Subgroups — C₅×#66161 #13106976

Order 1600.

752 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#41	C₂	1	1	yes	maximal, characteristic
2	C₅×Q₈×Dic₅	C₂	1	2	yes	maximal, complemented
3	C₅×D₄×Dic₅	C₂	1	6	yes	maximal, complemented
4	C₅×#4072	C₂	1	3	yes	maximal, complemented
5	C₂×C₅×C₄⋊Dic₅	C₂	1	3	yes	maximal, complemented
6	C₅²×#41	C₄	1	8	yes	complemented
7	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₅²×D₄	C₂²	1	3	yes	—
9	C₂²×C₄×C₅²	C₂²	1	3	yes	—
10	C₂²×C₅×Dic₅	C₂²	1	6	yes	—
11	C₅×C₂²⋊Dic₅	C₂²	1	6	yes	—
12	C₅×C₄⋊Dic₅	C₂²	1	3	yes	—
13	C₅×C₄⋊Dic₅	C₂²	1	6	yes	—
14	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
15	C₄×C₅×Dic₅	C₂²	1	6	yes	—
16	C₅×#520	—	5	1	no	maximal
17	#66161	C₅	1	1	yes	maximal, characteristic, direct product factor
18	C₂³×C₅²	C₂³	1	3	yes	—
19	C₅²×Q₈	C₂×C₄	1	4	yes	complemented
20	C₅²×D₄	C₂×C₄	1	12	yes	complemented
21	C₂×C₄×C₅²	C₂×C₄	1	12	yes	complemented
22	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₅²	C₂³	1	3	yes	—
24	C₂×C₅×Dic₅	—	4	6	no	—
25	C₂×C₅×Dic₅	C₂³	1	6	yes	—
26	C₂×C₅×Dic₅	C₂³	1	2	yes	—
27	C₂×C₅×#41	—	2	2	no	—
28	C₂×C₅×#41	D₅	1	1	yes	characteristic
29	C₂×C₅×#41	C₂×C₅	1	1	yes	characteristic
30	C₄×C₅×Q₈	—	5	2	no	—
31	C₄×C₅×D₄	—	5	6	no	—
32	C₅×#117	—	5	3	no	—
33	C₂×C₅×C₄⋊C₄	—	5	3	no	—
34	Q₈×Dic₅	C₂×C₅	1	2	yes	complemented
35	D₄×Dic₅	C₂×C₅	1	6	yes	complemented
36	#4072	C₂×C₅	1	3	yes	complemented
37	C₂×C₄⋊Dic₅	C₂×C₅	1	3	yes	complemented
38	C₂²×C₅²	—	2	3	no	—
39	C₂²×C₅²	—	2	3	no	—
40	C₂²×C₅²	C₂²×C₄	1	6	yes	—
41	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
42	C₄×C₅²	C₂²×C₄	1	2	yes	—
43	C₄×C₅²	C₂²×C₄	1	6	yes	—
44	C₅×Dic₅	—	2	6	no	—
45	C₅×Dic₅	—	2	2	no	—
46	C₅×#41	—	2	16	no	—
47	C₅×#41	Dic₅	1	8	yes	complemented
48	C₅×#41	C₄×C₅	1	8	yes	complemented
49	C₂×C₅×Q₈	—	2	2	no	—
50	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
51	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
52	C₂×C₅×D₄	—	2	6	no	—
53	C₂×C₅×D₄	C₂×D₅	1	3	yes	—
54	C₂×C₅×D₄	C₂²×C₅	1	3	yes	—
55	C₂²×C₄×C₅	—	2	6	no	—
56	C₂²×C₄×C₅	—	5	6	no	—
57	C₂²×C₄×C₅	C₂×D₅	1	3	yes	—
58	C₂²×C₄×C₅	C₂²×C₅	1	3	yes	—
59	C₂²×Dic₅	C₂²×C₅	1	6	yes	—
60	C₅×C₄⋊C₄	—	5	3	no	—
61	C₅×C₄⋊C₄	—	5	6	no	—
62	C₅×C₄⋊C₄	—	5	1	no	—
63	C₅×C₂²⋊C₄	—	5	6	no	—
64	C₄²×C₅	—	5	6	no	—
65	C₂²⋊Dic₅	C₂²×C₅	1	6	yes	—
66	C₄⋊Dic₅	C₂²×C₅	1	3	yes	—
67	C₄⋊Dic₅	C₂²×C₅	1	6	yes	—
68	C₄⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
69	C₄×Dic₅	C₂²×C₅	1	6	yes	—
70	#520	—	5	1	no	Sylow
71	C₂×C₅²	—	2	6	no	—
72	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
74	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
75	C₂³×C₅	—	2	6	no	—
76	C₂³×C₅	C₂²×D₅	1	3	yes	—
77	C₂³×C₅	C₂³×C₅	1	3	yes	—
78	C₅×Q₈	—	2	8	no	—
79	C₅×Q₈	C₂×Dic₅	1	4	yes	complemented
80	C₅×Q₈	C₂×C₄×C₅	1	4	yes	complemented
81	C₅×D₄	—	2	24	no	—
82	C₅×D₄	C₂×Dic₅	1	12	yes	complemented
83	C₅×D₄	C₂×C₄×C₅	1	12	yes	complemented
84	C₂×C₄×C₅	—	2	24	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	—	2	6	no	—
87	C₂×C₄×C₅	—	5	2	no	—
88	C₂×C₄×C₅	—	5	6	no	—
