Subgroups — C₅×#11364 #11934084

Order 1280.

606 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic
2	C₅×#1740	C₂	1	2	yes	maximal
3	C₅×#1735	C₂	1	1	yes	maximal, characteristic
4	C₅×#1733	C₂	1	2	yes	maximal
5	C₅×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#117	C₂²	1	2	yes	—
9	C₂×C₅×#117	C₂²	1	2	yes	—
10	C₂×C₅×#117	C₂²	1	1	yes	characteristic
11	C₅×#427	—	2	2	no	—
12	C₅×#427	—	2	2	no	—
13	C₂×C₅×#107	C₄	1	2	yes	complemented
14	C₂×C₅×#106	C₄	1	2	yes	complemented
15	C₅×#342	—	2	4	no	—
16	C₅×#340	—	2	2	no	—
17	C₅×#339	—	2	4	no	—
18	C₅×#336	—	2	2	no	—
19	#11364	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂×C₅×M₁₆	—	2	1	no	—
22	C₂×C₅×M₁₆	—	2	2	no	—
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	D₄	1	2	yes	—
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂²×C₅×C₈	—	2	2	no	—
27	C₂²×C₅×C₈	—	2	1	no	—
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	D₄	1	2	yes	—
31	C₂²×C₅×C₈	D₄	1	2	yes	—
32	C₅×#117	—	2	2	no	—
33	C₅×#117	—	2	2	no	—
34	C₅×#117	—	2	2	no	—
35	C₅×#117	—	2	2	no	—
36	C₅×#117	—	2	2	no	—
37	C₅×#117	—	2	2	no	—
38	C₅×#117	—	2	2	no	—
39	C₅×#117	—	2	2	no	—
40	C₅×#117	—	2	2	no	—
41	C₅×#117	Q₈	1	4	yes	—
42	C₂×C₅×C₄⋊C₄	—	2	2	no	—
43	C₂×C₅×C₄⋊C₄	—	2	1	no	—
44	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
45	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
46	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
47	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
48	C₂×C₄²×C₅	—	2	2	no	—
49	C₂×C₄²×C₅	—	2	1	no	—
50	C₂×C₄²×C₅	—	2	2	no	—
51	C₅×#107	—	2	4	no	—
52	C₅×#106	—	2	4	no	—
53	C₂×#427	C₂×C₅	1	1	yes	characteristic
54	#1740	C₂×C₅	1	2	yes	—
55	#1735	C₂×C₅	1	1	yes	characteristic
56	#1733	C₂×C₅	1	2	yes	—
57	#1729	C₂×C₅	1	1	yes	characteristic
58	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	—	2	2	no	—
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	—	4	2	no	—
62	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₅	#41	1	2	yes	—
68	C₂²×C₄×C₅	#41	1	2	yes	—
69	C₂²×C₄×C₅	#41	1	1	yes	characteristic
70	C₂²×C₄×C₅	#41	1	2	yes	—
71	C₂²×C₄×C₅	#41	1	1	yes	characteristic
72	C₅×M₁₆	—	2	4	no	—
73	C₅×M₁₆	—	4	2	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	4	1	no	—
81	C₂×C₅×C₈	—	4	1	no	—
82	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
83	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
84	C₅×C₄⋊C₄	—	2	4	no	—
85	C₅×C₄⋊C₄	—	2	4	no	—
86	C₅×C₄⋊C₄	—	2	4	no	—
87	C₅×C₄⋊C₄	—	4	2	no	—
88	C₅×C₄⋊C₄	—	4	2	no	—
89	C₅×C₂²⋊C₄	—	2	4	no	—
90	C₅×C₂²⋊C₄	—	2	4	no	—
91	C₅×C₂²⋊C₄	—	4	2	no	—
92	C₅×C₂²⋊C₄	—	4	2	no	—
93	C₅×C₂²⋊C₄	—	4	2	no	—
94	C₄²×C₅	—	2	4	no	—
95	C₄²×C₅	—	4	2	no	—
96	C₄²×C₅	—	4	1	no	—
97	C₄²×C₅	—	4	2	no	—
98	C₄²×C₅	—	4	2	no	—
99	C₄²×C₅	—	4	1	no	—
100	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
