Subgroups — #12065002

Order 1280.

734 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2965	C₂	1	1	yes	maximal, characteristic
2	#41×#168	C₂	1	1	yes	maximal, characteristic
3	#10614244	C₂	1	1	yes	maximal, characteristic
4	#10601537	C₂	1	2	yes	maximal, complemented
5	#10601535	C₂	1	2	yes	maximal, complemented
6	#10601534	C₂	1	1	yes	maximal, characteristic, complemented
7	#10601533	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600641	C₂	1	2	yes	maximal, complemented
9	M₁₆×#168	C₂	1	2	yes	maximal, complemented
10	C₈×#681	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#64675	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₅×#41	C₂²	1	1	yes	characteristic
13	C₅×#446	C₄	1	2	yes	—
14	C₅×#436	C₄	1	2	yes	—
15	C₅×#435	C₄	1	4	yes	—
16	C₅×#431	C₂²	1	1	yes	characteristic
17	C₅×#431	C₂²	1	2	yes	—
18	C₄×C₅×M₁₆	C₂²	1	2	yes	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	#65523	C₄	1	2	yes	—
21	Q₈×#168	C₄	1	2	yes	—
22	D₄×#168	C₄	1	2	yes	—
23	D₄×#168	C₄	1	4	yes	—
24	#65289	C₄	1	4	yes	—
25	#65288	C₄	1	2	yes	—
26	#65214	C₂²	1	1	yes	characteristic
27	#65214	C₂²	1	2	yes	—
28	#65214	C₂²	1	2	yes	—
29	#65214	C₂²	1	2	yes	—
30	C₄×#681	C₂²	1	1	yes	characteristic
31	C₂×C₄×#168	C₂²	1	1	yes	characteristic
32	C₂×C₄×#168	C₂²	1	1	yes	characteristic
33	C₂×C₄×#168	C₂²	1	2	yes	—
34	#64675	C₂²	1	2	yes	—
35	#64675	C₂²	1	1	yes	characteristic
36	#64675	C₂²	1	1	yes	characteristic
37	#64675	C₂²	1	1	yes	characteristic
38	#64675	C₂²	1	1	yes	characteristic
39	#64674	C₂²	1	2	yes	—
40	#64674	C₂²	1	2	yes	—
41	#64674	C₂²	1	2	yes	—
42	C₈×#168	C₂²	1	2	yes	—
43	C₈×#168	C₂²	1	2	yes	—
44	#17331	—	5	1	no	maximal, Sylow
45	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
46	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
47	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
48	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
51	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
52	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
53	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
54	C₅×#117	C₂×C₄	1	1	yes	characteristic
55	C₅×#117	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₅	C₂³	1	2	yes	—
58	C₂×#681	C₂×C₄	1	2	yes	—
59	C₂²×#168	C₂×C₄	1	2	yes	—
60	C₂²×#168	C₂×C₄	1	2	yes	—
61	C₂²×#168	C₂×C₄	1	4	yes	—
62	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
63	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
64	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
65	C₅×C₂²⋊C₈	C₂×C₄	1	4	yes	—
66	C₅×#97	C₂×C₄	1	4	yes	—
67	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
68	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
69	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂³	1	2	yes	—
71	C₂²⋊#168	C₂×C₄	1	4	yes	—
72	C₂²⋊#168	C₂×C₄	1	4	yes	—
73	C₂²⋊#168	C₂×C₄	1	2	yes	—
74	C₂²⋊#168	C₂×C₄	1	4	yes	—
75	C₄⋊#168	C₂×C₄	1	4	yes	—
