Subgroups — C₅×#21330 #11944050

Order 1280.

1114 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3458	C₂	1	1	yes	maximal, characteristic
2	C₅×#2905	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
4	C₅×#1995	C₂	1	4	yes	maximal, complemented
5	C₅×#1994	C₂	1	4	yes	maximal, complemented
6	C₅×#1733	C₂	1	2	yes	maximal
7	C₂×C₅×#339	C₂	1	2	yes	maximal
8	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
9	C₅×#544	—	2	2	no	—
10	C₅×#544	C₄	1	8	yes	complemented
11	C₅×#543	—	2	2	no	—
12	C₅×#533	—	2	4	no	—
13	C₅×#533	C₂²	1	4	yes	—
14	C₂×C₅×#117	C₂²	1	2	yes	—
15	C₂×C₅×#117	C₂²	1	1	yes	characteristic
16	C₂×C₅×#117	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₅×#424	—	2	4	no	—
19	C₅×#423	C₂²	1	4	yes	—
20	C₄×C₅×C₄⋊C₄	—	2	2	no	—
21	C₄×C₅×C₂²⋊C₄	—	2	2	no	—
22	C₅×#376	C₂²	1	2	yes	—
23	C₄³×C₅	C₂²	1	2	yes	—
24	C₅×#339	C₂²	1	4	yes	—
25	C₅×#339	C₂²	1	4	yes	—
26	C₅×#339	C₂²	1	8	yes	—
27	#21330	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂³×C₄×C₅	—	2	1	no	—
29	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₅×M₁₆	—	2	2	no	—
31	C₂×C₅×M₁₆	—	2	4	no	—
32	C₂×C₅×M₁₆	—	2	4	no	—
33	C₂×C₅×M₁₆	C₂³	1	4	yes	—
34	C₂²×C₅×C₈	—	2	2	no	—
35	C₅×C₄⋊Q₈	—	2	8	no	—
36	C₅×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
37	C₅×#125	—	2	8	no	—
38	C₅×#125	C₂×C₄	1	8	yes	complemented
39	C₅×C₂²⋊Q₈	—	2	8	no	—
40	C₅×C₂²⋊Q₈	—	2	16	no	—
41	C₄×C₅×Q₈	—	2	8	no	—
42	C₅×#117	—	2	2	no	—
43	C₅×#117	—	2	2	no	—
44	C₅×#117	—	2	2	no	—
45	C₅×#117	C₂×C₄	1	4	yes	—
46	C₅×#117	C₂×C₄	1	4	yes	—
47	C₅×#117	D₄	1	4	yes	—
48	C₅×#117	Q₈	1	4	yes	complemented
49	C₅×#117	C₂³	1	4	yes	—
50	C₂×C₅×C₄⋊C₄	—	2	1	no	—
51	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
52	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
53	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
54	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
55	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
56	C₂×C₄²×C₅	—	2	1	no	—
57	C₂×C₄²×C₅	—	2	2	no	—
58	C₂×C₄²×C₅	—	2	2	no	—
59	C₂×C₄²×C₅	—	2	4	no	—
60	C₂×C₄²×C₅	Q₈	1	4	yes	complemented
61	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₅	C₂³	1	2	yes	—
63	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₅	C₂³	1	2	yes	—
65	C₅×C₄⋊C₈	—	2	8	no	—
66	C₅×#95	—	2	4	no	—
67	#3458	C₂×C₅	1	1	yes	characteristic
68	#2905	C₂×C₅	1	1	yes	characteristic
69	C₄×#117	C₂×C₅	1	1	yes	characteristic
70	#1995	C₂×C₅	1	4	yes	complemented
71	#1994	C₂×C₅	1	4	yes	complemented
72	#1733	C₂×C₅	1	2	yes	—
73	C₂×#339	C₂×C₅	1	2	yes	—
74	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂×C₅×Q₈	—	2	8	no	—
76	C₂²×C₄×C₅	—	2	1	no	—
77	C₂²×C₄×C₅	—	2	4	no	—
78	C₂²×C₄×C₅	—	2	2	no	—
79	C₂²×C₄×C₅	—	2	1	no	—
80	C₂²×C₄×C₅	—	2	4	no	—
81	C₂²×C₄×C₅	—	4	4	no	—
82	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
87	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
88	C₂²×C₄×C₅	#41	1	2	yes	—
89	C₂²×C₄×C₅	#41	1	2	yes	—
90	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	C₅×M₁₆	—	2	4	no	—
92	C₅×M₁₆	—	2	4	no	—
93	C₅×M₁₆	—	4	4	no	—
94	C₂×C₅×C₈	—	2	4	no	—
95	C₂×C₅×C₈	—	4	4	no	—
96	C₅×C₄⋊C₄	—	2	4	no	—
97	C₅×C₄⋊C₄	—	2	8	no	—
98	C₅×C₄⋊C₄	—	2	16	no	—
99	C₅×C₄⋊C₄	—	2	4	no	—
100	C₅×C₄⋊C₄	C₂²⋊C₄	1	8	yes	—
101	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
102	C₅×C₂²⋊C₄	—	2	4	no	—
103	C₅×C₂²⋊C₄	—	2	4	no	—
104	C₅×C₂²⋊C₄	—	2	4	no	—
105	C₅×C₂²⋊C₄	C₂²⋊C₄	1	8	yes	—
106	C₄²×C₅	—	2	4	no	—
107	C₄²×C₅	—	2	4	no	—
108	C₄²×C₅	—	2	4	no	—
109	C₄²×C₅	—	2	1	no	—
110	C₄²×C₅	—	2	1	no	—
111	C₄²×C₅	—	2	2	no	—
112	C₄²×C₅	—	4	8	no	—
