Subgroups — #12005248

Order 1280.

714 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3951	C₂	1	1	yes	maximal, characteristic
3	C₂×#65390	C₂	1	1	yes	maximal, characteristic
4	#10602787	C₂	1	2	yes	maximal
5	C₂×#64776	C₂	1	2	yes	maximal
6	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#4072	C₂²	1	1	yes	characteristic
9	C₂²×#681	C₂²	1	2	yes	—
10	C₂²×#681	C₂²	1	1	yes	characteristic
11	C₂×C₅×#108	—	2	1	no	—
12	C₂×C₅×#108	—	2	1	no	—
13	C₂×C₅×#108	—	2	2	no	—
14	C₂×C₅×#108	C₄	1	4	yes	complemented
15	C₂×#3951	—	2	2	no	—
16	C₂×#3951	—	2	2	no	—
17	C₂×#3951	C₄	1	4	yes	complemented
18	#65390	—	2	2	no	—
19	#65390	C₄	1	4	yes	—
20	C₂×#3947	—	2	1	no	—
21	C₂×C₈×Dic₅	—	2	1	no	—
22	#64778	—	2	4	no	—
23	#64776	—	2	4	no	—
24	#10873	—	5	1	no	maximal, Sylow
25	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	—	2	2	no	—
31	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
32	C₂²×C₅×C₈	—	2	2	no	—
33	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₅×C₈	D₄	1	2	yes	—
37	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₅×C₈	D₄	1	2	yes	—
40	C₂²×C₅×C₈	Q₈	1	2	yes	—
41	C₂×C₂²⋊Dic₅	Q₈	1	2	yes	—
42	#4072	—	2	1	no	—
43	#4072	—	2	1	no	—
44	#4072	C₂×C₄	1	2	yes	—
45	#4072	D₄	1	2	yes	—
46	C₂×C₄⋊Dic₅	—	2	1	no	—
47	C₂×C₄×Dic₅	—	2	1	no	—
48	C₂×#681	—	2	1	no	—
49	C₂×#681	—	2	1	no	—
50	C₂×#681	—	2	1	no	—
51	C₂×#681	—	2	2	no	—
52	C₂×#681	—	2	2	no	—
53	C₂×#681	—	2	2	no	—
54	C₂×#681	—	2	2	no	—
55	C₂×#681	—	2	2	no	—
56	C₂×#681	C₂×C₄	1	4	yes	—
57	C₂×#681	C₂×C₄	1	2	yes	—
58	C₂×#681	D₄	1	2	yes	—
59	C₂×#681	Q₈	1	2	yes	—
60	C₂²×#168	—	2	1	no	—
61	C₂²×#168	—	2	2	no	—
62	C₅×#108	—	2	4	no	—
63	C₅×#108	—	2	4	no	—
64	#3951	—	2	8	no	—
65	#3947	—	2	4	no	—
66	C₈×Dic₅	—	2	4	no	—
67	C₂²×#108	—	5	1	no	—
68	C₂²×#108	D₅	1	1	yes	characteristic
69	C₂×#405	—	5	1	no	—
70	C₂×#344	—	5	2	no	—
71	#1738	—	5	2	no	—
72	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₅	#41	1	1	yes	characteristic
84	C₂²×C₄×C₅	#41	1	1	yes	characteristic
85	C₂²×C₄×C₅	#41	1	1	yes	characteristic
86	C₂²×C₄×C₅	#41	1	1	yes	characteristic
87	C₂²×Dic₅	—	2	2	no	—
88	C₅×M₁₆	—	2	4	no	—
89	C₅×M₁₆	—	2	4	no	—
90	C₅×M₁₆	—	4	2	no	—
91	C₅×M₁₆	—	4	2	no	—
92	C₂×C₅×C₈	—	2	4	no	—
93	C₂×C₅×C₈	—	4	2	no	—
94	C₂×C₅×C₈	—	4	2	no	—
95	C₂×C₅×C₈	C₄²	1	4	yes	—
96	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
97	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
98	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
99	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
100	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
101	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
102	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
103	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
104	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
105	C₂²⋊Dic₅	—	2	2	no	—
106	C₂²⋊Dic₅	C₄⋊C₄	1	4	yes	—
107	C₄⋊Dic₅	—	2	2	no	—
108	C₄⋊Dic₅	—	2	2	no	—