76	C₄⋊#168	C₂×C₄	1	4	yes	—
77	C₄⋊#168	C₂×C₄	1	2	yes	—
78	C₄⋊#168	C₂×C₄	1	4	yes	—
79	#3935	C₂×C₄	1	2	yes	—
80	C₄×#168	C₂×C₄	1	2	yes	—
81	C₄×#168	C₂×C₄	1	2	yes	—
82	C₄×#168	C₂×C₄	1	4	yes	—
83	C₄×#168	C₂³	1	2	yes	—
84	C₄×#168	C₂³	1	1	yes	characteristic
85	C₄×#168	C₂³	1	2	yes	—
86	C₄×#168	C₂³	1	1	yes	characteristic
87	C₄×#168	C₂³	1	1	yes	characteristic
88	C₄×#168	C₂³	1	1	yes	characteristic
89	#2965	D₅	1	1	yes	characteristic
90	C₈×#41	—	5	1	no	—
91	#2922	—	5	1	no	—
92	#1454	—	5	2	no	—
93	#1453	—	5	2	no	—
94	#1452	—	5	2	no	—
95	#1451	—	5	1	no	—
96	#1450	—	5	1	no	—
97	C₈×M₁₆	—	5	2	no	—
98	C₈×M₁₆	—	5	1	no	—
99	C₂×#322	—	5	1	no	—
100	C₅×#41	—	2	4	no	—
101	C₂×C₅×Q₈	C₄²	1	1	yes	characteristic
102	C₂×C₅×D₄	C₄²	1	2	yes	—
103	C₂×C₅×D₄	C₄²	1	1	yes	characteristic
104	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
110	C₅×M₁₆	—	2	8	no	—
111	C₂×C₅×C₈	—	2	4	no	—
112	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
113	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
114	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
115	C₅×C₄⋊C₄	C₄²	1	2	yes	—
116	C₅×C₄⋊C₄	C₄²	1	1	yes	characteristic
117	C₅×C₄⋊C₄	C₄²	1	2	yes	—
118	C₅×C₄⋊C₄	C₄²	1	1	yes	characteristic
119	C₅×C₂²⋊C₄	C₄²	1	2	yes	—
120	C₅×C₂²⋊C₄	C₄²	1	2	yes	—
121	C₅×C₂²⋊C₄	C₄²	1	2	yes	—
122	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₅	C₂²×C₄	1	2	yes	—
124	C₄²×C₅	C₂²×C₄	1	2	yes	—
125	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₅	C₂²×C₄	1	2	yes	—
128	C₄²×C₅	C₂⁴	1	1	yes	characteristic
129	#681	—	2	4	no	—
130	C₂×#168	—	2	4	no	—
131	C₂×#168	—	4	2	no	—
132	C₂×#168	—	4	4	no	—
133	C₂×#168	C₂²×C₄	1	2	yes	—
134	C₂×#168	C₂²×C₄	1	2	yes	—
135	C₂×#168	C₂²×C₄	1	2	yes	—
136	C₂×#168	C₂²×C₄	1	4	yes	—
137	C₂×#168	C₂²×C₄	1	2	yes	—
138	C₂×#168	C₂²×C₄	1	4	yes	—
139	C₄×#41	C₂×D₅	1	1	yes	characteristic
140	#446	Dic₅	1	2	yes	—
141	C₈×Q₈	—	5	2	no	—
142	#436	Dic₅	1	2	yes	—
143	#435	Dic₅	1	4	yes	—
144	C₈×D₄	—	5	2	no	—
145	C₈×D₄	—	5	4	no	—
146	#433	—	5	4	no	—
147	#431	—	5	2	no	—
148	#431	—	5	2	no	—
149	#431	—	5	1	no	—
150	#431	—	5	2	no	—
151	#431	C₂×D₅	1	1	yes	characteristic
152	#431	C₂×D₅	1	2	yes	—
153	#424	—	5	2	no	—
154	#408	—	5	2	no	—
155	C₄×M₁₆	—	5	1	no	—
156	C₄×M₁₆	C₂×D₅	1	2	yes	—
157	C₂×C₄×C₈	—	5	1	no	—
158	C₂×C₄×C₈	—	5	1	no	—
159	C₂×C₄×C₈	—	5	2	no	—
160	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
161	#322	—	5	2	no	—
162	#322	—	5	2	no	—
163	#322	—	5	2	no	—
164	#322	—	5	1	no	—
165	#322	—	5	1	no	—
166	#322	—	5	1	no	—
167	#322	—	5	1	no	—
168	#322	—	5	2	no	—
169	C₈²	—	5	2	no	—
170	C₈²	—	5	2	no	—
171	C₂³×C₅	C₂×C₄²	1	2	yes	—
172	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
173	C₅×Q₈	—	2	2	no	—
174	C₅×D₄	—	2	4	no	—
175	C₅×D₄	—	2	2	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	2	4	no	—
178	C₂×C₄×C₅	—	2	4	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	4	no	—
