Subgroups — C₅×#11330 #11934050

Order 1280.

722 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#428	C₂	1	2	yes	maximal
2	C₂²×C₅×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#343	C₂	1	2	yes	maximal
4	C₂×C₅×#341	C₂	1	1	yes	maximal, characteristic
5	C₅×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₅×#428	—	2	4	no	—
12	C₅×#428	—	2	4	no	—
13	C₂×C₅×#107	—	2	2	no	—
14	C₂×C₅×#106	—	2	2	no	—
15	C₂×C₅×#106	—	2	4	no	—
16	C₂×C₅×#106	—	2	2	no	—
17	C₅×#343	—	2	4	no	—
18	C₅×#341	C₄	1	4	yes	complemented
19	C₅×#336	—	2	2	no	—
20	#11330	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₅	—	2	2	no	—
22	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
30	C₂×C₅×M₁₆	D₄	1	4	yes	—
31	C₂×C₅×M₁₆	Q₈	1	2	yes	—
32	C₂×C₅×M₁₆	Q₈	1	4	yes	—
33	C₂²×C₅×C₈	—	2	2	no	—
34	C₂²×C₅×C₈	—	2	2	no	—
35	C₂²×C₅×C₈	—	2	2	no	—
36	C₂²×C₅×C₈	—	2	1	no	—
37	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
38	C₂²×C₅×C₈	D₄	1	2	yes	—
39	C₂²×C₅×C₈	Q₈	1	2	yes	—
40	C₅×#117	—	2	2	no	—
41	C₅×#117	—	2	2	no	—
42	C₂×C₅×C₄⋊C₄	—	2	2	no	—
43	C₂×C₅×C₄⋊C₄	—	2	2	no	—
44	C₂×C₅×C₄⋊C₄	—	2	2	no	—
45	C₂×C₅×C₄⋊C₄	—	2	2	no	—
46	C₂×C₅×C₄⋊C₄	—	2	2	no	—
47	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
48	C₂×C₅×C₄⋊C₄	D₄	1	4	yes	—
49	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
50	C₂×C₄²×C₅	—	2	1	no	—
51	C₅×#107	—	2	8	no	—
52	C₅×#106	—	2	8	no	—
53	C₅×#106	—	2	8	no	—
54	C₂×#428	C₂×C₅	1	2	yes	—
55	C₂²×#106	C₂×C₅	1	1	yes	characteristic
56	C₂×#343	C₂×C₅	1	2	yes	—
57	C₂×#341	C₂×C₅	1	1	yes	characteristic
58	#1729	C₂×C₅	1	1	yes	characteristic
59	C₂⁴×C₅	#41	1	1	yes	characteristic
60	C₂²×C₄×C₅	—	2	2	no	—
61	C₂²×C₄×C₅	—	2	4	no	—
62	C₂²×C₄×C₅	—	4	2	no	—
63	C₂²×C₄×C₅	—	4	2	no	—
64	C₂²×C₄×C₅	—	8	4	no	—
65	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	#41	1	2	yes	—
74	C₂²×C₄×C₅	#41	1	1	yes	characteristic
75	C₅×M₁₆	—	2	4	no	—
76	C₅×M₁₆	—	2	4	no	—
77	C₅×M₁₆	—	2	4	no	—
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	—	4	2	no	—
80	C₅×M₁₆	C₄⋊C₄	1	8	yes	—
81	C₂×C₅×C₈	—	2	4	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	—	2	4	no	—
85	C₂×C₅×C₈	—	2	4	no	—
86	C₂×C₅×C₈	—	2	4	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	—	2	2	no	—
89	C₂×C₅×C₈	—	4	1	no	—
90	C₂×C₅×C₈	—	4	1	no	—
91	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₅×C₄⋊C₄	—	2	4	no	—
94	C₅×C₄⋊C₄	—	2	2	no	—
95	C₅×C₄⋊C₄	—	2	2	no	—
96	C₅×C₄⋊C₄	—	4	2	no	—
97	C₅×C₄⋊C₄	—	4	2	no	—
98	C₅×C₂²⋊C₄	—	4	2	no	—
99	C₄²×C₅	—	4	1	no	—
100	C₄²×C₅	—	4	1	no	—
101	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
102	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
103	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
104	C₂×#117	C₂²×C₅	1	1	yes	characteristic
105	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
106	#428	—	2	4	no	—
107	#428	—	2	4	no	—
108	C₂×#107	—	2	2	no	—
109	C₂×#106	—	2	2	no	—
110	C₂×#106	—	2	4	no	—
111	C₂×#106	—	2	2	no	—
112	#343	—	2	4	no	—
113	#341	C₄×C₅	1	4	yes	complemented
114	#336	—	2	2	no	—
115	C₂³×C₅	—	2	4	no	—
116	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
117	C₂³×C₅	#121	1	2	yes	—
118	C₂³×C₅	#121	1	1	yes	characteristic
119	C₂³×C₅	#125	1	2	yes	—
120	C₂³×C₅	#125	1	1	yes	characteristic
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	—	4	4	no	—
123	C₂×C₄×C₅	—	4	4	no	—
124	C₂×C₄×C₅	—	8	4	no	—
125	C₂×C₄×C₅	—	8	4	no	—
126	C₂×C₄×C₅	—	8	4	no	—
127	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₅	#120	1	1	yes	characteristic
132	C₂×C₄×C₅	#120	1	2	yes	—
133	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
136	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₅	#123	1	1	yes	characteristic
139	C₂×C₄×C₅	#123	1	2	yes	—
140	C₂×C₄×C₅	#127	1	1	yes	characteristic
141	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
142	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
143	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
144	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
145	C₅×C₈	—	2	4	no	—
146	C₅×C₈	—	2	8	no	—
147	C₅×C₈	—	4	2	no	—
148	C₂³×C₄	—	2	2	no	—
149	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	1	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
157	C₂×M₁₆	C₅×D₄	1	4	yes	—
