Subgroups — C₅×#11393 #11934113

Order 1280.

850 conjugacy classes of subgroups.

The subgroup lattice has 318 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×#1740	C₂	1	1	yes	maximal, characteristic
3	C₅×#1735	C₂	1	2	yes	maximal
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	C₅×#544	—	2	4	no	—
9	C₅×#544	C₄	1	4	yes	complemented
10	C₅×#543	—	2	2	no	—
11	C₅×#533	—	2	4	no	—
12	C₅×#533	—	2	2	no	—
13	C₂×C₅×#117	C₂²	1	2	yes	—
14	C₂×C₅×#117	C₂²	1	2	yes	—
15	C₂×C₅×#117	C₂²	1	1	yes	characteristic
16	C₅×#342	—	2	2	no	—
17	C₅×#340	—	2	4	no	—
18	C₅×#339	—	2	4	no	—
19	C₅×#336	—	2	2	no	—
20	#11393	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂×C₅×M₁₆	—	2	1	no	—
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	Q₈	1	2	yes	complemented
26	C₂²×C₅×C₈	—	2	1	no	—
27	C₂²×C₅×C₈	—	2	1	no	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₂²×C₅×C₈	—	2	1	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	Q₈	1	2	yes	complemented
32	C₅×C₄⋊Q₈	—	2	4	no	—
33	C₅×C₄⋊Q₈	—	2	8	no	—
34	C₅×#125	—	2	4	no	—
35	C₅×#125	—	2	8	no	—
36	C₅×C₂²⋊Q₈	—	2	16	no	—
37	C₅×C₂²⋊Q₈	—	2	8	no	—
38	C₄×C₅×Q₈	—	2	8	no	—
39	C₅×#117	—	2	2	no	—
40	C₅×#117	—	2	2	no	—
41	C₅×#117	—	2	2	no	—
42	C₅×#117	—	2	2	no	—
43	C₅×#117	—	2	2	no	—
44	C₅×#117	—	2	2	no	—
45	C₅×#117	—	2	2	no	—
46	C₅×#117	C₂×C₄	1	4	yes	—
47	C₅×#117	D₄	1	4	yes	—
48	C₅×#117	Q₈	1	4	yes	complemented
49	C₂×C₅×C₄⋊C₄	—	2	2	no	—
50	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
51	C₂×C₅×C₄⋊C₄	D₄	1	4	yes	—
52	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
53	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
54	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
55	C₂×C₄²×C₅	—	2	2	no	—
56	C₂×C₄²×C₅	—	2	2	no	—
57	C₂×C₄²×C₅	D₄	1	2	yes	—
58	#3458	C₂×C₅	1	1	yes	characteristic
59	#1740	C₂×C₅	1	1	yes	characteristic
60	#1735	C₂×C₅	1	2	yes	—
61	#1733	C₂×C₅	1	2	yes	—
62	#1729	C₂×C₅	1	1	yes	characteristic
63	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₅×Q₈	—	2	8	no	—
65	C₂²×C₄×C₅	—	2	4	no	—
66	C₂²×C₄×C₅	—	4	2	no	—
67	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₅	#41	1	1	yes	characteristic
76	C₂²×C₄×C₅	#41	1	1	yes	characteristic
77	C₂²×C₄×C₅	#41	1	2	yes	—
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	—	4	2	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	4	1	no	—
84	C₂×C₅×C₈	—	4	1	no	—
85	C₂×C₅×C₈	—	4	2	no	—
86	C₂×C₅×C₈	—	4	1	no	—
87	C₂×C₅×C₈	—	4	1	no	—
88	C₅×C₄⋊C₄	—	2	16	no	—
89	C₅×C₄⋊C₄	—	2	8	no	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₄⋊C₄	—	2	4	no	—
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₅×C₄⋊C₄	—	4	2	no	—
94	C₅×C₄⋊C₄	C₂²⋊C₄	1	8	yes	—
95	C₅×C₂²⋊C₄	—	2	4	no	—
96	C₅×C₂²⋊C₄	—	2	4	no	—
97	C₅×C₂²⋊C₄	—	2	4	no	—
98	C₅×C₂²⋊C₄	—	2	4	no	—
99	C₅×C₂²⋊C₄	—	4	2	no	—
100	C₄²×C₅	—	2	4	no	—
101	C₄²×C₅	—	2	2	no	—
102	C₄²×C₅	—	2	4	no	—
103	C₄²×C₅	—	2	2	no	—
104	C₄²×C₅	—	2	4	no	—
105	C₄²×C₅	—	4	2	no	—
106	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
107	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
108	#544	—	2	4	no	—
109	#544	C₄×C₅	1	4	yes	complemented
110	#543	—	2	2	no	—
111	#533	—	2	4	no	—
