Subgroups — #34203006

Order 20224.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765675	C₂	1	4	yes	maximal
2	#23763018	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
4	#19098968	—	2	2	no	—
5	#19098968	C₂²	1	4	yes	—
6	#19098943	—	2	2	no	—
7	#19098389	—	2	8	no	—
8	C₄×C₁₆×C₇₉	C₂²	1	2	yes	—
9	C₈²×C₇₉	C₂²	1	1	yes	characteristic
10	C₂×Dic₃₁₆	—	4	4	no	—
11	#14579161	—	2	4	no	—
12	#14579095	—	2	8	no	—
13	#14579095	—	4	4	no	—
14	C₂×C₁₆×C₇₉	D₄	1	4	yes	—
15	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
16	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
17	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
18	C₂×Dic₁₅₈	—	4	4	no	—
19	Dic₃₁₆	—	8	8	no	—
20	C₄⋊Dic₇₉	—	4	4	no	—
21	C₄⋊Dic₇₉	—	4	4	no	—
22	C₂×C₈×C₇₉	D₈	1	2	yes	—
23	C₂×C₈×C₇₉	D₈	1	2	yes	—
24	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
25	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
26	C₁₆×C₇₉	Q₁₆	1	8	yes	complemented
27	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
28	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
29	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
30	C₂×Dic₇₉	—	8	4	no	—
31	Dic₁₅₈	—	8	8	no	—
32	C₈×C₇₉	D₁₆	1	4	yes	—
33	C₈×C₇₉	Q₃₂	1	4	yes	—
34	C₈×C₇₉	C₂×Q₁₆	1	4	yes	—
35	C₂²×C₇₉	#493	1	1	yes	characteristic
36	C₄×C₇₉	#494	1	2	yes	—
37	C₄×C₇₉	C₂×D₁₆	1	2	yes	—
38	C₄×C₇₉	C₂×Q₃₂	1	2	yes	—
39	Dic₇₉	—	16	4	no	—
40	#12521	—	79	1	no	maximal, Sylow
41	C₂×C₇₉	#1713	1	1	yes	characteristic
42	C₂×C₇₉	#2246	1	1	yes	characteristic
43	C₂×C₇₉	#2247	1	1	yes	characteristic
44	#2252	—	79	4	no	—
45	#1713	—	79	2	no	—
46	C₈×C₁₆	D₇₉	1	1	yes	characteristic
47	C₇₉	#12521	1	1	yes	characteristic, complemented, Sylow
48	#500	—	79	4	no	—
49	#500	—	158	2	no	—
50	#494	—	158	2	no	—
51	#366	—	158	8	no	—
52	C₄×C₁₆	C₂×D₇₉	1	2	yes	—
53	C₈²	C₂×D₇₉	1	1	yes	characteristic
54	C₂×Q₁₆	—	316	4	no	—
55	C₄⋊Q₈	—	158	4	no	—
56	C₂×C₁₆	D₃₁₆	1	4	yes	—
57	#107	—	316	4	no	—
58	#107	—	158	8	no	—
59	C₄×C₈	D₃₁₆	1	1	yes	characteristic
60	C₄×C₈	D₃₁₆	1	1	yes	characteristic
61	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
62	C₂×Q₈	—	316	4	no	—
63	Q₁₆	—	632	8	no	—
64	C₂×C₈	D₆₃₂	1	2	yes	—
65	C₂×C₈	D₆₃₂	1	2	yes	—
66	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
67	C₄⋊C₄	—	316	4	no	—
68	C₄⋊C₄	—	316	4	no	—
69	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
70	C₁₆	Dic₃₁₆	1	8	yes	complemented
71	Q₈	—	632	8	no	—
72	C₂×C₄	—	632	4	no	—
73	C₂×C₄	#14579158	1	1	yes	characteristic
74	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
75	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
76	C₈	D₁₂₆₄	1	4	yes	—
77	C₈	Dic₆₃₂	1	4	yes	—
78	C₈	C₂×Dic₃₁₆	1	4	yes	—
79	C₂²	#19098940	1	1	yes	characteristic
80	C₄	—	1264	4	no	—
81	C₄	#19098943	1	2	yes	—
82	C₄	C₂×D₁₂₆₄	1	2	yes	—
83	C₄	C₂×Dic₆₃₂	1	2	yes	—
84	C₂	#23763018	1	1	yes	characteristic
85	C₂	#23765650	1	1	yes	characteristic
86	C₂	#23765653	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098940
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₈×C₁₆×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	#493
hypercenter	C₈×C₁₆	D₇₉
hypocenter	C₇₉	#12521
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
26. C₁₆×C₇₉	Q₁₆	semidirect
47. C₇₉	#12521	semidirect
70. C₁₆	Dic₃₁₆	semidirect

Maximal subgroups

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

subgroup	index
#23765675	2
#23763018	2
C₈×C₁₆×C₇₉	2
#12521	79