Subgroups — #13107098

Order 1600.

800 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×#1039	C₂	1	2	yes	maximal
3	#11717210	C₂	1	6	yes	maximal
4	#11717189	C₂	1	3	yes	maximal
5	C₄×#6045	C₂	1	3	yes	maximal
6	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
7	C₂×#6045	C₂²	1	3	yes	—
8	C₅²×C₄⋊C₄	C₂²	1	3	yes	—
9	C₄²×C₅²	C₂²	1	3	yes	—
10	C₄⋊#1039	C₂²	1	1	yes	characteristic
11	C₄⋊#1039	C₂²	1	6	yes	—
12	C₄⋊#1039	C₂²	1	6	yes	—
13	#87806	C₂²	1	6	yes	—
14	C₄×#1039	C₂²	1	6	yes	—
15	#65904	—	5	6	no	maximal
16	C₅²×Q₈	Q₈	1	4	yes	complemented
17	C₂×C₄×C₅²	C₂³	1	3	yes	—
18	C₂×C₄×C₅²	C₂³	1	3	yes	—
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	C₂×#1039	C₂³	1	6	yes	—
21	C₂×#1039	C₂³	1	2	yes	—
22	#6045	—	2	6	no	—
23	C₄×C₅×Q₈	D₅	1	6	yes	—
24	Q₈×Dic₅	—	5	12	no	—
25	#4036	—	5	36	no	—
26	#4015	—	5	18	no	—
27	C₄×Dic₁₀	—	5	18	no	—
28	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
29	C₄×C₅²	—	2	3	no	—
30	C₄×C₅²	C₂×Q₈	1	6	yes	—
31	C₄×C₅²	#41	1	2	yes	—
32	#1039	—	2	6	no	—
33	#1039	—	2	2	no	—
34	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
35	C₂×Dic₁₀	—	5	18	no	—
36	C₅×C₄⋊C₄	C₂×D₅	1	18	yes	—
37	C₄²×C₅	C₂×D₅	1	18	yes	—
38	C₄⋊Dic₅	—	5	6	no	—
39	C₄⋊Dic₅	—	5	36	no	—
40	C₄⋊Dic₅	—	5	36	no	—
41	#683	—	5	36	no	—
42	C₄×Dic₅	—	5	36	no	—
43	#557	—	25	1	no	Sylow
44	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
45	C₂×C₅²	C₂×#41	1	1	yes	characteristic
46	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
47	C₅×Q₈	Dic₁₀	1	24	yes	complemented
48	C₂×C₄×C₅	C₂²×D₅	1	18	yes	—
49	C₂×C₄×C₅	C₂²×D₅	1	18	yes	—
50	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
51	C₂×Dic₅	—	5	12	no	—
52	C₂×Dic₅	—	5	36	no	—
53	Dic₁₀	—	10	36	no	—
54	C₄⋊Q₈	—	25	3	no	—
55	#125	—	25	6	no	—
56	C₄×Q₈	—	25	2	no	—
57	C₄×Q₈	—	25	3	no	—
58	C₄×Q₈	C₅⋊D₅	1	1	yes	characteristic
59	C₅²	#557	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₅	C₂³×D₅	1	6	yes	—
61	C₄×C₅	—	2	18	no	—
62	C₄×C₅	C₂×Dic₁₀	1	36	yes	—
63	C₄×C₅	#713	1	12	yes	—
64	Dic₅	—	10	36	no	—
65	Dic₅	—	10	12	no	—
66	C₂×Q₈	—	25	3	no	—
67	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
68	C₄⋊C₄	—	25	1	no	—
69	C₄⋊C₄	—	25	6	no	—
70	C₄⋊C₄	—	25	6	no	—
71	C₄⋊C₄	—	25	6	no	—
72	C₄⋊C₄	C₂×C₅⋊D₅	1	3	yes	—
73	C₄²	—	25	6	no	—
74	C₄²	C₂×C₅⋊D₅	1	3	yes	—
75	C₂×C₅	C₂²×Dic₁₀	1	6	yes	—
76	C₂×C₅	C₂×#713	1	6	yes	—
77	C₂×C₅	#4150	1	6	yes	—
78	Q₈	—	50	6	no	—
79	Q₈	#6045	1	4	yes	complemented
80	C₂×C₄	—	25	2	no	—
81	C₂×C₄	—	25	6	no	—
82	C₂×C₄	C₂²×C₅⋊D₅	1	3	yes	—
83	C₂×C₄	C₂²×C₅⋊D₅	1	3	yes	—
84	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
85	C₅	#65904	1	6	yes	complemented
86	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
87	C₄	—	2	3	no	—
88	C₄	—	50	6	no	—
89	C₄	—	50	2	no	—
90	C₄	C₂×#6045	1	6	yes	—
91	C₄	#87904	1	2	yes	—
92	C₂	C₂²×#6045	1	1	yes	characteristic
93	C₂	C₂×#87904	1	1	yes	characteristic
94	C₂	#11717523	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, prime radical, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₄×C₅²×Q₈	C₂
hypercenter	C₄×Q₈	C₅⋊D₅
hypocenter	C₅²	#557
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
16. C₅²×Q₈	Q₈	semidirect
47. C₅×Q₈	Dic₁₀	semidirect
59. C₅²	#557	semidirect
79. Q₈	#6045	semidirect
85. C₅	#65904	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅²×Q₈	2
Q₈×#1039	2
#11717210	2
#11717189	2
C₄×#6045	2
#65904	5