Subgroups — #13098274

Order 1600.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊#6014	C₂	1	2	yes	maximal
3	C₄⋊#6014	C₂	1	2	yes	maximal, complemented
4	#11716351	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#6014	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
7	C₄⋊C₄×C₂₅	C₄	1	2	yes	—
8	C₂²⋊C₄×C₂₅	C₄	1	2	yes	—
9	C₄²×C₂₅	C₂²	1	1	yes	characteristic
10	C₄²×C₂₅	C₂²	1	1	yes	characteristic
11	C₂×#6014	C₂²	1	2	yes	—
12	C₂×#6014	C₂²	1	2	yes	—
13	#65289	—	5	1	no	maximal
14	C₂³×C₂₅	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₅	—	2	2	no	—
16	C₂×C₄×C₂₅	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₂₅	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₂₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
20	#6014	—	2	2	no	—
21	#6014	—	2	2	no	—
22	C₅×#117	D₅	1	1	yes	characteristic
23	C₂²⋊#168	—	5	2	no	—
24	C₄⋊#168	—	5	2	no	—
25	#3935	—	5	1	no	—
26	C₄×#168	—	5	1	no	—
27	C₂²×C₂₅	—	2	1	no	—
28	C₂²×C₂₅	—	2	2	no	—
29	C₂²×C₂₅	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₂₅	—	2	1	no	—
31	C₄×C₂₅	—	2	2	no	—
32	C₄×C₂₅	—	2	2	no	—
33	C₄×C₂₅	#41	1	2	yes	—
34	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
35	C₅×C₄⋊C₄	Dic₅	1	2	yes	—
36	C₅×C₂²⋊C₄	Dic₅	1	2	yes	—
37	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
38	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
39	C₂×#168	—	5	2	no	—
40	C₂×#168	—	5	2	no	—
41	#433	—	25	1	no	Sylow
42	C₂×C₂₅	—	4	1	no	—
43	C₂×C₂₅	#117	1	1	yes	characteristic
44	C₂×C₂₅	#131	1	2	yes	—
45	C₂³×C₅	C₂×Dic₅	1	1	yes	characteristic
46	C₂×C₄×C₅	—	2	2	no	—
47	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
48	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
49	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
50	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
51	#168	—	10	2	no	—
52	#168	—	10	2	no	—
53	#117	D₂₅	1	1	yes	characteristic
54	C₄⋊C₈	—	25	2	no	—
55	C₂²⋊C₈	—	25	2	no	—
56	#97	—	25	1	no	—
57	C₄×C₈	—	25	1	no	—
58	C₂₅	#433	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₅	—	2	1	no	—
60	C₂²×C₅	—	2	2	no	—
61	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
62	C₄×C₅	—	2	1	no	—
63	C₄×C₅	—	2	2	no	—
64	C₄×C₅	—	2	2	no	—
65	C₄×C₅	#709	1	2	yes	—
66	C₂²×C₄	C₂×D₂₅	1	1	yes	characteristic
67	C₂×C₈	—	25	2	no	—
68	C₂×C₈	—	25	2	no	—
69	C₄⋊C₄	Dic₂₅	1	2	yes	—
70	C₂²⋊C₄	Dic₂₅	1	2	yes	—
71	C₄²	C₂×D₂₅	1	1	yes	characteristic
72	C₄²	C₂×D₂₅	1	1	yes	characteristic
73	C₂×C₅	—	4	1	no	—
74	C₂×C₅	#4072	1	1	yes	characteristic
75	C₂×C₅	#4094	1	2	yes	—
76	C₂³	C₂×Dic₂₅	1	1	yes	characteristic
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	C₂×Dic₂₅	1	2	yes	—
79	C₂×C₄	C₂×Dic₂₅	1	2	yes	—
80	C₂×C₄	C₂×Dic₂₅	1	1	yes	characteristic
81	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
82	C₈	—	50	2	no	—
83	C₈	—	50	2	no	—
84	C₅	#65289	1	1	yes	characteristic
85	C₂²	—	2	1	no	—
86	C₂²	—	2	2	no	—
87	C₂²	C₂²×Dic₂₅	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	2	no	—
90	C₄	—	2	2	no	—
91	C₄	#87744	1	2	yes	—
92	C₂	—	4	1	no	—
93	C₂	#11716488	1	1	yes	characteristic
94	C₂	#11716510	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₅
commutator	C₂²×C₂₅	C₂²×C₄
Frattini	C₂×C₄×C₅	C₂²×D₅
Fitting	C₂₅×#117	C₂
socle	C₂²×C₅	C₂²×Dic₅
prime radical	C₂³×C₅	C₂×Dic₅
cosocle	C₂×C₄×C₂₅	C₂³
hypercenter	#117	D₂₅
hypocenter	C₂₅	#433
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. C₄⋊#6014	C₂	semidirect
4. #11716351	C₂	semidirect
5. C₄×#6014	C₂	semidirect
58. C₂₅	#433	semidirect

Maximal subgroups

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

subgroup	index
C₂₅×#117	2
C₂²⋊#6014	2
C₄⋊#6014	2
#11716351	2
C₄×#6014	2
#65289	5