Subgroups — C₅×#17834 #11940554

Order 1280.

616 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2990	C₂	1	2	yes	maximal
2	C₅×C₈×#41	C₂	1	1	yes	maximal, characteristic
3	C₅×#1574	C₂	1	2	yes	maximal, complemented
4	C₅×#1573	C₂	1	4	yes	maximal, complemented
5	C₅×#1568	C₂	1	2	yes	maximal, complemented
6	C₅×#1566	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#334	C₂	1	2	yes	maximal, complemented
9	C₄×C₅×#41	C₂²	1	1	yes	characteristic
10	C₅×#446	—	2	2	no	—
11	C₅×C₈×Q₈	C₄	1	1	yes	characteristic
12	C₅×C₈×Q₈	C₄	1	1	yes	characteristic
13	C₅×#435	—	2	4	no	—
14	C₅×C₈×D₄	—	2	2	no	—
15	C₅×C₈×D₄	C₄	1	2	yes	—
16	C₅×C₈×D₄	C₄	1	1	yes	characteristic
17	C₅×C₈×D₄	C₄	1	2	yes	—
18	C₅×C₈×D₄	C₄	1	1	yes	characteristic
19	C₅×#432	C₂²	1	4	yes	—
20	C₅×#431	C₂²	1	2	yes	—
21	C₅×#431	C₂²	1	2	yes	—
22	C₅×#431	C₂²	1	1	yes	characteristic
23	C₅×#423	C₂²	1	2	yes	—
24	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
25	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
26	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
27	C₅×#335	C₂²	1	2	yes	—
28	C₅×#335	C₂²	1	4	yes	—
29	C₅×#334	C₂²	1	1	yes	characteristic
30	C₅×#334	C₂²	1	2	yes	—
31	C₅×#334	C₂²	1	4	yes	—
32	C₅×#334	C₂²	1	1	yes	characteristic
33	C₅×#334	C₂²	1	2	yes	—
34	#17834	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂×C₅×#41	Q₈	1	1	yes	characteristic
36	C₂×C₅×M₁₆	—	2	2	no	—
37	C₂²×C₅×C₈	—	2	4	no	—
38	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
39	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
41	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
44	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
45	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
46	C₄×C₅×D₄	D₄	1	1	yes	characteristic
47	C₄×C₅×D₄	D₄	1	2	yes	—
48	C₅×#117	Q₈	1	1	yes	characteristic
49	C₅×#117	Q₈	1	2	yes	—
50	C₂×C₄²×C₅	C₂³	1	2	yes	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₅×C₄⋊C₈	—	2	4	no	—
53	C₅×C₄⋊C₈	—	2	2	no	—
54	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
55	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
56	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₅×C₄⋊C₈	C₂³	1	2	yes	—
59	C₅×C₄⋊C₈	C₂³	1	4	yes	—
60	C₅×C₄⋊C₈	C₂³	1	2	yes	—
61	C₅×C₂²⋊C₈	—	2	2	no	—
62	C₅×C₂²⋊C₈	—	2	4	no	—
63	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
64	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
65	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
66	C₅×#97	—	2	4	no	—
67	C₄×C₅×C₈	—	2	2	no	—
68	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
69	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
71	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₄×C₅×C₈	C₂³	1	2	yes	—
74	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
75	#2990	C₂×C₅	1	2	yes	—
76	C₈×#41	C₂×C₅	1	1	yes	characteristic
77	#1574	C₂×C₅	1	2	yes	complemented
78	#1573	C₂×C₅	1	4	yes	complemented
79	#1568	C₂×C₅	1	2	yes	complemented
80	#1566	C₂×C₅	1	1	yes	characteristic, complemented
81	#1564	C₂×C₅	1	1	yes	characteristic, complemented
82	C₂×#334	C₂×C₅	1	2	yes	complemented
83	C₅×#41	—	2	4	no	—
84	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
86	C₂×C₅×D₄	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
92	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
93	C₅×M₁₆	—	8	2	no	—
94	C₂×C₅×C₈	—	2	4	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	4	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	4	2	no	—
100	C₂×C₅×C₈	—	4	1	no	—
101	C₂×C₅×C₈	—	8	4	no	—
102	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
109	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
110	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
111	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
112	C₅×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
113	C₅×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
114	C₅×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
115	C₄²×C₅	C₂²×C₄	1	2	yes	—
