Subgroups — #34203093

Order 20224.

478 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098339	C₂	1	3	yes	maximal
2	C₂×C₄×C₇₉⋊C₁₆	C₂	1	3	yes	maximal
3	C₇₉×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
5	C₂²×C₇₉⋊C₁₆	C₂²	1	3	yes	—
6	#19098339	C₄	1	12	yes	complemented
7	C₄×C₇₉⋊C₁₆	C₄	1	12	yes	complemented
8	C₂×#97×C₇₉	C₄	1	3	yes	—
9	C₂×C₄×C₈×C₇₉	C₂²	1	3	yes	—
10	#376×C₇₉	C₄	1	1	yes	characteristic
11	C₂×C₇₉⋊C₁₆	C₂×C₄	1	6	yes	—
12	C₂×C₇₉⋊C₁₆	C₂×C₄	1	6	yes	—
13	C₂×C₇₉⋊C₁₆	C₂×C₄	1	12	yes	—
14	C₂²×C₈×C₇₉	C₂×C₄	1	3	yes	—
15	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₇₉	C₂×C₄	1	3	yes	—
17	#97×C₇₉	C₈	1	12	yes	—
18	C₄×C₈×C₇₉	C₂×C₄	1	6	yes	—
19	C₄×C₈×C₇₉	C₂×C₄	1	6	yes	—
20	#95×C₇₉	C₈	1	4	yes	—
21	C₇₉⋊C₁₆	C₄²	1	4	yes	—
22	C₇₉⋊C₁₆	C₄²	1	12	yes	—
23	C₂²×C₄×C₇₉	C₂×C₈	1	6	yes	—
24	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₈×C₇₉	C₄²	1	6	yes	—
26	C₂×C₈×C₇₉	C₂×C₈	1	12	yes	—
27	C₂×C₈×C₇₉	C₂²×C₄	1	3	yes	—
28	C₂×C₈×C₇₉	C₂²×C₄	1	3	yes	—
29	C₄²×C₇₉	C₄²	1	6	yes	—
30	C₄²×C₇₉	C₂×C₈	1	6	yes	—
31	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₄×C₇₉	—	2	12	no	—
33	C₂×C₄×C₇₉	C₄×C₈	1	12	yes	—
34	C₂×C₄×C₇₉	C₂×C₄²	1	3	yes	—
35	C₂×C₄×C₇₉	C₂²×C₈	1	3	yes	—
36	C₈×C₇₉	C₄×C₈	1	12	yes	—
37	C₈×C₇₉	C₂×C₄²	1	1	yes	characteristic
38	C₈×C₇₉	C₂×C₄²	1	3	yes	—
39	C₂²×C₇₉	C₂×C₄×C₈	1	3	yes	—
40	C₂²×C₇₉	#504	1	4	yes	—
41	C₄×C₇₉	—	2	12	no	—
42	C₄×C₇₉	C₄³	1	1	yes	characteristic
43	C₄×C₇₉	C₂×C₄×C₈	1	3	yes	—
44	#12539	—	79	1	no	maximal, Sylow
45	C₂×C₇₉	C₄²×C₈	1	1	yes	characteristic
46	C₂×C₇₉	#2108	1	6	yes	—
47	C₂×#346	—	79	3	no	—
48	C₂×C₄×C₁₆	—	79	3	no	—
49	#1726	D₇₉	1	1	yes	characteristic
50	C₇₉	#12539	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₁₆	—	79	1	no	—
52	C₂²×C₁₆	—	79	3	no	—
53	C₂×#97	Dic₇₉	1	3	yes	—
54	C₂×C₄×C₈	C₂×D₇₉	1	3	yes	—
55	#376	Dic₇₉	1	1	yes	characteristic
56	#346	—	79	12	no	—
57	C₄×C₁₆	—	79	12	no	—
58	C₂²×C₈	C₂×Dic₇₉	1	3	yes	—
59	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
60	C₂×C₄²	C₂×Dic₇₉	1	3	yes	—
61	C₂×C₁₆	—	79	6	no	—
62	C₂×C₁₆	—	79	6	no	—
63	C₂×C₁₆	—	79	12	no	—
64	#97	C₇₉⋊C₈	1	12	yes	—
65	C₄×C₈	C₄×D₇₉	1	6	yes	—
66	C₄×C₈	C₄×D₇₉	1	6	yes	—
67	#95	C₇₉⋊C₈	1	4	yes	—
68	C₂²×C₄	C₂×C₇₉⋊C₈	1	6	yes	—
69	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
70	C₂×C₈	C₄×Dic₇₉	1	6	yes	—
71	C₂×C₈	C₂×C₇₉⋊C₈	1	12	yes	—
72	C₂×C₈	C₂×C₄×D₇₉	1	3	yes	—
73	C₂×C₈	C₂×C₄×D₇₉	1	3	yes	—
74	C₄²	C₄×Dic₇₉	1	6	yes	—
75	C₄²	C₂×C₇₉⋊C₈	1	6	yes	—
76	C₁₆	—	79	4	no	—
77	C₁₆	—	79	12	no	—
78	C₂³	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
79	C₂×C₄	—	2	12	no	—
80	C₂×C₄	C₄×C₇₉⋊C₈	1	12	yes	—
81	C₂×C₄	C₂×C₄×Dic₇₉	1	3	yes	—
82	C₂×C₄	C₂²×C₇₉⋊C₈	1	3	yes	—
83	C₈	C₄×C₇₉⋊C₈	1	12	yes	—
84	C₈	C₄²×D₇₉	1	1	yes	characteristic
85	C₈	C₄²×D₇₉	1	3	yes	—
86	C₂²	C₂×C₄×C₇₉⋊C₈	1	3	yes	—
87	C₂²	#19098980	1	4	yes	—
88	C₄	—	2	12	no	—
89	C₄	C₄²×Dic₇₉	1	1	yes	characteristic
90	C₄	C₂×C₄×C₇₉⋊C₈	1	3	yes	—
91	C₂	C₄²×C₇₉⋊C₈	1	1	yes	characteristic
92	C₂	#23764997	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₉
commutator	C₂×C₇₉	C₄²×C₈
Fitting	C₇₉×#1726	C₂
socle, prime radical	C₂³×C₇₉	C₂²×C₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1726	D₇₉
hypocenter	C₇₉	#12539
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
6. #19098339	C₄	semidirect
7. C₄×C₇₉⋊C₁₆	C₄	semidirect
50. C₇₉	#12539	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098339	2
C₂×C₄×C₇₉⋊C₁₆	2
C₇₉×#1726	2
#12539	79