Subgroups — Q₈×Dic₅₀ #13098889

Order 1600.

318 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×C₂₅	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₂₅	C₂	1	2	yes	maximal
3	#11716450	C₂	1	6	yes	maximal
4	#11716431	C₂	1	3	yes	maximal
5	C₄×Dic₅₀	C₂	1	3	yes	maximal
6	C₂×Q₈×C₂₅	C₂²	1	1	yes	characteristic
7	C₂×Dic₅₀	C₂²	1	6	yes	—
8	C₂×Dic₅₀	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₂₅	C₂²	1	3	yes	—
10	C₄²×C₂₅	C₂²	1	3	yes	—
11	C₄⋊Dic₂₅	C₂²	1	3	yes	—
12	C₄⋊Dic₂₅	C₂²	1	6	yes	—
13	#87718	C₂²	1	6	yes	—
14	C₄×Dic₂₅	C₂²	1	6	yes	—
15	Q₈×Dic₁₀	—	5	1	no	maximal
16	Q₈×C₂₅	Q₈	1	4	yes	complemented
17	C₂×C₄×C₂₅	C₂³	1	3	yes	—
18	C₂×C₄×C₂₅	C₂³	1	3	yes	—
19	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
20	C₂×Dic₂₅	C₂³	1	6	yes	—
21	C₂×Dic₂₅	C₂³	1	2	yes	—
22	Dic₅₀	—	4	6	no	—
23	Dic₅₀	Q₈	1	4	yes	direct product factor
24	C₄×C₅×Q₈	D₅	1	1	yes	characteristic
25	Q₈×Dic₅	—	5	2	no	—
26	#4034	—	5	6	no	—
27	#4015	—	5	3	no	—
28	C₄×Dic₁₀	—	5	3	no	—
29	C₂²×C₂₅	C₂⁴	1	1	yes	characteristic
30	C₄×C₂₅	—	2	3	no	—
31	C₄×C₂₅	C₂×Q₈	1	6	yes	—
32	C₄×C₂₅	C₂×Q₈	1	2	yes	—
33	Dic₂₅	—	2	6	no	—
34	Dic₂₅	C₂×Q₈	1	4	yes	—
35	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
36	C₂×Dic₁₀	—	5	1	no	—
37	C₂×Dic₁₀	—	5	6	no	—
38	C₅×C₄⋊C₄	C₂×D₅	1	3	yes	—
39	C₄²×C₅	C₂×D₅	1	3	yes	—
40	C₄⋊Dic₅	—	5	3	no	—
41	C₄⋊Dic₅	—	5	6	no	—
42	#683	—	5	6	no	—
43	C₄×Dic₅	—	5	6	no	—
44	Q₈²	—	25	1	no	Sylow
45	C₂×C₂₅	C₂²×Q₈	1	1	yes	characteristic
46	C₂×C₂₅	C₂²×Q₈	1	1	yes	characteristic
47	C₂×C₂₅	H(5,𝔽₂)	1	1	yes	characteristic
48	C₅×Q₈	Dic₁₀	1	4	yes	—
49	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
50	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
51	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
52	C₂×Dic₅	—	5	2	no	—
53	C₂×Dic₅	—	5	6	no	—
54	Dic₁₀	—	5	4	no	—
55	Dic₁₀	—	20	6	no	—
56	C₄⋊Q₈	—	25	6	no	—
57	C₄⋊Q₈	—	25	3	no	—
58	C₄×Q₈	—	25	2	no	—
59	C₄×Q₈	—	25	3	no	—
60	C₄×Q₈	D₂₅	1	1	yes	characteristic
61	C₂₅	Q₈²	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
63	C₄×C₅	—	2	3	no	—
64	C₄×C₅	C₂×Dic₁₀	1	6	yes	—
65	C₄×C₅	Q₈×D₅	1	2	yes	—
66	Dic₅	—	5	4	no	—
67	Dic₅	—	10	6	no	—
68	C₂×Q₈	—	25	1	no	—
69	C₂×Q₈	—	25	6	no	—
70	C₂×Q₈	C₂×D₂₅	1	1	yes	characteristic
71	C₄⋊C₄	—	25	6	no	—
72	C₄⋊C₄	—	25	6	no	—
73	C₄⋊C₄	—	25	3	no	—
74	C₄⋊C₄	C₂×D₂₅	1	3	yes	—
75	C₄²	—	25	6	no	—
76	C₄²	C₂×D₂₅	1	3	yes	—
77	C₂×C₅	C₂²×Dic₁₀	1	1	yes	characteristic
78	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
79	C₂×C₅	#4149	1	1	yes	characteristic
80	Q₈	—	25	4	no	—
81	Q₈	—	100	6	no	—
82	Q₈	Dic₅₀	1	4	yes	direct product factor
83	C₂×C₄	—	25	2	no	—
84	C₂×C₄	—	25	6	no	—
85	C₂×C₄	C₂²×D₂₅	1	3	yes	—
86	C₂×C₄	C₂²×D₂₅	1	3	yes	—
87	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
88	C₅	Q₈×Dic₁₀	1	1	yes	characteristic
89	C₂²	C₂³×D₂₅	1	1	yes	characteristic
90	C₄	—	2	3	no	—
91	C₄	—	25	4	no	—
92	C₄	—	50	6	no	—
93	C₄	C₂×Dic₅₀	1	6	yes	—
94	C₄	Q₈×D₂₅	1	2	yes	—
95	C₂	C₂²×Dic₅₀	1	1	yes	characteristic
96	C₂	C₂×Q₈×D₂₅	1	1	yes	characteristic
97	C₂	#11716568	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₂₅
commutator, cosocle	C₂²×C₂₅	C₂⁴
Frattini, socle, prime radical	C₂²×C₅	C₂³×D₅
Fitting	C₄×Q₈×C₂₅	C₂
hypercenter	C₄×Q₈	D₂₅
hypocenter	C₂₅	Q₈²
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
16. Q₈×C₂₅	Q₈	semidirect
23. Dic₅₀	Q₈	direct (N × H)
61. C₂₅	Q₈²	semidirect
82. Q₈	Dic₅₀	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×Q₈×C₂₅	2
Q₈×Dic₂₅	2
#11716450	2
#11716431	2
C₄×Dic₅₀	2
Q₈×Dic₁₀	5

Subgroups — Q8×Dic50 #13098889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Dic₅₀ #13098889