Subgroups — #11988975

Order 1280.

666 conjugacy classes of subgroups.

The subgroup lattice has 370 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	C₄×#4072	C₂	1	1	yes	maximal, characteristic
3	C₂×#65215	C₂	1	1	yes	maximal, characteristic
4	#10598169	C₂	1	4	yes	maximal
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₂×#4072	C₄	1	2	yes	—
7	C₅×#432	—	2	1	no	—
8	C₅×#432	—	2	1	no	—
9	C₅×#432	C₄	1	2	yes	complemented
10	C₅×#432	C₄	1	2	yes	complemented
11	C₂×C₅×C₄⋊C₈	—	2	1	no	—
12	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×#97	—	2	1	no	—
14	C₄×C₂²⋊Dic₅	C₄	1	2	yes	—
15	C₄×C₂²⋊Dic₅	C₂²	1	2	yes	—
16	C₂×C₂²⋊#168	C₂²	1	2	yes	—
17	C₄×C₄⋊Dic₅	—	2	1	no	—
18	C₄×C₄⋊Dic₅	—	2	1	no	—
19	#65221	—	2	1	no	—
20	C₄²×Dic₅	—	2	1	no	—
21	#65215	—	2	1	no	—
22	#65215	—	2	1	no	—
23	#65215	—	2	1	no	—
24	#65215	—	2	1	no	—
25	C₂×C₄⋊#168	C₄	1	2	yes	complemented
26	C₂×#3935	C₄	1	2	yes	complemented
27	#7930	—	5	1	no	maximal, Sylow
28	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	D₄	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×C₂²⋊Dic₅	C₂×C₄	1	4	yes	—
45	#4072	—	2	8	no	—
46	C₂×C₄⋊Dic₅	—	2	2	no	—
47	C₂×C₄×Dic₅	—	2	2	no	—
48	C₂×C₄×Dic₅	—	2	2	no	—
49	C₂×C₄×Dic₅	—	2	2	no	—
50	C₂²×#168	C₂×C₄	1	4	yes	—
51	C₅×C₄⋊C₈	—	2	2	no	—
52	C₅×C₄⋊C₈	—	2	2	no	—
53	C₅×C₂²⋊C₈	—	2	2	no	—
54	C₅×C₂²⋊C₈	C₂×C₄	1	4	yes	—
55	C₅×#97	—	2	2	no	—
56	C₅×#97	—	2	2	no	—
57	#3963	—	2	2	no	—
58	#3963	—	2	2	no	—
59	C₂²⋊#168	—	2	2	no	—
60	C₂²⋊#168	—	2	2	no	—
61	C₄⋊#168	—	2	2	no	—
62	C₄⋊#168	—	2	2	no	—
63	#3935	—	2	2	no	—
64	#3935	—	2	2	no	—
65	C₂×#432	—	5	1	no	—
66	C₂×#432	D₅	1	1	yes	characteristic
67	C₄×#117	—	5	1	no	—
68	#1282	—	5	4	no	—
69	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	—	2	4	no	—
71	C₂²×C₄×C₅	—	4	2	no	—
72	C₂²×C₄×C₅	—	4	2	no	—
73	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂²⋊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	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
91	C₂²×Dic₅	—	2	4	no	—
92	C₂×C₅×C₈	—	2	4	no	—
93	C₂×C₅×C₈	—	4	4	no	—
94	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
95	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₅	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
110	C₂²⋊Dic₅	—	2	4	no	—
111	C₂²⋊Dic₅	M₁₆	1	8	yes	—
112	C₄⋊Dic₅	—	2	8	no	—
113	C₄×Dic₅	—	2	8	no	—
114	C₄×Dic₅	—	4	4	no	—
115	C₄×Dic₅	—	4	4	no	—
116	C₂×#168	—	2	4	no	—
117	C₂×#168	—	4	4	no	—
118	C₂×#117	—	5	2	no	—
119	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
120	#432	—	2	1	no	—
121	#432	—	2	1	no	—
122	#432	—	10	1	no	—
123	#432	—	10	1	no	—
124	#432	—	10	1	no	—
125	#432	—	10	1	no	—
126	#432	Dic₅	1	2	yes	complemented
127	#432	Dic₅	1	2	yes	complemented
128	C₂×C₄⋊C₈	—	2	1	no	—
129	C₂×C₄⋊C₈	—	5	2	no	—
130	C₂×C₂²⋊C₈	—	5	2	no	—
131	C₂×C₂²⋊C₈	C₂×D₅	1	2	yes	—
132	C₂×#97	—	2	1	no	—
133	C₂×#97	—	5	2	no	—
134	C₄×C₄⋊C₄	—	10	1	no	—
135	C₄×C₄⋊C₄	—	10	1	no	—
136	C₄×C₂²⋊C₄	—	5	2	no	—
137	C₄×C₂²⋊C₄	—	5	2	no	—
