Subgroups — #34176163

Order 20224.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₂₆₄	C₂	1	1	yes	maximal, characteristic
2	#23761604	C₂	1	1	yes	maximal, characteristic
3	C₇₉×M₁₂₈	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₃₂	—	2	1	no	—
5	Dic₁₂₆₄	—	2	1	no	—
6	Dic₁₂₆₄	C₄	1	2	yes	—
7	#19098393	—	2	1	no	—
8	C₂×C₃₂×C₇₉	C₂²	1	1	yes	characteristic
9	C₆₄×C₇₉	—	2	1	no	—
10	C₂×Dic₃₁₆	—	4	1	no	—
11	Dic₆₃₂	—	2	2	no	—
12	Dic₆₃₂	—	4	1	no	—
13	#14579097	—	4	1	no	—
14	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
15	C₃₂×C₇₉	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₇₉	D₄	1	1	yes	characteristic
17	C₂×Dic₁₅₈	—	8	1	no	—
18	Dic₃₁₆	—	4	2	no	—
19	Dic₃₁₆	—	8	1	no	—
20	#11929608	—	8	1	no	—
21	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
22	C₁₆×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₇₉	SD₁₆	1	1	yes	characteristic
24	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
25	C₂×Dic₇₉	—	16	1	no	—
26	Dic₁₅₈	—	8	2	no	—
27	Dic₁₅₈	—	16	1	no	—
28	C₈×C₇₉	#102	1	1	yes	characteristic
29	C₈×C₇₉	SD₃₂	1	1	yes	characteristic
30	C₇₉⋊C₈	—	16	1	no	—
31	C₂²×C₇₉	SD₆₄	1	1	yes	characteristic
32	C₄×C₇₉	#357	1	1	yes	characteristic
33	C₄×C₇₉	D₃₂	1	1	yes	characteristic
34	Dic₇₉	—	16	2	no	—
35	#7542	—	79	1	no	maximal, Sylow
36	C₂×C₇₉	—	2	1	no	—
37	C₂×C₇₉	#1415	1	1	yes	characteristic
38	C₂×Q₆₄	—	79	1	no	—
39	M₁₂₈	D₇₉	1	1	yes	characteristic
40	#1425	—	79	1	no	—
41	C₇₉	#7542	1	1	yes	characteristic, complemented, Sylow
42	C₂×Q₃₂	—	158	1	no	—
43	Q₆₄	—	79	2	no	—
44	Q₆₄	—	158	1	no	—
45	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
46	#368	—	158	1	no	—
47	C₆₄	—	2	1	no	—
48	C₂×Q₁₆	—	316	1	no	—
49	Q₃₂	—	316	1	no	—
50	Q₃₂	—	158	2	no	—
51	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
52	#108	—	316	1	no	—
53	C₃₂	C₄×D₇₉	1	1	yes	characteristic
54	C₃₂	C₂²⋊D₇₉	1	1	yes	characteristic
55	C₂×Q₈	—	632	1	no	—
56	Q₁₆	—	316	2	no	—
57	Q₁₆	—	632	1	no	—
58	M₁₆	—	632	1	no	—
59	C₂×C₈	D₆₃₂	1	1	yes	characteristic
60	C₁₆	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
61	C₁₆	#11929613	1	1	yes	characteristic
62	Q₈	—	632	2	no	—
63	Q₈	—	1264	1	no	—
64	C₂×C₄	—	1264	1	no	—
65	C₂×C₄	D₁₂₆₄	1	1	yes	characteristic
66	C₈	—	1264	1	no	—
67	C₈	#14579083	1	1	yes	characteristic
68	C₈	#14579106	1	1	yes	characteristic
69	C₂²	#19098402	1	1	yes	characteristic
70	C₄	—	1264	2	no	—
71	C₄	#19098365	1	1	yes	characteristic
72	C₄	D₂₅₂₈	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	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	C₂²×C₇₉	SD₆₄
cosocle	C₂×C₃₂×C₇₉	C₂²
hypercenter	M₁₂₈	D₇₉
hypocenter	C₇₉	#7542
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
41. C₇₉	#7542	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₂₆₄	2
#23761604	2
C₇₉×M₁₂₈	2
#7542	79