Subgroups — C₅×#11345 #11934065

Order 1280.

698 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2905	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#424	C₂	1	2	yes	maximal
3	C₂×C₅×#344	C₂	1	2	yes	maximal
4	C₅×#1740	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	2	yes	—
10	C₂×C₅×#117	C₂²	1	1	yes	characteristic
11	C₅×#424	—	2	4	no	—
12	C₅×#424	—	2	4	no	—
13	C₅×#424	—	2	4	no	—
14	C₅×#423	—	2	2	no	—
15	C₂×C₅×C₄⋊C₈	—	2	4	no	—
16	C₅×#344	—	2	4	no	—
17	C₅×#342	C₄	1	4	yes	—
18	C₅×#336	—	2	2	no	—
19	#11345	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂×C₅×M₁₆	—	2	2	no	—
22	C₂×C₅×M₁₆	—	2	2	no	—
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	2	no	—
27	C₂×C₅×M₁₆	—	2	1	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	Q₈	1	2	yes	—
31	C₂²×C₅×C₈	—	2	1	no	—
32	C₂²×C₅×C₈	—	2	1	no	—
33	C₂²×C₅×C₈	—	2	2	no	—
34	C₂²×C₅×C₈	—	2	2	no	—
35	C₂²×C₅×C₈	—	2	1	no	—
36	C₂²×C₅×C₈	D₄	1	4	yes	—
37	C₂²×C₅×C₈	D₄	1	2	yes	—
38	C₅×#117	—	2	2	no	—
39	C₅×#117	—	2	2	no	—
40	C₅×#117	—	2	2	no	—
41	C₅×#117	—	2	2	no	—
42	C₅×#117	D₄	1	4	yes	—
43	C₅×#117	Q₈	1	4	yes	—
44	C₂×C₅×C₄⋊C₄	—	2	2	no	—
45	C₂×C₅×C₄⋊C₄	Q₈	1	2	yes	—
46	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₅	—	2	2	no	—
49	C₂×C₄²×C₅	D₄	1	2	yes	—
50	C₅×C₄⋊C₈	—	2	16	no	—
51	C₅×C₄⋊C₈	—	2	8	no	—
52	#2905	C₂×C₅	1	1	yes	characteristic
53	C₂×#424	C₂×C₅	1	2	yes	—
54	C₂×#344	C₂×C₅	1	2	yes	—
55	#1740	C₂×C₅	1	1	yes	characteristic
56	#1729	C₂×C₅	1	1	yes	characteristic
57	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	—	2	4	no	—
59	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₅	#41	1	1	yes	characteristic
68	C₂²×C₄×C₅	#41	1	2	yes	—
69	C₂²×C₄×C₅	#41	1	1	yes	characteristic
70	C₅×M₁₆	—	2	4	no	—
71	C₅×M₁₆	—	4	2	no	—
72	C₅×M₁₆	—	4	2	no	—
73	C₅×M₁₆	—	4	2	no	—
74	C₅×M₁₆	—	4	2	no	—
75	C₅×M₁₆	—	4	2	no	—
76	C₂×C₅×C₈	—	2	4	no	—
77	C₂×C₅×C₈	—	2	4	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	4	1	no	—
83	C₂×C₅×C₈	—	4	2	no	—
84	C₂×C₅×C₈	—	4	1	no	—
85	C₂×C₅×C₈	—	4	1	no	—
86	C₂×C₅×C₈	—	4	1	no	—
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₄⋊C₄	—	2	4	no	—
89	C₅×C₄⋊C₄	—	2	4	no	—
90	C₅×C₂²⋊C₄	—	2	4	no	—
91	C₅×C₂²⋊C₄	—	2	4	no	—
92	C₅×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
93	C₄²×C₅	—	2	4	no	—
94	C₄²×C₅	—	2	2	no	—
95	C₄²×C₅	—	2	4	no	—
96	C₄²×C₅	—	2	2	no	—
97	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
98	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
99	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×#117	C₂²×C₅	1	2	yes	—
101	C₂×#117	C₂²×C₅	1	1	yes	characteristic
102	#424	—	2	4	no	—
103	#424	—	2	4	no	—
104	#424	—	2	4	no	—
105	#423	—	2	2	no	—
106	C₂×C₄⋊C₈	—	2	4	no	—
107	#344	—	2	4	no	—
108	#342	C₄×C₅	1	4	yes	—
109	#336	—	2	2	no	—
110	C₂³×C₅	—	4	2	no	—
111	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³×C₅	C₄×D₄	1	2	yes	—
114	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
115	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	2	8	no	—
118	C₂×C₄×C₅	—	4	8	no	—
119	C₂×C₄×C₅	—	4	4	no	—
120	C₂×C₄×C₅	#120	1	2	yes	—
121	C₂×C₄×C₅	#120	1	1	yes	characteristic
122	C₂×C₄×C₅	#121	1	2	yes	—
123	C₂×C₄×C₅	#121	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₅	#123	1	1	yes	characteristic
130	C₂×C₄×C₅	#123	1	2	yes	—
131	C₂×C₄×C₅	#125	1	1	yes	characteristic
132	C₂×C₄×C₅	#125	1	2	yes	—
133	C₂×C₄×C₅	#127	1	1	yes	characteristic
134	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
136	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
137	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
138	C₅×C₈	—	4	2	no	—
139	C₅×C₈	—	4	4	no	—
140	C₅×C₈	—	4	2	no	—
141	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	—	2	2	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	—	2	1	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	C₅×Q₈	1	2	yes	—
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	C₅×D₄	1	2	yes	—
158	C₂²×C₈	C₅×D₄	1	4	yes	—
159	#117	—	2	2	no	—
160	#117	—	2	2	no	—
161	#117	—	2	2	no	—
162	#117	—	2	2	no	—
