Subgroups — C₅×#21710 #11944430

Order 1280.

1014 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3150	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#444	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#443	C₂	1	2	yes	maximal, complemented
4	C₅×#2932	C₂	1	2	yes	maximal, complemented
5	C₅×#1975	C₂	1	2	yes	maximal, complemented
6	C₅×#1974	C₂	1	2	yes	maximal, complemented
7	C₅×#1962	C₂	1	2	yes	maximal, complemented
8	C₅×#1849	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#135	C₂²	1	2	yes	—
11	C₂×C₅×#135	C₂²	1	2	yes	—
12	C₂×C₅×#131	C₂²	1	4	yes	—
13	C₅×#532	C₂²	1	2	yes	—
14	C₅×#497	C₄	1	4	yes	complemented
15	C₅×#496	C₄	1	4	yes	complemented
16	C₅×#444	—	2	8	no	—
17	C₅×#443	—	2	8	no	—
18	C₅×#435	—	2	8	no	—
19	C₂×C₅×#108	C₂²	1	2	yes	—
20	C₂×C₅×#108	C₂²	1	2	yes	—
21	C₅×#423	C₂²	1	2	yes	—
22	C₂×C₅×#104	C₂²	1	4	yes	—
23	C₂×C₅×#104	C₂²	1	4	yes	—
24	C₅×#408	C₂²	1	4	yes	—
25	C₂×C₅×#97	C₂²	1	1	yes	characteristic
26	C₂×C₅×#97	C₂²	1	1	yes	characteristic
27	C₂×C₅×#97	C₂²	1	1	yes	characteristic
28	C₂×C₅×#97	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
31	C₄³×C₅	C₂²	1	1	yes	characteristic
32	#21710	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×C₅×#41	C₂³	1	4	yes	—
34	C₅×#135	—	2	4	no	—
35	C₅×#135	—	2	4	no	—
36	C₅×#135	—	2	4	no	—
37	C₅×#135	—	2	4	no	—
38	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
40	C₂×C₅×SD₁₆	C₂×C₄	1	4	yes	—
41	C₂×C₅×SD₁₆	C₂×C₄	1	4	yes	—
42	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
43	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
44	C₅×#131	—	2	8	no	—
45	C₅×#131	—	2	8	no	—
46	C₂×C₅×M₁₆	—	2	4	no	—
47	C₂×C₅×M₁₆	C₂³	1	4	yes	—
48	C₂²×C₅×C₈	—	2	4	no	—
49	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
53	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
54	C₅×#127	C₂×C₄	1	2	yes	—
55	C₅×#124	C₂×C₄	1	4	yes	—
56	C₅×#121	—	2	4	no	—
57	C₄×C₅×D₄	—	2	4	no	—
58	C₂×C₄²×C₅	—	2	2	no	—
59	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
62	C₅×#108	—	2	4	no	—
63	C₅×#108	—	2	4	no	—
64	C₅×C₄⋊C₈	—	2	4	no	—
65	C₅×#104	—	2	8	no	—
66	C₅×#104	—	2	8	no	—
67	C₅×C₂²⋊C₈	—	2	8	no	—
68	C₅×#97	—	2	2	no	—
69	C₅×#97	C₂×C₄	1	4	yes	—
70	C₅×#97	D₄	1	4	yes	—
71	C₅×#97	D₄	1	4	yes	complemented
72	C₄×C₅×C₈	—	2	2	no	—
73	C₄×C₅×C₈	D₄	1	4	yes	complemented
74	#3150	C₂×C₅	1	1	yes	characteristic
75	C₂×#444	C₂×C₅	1	2	yes	complemented
76	C₂×#443	C₂×C₅	1	2	yes	complemented
77	#2932	C₂×C₅	1	2	yes	complemented
78	#1975	C₂×C₅	1	2	yes	complemented
79	#1974	C₂×C₅	1	2	yes	complemented
80	#1962	C₂×C₅	1	2	yes	complemented
81	#1849	C₂×C₅	1	1	yes	characteristic, complemented
82	C₄×#97	C₂×C₅	1	1	yes	characteristic, complemented
83	C₅×#41	—	2	4	no	—
84	C₅×#41	—	2	4	no	—
85	C₅×#41	—	2	4	no	—
86	C₅×#41	—	2	4	no	—
87	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
88	C₂×C₅×D₄	—	2	4	no	—
89	C₂×C₅×D₄	—	2	4	no	—
90	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
91	C₂²×C₄×C₅	—	2	1	no	—
92	C₂²×C₄×C₅	—	2	1	no	—
93	C₂²×C₄×C₅	—	2	4	no	—
94	C₂²×C₄×C₅	#41	1	2	yes	—
95	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
96	C₅×Q₁₆	—	2	4	no	—
97	C₅×Q₁₆	—	2	4	no	—
98	C₅×SD₁₆	—	2	8	no	—
99	C₅×SD₁₆	—	2	8	no	—
100	C₅×D₈	—	2	4	no	—
101	C₅×D₈	—	2	4	no	—
102	C₅×M₁₆	—	2	4	no	—
103	C₅×M₁₆	—	2	4	no	—
104	C₅×M₁₆	—	4	8	no	—
105	C₂×C₅×C₈	—	2	1	no	—
106	C₂×C₅×C₈	—	2	1	no	—
107	C₂×C₅×C₈	—	2	4	no	—
108	C₂×C₅×C₈	—	4	8	no	—
109	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
110	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
112	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
