Subgroups — C₅×#65411 #13102381

Order 1600.

328 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₅×#4072	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅×#168	C₂	1	1	yes	maximal, characteristic
4	C₅×#3947	C₂	1	2	yes	maximal, complemented
5	C₅×C₈×Dic₅	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
7	C₅²×M₁₆	C₄	1	4	yes	complemented
8	C₂×C₅²×C₈	C₂²	1	2	yes	—
9	C₅×C₂²⋊Dic₅	C₄	1	2	yes	—
10	C₅×C₄⋊Dic₅	C₄	1	2	yes	—
11	C₄×C₅×Dic₅	C₂²	1	2	yes	—
12	C₂×C₅×#168	C₄	1	4	yes	complemented
13	C₂×C₅×#168	C₂²	1	1	yes	characteristic
14	C₂×C₅×#168	C₂²	1	1	yes	characteristic
15	C₅×#405	—	5	1	no	maximal
16	#65411	C₅	1	1	yes	maximal, characteristic, direct product factor
17	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
22	C₂×C₅×Dic₅	C₂×C₄	1	4	yes	—
23	C₅²×C₈	C₂×C₄	1	4	yes	—
24	C₅×#168	C₂×C₄	1	2	yes	—
25	C₅×#168	C₂×C₄	1	2	yes	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	D₅	1	1	yes	characteristic
28	C₂×C₅×M₁₆	C₂×C₅	1	1	yes	characteristic
29	C₂²×C₅×C₈	—	5	1	no	—
30	C₅×#117	—	5	1	no	—
31	#4072	C₂×C₅	1	1	yes	characteristic
32	C₂²×#168	C₂×C₅	1	1	yes	characteristic
33	C₅×#97	—	5	2	no	—
34	C₄×C₅×C₈	—	5	2	no	—
35	#3947	C₂×C₅	1	2	yes	complemented
36	C₈×Dic₅	C₂×C₅	1	2	yes	complemented
37	C₂²×C₅²	—	2	2	no	—
38	C₂²×C₅²	C₄²	1	2	yes	—
39	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
40	C₄×C₅²	C₄²	1	2	yes	—
41	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
42	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
43	C₅×Dic₅	—	2	4	no	—
44	C₂²×C₄×C₅	—	2	2	no	—
45	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
47	C₅×M₁₆	—	2	8	no	—
48	C₅×M₁₆	Dic₅	1	4	yes	complemented
49	C₅×M₁₆	C₄×C₅	1	4	yes	complemented
50	C₂×C₅×C₈	—	2	4	no	—
51	C₂×C₅×C₈	—	5	4	no	—
52	C₂×C₅×C₈	—	5	1	no	—
53	C₂×C₅×C₈	—	5	1	no	—
54	C₂×C₅×C₈	C₂×D₅	1	2	yes	—
55	C₂×C₅×C₈	C₂²×C₅	1	2	yes	—
56	C₅×C₄⋊C₄	—	5	2	no	—
57	C₅×C₂²⋊C₄	—	5	2	no	—
58	C₄²×C₅	—	5	2	no	—
59	C₂²⋊Dic₅	C₄×C₅	1	2	yes	—
60	C₄⋊Dic₅	C₄×C₅	1	2	yes	—
61	C₄×Dic₅	C₂²×C₅	1	2	yes	—
62	C₂×#168	C₄×C₅	1	4	yes	complemented
63	C₂×#168	C₂²×C₅	1	1	yes	characteristic
64	C₂×#168	C₂²×C₅	1	1	yes	characteristic
65	#405	—	5	1	no	Sylow
66	C₂×C₅²	—	2	2	no	—
67	C₂×C₅²	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₅²	#131	1	2	yes	—
69	C₂³×C₅	—	2	2	no	—
70	C₂³×C₅	C₄×D₅	1	1	yes	characteristic
71	C₂³×C₅	C₂×C₄×C₅	1	1	yes	characteristic
72	C₂×C₄×C₅	—	2	4	no	—
73	C₂×C₄×C₅	—	2	2	no	—
74	C₂×C₄×C₅	—	2	4	no	—
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	—	5	4	no	—
77	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
78	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
79	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
80	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
82	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
83	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
84	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
85	C₂×Dic₅	C₂×C₄×C₅	1	4	yes	—
86	C₅×C₈	—	2	8	no	—
87	C₅×C₈	—	5	2	no	—
88	C₅×C₈	—	5	2	no	—
89	C₅×C₈	C₂×Dic₅	1	4	yes	—
90	C₅×C₈	C₂×C₄×C₅	1	4	yes	—
91	#168	C₂×C₄×C₅	1	2	yes	—
