Subgroups — #12068501

Order 1280.

616 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2920	C₂	1	1	yes	maximal, characteristic
2	#10613780	C₂	1	1	yes	maximal, characteristic
3	#10613780	C₂	1	1	yes	maximal, characteristic
4	#10601546	C₂	1	4	yes	maximal, complemented
5	#10601544	C₂	1	4	yes	maximal, complemented
6	#10600644	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₅×#432	C₄	1	2	yes	—
9	C₅×#432	C₄	1	2	yes	—
10	C₅×#431	C₂²	1	2	yes	—
11	C₅×#431	C₂²	1	2	yes	—
12	C₅×#423	C₄	1	2	yes	—
13	C₅×#407	C₄	1	2	yes	—
14	C₄×C₅×M₁₆	C₂²	1	2	yes	—
15	#65496	—	2	1	no	—
16	#65496	—	2	1	no	—
17	#65215	—	2	1	no	—
18	#65215	—	2	1	no	—
19	#65215	C₂²	1	2	yes	—
20	#65215	C₂²	1	2	yes	—
21	#65214	—	2	1	no	—
22	#65214	—	2	1	no	—
23	#65214	C₂²	1	2	yes	—
24	#65214	C₂²	1	2	yes	—
25	#65213	C₂²	1	2	yes	—
26	#65213	C₂²	1	2	yes	—
27	C₄×#681	—	2	1	no	—
28	C₄×#681	—	2	1	no	—
29	#64678	C₂²	1	8	yes	complemented
30	#64677	C₂²	1	8	yes	complemented
31	#17823	—	5	1	no	maximal, Sylow
32	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
35	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
44	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₂×#681	—	2	2	no	—
49	C₂×#681	—	2	2	no	—
50	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
51	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₅×C₂²⋊C₈	C₂×C₄	1	4	yes	—
53	C₅×C₂²⋊C₈	C₂×C₄	1	4	yes	—
54	C₅×#97	C₂×C₄	1	4	yes	—
55	C₄×C₅×C₈	C₂³	1	4	yes	—
56	C₂²⋊#168	—	2	2	no	—
57	C₂²⋊#168	—	2	2	no	—
58	C₂²⋊#168	—	2	2	no	—
59	C₂²⋊#168	—	2	2	no	—
60	C₄⋊#168	—	2	2	no	—
61	C₄⋊#168	—	2	2	no	—
62	C₄⋊#168	C₂³	1	4	yes	—
63	C₄⋊#168	C₂³	1	4	yes	—
64	#3935	—	2	2	no	—
65	#3935	—	2	2	no	—
66	C₄×#168	—	2	2	no	—
67	C₄×#168	—	2	2	no	—
68	#2920	—	5	1	no	—
69	#2920	—	5	1	no	—
70	#2920	D₅	1	1	yes	characteristic
71	#1574	—	5	4	no	—
72	#1572	—	5	4	no	—
73	#1569	—	5	4	no	—
74	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	—	4	2	no	—
76	C₂²×C₄×C₅	—	4	2	no	—
77	C₂²×C₄×C₅	—	4	2	no	—
78	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
96	C₅×M₁₆	—	4	4	no	—
97	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
98	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
99	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
111	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
112	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
113	C₄²×C₅	C₂⁴	1	1	yes	characteristic
114	#681	—	4	4	no	—
115	#681	—	4	4	no	—
116	C₂×#168	—	2	4	no	—
117	C₂×#168	—	2	4	no	—
118	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
119	#432	—	5	2	no	—
120	#432	—	5	2	no	—
121	#432	—	10	1	no	—
122	#432	—	10	1	no	—
123	#432	Dic₅	1	2	yes	—
124	#432	Dic₅	1	2	yes	—
125	#431	—	5	2	no	—
126	#431	—	5	2	no	—
127	#431	—	10	1	no	—
128	#431	—	10	1	no	—
129	#431	C₂×D₅	1	2	yes	—
130	#431	C₂×D₅	1	2	yes	—
131	#423	—	5	2	no	—
132	#423	—	5	2	no	—
133	#423	Dic₅	1	2	yes	—
134	#407	—	10	1	no	—
135	#407	—	10	1	no	—
136	#407	Dic₅	1	2	yes	—
137	C₄×M₁₆	—	10	1	no	—
138	C₄×M₁₆	—	10	1	no	—
139	C₄×M₁₆	C₂×D₅	1	2	yes	—
140	#335	—	5	8	no	—
141	#334	—	5	8	no	—
142	C₂³×C₅	—	4	1	no	—
143	C₂³×C₅	—	4	1	no	—
144	C₂³×C₅	—	4	1	no	—
145	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	4	1	no	—
158	C₂×C₄×C₅	—	4	1	no	—
159	C₂×C₄×C₅	—	4	1	no	—
160	C₂×C₄×C₅	—	4	1	no	—
161	C₂×C₄×C₅	—	4	1	no	—
162	C₂×C₄×C₅	—	4	1	no	—
163	C₂×C₄×C₅	—	4	1	no	—
164	C₂×C₄×C₅	—	4	1	no	—
165	C₂×C₄×C₅	—	4	1	no	—
166	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
178	C₅×C₈	—	2	8	no	—
179	#168	—	4	4	no	—
180	#168	—	4	4	no	—
181	C₂³×C₄	Dic₁₀	1	1	yes	characteristic
182	C₂³×C₄	D₂₀	1	1	yes	characteristic
183	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
184	C₂×M₁₆	—	10	2	no	—
185	C₂×M₁₆	—	10	2	no	—
186	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
187	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
188	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
189	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
190	C₂×C₄²	D₂₀	1	1	yes	characteristic
191	C₂×C₄²	D₂₀	1	1	yes	characteristic
192	C₂×C₄²	D₂₀	1	1	yes	characteristic
193	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
194	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
195	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
196	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
197	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
199	C₄⋊C₈	—	5	4	no	—
200	C₄⋊C₈	—	5	4	no	—
201	C₄⋊C₈	—	10	2	no	—
202	C₄⋊C₈	—	10	2	no	—
203	C₄⋊C₈	C₂×Dic₅	1	4	yes	—
204	C₄⋊C₈	C₂×Dic₅	1	4	yes	—
205	C₂²⋊C₈	—	10	2	no	—
