Subgroups — #12252411

Order 1280.

590 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₅×#131	C₂	1	1	yes	maximal, characteristic
2	C₅×#2113	C₂	1	1	yes	maximal, characteristic
3	#10602797	C₂	1	1	yes	maximal, characteristic
4	#10602213	C₂	1	1	yes	maximal, characteristic, complemented
5	#10602212	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#64721	C₂	1	1	yes	maximal, characteristic, complemented
7	#10602192	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599887	C₂	1	2	yes	maximal, complemented
9	#10599887	C₂	1	2	yes	maximal, complemented
10	#10599886	C₂	1	2	yes	maximal, complemented
11	#10599885	C₂	1	2	yes	maximal, complemented
12	C₂×C₅×#131	C₂²	1	1	yes	characteristic
13	#41×Dic₅	C₄	1	1	yes	characteristic
14	C₂×#4094	C₄	1	1	yes	characteristic
15	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
17	#65411	C₄	1	1	yes	characteristic
18	#65411	C₄	1	2	yes	—
19	M₁₆×Dic₅	C₄	1	2	yes	—
20	M₁₆×Dic₅	C₄	1	1	yes	characteristic
21	#65390	C₂²	1	2	yes	—
22	#65390	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₅	C₂²	1	2	yes	—
24	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
25	C₂×#3944	C₂²	1	1	yes	characteristic
26	C₂²×#672	C₂²	1	1	yes	characteristic
27	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
29	C₂²⋊#672	C₂²	1	2	yes	—
30	C₂²⋊#672	C₂²	1	2	yes	—
31	#64722	C₂²	1	2	yes	—
32	#64722	C₂²	1	2	yes	—
33	#64721	C₂²	1	1	yes	characteristic
34	#64721	C₂²	1	2	yes	—
35	#64721	C₂²	1	1	yes	characteristic
36	#64721	C₂²	1	1	yes	characteristic
37	#64721	C₂²	1	1	yes	characteristic
38	#64721	C₂²	1	2	yes	—
39	C₁₆×Dic₅	C₂²	1	2	yes	—
40	C₁₆×Dic₅	C₂²	1	2	yes	—
41	#33423	—	5	1	no	maximal, Sylow
42	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
43	C₅×#131	—	2	2	no	—
44	C₅×#131	—	2	2	no	—
45	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
47	C₂²×C₅×C₈	C₂³	1	2	yes	—
48	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
49	#4094	—	2	4	no	—
50	Q₈×Dic₅	C₈	1	1	yes	characteristic
51	Q₈×Dic₅	C₈	1	1	yes	characteristic
52	D₄×Dic₅	C₈	1	2	yes	—
53	D₄×Dic₅	C₈	1	1	yes	characteristic
54	D₄×Dic₅	C₈	1	2	yes	—
55	D₄×Dic₅	C₈	1	1	yes	characteristic
56	#4072	C₂×C₄	1	2	yes	—
57	#4072	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
59	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
60	C₂×#681	C₂×C₄	1	2	yes	—
61	C₂×#681	C₂×C₄	1	1	yes	characteristic
62	C₂²×#168	C₂×C₄	1	1	yes	characteristic
63	C₂²×#168	C₂×C₄	1	2	yes	—
64	C₅×M₃₂	—	2	2	no	—
65	C₂×C₅×C₁₆	—	2	2	no	—
66	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
68	C₂×C₅×C₁₆	C₂³	1	2	yes	—
69	#3947	C₂×C₄	1	2	yes	—
70	#3947	C₂×C₄	1	2	yes	—
71	#3947	C₂×C₄	1	2	yes	—
72	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₅	C₂×C₄	1	2	yes	—
74	C₈×Dic₅	C₂×C₄	1	2	yes	—
75	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
76	C₈×Dic₅	C₂³	1	1	yes	characteristic
77	C₈×Dic₅	C₂³	1	2	yes	—
78	C₈×Dic₅	C₂³	1	1	yes	characteristic
79	#3944	—	2	2	no	—