116	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₅	C₂×D₄	1	2	yes	—
119	C₄²×C₅	C₂×Q₈	1	2	yes	—
120	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
121	C₄²×C₅	C₂⁴	1	1	yes	characteristic
122	C₄×#41	C₂²×C₅	1	1	yes	characteristic
123	#446	—	2	2	no	—
124	C₈×Q₈	C₄×C₅	1	1	yes	characteristic
125	C₈×Q₈	C₄×C₅	1	1	yes	characteristic
126	#435	—	2	4	no	—
127	C₈×D₄	—	2	2	no	—
128	C₈×D₄	C₄×C₅	1	2	yes	—
129	C₈×D₄	C₄×C₅	1	1	yes	characteristic
130	C₈×D₄	C₄×C₅	1	2	yes	—
131	C₈×D₄	C₄×C₅	1	1	yes	characteristic
132	#432	C₂²×C₅	1	4	yes	—
133	#431	C₂²×C₅	1	2	yes	—
134	#431	C₂²×C₅	1	2	yes	—
135	#431	C₂²×C₅	1	1	yes	characteristic
136	#423	C₂²×C₅	1	2	yes	—
137	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
138	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
139	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
140	#335	C₂²×C₅	1	2	yes	—
141	#335	C₂²×C₅	1	4	yes	—
142	#334	C₂²×C₅	1	1	yes	characteristic
143	#334	C₂²×C₅	1	2	yes	—
144	#334	C₂²×C₅	1	4	yes	—
145	#334	C₂²×C₅	1	1	yes	characteristic
146	#334	C₂²×C₅	1	2	yes	—
147	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
149	C₅×Q₈	—	2	2	no	—
150	C₅×D₄	—	2	4	no	—
151	C₅×D₄	#108	1	4	yes	—
152	C₂×C₄×C₅	—	2	4	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	1	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	1	no	—
157	C₂×C₄×C₅	—	2	4	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
161	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
162	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
165	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
169	C₅×C₈	—	2	2	no	—
170	C₅×C₈	—	2	1	no	—
171	C₅×C₈	—	2	1	no	—
172	C₅×C₈	—	2	2	no	—
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	—	4	2	no	—
175	C₅×C₈	—	4	2	no	—
176	C₅×C₈	—	4	4	no	—
177	C₅×C₈	#131	1	2	yes	—
178	C₂×#41	C₅×Q₈	1	1	yes	characteristic
179	C₂×M₁₆	—	2	2	no	—
180	C₂²×C₈	—	2	4	no	—
181	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
182	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
183	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
184	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
185	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
186	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
187	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
188	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
189	C₄×D₄	C₅×D₄	1	1	yes	characteristic
190	C₄×D₄	C₅×D₄	1	2	yes	—
191	#117	C₅×Q₈	1	1	yes	characteristic
192	#117	C₅×Q₈	1	2	yes	—
193	C₂×C₄²	C₂³×C₅	1	2	yes	—
194	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
195	C₄⋊C₈	—	2	4	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
198	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
199	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
200	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
201	C₄⋊C₈	C₂³×C₅	1	2	yes	—
202	C₄⋊C₈	C₂³×C₅	1	4	yes	—
203	C₄⋊C₈	C₂³×C₅	1	2	yes	—
204	C₂²⋊C₈	—	2	2	no	—
205	C₂²⋊C₈	—	2	4	no	—
206	C₂²⋊C₈	C₂×C₄×C₅	1	2	yes	—
207	C₂²⋊C₈	C₂×C₄×C₅	1	2	yes	—
208	C₂²⋊C₈	C₂×C₄×C₅	1	2	yes	—
209	#97	—	2	4	no	—
210	C₄×C₈	—	2	2	no	—
211	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
212	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
213	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
214	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
215	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
216	C₄×C₈	C₂³×C₅	1	2	yes	—
217	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
218	C₂²×C₅	—	2	2	no	—
219	C₂²×C₅	—	2	2	no	—
220	C₂²×C₅	—	2	1	no	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	C₂×#108	1	4	yes	—
224	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
225	C₄×C₅	—	2	2	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	2	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	2	2	no	—
231	C₄×C₅	C₂×#108	1	2	yes	—
232	C₄×C₅	C₂×#131	1	1	yes	characteristic
233	C₄×C₅	#568	1	1	yes	characteristic
