Subgroups — C₅×#66167 #13106986

Order 1600.

752 conjugacy classes of subgroups.

The subgroup lattice has 305 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, complemented
2	C₂×C₅×#709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4092	C₂	1	2	yes	maximal, complemented
4	C₅×#4085	C₂	1	4	yes	maximal, complemented
5	C₅×#4084	C₂	1	2	yes	maximal, complemented
6	C₅×#4076	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4070	C₂	1	2	yes	maximal, complemented
9	C₅²×#41	—	2	4	no	—
10	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
12	C₂×C₅²×D₄	C₂²	1	2	yes	—
13	C₂²×C₄×C₅²	C₂²	1	2	yes	—
14	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₂²⋊D₅	C₂²	1	2	yes	—
16	C₂×C₅×C₂²⋊D₅	C₂²	1	4	yes	—
17	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
18	C₅×#709	—	2	4	no	—
19	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
21	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
22	C₅×C₂²⋊Dic₅	C₂²	1	4	yes	—
23	C₅×#685	C₂²	1	4	yes	—
24	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
25	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
26	C₅×#683	C₂²	1	4	yes	—
27	C₅×#537	—	5	1	no	maximal
28	#66167	C₅	1	1	yes	maximal, characteristic, direct product factor
29	C₂³×C₅²	C₂³	1	2	yes	—
30	C₂³×C₅²	C₂³	1	1	yes	characteristic
31	C₂²×C₅×D₅	C₂³	1	2	yes	—
32	C₅²×Q₈	—	2	2	no	—
33	C₅²×D₄	—	2	4	no	—
34	C₅²×D₄	—	2	2	no	—
35	C₂×C₄×C₅²	—	2	4	no	—
36	C₂×C₄×C₅²	D₄	1	4	yes	complemented
37	C₂×C₄×C₅²	C₂³	1	2	yes	—
38	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
40	C₅×C₂²⋊D₅	—	2	4	no	—
41	C₅×C₂²⋊D₅	—	4	4	no	—
42	C₂×C₅×Dic₅	—	4	2	no	—
43	C₂×C₅×Dic₅	C₂³	1	2	yes	—
44	C₂×C₅×Dic₅	C₂³	1	4	yes	—
45	C₅×D₂₀	—	2	2	no	—
46	C₄×C₅×D₅	—	2	4	no	—
47	C₅×Dic₁₀	—	2	2	no	—
48	C₂×C₅×#41	—	2	2	no	—
49	C₂×C₅×#41	—	5	1	no	—
50	C₂×C₅×#41	D₅	1	1	yes	characteristic, complemented
51	C₂×C₅×#41	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂×#709	C₂×C₅	1	1	yes	characteristic, complemented
53	C₅×C₂²⋊Q₈	—	5	2	no	—
54	C₅×C₂²⋊Q₈	—	5	2	no	—
55	C₅×#121	—	5	2	no	—
56	C₅×#121	—	5	4	no	—
57	C₅×#121	—	5	2	no	—
58	C₂×C₅×C₄⋊C₄	—	5	1	no	—
59	#4092	C₂×C₅	1	2	yes	complemented
60	#4085	C₂×C₅	1	4	yes	complemented
61	#4084	C₂×C₅	1	2	yes	complemented
62	#4076	C₂×C₅	1	2	yes	complemented
63	C₂×C₄⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
64	#4070	C₂×C₅	1	2	yes	complemented
65	C₂²×C₅²	—	2	1	no	—
66	C₂²×C₅²	—	2	1	no	—
67	C₂²×C₅²	—	2	2	no	—
68	C₂²×C₅²	—	2	2	no	—
69	C₂²×C₅²	—	2	2	no	—
70	C₂²×C₅²	C₂×D₄	1	2	yes	—
71	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
72	C₂×C₅×D₅	—	2	2	no	—
73	C₂×C₅×D₅	—	2	2	no	—
74	C₂×C₅×D₅	—	2	2	no	—
75	C₄×C₅²	—	2	2	no	—
76	C₄×C₅²	C₂×D₄	1	2	yes	—
77	C₄×C₅²	C₂×D₄	1	2	yes	—
78	C₅×Dic₅	—	2	2	no	—
79	C₅×Dic₅	—	2	4	no	—
80	C₅×#41	—	2	16	no	—
81	C₅×#41	—	2	4	no	—
82	C₅×#41	—	2	4	no	—
83	C₅×#41	—	10	4	no	—
84	C₂×C₅×Q₈	—	2	2	no	—
85	C₂×C₅×Q₈	—	5	1	no	—
86	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
87	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
88	C₂×C₅×D₄	—	2	2	no	—
89	C₂×C₅×D₄	—	2	4	no	—
90	C₂×C₅×D₄	—	5	2	no	—
91	C₂×C₅×D₄	—	5	4	no	—
92	C₂×C₅×D₄	—	5	1	no	—
93	C₂×C₅×D₄	C₂×D₅	1	2	yes	—
94	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic
95	C₂×C₅×D₄	C₂²×C₅	1	1	yes	characteristic
96	C₂×C₅×D₄	C₂²×C₅	1	2	yes	—
97	C₂²×C₄×C₅	—	2	4	no	—
98	C₂²×C₄×C₅	—	2	2	no	—
99	C₂²×C₄×C₅	—	5	2	no	—
100	C₂²×C₄×C₅	—	5	2	no	—
101	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
102	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
