Subgroups — C₅×#7904 #11930624

Order 1280.

838 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₅×#1279	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×M₁₆	—	2	1	no	—
6	C₂²×C₅×C₄⋊C₄	C₄	1	2	yes	—
7	C₂²×C₄²×C₅	—	2	1	no	—
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₅×#432	—	2	2	no	—
10	C₅×#432	C₄	1	4	yes	complemented
11	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
13	C₄×C₅×M₁₆	—	2	2	no	—
14	C₄×C₅×M₁₆	C₄	1	4	yes	complemented
15	C₂×C₅×#97	—	2	1	no	—
16	C₂×C₅×#97	—	2	1	no	—
17	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
18	C₄×C₅×C₄⋊C₄	—	2	1	no	—
19	C₄×C₅×C₄⋊C₄	—	2	1	no	—
20	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
21	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₅×#95	—	2	1	no	—
23	#7904	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₅	—	2	2	no	—
25	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂×C₅×M₁₆	—	2	8	no	—
29	C₂×C₅×M₁₆	—	2	4	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂×C₅×C₄⋊C₄	—	2	2	no	—
33	C₂×C₅×C₄⋊C₄	—	2	8	no	—
34	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₅	—	2	2	no	—
36	C₂×C₄²×C₅	—	2	2	no	—
37	C₂×C₄²×C₅	—	4	4	no	—
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	D₄	1	2	yes	—
44	C₂×C₄²×C₅	Q₈	1	2	yes	—
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₅×C₄⋊C₈	—	2	2	no	—
47	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
48	C₅×C₂²⋊C₈	—	2	4	no	—
49	C₅×#97	—	2	4	no	—
50	C₅×#97	—	2	4	no	—
51	C₄×C₅×C₈	—	2	2	no	—
52	C₄×C₅×C₈	C₂×C₄	1	4	yes	complemented
53	C₅×#95	—	2	2	no	—
54	C₅×#95	—	2	2	no	—
55	C₂×#432	C₂×C₅	1	1	yes	characteristic
56	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
57	C₂×C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
58	#1279	C₂×C₅	1	4	yes	complemented
59	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	—	2	4	no	—
62	C₂²×C₄×C₅	—	2	2	no	—
63	C₂²×C₄×C₅	—	2	2	no	—
64	C₂²×C₄×C₅	—	2	8	no	—
65	C₂²×C₄×C₅	—	2	4	no	—
66	C₂²×C₄×C₅	—	4	4	no	—
67	C₂²×C₄×C₅	—	4	4	no	—
68	C₂²×C₄×C₅	C₄²	1	2	yes	—
69	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₅×M₁₆	—	2	16	no	—
81	C₂×C₅×C₈	—	2	8	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	—	4	2	no	—
85	C₂×C₅×C₈	—	4	2	no	—
86	C₅×C₄⋊C₄	—	2	8	no	—
87	C₅×C₄⋊C₄	—	2	4	no	—
88	C₅×C₄⋊C₄	M₁₆	1	8	yes	complemented
89	C₄²×C₅	—	4	4	no	—
90	C₄²×C₅	—	4	4	no	—
91	C₄²×C₅	C₄²	1	2	yes	—
92	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
94	C₄²×C₅	C₄⋊C₄	1	2	yes	—
95	C₄²×C₅	C₄⋊C₄	1	2	yes	—
96	C₄²×C₅	C₄⋊C₄	1	2	yes	—
97	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
101	C₂²×M₁₆	—	2	1	no	—
102	C₂²×C₄⋊C₄	C₄×C₅	1	2	yes	—
103	C₂²×C₄²	—	2	1	no	—
104	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
105	#432	—	2	2	no	—
106	#432	C₄×C₅	1	4	yes	complemented
107	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
108	C₂×C₂²⋊C₈	—	2	1	no	—
109	C₄×M₁₆	—	2	2	no	—
110	C₄×M₁₆	C₄×C₅	1	4	yes	complemented
111	C₂×#97	—	2	1	no	—
112	C₂×#97	—	2	1	no	—
113	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
114	C₄×C₄⋊C₄	—	2	1	no	—
115	C₄×C₄⋊C₄	—	2	1	no	—
116	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
117	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
118	C₂×#95	—	2	1	no	—
119	C₂³×C₅	—	2	2	no	—
120	C₂³×C₅	—	2	1	no	—
121	C₂³×C₅	—	2	1	no	—
122	C₂³×C₅	#95	1	2	yes	—
123	C₂³×C₅	#104	1	4	yes	—
124	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	—	2	4	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	2	4	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	—	2	8	no	—
137	C₂×C₄×C₅	—	4	16	no	—
138	C₂×C₄×C₅	—	4	4	no	—
139	C₂×C₄×C₅	—	4	4	no	—
140	C₂×C₄×C₅	#95	1	2	yes	—
141	C₂×C₄×C₅	#95	1	2	yes	—
142	C₂×C₄×C₅	#95	1	2	yes	—
143	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
150	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
151	C₂×C₄×C₅	#137	1	2	yes	—
152	C₅×C₈	—	2	8	no	—
153	C₅×C₈	—	4	4	no	—
154	C₂³×C₄	—	2	2	no	—
155	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
157	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
158	C₂×M₁₆	—	2	8	no	—
159	C₂×M₁₆	—	2	4	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	8	no	—
164	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	—	4	4	no	—
168	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
169	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
