Subgroups — #34175972

Order 20224.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098393	C₂	1	2	yes	maximal
2	C₂²×C₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#14579097	—	2	2	no	—
4	#19098393	—	2	2	no	—
5	#19098393	C₄	1	4	yes	—
6	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
7	C₂×C₃₂×C₇₉	—	2	2	no	—
8	C₂×C₃₂×C₇₉	C₄	1	2	yes	—
9	C₂×#11929608	—	4	2	no	—
10	#14579097	—	2	4	no	—
11	#14579097	—	4	2	no	—
12	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
14	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
16	C₂×C₁₆×C₇₉	Q₈	1	1	yes	characteristic
17	C₃₂×C₇₉	—	2	2	no	—
18	#11929608	—	4	4	no	—
19	#11929608	—	8	2	no	—
20	C₂×C₇₉⋊C₈	—	8	2	no	—
21	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
22	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
23	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
25	C₂×C₈×C₇₉	Q₁₆	1	1	yes	characteristic
26	C₁₆×C₇₉	C₄²	1	1	yes	characteristic
27	C₁₆×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₁₆×C₇₉	C₄⋊C₄	1	2	yes	—
29	C₂³×C₇₉	Q₃₂	1	1	yes	characteristic
30	C₂×C₄×C₇₉	#102	1	2	yes	—
31	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
32	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
33	C₂×C₄×C₇₉	SD₃₂	1	2	yes	—
34	C₈×C₇₉	#95	1	1	yes	characteristic
35	C₈×C₇₉	#103	1	2	yes	—
36	C₈×C₇₉	#106	1	1	yes	characteristic
37	C₇₉⋊C₈	—	8	4	no	—
38	C₂²×C₇₉	—	2	2	no	—
39	C₂²×C₇₉	#358	1	2	yes	—
40	C₂²×C₇₉	#366	1	1	yes	characteristic
41	C₄×C₇₉	#340	1	1	yes	characteristic
42	C₄×C₇₉	#357	1	2	yes	—
43	C₄×C₇₉	#367	1	1	yes	characteristic
44	#7472	—	79	1	no	maximal, Sylow
45	C₂×C₇₉	—	2	2	no	—
46	C₂×C₇₉	#1380	1	1	yes	characteristic
47	C₂×C₇₉	#1425	1	2	yes	—
48	C₂²×C₃₂	D₇₉	1	1	yes	characteristic
49	C₂×#368	—	79	2	no	—
50	C₇₉	#7472	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
52	C₂×#108	—	158	2	no	—
53	C₂×C₃₂	—	2	2	no	—
54	C₂×C₃₂	Dic₇₉	1	2	yes	—
55	#368	—	79	4	no	—
56	#368	—	158	2	no	—
57	C₂×M₁₆	—	316	2	no	—
58	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
59	C₂×C₁₆	Dic₁₅₈	1	1	yes	characteristic
60	C₂×C₁₆	C₄×D₇₉	1	2	yes	—
61	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
62	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
63	#108	—	316	2	no	—
64	#108	—	158	4	no	—
65	C₃₂	—	2	2	no	—
66	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
67	M₁₆	—	316	4	no	—
68	M₁₆	—	632	2	no	—
69	C₂×C₈	—	632	2	no	—
70	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
71	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
72	C₂×C₈	#11929613	1	2	yes	—
73	C₂×C₈	Dic₃₁₆	1	1	yes	characteristic
74	C₁₆	C₄×Dic₇₉	1	1	yes	characteristic
75	C₁₆	C₂²⋊Dic₇₉	1	1	yes	characteristic
76	C₁₆	Dic₇₉⋊C₄	1	2	yes	—
77	C₂³	Dic₆₃₂	1	1	yes	characteristic
78	C₂×C₄	#14579083	1	2	yes	—
79	C₂×C₄	#14579095	1	1	yes	characteristic
80	C₂×C₄	D₁₂₆₄	1	1	yes	characteristic
81	C₂×C₄	#14579106	1	2	yes	—
82	C₈	—	632	4	no	—
83	C₈	C₇₉⋊#95	1	1	yes	characteristic
84	C₈	#14579086	1	2	yes	—
85	C₈	#14579093	1	1	yes	characteristic
86	C₂²	—	2	2	no	—
87	C₂²	#19098368	1	2	yes	—
88	C₂²	#19098389	1	1	yes	characteristic
89	C₄	#19098326	1	1	yes	characteristic
90	C₄	#19098365	1	2	yes	—
91	C₄	#19098391	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	#23761492	1	1	yes	characteristic
94	C₂	#23761604	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₁₂₆₄
commutator	C₁₆×C₇₉	C₄²
Frattini	C₂²×C₁₆	C₂×D₇₉
Fitting	C₂²×C₃₂×C₇₉	C₂
socle	C₂²×C₇₉	#366
prime radical	C₂³×C₇₉	Q₃₂
cosocle	C₂²×C₁₆×C₇₉	C₂²
hypercenter	C₂²×C₃₂	D₇₉
hypocenter	C₇₉	#7472
LCS	G ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

Complements

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

N	H ≅ G/N	type
50. C₇₉	#7472	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098393	2
C₂²×C₃₂×C₇₉	2
#7472	79