Subgroups — #13102863

Order 1600.

384 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#117	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊#6029	C₂	1	2	yes	maximal
3	C₄⋊#6029	C₂	1	2	yes	maximal, complemented
4	#11716670	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#6029	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
7	C₅²×C₄⋊C₄	C₄	1	2	yes	—
8	C₅²×C₂²⋊C₄	C₄	1	2	yes	—
9	C₄²×C₅²	C₂²	1	1	yes	characteristic
10	C₄²×C₅²	C₂²	1	1	yes	characteristic
11	C₂×#6029	C₂²	1	2	yes	—
12	C₂×#6029	C₂²	1	2	yes	—
13	#65289	—	5	6	no	maximal
14	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₅²	—	2	2	no	—
16	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	#6029	—	2	2	no	—
21	#6029	—	2	2	no	—
22	C₅×#117	D₅	1	6	yes	—
23	C₂²⋊#168	—	5	12	no	—
24	C₄⋊#168	—	5	12	no	—
25	#3935	—	5	6	no	—
26	C₄×#168	—	5	6	no	—
27	C₂²×C₅²	—	2	1	no	—
28	C₂²×C₅²	—	2	2	no	—
29	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₅²	—	2	1	no	—
31	C₄×C₅²	—	2	2	no	—
32	C₄×C₅²	—	2	2	no	—
33	C₄×C₅²	#41	1	2	yes	—
34	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
35	C₅×C₄⋊C₄	Dic₅	1	12	yes	—
36	C₅×C₂²⋊C₄	Dic₅	1	12	yes	—
37	C₄²×C₅	C₂×D₅	1	6	yes	—
38	C₄²×C₅	C₂×D₅	1	6	yes	—
39	C₂×#168	—	5	12	no	—
40	C₂×#168	—	5	12	no	—
41	#433	—	25	1	no	Sylow
42	C₂×C₅²	—	4	1	no	—
43	C₂×C₅²	#117	1	1	yes	characteristic
44	C₂×C₅²	#131	1	2	yes	—
45	C₂³×C₅	C₂×Dic₅	1	6	yes	—
46	C₂×C₄×C₅	—	2	12	no	—
47	C₂×C₄×C₅	C₂×Dic₅	1	12	yes	—
48	C₂×C₄×C₅	C₂×Dic₅	1	12	yes	—
49	C₂×C₄×C₅	C₂×Dic₅	1	6	yes	—
50	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
51	#168	—	10	12	no	—
52	#168	—	10	12	no	—
53	#117	C₅⋊D₅	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	C₅²	#433	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₅	—	2	6	no	—
60	C₂²×C₅	—	2	12	no	—
61	C₂²×C₅	C₂²×Dic₅	1	6	yes	—
62	C₄×C₅	—	2	6	no	—
63	C₄×C₅	—	2	12	no	—
64	C₄×C₅	—	2	12	no	—
65	C₄×C₅	#709	1	12	yes	—
66	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
67	C₂×C₈	—	25	2	no	—
68	C₂×C₈	—	25	2	no	—
69	C₄⋊C₄	#1039	1	2	yes	—
70	C₂²⋊C₄	#1039	1	2	yes	—
71	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
72	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
73	C₂×C₅	—	4	6	no	—
74	C₂×C₅	#4072	1	6	yes	—
75	C₂×C₅	#4094	1	12	yes	—
76	C₂³	C₂×#1039	1	1	yes	characteristic
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	C₂×#1039	1	2	yes	—
79	C₂×C₄	C₂×#1039	1	2	yes	—
80	C₂×C₄	C₂×#1039	1	1	yes	characteristic
81	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
82	C₈	—	50	2	no	—
83	C₈	—	50	2	no	—
84	C₅	#65289	1	6	yes	complemented
85	C₂²	—	2	1	no	—
86	C₂²	—	2	2	no	—
87	C₂²	C₂²×#1039	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	2	no	—
90	C₄	—	2	2	no	—
91	C₄	#87900	1	2	yes	—
92	C₂	—	4	1	no	—
93	C₂	#11717246	1	1	yes	characteristic
94	C₂	#11717268	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×C₅⋊D₅
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₅²×#117	C₂
prime radical	C₂³×C₅²	C₂×C₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	#117	C₅⋊D₅
hypocenter	C₅²	#433
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. C₄⋊#6029	C₂	semidirect
4. #11716670	C₂	semidirect
5. C₄×#6029	C₂	semidirect
58. C₅²	#433	semidirect
84. C₅	#65289	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#117	2
C₂²⋊#6029	2
C₄⋊#6029	2
#11716670	2
C₄×#6029	2
#65289	5