Subgroups — #12068786

Order 1280.

714 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611637	C₂	1	2	yes	maximal, complemented
2	C₄×#4065	C₂	1	1	yes	maximal, characteristic
3	C₅×#1579	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#688	C₂	1	2	yes	maximal, complemented
5	#10601695	C₂	1	1	yes	maximal, characteristic, complemented
6	D₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
7	#10599311	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599023	C₂	1	2	yes	maximal, complemented
9	#10599005	C₂	1	2	yes	maximal, complemented
10	#10598870	C₂	1	1	yes	maximal, characteristic, complemented
11	#10598861	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#4065	C₄	1	1	yes	characteristic
13	C₄×#713	C₂²	1	2	yes	—
14	C₅×C₈×Q₈	C₂²	1	2	yes	—
15	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
16	Q₁₆×Dic₅	C₄	1	1	yes	characteristic
17	C₄×#688	C₂²	1	2	yes	—
18	Q₈×#168	C₂²	1	2	yes	—
19	#65169	C₄	1	1	yes	characteristic
20	#65161	C₄	1	1	yes	characteristic
21	#65123	—	2	2	no	—
22	#65123	C₂²	1	2	yes	—
23	D₅×C₄⋊C₈	—	2	2	no	—
24	D₅×C₄⋊C₈	C₂²	1	2	yes	—
25	#65093	C₄	1	2	yes	—
26	#65077	C₄	1	2	yes	—
27	C₄×D₄₀	C₂²	1	1	yes	characteristic
28	C₈×D₂₀	—	2	2	no	—
29	C₈×D₂₀	C₂²	1	2	yes	—
30	#64974	—	2	2	no	—
31	#64974	C₂²	1	1	yes	characteristic
32	#64974	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
34	C₅×#334	C₂²	1	1	yes	characteristic
35	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₅×#322	C₂²	1	1	yes	characteristic
37	Q₈⋊#168	C₂²	1	2	yes	—
38	#64703	C₂²	1	2	yes	—
39	#64700	C₂²	1	2	yes	—
40	#64679	C₂²	1	2	yes	—
41	#64677	C₂²	1	1	yes	characteristic
42	#64675	C₂²	1	1	yes	characteristic
43	C₈×#168	C₂²	1	1	yes	characteristic
44	C₈×#168	C₂²	1	1	yes	characteristic
45	#17858	—	5	1	no	maximal, Sylow
46	C₂×#713	C₂×C₄	1	2	yes	—
47	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
48	C₄×C₅×Q₈	C₂³	1	2	yes	—
49	Q₈×Dic₅	C₂×C₄	1	2	yes	—
50	C₂×#688	C₂×C₄	1	2	yes	—
51	#4065	C₈	1	8	yes	complemented
52	C₂×D₄₀	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₅	—	2	2	no	—
54	C₂×C₈×D₅	—	2	2	no	—
55	C₂×C₈×D₅	—	2	2	no	—
56	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
58	#4039	—	2	2	no	—
59	#4039	C₂×C₄	1	2	yes	—
60	#4038	—	2	2	no	—
61	#4038	C₂×C₄	1	2	yes	—
62	C₄×D₂₀	—	2	2	no	—
63	C₄×D₂₀	C₂³	1	2	yes	—
64	C₄²×D₅	—	2	2	no	—
65	C₄²×D₅	C₂³	1	1	yes	characteristic
