Subgroups — #12252404

Order 1280.

600 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₅×#2108	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602224	C₂	1	1	yes	maximal, characteristic
4	#10602214	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#64727	C₂	1	1	yes	maximal, characteristic
6	#10602194	C₂	1	1	yes	maximal, characteristic, complemented
7	#10600674	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599867	C₂	1	2	yes	maximal, complemented
9	#10599866	C₂	1	2	yes	maximal, complemented
10	#10599846	C₂	1	2	yes	maximal, complemented
11	#10599845	C₂	1	2	yes	maximal, complemented
12	C₂×D₅×M₁₆	C₄	1	1	yes	characteristic
13	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
14	D₅×#117	C₄	1	1	yes	characteristic
15	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
17	C₅×#405	C₂²	1	1	yes	characteristic
18	#65411	C₄	1	1	yes	characteristic
19	#65390	C₂²	1	1	yes	characteristic
20	C₂×#3944	C₂²	1	1	yes	characteristic
21	C₂²×#672	C₂²	1	1	yes	characteristic
22	#65286	C₄	1	1	yes	characteristic
23	#64997	C₄	1	2	yes	—
24	D₅×#97	C₄	1	2	yes	—
25	#64978	C₂²	1	2	yes	—
26	C₄×C₈×D₅	C₂²	1	2	yes	—
27	C₅×#346	C₂²	1	2	yes	complemented
28	C₄×C₅×C₁₆	C₂²	1	2	yes	complemented
29	#64727	C₂²	1	1	yes	characteristic
30	#64727	C₂²	1	2	yes	—
31	#64727	C₂²	1	1	yes	characteristic
32	#64727	C₂²	1	2	yes	—
33	#64727	C₂²	1	1	yes	characteristic
34	#64727	C₂²	1	1	yes	characteristic
35	#64721	C₂²	1	2	yes	complemented
36	#64721	C₂²	1	2	yes	complemented
37	#64721	C₂²	1	2	yes	complemented
38	#64721	C₂²	1	2	yes	complemented
39	#64681	C₂²	1	2	yes	complemented
40	C₄×#672	C₂²	1	2	yes	complemented
41	#33422	—	5	1	no	maximal, Sylow
42	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₅×#117	C₂×C₄	1	1	yes	characteristic
46	#4072	C₂×C₄	1	1	yes	characteristic
47	C₂×#681	C₂×C₄	1	1	yes	characteristic
48	C₂²×#168	C₂×C₄	1	1	yes	characteristic
49	D₅×M₁₆	—	2	4	no	—
50	C₂×#676	C₂×C₄	1	2	yes	—
51	C₂×C₈×D₅	—	2	2	no	—
52	C₂×C₈×D₅	—	2	2	no	—
53	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
56	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
57	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
58	#4038	C₈	1	1	yes	characteristic
59	#4038	C₈	1	1	yes	characteristic
60	D₅×C₄⋊C₄	C₈	1	1	yes	characteristic
61	D₅×C₄⋊C₄	C₈	1	1	yes	characteristic
62	D₅×C₂²⋊C₄	C₈	1	2	yes	—
63	#4023	C₈	1	2	yes	—
64	#4018	C₂×C₄	1	2	yes	—
65	C₄²×D₅	C₂×C₄	1	2	yes	—
66	C₅×M₃₂	—	2	2	no	—
67	C₂×C₅×C₁₆	—	2	2	no	—
68	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₅×C₁₆	C₂³	1	2	yes	—
70	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
71	C₅×#97	C₂×C₄	1	2	yes	—
72	C₄×C₅×C₈	C₂³	1	2	yes	—
73	#3947	C₂×C₄	1	2	yes	—
74	#3947	C₂×C₄	1	2	yes	—
75	C₈×Dic₅	C₂×C₄	1	2	yes	—
76	C₈×Dic₅	C₂³	1	2	yes	—
77	#3944	—	2	2	no	—
78	C₂×#672	—	2	2	no	—
79	C₂×#672	C₂³	1	1	yes	characteristic
80	C₂×#672	C₂³	1	1	yes	characteristic
81	C₂×#672	C₂³	1	2	yes	—
82	#3935	C₂×C₄	1	2	yes	—
83	C₄×#168	C₂×C₄	1	2	yes	—
84	C₂×#405	—	5	1	no	—
85	#2164	—	5	2	no	—
86	#2164	—	5	2	no	—
87	#2163	—	5	2	no	—
88	#2162	—	5	2	no	—
89	#2151	—	5	1	no	—
90	#2151	—	5	1	no	—
91	#2112	—	5	1	no	—