181	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
183	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
185	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
186	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
187	C₂×C₄×C₅	#131	1	4	yes	—
188	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
191	C₅×C₈	—	2	2	no	—
192	C₅×C₈	—	2	4	no	—
193	C₅×C₈	#131	1	4	yes	—
194	#168	—	2	2	no	—
195	#168	—	2	2	no	—
196	#168	—	2	4	no	—
197	#168	—	2	2	no	—
198	#168	—	2	4	no	—
199	#168	—	2	2	no	—
200	C₂×#41	C₄×D₅	1	1	yes	characteristic
201	C₂×M₁₆	—	5	2	no	—
202	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
203	C₂²×C₈	—	5	2	no	—
204	C₂²×C₈	—	5	2	no	—
205	C₂²×C₈	—	5	4	no	—
206	C₂²×C₈	C₂×Dic₅	1	2	yes	—
207	C₄×Q₈	C₄×D₅	1	1	yes	characteristic
208	C₄×Q₈	C₂×Dic₅	1	1	yes	characteristic
209	C₄×D₄	C₄×D₅	1	2	yes	—
210	C₄×D₄	C₄×D₅	1	1	yes	characteristic
211	C₄×D₄	C₂×Dic₅	1	2	yes	—
212	C₄×D₄	C₂×Dic₅	1	1	yes	characteristic
213	#117	C₄×D₅	1	1	yes	characteristic
214	#117	C₄×D₅	1	2	yes	—
215	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
216	C₂×C₄²	C₂²×D₅	1	2	yes	—
217	C₄⋊C₈	—	5	4	no	—
218	C₄⋊C₈	—	5	4	no	—
219	C₄⋊C₈	—	5	2	no	—
220	C₄⋊C₈	—	5	4	no	—
221	C₄⋊C₈	C₂×Dic₅	1	4	yes	—
222	C₄⋊C₈	C₂×Dic₅	1	2	yes	—
223	C₂²⋊C₈	—	5	4	no	—
224	C₂²⋊C₈	—	5	4	no	—
225	C₂²⋊C₈	—	5	2	no	—
226	C₂²⋊C₈	—	5	4	no	—
227	C₂²⋊C₈	C₂×Dic₅	1	2	yes	—
228	C₂²⋊C₈	C₂×Dic₅	1	4	yes	—
229	#97	—	5	2	no	—
230	#97	C₂×Dic₅	1	4	yes	—
231	C₄×C₈	—	5	2	no	—
232	C₄×C₈	—	5	2	no	—
233	C₄×C₈	—	5	4	no	—
234	C₄×C₈	—	5	2	no	—
235	C₄×C₈	—	5	1	no	—
236	C₄×C₈	—	5	2	no	—
237	C₄×C₈	—	5	1	no	—
238	C₄×C₈	—	5	1	no	—
239	C₄×C₈	—	5	1	no	—
240	C₄×C₈	C₂×Dic₅	1	2	yes	—
241	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₅	1	2	yes	—
243	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
244	C₂²×C₅	—	2	1	no	—
245	C₂²×C₅	—	2	1	no	—
246	C₂²×C₅	—	2	2	no	—
247	C₂²×C₅	—	2	2	no	—
248	C₂²×C₅	—	2	2	no	—
249	C₂²×C₅	#405	1	2	yes	—
250	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
251	C₄×C₅	—	2	2	no	—
252	C₄×C₅	—	2	1	no	—
253	C₄×C₅	—	2	2	no	—
254	C₄×C₅	—	2	1	no	—
255	C₄×C₅	—	2	2	no	—
256	C₄×C₅	#405	1	2	yes	—
257	C₄×C₅	C₂×#131	1	2	yes	—
258	C₄×C₅	C₂×#131	1	2	yes	—
259	C₄×C₅	#568	1	2	yes	—
260	#41	—	2	4	no	—
261	C₂×Q₈	C₄×Dic₅	1	1	yes	characteristic
262	C₂×D₄	C₄×Dic₅	1	2	yes	—
263	C₂×D₄	C₄×Dic₅	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
265	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
266	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
268	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
269	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
270	M₁₆	—	2	8	no	—
271	M₁₆	—	10	4	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	5	2	no	—
274	C₂×C₈	—	5	2	no	—
275	C₂×C₈	—	5	2	no	—
276	C₂×C₈	—	5	4	no	—
277	C₂×C₈	—	5	2	no	—
278	C₂×C₈	—	5	4	no	—
279	C₂×C₈	—	10	4	no	—
