Subgroups — C₅²×#487 #13103224

Order 1600.

392 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₅²×#103	C₂	1	4	yes	maximal
4	C₄×C₅²×C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅²×Q₈	—	2	1	no	—
6	C₂×C₅²×Q₈	C₂²	1	2	yes	—
7	C₂×C₅²×C₈	C₂²	1	2	yes	—
8	C₅²×C₄⋊C₄	—	2	1	no	—
9	C₅²×C₄⋊C₄	—	2	1	no	—
10	C₅²×C₄⋊C₄	—	2	1	no	—
11	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
12	C₄²×C₅²	C₂²	1	1	yes	characteristic
13	C₅×#487	C₅	1	6	yes	maximal, direct product factor
14	C₅²×Q₈	—	2	4	no	—
15	C₅²×Q₈	—	4	2	no	—
16	C₂×C₄×C₅²	—	2	2	no	—
17	C₂×C₄×C₅²	—	2	2	no	—
18	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
20	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
21	C₅²×C₈	—	2	2	no	—
22	C₅×C₄⋊Q₈	C₂×C₅	1	6	yes	—
23	C₅×C₄⋊Q₈	C₂×C₅	1	6	yes	—
24	C₅×#103	C₂×C₅	1	24	yes	—
25	C₄×C₅×C₈	C₂×C₅	1	6	yes	—
26	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
27	C₄×C₅²	—	4	2	no	—
28	C₄×C₅²	—	4	2	no	—
29	C₄×C₅²	SD₁₆	1	2	yes	—
30	C₄×C₅²	Q₁₆	1	2	yes	—
31	C₄×C₅²	#41	1	2	yes	—
32	C₂×C₅×Q₈	—	2	6	no	—
33	C₂×C₅×Q₈	C₂²×C₅	1	12	yes	—
34	C₂×C₅×C₈	C₂²×C₅	1	12	yes	—
35	C₅×C₄⋊C₄	—	2	6	no	—
36	C₅×C₄⋊C₄	—	2	6	no	—
37	C₅×C₄⋊C₄	—	2	6	no	—
38	C₅×C₄⋊C₄	C₂²×C₅	1	12	yes	—
39	C₄²×C₅	C₂²×C₅	1	6	yes	—
40	#487	C₅²	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×C₅²	#124	1	1	yes	characteristic
42	C₂×C₅²	C₂×SD₁₆	1	1	yes	characteristic
43	C₂×C₅²	C₂×Q₁₆	1	1	yes	characteristic
44	C₅×Q₈	—	2	24	no	—
45	C₅×Q₈	—	4	12	no	—
46	C₂×C₄×C₅	—	2	12	no	—
47	C₂×C₄×C₅	—	2	12	no	—
48	C₂×C₄×C₅	C₅×D₄	1	6	yes	—
49	C₂×C₄×C₅	C₅×D₄	1	6	yes	—
50	C₂×C₄×C₅	C₂³×C₅	1	6	yes	—
51	C₅×C₈	—	2	12	no	—
52	C₄⋊Q₈	C₂×C₅²	1	1	yes	characteristic
53	C₄⋊Q₈	C₂×C₅²	1	1	yes	characteristic
54	#103	C₂×C₅²	1	4	yes	—
55	C₄×C₈	C₂×C₅²	1	1	yes	characteristic
56	C₅²	#487	1	1	yes	characteristic, direct product factor, Sylow
57	C₂²×C₅	C₂×C₅×D₄	1	6	yes	—
58	C₄×C₅	—	4	12	no	—
59	C₄×C₅	—	4	12	no	—
60	C₄×C₅	C₅×SD₁₆	1	12	yes	—
61	C₄×C₅	C₅×Q₁₆	1	12	yes	—
62	C₄×C₅	C₅×#41	1	12	yes	—
63	C₂×Q₈	—	2	1	no	—
64	C₂×Q₈	C₂²×C₅²	1	2	yes	—
65	C₂×C₈	C₂²×C₅²	1	2	yes	—
66	C₄⋊C₄	—	2	1	no	—
67	C₄⋊C₄	—	2	1	no	—
68	C₄⋊C₄	—	2	1	no	—
69	C₄⋊C₄	C₂²×C₅²	1	2	yes	—
70	C₄²	C₂²×C₅²	1	1	yes	characteristic
71	C₂×C₅	C₅×#124	1	6	yes	—
72	C₂×C₅	C₂×C₅×SD₁₆	1	6	yes	—
73	C₂×C₅	C₂×C₅×Q₁₆	1	6	yes	—
74	Q₈	—	2	4	no	—
75	Q₈	—	4	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	C₅²×D₄	1	1	yes	characteristic
79	C₂×C₄	C₅²×D₄	1	1	yes	characteristic
80	C₂×C₄	C₂³×C₅²	1	1	yes	characteristic
81	C₈	—	2	2	no	—
82	C₅	C₅×#487	1	6	yes	direct product factor
83	C₂²	C₂×C₅²×D₄	1	1	yes	characteristic
84	C₄	—	4	2	no	—
85	C₄	—	4	2	no	—
86	C₄	C₅²×SD₁₆	1	2	yes	—
87	C₄	C₅²×Q₁₆	1	2	yes	—
88	C₄	C₅²×#41	1	2	yes	—
89	C₂	C₅²×#124	1	1	yes	characteristic
90	C₂	C₂×C₅²×SD₁₆	1	1	yes	characteristic
91	C₂	C₂×C₅²×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₅²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₅²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅² ◁ C₄²×C₅² ◁ G

Complements

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

N	H ≅ G/N	type
13. C₅×#487	C₅	direct (N × H)
40. #487	C₅²	direct (N × H)
56. C₅²	#487	direct (N × H)
82. C₅	C₅×#487	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²×C₄⋊Q₈	2
C₅²×#103	2
C₄×C₅²×C₈	2
C₅×#487	5

Subgroups — C52×#487 #13103224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#487 #13103224