Subgroups — #12069508

Order 1280.

666 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2960	C₂	1	1	yes	maximal, characteristic, complemented
2	#10615082	C₂	1	1	yes	maximal, characteristic
3	C₂×#64977	C₂	1	1	yes	maximal, characteristic
4	#10602617	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601743	C₂	1	4	yes	maximal, complemented
6	#10601726	C₂	1	4	yes	maximal, complemented
7	#10601289	C₂	1	1	yes	maximal, characteristic
8	#10601286	C₂	1	1	yes	maximal, characteristic, complemented
9	#10600517	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#533	C₂²	1	1	yes	characteristic
11	C₂²×#676	—	2	1	no	—
12	C₂×C₄×D₂₀	C₂²	1	1	yes	characteristic
13	C₅×#446	—	2	1	no	—
14	C₅×#446	C₂²	1	2	yes	complemented
15	C₅×C₈×Q₈	—	2	1	no	—
16	C₅×C₈×Q₈	C₂²	1	2	yes	complemented
17	C₅×#424	—	2	1	no	—
18	C₅×#423	C₂²	1	1	yes	characteristic
19	C₅×#405	—	2	1	no	—
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₂×#3952	—	2	1	no	—
22	#65373	C₄	1	2	yes	—
23	#65372	C₄	1	2	yes	—
24	#65335	C₄	1	2	yes	—
25	#65332	C₄	1	2	yes	—
26	C₄×#688	C₂²	1	4	yes	—
27	#65214	C₂²	1	1	yes	characteristic
28	#65214	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
30	#64977	—	2	1	no	—
31	#64977	—	2	1	no	—
32	#64977	C₂²	1	2	yes	—
33	#64977	C₂²	1	2	yes	—
34	Q₈⋊#168	C₂²	1	4	yes	complemented
35	#64703	C₂²	1	4	yes	—
36	#64700	C₂²	1	4	yes	complemented
37	#64675	C₂²	1	2	yes	complemented
38	C₈×#168	C₂²	1	2	yes	complemented
39	#17930	—	5	1	no	maximal, Sylow
40	C₂²×D₂₀	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×D₅	—	2	1	no	—
42	C₂×C₅×M₁₆	—	2	1	no	—
43	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
44	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
45	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
46	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
47	C₅×#125	—	2	1	no	—
48	C₅×C₂²⋊Q₈	—	2	1	no	—
49	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
50	C₄×C₅×Q₈	—	2	1	no	—
51	C₄×C₅×Q₈	C₂³	1	2	yes	—
52	C₅×#117	—	2	1	no	—
53	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
54	C₂×#688	C₂×C₄	1	4	yes	—
55	C₂×#685	—	2	1	no	—
56	C₂×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
57	C₂²×#168	—	2	1	no	—
58	C₂×#676	—	2	2	no	—
59	C₂×#676	—	2	2	no	—
60	C₂×#676	—	4	4	no	—
61	C₄×D₂₀	—	2	1	no	—
62	C₄×D₂₀	—	2	1	no	—
63	C₄×D₂₀	C₂×C₄	1	2	yes	—
64	C₄×D₂₀	C₂³	1	2	yes	—
65	C₅×C₄⋊C₈	—	2	2	no	—
66	C₅×C₄⋊C₈	—	2	2	no	—
67	C₅×C₄⋊C₈	—	2	1	no	—
68	C₅×C₄⋊C₈	C₂³	1	2	yes	—
69	C₅×#97	—	2	2	no	—
70	C₄×C₅×C₈	—	2	2	no	—
71	C₄×C₅×C₈	D₄	1	1	yes	characteristic
72	C₄×C₅×C₈	D₄	1	1	yes	characteristic
73	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
74	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
75	#3967	C₂×C₄	1	4	yes	—
76	C₂²⋊#168	—	2	1	no	—
77	C₂²⋊#168	C₂×C₄	1	2	yes	—
78	#3952	—	2	2	no	—
79	#3952	—	2	2	no	—
80	#3941	C₂×C₄	1	4	yes	—
81	#3940	C₂×C₄	1	4	yes	—
82	C₄⋊#168	—	2	1	no	—
83	C₄⋊#168	—	2	1	no	—
84	C₄⋊#168	C₂×C₄	1	2	yes	—