101	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
102	C₂×#117	C₂²×C₅	1	2	yes	—
103	C₂×#117	C₂²×C₅	1	2	yes	—
104	C₂×#117	C₂²×C₅	1	1	yes	characteristic
105	#427	—	2	2	no	—
106	#427	—	2	2	no	—
107	C₂×#107	C₄×C₅	1	2	yes	complemented
108	C₂×#106	C₄×C₅	1	2	yes	complemented
109	#342	—	2	4	no	—
110	#340	—	2	2	no	—
111	#339	—	2	4	no	—
112	#336	—	2	2	no	—
113	C₂³×C₅	—	4	2	no	—
114	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
115	C₂³×C₅	#121	1	1	yes	characteristic
116	C₂³×C₅	C₂²⋊Q₈	1	2	yes	—
117	C₂³×C₅	#123	1	2	yes	—
118	C₂³×C₅	#127	1	1	yes	characteristic
119	C₂×C₄×C₅	—	4	4	no	—
120	C₂×C₄×C₅	—	4	4	no	—
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	—	4	4	no	—
123	C₂×C₄×C₅	—	8	4	no	—
124	C₂×C₄×C₅	—	8	4	no	—
125	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	#117	1	2	yes	—
127	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₅	#121	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₅	#123	1	1	yes	characteristic
133	C₂×C₄×C₅	#123	1	1	yes	characteristic
134	C₂×C₄×C₅	#123	1	2	yes	—
135	C₂×C₄×C₅	#123	1	2	yes	—
136	C₂×C₄×C₅	#124	1	1	yes	characteristic
137	C₂×C₄×C₅	#124	1	1	yes	characteristic
138	C₂×C₄×C₅	#125	1	2	yes	—
139	C₂×C₄×C₅	#125	1	2	yes	—
140	C₂×C₄×C₅	#126	1	2	yes	—
141	C₂×C₄×C₅	#126	1	2	yes	—
142	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
143	C₅×C₈	—	2	2	no	—
144	C₅×C₈	—	2	2	no	—
145	C₅×C₈	—	4	2	no	—
146	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
147	C₂×M₁₆	—	2	1	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	C₅×D₄	1	2	yes	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
155	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂²×C₈	C₅×D₄	1	2	yes	—
157	C₂²×C₈	C₅×D₄	1	2	yes	—
158	#117	—	2	2	no	—
159	#117	—	2	2	no	—
160	#117	—	2	2	no	—
161	#117	—	2	2	no	—
162	#117	—	2	2	no	—
163	#117	—	2	2	no	—
164	#117	—	2	2	no	—
165	#117	—	2	2	no	—
166	#117	—	2	2	no	—
167	#117	C₅×Q₈	1	4	yes	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	—	2	2	no	—
177	#107	—	2	4	no	—
178	#106	—	2	4	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	2	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	4	2	no	—
183	C₂²×C₅	—	4	2	no	—
184	C₂²×C₅	#387	1	2	yes	—
185	C₂²×C₅	#390	1	1	yes	characteristic
186	C₂²×C₅	#397	1	1	yes	characteristic
187	C₂²×C₅	#398	1	1	yes	characteristic
188	C₂²×C₅	#399	1	2	yes	—
189	C₄×C₅	—	8	2	no	—
190	C₄×C₅	—	8	4	no	—
191	C₄×C₅	—	8	4	no	—
192	C₄×C₅	#381	1	1	yes	characteristic
193	C₄×C₅	#387	1	2	yes	—
194	C₄×C₅	#391	1	1	yes	characteristic
195	C₄×C₅	#397	1	1	yes	characteristic
196	C₄×C₅	#398	1	1	yes	characteristic
197	C₄×C₅	#400	1	2	yes	—
198	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
207	C₂²×C₄	C₅×#41	1	2	yes	—
208	C₂²×C₄	C₅×#41	1	2	yes	—
209	C₂²×C₄	C₅×#41	1	1	yes	characteristic
210	C₂²×C₄	C₅×#41	1	2	yes	—
211	C₂²×C₄	C₅×#41	1	1	yes	characteristic
