Subgroups — C₅×#66151 #13106971

Order 1600.

736 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#713	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4093	C₂	1	3	yes	maximal, complemented
4	C₅×#4092	C₂	1	3	yes	maximal, complemented
5	C₅×Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4076	C₂	1	3	yes	maximal, complemented
7	C₄×C₅×C₂²⋊D₅	C₂	1	3	yes	maximal, complemented
8	C₂×C₅²×Q₈	—	2	4	no	—
9	C₂×C₅²×Q₈	C₂²	1	3	yes	—
10	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅²	C₂²	1	3	yes	—
12	C₂×C₅×C₂²⋊D₅	C₂²	1	3	yes	—
13	C₅×#713	—	2	4	no	—
14	C₂×C₅×D₂₀	C₂²	1	3	yes	—
15	C₂×C₄×C₅×D₅	C₂²	1	3	yes	—
16	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
17	C₅×#685	C₂²	1	3	yes	—
18	C₅×#685	C₂²	1	6	yes	—
19	C₅×C₄⋊Dic₅	C₂²	1	3	yes	—
20	C₅×#683	C₂²	1	3	yes	—
21	C₄×C₅×Dic₅	C₂²	1	3	yes	—
22	C₅×#548	—	5	1	no	maximal
23	#66151	C₅	1	1	yes	maximal, characteristic, direct product factor
24	C₂³×C₅²	C₂³	1	1	yes	characteristic
25	C₂²×C₅×D₅	C₂³	1	3	yes	—
26	C₅²×Q₈	—	2	6	no	—
27	C₅²×Q₈	D₄	1	4	yes	complemented
28	C₂×C₄×C₅²	—	2	6	no	—
29	C₂×C₄×C₅²	C₂³	1	3	yes	—
30	C₂×C₄×C₅²	C₂³	1	3	yes	—
31	C₅×C₂²⋊D₅	—	2	6	no	—
32	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
33	C₂×C₅×Dic₅	C₂³	1	3	yes	—
34	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
35	C₅×D₂₀	—	2	6	no	—
36	C₄×C₅×D₅	—	2	6	no	—
37	C₂×C₅×#41	—	5	1	no	—
38	C₂²×C₅×Q₈	—	2	2	no	—
39	C₂²×C₅×Q₈	D₅	1	1	yes	characteristic, complemented
40	C₂²×C₅×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂×#713	C₂×C₅	1	1	yes	characteristic, complemented
42	C₅×#124	—	5	3	no	—
43	C₅×C₂²⋊Q₈	—	5	3	no	—
44	C₅×#121	—	5	3	no	—
45	C₄×C₅×Q₈	—	5	1	no	—
46	C₄×C₅×D₄	—	5	3	no	—
47	#4093	C₂×C₅	1	3	yes	complemented
48	#4092	C₂×C₅	1	3	yes	complemented
49	Q₈×Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
50	#4076	C₂×C₅	1	3	yes	complemented
51	C₄×C₂²⋊D₅	C₂×C₅	1	3	yes	complemented
52	C₂²×C₅²	—	2	1	no	—
53	C₂²×C₅²	—	2	1	no	—
54	C₂²×C₅²	#41	1	2	yes	complemented
55	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
56	C₂×C₅×D₅	—	2	3	no	—
57	C₂×C₅×D₅	—	2	3	no	—
58	C₂×C₅×D₅	—	2	3	no	—
59	C₄×C₅²	—	2	3	no	—
60	C₄×C₅²	C₂×D₄	1	6	yes	—
61	C₅×Dic₅	—	2	1	no	—
62	C₅×Dic₅	—	2	3	no	—
63	C₅×Dic₅	—	2	1	no	—
64	C₅×#41	—	10	4	no	—
65	C₂×C₅×Q₈	—	2	16	no	—
66	C₂×C₅×Q₈	—	2	4	no	—
67	C₂×C₅×Q₈	—	2	4	no	—
68	C₂×C₅×Q₈	—	2	6	no	—
69	C₂×C₅×Q₈	—	2	2	no	—
70	C₂×C₅×Q₈	C₂×D₅	1	3	yes	—
71	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
72	C₂×C₅×Q₈	C₂²×C₅	1	3	yes	—
73	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₅×D₄	—	5	3	no	—
75	C₂×C₅×D₄	—	5	3	no	—
76	C₂²×C₄×C₅	—	2	6	no	—
77	C₂²×C₄×C₅	—	5	3	no	—
78	C₂²×C₄×C₅	C₂×D₅	1	3	yes	—
79	C₂²×C₄×C₅	C₂²×C₅	1	3	yes	—
80	C₂×C₂²⋊D₅	C₂²×C₅	1	3	yes	—
81	#713	—	2	4	no	—
82	C₂×D₂₀	C₂²×C₅	1	3	yes	—
83	C₂×C₄×D₅	C₂²×C₅	1	3	yes	—
84	C₅×C₄⋊C₄	—	5	3	no	—