109	C₄×Dic₅	—	2	2	no	—
110	C₄×Dic₅	—	2	2	no	—
111	#681	—	2	2	no	—
112	#681	—	2	2	no	—
113	#681	—	2	2	no	—
114	#681	—	2	4	no	—
115	#681	—	2	4	no	—
116	#681	—	4	2	no	—
117	#681	—	4	2	no	—
118	#681	C₄⋊C₄	1	4	yes	—
119	C₂×#168	—	2	2	no	—
120	C₂×#168	—	2	2	no	—
121	C₂×#168	—	2	2	no	—
122	C₂×#168	—	2	4	no	—
123	C₂×#168	—	4	2	no	—
124	C₂×#168	—	4	2	no	—
125	C₂²×M₁₆	—	5	1	no	—
126	C₂²×M₁₆	—	5	2	no	—
127	C₂²×M₁₆	C₂×D₅	1	2	yes	—
128	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
129	C₂×#117	—	5	1	no	—
130	C₂×#108	—	2	1	no	—
131	C₂×#108	—	2	1	no	—
132	C₂×#108	—	2	2	no	—
133	C₂×#108	—	5	4	no	—
134	C₂×#108	—	10	2	no	—
135	C₂×#108	—	10	2	no	—
136	C₂×#108	Dic₅	1	4	yes	complemented
137	#405	—	5	4	no	—
138	#405	—	10	2	no	—
139	C₂×#97	—	10	1	no	—
140	C₂×C₄×C₈	—	10	1	no	—
141	#344	—	10	4	no	—
142	#341	—	10	4	no	—
143	C₂³×C₅	—	4	2	no	—
144	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
146	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
147	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
148	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
149	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
150	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅	#117	1	1	yes	characteristic
160	C₂×C₄×C₅	#117	1	1	yes	characteristic
161	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
166	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
167	C₂×C₄×C₅	#121	1	1	yes	characteristic
168	C₂×C₄×C₅	#121	1	1	yes	characteristic
169	C₂×C₄×C₅	#121	1	1	yes	characteristic
170	C₂×C₄×C₅	#121	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
173	C₂×C₄×C₅	#124	1	1	yes	characteristic
174	C₂×C₄×C₅	#124	1	1	yes	characteristic
175	C₂×C₄×C₅	#125	1	1	yes	characteristic
176	C₂×C₄×C₅	#125	1	1	yes	characteristic
177	C₂×C₄×C₅	#127	1	1	yes	characteristic
178	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
179	C₂×Dic₅	—	2	4	no	—
180	C₂×Dic₅	—	4	4	no	—
181	C₅×C₈	—	4	4	no	—
182	C₅×C₈	#95	1	4	yes	—
183	C₅×C₈	#95	1	4	yes	—
184	#168	—	2	4	no	—
185	#168	—	4	4	no	—
186	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	5	2	no	—
193	C₂×M₁₆	—	5	2	no	—
194	C₂×M₁₆	—	5	2	no	—
195	C₂×M₁₆	—	5	4	no	—
196	C₂×M₁₆	—	10	1	no	—
197	C₂×M₁₆	—	10	1	no	—
198	C₂×M₁₆	—	10	1	no	—
199	C₂×M₁₆	—	10	2	no	—
200	C₂×M₁₆	—	10	2	no	—
201	C₂×M₁₆	—	10	2	no	—
202	C₂×M₁₆	—	10	2	no	—
203	C₂×M₁₆	—	10	2	no	—
204	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	10	1	no	—
207	C₂²×C₈	—	10	2	no	—
208	C₂²×C₈	Dic₁₀	1	2	yes	—
209	C₂²×C₈	C₄×D₅	1	2	yes	—
210	C₂²×C₈	C₄×D₅	1	2	yes	—
211	C₂²×C₈	D₂₀	1	2	yes	—
212	C₂²×C₈	C₂×Dic₅	1	2	yes	—
213	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
214	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
215	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
216	#117	—	5	2	no	—
217	#117	—	5	2	no	—
218	#117	—	10	1	no	—
219	#117	—	10	1	no	—
220	C₂×C₄⋊C₄	—	10	1	no	—