170	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
171	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
172	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
173	C₂×C₄²	C₅×D₄	1	2	yes	—
174	C₂×C₄²	C₅×Q₈	1	2	yes	—
175	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	complemented
178	C₂²⋊C₈	—	2	4	no	—
179	#97	—	2	4	no	—
180	#97	—	2	4	no	—
181	C₄×C₈	—	2	2	no	—
182	C₄×C₈	C₂×C₄×C₅	1	4	yes	complemented
183	#95	—	2	2	no	—
184	#95	—	2	2	no	—
185	C₂²×C₅	—	2	2	no	—
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	2	4	no	—
188	C₂²×C₅	—	2	4	no	—
189	C₂²×C₅	#339	1	4	yes	—
190	C₂²×C₅	C₂×#95	1	1	yes	characteristic
191	C₂²×C₅	#407	1	2	yes	—
192	C₂²×C₅	#418	1	1	yes	characteristic
193	C₂²×C₅	#419	1	1	yes	characteristic
194	C₂²×C₅	C₂×#104	1	2	yes	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	2	2	no	—
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	4	8	no	—
201	C₄×C₅	C₄×M₁₆	1	2	yes	—
202	C₄×C₅	#417	1	2	yes	—
203	C₄×C₅	#423	1	2	yes	—
204	C₄×C₅	#428	1	2	yes	—
205	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	C₄²×C₅	1	2	yes	—
215	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
216	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
217	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
221	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
222	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
223	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
226	M₁₆	—	2	16	no	—
227	C₂×C₈	—	2	8	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	C₅×M₁₆	1	8	yes	complemented
235	C₄²	—	4	4	no	—
236	C₄²	—	4	4	no	—
237	C₄²	C₄²×C₅	1	2	yes	—
238	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
239	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
240	C₄²	C₅×C₄⋊C₄	1	2	yes	—
241	C₄²	C₅×C₄⋊C₄	1	2	yes	—
242	C₄²	C₅×C₄⋊C₄	1	2	yes	—
243	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
244	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
245	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
246	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
247	C₂×C₅	—	2	4	no	—
248	C₂×C₅	#1490	1	2	yes	—
249	C₂×C₅	#1491	1	2	yes	—
250	C₂×C₅	#1728	1	1	yes	characteristic
251	C₂×C₅	C₂×#339	1	1	yes	characteristic
252	C₂×C₅	#1736	1	1	yes	characteristic
253	C₂³	—	2	2	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	C₅×#95	1	2	yes	—
257	C₂³	C₅×#104	1	4	yes	—
258	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	4	16	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	C₅×#95	1	2	yes	—
275	C₂×C₄	C₅×#95	1	2	yes	—
276	C₂×C₄	C₅×#95	1	2	yes	—
277	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
278	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
282	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
283	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
284	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
285	C₂×C₄	C₅×#137	1	2	yes	—
286	C₈	—	2	8	no	—
287	C₈	—	4	4	no	—
288	C₅	#7904	1	1	yes	characteristic, direct product factor, Sylow
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	2	4	no	—
293	C₂²	C₅×#339	1	4	yes	—
294	C₂²	C₂×C₅×#95	1	1	yes	characteristic
295	C₂²	C₅×#407	1	2	yes	—
296	C₂²	C₅×#418	1	1	yes	characteristic
297	C₂²	C₅×#419	1	1	yes	characteristic
298	C₂²	C₂×C₅×#104	1	2	yes	—
299	C₄	—	2	2	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	8	no	—
305	C₄	C₄×C₅×M₁₆	1	2	yes	—
306	C₄	C₅×#417	1	2	yes	—
307	C₄	C₅×#423	1	2	yes	—
308	C₄	C₅×#428	1	2	yes	—
309	C₂	—	2	4	no	—
310	C₂	C₅×#1490	1	2	yes	—
311	C₂	C₅×#1491	1	2	yes	—
312	C₂	C₅×#1728	1	1	yes	characteristic
313	C₂	C₂×C₅×#339	1	1	yes	characteristic
314	C₂	C₅×#1736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1279	C₂	semidirect
10. C₅×#432	C₄	semidirect
14. C₄×C₅×M₁₆	C₄	semidirect
23. #7904	C₅	direct (N × H)
47. C₅×C₄⋊C₈	C₂×C₄	semidirect
52. C₄×C₅×C₈	C₂×C₄	semidirect
58. #1279	C₂×C₅	semidirect
88. C₅×C₄⋊C₄	M₁₆	semidirect
106. #432	C₄×C₅	semidirect
110. C₄×M₁₆	C₄×C₅	semidirect
177. C₄⋊C₈	C₂×C₄×C₅	semidirect
182. C₄×C₈	C₂×C₄×C₅	semidirect
234. C₄⋊C₄	C₅×M₁₆	semidirect
288. C₅	#7904	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×#432	2
C₂×C₄×C₅×M₁₆	2
C₂×C₄×C₅×C₄⋊C₄	2
C₅×#1279	2
#7904	5

Subgroups — C5×#7904 #11930624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7904 #11930624