66	C₅×#107	C₂×C₄	1	1	yes	characteristic
67	C₅×C₄⋊C₈	—	2	2	no	—
68	C₅×C₄⋊C₈	C₂³	1	2	yes	—
69	C₅×#103	C₂×C₄	1	2	yes	—
70	C₄×C₅×C₈	—	2	2	no	—
71	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
74	#3967	C₂×C₄	1	2	yes	—
75	#3953	C₂×C₄	1	2	yes	—
76	#3952	—	2	1	no	—
77	#3952	—	2	2	no	—
78	#3952	—	2	2	no	—
79	#3952	D₄	1	2	yes	—
80	#3950	C₂×C₄	1	1	yes	characteristic
81	#3946	—	2	2	no	—
82	#3946	—	2	1	no	—
83	#3946	D₄	1	2	yes	—
84	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
85	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
86	#3941	C₂×C₄	1	2	yes	—
87	#3940	C₂×C₄	1	2	yes	—
88	C₄⋊#168	—	2	2	no	—
89	C₄⋊#168	C₂³	1	2	yes	—
90	C₄×#168	—	2	2	no	—
91	C₄×#168	C₂³	1	1	yes	characteristic
92	C₄×#168	C₂³	1	1	yes	characteristic
93	C₄×#168	C₂³	1	1	yes	characteristic
94	#2988	—	5	2	no	—
95	C₄×#135	—	5	1	no	—
96	#1580	—	5	1	no	—
97	#1580	—	5	1	no	—
98	#1579	—	5	1	no	—
99	#1579	D₅	1	1	yes	characteristic, complemented
100	C₈×SD₁₆	—	5	2	no	—
101	C₈×SD₁₆	—	5	2	no	—
102	C₂×#334	—	5	1	no	—
103	#1477	—	5	2	no	—
104	#1450	—	5	1	no	—
105	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
106	#713	—	2	4	no	—
107	#713	C₂×C₈	1	8	yes	complemented
108	C₂×D₂₀	—	2	2	no	—
109	C₂×D₂₀	C₂²×C₄	1	2	yes	—
110	C₂×C₄×D₅	—	2	2	no	—
111	C₂×C₄×D₅	—	2	2	no	—
112	C₂×C₄×D₅	—	2	2	no	—
113	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×D₅	#41	1	1	yes	characteristic
116	C₅×Q₁₆	C₂×C₈	1	4	yes	complemented
117	C₂×C₅×C₈	—	2	2	no	—
118	C₂×C₅×C₈	—	2	1	no	—
119	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
122	C₅×C₄⋊C₄	—	2	2	no	—
123	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
124	C₄²×C₅	—	2	2	no	—
125	C₄²×C₅	C₂⁴	1	1	yes	characteristic
126	#688	C₂×C₈	1	8	yes	complemented
127	#685	—	2	2	no	—
128	#685	—	2	2	no	—
129	#685	—	2	2	no	—
130	C₄⋊Dic₅	—	2	2	no	—
131	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
132	C₄×Dic₅	—	2	2	no	—
133	C₄×Dic₅	—	2	2	no	—
134	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
135	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
136	C₄×Dic₅	#41	1	1	yes	characteristic
137	C₂×#168	—	2	2	no	—
138	C₂×#168	—	2	1	no	—
139	C₂×#168	C₂²×C₄	1	1	yes	characteristic
140	C₂×#168	C₂²×C₄	1	1	yes	characteristic
141	C₂×#168	C₂×D₄	1	2	yes	—
142	D₄₀	C₂×C₈	1	4	yes	complemented
143	C₈×D₅	—	2	2	no	—
144	C₈×D₅	—	8	2	no	—
145	C₈×D₅	—	8	2	no	—
146	C₈×D₅	—	8	2	no	—