206	C₂²⋊C₈	—	10	2	no	—
207	C₂²⋊C₈	—	10	2	no	—
208	C₂²⋊C₈	—	10	2	no	—
209	C₂²⋊C₈	C₂×Dic₅	1	4	yes	—
210	C₂²⋊C₈	C₂×Dic₅	1	4	yes	—
211	#97	—	10	2	no	—
212	#97	—	10	2	no	—
213	#97	C₂×Dic₅	1	4	yes	—
214	C₄×C₈	—	10	2	no	—
215	C₄×C₈	—	10	2	no	—
216	C₄×C₈	C₂²×D₅	1	4	yes	—
217	C₂²×C₅	—	2	1	no	—
218	C₂²×C₅	—	2	1	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	—	2	1	no	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	2	1	no	—
225	C₂²×C₅	—	2	1	no	—
226	C₂²×C₅	—	4	4	no	—
227	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	2	1	no	—
236	C₄×C₅	—	2	1	no	—
237	C₄×C₅	#430	1	2	yes	—
238	C₄×C₅	#568	1	2	yes	—
239	C₄×C₅	#575	1	2	yes	—
240	C₂⁴	C₄⋊Dic₅	1	1	yes	characteristic
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
245	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
246	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
247	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
248	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
249	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
250	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
251	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
252	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
253	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
254	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
255	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
259	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
260	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
261	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
262	M₁₆	—	4	4	no	—
263	M₁₆	—	20	4	no	—
264	M₁₆	—	20	4	no	—
265	C₂×C₈	—	10	4	no	—
266	C₂×C₈	—	10	4	no	—
267	C₂×C₈	C₂²×Dic₅	1	8	yes	—
268	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
269	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
270	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
271	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
272	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
273	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
274	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
275	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
276	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
277	C₄²	C₂×D₂₀	1	1	yes	characteristic
278	C₄²	C₂×D₂₀	1	1	yes	characteristic
279	C₄²	C₂×D₂₀	1	1	yes	characteristic
280	C₄²	C₂²×Dic₅	1	1	yes	characteristic
281	C₄²	C₂²×Dic₅	1	1	yes	characteristic
282	C₄²	C₂²×Dic₅	1	1	yes	characteristic
283	C₄²	C₂³×D₅	1	1	yes	characteristic
284	C₂×C₅	—	4	1	no	—
285	C₂×C₅	—	4	1	no	—
286	C₂×C₅	—	4	1	no	—
287	C₂×C₅	#2905	1	1	yes	characteristic
288	C₂×C₅	#2911	1	1	yes	characteristic
289	C₂×C₅	C₂×#430	1	1	yes	characteristic
290	C₂³	—	4	1	no	—
291	C₂³	—	4	1	no	—
292	C₂³	—	4	1	no	—
293	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
294	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
295	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
317	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
318	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
319	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
320	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
321	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
322	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
323	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
324	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
325	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
326	C₈	—	2	8	no	—
327	C₈	—	20	4	no	—
328	C₈	—	20	4	no	—
329	C₅	#17823	1	1	yes	characteristic, complemented, Sylow
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	4	4	no	—
340	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	#65414	1	2	yes	—
351	C₄	#66082	1	2	yes	—
352	C₄	#66153	1	2	yes	—
353	C₂	—	4	1	no	—
354	C₂	—	4	1	no	—
355	C₂	—	4	1	no	—
356	C₂	#10613935	1	1	yes	characteristic
357	C₂	#10615483	1	1	yes	characteristic
358	C₂	C₂×#65414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#2920	C₂
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅	C₄⋊C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#2920	D₅
hypocenter	C₅	#17823
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
4. #10601546	C₂	semidirect
5. #10601544	C₂	semidirect
6. #10600644	C₂	semidirect
29. #64678	C₂²	semidirect
30. #64677	C₂²	semidirect
329. C₅	#17823	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#2920	2
#10613780	2
#10601546	2
#10601544	2
#10600644	2
#17823	5