Subgroups — #11716458

Order 800.

111 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₅₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₄×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#87720	C₂	1	1	yes	maximal, characteristic
5	#87720	C₂	1	1	yes	maximal, characteristic
6	#87718	C₂	1	1	yes	maximal, characteristic, complemented
7	#87718	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₅	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₂₅	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₂₅	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₂₅	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₅	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₅	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₅	C₂²	1	1	yes	characteristic
15	C₄×D₂₅	—	2	2	no	—
16	Dic₅₀	—	2	2	no	—
17	#4042	—	5	1	no	maximal
18	C₂²×C₂₅	C₂³	1	1	yes	characteristic
19	C₂×D₂₅	—	2	1	no	—
20	C₂×D₂₅	—	2	1	no	—
21	C₂×D₂₅	Q₈	1	2	yes	complemented
22	C₄×C₂₅	—	2	1	no	—
23	C₄×C₂₅	—	2	1	no	—
24	C₄×C₂₅	—	2	1	no	—
25	Dic₂₅	—	2	1	no	—
26	Dic₂₅	—	2	1	no	—
27	Dic₂₅	D₄	1	2	yes	—
28	C₂×C₄×D₅	—	5	1	no	—
29	C₂×Dic₁₀	—	5	1	no	—
30	C₅×C₄⋊C₄	D₅	1	1	yes	characteristic
31	#685	—	5	1	no	—
32	#685	—	5	1	no	—
33	#683	—	5	1	no	—
34	#683	—	5	1	no	—
35	C₂×C₂₅	C₂×D₄	1	1	yes	characteristic
36	C₂×C₂₅	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₂₅	#41	1	1	yes	characteristic
38	D₂₅	—	2	2	no	—
39	C₂²×D₅	—	5	1	no	—
40	C₂×C₄×C₅	C₂×D₅	1	1	yes	characteristic
41	C₂×C₄×C₅	C₂×D₅	1	1	yes	characteristic
42	C₂×C₄×C₅	C₂×D₅	1	1	yes	characteristic
43	C₂×Dic₅	—	5	1	no	—
44	C₂×Dic₅	—	5	1	no	—
45	C₂×Dic₅	—	5	1	no	—
46	C₄×D₅	—	10	2	no	—
47	Dic₁₀	—	10	2	no	—
48	C₂²⋊Q₈	—	25	1	no	Sylow
49	C₂₅	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
50	C₂²×C₅	C₂²×D₅	1	1	yes	characteristic
51	C₂×D₅	—	5	2	no	—
52	C₂×D₅	—	10	1	no	—
53	C₂×D₅	—	10	1	no	—
54	C₄×C₅	—	2	1	no	—
55	C₄×C₅	—	2	1	no	—
56	C₄×C₅	—	2	1	no	—
57	Dic₅	—	5	2	no	—
58	Dic₅	—	10	1	no	—
59	Dic₅	—	10	1	no	—
60	C₂×Q₈	—	25	1	no	—
61	C₂²×C₄	—	25	1	no	—
62	C₄⋊C₄	—	25	1	no	—
63	C₄⋊C₄	—	25	1	no	—
64	C₄⋊C₄	D₂₅	1	1	yes	characteristic, complemented
65	C₂²⋊C₄	—	25	1	no	—
66	C₂²⋊C₄	—	25	1	no	—
67	C₂×C₅	#709	1	1	yes	characteristic
68	C₂×C₅	D₄×D₅	1	1	yes	characteristic
69	C₂×C₅	Q₈×D₅	1	1	yes	characteristic
70	D₅	—	10	2	no	—
71	C₂³	—	25	1	no	—
72	Q₈	—	50	2	no	—
73	C₂×C₄	—	25	1	no	—
74	C₂×C₄	—	25	1	no	—
75	C₂×C₄	—	25	1	no	—
76	C₂×C₄	—	50	2	no	—
77	C₂×C₄	C₂×D₂₅	1	1	yes	characteristic
78	C₂×C₄	C₂×D₂₅	1	1	yes	characteristic
79	C₂×C₄	C₂×D₂₅	1	1	yes	characteristic
80	C₅	#4042	1	1	yes	characteristic
81	C₂²	—	25	2	no	—
82	C₂²	—	50	1	no	—
83	C₂²	—	50	1	no	—
84	C₂²	C₂²×D₂₅	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	25	2	no	—
89	C₄	—	50	1	no	—
90	C₄	—	50	1	no	—
91	C₂	—	50	2	no	—
92	C₂	#87744	1	1	yes	characteristic
93	C₂	D₄×D₂₅	1	1	yes	characteristic
94	C₂	Q₈×D₂₅	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₅₀	C₂	semidirect
3. C₄⋊C₄×C₂₅	C₂	semidirect
6. #87718	C₂	semidirect
7. #87718	C₂	semidirect
21. C₂×D₂₅	Q₈	semidirect
49. C₂₅	C₂²⋊Q₈	semidirect
64. C₄⋊C₄	D₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₂₅	2
C₂×Dic₅₀	2
C₄⋊C₄×C₂₅	2
#87720	2
#87718	2
#4042	5