80	C₂×#672	—	2	2	no	—
81	C₂×#672	C₂³	1	1	yes	characteristic
82	C₂×#672	C₂³	1	1	yes	characteristic
83	C₂×#672	C₂³	1	2	yes	—
84	C₄×#131	—	5	1	no	—
85	#2164	—	5	2	no	—
86	#2164	—	5	2	no	—
87	#2163	—	5	2	no	—
88	#2162	—	5	2	no	—
89	#2150	—	5	1	no	—
90	C₂×C₄⋊C₁₆	—	5	1	no	—
91	#2113	—	5	1	no	—
92	#2113	D₅	1	1	yes	characteristic
93	#2108	—	5	1	no	—
94	#2108	—	5	1	no	—
95	C₅×#41	—	2	2	no	—
96	C₅×#41	—	2	2	no	—
97	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
98	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
99	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
102	C₂²×Dic₅	C₂×C₈	1	2	yes	—
103	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
104	C₂²×Dic₅	C₂×C₈	1	2	yes	—
105	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
106	C₅×M₁₆	—	2	2	no	—
107	C₅×M₁₆	—	2	4	no	—
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	—	2	2	no	—
110	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₅×C₈	#41	1	2	yes	—
113	C₂×C₅×C₈	#41	1	2	yes	—
114	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
115	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
116	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
117	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
118	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
119	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
120	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
121	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
122	C₄×Dic₅	C₂×C₈	1	2	yes	—
123	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
124	C₄×Dic₅	C₂×C₈	1	2	yes	—
125	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
126	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
127	C₄×Dic₅	C₂²×C₄	1	2	yes	—
128	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
129	#681	—	2	4	no	—
130	#681	M₁₆	1	4	yes	—
131	C₂×#168	—	2	2	no	—
132	C₂×#168	—	2	4	no	—
133	C₂×#168	C₂²×C₄	1	1	yes	characteristic
134	C₂×#168	C₂²×C₄	1	1	yes	characteristic
135	C₂×#168	C₂²×C₄	1	2	yes	—
136	C₅×C₁₆	—	2	1	no	—
137	C₅×C₁₆	—	2	1	no	—
138	C₅×C₁₆	—	2	2	no	—
139	#672	—	2	1	no	—
140	#672	—	2	1	no	—
141	#672	—	2	2	no	—
142	C₂×#131	—	5	1	no	—
143	C₂×#131	C₂×D₅	1	1	yes	characteristic
144	C₄×#41	—	5	1	no	—
145	C₂×M₃₂	—	5	1	no	—
146	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
147	C₂²×C₁₆	—	5	1	no	—
148	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
149	#405	—	5	1	no	—
150	#405	—	5	2	no	—
151	#405	—	5	1	no	—
152	#405	—	5	2	no	—
153	C₄×M₁₆	—	5	1	no	—
154	C₄×M₁₆	—	5	2	no	—
155	C₂×C₄×C₈	—	5	1	no	—
156	C₂×C₄×C₈	—	5	2	no	—
157	C₄⋊C₁₆	—	5	1	no	—
158	C₄⋊C₁₆	—	5	1	no	—
159	C₄⋊C₁₆	—	5	1	no	—
160	C₄⋊C₁₆	—	5	1	no	—
161	C₄⋊C₁₆	—	5	2	no	—
162	C₄⋊C₁₆	—	5	2	no	—
163	C₂²⋊C₁₆	—	5	2	no	—
164	C₂²⋊C₁₆	—	5	2	no	—
165	C₂²⋊C₁₆	C₂×D₅	1	2	yes	—
166	C₂²⋊C₁₆	C₂×D₅	1	2	yes	—
167	#346	—	5	2	no	—
168	#346	—	5	2	no	—
169	C₄×C₁₆	—	5	2	no	—
170	C₄×C₁₆	—	5	2	no	—
171	C₂³×C₅	C₂²×C₈	1	2	yes	—