104	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
105	C₂×C₂²⋊D₅	C₂²×C₅	1	2	yes	—
106	C₂×C₂²⋊D₅	C₂²×C₅	1	4	yes	—
107	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
108	#709	—	2	4	no	—
109	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic
110	C₂×C₄×D₅	C₂²×C₅	1	2	yes	—
111	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic
112	C₅×C₄⋊C₄	—	5	2	no	—
113	C₅×C₄⋊C₄	—	5	2	no	—
114	C₅×C₄⋊C₄	—	5	4	no	—
115	C₅×C₂²⋊C₄	—	5	4	no	—
116	C₅×C₂²⋊C₄	—	5	4	no	—
117	C₂²⋊Dic₅	C₂²×C₅	1	4	yes	—
118	#685	C₂²×C₅	1	4	yes	—
119	C₄⋊Dic₅	C₂²×C₅	1	2	yes	—
120	C₄⋊Dic₅	C₂²×C₅	1	2	yes	—
121	#683	C₂²×C₅	1	4	yes	—
122	#537	—	5	1	no	Sylow
123	C₂×C₅²	—	2	2	no	—
124	C₂×C₅²	—	4	2	no	—
125	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
128	C₅×D₅	—	4	2	no	—
129	C₂³×C₅	—	2	4	no	—
130	C₂³×C₅	—	2	2	no	—
131	C₂³×C₅	—	5	2	no	—
132	C₂³×C₅	C₂²×D₅	1	2	yes	—
133	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
134	C₂³×C₅	C₂³×C₅	1	2	yes	—
135	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
136	C₂²×D₅	C₂³×C₅	1	2	yes	—
137	C₅×Q₈	—	2	8	no	—
138	C₅×Q₈	—	2	2	no	—
139	C₅×Q₈	—	2	2	no	—
140	C₅×Q₈	—	10	2	no	—
141	C₅×D₄	—	2	16	no	—
142	C₅×D₄	—	2	8	no	—
143	C₅×D₄	—	2	4	no	—
144	C₅×D₄	—	2	4	no	—
145	C₅×D₄	—	2	2	no	—
146	C₅×D₄	—	2	2	no	—
147	C₅×D₄	—	10	4	no	—
148	C₅×D₄	—	10	2	no	—
149	C₅×D₄	—	20	4	no	—
150	C₂×C₄×C₅	—	2	16	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	—	2	4	no	—
153	C₂×C₄×C₅	—	2	8	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	4	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	5	4	no	—
158	C₂×C₄×C₅	—	5	2	no	—
159	C₂×C₄×C₅	—	10	4	no	—
160	C₂×C₄×C₅	—	20	2	no	—
161	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	complemented
162	C₂×C₄×C₅	C₅×D₄	1	4	yes	complemented
163	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
164	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
167	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
169	C₂²⋊D₅	—	2	4	no	—
170	C₂²⋊D₅	—	4	4	no	—
171	C₂×Dic₅	—	4	2	no	—
172	C₂×Dic₅	C₂³×C₅	1	2	yes	—
173	C₂×Dic₅	C₂³×C₅	1	4	yes	—
174	D₂₀	—	2	2	no	—
175	C₄×D₅	—	2	4	no	—
176	Dic₁₀	—	2	2	no	—
177	C₂×#41	—	5	1	no	—
178	C₂×#41	C₅×D₅	1	1	yes	characteristic, complemented
179	C₂²⋊Q₈	—	5	2	no	—
180	C₂²⋊Q₈	—	5	2	no	—
181	#121	—	5	2	no	—
182	#121	—	5	4	no	—
183	#121	—	5	2	no	—
184	C₂×C₄⋊C₄	—	5	1	no	—
185	C₅²	#537	1	1	yes	characteristic, complemented, Sylow
186	C₂²×C₅	—	2	1	no	—
187	C₂²×C₅	—	2	1	no	—
188	C₂²×C₅	—	2	1	no	—
189	C₂²×C₅	—	2	1	no	—
190	C₂²×C₅	—	2	8	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	2	4	no	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	4	2	no	—
200	C₂²×C₅	—	4	2	no	—
201	C₂²×C₅	—	4	4	no	—
202	C₂²×C₅	—	4	4	no	—
203	C₂²×C₅	—	10	2	no	—
204	C₂²×C₅	—	10	2	no	—
205	C₂²×C₅	—	10	2	no	—
206	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
207	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
208	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
209	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
210	C₂×D₅	—	2	2	no	—
211	C₂×D₅	—	2	2	no	—
212	C₂×D₅	—	2	2	no	—
213	C₄×C₅	—	2	8	no	—
214	C₄×C₅	—	2	2	no	—
215	C₄×C₅	—	2	2	no	—
216	C₄×C₅	—	2	4	no	—
217	C₄×C₅	—	2	4	no	—
218	C₄×C₅	—	10	4	no	—
219	C₄×C₅	—	10	2	no	—
220	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
221	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
222	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
223	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