92	#168	C₂×C₄×C₅	1	2	yes	—
93	C₂×M₁₆	C₅×D₅	1	1	yes	characteristic
94	C₂²×C₈	—	5	1	no	—
95	#117	—	5	1	no	—
96	#97	—	5	2	no	—
97	C₄×C₈	—	5	2	no	—
98	C₅²	#405	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₅	—	2	2	no	—
100	C₂²×C₅	—	2	2	no	—
101	C₂²×C₅	—	2	2	no	—
102	C₂²×C₅	—	2	4	no	—
103	C₂²×C₅	—	4	4	no	—
104	C₂²×C₅	C₄×Dic₅	1	2	yes	—
105	C₂²×C₅	C₄²×C₅	1	2	yes	—
106	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
107	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
108	C₄×C₅	—	2	2	no	—
109	C₄×C₅	—	2	2	no	—
110	C₄×C₅	—	2	4	no	—
111	C₄×C₅	—	10	4	no	—
112	C₄×C₅	C₄×Dic₅	1	2	yes	—
113	C₄×C₅	C₄²×C₅	1	2	yes	—
114	C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
115	C₄×C₅	C₂²×Dic₅	1	1	yes	characteristic
116	C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
117	C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
118	Dic₅	—	2	4	no	—
119	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
120	M₁₆	C₅×Dic₅	1	4	yes	complemented
121	C₂×C₈	—	5	4	no	—
122	C₂×C₈	—	5	1	no	—
123	C₂×C₈	—	5	1	no	—
124	C₂×C₈	C₂×C₅×D₅	1	2	yes	—
125	C₄⋊C₄	—	5	2	no	—
126	C₂²⋊C₄	—	5	2	no	—
127	C₄²	—	5	2	no	—
128	C₂×C₅	—	2	2	no	—
129	C₂×C₅	—	2	2	no	—
130	C₂×C₅	—	2	4	no	—
131	C₂×C₅	—	2	2	no	—
132	C₂×C₅	—	4	4	no	—
133	C₂×C₅	#4053	1	2	yes	—
134	C₂×C₅	C₂×C₄×Dic₅	1	1	yes	characteristic
135	C₂×C₅	C₂×C₄²×C₅	1	1	yes	characteristic
136	C₂×C₅	C₅×#131	1	2	yes	—
137	C₂³	C₄×C₅×D₅	1	1	yes	characteristic
138	C₂×C₄	—	5	4	no	—
139	C₂×C₄	C₄×C₅×D₅	1	2	yes	—
140	C₂×C₄	C₄×C₅×D₅	1	1	yes	characteristic
141	C₂×C₄	C₂×C₅×Dic₅	1	2	yes	—
142	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
143	C₈	—	5	2	no	—
144	C₈	—	5	2	no	—
145	C₈	C₂×C₅×Dic₅	1	4	yes	—
146	C₅	—	2	2	no	—
147	C₅	#65411	1	1	yes	characteristic, direct product factor
148	C₅	C₅×#405	1	1	yes	characteristic, complemented
149	C₂²	—	2	2	no	—
150	C₂²	C₄×C₅×Dic₅	1	2	yes	—
151	C₂²	C₂×C₄×C₅×D₅	1	1	yes	characteristic
152	C₄	—	10	4	no	—
153	C₄	C₄×C₅×Dic₅	1	2	yes	—
154	C₄	C₂×C₄×C₅×D₅	1	1	yes	characteristic
155	C₄	C₂²×C₅×Dic₅	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	C₅×#4053	1	2	yes	—
158	C₂	C₂×C₄×C₅×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₅
commutator	C₂×C₅	C₂×C₄²×C₅
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting	C₂×C₅²×M₁₆	C₂
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂³×C₅²	C₂×C₄
cosocle	C₂×C₄×C₅	C₂³×C₅
hypercenter	C₂×C₅×M₁₆	D₅
hypocenter	C₅	C₅×#405
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₅×#3947	C₂	semidirect
5. C₅×C₈×Dic₅	C₂	semidirect
7. C₅²×M₁₆	C₄	semidirect
12. C₂×C₅×#168	C₄	semidirect
16. #65411	C₅	direct (N × H)
35. #3947	C₂×C₅	semidirect
36. C₈×Dic₅	C₂×C₅	semidirect
48. C₅×M₁₆	Dic₅	semidirect
49. C₅×M₁₆	C₄×C₅	semidirect
62. C₂×#168	C₄×C₅	semidirect
98. C₅²	#405	semidirect
120. M₁₆	C₅×Dic₅	semidirect
147. C₅	#65411	direct (N × H)
148. C₅	C₅×#405	semidirect

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
C₅×#4072	2
C₂²×C₅×#168	2
C₅×#3947	2
C₅×C₈×Dic₅	2
C₅×#405	5
#65411	5

Subgroups — C5×#65411 #13102381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65411 #13102381