Subgroups — C₅×#21184 #11943904

Order 1280.

844 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×#430	C₂	1	2	yes	maximal
2	C₂×C₅×#430	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#430	C₂	1	1	yes	maximal, characteristic
4	C₅×#2873	C₂	1	1	yes	maximal, characteristic
5	C₅×#1953	C₂	1	8	yes	maximal, complemented
6	C₂×C₅×#344	C₂	1	2	yes	maximal
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₅×#117	C₂²	1	1	yes	characteristic
12	C₅×#430	—	2	4	no	—
13	C₅×#430	—	2	2	no	—
14	C₅×#430	—	2	2	no	—
15	C₅×#430	C₄	1	8	yes	complemented
16	C₅×#430	C₂²	1	8	yes	—
17	C₂×C₅×#108	—	2	4	no	—
18	C₂×C₅×#108	C₂²	1	4	yes	—
19	C₂×C₅×#108	C₂²	1	4	yes	—
20	C₅×#405	—	2	2	no	—
21	C₅×#405	C₂²	1	4	yes	—
22	C₄×C₅×M₁₆	—	2	2	no	—
23	C₅×#344	C₂²	1	8	yes	—
24	#21184	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₅×M₁₆	—	2	1	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	2	no	—
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	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
37	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
38	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
39	C₂×C₅×M₁₆	D₄	1	4	yes	—
40	C₂×C₅×M₁₆	D₄	1	2	yes	—
41	C₂×C₅×M₁₆	D₄	1	2	yes	—
42	C₂×C₅×M₁₆	Q₈	1	4	yes	—
43	C₂×C₅×M₁₆	C₂³	1	4	yes	—
44	C₂×C₅×M₁₆	C₂³	1	4	yes	—
45	C₂×C₅×M₁₆	C₂³	1	2	yes	—
46	C₂²×C₅×C₈	—	2	1	no	—
47	C₂²×C₅×C₈	—	2	2	no	—
48	C₂²×C₅×C₈	—	2	2	no	—
49	C₂²×C₅×C₈	C₂³	1	2	yes	—
50	C₅×#117	—	2	1	no	—
51	C₅×#117	—	2	1	no	—
52	C₅×#117	Q₈	1	2	yes	—
53	C₅×#117	C₂³	1	2	yes	—
54	C₂×C₅×C₄⋊C₄	—	2	1	no	—
55	C₂×C₅×C₂²⋊C₄	Q₈	1	2	yes	—
56	C₂×C₄²×C₅	—	2	1	no	—
57	C₅×#108	—	2	16	no	—
58	C₅×#108	—	2	8	no	—
59	C₅×#108	C₂×C₄	1	16	yes	complemented
60	C₅×#97	—	2	4	no	—
61	C₄×C₅×C₈	—	2	4	no	—
62	C₂×#430	C₂×C₅	1	2	yes	—
63	C₂×#430	C₂×C₅	1	1	yes	characteristic
64	C₂×#430	C₂×C₅	1	1	yes	characteristic
65	#2873	C₂×C₅	1	1	yes	characteristic
66	#1953	C₂×C₅	1	8	yes	complemented
67	C₂×#344	C₂×C₅	1	2	yes	—
68	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₅	—	2	2	no	—
70	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₅	#41	1	1	yes	characteristic
77	C₂²×C₄×C₅	#41	1	1	yes	characteristic
78	C₂²×C₄×C₅	#41	1	2	yes	—
79	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
80	C₅×M₁₆	—	2	4	no	—
81	C₅×M₁₆	—	2	4	no	—
82	C₅×M₁₆	—	2	4	no	—
83	C₅×M₁₆	—	2	4	no	—
84	C₅×M₁₆	—	2	2	no	—
85	C₅×M₁₆	—	2	4	no	—
86	C₅×M₁₆	—	2	4	no	—
87	C₅×M₁₆	—	4	2	no	—
88	C₅×M₁₆	—	4	2	no	—
89	C₅×M₁₆	C₄⋊C₄	1	8	yes	—
90	C₅×M₁₆	C₄⋊C₄	1	8	yes	—
91	C₅×M₁₆	C₂²×C₄	1	8	yes	—
92	C₅×M₁₆	C₂×D₄	1	4	yes	—
93	C₂×C₅×C₈	—	2	4	no	—
94	C₂×C₅×C₈	—	2	4	no	—
95	C₂×C₅×C₈	—	2	4	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	4	no	—
98	C₂×C₅×C₈	—	2	4	no	—
99	C₂×C₅×C₈	—	4	2	no	—
100	C₂×C₅×C₈	—	4	2	no	—
101	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
102	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
103	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
104	C₅×C₄⋊C₄	—	2	2	no	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₂²⋊C₄	—	2	2	no	—
107	C₅×C₂²⋊C₄	C₂×Q₈	1	4	yes	—
108	C₄²×C₅	—	2	2	no	—
109	C₄²×C₅	—	2	2	no	—
110	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
111	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
112	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
113	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
114	C₂×#117	C₂²×C₅	1	1	yes	characteristic
115	#430	—	2	4	no	—
116	#430	—	2	2	no	—
117	#430	—	2	2	no	—
118	#430	C₄×C₅	1	8	yes	complemented
119	#430	C₂²×C₅	1	8	yes	—
120	C₂×#108	—	2	4	no	—
121	C₂×#108	C₂²×C₅	1	4	yes	—
122	C₂×#108	C₂²×C₅	1	4	yes	—
123	#405	—	2	2	no	—
124	#405	C₂²×C₅	1	4	yes	—
125	C₄×M₁₆	—	2	2	no	—
126	#344	C₂²×C₅	1	8	yes	—
127	C₂³×C₅	—	8	1	no	—
128	C₂³×C₅	C₄×Q₈	1	2	yes	—
129	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
132	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	8	2	no	—
135	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₅	C₂×C₄⋊C₄	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	#121	1	2	yes	—
141	C₂×C₄×C₅	#121	1	2	yes	—
142	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
144	C₂×C₄×C₅	#125	1	2	yes	—
145	C₂×C₄×C₅	#125	1	2	yes	—
146	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
147	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
153	C₅×C₈	—	2	8	no	—
154	C₅×C₈	—	2	4	no	—
155	C₅×C₈	—	4	4	no	—
156	C₅×C₈	C₂×C₄⋊C₄	1	8	yes	—
157	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
158	C₂×M₁₆	—	2	1	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
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	C₂×C₄×C₅	1	4	yes	—
