Subgroups — C₅×#10839 #11933559

Order 1280.

1110 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×#423	C₂	1	1	yes	maximal, characteristic
2	C₅×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#339	C₂	1	4	yes	maximal
5	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
6	C₂×C₅×#117	C₂²	1	2	yes	—
7	C₂²×C₄²×C₅	—	2	2	no	—
8	C₂²×C₄²×C₅	C₂²	1	2	yes	—
9	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₅×#423	—	2	2	no	—
11	C₅×#423	C₄	1	4	yes	complemented
12	C₂×C₅×C₄⋊C₈	—	2	2	no	—
13	C₅×#384	C₄	1	2	yes	complemented
14	C₅×#382	C₄	1	2	yes	complemented
15	C₅×#380	—	2	2	no	—
16	C₄×C₅×C₄⋊C₄	—	2	2	no	—
17	C₄³×C₅	—	2	2	no	—
18	C₄³×C₅	C₄	1	4	yes	complemented
19	C₅×#339	—	2	8	no	—
20	#10839	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₅	—	2	2	no	—
22	C₂³×C₄×C₅	D₄	1	2	yes	—
23	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	—	2	4	no	—
26	C₂×C₅×M₁₆	—	2	4	no	—
27	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₅×#117	—	2	4	no	—
30	C₅×#117	—	2	4	no	—
31	C₂×C₅×C₄⋊C₄	—	2	4	no	—
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
33	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
34	C₂×C₄²×C₅	—	2	16	no	—
35	C₂×C₄²×C₅	—	2	8	no	—
36	C₂×C₄²×C₅	—	2	2	no	—
37	C₂×C₄²×C₅	—	2	2	no	—
38	C₂×C₄²×C₅	—	2	2	no	—
39	C₂×C₄²×C₅	—	2	2	no	—
40	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	D₄	1	4	yes	—
46	C₂×C₄²×C₅	D₄	1	2	yes	—
47	C₂×C₄²×C₅	D₄	1	2	yes	—
48	C₂×C₄²×C₅	Q₈	1	4	yes	—
49	C₂×C₄²×C₅	Q₈	1	2	yes	—
50	C₅×C₄⋊C₈	—	2	8	no	—
51	C₅×#95	—	2	4	no	—
52	C₂×#423	C₂×C₅	1	1	yes	characteristic
53	#2285	C₂×C₅	1	1	yes	characteristic
54	C₂×C₄³	C₂×C₅	1	1	yes	characteristic
55	C₂×#339	C₂×C₅	1	4	yes	—
56	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	—	2	16	no	—
58	C₂²×C₄×C₅	—	2	4	no	—
59	C₂²×C₄×C₅	—	2	4	no	—
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	—	2	8	no	—
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
64	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₅	#41	1	2	yes	—
76	C₂²×C₄×C₅	#41	1	2	yes	—
77	C₅×M₁₆	—	2	8	no	—
78	C₅×M₁₆	—	4	4	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	4	2	no	—
81	C₂×C₅×C₈	—	4	2	no	—
82	C₅×C₄⋊C₄	—	2	8	no	—
83	C₅×C₄⋊C₄	—	4	8	no	—
84	C₅×C₂²⋊C₄	—	4	4	no	—
85	C₄²×C₅	—	2	32	no	—
86	C₄²×C₅	—	2	4	no	—
87	C₄²×C₅	—	2	4	no	—
88	C₄²×C₅	—	2	4	no	—
89	C₄²×C₅	—	4	2	no	—
90	C₄²×C₅	—	4	2	no	—
91	C₄²×C₅	C₄²	1	2	yes	—
92	C₄²×C₅	C₄²	1	2	yes	—
93	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
94	C₄²×C₅	C₄⋊C₄	1	2	yes	—
95	C₄²×C₅	C₄⋊C₄	1	4	yes	—
96	C₄²×C₅	C₄⋊C₄	1	2	yes	—
97	C₄²×C₅	C₄⋊C₄	1	8	yes	complemented
98	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
99	C₂×#117	C₂²×C₅	1	2	yes	—
100	C₂²×C₄²	—	2	2	no	—
101	C₂²×C₄²	C₂²×C₅	1	2	yes	—
102	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
103	#423	—	2	2	no	—
104	#423	C₄×C₅	1	4	yes	complemented
105	C₂×C₄⋊C₈	—	2	2	no	—
106	#384	C₄×C₅	1	2	yes	complemented
107	#382	C₄×C₅	1	2	yes	complemented
108	#380	—	2	2	no	—
109	C₄×C₄⋊C₄	—	2	2	no	—
110	C₄³	—	2	2	no	—
111	C₄³	C₄×C₅	1	4	yes	complemented
112	#339	—	2	8	no	—
113	C₂³×C₅	—	2	4	no	—
114	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂³×C₅	#121	1	2	yes	—