172	C₂³×C₅	C₂²×C₈	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	2	no	—
178	C₂×C₄×C₅	#117	1	2	yes	—
179	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
180	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
182	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
183	C₂×Dic₅	—	2	2	no	—
184	C₂×Dic₅	—	2	4	no	—
185	C₂×Dic₅	—	4	1	no	—
186	C₂×Dic₅	—	4	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	1	yes	characteristic
190	C₂×Dic₅	C₂²×C₈	1	2	yes	—
191	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
192	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
193	C₅×C₈	—	2	2	no	—
194	C₅×C₈	#117	1	2	yes	—
195	C₅×C₈	C₂×#41	1	1	yes	characteristic
196	C₅×C₈	C₂×#41	1	1	yes	characteristic
197	#168	—	2	1	no	—
198	#168	—	2	1	no	—
199	#168	C₂×M₁₆	1	4	yes	—
200	C₂×#41	C₄×D₅	1	1	yes	characteristic
201	#131	—	2	2	no	—
202	#131	—	2	2	no	—
203	#131	—	10	4	no	—
204	C₂×M₁₆	—	5	1	no	—
205	C₂×M₁₆	—	5	2	no	—
206	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
207	C₂×M₁₆	C₄×D₅	1	2	yes	—
208	C₂²×C₈	—	5	2	no	—
209	C₂²×C₈	—	5	1	no	—
210	C₂²×C₈	C₂²×D₅	1	2	yes	—
211	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
212	C₄×Q₈	—	5	1	no	—
213	C₄×Q₈	—	5	1	no	—
214	C₄×D₄	—	5	2	no	—
215	C₄×D₄	—	5	1	no	—
216	C₄×D₄	—	5	2	no	—
217	C₄×D₄	—	5	1	no	—
218	#117	—	5	2	no	—
219	#117	—	5	1	no	—
220	C₂×C₄²	—	5	2	no	—
221	C₂×C₄²	—	5	1	no	—
222	M₃₂	—	2	2	no	—
223	M₃₂	—	10	2	no	—
224	C₂×C₁₆	—	2	2	no	—
225	C₂×C₁₆	—	5	1	no	—
226	C₂×C₁₆	—	5	2	no	—
227	C₂×C₁₆	—	5	1	no	—
228	C₂×C₁₆	—	10	2	no	—
229	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
230	C₂×C₁₆	C₂²×D₅	1	2	yes	—
231	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
232	#97	—	5	2	no	—
233	#97	—	5	2	no	—
234	#97	—	5	2	no	—
235	C₄×C₈	—	5	2	no	—
236	C₄×C₈	—	5	1	no	—
237	C₄×C₈	—	5	2	no	—
238	C₄×C₈	—	5	1	no	—
239	C₄×C₈	—	5	2	no	—
240	C₄×C₈	—	5	1	no	—
241	C₄×C₈	—	5	1	no	—
242	C₂²×C₅	—	2	1	no	—
243	C₂²×C₅	—	2	1	no	—
244	C₂²×C₅	—	2	2	no	—
245	C₂²×C₅	—	2	2	no	—
246	C₂²×C₅	—	2	2	no	—
247	C₂²×C₅	#431	1	2	yes	—
248	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
249	C₄×C₅	—	2	2	no	—
250	C₄×C₅	#431	1	2	yes	—
251	C₄×C₅	C₂×#117	1	1	yes	characteristic
252	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
253	Dic₅	—	2	1	no	—
254	Dic₅	—	2	2	no	—
255	Dic₅	—	2	1	no	—
256	Dic₅	—	2	1	no	—
257	Dic₅	—	2	2	no	—
258	Dic₅	#504	1	2	yes	—
259	#41	—	2	2	no	—
260	#41	—	2	2	no	—
261	C₂×Q₈	C₈×D₅	1	1	yes	characteristic
262	C₂×D₄	C₈×D₅	1	2	yes	—
263	C₂×D₄	C₈×D₅	1	1	yes	characteristic
264	C₂²×C₄	—	5	2	no	—
265	C₂²×C₄	—	5	1	no	—
266	C₂²×C₄	—	5	2	no	—
267	C₂²×C₄	—	5	1	no	—
268	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
269	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
270	M₁₆	—	2	2	no	—
271	M₁₆	—	2	4	no	—
272	M₁₆	—	5	4	no	—
273	M₁₆	—	10	4	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	5	2	no	—
277	C₂×C₈	—	5	1	no	—
278	C₂×C₈	—	5	1	no	—
279	C₂×C₈	—	10	4	no	—
