Subgroups — C₅×#21021 #11943741

Order 1280.

886 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#424	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic
3	C₅×#2296	C₂	1	1	yes	maximal, characteristic
4	C₅×#1917	C₂	1	8	yes	maximal, complemented
5	C₅×#1729	C₂	1	1	yes	maximal, characteristic
6	C₅×#1729	C₂	1	1	yes	maximal, characteristic
7	C₅×#1729	C₂	1	1	yes	maximal, characteristic
8	C₅×#1728	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×#117	C₂²	1	1	yes	characteristic
14	C₂×C₅×#117	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₅×#424	—	2	4	no	—
17	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
18	C₂×C₅×#97	—	2	4	no	—
19	C₂×C₅×#97	C₂²	1	4	yes	—
20	C₅×#389	C₄	1	4	yes	—
21	C₅×#385	C₄	1	4	yes	—
22	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
23	C₅×#336	C₂²	1	4	yes	—
24	C₅×#336	C₂²	1	4	yes	—
25	C₅×#336	C₂²	1	4	yes	—
26	C₅×#336	C₂²	1	4	yes	—
27	#21021	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₅×M₁₆	—	2	2	no	—
31	C₂×C₅×M₁₆	—	2	2	no	—
32	C₂×C₅×M₁₆	—	2	2	no	—
33	C₂×C₅×M₁₆	—	2	2	no	—
34	C₂×C₅×M₁₆	—	2	2	no	—
35	C₂×C₅×M₁₆	—	2	2	no	—
36	C₂²×C₅×C₈	—	2	2	no	—
37	C₂²×C₅×C₈	—	2	2	no	—
38	C₂²×C₅×C₈	—	2	2	no	—
39	C₂²×C₅×C₈	—	2	2	no	—
40	C₂²×C₅×C₈	D₄	1	4	yes	—
41	C₂²×C₅×C₈	Q₈	1	4	yes	—
42	C₂²×C₅×C₈	C₂³	1	2	yes	—
43	C₂²×C₅×C₈	C₂³	1	2	yes	—
44	C₂²×C₅×C₈	C₂³	1	2	yes	—
45	C₂²×C₅×C₈	C₂³	1	2	yes	—
46	C₅×#117	—	2	2	no	—
47	C₅×#117	—	2	2	no	—
48	C₅×#117	—	2	2	no	—
49	C₅×#117	—	2	2	no	—
50	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
51	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
53	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
54	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₅	—	2	2	no	—
56	C₂×C₄²×C₅	—	2	2	no	—
57	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₅	C₂³	1	2	yes	—
59	C₂×C₄²×C₅	C₂³	1	2	yes	—
60	C₂×C₄²×C₅	C₂³	1	2	yes	—
61	C₅×C₄⋊C₈	—	2	8	no	—
62	C₅×#97	—	2	8	no	—
63	C₅×#95	C₂×C₄	1	8	yes	—
64	C₂×#424	C₂×C₅	1	1	yes	characteristic
65	C₂²×#97	C₂×C₅	1	1	yes	characteristic
66	#2296	C₂×C₅	1	1	yes	characteristic
67	#1917	C₂×C₅	1	8	yes	complemented
68	#1729	C₂×C₅	1	1	yes	characteristic
69	#1729	C₂×C₅	1	1	yes	characteristic
70	#1729	C₂×C₅	1	1	yes	characteristic
71	#1728	C₂×C₅	1	1	yes	characteristic
72	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	—	2	4	no	—
74	C₂²×C₄×C₅	—	4	4	no	—
75	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₅	#41	1	2	yes	—
82	C₂²×C₄×C₅	#41	1	2	yes	—
83	C₂²×C₄×C₅	#41	1	2	yes	—
84	C₂²×C₄×C₅	#41	1	2	yes	—
85	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
86	C₅×M₁₆	—	4	4	no	—
87	C₅×M₁₆	—	4	4	no	—
88	C₂×C₅×C₈	—	2	8	no	—
89	C₂×C₅×C₈	—	2	8	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
101	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
102	C₅×C₄⋊C₄	—	2	4	no	—
103	C₅×C₄⋊C₄	—	2	4	no	—
104	C₅×C₄⋊C₄	—	4	8	no	—
105	C₅×C₂²⋊C₄	—	2	4	no	—
106	C₅×C₂²⋊C₄	—	2	4	no	—
107	C₄²×C₅	—	2	4	no	—
108	C₄²×C₅	—	2	2	no	—
109	C₄²×C₅	—	2	2	no	—
110	C₄²×C₅	—	2	2	no	—
111	C₄²×C₅	—	2	2	no	—
112	C₄²×C₅	—	2	2	no	—
113	C₄²×C₅	—	2	2	no	—
114	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
115	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
116	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
117	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
118	C₂×#117	C₂²×C₅	1	1	yes	characteristic
119	C₂×#117	C₂²×C₅	1	1	yes	characteristic
120	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
121	#424	—	2	4	no	—
122	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
123	C₂×#97	—	2	4	no	—
124	C₂×#97	C₂²×C₅	1	4	yes	—
125	#389	C₄×C₅	1	4	yes	—
126	#385	C₄×C₅	1	4	yes	—
127	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
128	#336	C₂²×C₅	1	4	yes	—
129	#336	C₂²×C₅	1	4	yes	—
130	#336	C₂²×C₅	1	4	yes	—
131	#336	C₂²×C₅	1	4	yes	—
132	C₂³×C₅	—	2	4	no	—
133	C₂³×C₅	#117	1	2	yes	—
134	C₂³×C₅	C₄×D₄	1	2	yes	—
135	C₂³×C₅	C₄×Q₈	1	2	yes	—
136	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	—	2	4	no	—
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	—	2	8	no	—
142	C₂×C₄×C₅	—	2	8	no	—
143	C₂×C₄×C₅	—	2	4	no	—
144	C₂×C₄×C₅	—	4	8	no	—
145	C₂×C₄×C₅	#117	1	2	yes	—
146	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
147	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
148	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
149	C₂×C₄×C₅	#123	1	4	yes	—
150	C₂×C₄×C₅	#125	1	4	yes	—
151	C₂×C₄×C₅	#126	1	4	yes	—
152	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