118	C₂³×C₅	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₅	—	2	16	no	—
120	C₂×C₄×C₅	—	2	16	no	—
121	C₂×C₄×C₅	—	2	16	no	—
122	C₂×C₄×C₅	—	2	8	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	8	no	—
125	C₂×C₄×C₅	—	2	8	no	—
126	C₂×C₄×C₅	—	4	4	no	—
127	C₂×C₄×C₅	—	4	8	no	—
128	C₂×C₄×C₅	#95	1	4	yes	—
129	C₂×C₄×C₅	#95	1	4	yes	—
130	C₂×C₄×C₅	#104	1	8	yes	—
131	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	#117	1	2	yes	—
137	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
140	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
141	C₂×C₄×C₅	#121	1	2	yes	—
142	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₅	#124	1	1	yes	characteristic
144	C₂×C₄×C₅	#124	1	1	yes	characteristic
145	C₂×C₄×C₅	#125	1	2	yes	—
146	C₂×C₄×C₅	#127	1	1	yes	characteristic
147	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
148	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
149	C₅×C₈	—	4	4	no	—
150	C₂³×C₄	—	2	2	no	—
151	C₂³×C₄	C₅×D₄	1	2	yes	—
152	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	4	no	—
155	C₂×M₁₆	—	2	4	no	—
156	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
157	C₂²×C₈	—	2	2	no	—
158	#117	—	2	4	no	—
159	#117	—	2	4	no	—
160	C₂×C₄⋊C₄	—	2	4	no	—
161	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
162	C₂×C₂²⋊C₄	—	2	2	no	—
163	C₂×C₄²	—	2	16	no	—
164	C₂×C₄²	—	2	8	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
170	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
171	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
172	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
173	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
174	C₂×C₄²	C₅×D₄	1	4	yes	—
175	C₂×C₄²	C₅×D₄	1	2	yes	—
176	C₂×C₄²	C₅×D₄	1	2	yes	—
177	C₂×C₄²	C₅×Q₈	1	4	yes	—
178	C₂×C₄²	C₅×Q₈	1	2	yes	—
179	C₄⋊C₈	—	2	8	no	—
180	#95	—	2	4	no	—
181	C₂²×C₅	—	2	4	no	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	—	2	4	no	—
185	C₂²×C₅	C₂×#95	1	1	yes	characteristic
186	C₂²×C₅	#380	1	2	yes	—
187	C₂²×C₅	#389	1	2	yes	—
188	C₂²×C₅	#390	1	1	yes	characteristic
189	C₂²×C₅	#391	1	1	yes	characteristic
190	C₂²×C₅	C₂×#104	1	4	yes	—
191	C₄×C₅	—	2	16	no	—
192	C₄×C₅	—	2	4	no	—
193	C₄×C₅	—	8	4	no	—
194	C₄×C₅	#339	1	8	yes	complemented
195	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
196	C₄×C₅	#382	1	1	yes	characteristic
197	C₄×C₅	#384	1	1	yes	characteristic
198	C₄×C₅	#389	1	2	yes	—
199	C₄×C₅	#390	1	1	yes	characteristic
200	C₄×C₅	#391	1	1	yes	characteristic
201	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
202	C₂²×C₄	—	2	16	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
209	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
210	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
211	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
212	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
218	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
220	C₂²×C₄	C₅×#41	1	2	yes	—
221	C₂²×C₄	C₅×#41	1	2	yes	—
222	M₁₆	—	2	8	no	—
223	M₁₆	—	4	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	4	8	no	—
229	C₂²⋊C₄	—	4	4	no	—