163	#117	C₅×D₄	1	4	yes	—
164	#117	C₅×Q₈	1	4	yes	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	C₅×Q₈	1	2	yes	—
167	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
168	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
169	C₂×C₄²	—	2	2	no	—
170	C₂×C₄²	C₅×D₄	1	2	yes	—
171	C₄⋊C₈	—	2	16	no	—
172	C₄⋊C₈	—	2	8	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	2	4	no	—
175	C₂²×C₅	—	4	2	no	—
176	C₂²×C₅	—	4	1	no	—
177	C₂²×C₅	—	4	1	no	—
178	C₂²×C₅	#384	1	1	yes	characteristic
179	C₂²×C₅	#385	1	2	yes	—
180	C₂²×C₅	#385	1	1	yes	characteristic
181	C₂²×C₅	#389	1	2	yes	—
182	C₂²×C₅	#389	1	1	yes	characteristic
183	C₄×C₅	—	4	8	no	—
184	C₄×C₅	—	4	4	no	—
185	C₄×C₅	#392	1	1	yes	characteristic
186	C₄×C₅	#395	1	2	yes	—
187	C₄×C₅	#395	1	1	yes	characteristic
188	C₄×C₅	#398	1	2	yes	—
189	C₄×C₅	#398	1	1	yes	characteristic
190	C₄×C₅	#399	1	1	yes	characteristic
191	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
194	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
195	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
198	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
201	C₂²×C₄	C₅×#41	1	1	yes	characteristic
202	C₂²×C₄	C₅×#41	1	2	yes	—
203	C₂²×C₄	C₅×#41	1	1	yes	characteristic
204	M₁₆	—	2	4	no	—
205	M₁₆	—	4	2	no	—
206	M₁₆	—	4	2	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	C₅×C₄⋊C₄	1	8	yes	—
227	C₄²	—	2	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	4	no	—
230	C₄²	—	2	2	no	—
231	C₂×C₅	—	2	2	no	—
232	C₂×C₅	—	2	1	no	—
233	C₂×C₅	—	2	1	no	—
234	C₂×C₅	—	2	2	no	—
235	C₂×C₅	#1444	1	1	yes	characteristic
236	C₂×C₅	#1868	1	2	yes	—
237	C₂³	—	4	2	no	—
238	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
239	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
240	C₂³	C₄×C₅×D₄	1	2	yes	—
241	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
242	C₂³	C₄×C₅×Q₈	1	1	yes	characteristic
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	4	8	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	C₅×#120	1	1	yes	characteristic
248	C₂×C₄	C₅×#120	1	2	yes	—
249	C₂×C₄	C₅×#121	1	2	yes	—
250	C₂×C₄	C₅×#121	1	1	yes	characteristic
251	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
252	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
253	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
254	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
255	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
256	C₂×C₄	C₅×#123	1	1	yes	characteristic
257	C₂×C₄	C₅×#123	1	2	yes	—
258	C₂×C₄	C₅×#125	1	1	yes	characteristic
259	C₂×C₄	C₅×#125	1	2	yes	—
260	C₂×C₄	C₅×#127	1	1	yes	characteristic
261	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
262	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
263	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
264	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
265	C₈	—	4	2	no	—
266	C₈	—	4	4	no	—
267	C₈	—	4	2	no	—
268	C₅	#11345	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	C₅×#384	1	1	yes	characteristic
275	C₂²	C₅×#385	1	2	yes	—
276	C₂²	C₅×#385	1	1	yes	characteristic
277	C₂²	C₅×#389	1	2	yes	—
278	C₂²	C₅×#389	1	1	yes	characteristic
279	C₄	—	4	8	no	—
280	C₄	—	4	4	no	—
281	C₄	C₅×#392	1	1	yes	characteristic
282	C₄	C₅×#395	1	1	yes	characteristic
283	C₄	C₅×#395	1	2	yes	—
284	C₄	C₅×#398	1	2	yes	—
285	C₄	C₅×#398	1	1	yes	characteristic
286	C₄	C₅×#399	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	C₅×#1444	1	1	yes	characteristic
292	C₂	C₅×#1868	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	#120
commutator	C₂⁴	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂²×C₅	#385
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ 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
19. #11345	C₅	direct (N × H)
268. C₅	#11345	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×#2905	2
C₂×C₅×#424	2
C₂×C₅×#344	2
C₅×#1740	2
C₅×#1729	2
#11345	5

Subgroups — C5×#11345 #11934065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11345 #11934065