138	#376	—	10	1	no	—
139	C₄³	—	10	1	no	—
140	C₂³×C₅	—	4	1	no	—
141	C₂³×C₅	—	4	1	no	—
142	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
143	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂³×C₅	C₂×M₁₆	1	4	yes	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
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	—	2	2	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	#95	1	1	yes	characteristic
165	C₂×C₄×C₅	#95	1	1	yes	characteristic
166	C₂×C₄×C₅	#95	1	1	yes	characteristic
167	C₂×C₄×C₅	#95	1	1	yes	characteristic
168	C₂×C₄×C₅	#95	1	1	yes	characteristic
169	C₂×C₄×C₅	#95	1	1	yes	characteristic
170	C₂×C₄×C₅	#95	1	1	yes	characteristic
171	C₂×C₄×C₅	#95	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×Dic₅	—	2	8	no	—
177	C₂×Dic₅	—	4	4	no	—
178	C₂×Dic₅	—	4	4	no	—
179	C₅×C₈	—	4	4	no	—
180	#168	—	4	4	no	—
181	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
182	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
183	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	5	4	no	—
186	#117	—	10	8	no	—
187	C₂×C₄⋊C₄	—	10	2	no	—
188	C₂×C₂²⋊C₄	—	5	4	no	—
189	C₂×C₄²	—	10	2	no	—
190	C₂×C₄²	—	10	2	no	—
191	C₂×C₄²	—	10	2	no	—
192	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
193	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
194	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
196	C₂×C₄²	D₂₀	1	1	yes	characteristic
197	C₂×C₄²	D₂₀	1	1	yes	characteristic
198	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
199	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
202	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	—	10	2	no	—
207	C₄⋊C₈	—	10	2	no	—
208	C₂²⋊C₈	—	2	2	no	—
209	C₂²⋊C₈	—	10	2	no	—
210	C₂²⋊C₈	—	10	2	no	—
211	C₂²⋊C₈	C₂×Dic₅	1	4	yes	—
212	#97	—	2	2	no	—
213	#97	—	2	2	no	—
214	#97	—	10	2	no	—
215	#97	—	10	2	no	—
216	#95	—	10	2	no	—
217	#95	—	10	2	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	—	4	4	no	—
225	C₂²×C₅	C₂×#95	1	1	yes	characteristic
226	C₂²×C₅	C₄×M₁₆	1	2	yes	—
227	C₂²×C₅	#407	1	2	yes	—
228	C₂²×C₅	#423	1	2	yes	—
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	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	#344	1	2	yes	—
240	C₄×C₅	#410	1	2	yes	—
241	Dic₅	—	4	8	no	—
242	C₂⁴	C₂×C₄×D₅	1	1	yes	characteristic
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	10	4	no	—
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	#683	1	1	yes	characteristic
252	C₂²×C₄	#683	1	1	yes	characteristic
253	C₂²×C₄	#683	1	1	yes	characteristic
254	C₂²×C₄	#683	1	1	yes	characteristic
255	C₂²×C₄	#685	1	1	yes	characteristic
256	C₂²×C₄	#685	1	1	yes	characteristic
257	C₂²×C₄	#685	1	1	yes	characteristic
258	C₂²×C₄	#685	1	1	yes	characteristic
259	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
262	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	4	4	no	—
267	C₂×C₈	—	10	4	no	—
268	C₂×C₈	—	20	4	no	—
269	C₄⋊C₄	—	10	8	no	—
270	C₂²⋊C₄	—	5	8	no	—
271	C₂²⋊C₄	—	10	4	no	—