280	C₂×C₈	—	10	2	no	—
281	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
282	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
283	C₂×C₈	#711	1	2	yes	—
284	C₂×C₈	#711	1	2	yes	—
285	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
286	C₄⋊C₄	—	5	2	no	—
287	C₄⋊C₄	—	5	2	no	—
288	C₄⋊C₄	—	5	1	no	—
289	C₄⋊C₄	—	5	1	no	—
290	C₂²⋊C₄	—	5	2	no	—
291	C₂²⋊C₄	—	5	2	no	—
292	C₂²⋊C₄	—	5	2	no	—
293	C₄²	—	5	1	no	—
294	C₄²	—	5	2	no	—
295	C₄²	—	5	1	no	—
296	C₄²	—	5	2	no	—
297	C₄²	—	5	1	no	—
298	C₄²	—	5	2	no	—
299	C₄²	—	5	1	no	—
300	C₁₆	—	2	1	no	—
301	C₁₆	—	2	1	no	—
302	C₁₆	—	2	2	no	—
303	C₁₆	—	10	1	no	—
304	C₁₆	—	10	1	no	—
305	C₁₆	—	10	2	no	—
306	C₂×C₅	—	2	2	no	—
307	C₂×C₅	—	4	2	no	—
308	C₂×C₅	C₂×#431	1	1	yes	characteristic
309	C₂×C₅	C₂×#504	1	1	yes	characteristic
310	C₂×C₅	#3407	1	1	yes	characteristic
311	C₂³	C₂×C₈×D₅	1	2	yes	—
312	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
313	Q₈	—	2	2	no	—
314	D₄	—	2	4	no	—
315	D₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	5	1	no	—
319	C₂×C₄	—	5	1	no	—
320	C₂×C₄	—	5	2	no	—
321	C₂×C₄	—	5	1	no	—
322	C₂×C₄	—	5	2	no	—
323	C₂×C₄	—	5	1	no	—
324	C₂×C₄	—	10	2	no	—
325	C₂×C₄	—	10	4	no	—
326	C₂×C₄	—	20	1	no	—
327	C₂×C₄	—	20	2	no	—
328	C₂×C₄	#4023	1	2	yes	—
329	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
330	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
331	C₂×C₄	D₅×M₁₆	1	2	yes	—
332	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	5	4	no	—
335	C₈	—	10	1	no	—
336	C₈	—	10	1	no	—
337	C₈	#4023	1	2	yes	—
338	C₈	C₂×#711	1	1	yes	characteristic
339	C₈	C₂×#711	1	1	yes	characteristic
340	C₅	#33423	1	1	yes	characteristic, complemented, Sylow
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	#65007	1	2	yes	—
347	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	5	2	no	—
350	C₄	—	10	1	no	—
351	C₄	—	10	2	no	—
352	C₄	—	10	1	no	—
353	C₄	—	10	1	no	—
354	C₄	—	10	2	no	—
355	C₄	#65007	1	2	yes	—
356	C₄	C₂×#4023	1	1	yes	characteristic
357	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
358	C₂	—	2	2	no	—
359	C₂	—	4	2	no	—
360	C₂	C₂×#65007	1	1	yes	characteristic
361	C₂	D₅×#504	1	1	yes	characteristic
362	C₂	#10613743	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#2113	C₂
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2113	D₅
hypocenter	C₅	#33423
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ #2113

Complements

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

N	H ≅ G/N	type
4. #10602213	C₂	semidirect
5. #10602212	C₂	semidirect
6. C₂×#64721	C₂	semidirect
7. #10602192	C₂	semidirect
8. #10599887	C₂	semidirect
9. #10599887	C₂	semidirect
10. #10599886	C₂	semidirect
11. #10599885	C₂	semidirect
340. C₅	#33423	semidirect

Maximal subgroups

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

subgroup	index
Dic₅×#131	2
C₅×#2113	2
#10602797	2
#10602213	2
#10602212	2
C₂×#64721	2
#10602192	2
#10599887	2
#10599886	2
#10599885	2
#33423	5