89	C₂×C₄×C₅	—	20	6	no	—
90	C₂×C₄×C₅	C₂×Dic₅	1	12	yes	complemented
91	C₂×C₄×C₅	C₂×C₄×C₅	1	12	yes	complemented
92	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
93	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂³×C₅	1	3	yes	—
95	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
96	C₂×Dic₅	—	4	6	no	—
97	C₂×Dic₅	C₂³×C₅	1	2	yes	—
98	C₂×Dic₅	C₂³×C₅	1	6	yes	—
99	C₂×#41	C₅×D₅	1	1	yes	characteristic
100	C₄×Q₈	—	5	2	no	—
101	C₄×D₄	—	5	6	no	—
102	#117	—	5	3	no	—
103	C₂×C₄⋊C₄	—	5	3	no	—
104	C₅²	#520	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₅	—	2	3	no	—
106	C₂²×C₅	—	2	3	no	—
107	C₂²×C₅	—	2	3	no	—
108	C₂²×C₅	—	2	3	no	—
109	C₂²×C₅	—	2	12	no	—
110	C₂²×C₅	—	2	2	no	—
111	C₂²×C₅	—	4	6	no	—
112	C₂²×C₅	—	4	6	no	—
113	C₂²×C₅	C₂²×Dic₅	1	6	yes	—
114	C₂²×C₅	C₂²×C₄×C₅	1	6	yes	—
115	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
116	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
117	C₄×C₅	—	2	4	no	—
118	C₄×C₅	—	2	12	no	—
119	C₄×C₅	—	10	6	no	—
120	C₄×C₅	—	10	2	no	—
121	C₄×C₅	C₂²×Dic₅	1	2	yes	—
122	C₄×C₅	C₂²×Dic₅	1	6	yes	—
123	C₄×C₅	C₂²×C₄×C₅	1	2	yes	—
124	C₄×C₅	C₂²×C₄×C₅	1	6	yes	—
125	Dic₅	—	2	6	no	—
126	Dic₅	—	2	2	no	—
127	#41	C₅×Dic₅	1	8	yes	complemented
128	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
129	C₂×D₄	C₂×C₅×D₅	1	3	yes	—
130	C₂²×C₄	—	5	6	no	—
131	C₂²×C₄	C₂×C₅×D₅	1	3	yes	—
132	C₄⋊C₄	—	5	3	no	—
133	C₄⋊C₄	—	5	6	no	—
134	C₄⋊C₄	—	5	1	no	—
135	C₂²⋊C₄	—	5	6	no	—
136	C₄²	—	5	6	no	—
137	C₂×C₅	—	2	6	no	—
138	C₂×C₅	—	2	6	no	—
139	C₂×C₅	—	2	2	no	—
140	C₂×C₅	—	2	2	no	—
141	C₂×C₅	—	2	2	no	—
142	C₂×C₅	—	4	12	no	—
143	C₂×C₅	#4149	1	1	yes	characteristic
144	C₂×C₅	#4150	1	1	yes	characteristic
145	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
146	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
147	C₂×C₅	C₅×H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₅	C₅×2₋¹⁺⁴	1	1	yes	characteristic
149	C₂³	C₂²×C₅×D₅	1	3	yes	—
150	Q₈	C₂×C₅×Dic₅	1	4	yes	complemented
151	D₄	C₂×C₅×Dic₅	1	12	yes	complemented
152	C₂×C₄	—	5	2	no	—
153	C₂×C₄	—	5	6	no	—
154	C₂×C₄	—	20	6	no	—
155	C₂×C₄	C₂×C₅×Dic₅	1	12	yes	complemented
156	C₂×C₄	C₂²×C₅×D₅	1	3	yes	—
157	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
158	C₅	—	2	2	no	—
159	C₅	#66161	1	1	yes	characteristic, direct product factor
160	C₅	C₅×#520	1	1	yes	characteristic, complemented
161	C₂²	—	2	3	no	—
162	C₂²	—	2	3	no	—
163	C₂²	C₂²×C₅×Dic₅	1	6	yes	—
164	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
165	C₄	—	10	6	no	—
166	C₄	—	10	2	no	—
167	C₄	C₂²×C₅×Dic₅	1	2	yes	—
168	C₄	C₂²×C₅×Dic₅	1	6	yes	—
169	C₂	—	2	6	no	—
170	C₂	C₅×#4149	1	1	yes	characteristic
171	C₂	C₅×#4150	1	1	yes	characteristic
172	C₂	C₂³×C₅×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₅
commutator	C₂×C₅	C₂³×C₄×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting, prime radical	C₂×C₅²×#41	C₂
socle	C₂²×C₅²	C₂⁴
cosocle	C₂²×C₅	C₂⁴×C₅
hypercenter	C₂×C₅×#41	D₅
hypocenter	C₅	C₅×#520
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×#41