221	C₂×C₂²⋊C₄	—	5	2	no	—
222	C₂×C₄²	—	10	1	no	—
223	#108	—	2	4	no	—
224	#108	—	2	4	no	—
225	#108	—	10	8	no	—
226	#97	—	10	4	no	—
227	C₄×C₈	—	10	4	no	—
228	C₂²×C₅	—	2	2	no	—
229	C₂²×C₅	—	2	2	no	—
230	C₂²×C₅	—	2	2	no	—
231	C₂²×C₅	—	4	1	no	—
232	C₂²×C₅	—	4	1	no	—
233	C₂²×C₅	—	4	1	no	—
234	C₂²×C₅	—	4	1	no	—
235	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
236	C₂²×C₅	#380	1	1	yes	characteristic
237	C₂²×C₅	#384	1	1	yes	characteristic
238	C₂²×C₅	#389	1	1	yes	characteristic
239	C₂²×C₅	#389	1	1	yes	characteristic
240	C₂²×C₅	#391	1	1	yes	characteristic
241	C₂²×C₅	#391	1	1	yes	characteristic
242	C₄×C₅	C₂×#95	1	1	yes	characteristic
243	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
244	C₄×C₅	#380	1	1	yes	characteristic
245	C₄×C₅	#382	1	1	yes	characteristic
246	C₄×C₅	#389	1	1	yes	characteristic
247	C₄×C₅	#389	1	1	yes	characteristic
248	C₄×C₅	#390	1	1	yes	characteristic
249	C₄×C₅	#390	1	1	yes	characteristic
250	Dic₅	—	4	4	no	—
251	C₂⁴	Q₈×D₅	1	1	yes	characteristic
252	C₂²×C₄	—	10	2	no	—
253	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
257	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
259	C₂²×C₄	#711	1	1	yes	characteristic
260	C₂²×C₄	#711	1	1	yes	characteristic
261	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
262	C₂²×C₄	#713	1	1	yes	characteristic
263	C₂²×C₄	#713	1	1	yes	characteristic
264	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
267	M₁₆	—	2	4	no	—
268	M₁₆	—	2	4	no	—
269	M₁₆	—	4	2	no	—
270	M₁₆	—	4	2	no	—
271	M₁₆	—	5	4	no	—
272	M₁₆	—	10	2	no	—
273	M₁₆	—	10	2	no	—
274	M₁₆	—	10	2	no	—
275	M₁₆	—	10	4	no	—
276	M₁₆	—	10	4	no	—
277	M₁₆	—	20	2	no	—
278	M₁₆	—	20	2	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	4	2	no	—
281	C₂×C₈	—	4	2	no	—
282	C₂×C₈	—	10	2	no	—
283	C₂×C₈	—	10	2	no	—
284	C₂×C₈	—	10	2	no	—
285	C₂×C₈	—	10	4	no	—
286	C₂×C₈	—	20	2	no	—
287	C₂×C₈	—	20	2	no	—
288	C₂×C₈	C₄×Dic₅	1	4	yes	—
289	C₂×C₈	#683	1	2	yes	—
290	C₂×C₈	#683	1	2	yes	—
291	C₂×C₈	#683	1	4	yes	—
292	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
293	C₂×C₈	#685	1	2	yes	—
294	C₂×C₈	#685	1	2	yes	—
295	C₂×C₈	#685	1	4	yes	—
296	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
297	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
298	C₄⋊C₄	—	10	2	no	—
299	C₄⋊C₄	—	10	2	no	—
300	C₂²⋊C₄	—	5	4	no	—
301	C₂²⋊C₄	—	10	2	no	—
302	C₄²	—	10	2	no	—
303	C₄²	—	10	2	no	—
304	C₂×C₅	—	2	1	no	—
305	C₂×C₅	—	2	1	no	—
306	C₂×C₅	—	2	1	no	—
307	C₂×C₅	—	2	1	no	—
308	C₂×C₅	—	2	1	no	—
309	C₂×C₅	—	2	1	no	—
310	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
311	C₂×C₅	#1953	1	2	yes	—
312	C₂³	—	4	2	no	—
313	C₂³	#4033	1	1	yes	characteristic
314	C₂³	#4034	1	1	yes	characteristic
315	C₂³	D₅×C₄⋊C₄	1	1	yes	characteristic
316	C₂³	#4043	1	1	yes	characteristic
317	C₂³	#4090	1	1	yes	characteristic
318	C₂³	Q₈×Dic₅	1	1	yes	characteristic
319	C₂³	#4092	1	1	yes	characteristic