85	C₅×C₄⋊C₄	—	5	3	no	—
86	C₅×C₂²⋊C₄	—	5	1	no	—
87	C₅×C₂²⋊C₄	—	5	3	no	—
88	C₅×C₂²⋊C₄	—	5	6	no	—
89	C₄²×C₅	—	5	3	no	—
90	C₂²⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
91	#685	C₂²×C₅	1	3	yes	—
92	#685	C₂²×C₅	1	6	yes	—
93	C₄⋊Dic₅	C₂²×C₅	1	3	yes	—
94	#683	C₂²×C₅	1	3	yes	—
95	C₄×Dic₅	C₂²×C₅	1	3	yes	—
96	#548	—	5	1	no	Sylow
97	C₂×C₅²	—	2	2	no	—
98	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₅²	C₂×#41	1	1	yes	characteristic
100	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
101	C₅×D₅	—	4	3	no	—
102	C₂³×C₅	—	2	2	no	—
103	C₂³×C₅	—	5	3	no	—
104	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
105	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
106	C₂²×D₅	C₂³×C₅	1	3	yes	—
107	C₅×Q₈	—	2	24	no	—
108	C₅×Q₈	—	2	6	no	—
109	C₅×Q₈	—	2	6	no	—
110	C₅×Q₈	—	2	8	no	—
111	C₅×Q₈	C₂²⋊D₅	1	4	yes	complemented
112	C₅×Q₈	C₅×D₄	1	4	yes	complemented
113	C₅×D₄	—	10	6	no	—
114	C₅×D₄	—	10	6	no	—
115	C₂×C₄×C₅	—	2	24	no	—
116	C₂×C₄×C₅	—	2	6	no	—
117	C₂×C₄×C₅	—	2	6	no	—
118	C₂×C₄×C₅	—	2	6	no	—
119	C₂×C₄×C₅	—	2	6	no	—
120	C₂×C₄×C₅	—	5	1	no	—
121	C₂×C₄×C₅	—	5	3	no	—
122	C₂×C₄×C₅	—	5	1	no	—
123	C₂×C₄×C₅	—	10	6	no	—
124	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
125	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
126	C₂×C₄×C₅	C₂³×C₅	1	3	yes	—
127	C₂×C₄×C₅	C₂³×C₅	1	3	yes	—
128	C₂²⋊D₅	—	2	6	no	—
129	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
130	C₂×Dic₅	C₂³×C₅	1	3	yes	—
131	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
132	D₂₀	—	2	6	no	—
133	C₄×D₅	—	2	6	no	—
134	C₂×#41	—	5	1	no	—
135	C₂²×Q₈	C₅×D₅	1	1	yes	characteristic, complemented
136	#124	—	5	3	no	—
137	C₂²⋊Q₈	—	5	3	no	—
138	#121	—	5	3	no	—
139	C₄×Q₈	—	5	1	no	—
140	C₄×D₄	—	5	3	no	—
141	C₅²	#548	1	1	yes	characteristic, complemented, Sylow
142	C₂²×C₅	—	2	4	no	—
143	C₂²×C₅	—	2	4	no	—
144	C₂²×C₅	—	2	1	no	—
145	C₂²×C₅	—	2	1	no	—
146	C₂²×C₅	—	2	1	no	—
147	C₂²×C₅	—	2	1	no	—
148	C₂²×C₅	—	2	4	no	—
149	C₂²×C₅	—	2	2	no	—
150	C₂²×C₅	—	10	3	no	—
151	C₂²×C₅	—	10	3	no	—
152	C₂²×C₅	—	10	3	no	—
153	C₂²×C₅	#713	1	2	yes	complemented
154	C₂²×C₅	C₅×#41	1	2	yes	complemented
155	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
156	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
157	C₂×D₅	—	2	3	no	—
158	C₂×D₅	—	2	3	no	—
159	C₂×D₅	—	2	3	no	—
160	C₄×C₅	—	2	12	no	—
161	C₄×C₅	—	2	3	no	—
162	C₄×C₅	—	2	3	no	—
163	C₄×C₅	—	2	12	no	—
164	C₄×C₅	—	10	3	no	—
165	C₄×C₅	—	10	1	no	—
166	C₄×C₅	—	10	1	no	—
167	C₄×C₅	C₂×C₂²⋊D₅	1	6	yes	—
168	C₄×C₅	C₂×C₅×D₄	1	6	yes	—
169	Dic₅	—	2	1	no	—
170	Dic₅	—	2	3	no	—
171	Dic₅	—	2	1	no	—
172	#41	—	10	4	no	—
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	C₂×C₅×D₅	1	3	yes	—
175	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
176	C₂×D₄	—	5	3	no	—
177	C₂×D₄	—	5	3	no	—
178	C₂²×C₄	—	5	3	no	—