169	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
170	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
171	C₂×M₁₆	C₅×D₄	1	4	yes	—
172	C₂×M₁₆	C₅×D₄	1	2	yes	—
173	C₂×M₁₆	C₅×D₄	1	2	yes	—
174	C₂×M₁₆	C₅×Q₈	1	4	yes	—
175	C₂×M₁₆	C₂³×C₅	1	4	yes	—
176	C₂×M₁₆	C₂³×C₅	1	4	yes	—
177	C₂×M₁₆	C₂³×C₅	1	2	yes	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	C₂³×C₅	1	2	yes	—
182	#117	—	2	1	no	—
183	#117	—	2	1	no	—
184	#117	C₅×Q₈	1	2	yes	—
185	#117	C₂³×C₅	1	2	yes	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₂²⋊C₄	C₅×Q₈	1	2	yes	—
188	C₂×C₄²	—	2	1	no	—
189	#108	—	2	16	no	—
190	#108	—	2	8	no	—
191	#108	C₂×C₄×C₅	1	16	yes	complemented
192	#97	—	2	4	no	—
193	C₄×C₈	—	2	4	no	—
194	C₂²×C₅	—	4	1	no	—
195	C₂²×C₅	—	4	2	no	—
196	C₂²×C₅	—	4	1	no	—
197	C₂²×C₅	—	4	2	no	—
198	C₂²×C₅	—	4	1	no	—
199	C₂²×C₅	#389	1	2	yes	—
200	C₂²×C₅	#389	1	2	yes	—
201	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
202	C₂²×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
203	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
204	C₄×C₅	—	4	4	no	—
205	C₄×C₅	#389	1	2	yes	—
206	C₄×C₅	#389	1	2	yes	—
207	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
208	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
209	C₄×C₅	C₂×#121	1	1	yes	characteristic
210	C₄×C₅	C₂×#125	1	1	yes	characteristic
211	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
214	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
217	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
218	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
219	C₂²×C₄	C₅×#41	1	1	yes	characteristic
220	C₂²×C₄	C₅×#41	1	1	yes	characteristic
221	C₂²×C₄	C₅×#41	1	2	yes	—
222	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	4	2	no	—
231	M₁₆	—	4	2	no	—
232	M₁₆	C₅×C₄⋊C₄	1	8	yes	—
233	M₁₆	C₅×C₄⋊C₄	1	8	yes	—
234	M₁₆	C₂²×C₄×C₅	1	8	yes	—
235	M₁₆	C₂×C₅×D₄	1	4	yes	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
245	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
246	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	C₂×C₅×Q₈	1	4	yes	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₂×C₅	—	2	1	no	—
254	C₂×C₅	—	2	1	no	—
255	C₂×C₅	—	2	1	no	—
256	C₂×C₅	—	2	2	no	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	C₂×#389	1	1	yes	characteristic
259	C₂³	—	8	1	no	—
260	C₂³	C₄×C₅×Q₈	1	2	yes	—
261	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
262	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
263	C₂³	C₅×C₄⋊Q₈	1	2	yes	—
264	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
268	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
269	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
270	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
271	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
272	C₂×C₄	C₅×#121	1	2	yes	—
273	C₂×C₄	C₅×#121	1	2	yes	—
274	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
275	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
276	C₂×C₄	C₅×#125	1	2	yes	—
277	C₂×C₄	C₅×#125	1	2	yes	—
278	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
279	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
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	C₈	—	2	8	no	—
286	C₈	—	2	4	no	—
287	C₈	—	4	4	no	—
288	C₈	C₂×C₅×C₄⋊C₄	1	8	yes	—
289	C₅	#21184	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²	—	4	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	1	no	—
295	C₂²	C₅×#389	1	2	yes	—
296	C₂²	C₅×#389	1	2	yes	—
297	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
298	C₂²	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
299	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
300	C₄	—	4	4	no	—
301	C₄	C₅×#389	1	2	yes	—
302	C₄	C₅×#389	1	2	yes	—
303	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
304	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
305	C₄	C₂×C₅×#121	1	1	yes	characteristic
306	C₄	C₂×C₅×#125	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	C₂×C₅×#389	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂×#121
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂×C₅	C₂×#389
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅×#1953	C₂	semidirect
15. C₅×#430	C₄	semidirect
24. #21184	C₅	direct (N × H)
59. C₅×#108	C₂×C₄	semidirect
66. #1953	C₂×C₅	semidirect
118. #430	C₄×C₅	semidirect
191. #108	C₂×C₄×C₅	semidirect
289. C₅	#21184	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×#430	2
C₅×#2873	2
C₅×#1953	2
C₂×C₅×#344	2
#21184	5

Subgroups — C5×#21184 #11943904

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21184 #11943904