no	—
74	C₂×C₄	D₁₂₆₄	1	1	yes	characteristic
75	C₈	D₁₂₆₄	1	1	yes	characteristic
76	C₈	C₂×D₆₃₂	1	1	yes	characteristic
77	C₂²	—	1264	2	no	—
78	C₂²	D₂₅₂₈	1	1	yes	characteristic
79	C₄	—	1264	2	no	—
80	C₄	D₂₅₂₈	1	1	yes	characteristic
81	C₄	C₂×D₁₂₆₄	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	2528	2	no	—
84	C₂	C₂×D₂₅₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₂₅₂₈
commutator, cosocle	C₃₂×C₇₉	C₂³
Frattini	C₃₂	C₂²×D₇₉
Fitting	C₂×C₆₄×C₇₉	C₂
socle	C₂×C₇₉	C₂×D₃₂
hypercenter	C₂×C₆₄	D₇₉
hypocenter	C₇₉	#13741
LCS	G ▷ C₃₂×C₇₉ ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
1. #23765709	C₂	semidirect
2. Dic₂₅₂₈	C₂	semidirect
3. #23761616	C₂	semidirect
4. D₅₀₅₆	C₂	semidirect
5. C₂×C₆₄×C₇₉	C₂	semidirect
47. C₂×C₆₄	D₇₉	semidirect
48. C₇₉	#13741	semidirect

Maximal subgroups

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

subgroup	index
#23765709	2
Dic₂₅₂₈	2
#23761616	2
D₅₀₅₆	2
C₂×C₆₄×C₇₉	2
#13741	79