Subgroups — #34175601

Order 20224.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₁₆×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇₉⋊C₃₂	C₄	1	2	yes	—
5	C₄×C₇₉⋊C₁₆	C₄	1	1	yes	characteristic
6	C₈×C₇₉⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₇₉	C₄	1	2	yes	—
8	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
9	C₂×C₇₉⋊C₁₆	C₈	1	2	yes	—
10	C₄×C₇₉⋊C₈	C₈	1	1	yes	characteristic
11	C₄×C₇₉⋊C₈	C₈	1	1	yes	characteristic
12	C₇₉⋊C₃₂	—	2	2	no	—
13	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₇₉	—	2	2	no	—
17	C₂×C₇₉⋊C₈	C₁₆	1	2	yes	—
18	C₂×C₇₉⋊C₈	C₁₆	1	2	yes	—
19	C₇₉⋊C₁₆	—	2	2	no	—
20	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
21	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
22	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₇₉	M₁₆	1	2	yes	—
24	C₁₆×C₇₉	M₁₆	1	2	yes	—
25	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
26	C₂×C₄×C₇₉	C₂×C₁₆	1	1	yes	characteristic
27	C₂×C₄×C₇₉	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₇₉	#97	1	1	yes	characteristic
29	C₈×C₇₉	#97	1	1	yes	characteristic
30	C₈×C₇₉	M₃₂	1	2	yes	—
31	C₇₉⋊C₈	—	2	2	no	—
32	C₇₉⋊C₈	C₃₂	1	4	yes	complemented
33	C₂²×C₇₉	C₄×C₁₆	1	1	yes	characteristic
34	C₄×C₇₉	#322	1	1	yes	characteristic
35	C₄×C₇₉	#346	1	1	yes	characteristic
36	C₄×C₇₉	C₂×C₃₂	1	2	yes	—
37	C₄×C₇₉	M₆₄	1	2	yes	—
38	#7331	—	79	1	no	maximal, Sylow
39	C₂×C₇₉	#1312	1	1	yes	characteristic
40	C₂×C₇₉	C₄×C₃₂	1	1	yes	characteristic
41	C₂×C₇₉	#1397	1	1	yes	characteristic
42	C₄×C₃₂	—	79	1	no	—
43	C₄×C₃₂	D₇₉	1	1	yes	characteristic
44	C₈×C₁₆	—	79	1	no	—
45	C₇₉	#7331	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₃₂	—	79	2	no	—
47	C₂×C₃₂	Dic₇₉	1	2	yes	—
48	C₄×C₁₆	—	79	1	no	—
49	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
50	C₈²	—	79	1	no	—
51	C₂×C₁₆	—	79	2	no	—
52	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
53	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
54	C₄×C₈	—	79	1	no	—
55	C₄×C₈	—	79	1	no	—
56	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
57	C₃₂	—	2	2	no	—
58	C₃₂	—	158	2	no	—
59	C₂×C₈	—	79	2	no	—
60	C₂×C₈	—	79	2	no	—
61	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
62	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
63	C₄²	C₈×D₇₉	1	1	yes	characteristic
64	C₁₆	—	158	2	no	—
65	C₁₆	#11929607	1	2	yes	—
66	C₁₆	#11929608	1	2	yes	—
67	C₂×C₄	C₈×Dic₇₉	1	1	yes	characteristic
68	C₂×C₄	C₁₆×D₇₉	1	1	yes	characteristic
69	C₂×C₄	C₁₆×D₇₉	1	1	yes	characteristic
70	C₈	—	79	4	no	—
71	C₈	—	158	2	no	—
72	C₈	#14579071	1	1	yes	characteristic
73	C₈	#14579072	1	1	yes	characteristic
74	C₈	#14579100	1	2	yes	—
75	C₂²	C₁₆×Dic₇₉	1	1	yes	characteristic
76	C₄	#19098285	1	1	yes	characteristic
77	C₄	#19098338	1	1	yes	characteristic
78	C₄	C₃₂×D₇₉	1	2	yes	—
79	C₄	#19098396	1	2	yes	—
80	C₂	#23761319	1	1	yes	characteristic
81	C₂	C₃₂×Dic₇₉	1	1	yes	characteristic
82	C₂	#23761533	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₁₆	C₂×D₇₉
commutator	C₂×C₇₉	C₄×C₃₂
Fitting	C₄×C₃₂×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	C₄×C₁₆
cosocle	C₄×C₁₆×C₇₉	C₂²
hypercenter	C₄×C₃₂	D₇₉
hypocenter	C₇₉	#7331
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
32. C₇₉⋊C₈	C₃₂	semidirect
45. C₇₉	#7331	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇₉⋊C₃₂	2
C₁₆×C₇₉⋊C₈	2
C₄×C₃₂×C₇₉	2
#7331	79