92	C₂×C₂²⋊C₁₆	—	5	1	no	—
93	#2108	—	5	1	no	—
94	#2108	D₅	1	1	yes	characteristic, complemented
95	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
98	C₂×C₄×D₅	—	2	2	no	—
99	C₂×C₄×D₅	—	2	2	no	—
100	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
101	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
102	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
103	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
104	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
106	C₅×M₁₆	M₁₆	1	4	yes	—
107	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₅×C₈	#41	1	2	yes	—
110	C₂×C₅×C₈	#41	1	2	yes	—
111	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
112	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
113	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
114	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
115	C₄²×C₅	C₂²×C₄	1	2	yes	—
116	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
117	#685	C₂×C₈	1	2	yes	—
118	#685	C₂×C₈	1	2	yes	—
119	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
120	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
121	#683	C₂×C₈	1	2	yes	—
122	#683	C₂×C₈	1	2	yes	—
123	C₄×Dic₅	C₂×C₈	1	2	yes	—
124	C₄×Dic₅	C₂×C₈	1	2	yes	—
125	C₄×Dic₅	C₂²×C₄	1	2	yes	—
126	#681	—	2	2	no	—
127	C₂×#168	—	2	2	no	—
128	C₂×#168	C₂²×C₄	1	1	yes	characteristic
129	C₂×#168	C₂²×C₄	1	2	yes	—
130	C₂×#168	C₂²×C₄	1	1	yes	characteristic
131	#676	—	2	4	no	—
132	C₈×D₅	—	2	2	no	—
133	C₈×D₅	—	2	2	no	—
134	C₈×D₅	—	2	4	no	—
135	C₈×D₅	—	2	1	no	—
136	C₈×D₅	—	2	1	no	—
137	C₈×D₅	—	2	1	no	—
138	C₈×D₅	—	2	1	no	—
139	C₅×C₁₆	—	2	1	no	—
140	C₅×C₁₆	—	2	1	no	—
141	C₅×C₁₆	—	2	2	no	—
142	#672	—	2	1	no	—
143	#672	—	2	1	no	—
144	#672	—	2	2	no	—
145	C₂²×M₁₆	—	5	1	no	—
146	C₂³×C₈	—	5	1	no	—
147	C₂×#117	—	5	1	no	—
148	C₂×M₃₂	—	5	1	no	—
149	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
150	C₂²×C₁₆	—	5	1	no	—
151	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
152	#405	—	5	1	no	—
153	#405	—	5	1	no	—
154	#405	—	5	1	no	—
155	#405	—	5	2	no	—
156	#405	—	5	2	no	—
157	#405	C₂×D₅	1	1	yes	characteristic
158	C₂×#97	—	5	2	no	—
159	C₂×C₄×C₈	—	5	2	no	—
160	C₄⋊C₁₆	—	5	2	no	—
161	C₄⋊C₁₆	—	5	2	no	—
162	C₄⋊C₁₆	—	5	2	no	—
163	C₄⋊C₁₆	—	5	2	no	—
164	C₂²⋊C₁₆	—	5	1	no	—
165	C₂²⋊C₁₆	—	5	1	no	—
166	C₂²⋊C₁₆	—	5	1	no	—
167	C₂²⋊C₁₆	—	5	1	no	—
168	C₂²⋊C₁₆	—	5	2	no	—
169	C₂²⋊C₁₆	—	5	2	no	—
170	#346	—	5	2	no	—
171	#346	C₂×D₅	1	2	yes	complemented
172	C₄×C₁₆	—	5	2	no	—
173	C₄×C₁₆	C₂×D₅	1	2	yes	complemented
174	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
175	C₂²×D₅	—	2	2	no	—
176	C₂²×D₅	—	2	2	no	—
177	C₂²×D₅	—	4	1	no	—
178	C₂²×D₅	C₂²×C₈	1	1	yes	characteristic
179	C₂²×D₅	C₂²×C₈	1	1	yes	characteristic
180	C₂×C₄×C₅	#117	1	2	yes	—
181	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
182	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
183	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
185	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
186	C₂×Dic₅	—	2	2	no	—
187	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
188	C₂×Dic₅	C₂²×C₈	1	2	yes	—
189	C₂×Dic₅	C₂²×C₈	1	2	yes	—