179	C₂²×C₄	C₂×C₅×D₅	1	3	yes	—
180	C₄⋊C₄	—	5	3	no	—
181	C₄⋊C₄	—	5	3	no	—
182	C₂²⋊C₄	—	5	1	no	—
183	C₂²⋊C₄	—	5	3	no	—
184	C₂²⋊C₄	—	5	6	no	—
185	C₄²	—	5	3	no	—
186	C₂×C₅	—	2	8	no	—
187	C₂×C₅	—	2	2	no	—
188	C₂×C₅	—	2	2	no	—
189	C₂×C₅	—	2	2	no	—
190	C₂×C₅	—	2	2	no	—
191	C₂×C₅	—	2	2	no	—
192	C₂×C₅	—	20	3	no	—
193	C₂×C₅	C₂×#713	1	1	yes	characteristic
194	C₂×C₅	#4147	1	1	yes	characteristic
195	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
196	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
197	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
198	C₂×C₅	C₅×2₋¹⁺⁴	1	1	yes	characteristic
199	D₅	—	4	3	no	—
200	C₂³	—	5	3	no	—
201	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
202	Q₈	—	2	6	no	—
203	Q₈	C₅×C₂²⋊D₅	1	4	yes	complemented
204	D₄	—	10	6	no	—
205	D₄	—	10	6	no	—
206	C₂×C₄	—	2	6	no	—
207	C₂×C₄	—	5	1	no	—
208	C₂×C₄	—	5	3	no	—
209	C₂×C₄	—	5	1	no	—
210	C₂×C₄	—	10	6	no	—
211	C₂×C₄	C₂²×C₅×D₅	1	3	yes	—
212	C₂×C₄	C₂²×C₅×D₅	1	3	yes	—
213	C₅	—	2	2	no	—
214	C₅	#66151	1	1	yes	characteristic, direct product factor
215	C₅	C₅×#548	1	1	yes	characteristic, complemented
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	10	3	no	—
219	C₂²	—	10	3	no	—
220	C₂²	—	10	3	no	—
221	C₂²	C₅×#713	1	2	yes	complemented
222	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
223	C₄	—	2	3	no	—
224	C₄	—	10	3	no	—
225	C₄	—	10	1	no	—
226	C₄	—	10	1	no	—
227	C₄	C₂×C₅×C₂²⋊D₅	1	6	yes	—
228	C₂	—	2	2	no	—
229	C₂	—	20	3	no	—
230	C₂	C₂×C₅×#713	1	1	yes	characteristic
231	C₂	C₅×#4147	1	1	yes	characteristic
232	C₂	C₂²×C₅×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂²×C₅²×Q₈	C₂
socle	C₂²×C₅²	C₂⁴
hypercenter	C₂²×C₅×Q₈	D₅
hypocenter	C₅	C₅×#548
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅²×Q₈	C₂	semidirect
2. C₂×C₅×#713	C₂	semidirect
3. C₅×#4093	C₂	semidirect
4. C₅×#4092	C₂	semidirect
5. C₅×Q₈×Dic₅	C₂	semidirect
6. C₅×#4076	C₂	semidirect
7. C₄×C₅×C₂²⋊D₅	C₂	semidirect
23. #66151	C₅	direct (N × H)
27. C₅²×Q₈	D₄	semidirect
39. C₂²×C₅×Q₈	D₅	semidirect
40. C₂²×C₅×Q₈	C₂×C₅	semidirect
41. C₂×#713	C₂×C₅	semidirect
47. #4093	C₂×C₅	semidirect
48. #4092	C₂×C₅	semidirect
49. Q₈×Dic₅	C₂×C₅	semidirect
50. #4076	C₂×C₅	semidirect
51. C₄×C₂²⋊D₅	C₂×C₅	semidirect
54. C₂²×C₅²	#41	semidirect
111. C₅×Q₈	C₂²⋊D₅	semidirect
112. C₅×Q₈	C₅×D₄	semidirect
135. C₂²×Q₈	C₅×D₅	semidirect
141. C₅²	#548	semidirect
153. C₂²×C₅	#713	semidirect
154. C₂²×C₅	C₅×#41	semidirect
203. Q₈	C₅×C₂²⋊D₅	semidirect
214. C₅	#66151	direct (N × H)
215. C₅	C₅×#548	semidirect
221. C₂²	C₅×#713	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅²×Q₈	2
C₂×C₅×#713	2
C₅×#4093	2
C₅×#4092	2
C₅×Q₈×Dic₅	2
C₅×#4076	2
C₄×C₅×C₂²⋊D₅	2
C₅×#548	5
#66151	5

Subgroups — C5×#66151 #13106971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66151 #13106971