112	#533	—	2	2	no	—
113	C₂×#117	C₂²×C₅	1	2	yes	—
114	C₂×#117	C₂²×C₅	1	2	yes	—
115	C₂×#117	C₂²×C₅	1	1	yes	characteristic
116	#342	—	2	2	no	—
117	#340	—	2	4	no	—
118	#339	—	2	4	no	—
119	#336	—	2	2	no	—
120	C₂³×C₅	—	4	2	no	—
121	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂³×C₅	#120	1	2	yes	—
123	C₂³×C₅	C₂²⋊Q₈	1	2	yes	—
124	C₂³×C₅	#124	1	1	yes	characteristic
125	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
126	C₅×Q₈	—	8	8	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	8	no	—
129	C₂×C₄×C₅	—	4	4	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	8	4	no	—
132	C₂×C₄×C₅	—	8	4	no	—
133	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
137	C₂×C₄×C₅	#120	1	1	yes	characteristic
138	C₂×C₄×C₅	#120	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
140	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
141	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
142	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
145	C₂×C₄×C₅	#124	1	2	yes	—
146	C₂×C₄×C₅	#124	1	2	yes	—
147	C₂×C₄×C₅	#124	1	1	yes	characteristic
148	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
149	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
150	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
151	C₅×C₈	—	4	2	no	—
152	C₅×C₈	—	4	2	no	—
153	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
154	C₂×M₁₆	—	2	1	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	C₅×Q₈	1	2	yes	complemented
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	C₅×Q₈	1	2	yes	complemented
164	C₄⋊Q₈	—	2	4	no	—
165	C₄⋊Q₈	—	2	8	no	—
166	#125	—	2	4	no	—
167	#125	—	2	8	no	—
168	C₂²⋊Q₈	—	2	16	no	—
169	C₂²⋊Q₈	—	2	8	no	—
170	C₄×Q₈	—	2	8	no	—
171	#117	—	2	2	no	—
172	#117	—	2	2	no	—
173	#117	—	2	2	no	—
174	#117	—	2	2	no	—
175	#117	—	2	2	no	—
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	#117	C₂×C₄×C₅	1	4	yes	—
179	#117	C₅×D₄	1	4	yes	—
180	#117	C₅×Q₈	1	4	yes	complemented
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
183	C₂×C₄⋊C₄	C₅×D₄	1	4	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	C₅×D₄	1	2	yes	—
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	4	no	—
192	C₂²×C₅	—	4	2	no	—
193	C₂²×C₅	—	4	1	no	—
194	C₂²×C₅	—	4	1	no	—
195	C₂²×C₅	#379	1	1	yes	characteristic
196	C₂²×C₅	#391	1	1	yes	characteristic
197	C₂²×C₅	#391	1	1	yes	characteristic
198	C₂²×C₅	#393	1	2	yes	—
199	C₂²×C₅	#395	1	2	yes	—
200	C₄×C₅	—	4	4	no	—
201	C₄×C₅	—	4	8	no	—
202	C₄×C₅	—	8	4	no	—
203	C₄×C₅	#391	1	2	yes	—
204	C₄×C₅	#391	1	2	yes	—
205	C₄×C₅	#393	1	1	yes	characteristic
206	C₄×C₅	#393	1	1	yes	characteristic
207	C₄×C₅	#395	1	1	yes	characteristic
208	C₄×C₅	#395	1	1	yes	characteristic
209	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
210	C₂×Q₈	—	2	8	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
214	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
216	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
217	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
221	C₂²×C₄	C₅×#41	1	1	yes	characteristic
222	C₂²×C₄	C₅×#41	1	1	yes	characteristic
223	C₂²×C₄	C₅×#41	1	2	yes	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₄⋊C₄	—	2	16	no	—
