Subgroups — #13102918

Order 1600.

384 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#124	C₂	1	1	yes	maximal, characteristic
2	#11717190	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716702	C₂	1	2	yes	maximal, complemented
4	#11716699	C₂	1	2	yes	maximal
5	C₄×#6029	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
8	C₅²×C₂²⋊C₄	—	2	2	no	—
9	C₄²×C₅²	C₂²	1	1	yes	characteristic
10	C₄⋊#1039	C₂²	1	2	yes	—
11	#87806	—	2	2	no	—
12	C₂×#6029	C₂²	1	2	yes	—
13	#65345	—	5	6	no	maximal
14	C₂³×C₅²	—	2	1	no	—
15	C₅²×Q₈	—	2	2	no	—
16	C₅²×D₄	—	2	2	no	—
17	C₂×C₄×C₅²	—	2	1	no	—
18	C₂×C₄×C₅²	D₄	1	2	yes	—
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	C₂×#1039	—	2	2	no	—
21	#6029	—	2	2	no	—
22	C₅×#124	D₅	1	6	yes	—
23	#4016	—	5	6	no	—
24	#3967	—	5	12	no	—
25	#3964	—	5	12	no	—
26	C₄×#168	—	5	6	no	—
27	C₂²×C₅²	—	4	1	no	—
28	C₂²×C₅²	—	4	2	no	—
29	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
30	C₄×C₅²	—	2	2	no	—
31	C₄×C₅²	—	4	1	no	—
32	C₄×C₅²	#41	1	2	yes	—
33	#1039	—	4	2	no	—
34	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
35	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
36	C₅×C₂²⋊C₄	—	2	12	no	—
37	C₄²×C₅	C₂×D₅	1	6	yes	—
38	C₄⋊Dic₅	—	5	12	no	—
39	#683	—	10	12	no	—
40	C₂×#168	—	5	12	no	—
41	#488	—	25	1	no	Sylow
42	C₂×C₅²	—	8	1	no	—
43	C₂×C₅²	#124	1	1	yes	characteristic
44	C₂×C₅²	#135	1	2	yes	—
45	C₂³×C₅	—	2	6	no	—
46	C₅×Q₈	—	2	12	no	—
47	C₅×D₄	—	2	12	no	—
48	C₂×C₄×C₅	—	2	6	no	—
49	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
50	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
51	C₂×Dic₅	—	10	12	no	—
52	#168	—	10	12	no	—
53	#125	—	25	1	no	—
54	#124	C₅⋊D₅	1	1	yes	characteristic
55	#103	—	25	2	no	—
56	#102	—	25	2	no	—
57	C₄×C₈	—	25	1	no	—
58	C₅²	#488	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₅	—	4	12	no	—
60	C₂²×C₅	—	4	6	no	—
61	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
62	C₄×C₅	—	2	12	no	—
63	C₄×C₅	—	4	6	no	—
64	C₄×C₅	#711	1	12	yes	—
65	Dic₅	—	20	12	no	—
66	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
67	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
68	C₂×C₈	—	25	2	no	—
69	C₄⋊C₄	—	25	2	no	—
70	C₄⋊C₄	—	50	2	no	—
71	C₂²⋊C₄	—	2	2	no	—
72	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
73	C₂×C₅	—	8	6	no	—
74	C₂×C₅	#4082	1	6	yes	—
75	C₂×C₅	#4096	1	12	yes	—
76	C₂³	—	2	1	no	—
77	Q₈	—	2	2	no	—
78	D₄	—	2	2	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	50	2	no	—
81	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
82	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
83	C₈	—	50	2	no	—
84	C₅	#65345	1	6	yes	complemented
85	C₂²	—	4	1	no	—
86	C₂²	—	4	2	no	—
87	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	4	1	no	—
90	C₄	—	100	2	no	—
91	C₄	#87902	1	2	yes	—
92	C₂	—	8	1	no	—
93	C₂	#11717256	1	1	yes	characteristic
94	C₂	#11717270	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#124	C₂
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂×C₅²×D₄	C₂²
hypercenter	#124	C₅⋊D₅
hypocenter	C₅²	#488
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
2. #11717190	C₂	semidirect
3. #11716702	C₂	semidirect
5. C₄×#6029	C₂	semidirect
58. C₅²	#488	semidirect
84. C₅	#65345	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#124	2
#11717190	2
#11716702	2
#11716699	2
C₄×#6029	2
#65345	5