Subgroups — #34204669

Order 20224.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23761478	C₂	1	6	yes	maximal
2	C₄²×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇₉⋊C₁₆	C₂²	1	4	yes	—
4	C₂×C₄×C₈×C₇₉	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₇₉	C₂²	1	3	yes	—
6	C₄³×C₇₉	C₄	1	1	yes	characteristic
7	C₂×C₇₉⋊C₁₆	—	2	12	no	—
8	C₂²×C₈×C₇₉	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₇₉	C₈	1	3	yes	—
11	C₂×C₄²×C₇₉	C₈	1	1	yes	characteristic
12	C₂×C₄²×C₇₉	C₂×C₄	1	3	yes	—
13	C₄×C₈×C₇₉	—	2	6	no	—
14	C₄×C₈×C₇₉	—	2	6	no	—
15	C₇₉⋊C₁₆	—	4	4	no	—
16	C₂²×C₄×C₇₉	C₂×C₈	1	3	yes	—
17	C₂²×C₄×C₇₉	C₂×C₈	1	3	yes	—
18	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₈×C₇₉	—	2	3	no	—
20	C₂×C₈×C₇₉	—	2	3	no	—
21	C₂×C₈×C₇₉	M₁₆	1	6	yes	—
22	C₂×C₈×C₇₉	#41	1	6	yes	—
23	C₄²×C₇₉	—	2	6	no	—
24	C₄²×C₇₉	—	2	3	no	—
25	C₄²×C₇₉	—	2	3	no	—
26	C₄²×C₇₉	C₁₆	1	4	yes	complemented
27	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
28	C₂×C₄×C₇₉	—	2	3	no	—
29	C₂×C₄×C₇₉	—	2	3	no	—
30	C₂×C₄×C₇₉	—	2	3	no	—
31	C₂×C₄×C₇₉	—	2	3	no	—
32	C₂×C₄×C₇₉	C₂×C₁₆	1	6	yes	—
33	C₂×C₄×C₇₉	M₃₂	1	6	yes	—
34	C₂×C₄×C₇₉	#117	1	3	yes	—
35	C₂×C₄×C₇₉	C₂×M₁₆	1	3	yes	—
36	C₈×C₇₉	—	2	6	no	—
37	C₈×C₇₉	#126	1	4	yes	—
38	C₂²×C₇₉	#431	1	3	yes	—
39	C₂²×C₇₉	C₂²×C₁₆	1	1	yes	characteristic
40	C₂²×C₇₉	C₂×M₃₂	1	3	yes	—
41	C₄×C₇₉	—	2	6	no	—
42	C₄×C₇₉	—	2	6	no	—
43	C₄×C₇₉	#383	1	1	yes	characteristic
44	C₄×C₇₉	#432	1	3	yes	—
45	#12894	—	79	1	no	maximal, Sylow
46	C₂×C₇₉	#1839	1	1	yes	characteristic
47	C₂×C₇₉	#2162	1	3	yes	—
48	C₂×C₇₉	#2163	1	3	yes	—
49	C₄²×C₈	D₇₉	1	1	yes	characteristic
50	#1374	—	79	6	no	—
51	C₇₉	#12894	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₁₆	—	79	4	no	—
53	C₂×C₄×C₈	Dic₇₉	1	3	yes	—
54	C₂×C₄×C₈	C₂×D₇₉	1	3	yes	—
55	C₄³	Dic₇₉	1	1	yes	characteristic
56	C₂²×C₈	C₂×Dic₇₉	1	3	yes	—
57	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
58	C₂×C₄²	C₇₉⋊C₈	1	3	yes	—
59	C₂×C₄²	C₇₉⋊C₈	1	1	yes	characteristic
60	C₂×C₄²	C₂×Dic₇₉	1	3	yes	—
61	C₂×C₁₆	—	158	12	no	—
62	C₄×C₈	—	2	6	no	—
63	C₄×C₈	—	2	6	no	—
64	C₂²×C₄	C₂×C₇₉⋊C₈	1	3	yes	—
65	C₂²×C₄	C₂×C₇₉⋊C₈	1	3	yes	—
66	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
67	C₂×C₈	—	2	3	no	—
68	C₂×C₈	—	2	3	no	—
69	C₂×C₈	#11929608	1	6	yes	—
70	C₂×C₈	#11929624	1	6	yes	—
71	C₄²	—	2	6	no	—
72	C₄²	—	2	3	no	—
73	C₄²	—	2	3	no	—
74	C₄²	C₇₉⋊C₁₆	1	4	yes	complemented
75	C₁₆	—	316	4	no	—
76	C₂³	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	C₂×C₇₉⋊C₁₆	1	6	yes	—
82	C₂×C₄	#14579101	1	6	yes	—
83	C₂×C₄	#14579120	1	3	yes	—
84	C₂×C₄	C₂×#11929608	1	3	yes	—
85	C₈	—	2	6	no	—
86	C₈	#14579155	1	4	yes	—
87	C₂²	#19098616	1	3	yes	—
88	C₂²	C₂²×C₇₉⋊C₁₆	1	1	yes	characteristic
89	C₂²	C₂×#14579101	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	#19098431	1	1	yes	characteristic
93	C₄	#19098620	1	3	yes	—
94	C₂	#23763543	1	1	yes	characteristic
95	C₂	#23765199	1	3	yes	—
96	C₂	#23765203	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂²×C₁₆
Fitting	C₄²×C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×C₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₄²×C₈	D₇₉
hypocenter	C₇₉	#12894
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
26. C₄²×C₇₉	C₁₆	semidirect
51. C₇₉	#12894	semidirect
74. C₄²	C₇₉⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#23761478	2
C₄²×C₈×C₇₉	2
#12894	79