Subgroups — #13098486

Order 1600.

327 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₂₅	C₂	1	2	yes	maximal
3	#11716379	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₂₅	C₂²	1	1	yes	characteristic
5	C₂²×C₄×C₂₅	—	2	4	no	—
6	C₂²×C₄×C₂₅	C₄	1	4	yes	complemented
7	C₂²×C₄×C₂₅	C₂²	1	2	yes	—
8	C₂²×Dic₂₅	C₂²	1	4	yes	—
9	C₂²⋊Dic₂₅	—	2	4	no	—
10	#65500	—	5	1	no	maximal
11	C₂³×C₂₅	—	2	4	no	—
12	C₂³×C₂₅	C₂×C₄	1	2	yes	—
13	C₂³×C₂₅	D₄	1	4	yes	—
14	C₂³×C₂₅	C₂³	1	1	yes	characteristic
15	C₂×C₄×C₂₅	—	2	8	no	—
16	C₂×C₄×C₂₅	—	2	4	no	—
17	C₂×C₄×C₂₅	C₂×C₄	1	4	yes	—
18	C₂×Dic₂₅	—	2	4	no	—
19	C₂×Dic₂₅	—	2	8	no	—
20	C₂³×C₄×C₅	D₅	1	1	yes	characteristic
21	C₂×C₂²⋊Dic₅	—	5	2	no	—
22	#3963	—	5	4	no	—
23	C₂²×C₂₅	—	2	4	no	—
24	C₂²×C₂₅	—	2	8	no	—
25	C₂²×C₂₅	C₂²⋊C₄	1	4	yes	—
26	C₂²×C₂₅	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₂₅	C₂×D₄	1	2	yes	—
28	C₂²×C₂₅	#41	1	4	yes	—
29	C₄×C₂₅	—	2	4	no	—
30	Dic₂₅	—	4	4	no	—
31	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
32	C₂²×C₄×C₅	—	2	4	no	—
33	C₂²×C₄×C₅	Dic₅	1	4	yes	—
34	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
35	C₂²×Dic₅	—	5	4	no	—
36	C₂²⋊Dic₅	—	10	4	no	—
37	#381	—	25	1	no	Sylow
38	C₂×C₂₅	—	2	4	no	—
39	C₂×C₂₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₂₅	#117	1	2	yes	—
41	C₂×C₂₅	#123	1	4	yes	—
42	C₂³×C₅	—	2	4	no	—
43	C₂³×C₅	C₂×Dic₅	1	2	yes	—
44	C₂³×C₅	C₂²⋊D₅	1	4	yes	—
45	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
46	C₂×C₄×C₅	—	2	8	no	—
47	C₂×C₄×C₅	—	2	4	no	—
48	C₂×C₄×C₅	C₂×Dic₅	1	4	yes	—
49	C₂×Dic₅	—	10	8	no	—
50	C₂×Dic₅	—	10	4	no	—
51	C₂³×C₄	D₂₅	1	1	yes	characteristic
52	C₂×C₂²⋊C₄	—	25	2	no	—
53	#95	—	25	4	no	—
54	C₂₅	#381	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₅	—	2	4	no	—
56	C₂²×C₅	—	2	8	no	—
57	C₂²×C₅	C₂²⋊Dic₅	1	4	yes	—
58	C₂²×C₅	#709	1	4	yes	—
59	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
60	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
61	C₄×C₅	—	2	4	no	—
62	Dic₅	—	20	4	no	—
63	C₂⁴	C₂×D₂₅	1	1	yes	characteristic
64	C₂²×C₄	—	2	4	no	—
65	C₂²×C₄	—	25	4	no	—
66	C₂²×C₄	Dic₂₅	1	4	yes	complemented
67	C₂²×C₄	C₂×D₂₅	1	2	yes	—
68	C₂²⋊C₄	—	50	4	no	—
69	C₂×C₅	—	2	4	no	—
70	C₂×C₅	#4072	1	2	yes	—
71	C₂×C₅	#4075	1	4	yes	—
72	C₂×C₅	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
73	C₂³	—	2	4	no	—
74	C₂³	C₂×Dic₂₅	1	2	yes	—
75	C₂³	C₂²⋊D₂₅	1	4	yes	—
76	C₂³	C₂²×D₂₅	1	1	yes	characteristic
77	C₂×C₄	—	2	8	no	—
78	C₂×C₄	—	2	4	no	—
79	C₂×C₄	—	50	8	no	—
80	C₂×C₄	—	50	4	no	—
81	C₂×C₄	C₂×Dic₂₅	1	4	yes	—
82	C₅	#65500	1	1	yes	characteristic
83	C₂²	—	2	4	no	—
84	C₂²	—	2	8	no	—
85	C₂²	C₂²⋊Dic₂₅	1	4	yes	—
86	C₂²	#87744	1	4	yes	—
87	C₂²	C₂²×Dic₂₅	1	1	yes	characteristic
88	C₂²	C₂×C₂²⋊D₂₅	1	2	yes	—
89	C₄	—	2	4	no	—
90	C₄	—	100	4	no	—
91	C₂	—	2	4	no	—
92	C₂	#11716488	1	2	yes	—
93	C₂	#11716491	1	4	yes	—
94	C₂	C₂×C₂²⋊Dic₂₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₅
commutator	C₂²×C₂₅	C₂²×C₄
Frattini, socle	C₂³×C₅	C₂²×D₅
Fitting	C₂³×C₄×C₂₅	C₂
prime radical	C₂⁴×C₅	C₂×D₅
cosocle	C₂³×C₂₅	C₂³
hypercenter	C₂³×C₄	D₂₅
hypocenter	C₂₅	#381
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. #11716379	C₂	semidirect
6. C₂²×C₄×C₂₅	C₄	semidirect
54. C₂₅	#381	semidirect
66. C₂²×C₄	Dic₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₂₅	2
C₂×C₂²⋊Dic₂₅	2
#11716379	2
#65500	5