158	C₅×C₈	—	2	4	no	—
159	C₅×C₈	—	2	4	no	—
160	C₅×C₈	—	4	2	no	—
161	C₅×C₈	—	4	2	no	—
162	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
163	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	C₅×D₄	1	4	yes	—
175	C₂²×C₈	C₅×Q₈	1	4	yes	—
176	C₂²×C₈	C₂³×C₅	1	2	yes	—
177	C₂²×C₈	C₂³×C₅	1	2	yes	—
178	C₂²×C₈	C₂³×C₅	1	2	yes	—
179	C₂²×C₈	C₂³×C₅	1	2	yes	—
180	#117	—	2	2	no	—
181	#117	—	2	2	no	—
182	#117	—	2	2	no	—
183	#117	—	2	2	no	—
184	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
185	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
186	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
187	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
188	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
192	C₂×C₄²	C₂³×C₅	1	2	yes	—
193	C₂×C₄²	C₂³×C₅	1	2	yes	—
194	C₂×C₄²	C₂³×C₅	1	2	yes	—
195	C₄⋊C₈	—	2	8	no	—
196	#97	—	2	8	no	—
197	#95	C₂×C₄×C₅	1	8	yes	—
198	C₂²×C₅	—	2	4	no	—
199	C₂²×C₅	—	2	4	no	—
200	C₂²×C₅	—	2	4	no	—
201	C₂²×C₅	#387	1	4	yes	—
202	C₂²×C₅	C₂×#117	1	1	yes	characteristic
203	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
204	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
205	C₂²×C₅	#568	1	4	yes	—
206	C₄×C₅	—	4	4	no	—
207	C₄×C₅	—	4	4	no	—
208	C₄×C₅	—	4	4	no	—
209	C₄×C₅	#387	1	4	yes	—
210	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
211	C₄×C₅	C₂×#123	1	1	yes	characteristic
212	C₄×C₅	C₂×#125	1	1	yes	characteristic
213	C₄×C₅	C₂×#126	1	1	yes	characteristic
214	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
218	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
219	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
220	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
222	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
223	C₂²×C₄	C₅×#41	1	2	yes	—
224	C₂²×C₄	C₅×#41	1	2	yes	—
225	C₂²×C₄	C₅×#41	1	2	yes	—
226	C₂²×C₄	C₅×#41	1	2	yes	—
227	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
228	M₁₆	—	4	4	no	—
229	M₁₆	—	4	4	no	—
230	C₂×C₈	—	2	8	no	—
231	C₂×C₈	—	2	8	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
243	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	8	no	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₄²	—	2	4	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₂×C₅	—	2	4	no	—
257	C₂×C₅	C₂×#387	1	1	yes	characteristic
258	C₂×C₅	#2923	1	2	yes	—
259	C₂×C₅	#2930	1	2	yes	—
260	C₂×C₅	#2961	1	2	yes	—
261	C₂³	—	2	4	no	—
262	C₂³	C₅×#117	1	2	yes	—
263	C₂³	C₄×C₅×D₄	1	2	yes	—
264	C₂³	C₄×C₅×Q₈	1	2	yes	—
265	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	8	no	—
274	C₂×C₄	C₅×#117	1	2	yes	—
275	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
276	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
277	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
278	C₂×C₄	C₅×#123	1	4	yes	—
279	C₂×C₄	C₅×#125	1	4	yes	—
280	C₂×C₄	C₅×#126	1	4	yes	—
281	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
283	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
284	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
285	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
286	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
287	C₈	—	2	4	no	—
288	C₈	—	2	4	no	—
289	C₈	—	4	2	no	—
290	C₈	—	4	2	no	—
291	C₅	#21021	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²	—	2	4	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	C₅×#387	1	4	yes	—
296	C₂²	C₂×C₅×#117	1	1	yes	characteristic
297	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
298	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
299	C₂²	C₅×#568	1	4	yes	—
300	C₄	—	4	4	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	4	no	—
303	C₄	C₅×#387	1	4	yes	—
304	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
305	C₄	C₂×C₅×#123	1	1	yes	characteristic
306	C₄	C₂×C₅×#125	1	1	yes	characteristic
307	C₄	C₂×C₅×#126	1	1	yes	characteristic
308	C₂	—	2	4	no	—
309	C₂	C₂×C₅×#387	1	1	yes	characteristic
310	C₂	C₅×#2923	1	2	yes	—
311	C₂	C₅×#2930	1	2	yes	—
312	C₂	C₅×#2961	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	C₂⁴×C₅	C₂²×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
4. C₅×#1917	C₂	semidirect
27. #21021	C₅	direct (N × H)
67. #1917	C₂×C₅	semidirect
291. C₅	#21021	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#424	2
C₂²×C₅×#97	2
C₅×#2296	2
C₅×#1917	2
C₅×#1729	2
C₅×#1728	2
#21021	5

Subgroups — C5×#21021 #11943741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21021 #11943741