Complements

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

N	H ≅ G/N	type
2. C₅×Q₈×Dic₅	C₂	semidirect
3. C₅×D₄×Dic₅	C₂	semidirect
4. C₅×#4072	C₂	semidirect
5. C₂×C₅×C₄⋊Dic₅	C₂	semidirect
6. C₅²×#41	C₄	semidirect
17. #66161	C₅	direct (N × H)
19. C₅²×Q₈	C₂×C₄	semidirect
20. C₅²×D₄	C₂×C₄	semidirect
21. C₂×C₄×C₅²	C₂×C₄	semidirect
34. Q₈×Dic₅	C₂×C₅	semidirect
35. D₄×Dic₅	C₂×C₅	semidirect
36. #4072	C₂×C₅	semidirect
37. C₂×C₄⋊Dic₅	C₂×C₅	semidirect
47. C₅×#41	Dic₅	semidirect
48. C₅×#41	C₄×C₅	semidirect
79. C₅×Q₈	C₂×Dic₅	semidirect
80. C₅×Q₈	C₂×C₄×C₅	semidirect
82. C₅×D₄	C₂×Dic₅	semidirect
83. C₅×D₄	C₂×C₄×C₅	semidirect
90. C₂×C₄×C₅	C₂×Dic₅	semidirect
91. C₂×C₄×C₅	C₂×C₄×C₅	semidirect
104. C₅²	#520	semidirect
127. #41	C₅×Dic₅	semidirect
150. Q₈	C₂×C₅×Dic₅	semidirect
151. D₄	C₂×C₅×Dic₅	semidirect
155. C₂×C₄	C₂×C₅×Dic₅	semidirect
159. C₅	#66161	direct (N × H)
160. C₅	C₅×#520	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×#41	2
C₅×Q₈×Dic₅	2
C₅×D₄×Dic₅	2
C₅×#4072	2
C₂×C₅×C₄⋊Dic₅	2
C₅×#520	5
#66161	5

Subgroups — C5×#66161 #13106976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66161 #13106976