Subgroups — #11717397

Order 800.

109 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6039	C₂	1	1	yes	maximal, characteristic, complemented
2	#87880	C₂	1	1	yes	maximal, characteristic
3	#87867	C₂	1	1	yes	maximal, characteristic, complemented
4	#87839	C₂	1	2	yes	maximal, complemented
5	#87838	C₂	1	2	yes	maximal, complemented
6	C₂×#1039	C₂²	1	1	yes	characteristic
7	C₅×C₂²⋊D₅	—	2	1	no	—
8	C₂×C₅×Dic₅	—	2	1	no	—
9	#6039	—	2	1	no	—
10	#6039	C₂²	1	2	yes	—
11	#6039	C₂²	1	1	yes	characteristic
12	#6037	C₂²	1	1	yes	characteristic
13	D₅×Dic₅	—	2	1	no	—
14	#6034	C₂²	1	2	yes	—
15	C₂²×C₅²	D₄	1	1	yes	characteristic
16	C₂×C₅×D₅	—	2	1	no	—
17	#1039	D₄	1	1	yes	characteristic
18	#1039	C₂³	1	1	yes	characteristic
19	C₅×Dic₅	—	2	1	no	—
20	C₅×Dic₅	—	2	2	no	—
21	#711	—	10	1	no	—
22	C₂×Dic₁₀	—	10	1	no	—
23	#704	—	10	1	no	—
24	C₂×C₅²	—	2	1	no	—
25	C₂×C₅²	C₂×D₄	1	1	yes	characteristic
26	C₅×D₅	—	4	1	no	—
27	C₅×D₄	—	10	1	no	—
28	C₂×C₄×C₅	—	10	1	no	—
29	C₂²⋊D₅	—	2	1	no	—
30	C₂²⋊D₅	—	10	1	no	—
31	C₂×Dic₅	—	2	1	no	—
32	C₂×Dic₅	—	10	1	no	—
33	C₂×Dic₅	—	10	1	no	—
34	C₂×Dic₅	—	10	1	no	—
35	C₂×Dic₅	—	10	1	no	—
36	C₄×D₅	—	10	1	no	—
37	Dic₁₀	—	10	2	no	—
38	Dic₁₀	—	10	1	no	—
39	Dic₁₀	—	10	2	no	—
40	#170	—	10	2	no	—
41	#137	—	25	1	no	maximal, Sylow
42	C₅²	#137	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₅	—	2	1	no	—
44	C₂²×C₅	—	2	1	no	—
45	C₂²×C₅	—	2	1	no	—
46	C₂²×C₅	—	10	1	no	—
47	C₂×D₅	—	2	1	no	—
48	C₄×C₅	—	10	2	no	—
49	C₄×C₅	—	10	1	no	—
50	Dic₅	—	2	2	no	—
51	Dic₅	—	2	1	no	—
52	Dic₅	—	10	1	no	—
53	Dic₅	—	10	1	no	—
54	Dic₅	—	10	1	no	—
55	Dic₅	—	10	1	no	—
56	Dic₅	—	10	1	no	—
57	Dic₅	—	10	1	no	—
58	#41	—	25	1	no	—
59	C₂×Q₈	—	25	1	no	—
60	Q₁₆	—	25	2	no	—
61	SD₁₆	—	25	2	no	—
62	M₁₆	—	25	1	no	—
63	C₂×C₅	—	2	2	no	—
64	C₂×C₅	—	2	1	no	—
65	C₂×C₅	—	2	1	no	—
66	C₂×C₅	—	2	1	no	—
67	C₂×C₅	—	4	1	no	—
68	C₂×C₅	—	4	1	no	—
69	C₂×C₅	—	20	1	no	—
70	D₅	—	4	1	no	—
71	Q₈	—	25	2	no	—
72	Q₈	—	25	1	no	—
73	Q₈	—	50	1	no	—
74	D₄	—	10	1	no	—
75	D₄	—	25	1	no	—
76	C₂×C₄	—	10	1	no	—
77	C₂×C₄	—	25	1	no	—
78	C₂×C₄	—	50	1	no	—
79	C₈	—	25	2	no	—
80	C₅	—	2	1	no	—
81	C₅	—	2	1	no	—
82	C₅	—	2	1	no	—
83	C₂²	—	10	1	no	—
84	C₂²	D₅²⋊C₂	1	1	yes	characteristic
85	C₄	—	10	2	no	—
86	C₄	—	10	1	no	—
87	C₄	—	25	1	no	—
88	C₄	—	25	1	no	—
89	C₂	—	2	1	no	—
90	C₂	—	20	1	no	—
91	C₂	C₂×D₅²⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₅²⋊C₂
commutator, cosocle	#1039	C₂³
Fitting	C₂²×C₅²	D₄
socle	C₂×C₅²	C₂×D₄
prime radical	#87880	C₂
hypercenter	C₂²	D₅²⋊C₂
hypocenter	C₅²	#137
LCS	G ▷ #1039 ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂×#6039	C₂	semidirect
3. #87867	C₂	semidirect
4. #87839	C₂	semidirect
5. #87838	C₂	semidirect
42. C₅²	#137	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6039	2
#87880	2
#87867	2
#87839	2
#87838	2
#137	25