85	C₄⋊#168	C₂³	1	2	yes	—
86	C₄×#168	C₂³	1	2	yes	—
87	C₄×#168	C₂³	1	2	yes	—
88	#3101	—	5	1	no	—
89	#2960	D₅	1	1	yes	characteristic, complemented
90	C₂×#436	—	5	1	no	—
91	#1590	—	5	4	no	—
92	#1583	—	5	4	no	—
93	#1570	—	5	1	no	—
94	#1486	—	5	1	no	—
95	#1481	—	5	1	no	—
96	#1450	—	5	1	no	—
97	C₂³×D₅	—	2	1	no	—
98	C₂×C₅×Q₈	—	2	1	no	—
99	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅	#41	1	1	yes	characteristic
103	C₂²×Dic₅	—	2	1	no	—
104	C₂×D₂₀	—	2	1	no	—
105	C₂×D₂₀	—	4	2	no	—
106	C₂×D₂₀	C₂²×C₄	1	2	yes	—
107	C₂×C₄×D₅	—	2	2	no	—
108	C₂×C₄×D₅	—	2	2	no	—
109	C₂×C₄×D₅	—	4	2	no	—
110	C₂×C₄×D₅	—	4	2	no	—
111	C₅×M₁₆	—	4	2	no	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	SD₁₆	1	4	yes	complemented
115	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
119	C₅×C₄⋊C₄	—	2	1	no	—
120	C₅×C₄⋊C₄	—	2	1	no	—
121	C₅×C₄⋊C₄	—	2	1	no	—
122	C₅×C₄⋊C₄	—	2	2	no	—
123	C₅×C₄⋊C₄	—	2	2	no	—
124	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
125	C₅×C₂²⋊C₄	—	2	2	no	—
126	C₄²×C₅	—	2	2	no	—
127	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₅	#41	1	1	yes	characteristic
130	C₄²×C₅	C₂⁴	1	1	yes	characteristic
131	#688	—	4	4	no	—
132	#685	—	2	2	no	—
133	#685	—	2	2	no	—
134	C₄⋊Dic₅	—	2	1	no	—
135	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
136	C₂×#168	—	2	2	no	—
137	C₂×#168	—	2	2	no	—
138	C₂×#168	—	4	2	no	—
139	C₂×#168	C₂²×C₄	1	4	yes	—
140	#676	—	4	4	no	—
141	#676	—	4	4	no	—
142	C₂²×M₁₆	—	10	1	no	—
143	#533	C₂×D₅	1	1	yes	characteristic
144	C₂×C₄×D₄	—	5	1	no	—
145	#483	—	5	2	no	—
146	#476	—	5	2	no	—
147	#461	—	5	2	no	—
148	#450	—	5	2	no	—
149	#446	—	2	1	no	—
150	#446	C₂×D₅	1	2	yes	complemented
151	C₈×Q₈	—	2	1	no	—
152	C₈×Q₈	C₂×D₅	1	2	yes	complemented
153	C₄×SD₁₆	—	5	4	no	—
154	#436	—	5	2	no	—
155	#436	—	5	2	no	—
156	#436	—	10	1	no	—
157	#436	—	10	1	no	—
158	#431	—	5	1	no	—
159	#431	—	5	1	no	—
160	#424	—	2	1	no	—
161	#423	C₂×D₅	1	1	yes	characteristic
162	C₂×C₄⋊C₈	—	5	1	no	—
163	C₂×C₂²⋊C₈	—	10	1	no	—
164	#405	—	2	1	no	—
165	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
166	#334	—	5	4	no	—
167	Q₈⋊C₈	—	5	4	no	—
168	D₄⋊C₈	—	5	4	no	—
169	#322	—	5	2	no	—
170	C₈²	—	5	2	no	—
171	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
172	C₂²×D₅	—	2	2	no	—
173	C₂²×D₅	—	4	2	no	—
174	C₂²×D₅	—	4	2	no	—
175	C₂²×D₅	—	4	2	no	—
176	C₅×Q₈	—	2	4	no	—
177	C₅×Q₈	—	4	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	4	1	no	—
182	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
186	C₂×C₄×C₅	#135	1	2	yes	—
187	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
190	C₂×Dic₅	—	2	2	no	—
191	C₂×Dic₅	—	4	2	no	—
192	D₂₀	—	2	2	no	—
193	D₂₀	—	2	2	no	—
194	D₂₀	—	4	2	no	—
195	C₄×D₅	—	4	2	no	—
196	C₄×D₅	—	4	2	no	—
197	C₄×D₅	—	4	4	no	—
198	C₅×C₈	—	2	1	no	—
199	C₅×C₈	—	2	1	no	—
200	C₅×C₈	—	4	2	no	—
201	C₅×C₈	C₂×SD₁₆	1	2	yes	—
202	C₅×C₈	C₂×SD₁₆	1	2	yes	—
203	#168	—	2	4	no	—
204	#168	—	4	2	no	—
205	#168	—	4	2	no	—