147	C₈×D₅	C₂×C₈	1	4	yes	complemented
148	C₂×#135	—	5	1	no	—
149	C₄×#41	—	5	2	no	—
150	C₈×Q₈	—	5	2	no	—
151	C₈×Q₈	C₂×D₅	1	2	yes	—
152	C₄×Q₁₆	—	5	1	no	—
153	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
154	C₄×SD₁₆	—	5	2	no	—
155	C₄×SD₁₆	—	5	2	no	—
156	C₄×D₈	—	5	1	no	—
157	C₄×D₈	—	5	1	no	—
158	C₈×D₄	—	5	2	no	—
159	C₈×D₄	—	5	2	no	—
160	C₈×D₄	—	10	2	no	—
161	C₈×D₄	—	10	2	no	—
162	#431	—	5	1	no	—
163	#431	—	5	1	no	—
164	#431	—	10	2	no	—
165	#427	—	5	1	no	—
166	C₂×C₄⋊C₈	—	5	2	no	—
167	C₂×C₄⋊C₈	—	10	2	no	—
168	#416	—	5	2	no	—
169	C₂×C₄×C₈	—	5	1	no	—
170	#334	—	5	2	no	—
171	#334	—	5	1	no	—
172	#334	C₂×D₅	1	1	yes	characteristic
173	Q₈⋊C₈	—	5	2	no	—
174	Q₈⋊C₈	C₂×D₅	1	2	yes	—
175	D₄⋊C₈	—	5	2	no	—
176	D₄⋊C₈	—	5	2	no	—
177	#322	—	5	1	no	—
178	#322	C₂×D₅	1	1	yes	characteristic
179	C₈²	—	5	1	no	—
180	C₈²	—	5	1	no	—
181	C₂²×D₅	—	2	2	no	—
182	C₂²×D₅	C₄×D₄	1	1	yes	characteristic
183	C₅×Q₈	—	2	2	no	—
184	C₅×Q₈	C₂²×C₈	1	4	yes	—
185	C₂×C₄×C₅	—	2	2	no	—
186	C₂×C₄×C₅	—	2	2	no	—
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	C₄×D₄	1	1	yes	characteristic
192	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
193	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
194	D₂₀	—	2	8	no	—
195	D₂₀	—	2	2	no	—
196	D₂₀	C₂²×C₈	1	4	yes	—
197	C₄×D₅	—	2	2	no	—
198	C₄×D₅	—	2	8	no	—
199	C₄×D₅	—	4	1	no	—
200	C₄×D₅	—	4	2	no	—
201	C₄×D₅	—	4	2	no	—
202	C₄×D₅	—	8	2	no	—
203	C₄×D₅	C₂²×C₈	1	2	yes	—
204	C₄×D₅	#131	1	2	yes	—
205	C₅×C₈	—	2	1	no	—
206	C₅×C₈	—	2	2	no	—
207	C₅×C₈	—	4	1	no	—
208	C₅×C₈	—	4	2	no	—
209	C₅×C₈	C₂²×C₈	1	2	yes	—
210	#168	—	2	1	no	—
211	#168	—	2	2	no	—
212	#168	—	4	1	no	—
213	#168	—	4	2	no	—
214	#168	C₂²×C₈	1	2	yes	—
215	C₂×#41	—	5	2	no	—
216	#135	—	5	8	no	—
217	C₂×Q₁₆	C₄×D₅	1	1	yes	characteristic
218	C₂×SD₁₆	—	5	2	no	—
219	C₂×D₈	—	5	1	no	—
220	C₂²×C₈	—	5	1	no	—
221	C₂²×C₈	—	5	1	no	—
222	C₂²×C₈	—	10	2	no	—
223	C₂²×C₈	—	10	2	no	—
224	C₂²×C₈	—	10	2	no	—
225	C₄×Q₈	—	5	2	no	—
226	C₄×Q₈	C₂²×D₅	1	2	yes	—
227	C₄×D₄	—	5	2	no	—
228	C₄×D₄	—	5	2	no	—
229	C₄×D₄	—	10	2	no	—
230	C₄×D₄	—	10	2	no	—
231	#117	—	5	2	no	—
232	#117	—	10	2	no	—
233	C₂×C₄²	—	5	1	no	—
234	C₂×C₄²	—	10	2	no	—
235	#107	—	5	1	no	—
236	#107	C₄×D₅	1	1	yes	characteristic
237	#106	—	5	2	no	—
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	—	5	2	no	—
240	C₄⋊C₈	—	5	2	no	—
241	C₄⋊C₈	—	10	2	no	—
242	C₄⋊C₈	—	10	2	no	—
243	C₄⋊C₈	—	10	1	no	—