280	C₂×C₈	—	20	2	no	—
281	C₂×C₈	—	20	4	no	—
282	C₂×C₈	C₂²×Dic₅	1	2	yes	—
283	C₂×C₈	C₂²×Dic₅	1	4	yes	—
284	C₂×C₈	C₂²×Dic₅	1	2	yes	—
285	C₄⋊C₄	C₄×Dic₅	1	2	yes	—
286	C₄⋊C₄	C₄×Dic₅	1	1	yes	characteristic
287	C₄⋊C₄	C₄×Dic₅	1	2	yes	—
288	C₄⋊C₄	C₄×Dic₅	1	1	yes	characteristic
289	C₂²⋊C₄	C₄×Dic₅	1	2	yes	—
290	C₂²⋊C₄	C₄×Dic₅	1	2	yes	—
291	C₂²⋊C₄	C₄×Dic₅	1	2	yes	—
292	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
293	C₄²	C₂×C₄×D₅	1	2	yes	—
294	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
295	C₄²	C₂×C₄×D₅	1	2	yes	—
296	C₄²	C₂²×Dic₅	1	1	yes	characteristic
297	C₄²	C₂²×Dic₅	1	2	yes	—
298	C₄²	C₂³×D₅	1	1	yes	characteristic
299	C₂×C₅	—	2	2	no	—
300	C₂×C₅	—	4	2	no	—
301	C₂×C₅	C₂×#405	1	1	yes	characteristic
302	C₂×C₅	C₄×#131	1	1	yes	characteristic
303	C₂×C₅	#2875	1	1	yes	characteristic
304	C₂³	C₂×C₄×Dic₅	1	2	yes	—
305	C₂³	C₂×C₄×Dic₅	1	1	yes	characteristic
306	Q₈	—	2	2	no	—
307	D₄	—	2	4	no	—
308	D₄	—	2	2	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	#4053	1	4	yes	—
315	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
317	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
318	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
319	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
320	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
321	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
322	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
323	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	4	no	—
326	C₈	—	10	2	no	—
327	C₈	—	10	2	no	—
328	C₈	—	10	4	no	—
329	C₈	—	10	2	no	—
330	C₈	—	10	4	no	—
331	C₈	—	10	2	no	—
332	C₈	#4094	1	4	yes	—
333	C₅	#17331	1	1	yes	characteristic, complemented, Sylow
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	#65411	1	2	yes	—
340	C₂²	C₂²×C₄×Dic₅	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	#65411	1	2	yes	—
347	C₄	C₂×#4053	1	2	yes	—
348	C₄	#66084	1	2	yes	—
349	C₄	C₂×#4094	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	4	2	no	—
352	C₂	C₄×#4094	1	1	yes	characteristic
353	C₂	C₂×#65411	1	1	yes	characteristic
354	C₂	#10615513	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅
commutator, socle	C₂²×C₅	C₂²×C₄²
Fitting	C₅×#2965	C₂
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#2965	D₅
hypocenter	C₅	#17331
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ #2965

Complements

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

N	H ≅ G/N	type
4. #10601537	C₂	semidirect
5. #10601535	C₂	semidirect
6. #10601534	C₂	semidirect
7. #10601533	C₂	semidirect
8. #10600641	C₂	semidirect
9. M₁₆×#168	C₂	semidirect
10. C₈×#681	C₂	semidirect
11. C₂×#64675	C₂	semidirect
333. C₅	#17331	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2965	2
#41×#168	2
#10614244	2
#10601537	2
#10601535	2
#10601534	2
#10601533	2
#10600641	2
M₁₆×#168	2
C₈×#681	2
C₂×#64675	2
#17331	5