206	C₂²×D₄	—	5	1	no	—
207	C₂³×C₄	—	10	1	no	—
208	C₂×SD₁₆	—	5	4	no	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	—	10	2	no	—
211	C₂×M₁₆	—	10	2	no	—
212	C₂×M₁₆	—	20	4	no	—
213	C₂²×C₈	—	10	1	no	—
214	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
215	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
216	C₄⋊Q₈	C₄×D₅	1	1	yes	characteristic
217	C₄⋊Q₈	C₄×D₅	1	1	yes	characteristic
218	#125	—	2	1	no	—
219	C₂²⋊Q₈	—	2	1	no	—
220	C₂²⋊Q₈	C₄×D₅	1	2	yes	—
221	C₄×Q₈	—	2	1	no	—
222	C₄×Q₈	C₂²×D₅	1	2	yes	—
223	C₄×D₄	—	5	2	no	—
224	C₄×D₄	—	5	2	no	—
225	C₄×D₄	—	10	1	no	—
226	C₄×D₄	—	10	1	no	—
227	#117	—	2	1	no	—
228	C₂×C₄⋊C₄	—	5	1	no	—
229	C₂×C₂²⋊C₄	—	10	1	no	—
230	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
231	#106	—	5	4	no	—
232	C₄⋊C₈	—	2	2	no	—
233	C₄⋊C₈	—	2	2	no	—
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	—	5	2	no	—
236	C₄⋊C₈	—	5	2	no	—
237	C₄⋊C₈	—	10	1	no	—
238	C₄⋊C₈	—	10	1	no	—
239	C₄⋊C₈	C₂²×D₅	1	2	yes	—
240	#103	—	5	4	no	—
241	#102	—	5	4	no	—
242	C₂²⋊C₈	—	5	2	no	—
243	C₂²⋊C₈	—	10	1	no	—
244	C₂²⋊C₈	—	10	2	no	—
245	C₂²⋊C₈	—	10	2	no	—
246	#97	—	2	2	no	—
247	C₄×C₈	—	2	2	no	—
248	C₄×C₈	—	5	2	no	—
249	C₄×C₈	—	5	2	no	—
250	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
251	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
252	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
253	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
254	C₂²×C₅	—	2	1	no	—
255	C₂²×C₅	—	2	1	no	—
256	C₂²×C₅	C₄×SD₁₆	1	2	yes	—
257	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
258	C₂×D₅	—	4	2	no	—
259	C₂×D₅	—	4	2	no	—
260	C₂×D₅	—	4	2	no	—
261	C₂×D₅	—	8	4	no	—
262	C₄×C₅	—	2	1	no	—
263	C₄×C₅	—	2	1	no	—
264	C₄×C₅	—	2	1	no	—
265	C₄×C₅	—	4	2	no	—
266	C₄×C₅	—	4	2	no	—
267	C₄×C₅	C₄×SD₁₆	1	2	yes	—
268	C₄×C₅	#568	1	2	yes	—
269	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
270	C₄×C₅	C₂×#135	1	1	yes	characteristic
271	Dic₅	—	4	2	no	—
272	C₂⁴	—	10	1	no	—
273	C₂×Q₈	—	2	1	no	—
274	C₂×Q₈	C₂×C₄×D₅	1	2	yes	—
275	C₂×D₄	—	5	2	no	—
276	C₂×D₄	—	10	1	no	—
277	C₂×D₄	—	20	2	no	—
278	C₂²×C₄	—	10	1	no	—
279	C₂²×C₄	—	10	2	no	—
280	C₂²×C₄	—	10	2	no	—
281	C₂²×C₄	—	20	2	no	—
282	C₂²×C₄	—	20	2	no	—
283	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
284	C₂²×C₄	#709	1	1	yes	characteristic
285	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
286	SD₁₆	—	20	4	no	—
287	M₁₆	—	4	2	no	—
288	M₁₆	—	20	4	no	—
289	M₁₆	—	20	4	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	5	4	no	—
293	C₂×C₈	—	10	2	no	—
294	C₂×C₈	—	10	2	no	—
295	C₂×C₈	—	20	2	no	—
296	C₂×C₈	#688	1	4	yes	complemented
297	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
298	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
299	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
300	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	2	1	no	—
304	C₄⋊C₄	—	2	1	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	5	2	no	—