244	C₄⋊C₈	C₂²×D₅	1	2	yes	—
245	#103	—	5	2	no	—
246	#103	C₄×D₅	1	2	yes	—
247	#102	—	5	2	no	—
248	#102	—	5	2	no	—
249	C₂²⋊C₈	—	5	2	no	—
250	C₂²⋊C₈	—	10	2	no	—
251	C₂²⋊C₈	—	10	1	no	—
252	C₂²⋊C₈	—	10	2	no	—
253	C₄×C₈	—	2	2	no	—
254	C₄×C₈	—	5	1	no	—
255	C₄×C₈	—	5	1	no	—
256	C₄×C₈	—	5	1	no	—
257	C₄×C₈	—	5	1	no	—
258	C₄×C₈	—	5	1	no	—
259	C₄×C₈	—	10	2	no	—
260	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
263	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
264	C₂×D₅	—	2	1	no	—
265	C₂×D₅	—	2	1	no	—
266	C₂×D₅	—	2	4	no	—
267	C₂×D₅	—	4	2	no	—
268	C₂×D₅	—	4	2	no	—
269	C₂×D₅	C₈×D₄	1	2	yes	—
270	C₄×C₅	—	2	4	no	—
271	C₄×C₅	—	2	1	no	—
272	C₄×C₅	—	4	2	no	—
273	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
274	C₄×C₅	C₂×#131	1	1	yes	characteristic
275	C₄×C₅	#577	1	2	yes	—
276	Dic₅	—	2	1	no	—
277	Dic₅	—	2	1	no	—
278	Dic₅	—	4	2	no	—
279	Dic₅	C₈×D₄	1	2	yes	—
280	#41	—	5	8	no	—
281	#41	—	10	4	no	—
282	C₂×Q₈	C₂×C₄×D₅	1	2	yes	—
283	C₂×D₄	—	5	2	no	—
284	C₂×D₄	—	10	2	no	—
285	C₂²×C₄	—	5	1	no	—
286	C₂²×C₄	—	5	1	no	—
287	C₂²×C₄	—	5	1	no	—
288	C₂²×C₄	—	10	2	no	—
289	C₂²×C₄	—	10	2	no	—
290	C₂²×C₄	—	10	2	no	—
291	Q₁₆	C₈×D₅	1	4	yes	complemented
292	SD₁₆	—	5	8	no	—
293	D₈	—	5	4	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	5	1	no	—
297	C₂×C₈	—	5	4	no	—
298	C₂×C₈	—	5	1	no	—
299	C₂×C₈	—	5	2	no	—
300	C₂×C₈	—	10	2	no	—
301	C₂×C₈	—	10	2	no	—
302	C₂×C₈	—	10	1	no	—
303	C₂×C₈	—	40	2	no	—
304	C₂×C₈	—	40	2	no	—
305	C₂×C₈	—	40	2	no	—
306	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
307	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
308	C₂×C₈	D₄×D₅	1	2	yes	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	5	2	no	—
311	C₄⋊C₄	—	10	2	no	—
312	C₄⋊C₄	C₂×C₄×D₅	1	2	yes	—
313	C₂²⋊C₄	—	10	2	no	—
314	C₂²⋊C₄	—	10	2	no	—
315	C₂²⋊C₄	—	10	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	5	1	no	—
318	C₄²	—	5	1	no	—
319	C₄²	—	5	1	no	—
320	C₄²	—	10	2	no	—
321	C₄²	—	10	2	no	—
322	C₄²	C₂³×D₅	1	1	yes	characteristic
323	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
324	C₂×C₅	#2952	1	1	yes	characteristic
325	C₂×C₅	#2957	1	1	yes	characteristic
326	D₅	—	2	2	no	—
327	D₅	—	4	4	no	—
328	C₂³	—	5	1	no	—
329	C₂³	—	10	2	no	—
330	Q₈	—	2	2	no	—
331	Q₈	C₂×C₈×D₅	1	4	yes	—
332	D₄	—	5	4	no	—
333	D₄	—	10	8	no	—
334	D₄	—	10	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	5	1	no	—