113	C₄²×C₅	C₂²×C₄	1	4	yes	—
114	C₄²×C₅	C₂×D₄	1	4	yes	—
115	C₄²×C₅	C₂×Q₈	1	2	yes	—
116	C₄²×C₅	C₂×Q₈	1	2	yes	—
117	C₄²×C₅	C₂×Q₈	1	4	yes	—
118	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
119	#544	—	2	2	no	—
120	#544	C₄×C₅	1	8	yes	complemented
121	#543	—	2	2	no	—
122	#533	—	2	4	no	—
123	#533	C₂²×C₅	1	4	yes	—
124	C₂×#117	C₂²×C₅	1	2	yes	—
125	C₂×#117	C₂²×C₅	1	1	yes	characteristic
126	C₂×#117	C₂²×C₅	1	1	yes	characteristic
127	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
128	#424	—	2	4	no	—
129	#423	C₂²×C₅	1	4	yes	—
130	C₄×C₄⋊C₄	—	2	2	no	—
131	C₄×C₂²⋊C₄	—	2	2	no	—
132	#376	C₂²×C₅	1	2	yes	—
133	C₄³	C₂²×C₅	1	2	yes	—
134	#339	C₂²×C₅	1	4	yes	—
135	#339	C₂²×C₅	1	4	yes	—
136	#339	C₂²×C₅	1	8	yes	—
137	C₂³×C₅	—	4	2	no	—
138	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
139	C₂³×C₅	C₂²⋊Q₈	1	2	yes	—
140	C₂³×C₅	C₂²⋊Q₈	1	2	yes	—
141	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
142	C₅×Q₈	—	8	8	no	—
143	C₂×C₄×C₅	—	2	4	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	8	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	4	no	—
148	C₂×C₄×C₅	—	4	4	no	—
149	C₂×C₄×C₅	—	4	8	no	—
150	C₂×C₄×C₅	—	4	4	no	—
151	C₂×C₄×C₅	—	4	2	no	—
152	C₂×C₄×C₅	—	8	4	no	—
153	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	8	yes	—
154	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
155	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
156	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
157	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
158	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
159	C₂×C₄×C₅	#124	1	4	yes	—
160	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
161	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
167	C₅×C₈	—	4	4	no	—
168	C₂³×C₄	—	2	1	no	—
169	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	4	no	—
172	C₂×M₁₆	—	2	4	no	—
173	C₂×M₁₆	C₂³×C₅	1	4	yes	—
174	C₂²×C₈	—	2	2	no	—
175	C₄⋊Q₈	—	2	8	no	—
176	C₄⋊Q₈	C₂×C₄×C₅	1	8	yes	complemented
177	#125	—	2	8	no	—
178	#125	C₂×C₄×C₅	1	8	yes	complemented
179	C₂²⋊Q₈	—	2	8	no	—
180	C₂²⋊Q₈	—	2	16	no	—
181	C₄×Q₈	—	2	8	no	—
182	#117	—	2	2	no	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	#117	C₂×C₄×C₅	1	4	yes	—
186	#117	C₂×C₄×C₅	1	4	yes	—
187	#117	C₅×D₄	1	4	yes	—
188	#117	C₅×Q₈	1	4	yes	complemented
189	#117	C₂³×C₅	1	4	yes	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
192	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₂²⋊C₄	—	2	1	no	—
195	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	2	1	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	2	4	no	—
200	C₂×C₄²	C₅×Q₈	1	4	yes	complemented
201	C₂×C₄²	C₂³×C₅	1	2	yes	—
202	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
203	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
204	C₂×C₄²	C₂³×C₅	1	2	yes	—
205	C₄⋊C₈	—	2	8	no	—
206	#95	—	2	4	no	—
207	C₂²×C₅	—	2	2	no	—
208	C₂²×C₅	—	2	4	no	—
209	C₂²×C₅	—	4	2	no	—
210	C₂²×C₅	—	4	2	no	—
211	C₂²×C₅	#391	1	4	yes	—
212	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
213	C₂²×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
214	C₂²×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
215	C₄×C₅	—	2	4	no	—
216	C₄×C₅	—	2	8	no	—
217	C₄×C₅	—	4	4	no	—
218	C₄×C₅	—	4	2	no	—
219	C₄×C₅	—	4	8	no	—
220	C₄×C₅	#391	1	4	yes	—
221	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
222	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
223	C₄×C₅	C₂×#124	1	1	yes	characteristic
224	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
225	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