212	M₁₆	—	2	4	no	—
213	M₁₆	—	4	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
223	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	2	4	no	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₂²⋊C₄	—	4	2	no	—
233	C₂²⋊C₄	—	4	2	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	4	2	no	—
236	C₄²	—	4	1	no	—
237	C₄²	—	4	2	no	—
238	C₄²	—	4	2	no	—
239	C₄²	—	4	1	no	—
240	C₂×C₅	—	2	1	no	—
241	C₂×C₅	—	2	2	no	—
242	C₂×C₅	—	2	2	no	—
243	C₂×C₅	—	2	1	no	—
244	C₂×C₅	#1445	1	1	yes	characteristic
245	C₂×C₅	#1976	1	2	yes	—
246	C₂³	—	4	2	no	—
247	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
248	C₂³	C₅×#121	1	1	yes	characteristic
249	C₂³	C₅×C₂²⋊Q₈	1	2	yes	—
250	C₂³	C₅×#123	1	2	yes	—
251	C₂³	C₅×#127	1	1	yes	characteristic
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₅×#117	1	2	yes	—
260	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
261	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
262	C₂×C₄	C₅×#121	1	1	yes	characteristic
263	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
264	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
265	C₂×C₄	C₅×#123	1	1	yes	characteristic
266	C₂×C₄	C₅×#123	1	1	yes	characteristic
267	C₂×C₄	C₅×#123	1	2	yes	—
268	C₂×C₄	C₅×#123	1	2	yes	—
269	C₂×C₄	C₅×#124	1	1	yes	characteristic
270	C₂×C₄	C₅×#124	1	1	yes	characteristic
271	C₂×C₄	C₅×#125	1	2	yes	—
272	C₂×C₄	C₅×#125	1	2	yes	—
273	C₂×C₄	C₅×#126	1	2	yes	—
274	C₂×C₄	C₅×#126	1	2	yes	—
275	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	2	2	no	—
278	C₈	—	4	2	no	—
279	C₅	#11364	1	1	yes	characteristic, direct product factor, Sylow
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₅×#387	1	2	yes	—
286	C₂²	C₅×#390	1	1	yes	characteristic
287	C₂²	C₅×#397	1	1	yes	characteristic
288	C₂²	C₅×#398	1	1	yes	characteristic
289	C₂²	C₅×#399	1	2	yes	—
290	C₄	—	8	2	no	—
291	C₄	—	8	4	no	—
292	C₄	—	8	4	no	—
293	C₄	C₅×#381	1	1	yes	characteristic
294	C₄	C₅×#387	1	2	yes	—
295	C₄	C₅×#391	1	1	yes	characteristic
296	C₄	C₅×#397	1	1	yes	characteristic
297	C₄	C₅×#398	1	1	yes	characteristic
298	C₄	C₅×#400	1	2	yes	—
299	C₂	—	2	1	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	2	no	—
302	C₂	—	2	1	no	—
303	C₂	C₅×#1445	1	1	yes	characteristic
304	C₂	C₅×#1976	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	#121
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂²×C₅	#390
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×C₅×#107	C₄	semidirect
14. C₂×C₅×#106	C₄	semidirect
19. #11364	C₅	direct (N × H)
107. C₂×#107	C₄×C₅	semidirect
108. C₂×#106	C₄×C₅	semidirect
279. C₅	#11364	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×#427	2
C₅×#1740	2
C₅×#1735	2
C₅×#1733	2
C₅×#1729	2
#11364	5

Subgroups — C5×#11364 #11934084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11364 #11934084