272	C₄²	—	10	8	no	—
273	C₄²	—	20	4	no	—
274	C₄²	—	20	4	no	—
275	C₄²	#683	1	1	yes	characteristic
276	C₄²	#683	1	1	yes	characteristic
277	C₄²	#683	1	1	yes	characteristic
278	C₄²	#683	1	1	yes	characteristic
279	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
280	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
281	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
282	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
283	C₄²	#685	1	1	yes	characteristic
284	C₄²	#685	1	1	yes	characteristic
285	C₄²	#685	1	1	yes	characteristic
286	C₄²	#685	1	1	yes	characteristic
287	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
288	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
289	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
290	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
291	C₂×C₅	—	2	2	no	—
292	C₂×C₅	—	2	2	no	—
293	C₂×C₅	—	2	2	no	—
294	C₂×C₅	#1728	1	1	yes	characteristic
295	C₂×C₅	#1740	1	1	yes	characteristic
296	C₂×C₅	C₂×#344	1	1	yes	characteristic
297	C₂³	—	4	1	no	—
298	C₂³	—	4	1	no	—
299	C₂³	D₅×M₁₆	1	4	yes	—
300	C₂³	C₂×C₄×Dic₅	1	1	yes	characteristic
301	C₂³	C₂×#683	1	1	yes	characteristic
302	C₂³	C₂×#685	1	1	yes	characteristic
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	10	8	no	—
322	C₂×C₄	—	20	4	no	—
323	C₂×C₄	—	20	4	no	—
324	C₂×C₄	#3963	1	1	yes	characteristic
325	C₂×C₄	#3963	1	1	yes	characteristic
326	C₂×C₄	#3963	1	1	yes	characteristic
327	C₂×C₄	#3963	1	1	yes	characteristic
328	C₂×C₄	#3963	1	1	yes	characteristic
329	C₂×C₄	#3963	1	1	yes	characteristic
330	C₂×C₄	#3963	1	1	yes	characteristic
331	C₂×C₄	#3963	1	1	yes	characteristic
332	C₂×C₄	C₂×#683	1	1	yes	characteristic
333	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
334	C₂×C₄	C₂×#685	1	1	yes	characteristic
335	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
336	C₈	—	4	4	no	—
337	C₈	—	20	4	no	—
338	C₅	#7930	1	1	yes	characteristic, complemented, Sylow
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	4	4	no	—
346	C₂²	M₁₆×Dic₅	1	2	yes	—
347	C₂²	#65407	1	2	yes	—
348	C₂²	#65415	1	2	yes	—
349	C₂²	C₂×#3963	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	20	8	no	—
361	C₄	#64779	1	2	yes	—
362	C₄	#65295	1	2	yes	—
363	C₂	—	2	2	no	—
364	C₂	—	2	2	no	—
365	C₂	—	2	2	no	—
366	C₂	#10602386	1	1	yes	characteristic
367	C₂	#10602772	1	1	yes	characteristic
368	C₂	C₂×#64779	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#685
commutator	C₂³×C₅	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂×C₅×#432	C₂
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂×#432	D₅
hypocenter	C₅	#7930
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
9. C₅×#432	C₄	semidirect
10. C₅×#432	C₄	semidirect
25. C₂×C₄⋊#168	C₄	semidirect
26. C₂×#3935	C₄	semidirect
126. #432	Dic₅	semidirect
127. #432	Dic₅	semidirect
338. C₅	#7930	semidirect

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
C₄×#4072	2
C₂×#65215	2
#10598169	2
#7930	5