234	C₄×C₅	#577	1	2	yes	—
235	#41	—	2	4	no	—
236	C₂×Q₈	C₅×C₄⋊C₄	1	1	yes	characteristic
237	C₂×D₄	C₅×C₄⋊C₄	1	2	yes	—
238	C₂×D₄	C₅×C₄⋊C₄	1	1	yes	characteristic
239	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
240	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
241	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
242	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
244	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
245	M₁₆	—	8	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	8	4	no	—
254	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
255	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
256	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
257	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
258	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
259	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
260	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
261	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
262	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
263	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
264	C₂²⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
265	C₂²⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
266	C₂²⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
267	C₄²	C₂²×C₄×C₅	1	2	yes	—
268	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
269	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
270	C₄²	C₂×C₅×D₄	1	2	yes	—
271	C₄²	C₂×C₅×Q₈	1	2	yes	—
272	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
273	C₄²	C₂⁴×C₅	1	1	yes	characteristic
274	C₂×C₅	—	2	4	no	—
275	C₂×C₅	—	4	1	no	—
276	C₂×C₅	#2906	1	1	yes	characteristic
277	C₂×C₅	#2912	1	1	yes	characteristic
278	C₂×C₅	C₂²×#108	1	1	yes	characteristic
279	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
280	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
281	Q₈	—	2	2	no	—
282	D₄	—	2	4	no	—
283	D₄	C₅×#108	1	4	yes	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
293	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
294	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
295	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
296	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
297	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
298	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
301	C₈	—	2	2	no	—
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	2	no	—
305	C₈	—	2	1	no	—
306	C₈	—	4	2	no	—
307	C₈	—	4	4	no	—
308	C₈	—	4	2	no	—
309	C₈	C₅×#131	1	2	yes	—
310	C₅	#17834	1	1	yes	characteristic, direct product factor, Sylow
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	C₂×C₅×#108	1	4	yes	—
317	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	2	no	—
324	C₄	C₂×C₅×#108	1	2	yes	—
325	C₄	C₂×C₅×#131	1	1	yes	characteristic
326	C₄	C₅×#568	1	1	yes	characteristic
327	C₄	C₅×#577	1	2	yes	—
328	C₂	—	2	4	no	—
329	C₂	—	4	1	no	—
330	C₂	C₅×#2906	1	1	yes	characteristic
331	C₂	C₅×#2912	1	1	yes	characteristic
332	C₂	C₂²×C₅×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₄²	C₂⁴×C₅
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂×C₅×#41	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄×C₅×#41 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1574	C₂	semidirect
4. C₅×#1573	C₂	semidirect
5. C₅×#1568	C₂	semidirect
6. C₅×#1566	C₂	semidirect
7. C₅×#1564	C₂	semidirect
8. C₂×C₅×#334	C₂	semidirect
34. #17834	C₅	direct (N × H)
77. #1574	C₂×C₅	semidirect
78. #1573	C₂×C₅	semidirect
79. #1568	C₂×C₅	semidirect
80. #1566	C₂×C₅	semidirect
81. #1564	C₂×C₅	semidirect
82. C₂×#334	C₂×C₅	semidirect
310. C₅	#17834	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×#2990	2
C₅×C₈×#41	2
C₅×#1574	2
C₅×#1573	2
C₅×#1568	2
C₅×#1566	2
C₅×#1564	2
C₂×C₅×#334	2
#17834	5

Subgroups — C5×#17834 #11940554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17834 #11940554