Subgroups — #12090902

Order 1280.

638 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611661	C₂	1	1	yes	maximal, characteristic, complemented
2	#10611641	C₂	1	1	yes	maximal, characteristic
3	#10611471	C₂	1	2	yes	maximal, complemented
4	C₅×C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601738	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601732	C₂	1	1	yes	maximal, characteristic, complemented
7	#10599377	C₂	1	2	yes	maximal, complemented
8	#10599054	C₂	1	2	yes	maximal, complemented
9	#10599037	C₂	1	2	yes	maximal, complemented
10	#10598851	C₂	1	1	yes	maximal, characteristic, complemented
11	#10598845	C₂	1	1	yes	maximal, characteristic, complemented
12	#65913	C₂²	1	2	yes	—
13	#65911	C₂²	1	2	yes	—
14	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
15	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
16	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
17	#65476	—	2	2	no	—
18	#65316	C₂²	1	2	yes	—
19	#65310	C₂²	1	2	yes	—
20	#65171	—	2	2	no	—
21	#65136	—	2	2	no	—
22	#65132	—	2	2	no	—
23	#65132	C₂²	1	2	yes	—
24	#65131	C₂²	1	2	yes	—
25	#65101	—	2	2	no	—
26	#65092	—	2	2	no	—
27	#64984	—	2	1	no	—
28	#64984	C₂²	1	1	yes	characteristic
29	#64983	C₂²	1	2	yes	—
30	#64982	C₂²	1	2	yes	—
31	C₄×D₄₀	—	2	1	no	—
32	C₄×D₄₀	—	2	1	no	—
33	C₄×D₄₀	C₂²	1	1	yes	characteristic
34	#64977	C₂²	1	2	yes	—
35	#64972	—	2	1	no	—
36	#64971	C₂²	1	1	yes	characteristic
37	C₅×#335	C₂²	1	1	yes	characteristic
38	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic
39	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic
40	C₅×C₈²	C₂²	1	1	yes	characteristic
41	#64705	C₂²	1	2	yes	—
42	#64701	C₂²	1	2	yes	—
43	#64679	C₂²	1	2	yes	—
44	#64678	C₂²	1	1	yes	characteristic
45	#19909	—	5	1	no	maximal, Sylow
46	C₂×#713	—	2	2	no	—
47	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
48	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
49	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
50	#4092	—	2	2	no	—
51	C₂×#688	—	2	2	no	—
52	C₂×D₄₀	—	2	1	no	—
53	C₂×D₄₀	—	2	2	no	—
54	C₂×D₄₀	—	2	1	no	—
55	C₂×D₄₀	D₄	1	1	yes	characteristic
56	C₂×#677	—	4	2	no	—
57	C₂×#676	—	2	2	no	—
58	#4042	—	4	2	no	—
59	#4041	—	2	2	no	—
60	#4041	—	4	2	no	—
61	D₅×C₄⋊C₄	—	2	2	no	—
62	#4020	—	2	2	no	—
63	#4020	C₂³	1	2	yes	—
64	C₄×D₂₀	—	2	2	no	—
65	C₄×D₂₀	—	2	2	no	—
66	C₄×D₂₀	C₂³	1	2	yes	—
67	#4016	—	2	2	no	—
68	#4015	C₂³	1	2	yes	—
69	C₅×#107	D₄	1	1	yes	characteristic
70	C₅×C₄⋊C₈	—	2	1	no	—
71	C₅×C₄⋊C₈	—	2	1	no	—
72	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
73	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
74	C₅×#103	D₄	1	1	yes	characteristic
75	C₅×#103	D₄	1	1	yes	characteristic
76	C₄×C₅×C₈	—	2	1	no	—
77	C₄×C₅×C₈	—	2	1	no	—
78	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
79	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
80	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