158	C₂×M₁₆	C₅×Q₈	1	2	yes	—
159	C₂×M₁₆	C₅×Q₈	1	4	yes	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
165	C₂²×C₈	C₅×D₄	1	2	yes	—
166	C₂²×C₈	C₅×Q₈	1	2	yes	—
167	#117	—	2	2	no	—
168	#117	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	C₅×D₄	1	4	yes	—
175	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
176	C₂×C₂²⋊C₄	—	2	1	no	—
177	C₂×C₄²	—	2	1	no	—
178	#107	—	2	8	no	—
179	#106	—	2	8	no	—
180	#106	—	2	8	no	—
181	C₂²×C₅	—	2	4	no	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	—	2	2	no	—
185	C₂²×C₅	—	2	2	no	—
186	C₂²×C₅	#389	1	2	yes	—
187	C₂²×C₅	#389	1	1	yes	characteristic
188	C₂²×C₅	#392	1	1	yes	characteristic
189	C₂²×C₅	#398	1	2	yes	—
190	C₂²×C₅	#398	1	1	yes	characteristic
191	C₂²×C₅	#471	1	4	yes	—
192	C₄×C₅	—	8	2	no	—
193	C₄×C₅	—	8	4	no	—
194	C₄×C₅	#384	1	1	yes	characteristic
195	C₄×C₅	#385	1	1	yes	characteristic
196	C₄×C₅	#385	1	2	yes	—
197	C₄×C₅	#395	1	1	yes	characteristic
198	C₄×C₅	#395	1	2	yes	—
199	C₄×C₅	#399	1	1	yes	characteristic
200	C₂⁴	C₅×#41	1	1	yes	characteristic
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	8	4	no	—
206	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
209	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
211	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
213	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
214	C₂²×C₄	C₅×#41	1	2	yes	—
215	C₂²×C₄	C₅×#41	1	1	yes	characteristic
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	C₅×C₄⋊C₄	1	8	yes	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₄²	—	4	1	no	—
241	C₄²	—	4	1	no	—
242	C₂×C₅	—	2	2	no	—
243	C₂×C₅	—	2	2	no	—
244	C₂×C₅	#1444	1	1	yes	characteristic
245	C₂×C₅	#1951	1	2	yes	—
246	C₂×C₅	#2019	1	2	yes	—
247	C₂×C₅	#2090	1	2	yes	—
248	C₂³	—	2	4	no	—
249	C₂³	C₄×C₅×Q₈	1	1	yes	characteristic
250	C₂³	C₅×#121	1	2	yes	—
251	C₂³	C₅×#121	1	1	yes	characteristic
252	C₂³	C₅×#125	1	2	yes	—
253	C₂³	C₅×#125	1	1	yes	characteristic
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	—	8	4	no	—
259	C₂×C₄	—	8	4	no	—
260	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
261	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
262	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
263	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
264	C₂×C₄	C₅×#120	1	1	yes	characteristic
265	C₂×C₄	C₅×#120	1	2	yes	—
266	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
267	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
268	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
269	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
270	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
271	C₂×C₄	C₅×#123	1	1	yes	characteristic
272	C₂×C₄	C₅×#123	1	2	yes	—
273	C₂×C₄	C₅×#127	1	1	yes	characteristic
274	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
275	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
276	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
277	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
278	C₈	—	2	4	no	—
279	C₈	—	2	8	no	—
280	C₈	—	4	2	no	—
281	C₅	#11330	1	1	yes	characteristic, direct product factor, Sylow
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	C₅×#389	1	2	yes	—
288	C₂²	C₅×#389	1	1	yes	characteristic
289	C₂²	C₅×#392	1	1	yes	characteristic
290	C₂²	C₅×#398	1	2	yes	—
291	C₂²	C₅×#398	1	1	yes	characteristic
292	C₂²	C₅×#471	1	4	yes	—
293	C₄	—	8	2	no	—
294	C₄	—	8	4	no	—
295	C₄	C₅×#384	1	1	yes	characteristic
296	C₄	C₅×#385	1	1	yes	characteristic
297	C₄	C₅×#385	1	2	yes	—
298	C₄	C₅×#395	1	1	yes	characteristic
299	C₄	C₅×#395	1	2	yes	—
300	C₄	C₅×#399	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	2	2	no	—
303	C₂	C₅×#1444	1	1	yes	characteristic
304	C₂	C₅×#1951	1	2	yes	—
305	C₂	C₅×#2019	1	2	yes	—
306	C₂	C₅×#2090	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	#121
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
18. C₅×#341	C₄	semidirect
20. #11330	C₅	direct (N × H)
113. #341	C₄×C₅	semidirect
281. C₅	#11330	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×#428	2
C₂²×C₅×#106	2
C₂×C₅×#343	2
C₂×C₅×#341	2
C₅×#1729	2
#11330	5

Subgroups — C5×#11330 #11934050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11330 #11934050