320	C₂×C₄	—	10	4	no	—
321	C₂×C₄	—	20	4	no	—
322	C₂×C₄	#4033	1	1	yes	characteristic
323	C₂×C₄	#4034	1	1	yes	characteristic
324	C₂×C₄	#4036	1	1	yes	characteristic
325	C₂×C₄	#4036	1	1	yes	characteristic
326	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
327	C₂×C₄	#4038	1	1	yes	characteristic
328	C₂×C₄	#4038	1	1	yes	characteristic
329	C₂×C₄	#4039	1	1	yes	characteristic
330	C₂×C₄	#4039	1	1	yes	characteristic
331	C₂×C₄	#4041	1	1	yes	characteristic
332	C₂×C₄	#4041	1	1	yes	characteristic
333	C₂×C₄	#4043	1	1	yes	characteristic
334	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
335	C₂×C₄	C₂×#683	1	1	yes	characteristic
336	C₂×C₄	C₂×#683	1	1	yes	characteristic
337	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
338	C₂×C₄	C₂×#685	1	1	yes	characteristic
339	C₂×C₄	C₂×#685	1	1	yes	characteristic
340	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
341	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
342	C₂×C₄	#4082	1	1	yes	characteristic
343	C₂×C₄	#4084	1	1	yes	characteristic
344	C₂×C₄	#4084	1	1	yes	characteristic
345	C₂×C₄	#4086	1	1	yes	characteristic
346	C₂×C₄	Q₈×Dic₅	1	1	yes	characteristic
347	C₂×C₄	#4092	1	1	yes	characteristic
348	C₂×C₄	#4093	1	1	yes	characteristic
349	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
350	C₈	—	4	4	no	—
351	C₈	—	10	4	no	—
352	C₈	—	20	4	no	—
353	C₈	#3963	1	4	yes	—
354	C₈	#3963	1	4	yes	—
355	C₅	#10873	1	1	yes	characteristic, complemented, Sylow
356	C₂²	—	2	2	no	—
357	C₂²	—	2	2	no	—
358	C₂²	—	2	2	no	—
359	C₂²	—	4	1	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	4	1	no	—
362	C₂²	—	4	1	no	—
363	C₂²	#65262	1	1	yes	characteristic
364	C₂²	#65263	1	1	yes	characteristic
365	C₂²	C₄⋊C₄×Dic₅	1	1	yes	characteristic
366	C₂²	#65265	1	1	yes	characteristic
367	C₂²	#65274	1	1	yes	characteristic
368	C₂²	#65276	1	1	yes	characteristic
369	C₂²	#65519	1	1	yes	characteristic
370	C₄	—	20	4	no	—
371	C₄	#65262	1	1	yes	characteristic
372	C₄	C₄⋊C₄×Dic₅	1	1	yes	characteristic
373	C₄	#65266	1	1	yes	characteristic
374	C₄	#65274	1	1	yes	characteristic
375	C₄	#65276	1	1	yes	characteristic
376	C₄	#65277	1	1	yes	characteristic
377	C₄	C₂×#3963	1	1	yes	characteristic
378	C₄	#65510	1	1	yes	characteristic
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	2	1	no	—
382	C₂	—	2	1	no	—
383	C₂	—	2	1	no	—
384	C₂	—	2	1	no	—
385	C₂	#10601499	1	2	yes	—
386	C₂	#10602356	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#4084
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂²×C₅×#108	C₂
socle	C₂²×C₅	#389
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂²×#108	D₅
hypocenter	C₅	#10873
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
14. C₂×C₅×#108	C₄	semidirect
17. C₂×#3951	C₄	semidirect
136. C₂×#108	Dic₅	semidirect
355. C₅	#10873	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×#108	2
C₂²×#3951	2
C₂×#65390	2
#10602787	2
C₂×#64776	2
#10873	5