Subgroups — #34202978

Order 20224.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765629	C₂	1	4	yes	maximal
2	#23763018	C₂	1	1	yes	maximal, characteristic
3	#23761317	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#1710	C₂	1	1	yes	maximal, characteristic
5	#19098968	—	2	2	no	—
6	#19098943	C₂²	1	2	yes	—
7	#19098366	—	2	8	no	—
8	C₄×C₇₉⋊C₁₆	C₂²	1	2	yes	—
9	#500×C₇₉	C₂²	1	2	yes	—
10	#487×C₇₉	—	2	2	no	—
11	C₈²×C₇₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₃₁₆	—	2	4	no	—
13	#14579161	—	2	2	no	—
14	C₂×C₇₉⋊C₁₆	D₄	1	4	yes	—
15	#14579095	—	4	4	no	—
16	C₄⋊Q₈×C₇₉	—	2	2	no	—
17	#107×C₇₉	—	2	4	no	—
18	#103×C₇₉	—	4	4	no	—
19	C₄×C₈×C₇₉	D₄	1	2	yes	—
20	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₅₈	—	4	2	no	—
22	Dic₃₁₆	—	4	8	no	—
23	C₄⋊Dic₇₉	—	4	4	no	—
24	C₇₉⋊C₁₆	—	2	4	no	—
25	C₂×Q₈×C₇₉	—	4	2	no	—
26	C₂×C₈×C₇₉	D₈	1	4	yes	—
27	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
28	C₄⋊C₄×C₇₉	—	4	2	no	—
29	C₄⋊C₄×C₇₉	—	4	2	no	—
30	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₇₉	—	8	4	no	—
32	C₂×C₄×C₇₉	—	8	2	no	—
33	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
34	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
35	C₂×Dic₇₉	—	8	2	no	—
36	Dic₁₅₈	—	8	4	no	—
37	C₈×C₇₉	—	2	4	no	—
38	C₈×C₇₉	#135	1	4	yes	—
39	C₂²×C₇₉	#493	1	1	yes	characteristic
40	C₄×C₇₉	—	16	2	no	—
41	C₄×C₇₉	#495	1	2	yes	—
42	C₄×C₇₉	#510	1	4	yes	—
43	Dic₇₉	—	16	2	no	—
44	#12514	—	79	1	no	maximal, Sylow
45	C₂×C₇₉	#1709	1	1	yes	characteristic
46	C₂×C₇₉	#2251	1	2	yes	—
47	#2241	—	79	4	no	—
48	#1713	—	79	1	no	—
49	#1710	D₇₉	1	1	yes	characteristic
50	#1311	—	79	1	no	—
51	C₇₉	#12514	1	1	yes	characteristic, complemented, Sylow
52	#500	—	158	2	no	—
53	#500	C₂×D₇₉	1	2	yes	—
54	#494	—	79	2	no	—
55	#487	—	2	2	no	—
56	#358	—	158	8	no	—
57	C₄×C₁₆	—	79	2	no	—
58	C₈²	C₂×D₇₉	1	1	yes	characteristic
59	C₂×Q₁₆	—	158	4	no	—
60	C₄⋊Q₈	—	2	2	no	—
61	C₄⋊Q₈	—	158	2	no	—
62	C₂×C₁₆	—	79	4	no	—
63	#107	—	2	4	no	—
64	#107	—	316	4	no	—
65	#103	—	4	4	no	—
66	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
67	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
68	C₂×Q₈	—	4	2	no	—
69	C₂×Q₈	—	316	2	no	—
70	Q₁₆	—	316	8	no	—
71	C₂×C₈	#11929609	1	4	yes	—
72	C₂×C₈	D₄×D₇₉	1	2	yes	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	316	4	no	—
76	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
77	C₁₆	—	158	4	no	—
78	Q₈	—	8	4	no	—
79	Q₈	—	632	4	no	—
80	C₂×C₄	—	8	2	no	—
81	C₂×C₄	—	632	2	no	—
82	C₂×C₄	#14579157	1	1	yes	characteristic
83	C₂×C₄	C₂×#11929609	1	2	yes	—
84	C₈	—	2	4	no	—
85	C₈	#14579184	1	4	yes	—
86	C₂²	#19098939	1	1	yes	characteristic
87	C₄	—	16	2	no	—
88	C₄	—	1264	2	no	—
89	C₄	#19098948	1	2	yes	—
90	C₄	#19099003	1	4	yes	—
91	C₂	#23763007	1	1	yes	characteristic
92	C₂	#23765672	1	2	yes	—

	subgroup	G/H
center	C₂²	#19098939
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1710	C₂
socle, prime radical	C₂²×C₇₉	#493
hypercenter	#1710	D₇₉
hypocenter	C₇₉	#12514
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

Complements

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

N	H ≅ G/N	type
51. C₇₉	#12514	semidirect

Maximal subgroups

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

subgroup	index
#23765629	2
#23763018	2
#23761317	2
C₇₉×#1710	2
#12514	79