226	C₂×Q₈	—	2	8	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	4	4	no	—
233	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
234	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
235	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
237	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
238	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
239	C₂²×C₄	C₅×#41	1	2	yes	—
240	C₂²×C₄	C₅×#41	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	4	no	—
244	M₁₆	—	4	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	4	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	16	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	C₅×C₂²⋊C₄	1	8	yes	—
252	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	C₅×C₂²⋊C₄	1	8	yes	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	4	8	no	—
264	C₄²	C₂²×C₄×C₅	1	4	yes	—
265	C₄²	C₂×C₅×D₄	1	4	yes	—
266	C₄²	C₂×C₅×Q₈	1	2	yes	—
267	C₄²	C₂×C₅×Q₈	1	2	yes	—
268	C₄²	C₂×C₅×Q₈	1	4	yes	—
269	C₂×C₅	—	2	1	no	—
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	—	2	4	no	—
272	C₂×C₅	C₂×#391	1	1	yes	characteristic
273	C₂×C₅	#2901	1	2	yes	—
274	C₂³	—	4	2	no	—
275	C₂³	C₂×C₅×C₂²⋊C₄	1	2	yes	—
276	C₂³	C₅×C₂²⋊Q₈	1	2	yes	—
277	C₂³	C₅×C₂²⋊Q₈	1	2	yes	—
278	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
279	Q₈	—	8	8	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	8	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	8	yes	—
291	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
292	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
293	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
294	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
295	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
296	C₂×C₄	C₅×#124	1	4	yes	—
297	C₂×C₄	C₅×C₄⋊Q₈	1	4	yes	—
298	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
302	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
303	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
304	C₈	—	4	4	no	—
305	C₅	#21330	1	1	yes	characteristic, direct product factor, Sylow
306	C₂²	—	2	2	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	C₅×#391	1	4	yes	—
311	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
312	C₂²	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
313	C₂²	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
314	C₄	—	2	4	no	—
315	C₄	—	2	8	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	4	no	—
318	C₄	—	4	8	no	—
319	C₄	C₅×#391	1	4	yes	—
320	C₄	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
321	C₄	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
322	C₄	C₂×C₅×#124	1	1	yes	characteristic
323	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	4	no	—
327	C₂	C₂×C₅×#391	1	1	yes	characteristic
328	C₂	C₅×#2901	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1995	C₂	semidirect
5. C₅×#1994	C₂	semidirect
10. C₅×#544	C₄	semidirect
27. #21330	C₅	direct (N × H)
36. C₅×C₄⋊Q₈	C₂×C₄	semidirect
38. C₅×#125	C₂×C₄	semidirect
48. C₅×#117	Q₈	semidirect
60. C₂×C₄²×C₅	Q₈	semidirect
70. #1995	C₂×C₅	semidirect
71. #1994	C₂×C₅	semidirect
120. #544	C₄×C₅	semidirect
176. C₄⋊Q₈	C₂×C₄×C₅	semidirect
178. #125	C₂×C₄×C₅	semidirect
188. #117	C₅×Q₈	semidirect
200. C₂×C₄²	C₅×Q₈	semidirect
305. C₅	#21330	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×#3458	2
C₅×#2905	2
C₄×C₅×#117	2
C₅×#1995	2
C₅×#1994	2
C₅×#1733	2
C₂×C₅×#339	2
#21330	5

Subgroups — C5×#21330 #11944050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21330 #11944050