Subgroups — #34176083

Order 20224.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765693	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇₉⋊C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₉×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098979	C₄	1	1	yes	characteristic
5	#19098397	C₄	1	1	yes	characteristic
6	#19098396	—	2	1	no	—
7	C₃₂×D₇₉	—	2	1	no	—
8	C₇₉⋊C₆₄	—	2	1	no	—
9	C₂×C₃₂×C₇₉	C₂²	1	1	yes	characteristic
10	C₆₄×C₇₉	—	2	1	no	—
11	#14579167	C₈	1	1	yes	characteristic
12	#14579101	C₈	1	1	yes	characteristic
13	#14579100	—	2	1	no	—
14	C₁₆×D₇₉	—	2	1	no	—
15	C₇₉⋊C₃₂	—	2	1	no	—
16	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
17	C₃₂×C₇₉	D₄	1	1	yes	characteristic
18	C₃₂×C₇₉	D₄	1	1	yes	characteristic
19	#11929624	C₁₆	1	1	yes	characteristic
20	#11929608	C₁₆	1	1	yes	characteristic
21	#11929607	—	2	1	no	—
22	C₈×D₇₉	—	2	1	no	—
23	C₇₉⋊C₁₆	—	2	1	no	—
24	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
25	C₁₆×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₇₉	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₇₉	C₂×C₁₆	1	1	yes	characteristic
28	C₂²⋊D₇₉	—	2	1	no	—
29	D₃₁₆	C₃₂	1	1	yes	characteristic
30	C₄×D₇₉	—	2	1	no	—
31	Dic₁₅₈	C₃₂	1	1	yes	characteristic
32	C₈×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
33	C₈×C₇₉	M₃₂	1	1	yes	characteristic
34	C₇₉⋊C₈	—	2	1	no	—
35	C₂²×C₇₉	M₆₄	1	1	yes	characteristic
36	C₂×D₇₉	—	2	1	no	—
37	C₄×C₇₉	C₂²⋊C₁₆	1	1	yes	characteristic
38	C₄×C₇₉	C₂×C₃₂	1	1	yes	characteristic
39	Dic₇₉	—	2	1	no	—
40	#7514	—	79	1	no	maximal, Sylow
41	C₂×C₇₉	—	2	1	no	—
42	C₂×C₇₉	C₂²⋊C₃₂	1	1	yes	characteristic
43	D₇₉	—	4	1	no	—
44	#2258	—	79	1	no	—
45	M₁₂₈	D₇₉	1	1	yes	characteristic, complemented
46	C₂×C₆₄	—	79	1	no	—
47	C₇₉	#7514	1	1	yes	characteristic, complemented, Sylow
48	#504	—	79	1	no	—
49	M₆₄	—	79	1	no	—
50	M₆₄	—	158	1	no	—
51	C₂×C₃₂	—	158	1	no	—
52	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
53	C₆₄	—	2	1	no	—
54	C₆₄	—	158	1	no	—
55	#131	—	79	1	no	—
56	M₃₂	—	79	1	no	—
57	M₃₂	—	158	1	no	—
58	C₂×C₁₆	—	158	1	no	—
59	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
60	C₃₂	—	158	1	no	—
61	C₃₂	D₃₁₆	1	1	yes	characteristic
62	C₃₂	C₂²⋊D₇₉	1	1	yes	characteristic
63	#41	—	79	1	no	—
64	M₁₆	—	79	1	no	—
65	M₁₆	—	158	1	no	—
66	C₂×C₈	—	158	1	no	—
67	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
68	C₁₆	—	158	1	no	—
69	C₁₆	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
70	C₁₆	#11929607	1	1	yes	characteristic
71	Q₈	—	79	1	no	—
72	D₄	—	79	1	no	—
73	D₄	—	158	1	no	—
74	C₂×C₄	—	158	1	no	—
75	C₂×C₄	C₁₆×D₇₉	1	1	yes	characteristic
76	C₈	—	158	1	no	—
77	C₈	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
78	C₈	#14579100	1	1	yes	characteristic
79	C₂²	—	158	1	no	—
80	C₂²	#19098396	1	1	yes	characteristic
81	C₄	—	158	1	no	—
82	C₄	(C₂×D₇₉)⋊C₁₆	1	1	yes	characteristic
83	C₄	C₃₂×D₇₉	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	316	1	no	—
86	C₂	(C₂×D₇₉)⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	C₂²⋊D₇₉
commutator	C₄×C₇₉	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₇₉
Fitting	C₇₉×M₁₂₈	C₂
socle	C₂×C₇₉	C₂²⋊C₃₂
prime radical	#11929624	C₁₆
cosocle	C₂×C₃₂×C₇₉	C₂²
hypercenter	M₁₂₈	D₇₉
hypocenter	C₇₉	#7514
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₇₉⋊C₆₄	C₂	semidirect
3. C₇₉×M₁₂₈	C₂	semidirect
45. M₁₂₈	D₇₉	semidirect
47. C₇₉	#7514	semidirect

Maximal subgroups

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

subgroup	index
#23765693	2
C₂×C₇₉⋊C₆₄	2
C₇₉×M₁₂₈	2
#7514	79