Subgroups — #34176160

Order 20224.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₅₂₈	C₂	1	1	yes	maximal, characteristic
2	#23761604	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₉×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₂₆₄	—	2	1	no	—
5	D₂₅₂₈	—	2	1	no	—
6	D₂₅₂₈	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×D₆₃₂	—	4	1	no	—
11	D₁₂₆₄	—	2	2	no	—
12	D₁₂₆₄	—	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×D₃₁₆	—	8	1	no	—
18	D₆₃₂	—	4	2	no	—
19	D₆₃₂	—	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	C₂²×D₇₉	—	16	1	no	—
25	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
26	D₃₁₆	—	8	2	no	—
27	D₃₁₆	—	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	C₂×D₇₉	—	16	2	no	—
33	C₂×D₇₉	—	32	2	no	—
34	C₄×C₇₉	#357	1	1	yes	characteristic
35	C₄×C₇₉	D₃₂	1	1	yes	characteristic
36	#7541	—	79	1	no	maximal, Sylow
37	C₂×C₇₉	—	2	1	no	—
38	C₂×C₇₉	#1415	1	1	yes	characteristic
39	D₇₉	—	32	2	no	—
40	C₂×D₃₂	—	79	1	no	—
41	M₁₂₈	D₇₉	1	1	yes	characteristic, complemented
42	#1425	—	79	1	no	—
43	C₇₉	#7541	1	1	yes	characteristic, complemented, Sylow
44	C₂×D₁₆	—	158	1	no	—
45	D₃₂	—	79	2	no	—
46	D₃₂	—	158	1	no	—
47	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
48	#368	—	158	1	no	—
49	C₆₄	—	2	1	no	—
50	C₂×D₈	—	316	1	no	—
51	D₁₆	—	316	1	no	—
52	D₁₆	—	158	2	no	—
53	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
54	#108	—	316	1	no	—
55	C₃₂	C₄×D₇₉	1	1	yes	characteristic
56	C₃₂	C₂²⋊D₇₉	1	1	yes	characteristic
57	C₂×D₄	—	632	1	no	—
58	D₈	—	316	2	no	—
59	D₈	—	632	1	no	—
60	M₁₆	—	632	1	no	—
61	C₂×C₈	D₆₃₂	1	1	yes	characteristic
62	C₁₆	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
63	C₁₆	#11929613	1	1	yes	characteristic
64	C₂³	—	1264	1	no	—
65	D₄	—	632	2	no	—
66	D₄	—	1264	1	no	—
67	C₂×C₄	D₁₂₆₄	1	1	yes	characteristic
68	C₈	—	1264	1	no	—
69	C₈	#14579083	1	1	yes	characteristic
70	C₈	#14579106	1	1	yes	characteristic
71	C₂²	—	2528	2	no	—
72	C₂²	—	1264	2	no	—
73	C₂²	#19098402	1	1	yes	characteristic
74	C₄	#19098365	1	1	yes	characteristic
75	C₄	D₂₅₂₈	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	2528	2	no	—
78	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	C₂×D₂₅₂₈	C₂
cosocle	C₂×C₃₂×C₇₉	C₂²
hypercenter	M₁₂₈	D₇₉
hypocenter	C₇₉	#7541
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. #23761604	C₂	semidirect
3. C₇₉×M₁₂₈	C₂	semidirect
41. M₁₂₈	D₇₉	semidirect
43. C₇₉	#7541	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₅₂₈	2
#23761604	2
C₇₉×M₁₂₈	2
#7541	79