338	C₂×C₄	—	5	2	no	—
339	C₂×C₄	—	5	2	no	—
340	C₂×C₄	—	5	1	no	—
341	C₂×C₄	—	5	1	no	—
342	C₂×C₄	—	10	8	no	—
343	C₂×C₄	—	10	2	no	—
344	C₂×C₄	—	10	2	no	—
345	C₂×C₄	—	20	1	no	—
346	C₂×C₄	—	20	2	no	—
347	C₂×C₄	—	20	2	no	—
348	C₂×C₄	—	40	2	no	—
349	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
350	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
351	C₂×C₄	D₅×#41	1	1	yes	characteristic
352	C₈	—	2	1	no	—
353	C₈	—	2	2	no	—
354	C₈	—	4	1	no	—
355	C₈	—	4	2	no	—
356	C₈	—	5	2	no	—
357	C₈	—	10	1	no	—
358	C₈	—	10	2	no	—
359	C₈	—	20	1	no	—
360	C₈	—	20	2	no	—
361	C₈	C₂×C₈×D₅	1	2	yes	—
362	C₅	#17858	1	1	yes	characteristic, complemented, Sylow
363	C₂²	—	5	2	no	—
364	C₂²	—	10	1	no	—
365	C₂²	—	10	1	no	—
366	C₂²	—	10	4	no	—
367	C₂²	—	20	2	no	—
368	C₂²	—	20	2	no	—
369	C₂²	C₄×D₄×D₅	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	2	4	no	—
372	C₄	—	4	2	no	—
373	C₄	—	5	2	no	—
374	C₄	—	10	1	no	—
375	C₄	—	10	1	no	—
376	C₄	—	20	2	no	—
377	C₄	C₂²×C₈×D₅	1	1	yes	characteristic
378	C₄	D₅×#131	1	1	yes	characteristic
379	C₄	#66112	1	2	yes	—
380	C₂	—	10	2	no	—
381	C₂	—	20	4	no	—
382	C₂	C₈×D₄×D₅	1	1	yes	characteristic
383	C₂	#10611563	1	1	yes	characteristic
384	C₂	#10611630	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅
commutator	C₄×C₅	C₂³×C₈
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#1579	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂×#4065	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1579	D₅
hypocenter	C₅	#17858
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

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

N	H ≅ G/N	type
1. #10611637	C₂	semidirect
3. C₅×#1579	C₂	semidirect
4. C₈×#688	C₂	semidirect
5. #10601695	C₂	semidirect
6. D₅×#334	C₂	semidirect
7. #10599311	C₂	semidirect
8. #10599023	C₂	semidirect
9. #10599005	C₂	semidirect
10. #10598870	C₂	semidirect
11. #10598861	C₂	semidirect
51. #4065	C₈	semidirect
99. #1579	D₅	semidirect
107. #713	C₂×C₈	semidirect
116. C₅×Q₁₆	C₂×C₈	semidirect
126. #688	C₂×C₈	semidirect
142. D₄₀	C₂×C₈	semidirect
147. C₈×D₅	C₂×C₈	semidirect
291. Q₁₆	C₈×D₅	semidirect
362. C₅	#17858	semidirect

Maximal subgroups

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

subgroup	index
#10611637	2
C₄×#4065	2
C₅×#1579	2
C₈×#688	2
#10601695	2
D₅×#334	2
#10599311	2
#10599023	2
#10599005	2
#10598870	2
#10598861	2
#17858	5