235	C₄⋊C₄	—	2	8	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	C₅×C₂²⋊C₄	1	8	yes	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	—	2	4	no	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₂²⋊C₄	—	2	4	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₄²	—	2	4	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	4	2	no	—
252	C₂×C₅	—	2	2	no	—
253	C₂×C₅	—	2	1	no	—
254	C₂×C₅	—	2	1	no	—
255	C₂×C₅	—	2	2	no	—
256	C₂×C₅	#1446	1	1	yes	characteristic
257	C₂×C₅	#1794	1	2	yes	—
258	C₂³	—	4	2	no	—
259	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
260	C₂³	C₅×#120	1	2	yes	—
261	C₂³	C₅×C₂²⋊Q₈	1	2	yes	—
262	C₂³	C₅×#124	1	1	yes	characteristic
263	C₂³	C₅×C₄⋊Q₈	1	1	yes	characteristic
264	Q₈	—	8	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	—	8	4	no	—
271	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
272	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
273	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
274	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
275	C₂×C₄	C₅×#120	1	1	yes	characteristic
276	C₂×C₄	C₅×#120	1	1	yes	characteristic
277	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
278	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
279	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
280	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
281	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
282	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
283	C₂×C₄	C₅×#124	1	2	yes	—
284	C₂×C₄	C₅×#124	1	2	yes	—
285	C₂×C₄	C₅×#124	1	1	yes	characteristic
286	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
287	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
288	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
289	C₈	—	4	2	no	—
290	C₈	—	4	2	no	—
291	C₅	#11393	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₅×#379	1	1	yes	characteristic
298	C₂²	C₅×#391	1	1	yes	characteristic
299	C₂²	C₅×#391	1	1	yes	characteristic
300	C₂²	C₅×#393	1	2	yes	—
301	C₂²	C₅×#395	1	2	yes	—
302	C₄	—	4	4	no	—
303	C₄	—	4	8	no	—
304	C₄	—	8	4	no	—
305	C₄	C₅×#391	1	2	yes	—
306	C₄	C₅×#391	1	2	yes	—
307	C₄	C₅×#393	1	1	yes	characteristic
308	C₄	C₅×#393	1	1	yes	characteristic
309	C₄	C₅×#395	1	1	yes	characteristic
310	C₄	C₅×#395	1	1	yes	characteristic
311	C₂	—	2	2	no	—
312	C₂	—	2	1	no	—
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	C₅×#1446	1	1	yes	characteristic
316	C₂	C₅×#1794	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	#120
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂²×C₅	#379
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
9. C₅×#544	C₄	semidirect
20. #11393	C₅	direct (N × H)
25. C₂×C₅×M₁₆	Q₈	semidirect
31. C₂²×C₅×C₈	Q₈	semidirect
48. C₅×#117	Q₈	semidirect
109. #544	C₄×C₅	semidirect
157. C₂×M₁₆	C₅×Q₈	semidirect
163. C₂²×C₈	C₅×Q₈	semidirect
180. #117	C₅×Q₈	semidirect
291. C₅	#11393	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×#1740	2
C₅×#1735	2
C₅×#1733	2
C₅×#1729	2
#11393	5

Subgroups — C5×#11393 #11934113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11393 #11934113