Subgroups — C₂²×#564×C₇₉ #34174381

Order 20224.

6374 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#564×C₇₉	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#125×C₇₉	C₂	1	15	yes	maximal
3	#564×C₇₉	C₂²	1	256	yes	direct product factor
4	C₂×#125×C₇₉	C₂²	1	360	yes	—
5	C₂²×C₄⋊C₄×C₇₉	C₂²	1	30	yes	—
6	C₂²×C₄²×C₇₉	C₂²	1	5	yes	—
7	C₂³×C₄×C₇₉	C₂³	1	15	yes	—
8	#125×C₇₉	C₂³	1	960	yes	—
9	C₂×C₄⋊C₄×C₇₉	C₂³	1	360	yes	—
10	C₂×C₄²×C₇₉	C₂³	1	60	yes	—
11	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
12	C₂²×C₄×C₇₉	—	2	60	no	—
13	C₂²×C₄×C₇₉	C₂⁴	1	90	yes	—
14	C₄⋊C₄×C₇₉	C₂⁴	1	480	yes	—
15	C₄²×C₇₉	C₂⁴	1	80	yes	—
16	C₂³×C₇₉	2₋¹⁺⁴	1	12	yes	—
17	C₂³×C₇₉	C₂⁵	1	3	yes	—
18	C₂×C₄×C₇₉	—	2	180	no	—
19	C₂×C₄×C₇₉	C₂⁵	1	60	yes	—
20	C₂²×C₇₉	#564	1	16	yes	direct product factor
21	C₂²×C₇₉	C₂×2₋¹⁺⁴	1	18	yes	—
22	C₂²×C₇₉	C₂⁶	1	1	yes	characteristic
23	C₄×C₇₉	—	2	60	no	—
24	C₂²×#564	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₇₉	C₂×#564	1	12	yes	direct product factor
26	C₂×C₇₉	C₂²×2₋¹⁺⁴	1	3	yes	—
27	C₂×#564	C₂×C₇₉	1	48	yes	direct product factor
28	C₂²×#125	C₂×C₇₉	1	15	yes	—
29	C₇₉	C₂²×#564	1	1	yes	characteristic, direct product factor, Sylow
30	#564	C₂²×C₇₉	1	256	yes	direct product factor
31	C₂×#125	C₂²×C₇₉	1	360	yes	—
32	C₂²×C₄⋊C₄	C₂²×C₇₉	1	30	yes	—
33	C₂²×C₄²	C₂²×C₇₉	1	5	yes	—
34	C₂³×C₄	C₂³×C₇₉	1	15	yes	—
35	#125	C₂³×C₇₉	1	960	yes	—
36	C₂×C₄⋊C₄	C₂³×C₇₉	1	360	yes	—
37	C₂×C₄²	C₂³×C₇₉	1	60	yes	—
38	C₂⁴	C₂⁴×C₇₉	1	1	yes	characteristic
39	C₂²×C₄	—	2	60	no	—
40	C₂²×C₄	C₂⁴×C₇₉	1	90	yes	—
41	C₄⋊C₄	C₂⁴×C₇₉	1	480	yes	—
42	C₄²	C₂⁴×C₇₉	1	80	yes	—
43	C₂³	2₋¹⁺⁴×C₇₉	1	12	yes	—
44	C₂³	C₂⁵×C₇₉	1	3	yes	—
45	C₂×C₄	—	2	180	no	—
46	C₂×C₄	C₂⁵×C₇₉	1	60	yes	—
47	C₂²	#564×C₇₉	1	16	yes	direct product factor
48	C₂²	C₂×2₋¹⁺⁴×C₇₉	1	18	yes	—
49	C₂²	C₂⁶×C₇₉	1	1	yes	characteristic
50	C₄	—	2	60	no	—
51	C₂	C₂×#564×C₇₉	1	12	yes	direct product factor
52	C₂	C₂²×2₋¹⁺⁴×C₇₉	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₇₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₇₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#564×C₇₉	C₂	direct (N × H)
3. #564×C₇₉	C₂²	direct (N × H)
20. C₂²×C₇₉	#564	direct (N × H)
24. C₂²×#564	C₇₉	direct (N × H)
25. C₂×C₇₉	C₂×#564	direct (N × H)
27. C₂×#564	C₂×C₇₉	direct (N × H)
29. C₇₉	C₂²×#564	direct (N × H)
30. #564	C₂²×C₇₉	direct (N × H)
47. C₂²	#564×C₇₉	direct (N × H)
51. C₂	C₂×#564×C₇₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#564×C₇₉	2
C₂²×#125×C₇₉	2
C₂²×#564	79

Subgroups — C22×#564×C79 #34174381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#564×C₇₉ #34174381