190	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
191	C₄×D₅	—	2	4	no	—
192	C₄×D₅	—	2	1	no	—
193	C₄×D₅	—	2	1	no	—
194	C₄×D₅	—	2	1	no	—
195	C₄×D₅	—	2	1	no	—
196	C₄×D₅	—	2	2	no	—
197	C₄×D₅	—	2	2	no	—
198	C₄×D₅	—	4	2	no	—
199	C₅×C₈	#117	1	2	yes	—
200	C₅×C₈	C₂×M₁₆	1	4	yes	—
201	C₅×C₈	C₂×#41	1	1	yes	characteristic
202	C₅×C₈	C₂×#41	1	1	yes	characteristic
203	#168	—	2	1	no	—
204	#168	—	2	1	no	—
205	#168	—	2	2	no	—
206	C₂³×C₄	—	5	1	no	—
207	C₂×M₁₆	—	5	1	no	—
208	C₂×M₁₆	—	5	2	no	—
209	C₂×M₁₆	—	10	4	no	—
210	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
211	C₂²×C₈	—	5	1	no	—
212	C₂²×C₈	—	5	1	no	—
213	C₂²×C₈	—	5	1	no	—
214	C₂²×C₈	—	5	2	no	—
215	C₂²×C₈	—	5	1	no	—
216	C₂²×C₈	—	5	1	no	—
217	C₂²×C₈	—	10	2	no	—
218	C₂²×C₈	—	10	2	no	—
219	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
220	#117	—	5	1	no	—
221	#117	—	5	1	no	—
222	#117	—	5	1	no	—
223	#117	—	5	2	no	—
224	#117	—	5	2	no	—
225	#117	C₄×D₅	1	1	yes	characteristic
226	C₂×C₄⋊C₄	—	5	1	no	—
227	C₂×C₄⋊C₄	—	5	1	no	—
228	C₂×C₂²⋊C₄	—	5	2	no	—
229	C₂×C₄²	—	5	2	no	—
230	M₃₂	—	2	2	no	—
231	M₃₂	—	10	2	no	—
232	C₂×C₁₆	—	2	2	no	—
233	C₂×C₁₆	—	5	1	no	—
234	C₂×C₁₆	—	5	2	no	—
235	C₂×C₁₆	—	5	1	no	—
236	C₂×C₁₆	—	10	2	no	—
237	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
238	C₂×C₁₆	C₂²×D₅	1	2	yes	—
239	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
240	#97	—	5	2	no	—
241	#97	—	5	2	no	—
242	#97	—	5	2	no	—
243	#97	C₄×D₅	1	2	yes	—
244	C₄×C₈	—	5	2	no	—
245	C₄×C₈	—	5	2	no	—
246	C₄×C₈	—	5	2	no	—
247	C₄×C₈	C₂²×D₅	1	2	yes	—
248	C₂²×C₅	—	2	1	no	—
249	C₂²×C₅	—	2	1	no	—
250	C₂²×C₅	#431	1	2	yes	—
251	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
252	C₂×D₅	—	2	1	no	—
253	C₂×D₅	—	2	1	no	—
254	C₂×D₅	—	2	1	no	—
255	C₂×D₅	—	2	1	no	—
256	C₂×D₅	—	2	1	no	—
257	C₂×D₅	—	4	4	no	—
258	C₂×D₅	#504	1	2	yes	—
259	C₄×C₅	—	2	2	no	—
260	C₄×C₅	—	2	2	no	—
261	C₄×C₅	#431	1	2	yes	—
262	C₄×C₅	C₂×#117	1	1	yes	characteristic
263	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
264	Dic₅	—	2	2	no	—
265	Dic₅	—	2	2	no	—
266	Dic₅	—	2	1	no	—
267	Dic₅	—	2	1	no	—
268	C₂⁴	—	5	1	no	—
269	C₂²×C₄	—	5	1	no	—
270	C₂²×C₄	—	5	1	no	—
271	C₂²×C₄	—	5	1	no	—
272	C₂²×C₄	—	5	2	no	—
273	C₂²×C₄	—	5	1	no	—
274	C₂²×C₄	—	5	1	no	—
275	C₂²×C₄	—	5	2	no	—
276	C₂²×C₄	—	10	2	no	—
277	C₂²×C₄	—	10	2	no	—
278	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
279	M₁₆	—	10	2	no	—
280	M₁₆	—	10	4	no	—
281	M₁₆	#676	1	4	yes	—
282	C₂×C₈	—	5	1	no	—
283	C₂×C₈	—	5	2	no	—
284	C₂×C₈	—	5	1	no	—
285	C₂×C₈	—	10	2	no	—
286	C₂×C₈	—	10	1	no	—
287	C₂×C₈	—	10	1	no	—
288	C₂×C₈	—	10	1	no	—
289	C₂×C₈	—	10	1	no	—
290	C₂×C₈	—	10	2	no	—
291	C₂×C₈	—	10	2	no	—
292	C₂×C₈	—	10	4	no	—
293	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
294	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
295	C₂×C₈	#709	1	2	yes	—
296	C₂×C₈	#709	1	2	yes	—
297	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
298	C₄⋊C₄	—	5	2	no	—
299	C₄⋊C₄	—	5	2	no	—
300	C₄⋊C₄	—	5	1	no	—
301	C₄⋊C₄	—	5	1	no	—
302	C₄⋊C₄	C₈×D₅	1	1	yes	characteristic
303	C₄⋊C₄	C₈×D₅	1	1	yes	characteristic
304	C₂²⋊C₄	—	5	2	no	—
305	C₂²⋊C₄	—	5	2	no	—
306	C₂²⋊C₄	—	5	2	no	—
