Subgroups — C₅²×#358 #13099639

Order 1600.

232 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅²×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅²×#107	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅²×Q₈	—	2	1	no	—
5	C₅²×Q₁₆	—	2	1	no	—
6	C₅²×Q₁₆	C₄	1	2	yes	complemented
7	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
8	C₅²×C₄⋊C₄	—	2	1	no	—
9	C₅²×C₁₆	—	2	1	no	—
10	C₅×#358	C₅	1	6	yes	maximal, direct product factor
11	C₅²×Q₈	—	2	2	no	—
12	C₅²×Q₈	—	4	1	no	—
13	C₂×C₄×C₅²	—	4	1	no	—
14	C₂×C₄×C₅²	—	4	1	no	—
15	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
16	C₅²×C₈	C₂×C₄	1	1	yes	characteristic
17	C₅²×C₈	D₄	1	1	yes	characteristic
18	C₂×C₅×Q₁₆	C₂×C₅	1	6	yes	—
19	C₂×C₅×C₁₆	C₂×C₅	1	6	yes	—
20	C₅×#107	C₂×C₅	1	6	yes	—
21	C₂²×C₅²	D₈	1	1	yes	characteristic
22	C₄×C₅²	—	4	2	no	—
23	C₄×C₅²	—	8	1	no	—
24	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
25	C₄×C₅²	SD₁₆	1	1	yes	characteristic
26	C₂×C₅×Q₈	—	2	6	no	—
27	C₅×Q₁₆	—	2	6	no	—
28	C₅×Q₁₆	C₄×C₅	1	12	yes	complemented
29	C₂×C₅×C₈	C₂²×C₅	1	6	yes	—
30	C₅×C₄⋊C₄	—	2	6	no	—
31	C₅×C₁₆	—	2	6	no	—
32	#358	C₅²	1	1	yes	characteristic, direct product factor, Sylow
33	C₂×C₅²	#102	1	1	yes	characteristic
34	C₂×C₅²	SD₃₂	1	1	yes	characteristic
35	C₂×C₅²	Q₃₂	1	1	yes	characteristic
36	C₅×Q₈	—	2	12	no	—
37	C₅×Q₈	—	4	6	no	—
38	C₂×C₄×C₅	—	4	6	no	—
39	C₂×C₄×C₅	—	4	6	no	—
40	C₂×C₄×C₅	C₅×D₄	1	6	yes	—
41	C₅×C₈	C₂×C₄×C₅	1	6	yes	—
42	C₅×C₈	C₅×D₄	1	6	yes	—
43	C₂×Q₁₆	C₂×C₅²	1	1	yes	characteristic
44	C₂×C₁₆	C₂×C₅²	1	1	yes	characteristic
45	#107	C₂×C₅²	1	1	yes	characteristic
46	C₅²	#358	1	1	yes	characteristic, direct product factor, Sylow
47	C₂²×C₅	C₅×D₈	1	6	yes	—
48	C₄×C₅	—	4	12	no	—
49	C₄×C₅	—	8	6	no	—
50	C₄×C₅	C₅×C₂²⋊C₄	1	6	yes	—
51	C₄×C₅	C₅×SD₁₆	1	6	yes	—
52	C₂×Q₈	—	2	1	no	—
53	Q₁₆	—	2	1	no	—
54	Q₁₆	C₄×C₅²	1	2	yes	complemented
55	C₂×C₈	C₂²×C₅²	1	1	yes	characteristic
56	C₄⋊C₄	—	2	1	no	—
57	C₁₆	—	2	1	no	—
58	C₂×C₅	C₅×#102	1	6	yes	—
59	C₂×C₅	C₅×SD₃₂	1	6	yes	—
60	C₂×C₅	C₅×Q₃₂	1	6	yes	—
61	Q₈	—	2	2	no	—
62	Q₈	—	4	1	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	—	4	1	no	—
65	C₂×C₄	C₅²×D₄	1	1	yes	characteristic
66	C₈	C₂×C₄×C₅²	1	1	yes	characteristic
67	C₈	C₅²×D₄	1	1	yes	characteristic
68	C₅	C₅×#358	1	6	yes	direct product factor
69	C₂²	C₅²×D₈	1	1	yes	characteristic
70	C₄	—	4	2	no	—
71	C₄	—	8	1	no	—
72	C₄	C₅²×C₂²⋊C₄	1	1	yes	characteristic
73	C₄	C₅²×SD₁₆	1	1	yes	characteristic
74	C₂	C₅²×#102	1	1	yes	characteristic
75	C₂	C₅²×SD₃₂	1	1	yes	characteristic
76	C₂	C₅²×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₅²	D₈
commutator	C₈	C₂×C₄×C₅²
Frattini, cosocle	C₂×C₈	C₂²×C₅²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅² ◁ C₂×C₄×C₅² ◁ C₂×C₅²×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅²×Q₁₆	C₄	semidirect
10. C₅×#358	C₅	direct (N × H)
28. C₅×Q₁₆	C₄×C₅	semidirect
32. #358	C₅²	direct (N × H)
46. C₅²	#358	direct (N × H)
54. Q₁₆	C₄×C₅²	semidirect
68. C₅	C₅×#358	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×C₅²×Q₁₆	2
C₂×C₅²×C₁₆	2
C₅²×#107	2
C₅×#358	5

Subgroups — C52×#358 #13099639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#358 #13099639