307	C₄⋊C₄	—	10	1	no	—
308	C₄⋊C₄	C₂×C₄×D₅	1	2	yes	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₂²⋊C₄	—	10	2	no	—
311	C₂²⋊C₄	—	10	2	no	—
312	C₄²	—	2	2	no	—
313	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
314	C₄²	#709	1	1	yes	characteristic
315	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
316	C₄²	C₂³×D₅	1	1	yes	characteristic
317	C₂×C₅	—	2	2	no	—
318	C₂×C₅	#2930	1	1	yes	characteristic
319	C₂×C₅	C₂×C₄×SD₁₆	1	1	yes	characteristic
320	C₂×C₅	#2955	1	1	yes	characteristic
321	D₅	—	8	2	no	—
322	C₂³	—	10	2	no	—
323	C₂³	—	20	2	no	—
324	C₂³	—	20	2	no	—
325	C₂³	—	20	2	no	—
326	C₂³	C₄×C₂²⋊D₅	1	1	yes	characteristic
327	Q₈	—	2	4	no	—
328	Q₈	—	4	2	no	—
329	D₄	—	10	2	no	—
330	D₄	—	10	2	no	—
331	D₄	—	20	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	10	2	no	—
337	C₂×C₄	—	20	2	no	—
338	C₂×C₄	—	20	4	no	—
339	C₂×C₄	—	20	2	no	—
340	C₂×C₄	—	20	2	no	—
341	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
342	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
343	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
344	C₂×C₄	C₂×#688	1	2	yes	—
345	C₂×C₄	#4096	1	2	yes	—
346	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
347	C₂×C₄	C₂×#709	1	1	yes	characteristic
348	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	4	2	no	—
352	C₈	—	10	4	no	—
353	C₈	—	20	2	no	—
354	C₈	—	20	2	no	—
355	C₈	C₂×#688	1	2	yes	—
356	C₈	C₂×#688	1	2	yes	—
357	C₅	#17930	1	1	yes	characteristic, complemented, Sylow
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	20	2	no	—
361	C₂²	—	20	2	no	—
362	C₂²	—	20	2	no	—
363	C₂²	—	40	4	no	—
364	C₂²	C₄×#688	1	2	yes	—
365	C₂²	C₂×C₄×C₂²⋊D₅	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	2	no	—
370	C₄	—	4	2	no	—
371	C₄	—	20	2	no	—
372	C₄	C₄×#688	1	2	yes	—
373	C₄	#66084	1	2	yes	—
374	C₄	C₂²×#688	1	1	yes	characteristic
375	C₄	C₂×#4096	1	1	yes	characteristic
376	C₂	—	2	2	no	—
377	C₂	—	40	2	no	—
378	C₂	C₂×C₄×#688	1	1	yes	characteristic
379	C₂	#10615523	1	1	yes	characteristic
380	C₂	#10615526	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#2960	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂²×D₂₀	C₂×C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#2960	D₅
hypocenter	C₅	#17930
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. C₅×#2960	C₂	semidirect
4. #10602617	C₂	semidirect
5. #10601743	C₂	semidirect
6. #10601726	C₂	semidirect
8. #10601286	C₂	semidirect
9. #10600517	C₂	semidirect
14. C₅×#446	C₂²	semidirect
16. C₅×C₈×Q₈	C₂²	semidirect
34. Q₈⋊#168	C₂²	semidirect
36. #64700	C₂²	semidirect
37. #64675	C₂²	semidirect
38. C₈×#168	C₂²	semidirect
89. #2960	D₅	semidirect
114. C₂×C₅×C₈	SD₁₆	semidirect
150. #446	C₂×D₅	semidirect
152. C₈×Q₈	C₂×D₅	semidirect
296. C₂×C₈	#688	semidirect
357. C₅	#17930	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2960	2
#10615082	2
C₂×#64977	2
#10602617	2
#10601743	2
#10601726	2
#10601289	2
#10601286	2
#10600517	2
#17930	5