Subgroups — C₅×#22015 #11944735

Order 1280.

702 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2293	C₂	1	1	yes	maximal, characteristic
2	C₅×#1955	C₂	1	2	yes	maximal
3	C₅×#1940	C₂	1	2	yes	maximal, complemented
4	C₅×#1924	C₂	1	4	yes	maximal, complemented
5	C₅×#1818	C₂	1	2	yes	maximal, complemented
6	C₅×#1798	C₂	1	2	yes	maximal, complemented
7	C₅×#1767	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1745	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
10	C₂×C₅×#117	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
13	C₂×C₅×C₂²⋊C₈	C₂²	1	4	yes	—
14	C₂×C₅×C₂²⋊C₈	C₂²	1	4	yes	—
15	C₂×C₅×#97	C₂²	1	2	yes	—
16	C₂×C₅×#97	C₂²	1	2	yes	—
17	C₅×#390	C₄	1	2	yes	—
18	C₅×#389	C₄	1	2	yes	—
19	C₅×#388	C₄	1	4	yes	—
20	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₄³×C₅	C₂²	1	1	yes	characteristic
23	C₅×#336	C₂²	1	2	yes	—
24	C₅×#336	C₂²	1	1	yes	characteristic
25	C₅×#336	C₂²	1	1	yes	characteristic
26	C₅×#336	C₂²	1	4	yes	—
27	C₅×#336	C₂²	1	2	yes	—
28	#22015	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂²×C₅×D₄	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	C₂³	1	2	yes	—
31	C₂²×C₅×C₈	C₂³	1	2	yes	—
32	C₂²×C₅×C₈	C₂³	1	2	yes	—
33	C₂²×C₅×C₈	C₂³	1	4	yes	—
34	C₄×C₅×D₄	—	2	4	no	—
35	C₅×#117	—	4	4	no	—
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
39	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
41	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂³	1	2	yes	—
47	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₅×C₄⋊C₈	—	2	4	no	—
49	C₅×C₄⋊C₈	—	4	4	no	—
50	C₅×C₂²⋊C₈	—	4	4	no	—
51	C₅×C₂²⋊C₈	—	4	4	no	—
52	C₅×#97	—	2	4	no	—
53	C₅×#97	—	4	2	no	—
54	C₅×#95	C₂×C₄	1	4	yes	—
55	C₅×#95	C₂×C₄	1	2	yes	—
56	#2293	C₂×C₅	1	1	yes	characteristic
57	#1955	C₂×C₅	1	2	yes	—
58	#1940	C₂×C₅	1	2	yes	complemented
59	#1924	C₂×C₅	1	4	yes	complemented
60	#1818	C₂×C₅	1	2	yes	complemented
61	#1798	C₂×C₅	1	2	yes	complemented
62	#1767	C₂×C₅	1	1	yes	characteristic, complemented
63	#1745	C₂×C₅	1	1	yes	characteristic, complemented
64	C₂⁴×C₅	C₂²×C₄	1	2	yes	—
65	C₂×C₅×D₄	—	2	2	no	—
66	C₂×C₅×D₄	—	4	2	no	—
67	C₂²×C₄×C₅	—	2	4	no	—
68	C₂²×C₄×C₅	—	2	4	no	—
69	C₂²×C₄×C₅	—	2	2	no	—
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	2	4	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	—	2	4	no	—
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₅×C₄⋊C₄	—	2	2	no	—
88	C₅×C₄⋊C₄	—	4	2	no	—
89	C₅×C₄⋊C₄	—	4	4	no	—
90	C₅×C₂²⋊C₄	—	2	4	no	—
91	C₅×C₂²⋊C₄	—	4	4	no	—
92	C₅×C₂²⋊C₄	—	4	4	no	—
93	C₄²×C₅	—	2	2	no	—
94	C₄²×C₅	—	2	4	no	—
95	C₄²×C₅	—	2	1	no	—
96	C₄²×C₅	—	2	1	no	—
97	C₄²×C₅	—	4	2	no	—
98	C₄²×C₅	—	4	2	no	—
99	C₄²×C₅	#41	1	4	yes	—
100	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
101	C₂×#117	C₂²×C₅	1	2	yes	—
102	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
103	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
104	C₂×C₂²⋊C₈	C₂²×C₅	1	4	yes	—
105	C₂×C₂²⋊C₈	C₂²×C₅	1	4	yes	—
106	C₂×#97	C₂²×C₅	1	2	yes	—
107	C₂×#97	C₂²×C₅	1	2	yes	—
108	#390	C₄×C₅	1	2	yes	—
109	#389	C₄×C₅	1	2	yes	—
110	#388	C₄×C₅	1	4	yes	—
111	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
112	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
113	C₄³	C₂²×C₅	1	1	yes	characteristic
114	#336	C₂²×C₅	1	1	yes	characteristic
115	#336	C₂²×C₅	1	2	yes	—
116	#336	C₂²×C₅	1	1	yes	characteristic
117	#336	C₂²×C₅	1	4	yes	—
118	#336	C₂²×C₅	1	2	yes	—
119	C₂³×C₅	—	2	2	no	—
120	C₂³×C₅	—	2	2	no	—
121	C₂³×C₅	—	2	2	no	—
122	C₂³×C₅	—	2	4	no	—
123	C₂³×C₅	—	2	4	no	—
124	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
125	C₅×D₄	—	4	4	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	—	4	4	no	—
141	C₂×C₄×C₅	—	4	4	no	—
142	C₂×C₄×C₅	#117	1	2	yes	—
143	C₂×C₄×C₅	#117	1	2	yes	—
144	C₂×C₄×C₅	C₂×#41	1	2	yes	—
145	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
147	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
148	C₅×C₈	—	4	2	no	—
149	C₅×C₈	—	4	4	no	—
150	C₅×C₈	—	4	2	no	—
151	C₂²×D₄	C₂×C₄×C₅	1	1	yes	characteristic
152	C₂³×C₄	C₂³×C₅	1	2	yes	—