230	C₄²	—	2	32	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	4	2	no	—
235	C₄²	—	4	2	no	—
236	C₄²	C₄²×C₅	1	2	yes	—
237	C₄²	C₄²×C₅	1	2	yes	—
238	C₄²	C₅×C₂²⋊C₄	1	4	yes	—
239	C₄²	C₅×C₄⋊C₄	1	4	yes	—
240	C₄²	C₅×C₄⋊C₄	1	2	yes	—
241	C₄²	C₅×C₄⋊C₄	1	2	yes	—
242	C₄²	C₅×C₄⋊C₄	1	8	yes	complemented
243	C₂×C₅	—	2	2	no	—
244	C₂×C₅	—	2	2	no	—
245	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
246	C₂×C₅	C₂×#339	1	2	yes	—
247	C₂×C₅	#1962	1	2	yes	—
248	C₂×C₅	#1994	1	2	yes	—
249	C₂³	—	2	4	no	—
250	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
251	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
252	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
253	C₂³	C₅×#121	1	2	yes	—
254	C₂³	C₅×C₂²⋊Q₈	1	2	yes	—
255	C₂×C₄	—	2	16	no	—
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	8	no	—
264	C₂×C₄	C₅×#95	1	4	yes	—
265	C₂×C₄	C₅×#95	1	4	yes	—
266	C₂×C₄	C₅×#104	1	8	yes	—
267	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
268	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
270	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
272	C₂×C₄	C₅×#117	1	2	yes	—
273	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
274	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
275	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
276	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
277	C₂×C₄	C₅×#121	1	2	yes	—
278	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
279	C₂×C₄	C₅×#124	1	1	yes	characteristic
280	C₂×C₄	C₅×#124	1	1	yes	characteristic
281	C₂×C₄	C₅×#125	1	2	yes	—
282	C₂×C₄	C₅×#127	1	1	yes	characteristic
283	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
284	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
285	C₈	—	4	4	no	—
286	C₅	#10839	1	1	yes	characteristic, direct product factor, Sylow
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	C₂×C₅×#95	1	1	yes	characteristic
292	C₂²	C₅×#380	1	2	yes	—
293	C₂²	C₅×#389	1	2	yes	—
294	C₂²	C₅×#390	1	1	yes	characteristic
295	C₂²	C₅×#391	1	1	yes	characteristic
296	C₂²	C₂×C₅×#104	1	4	yes	—
297	C₄	—	2	16	no	—
298	C₄	—	2	4	no	—
299	C₄	—	8	4	no	—
300	C₄	C₅×#339	1	8	yes	complemented
301	C₄	C₄×C₅×C₄⋊C₄	1	2	yes	—
302	C₄	C₅×#382	1	1	yes	characteristic
303	C₄	C₅×#384	1	1	yes	characteristic
304	C₄	C₅×#389	1	2	yes	—
305	C₄	C₅×#390	1	1	yes	characteristic
306	C₄	C₅×#391	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	C₅×C₄⋊#95	1	1	yes	characteristic
310	C₂	C₂×C₅×#339	1	2	yes	—
311	C₂	C₅×#1962	1	2	yes	—
312	C₂	C₅×#1994	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅×#423	C₄	semidirect
13. C₅×#384	C₄	semidirect
14. C₅×#382	C₄	semidirect
18. C₄³×C₅	C₄	semidirect
20. #10839	C₅	direct (N × H)
97. C₄²×C₅	C₄⋊C₄	semidirect
104. #423	C₄×C₅	semidirect
106. #384	C₄×C₅	semidirect
107. #382	C₄×C₅	semidirect
111. C₄³	C₄×C₅	semidirect
194. C₄×C₅	#339	semidirect
242. C₄²	C₅×C₄⋊C₄	semidirect
286. C₅	#10839	direct (N × H)
300. C₄	C₅×#339	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#423	2
C₅×#2285	2
C₂×C₄³×C₅	2
C₂×C₅×#339	2
#10839	5

Subgroups — C5×#10839 #11933559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10839 #11933559