307	C₂²⋊C₄	C₈×D₅	1	2	yes	—
308	C₄²	—	5	2	no	—
309	C₄²	—	5	2	no	—
310	C₄²	—	5	2	no	—
311	C₄²	C₂×C₄×D₅	1	2	yes	—
312	C₁₆	—	2	1	no	—
313	C₁₆	—	2	1	no	—
314	C₁₆	—	2	2	no	—
315	C₁₆	—	10	1	no	—
316	C₁₆	—	10	1	no	—
317	C₁₆	—	10	2	no	—
318	C₂×C₅	—	2	2	no	—
319	C₂×C₅	C₂×#431	1	1	yes	characteristic
320	C₂×C₅	C₂×#504	1	1	yes	characteristic
321	C₂×C₅	#3407	1	1	yes	characteristic
322	D₅	—	2	2	no	—
323	D₅	—	4	1	no	—
324	C₂³	—	5	1	no	—
325	C₂³	—	5	1	no	—
326	C₂³	—	10	2	no	—
327	C₂³	—	10	2	no	—
328	C₂³	—	20	1	no	—
329	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
330	C₂×C₄	—	5	1	no	—
331	C₂×C₄	—	5	2	no	—
332	C₂×C₄	—	5	2	no	—
333	C₂×C₄	—	5	1	no	—
334	C₂×C₄	—	10	4	no	—
335	C₂×C₄	—	10	2	no	—
336	C₂×C₄	—	10	1	no	—
337	C₂×C₄	—	10	1	no	—
338	C₂×C₄	—	10	1	no	—
339	C₂×C₄	—	10	1	no	—
340	C₂×C₄	—	10	2	no	—
341	C₂×C₄	—	10	2	no	—
342	C₂×C₄	—	20	2	no	—
343	C₂×C₄	#4018	1	2	yes	—
344	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
345	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
346	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
347	C₂×C₄	C₂×#676	1	2	yes	—
348	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
349	C₈	—	10	1	no	—
350	C₈	—	10	1	no	—
351	C₈	—	10	2	no	—
352	C₈	#4018	1	2	yes	—
353	C₈	C₂×#676	1	4	yes	—
354	C₈	C₂×#709	1	1	yes	characteristic
355	C₈	C₂×#709	1	1	yes	characteristic
356	C₅	#33422	1	1	yes	characteristic, complemented, Sylow
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	5	2	no	—
360	C₂²	—	10	1	no	—
361	C₂²	—	10	1	no	—
362	C₂²	—	10	1	no	—
363	C₂²	—	10	1	no	—
364	C₂²	—	10	1	no	—
365	C₂²	—	20	4	no	—
366	C₂²	#64974	1	2	yes	—
367	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
368	C₄	—	2	2	no	—
369	C₄	—	2	2	no	—
370	C₄	—	10	1	no	—
371	C₄	—	10	1	no	—
372	C₄	—	10	2	no	—
373	C₄	—	10	2	no	—
374	C₄	#64974	1	2	yes	—
375	C₄	C₂×#4018	1	1	yes	characteristic
376	C₄	C₂²×#676	1	1	yes	characteristic
377	C₂	—	2	2	no	—
378	C₂	—	10	2	no	—
379	C₂	—	20	1	no	—
380	C₂	C₂×#64974	1	1	yes	characteristic
381	C₂	D₅×#504	1	1	yes	characteristic
382	C₂	#10613743	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#2108	C₂
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2108	D₅
hypocenter	C₅	#33422
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ #2108

Complements

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

N	H ≅ G/N	type
2. C₅×#2108	C₂	semidirect
4. #10602214	C₂	semidirect
6. #10602194	C₂	semidirect
7. #10600674	C₂	semidirect
8. #10599867	C₂	semidirect
9. #10599866	C₂	semidirect
10. #10599846	C₂	semidirect
11. #10599845	C₂	semidirect
27. C₅×#346	C₂²	semidirect
28. C₄×C₅×C₁₆	C₂²	semidirect
35. #64721	C₂²	semidirect
36. #64721	C₂²	semidirect
37. #64721	C₂²	semidirect
38. #64721	C₂²	semidirect
39. #64681	C₂²	semidirect
40. C₄×#672	C₂²	semidirect
94. #2108	D₅	semidirect
171. #346	C₂×D₅	semidirect
173. C₄×C₁₆	C₂×D₅	semidirect
356. C₅	#33422	semidirect

Maximal subgroups

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

subgroup	index
D₅×#405	2
C₅×#2108	2
#10602224	2
#10602214	2
C₂×#64727	2
#10602194	2
#10600674	2
#10599867	2
#10599866	2
#10599846	2
#10599845	2
#33422	5