153	C₂²×C₈	C₂³×C₅	1	2	yes	—
154	C₂²×C₈	C₂³×C₅	1	2	yes	—
155	C₂²×C₈	C₂³×C₅	1	4	yes	—
156	C₄×D₄	—	2	4	no	—
157	#117	—	4	4	no	—
158	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
159	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
160	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
161	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
162	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
163	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
164	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
165	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
166	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
167	C₂×C₄²	C₂³×C₅	1	2	yes	—
168	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	—	4	4	no	—
172	C₂²⋊C₈	—	4	4	no	—
173	C₂²⋊C₈	—	4	4	no	—
174	#97	—	2	4	no	—
175	#97	—	4	2	no	—
176	#95	C₂×C₄×C₅	1	4	yes	—
177	#95	C₂×C₄×C₅	1	2	yes	—
178	C₂²×C₅	—	4	2	no	—
179	C₂²×C₅	—	4	2	no	—
180	C₂²×C₅	—	4	2	no	—
181	C₂²×C₅	—	4	4	no	—
182	C₂²×C₅	—	4	4	no	—
183	C₂²×C₅	C₂×#117	1	1	yes	characteristic
184	C₂²×C₅	#518	1	1	yes	characteristic
185	C₂²×C₅	#520	1	1	yes	characteristic
186	C₂²×C₅	#568	1	2	yes	—
187	C₂²×C₅	#568	1	2	yes	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	—	4	2	no	—
191	C₄×C₅	—	4	2	no	—
192	C₄×C₅	—	4	4	no	—
193	C₄×C₅	—	4	2	no	—
194	C₄×C₅	#540	1	2	yes	—
195	C₄×C₅	#542	1	2	yes	—
196	C₂⁴	C₂²×C₄×C₅	1	2	yes	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
205	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
206	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
207	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
209	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₂²⋊C₄	—	4	4	no	—
224	C₂²⋊C₄	—	4	4	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	4	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	4	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	C₅×#41	1	4	yes	—
232	C₂×C₅	—	8	2	no	—
233	C₂×C₅	#2319	1	1	yes	characteristic
234	C₂×C₅	#2921	1	1	yes	characteristic
235	C₂×C₅	#2923	1	1	yes	characteristic
236	C₂×C₅	#2980	1	2	yes	—
237	C₂×C₅	#2995	1	2	yes	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	4	no	—
242	C₂³	—	2	4	no	—
243	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
244	D₄	—	4	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	C₅×#117	1	2	yes	—
262	C₂×C₄	C₅×#117	1	2	yes	—
263	C₂×C₄	C₂×C₅×#41	1	2	yes	—
264	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
265	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
266	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
267	C₈	—	4	2	no	—
268	C₈	—	4	4	no	—
269	C₈	—	4	2	no	—
270	C₅	#22015	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²	—	4	4	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	C₂×C₅×#117	1	1	yes	characteristic
277	C₂²	C₅×#518	1	1	yes	characteristic
278	C₂²	C₅×#520	1	1	yes	characteristic
279	C₂²	C₅×#568	1	2	yes	—
280	C₂²	C₅×#568	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	2	no	—
285	C₄	—	4	2	no	—
286	C₄	—	4	4	no	—
287	C₄	C₅×#540	1	2	yes	—
288	C₄	C₅×#542	1	2	yes	—
289	C₂	—	8	2	no	—
290	C₂	C₅×#2319	1	1	yes	characteristic
291	C₂	C₅×#2921	1	1	yes	characteristic
292	C₂	C₅×#2923	1	1	yes	characteristic
293	C₂	C₅×#2980	1	2	yes	—
294	C₂	C₅×#2995	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂²×C₅×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1940	C₂	semidirect
4. C₅×#1924	C₂	semidirect
5. C₅×#1818	C₂	semidirect
6. C₅×#1798	C₂	semidirect
7. C₅×#1767	C₂	semidirect
8. C₅×#1745	C₂	semidirect
28. #22015	C₅	direct (N × H)
58. #1940	C₂×C₅	semidirect
59. #1924	C₂×C₅	semidirect
60. #1818	C₂×C₅	semidirect
61. #1798	C₂×C₅	semidirect
62. #1767	C₂×C₅	semidirect
63. #1745	C₂×C₅	semidirect
270. C₅	#22015	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#2293	2
C₅×#1955	2
C₅×#1940	2
C₅×#1924	2
C₅×#1818	2
C₅×#1798	2
C₅×#1767	2
C₅×#1745	2
#22015	5

Subgroups — C5×#22015 #11944735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22015 #11944735