Subgroups — #34176151

Order 20224.

90 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	#23761600	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₄×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₃₂	—	2	1	no	—
5	Dic₁₂₆₄	—	2	1	no	—
6	Dic₁₂₆₄	C₄	1	2	yes	complemented
7	#19098389	—	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	#14579095	—	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	C₄⋊Dic₇₉	—	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	C₂×Dic₇₉	—	16	1	no	—
27	Dic₁₅₈	—	8	2	no	—
28	Dic₁₅₈	—	16	1	no	—
29	C₈×C₇₉	#102	1	1	yes	characteristic
30	C₈×C₇₉	SD₃₂	1	1	yes	characteristic
31	C₂²×C₇₉	D₃₂	1	1	yes	characteristic
32	C₄×C₇₉	#357	1	1	yes	characteristic
33	C₄×C₇₉	SD₆₄	1	1	yes	characteristic
34	Dic₇₉	—	16	2	no	—
35	Dic₇₉	—	32	1	no	—
36	#7538	—	79	1	no	maximal, Sylow
37	C₂×C₇₉	#1415	1	1	yes	characteristic
38	C₂×C₇₉	SD₁₂₈	1	1	yes	characteristic
39	C₂×C₇₉	Q₁₂₈	1	1	yes	characteristic
40	C₂×Q₆₄	—	79	1	no	—
41	C₂×C₆₄	D₇₉	1	1	yes	characteristic
42	#1423	—	79	1	no	—
43	C₇₉	#7538	1	1	yes	characteristic, complemented, Sylow
44	C₂×Q₃₂	—	158	1	no	—
45	Q₆₄	—	79	2	no	—
46	Q₆₄	—	158	1	no	—
47	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
48	#366	—	158	1	no	—
49	C₆₄	—	2	1	no	—
50	C₂×Q₁₆	—	316	1	no	—
51	Q₃₂	—	316	1	no	—
52	Q₃₂	—	158	2	no	—
53	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
54	#107	—	316	1	no	—
55	C₃₂	C₄×D₇₉	1	1	yes	characteristic
56	C₃₂	C₂²⋊D₇₉	1	1	yes	characteristic
57	C₂×Q₈	—	632	1	no	—
58	Q₁₆	—	316	2	no	—
59	Q₁₆	—	632	1	no	—
60	C₂×C₈	D₆₃₂	1	1	yes	characteristic
61	C₄⋊C₄	—	632	1	no	—
62	C₁₆	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
63	C₁₆	#11929613	1	1	yes	characteristic
64	Q₈	—	632	2	no	—
65	Q₈	—	1264	1	no	—
66	C₂×C₄	—	1264	1	no	—
67	C₂×C₄	—	1264	1	no	—
68	C₂×C₄	D₁₂₆₄	1	1	yes	characteristic
69	C₈	#14579083	1	1	yes	characteristic
70	C₈	#14579106	1	1	yes	characteristic
71	C₂²	D₂₅₂₈	1	1	yes	characteristic
72	C₄	—	2528	1	no	—
73	C₄	—	1264	2	no	—
74	C₄	#19098365	1	1	yes	characteristic
75	C₄	#19098402	1	1	yes	characteristic
76	C₂	#23761579	1	1	yes	characteristic
77	C₂	#23761616	1	1	yes	characteristic
78	C₂	Dic₂₅₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₂₅₂₈
commutator	C₃₂×C₇₉	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₇₉
Fitting	C₂×C₆₄×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	D₃₂
cosocle	C₂×C₃₂×C₇₉	C₂²
hypercenter	C₂×C₆₄	D₇₉
hypocenter	C₇₉	#7538
LCS	G ▷ C₃₂×C₇₉ ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ 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
6. Dic₁₂₆₄	C₄	semidirect
43. C₇₉	#7538	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
#23761600	2
C₂×C₆₄×C₇₉	2
#7538	79