Subgroups — #13102984

Order 1600.

456 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#11717246	C₂	1	1	yes	maximal, characteristic
3	C₂²×#6029	C₂	1	1	yes	maximal, characteristic
4	#11716681	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
6	C₅²×M₁₆	—	2	2	no	—
7	C₂×C₅²×C₈	C₂²	1	2	yes	—
8	C₂²⋊#1039	C₄	1	2	yes	—
9	C₄⋊#1039	C₄	1	2	yes	—
10	C₄×#1039	C₂²	1	2	yes	—
11	C₂×#6029	—	2	2	no	—
12	C₂×#6029	C₂²	1	2	yes	—
13	#65408	—	5	6	no	maximal
14	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₅²	D₄	1	2	yes	—
17	C₂×C₄×C₅²	Q₈	1	2	yes	—
18	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
19	C₂×#1039	C₂×C₄	1	4	yes	—
20	C₅²×C₈	—	2	2	no	—
21	#6029	—	2	2	no	—
22	C₂×C₅×M₁₆	D₅	1	6	yes	—
23	#4072	—	5	6	no	—
24	C₂²×#168	—	5	6	no	—
25	#3946	—	5	24	no	—
26	C₂²×C₅²	—	2	2	no	—
27	C₂²×C₅²	C₄⋊C₄	1	2	yes	—
28	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₅²	C₄⋊C₄	1	2	yes	—
30	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
31	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
32	#1039	—	2	4	no	—
33	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
34	C₅×M₁₆	—	2	12	no	—
35	C₂×C₅×C₈	C₂×D₅	1	12	yes	—
36	C₂²⋊Dic₅	—	5	12	no	—
37	C₄⋊Dic₅	—	5	12	no	—
38	C₄×Dic₅	—	5	12	no	—
39	C₂×#168	—	5	12	no	—
40	C₂×#168	—	10	12	no	—
41	#424	—	25	1	no	Sylow
42	C₂×C₅²	—	2	2	no	—
43	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₅²	#131	1	2	yes	—
45	C₂³×C₅	C₄×D₅	1	6	yes	—
46	C₂×C₄×C₅	Dic₁₀	1	12	yes	—
47	C₂×C₄×C₅	C₄×D₅	1	6	yes	—
48	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
49	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
50	C₂×Dic₅	—	5	24	no	—
51	C₅×C₈	—	2	12	no	—
52	#168	—	10	12	no	—
53	C₂×M₁₆	C₅⋊D₅	1	1	yes	characteristic
54	C₂²×C₈	—	25	1	no	—
55	#117	—	25	1	no	—
56	C₄⋊C₈	—	25	4	no	—
57	C₅²	#424	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₅	—	2	12	no	—
59	C₂²×C₅	#683	1	12	yes	—
60	C₂²×C₅	C₂×C₄×D₅	1	6	yes	—
61	C₄×C₅	#683	1	12	yes	—
62	C₄×C₅	C₂×Dic₁₀	1	6	yes	—
63	C₄×C₅	C₂×C₂²⋊D₅	1	6	yes	—
64	Dic₅	—	10	24	no	—
65	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
66	M₁₆	—	2	2	no	—
67	C₂×C₈	—	25	2	no	—
68	C₂×C₈	—	50	2	no	—
69	C₂×C₈	C₂×C₅⋊D₅	1	2	yes	—
70	C₄⋊C₄	—	25	2	no	—
71	C₂²⋊C₄	—	25	2	no	—
72	C₄²	—	25	2	no	—
73	C₂×C₅	—	2	12	no	—
74	C₂×C₅	#4053	1	12	yes	—
75	C₂×C₅	C₂×#683	1	6	yes	—
76	C₂³	C₄×C₅⋊D₅	1	1	yes	characteristic
77	C₂×C₄	—	25	4	no	—
78	C₂×C₄	#6045	1	2	yes	—
79	C₂×C₄	C₄×C₅⋊D₅	1	1	yes	characteristic
80	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
81	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
82	C₈	—	2	2	no	—
83	C₈	—	50	2	no	—
84	C₅	#65408	1	6	yes	complemented
85	C₂²	—	2	2	no	—
86	C₂²	#87806	1	2	yes	—
87	C₂²	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
88	C₄	—	50	4	no	—
89	C₄	#87806	1	2	yes	—
90	C₄	C₂×#6045	1	1	yes	characteristic
91	C₄	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	#11717227	1	2	yes	—
94	C₂	C₂×#87806	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×C₅⋊D₅
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₂×C₅²×M₁₆	C₂
prime radical	C₂³×C₅²	C₂×C₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₂×M₁₆	C₅⋊D₅
hypocenter	C₅²	#424
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
4. #11716681	C₂	semidirect
57. C₅²	#424	semidirect
84. C₅	#65408	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×M₁₆	2
#11717246	2
C₂²×#6029	2
#11716681	2
#65408	5