113	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
114	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
115	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
116	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
117	C₂×C₅×C₈	#41	1	2	yes	—
118	C₂×C₅×C₈	#41	1	2	yes	—
119	C₅×C₄⋊C₄	—	2	4	no	—
120	C₅×C₂²⋊C₄	—	2	8	no	—
121	C₄²×C₅	—	2	1	no	—
122	C₄²×C₅	—	2	1	no	—
123	C₄²×C₅	—	4	4	no	—
124	C₄²×C₅	C₂²×C₄	1	2	yes	—
125	C₄²×C₅	C₂×D₄	1	2	yes	—
126	C₄²×C₅	C₂×D₄	1	2	yes	—
127	C₄²×C₅	C₂×D₄	1	2	yes	—
128	C₂×#135	C₂²×C₅	1	2	yes	—
129	C₂×#135	C₂²×C₅	1	2	yes	—
130	C₂×#131	C₂²×C₅	1	4	yes	—
131	#532	C₂²×C₅	1	2	yes	—
132	#497	C₄×C₅	1	4	yes	complemented
133	#496	C₄×C₅	1	4	yes	complemented
134	#444	—	2	8	no	—
135	#443	—	2	8	no	—
136	#435	—	2	8	no	—
137	C₂×#108	C₂²×C₅	1	2	yes	—
138	C₂×#108	C₂²×C₅	1	2	yes	—
139	#423	C₂²×C₅	1	2	yes	—
140	C₂×#104	C₂²×C₅	1	4	yes	—
141	C₂×#104	C₂²×C₅	1	4	yes	—
142	#408	C₂²×C₅	1	4	yes	—
143	C₂×#97	C₂²×C₅	1	1	yes	characteristic
144	C₂×#97	C₂²×C₅	1	1	yes	characteristic
145	C₂×#97	C₂²×C₅	1	1	yes	characteristic
146	C₂×#97	C₂²×C₅	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
149	C₄³	C₂²×C₅	1	1	yes	characteristic
150	C₂³×C₅	—	2	4	no	—
151	C₂³×C₅	C₂×#41	1	1	yes	characteristic
152	C₅×Q₈	—	2	4	no	—
153	C₅×Q₈	—	2	4	no	—
154	C₅×D₄	—	2	4	no	—
155	C₅×D₄	—	2	4	no	—
156	C₅×D₄	—	4	4	no	—
157	C₅×D₄	—	4	4	no	—
158	C₅×D₄	—	8	4	no	—
159	C₂×C₄×C₅	—	2	4	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	4	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	4	2	no	—
164	C₂×C₄×C₅	—	4	2	no	—
165	C₂×C₄×C₅	—	4	4	no	—
166	C₂×C₄×C₅	—	4	4	no	—
167	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
168	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
174	C₅×C₈	—	2	2	no	—
175	C₅×C₈	—	4	4	no	—
176	C₅×C₈	C₄×D₄	1	4	yes	—
177	C₅×C₈	C₄×D₄	1	4	yes	—
178	C₅×C₈	#127	1	4	yes	—
179	C₂×#41	C₂³×C₅	1	4	yes	—
180	#135	—	2	4	no	—
181	#135	—	2	4	no	—
182	#135	—	2	4	no	—
183	#135	—	2	4	no	—
184	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
185	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
186	C₂×SD₁₆	C₂×C₄×C₅	1	4	yes	—
187	C₂×SD₁₆	C₂×C₄×C₅	1	4	yes	—
188	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
189	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
190	#131	—	2	8	no	—
191	#131	—	2	8	no	—
192	C₂×M₁₆	—	2	4	no	—
193	C₂×M₁₆	C₂³×C₅	1	4	yes	—
194	C₂²×C₈	—	2	4	no	—
195	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
199	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
200	#127	C₂×C₄×C₅	1	2	yes	—
201	#124	C₂×C₄×C₅	1	4	yes	—
202	#121	—	2	4	no	—
203	C₄×D₄	—	2	4	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
208	#108	—	2	4	no	—
209	#108	—	2	4	no	—
210	C₄⋊C₈	—	2	4	no	—
211	#104	—	2	8	no	—
212	#104	—	2	8	no	—
213	C₂²⋊C₈	—	2	8	no	—
214	#97	—	2	2	no	—
215	#97	C₂×C₄×C₅	1	4	yes	—
216	#97	C₅×D₄	1	4	yes	—
217	#97	C₅×D₄	1	4	yes	complemented
218	C₄×C₈	—	2	2	no	—
219	C₄×C₈	C₅×D₄	1	4	yes	complemented
220	C₂²×C₅	—	2	2	no	—
221	C₂²×C₅	—	4	4	no	—
222	C₂²×C₅	—	4	8	no	—
223	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
224	C₂²×C₅	#532	1	1	yes	characteristic
225	C₂²×C₅	#532	1	1	yes	characteristic
226	C₄×C₅	—	2	4	no	—
227	C₄×C₅	—	2	4	no	—
228	C₄×C₅	—	4	2	no	—
229	C₄×C₅	—	4	4	no	—
230	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
231	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
232	C₄×C₅	C₂×#127	1	1	yes	characteristic
233	C₄×C₅	#532	1	1	yes	characteristic
234	#41	—	2	4	no	—
235	#41	—	2	4	no	—
236	#41	—	2	4	no	—
237	#41	—	2	4	no	—