224	Dic₅	—	2	2	no	—
225	Dic₅	—	2	4	no	—
226	#41	—	2	4	no	—
227	#41	—	10	4	no	—
228	C₂×Q₈	—	5	1	no	—
229	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
230	C₂×D₄	—	5	2	no	—
231	C₂×D₄	—	5	4	no	—
232	C₂×D₄	—	5	1	no	—
233	C₂×D₄	C₂×C₅×D₅	1	2	yes	—
234	C₂×D₄	C₂×C₅×D₅	1	1	yes	characteristic
235	C₂²×C₄	—	5	2	no	—
236	C₂²×C₄	—	5	2	no	—
237	C₂²×C₄	C₂×C₅×D₅	1	2	yes	—
238	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
239	C₄⋊C₄	—	5	2	no	—
240	C₄⋊C₄	—	5	2	no	—
241	C₄⋊C₄	—	5	4	no	—
242	C₂²⋊C₄	—	5	4	no	—
243	C₂²⋊C₄	—	5	4	no	—
244	C₂×C₅	—	2	2	no	—
245	C₂×C₅	—	2	2	no	—
246	C₂×C₅	—	2	2	no	—
247	C₂×C₅	—	2	2	no	—
248	C₂×C₅	—	2	2	no	—
249	C₂×C₅	—	4	4	no	—
250	C₂×C₅	—	4	8	no	—
251	C₂×C₅	—	4	2	no	—
252	C₂×C₅	—	4	2	no	—
253	C₂×C₅	—	20	2	no	—
254	C₂×C₅	#4149	1	1	yes	characteristic
255	C₂×C₅	#4150	1	1	yes	characteristic
256	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
257	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
258	C₂×C₅	C₅×H(5,𝔽₂)	1	1	yes	characteristic
259	C₂×C₅	C₅×2₋¹⁺⁴	1	1	yes	characteristic
260	D₅	—	4	2	no	—
261	C₂³	—	5	2	no	—
262	C₂³	C₂²×C₅×D₅	1	2	yes	—
263	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
264	Q₈	—	2	2	no	—
265	Q₈	—	10	2	no	—
266	D₄	—	2	4	no	—
267	D₄	—	2	2	no	—
268	D₄	—	10	4	no	—
269	D₄	—	10	2	no	—
270	D₄	—	20	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	5	4	no	—
273	C₂×C₄	—	5	2	no	—
274	C₂×C₄	—	10	4	no	—
275	C₂×C₄	—	20	2	no	—
276	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	complemented
277	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
278	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
280	C₅	—	2	2	no	—
281	C₅	#66167	1	1	yes	characteristic, direct product factor
282	C₅	C₅×#537	1	1	yes	characteristic, complemented
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	10	2	no	—
289	C₂²	—	10	2	no	—
290	C₂²	—	10	2	no	—
291	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
292	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	10	4	no	—
295	C₄	—	10	2	no	—
296	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
297	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	4	2	no	—
300	C₂	—	20	2	no	—
301	C₂	C₅×#4149	1	1	yes	characteristic
302	C₂	C₅×#4150	1	1	yes	characteristic
303	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	C₂×C₅²×#41	C₂
socle	C₂²×C₅²	C₂⁴
hypercenter	C₂×C₅×#41	D₅
hypocenter	C₅	C₅×#537
LCS	G ▷ C₂²×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
1. C₂×C₅²×#41	C₂	semidirect
2. C₂×C₅×#709	C₂	semidirect
3. C₅×#4092	C₂	semidirect
4. C₅×#4085	C₂	semidirect
5. C₅×#4084	C₂	semidirect
6. C₅×#4076	C₂	semidirect
7. C₂×C₅×C₄⋊Dic₅	C₂	semidirect
8. C₅×#4070	C₂	semidirect
28. #66167	C₅	direct (N × H)
36. C₂×C₄×C₅²	D₄	semidirect
50. C₂×C₅×#41	D₅	semidirect
51. C₂×C₅×#41	C₂×C₅	semidirect
52. C₂×#709	C₂×C₅	semidirect
59. #4092	C₂×C₅	semidirect
60. #4085	C₂×C₅	semidirect
61. #4084	C₂×C₅	semidirect
62. #4076	C₂×C₅	semidirect
63. C₂×C₄⋊Dic₅	C₂×C₅	semidirect
64. #4070	C₂×C₅	semidirect
161. C₂×C₄×C₅	C₂²⋊D₅	semidirect
162. C₂×C₄×C₅	C₅×D₄	semidirect
178. C₂×#41	C₅×D₅	semidirect
185. C₅²	#537	semidirect
276. C₂×C₄	C₅×C₂²⋊D₅	semidirect
281. C₅	#66167	direct (N × H)
282. C₅	C₅×#537	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
C₂×C₅×#709	2
C₅×#4092	2
C₅×#4085	2
C₅×#4084	2
C₅×#4076	2
C₂×C₅×C₄⋊Dic₅	2
C₅×#4070	2
C₅×#537	5
#66167	5

Subgroups — C5×#66167 #13106986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66167 #13106986