Subgroups — #12015700

Order 1280.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	#10598148	C₂	1	6	yes	maximal
3	C₂×C₄×C₅×C₈	C₄	1	3	yes	—
4	C₂×C₄×C₅×C₈	C₂²	1	3	yes	—
5	C₄³×C₅	C₄	1	1	yes	characteristic
6	C₂²×#672	C₂²	1	4	yes	—
7	#12894	—	5	1	no	maximal, Sylow
8	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
9	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₅	C₈	1	3	yes	—
11	C₂×C₄²×C₅	C₈	1	1	yes	characteristic
12	C₂×C₄²×C₅	C₂×C₄	1	3	yes	—
13	C₄×C₅×C₈	—	2	6	no	—
14	C₄×C₅×C₈	—	2	6	no	—
15	C₂×#672	—	2	12	no	—
16	C₄²×C₈	D₅	1	1	yes	characteristic
17	#1374	—	5	6	no	—
18	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
19	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
20	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₅×C₈	—	2	3	no	—
22	C₂×C₅×C₈	—	2	3	no	—
23	C₂×C₅×C₈	M₁₆	1	6	yes	—
24	C₂×C₅×C₈	#41	1	6	yes	—
25	C₄²×C₅	—	2	6	no	—
26	C₄²×C₅	—	2	3	no	—
27	C₄²×C₅	—	2	3	no	—
28	C₄²×C₅	C₁₆	1	4	yes	complemented
29	#672	—	4	4	no	—
30	C₂²×C₁₆	—	5	4	no	—
31	C₂×C₄×C₈	Dic₅	1	3	yes	—
32	C₂×C₄×C₈	C₂×D₅	1	3	yes	—
33	C₄³	Dic₅	1	1	yes	characteristic
34	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
35	C₂×C₄×C₅	—	2	3	no	—
36	C₂×C₄×C₅	—	2	3	no	—
37	C₂×C₄×C₅	—	2	3	no	—
38	C₂×C₄×C₅	—	2	3	no	—
39	C₂×C₄×C₅	C₂×C₁₆	1	6	yes	—
40	C₂×C₄×C₅	M₃₂	1	6	yes	—
41	C₂×C₄×C₅	#117	1	3	yes	—
42	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
43	C₅×C₈	—	2	6	no	—
44	C₅×C₈	#126	1	4	yes	—
45	C₂²×C₈	C₂×Dic₅	1	3	yes	—
46	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
47	C₂×C₄²	#168	1	3	yes	—
48	C₂×C₄²	#168	1	1	yes	characteristic
49	C₂×C₄²	C₂×Dic₅	1	3	yes	—
50	C₂×C₁₆	—	10	12	no	—
51	C₄×C₈	—	2	6	no	—
52	C₄×C₈	—	2	6	no	—
53	C₂²×C₅	#431	1	3	yes	—
54	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
55	C₂²×C₅	C₂×M₃₂	1	3	yes	—
56	C₄×C₅	—	2	6	no	—
57	C₄×C₅	—	2	6	no	—
58	C₄×C₅	#383	1	1	yes	characteristic
59	C₄×C₅	#432	1	3	yes	—
60	C₂²×C₄	C₂×#168	1	3	yes	—
61	C₂²×C₄	C₂×#168	1	3	yes	—
62	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
63	C₂×C₈	—	2	3	no	—
64	C₂×C₈	—	2	3	no	—
65	C₂×C₈	#681	1	6	yes	—
66	C₂×C₈	#709	1	6	yes	—
67	C₄²	—	2	6	no	—
68	C₄²	—	2	3	no	—
69	C₄²	—	2	3	no	—
70	C₄²	#672	1	4	yes	complemented
71	C₁₆	—	20	4	no	—
72	C₂×C₅	#1839	1	1	yes	characteristic
73	C₂×C₅	#2162	1	3	yes	—
74	C₂×C₅	#2163	1	3	yes	—
75	C₂³	C₂²×#168	1	1	yes	characteristic
76	C₂×C₄	—	2	3	no	—
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	C₂×#672	1	6	yes	—
81	C₂×C₄	#3944	1	6	yes	—
82	C₂×C₄	C₂×#681	1	3	yes	—
83	C₂×C₄	#4072	1	3	yes	—
84	C₈	—	2	6	no	—
85	C₈	#4022	1	4	yes	—
86	C₅	#12894	1	1	yes	characteristic, complemented, Sylow
87	C₂²	#65214	1	3	yes	—
88	C₂²	C₂²×#672	1	1	yes	characteristic
89	C₂²	C₂×#3944	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	#65215	1	3	yes	—
93	C₄	#65226	1	1	yes	characteristic
94	C₂	#10600186	1	1	yes	characteristic
95	C₂	#10600537	1	3	yes	—
96	C₂	#10600538	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₁₆
Fitting	C₄²×C₅×C₈	C₂
socle, prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₄²×C₈	D₅
hypocenter	C₅	#12894
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
28. C₄²×C₅	C₁₆	semidirect
70. C₄²	#672	semidirect
86. C₅	#12894	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₅×C₈	2
#10598148	2
#12894	5