Subgroups — C₅²×#424 #13103161

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	C₂²×C₅²×C₈	C₂	1	1	yes	maximal, characteristic
3	C₅²×#117	C₂	1	1	yes	maximal, characteristic
4	C₅²×C₄⋊C₈	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	—	2	2	no	—
8	C₂×C₅²×C₈	C₂²	1	2	yes	—
9	C₂×C₅²×C₈	C₂²	1	2	yes	—
10	C₅²×C₄⋊C₄	C₄	1	2	yes	—
11	C₅²×C₂²⋊C₄	C₄	1	2	yes	—
12	C₄²×C₅²	C₂²	1	2	yes	—
13	C₅×#424	C₅	1	6	yes	maximal, direct product factor
14	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₅²	C₂×C₄	1	4	yes	—
16	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅²	D₄	1	2	yes	—
18	C₂×C₄×C₅²	Q₈	1	2	yes	—
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	C₅²×C₈	—	2	2	no	—
21	C₅²×C₈	—	2	2	no	—
22	C₂×C₅×M₁₆	C₂×C₅	1	6	yes	—
23	C₂²×C₅×C₈	C₂×C₅	1	6	yes	—
24	C₅×#117	C₂×C₅	1	6	yes	—
25	C₅×C₄⋊C₈	C₂×C₅	1	24	yes	complemented
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	—	2	4	no	—
30	C₄×C₅²	C₄⋊C₄	1	2	yes	—
31	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
32	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅	C₂²×C₅	1	6	yes	—
34	C₅×M₁₆	—	2	12	no	—
35	C₂×C₅×C₈	—	2	12	no	—
36	C₂×C₅×C₈	C₂²×C₅	1	12	yes	—
37	C₂×C₅×C₈	C₂²×C₅	1	12	yes	—
38	C₅×C₄⋊C₄	C₄×C₅	1	12	yes	—
39	C₅×C₂²⋊C₄	C₄×C₅	1	12	yes	—
40	C₄²×C₅	C₂²×C₅	1	12	yes	—
41	#424	C₅²	1	1	yes	characteristic, direct product factor, 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	C₂×C₄×C₅	1	6	yes	—
46	C₂×C₄×C₅	C₂×C₄×C₅	1	24	yes	—
47	C₂×C₄×C₅	C₂×C₄×C₅	1	6	yes	—
48	C₂×C₄×C₅	C₅×D₄	1	12	yes	—
49	C₂×C₄×C₅	C₅×Q₈	1	12	yes	—
50	C₂×C₄×C₅	C₂³×C₅	1	6	yes	—
51	C₅×C₈	—	2	12	no	—
52	C₅×C₈	—	2	12	no	—
53	C₂×M₁₆	C₂×C₅²	1	1	yes	characteristic
54	C₂²×C₈	C₂×C₅²	1	1	yes	characteristic
55	#117	C₂×C₅²	1	1	yes	characteristic
56	C₄⋊C₈	C₂×C₅²	1	4	yes	complemented
57	C₅²	#424	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²×C₅	—	2	12	no	—
59	C₂²×C₅	C₅×C₄⋊C₄	1	12	yes	—
60	C₂²×C₅	C₂²×C₄×C₅	1	6	yes	—
61	C₄×C₅	—	2	24	no	—
62	C₄×C₅	C₅×C₄⋊C₄	1	12	yes	—
63	C₄×C₅	C₂×C₅×D₄	1	6	yes	—
64	C₄×C₅	C₂×C₅×Q₈	1	6	yes	—
65	C₂²×C₄	C₂²×C₅²	1	1	yes	characteristic
66	M₁₆	—	2	2	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	C₂²×C₅²	1	2	yes	—
69	C₂×C₈	C₂²×C₅²	1	2	yes	—
70	C₄⋊C₄	C₄×C₅²	1	2	yes	—
71	C₂²⋊C₄	C₄×C₅²	1	2	yes	—
72	C₄²	C₂²×C₅²	1	2	yes	—
73	C₂×C₅	—	2	12	no	—
74	C₂×C₅	C₂×C₅×C₄⋊C₄	1	6	yes	—
75	C₂×C₅	C₅×#131	1	12	yes	—
76	C₂³	C₂×C₄×C₅²	1	1	yes	characteristic
77	C₂×C₄	C₂×C₄×C₅²	1	4	yes	—
78	C₂×C₄	C₂×C₄×C₅²	1	1	yes	characteristic
79	C₂×C₄	C₅²×D₄	1	2	yes	—
80	C₂×C₄	C₅²×Q₈	1	2	yes	—
81	C₂×C₄	C₂³×C₅²	1	1	yes	characteristic
82	C₈	—	2	2	no	—
83	C₈	—	2	2	no	—
84	C₅	C₅×#424	1	6	yes	direct product factor
85	C₂²	—	2	2	no	—
86	C₂²	C₅²×C₄⋊C₄	1	2	yes	—
87	C₂²	C₂²×C₄×C₅²	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	C₅²×C₄⋊C₄	1	2	yes	—
90	C₄	C₂×C₅²×D₄	1	1	yes	characteristic
91	C₄	C₂×C₅²×Q₈	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	C₂×C₅²×C₄⋊C₄	1	1	yes	characteristic
94	C₂	C₅²×#131	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅²	C₂³
commutator	C₂²	C₂²×C₄×C₅²
Frattini, cosocle	C₂×C₄	C₂³×C₅²
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂³×C₅²	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅² ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅²×C₄⋊C₈	C₂	semidirect
13. C₅×#424	C₅	direct (N × H)
25. C₅×C₄⋊C₈	C₂×C₅	semidirect
41. #424	C₅²	direct (N × H)
56. C₄⋊C₈	C₂×C₅²	semidirect
57. C₅²	#424	direct (N × H)
84. C₅	C₅×#424	direct (N × H)

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
C₂²×C₅²×C₈	2
C₅²×#117	2
C₅²×C₄⋊C₈	2
C₅×#424	5

Subgroups — C52×#424 #13103161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#424 #13103161