238	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	C₅×#41	1	2	yes	—
246	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
247	Q₁₆	—	2	4	no	—
248	Q₁₆	—	2	4	no	—
249	SD₁₆	—	2	8	no	—
250	SD₁₆	—	2	8	no	—
251	D₈	—	2	4	no	—
252	D₈	—	2	4	no	—
253	M₁₆	—	2	4	no	—
254	M₁₆	—	2	4	no	—
255	M₁₆	—	4	8	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	4	8	no	—
260	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
261	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
262	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
263	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
264	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
265	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
266	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
267	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
268	C₂×C₈	C₅×#41	1	2	yes	—
269	C₂×C₈	C₅×#41	1	2	yes	—
270	C₄⋊C₄	—	2	4	no	—
271	C₂²⋊C₄	—	2	8	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	4	4	no	—
275	C₄²	C₂²×C₄×C₅	1	2	yes	—
276	C₄²	C₂×C₅×D₄	1	2	yes	—
277	C₄²	C₂×C₅×D₄	1	2	yes	—
278	C₄²	C₂×C₅×D₄	1	2	yes	—
279	C₂×C₅	—	2	1	no	—
280	C₂×C₅	—	2	1	no	—
281	C₂×C₅	—	8	4	no	—
282	C₂×C₅	C₄×#127	1	1	yes	characteristic
283	C₂×C₅	#2956	1	2	yes	—
284	C₂³	—	2	4	no	—
285	C₂³	C₂×C₅×#41	1	1	yes	characteristic
286	Q₈	—	2	4	no	—
287	Q₈	—	2	4	no	—
288	D₄	—	2	4	no	—
289	D₄	—	2	4	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	8	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
302	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
306	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
307	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	4	4	no	—
310	C₈	C₄×C₅×D₄	1	4	yes	—
311	C₈	C₄×C₅×D₄	1	4	yes	—
312	C₈	C₅×#127	1	4	yes	—
313	C₅	#21710	1	1	yes	characteristic, direct product factor, Sylow
314	C₂²	—	2	2	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	4	8	no	—
317	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
318	C₂²	C₅×#532	1	1	yes	characteristic
319	C₂²	C₅×#532	1	1	yes	characteristic
320	C₄	—	2	4	no	—
321	C₄	—	2	4	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	4	no	—
324	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
325	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
326	C₄	C₂×C₅×#127	1	1	yes	characteristic
327	C₄	C₅×#532	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	8	4	no	—
331	C₂	C₄×C₅×#127	1	1	yes	characteristic
332	C₂	C₅×#2956	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	C₂×C₄×D₄
prime radical	C₅×#3150	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#444	C₂	semidirect
3. C₂×C₅×#443	C₂	semidirect
4. C₅×#2932	C₂	semidirect
5. C₅×#1975	C₂	semidirect
6. C₅×#1974	C₂	semidirect
7. C₅×#1962	C₂	semidirect
8. C₅×#1849	C₂	semidirect
9. C₄×C₅×#97	C₂	semidirect
14. C₅×#497	C₄	semidirect
15. C₅×#496	C₄	semidirect
32. #21710	C₅	direct (N × H)
71. C₅×#97	D₄	semidirect
73. C₄×C₅×C₈	D₄	semidirect
75. C₂×#444	C₂×C₅	semidirect
76. C₂×#443	C₂×C₅	semidirect
77. #2932	C₂×C₅	semidirect
78. #1975	C₂×C₅	semidirect
79. #1974	C₂×C₅	semidirect
80. #1962	C₂×C₅	semidirect
81. #1849	C₂×C₅	semidirect
82. C₄×#97	C₂×C₅	semidirect
132. #497	C₄×C₅	semidirect
133. #496	C₄×C₅	semidirect
217. #97	C₅×D₄	semidirect
219. C₄×C₈	C₅×D₄	semidirect
313. C₅	#21710	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×#3150	2
C₂×C₅×#444	2
C₂×C₅×#443	2
C₅×#2932	2
C₅×#1975	2
C₅×#1974	2
C₅×#1962	2
C₅×#1849	2
C₄×C₅×#97	2
#21710	5

Subgroups — C5×#21710 #11944430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21710 #11944430