Subgroups — C₂²×#87804 #13107309

Order 1600.

1192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₅²	C₂	1	1	yes	maximal, characteristic
2	C₂×#87804	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#6029	C₂	1	2	yes	maximal, complemented
4	C₂⁴×C₅²	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₅²	C₄	1	6	yes	—
6	C₂²×C₄×C₅²	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅²	C₂²	1	6	yes	—
9	#87804	C₂²	1	16	yes	direct product factor
10	C₂×#6029	C₂²	1	12	yes	complemented
11	C₂²×#681	—	5	6	no	maximal
12	C₂³×C₅²	—	2	4	no	—
13	C₂³×C₅²	C₂×C₄	1	6	yes	—
14	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₅²	C₂×C₄	1	12	yes	—
16	C₂×C₄×C₅²	C₂×C₄	1	3	yes	—
17	C₂×C₄×C₅²	C₂×C₄	1	6	yes	—
18	C₂×C₄×C₅²	C₂³	1	3	yes	—
19	C₂×C₄×C₅²	C₂³	1	4	yes	—
20	#6029	C₂³	1	8	yes	complemented
21	C₂³×C₄×C₅	D₅	1	6	yes	—
22	C₂×#681	—	5	72	no	—
23	C₂²×#168	—	5	12	no	—
24	C₂²×C₅²	—	2	12	no	—
25	C₂²×C₅²	M₁₆	1	4	yes	complemented
26	C₂²×C₅²	C₂²×C₄	1	4	yes	—
27	C₂²×C₅²	C₂²×C₄	1	3	yes	—
28	C₄×C₅²	C₂²×C₄	1	3	yes	—
29	C₄×C₅²	C₂²×C₄	1	4	yes	—
30	C₄×C₅²	C₂⁴	1	1	yes	characteristic
31	C₂⁴×C₅	Dic₅	1	6	yes	—
32	C₂²×C₄×C₅	Dic₅	1	36	yes	—
33	C₂²×C₄×C₅	Dic₅	1	6	yes	—
34	C₂²×C₄×C₅	C₂×D₅	1	36	yes	—
35	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
36	#681	—	5	96	no	—
37	C₂×#168	—	5	72	no	—
38	C₂²×M₁₆	—	25	1	no	Sylow
39	C₂×C₅²	—	2	4	no	—
40	C₂×C₅²	C₂×M₁₆	1	6	yes	complemented
41	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
42	C₂³×C₅	—	2	24	no	—
43	C₂³×C₅	C₂×Dic₅	1	36	yes	—
44	C₂³×C₅	C₂×Dic₅	1	6	yes	—
45	C₂×C₄×C₅	C₂×Dic₅	1	72	yes	—
46	C₂×C₄×C₅	C₂×Dic₅	1	36	yes	—
47	C₂×C₄×C₅	C₂×Dic₅	1	18	yes	—
48	C₂×C₄×C₅	C₂²×D₅	1	18	yes	—
49	C₂×C₄×C₅	C₂²×D₅	1	24	yes	—
50	#168	—	5	48	no	—
51	C₂³×C₄	C₅⋊D₅	1	1	yes	characteristic
52	C₂×M₁₆	—	25	12	no	—
53	C₂²×C₈	—	25	2	no	—
54	C₅²	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₅	—	2	72	no	—
56	C₂²×C₅	#681	1	24	yes	complemented
57	C₂²×C₅	C₂²×Dic₅	1	24	yes	—
58	C₂²×C₅	C₂²×Dic₅	1	18	yes	—
59	C₄×C₅	C₂²×Dic₅	1	18	yes	—
60	C₄×C₅	C₂²×Dic₅	1	24	yes	—
61	C₄×C₅	C₂³×D₅	1	6	yes	—
62	C₂⁴	#1039	1	1	yes	characteristic
63	C₂²×C₄	#1039	1	1	yes	characteristic
64	C₂²×C₄	#1039	1	6	yes	—
65	C₂²×C₄	C₂×C₅⋊D₅	1	6	yes	—
66	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
67	M₁₆	—	25	16	no	—
68	C₂×C₈	—	25	12	no	—
69	C₂×C₅	—	2	24	no	—
70	C₂×C₅	C₂×#681	1	36	yes	complemented
71	C₂×C₅	C₂³×Dic₅	1	6	yes	—
72	C₂³	—	2	4	no	—
73	C₂³	C₂×#1039	1	6	yes	—
74	C₂³	C₂×#1039	1	1	yes	characteristic
75	C₂×C₄	C₂×#1039	1	12	yes	—
76	C₂×C₄	C₂×#1039	1	6	yes	—
77	C₂×C₄	C₂×#1039	1	3	yes	—
78	C₂×C₄	C₂²×C₅⋊D₅	1	3	yes	—
79	C₂×C₄	C₂²×C₅⋊D₅	1	4	yes	—
80	C₈	—	25	8	no	—
81	C₅	C₂²×#681	1	6	yes	complemented
82	C₂²	—	2	12	no	—
83	C₂²	#87804	1	4	yes	direct product factor
84	C₂²	C₂²×#1039	1	4	yes	—
85	C₂²	C₂²×#1039	1	3	yes	—
86	C₄	C₂²×#1039	1	4	yes	—
87	C₄	C₂²×#1039	1	3	yes	—
88	C₄	C₂³×C₅⋊D₅	1	1	yes	characteristic
89	C₂	—	2	4	no	—
90	C₂	C₂×#87804	1	6	yes	direct product factor
91	C₂	C₂³×#1039	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₅⋊D₅
commutator	C₂×C₅²	C₂³×C₄
Frattini	C₄	C₂³×C₅⋊D₅
Fitting	C₂³×C₄×C₅²	C₂
socle	C₂³×C₅²	C₂×C₄
prime radical	C₂⁴×C₅²	C₄
cosocle	C₄×C₅²	C₂⁴
hypercenter	C₂³×C₄	C₅⋊D₅
hypocenter	C₅²	C₂²×M₁₆
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#87804	C₂	direct (N × H)
3. C₂²×#6029	C₂	semidirect
9. #87804	C₂²	direct (N × H)
10. C₂×#6029	C₂²	semidirect
20. #6029	C₂³	semidirect
25. C₂²×C₅²	M₁₆	semidirect
40. C₂×C₅²	C₂×M₁₆	semidirect
54. C₅²	C₂²×M₁₆	semidirect
56. C₂²×C₅	#681	semidirect
70. C₂×C₅	C₂×#681	semidirect
81. C₅	C₂²×#681	semidirect
83. C₂²	#87804	direct (N × H)
90. C₂	C₂×#87804	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₅²	2
C₂×#87804	2
C₂²×#6029	2
C₂²×#681	5

Subgroups — C22×#87804 #13107309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#87804 #13107309