Subgroups — #34178706

Order 20224.

478 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579072	C₂	1	2	yes	maximal
2	C₇₉⋊#1273	C₂	1	4	yes	maximal
3	C₄×#97×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579072	C₄	1	8	yes	—
5	C₂×C₄×C₇₉⋊C₈	C₂²	1	4	yes	—
6	C₂×#97×C₇₉	C₄	1	4	yes	—
7	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
8	C₄³×C₇₉	C₂²	1	1	yes	characteristic
9	C₂²×C₇₉⋊C₈	C₂×C₄	1	8	yes	—
10	#14579072	—	2	16	no	—
11	C₄×C₇₉⋊C₈	—	2	8	no	—
12	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
16	#97×C₇₉	—	2	8	no	—
17	C₄×C₈×C₇₉	—	2	4	no	—
18	C₂×C₇₉⋊C₈	—	2	8	no	—
19	C₂×C₇₉⋊C₈	—	2	8	no	—
20	C₂×C₇₉⋊C₈	M₁₆	1	16	yes	—
21	C₂²×C₄×C₇₉	C₄²	1	4	yes	—
22	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
23	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₈×C₇₉	—	2	4	no	—
25	C₂×C₈×C₇₉	—	2	4	no	—
26	C₂×C₈×C₇₉	M₁₆	1	8	yes	—
27	C₄²×C₇₉	—	2	4	no	—
28	C₄²×C₇₉	—	2	8	no	—
29	C₄²×C₇₉	#41	1	4	yes	—
30	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₄×C₇₉	—	2	8	no	—
32	C₂×C₄×C₇₉	—	2	4	no	—
33	C₂×C₄×C₇₉	#117	1	4	yes	—
34	C₂×C₄×C₇₉	#117	1	2	yes	—
35	C₂×C₄×C₇₉	C₂×M₁₆	1	8	yes	—
36	C₂×C₄×C₇₉	C₂×M₁₆	1	4	yes	—
37	C₈×C₇₉	—	2	8	no	—
38	C₇₉⋊C₈	—	2	16	no	—
39	C₂²×C₇₉	#376	1	1	yes	characteristic
40	C₂²×C₇₉	C₄×M₁₆	1	4	yes	—
41	C₂²×C₇₉	C₄×M₁₆	1	2	yes	—
42	C₄×C₇₉	—	4	4	no	—
43	C₄×C₇₉	#432	1	8	yes	—
44	C₄×C₇₉	#432	1	4	yes	—
45	#8136	—	79	1	no	maximal, Sylow
46	C₂×C₇₉	#1455	1	4	yes	—
47	C₂×C₇₉	#1745	1	2	yes	—
48	C₂×C₇₉	#1745	1	1	yes	characteristic
49	C₄×#97	—	79	2	no	—
50	C₄×#97	D₇₉	1	1	yes	characteristic
51	#1273	—	79	4	no	—
52	C₇₉	#8136	1	1	yes	characteristic, complemented, Sylow
53	C₂×#97	—	79	8	no	—
54	C₂×#97	Dic₇₉	1	4	yes	—
55	C₂×C₄×C₈	—	79	4	no	—
56	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
57	C₄³	C₂×D₇₉	1	1	yes	characteristic
58	C₂²×C₈	—	79	8	no	—
59	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
60	C₂×C₄²	C₄×D₇₉	1	4	yes	—
61	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
62	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
63	#97	—	2	8	no	—
64	#97	—	158	16	no	—
65	C₄×C₈	—	2	4	no	—
66	C₄×C₈	—	158	8	no	—
67	C₂²×C₄	C₄×Dic₇₉	1	4	yes	—
68	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
69	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
70	C₂×C₈	—	2	4	no	—
71	C₂×C₈	—	2	4	no	—
72	C₂×C₈	—	79	16	no	—
73	C₂×C₈	—	158	8	no	—
74	C₂×C₈	—	158	8	no	—
75	C₂×C₈	#11929608	1	8	yes	—
76	C₄²	—	2	8	no	—
77	C₄²	—	2	4	no	—
78	C₄²	#11929624	1	4	yes	—
79	C₂³	C₂×C₄×Dic₇₉	1	1	yes	characteristic
80	C₂×C₄	—	2	8	no	—
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	#14579118	1	4	yes	—
83	C₂×C₄	#14579120	1	2	yes	—
84	C₂×C₄	M₁₆×D₇₉	1	8	yes	—
85	C₂×C₄	C₂×#11929608	1	4	yes	—
86	C₈	—	2	8	no	—
87	C₈	—	158	16	no	—
88	C₂²	#19098409	1	1	yes	characteristic
89	C₂²	M₁₆×Dic₇₉	1	4	yes	—
90	C₂²	C₄×#11929608	1	2	yes	—
91	C₄	—	4	4	no	—
92	C₄	#19098618	1	8	yes	—
93	C₄	#19098620	1	4	yes	—
94	C₂	#23761703	1	4	yes	—
95	C₂	#23763148	1	2	yes	—
96	C₂	#23763149	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₉
commutator, socle, prime radical	C₂³×C₇₉	C₂×C₄²
Fitting	C₄×#97×C₇₉	C₂
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×#97	D₇₉
hypocenter	C₇₉	#8136
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
52. C₇₉	#8136	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579072	2
C₇₉⋊#1273	2
C₄×#97×C₇₉	2
#8136	79