81	#3967	—	2	2	no	—
82	#3953	—	2	2	no	—
83	#3953	—	2	2	no	—
84	#3953	—	2	2	no	—
85	#3953	—	2	2	no	—
86	#3953	D₄	1	2	yes	—
87	#3952	—	2	2	no	—
88	#3950	—	2	1	no	—
89	#3950	—	2	1	no	—
90	#3950	D₄	1	1	yes	characteristic
91	#3949	—	2	2	no	—
92	#3941	—	2	2	no	—
93	#3939	—	2	2	no	—
94	C₄⋊#168	C₂³	1	2	yes	—
95	#3391	—	5	1	no	—
96	#3380	—	5	1	no	—
97	#3333	—	5	2	no	—
98	#1706	—	5	1	no	—
99	#1697	—	5	2	no	—
100	#1676	—	5	1	no	—
101	#1668	—	5	1	no	—
102	#1631	—	5	2	no	—
103	#1622	—	5	2	no	—
104	#1589	—	5	1	no	—
105	C₈×Q₁₆	D₅	1	1	yes	characteristic, complemented
106	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
108	#713	—	4	4	no	—
109	#713	—	8	2	no	—
110	C₂×D₂₀	—	2	2	no	—
111	C₂×D₂₀	—	2	2	no	—
112	C₂×D₂₀	—	2	2	no	—
113	C₂×D₂₀	—	4	2	no	—
114	C₂×D₂₀	—	4	2	no	—
115	C₂×D₂₀	C₂×D₄	1	2	yes	—
116	C₂×C₄×D₅	—	2	2	no	—
117	C₂×C₄×D₅	—	4	2	no	—
118	C₂×C₄×D₅	—	4	2	no	—
119	C₂×Dic₁₀	—	2	2	no	—
120	C₅×Q₁₆	—	2	2	no	—
121	C₂×C₅×C₈	—	2	1	no	—
122	C₂×C₅×C₈	—	2	1	no	—
123	C₂×C₅×C₈	—	2	1	no	—
124	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
127	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
128	C₅×C₄⋊C₄	—	2	1	no	—
129	C₅×C₄⋊C₄	—	2	1	no	—
130	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
131	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₅	—	2	1	no	—
133	C₄²×C₅	—	2	1	no	—
134	C₄²×C₅	C₂⁴	1	1	yes	characteristic
135	#688	—	8	2	no	—
136	#685	—	2	2	no	—
137	#685	—	4	2	no	—
138	#685	—	4	2	no	—
139	C₄⋊Dic₅	—	2	2	no	—
140	C₄⋊Dic₅	—	2	2	no	—
141	C₄⋊Dic₅	C₂×D₄	1	2	yes	—
142	#683	—	4	2	no	—
143	#683	—	4	2	no	—
144	C₂×#168	—	2	2	no	—
145	D₄₀	—	2	2	no	—
146	D₄₀	—	4	2	no	—
147	D₄₀	—	8	1	no	—
148	D₄₀	—	8	2	no	—
149	#677	—	8	4	no	—
150	#676	—	8	2	no	—
151	#554	—	5	2	no	—
152	#550	—	5	2	no	—
153	#500	—	5	1	no	—
154	#499	—	10	1	no	—
155	#493	—	5	1	no	—
156	#493	—	10	1	no	—
157	#492	—	5	2	no	—
158	#486	—	5	2	no	—
159	#480	—	10	2	no	—
160	#478	—	10	2	no	—
161	#477	—	5	2	no	—
162	#476	—	5	2	no	—
163	#468	—	10	2	no	—
164	#460	—	5	2	no	—
165	#459	—	5	2	no	—
166	#459	—	10	2	no	—
167	#452	—	10	2	no	—
168	C₈×Q₈	C₂×D₅	1	1	yes	characteristic
169	C₈×Q₈	C₂×D₅	1	1	yes	characteristic
170	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
171	C₄×D₈	—	5	1	no	—
172	C₄×D₈	—	10	1	no	—
173	C₄×D₈	—	10	1	no	—
174	#436	—	5	2	no	—
175	#417	—	10	2	no	—
176	#335	—	5	1	no	—
177	#335	C₂×D₅	1	1	yes	characteristic
178	#333	—	5	2	no	—
179	#331	—	5	2	no	—
180	Q₈⋊C₈	C₂×D₅	1	1	yes	characteristic
181	Q₈⋊C₈	C₂×D₅	1	1	yes	characteristic
182	D₄⋊C₈	—	5	2	no	—
183	C₈²	C₂×D₅	1	1	yes	characteristic
184	C₂²×D₅	—	2	2	no	—
185	C₂²×D₅	—	4	2	no	—
186	C₅×Q₈	—	2	1	no	—
187	C₅×Q₈	—	2	1	no	—
188	C₅×Q₈	—	2	1	no	—
189	C₅×Q₈	#135	1	2	yes	—
190	C₂×C₄×C₅	—	2	1	no	—
191	C₂×C₄×C₅	—	2	1	no	—
192	C₂×C₄×C₅	—	2	1	no	—
193	C₂×C₄×C₅	—	2	1	no	—
194	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
197	C₂×Dic₅	—	2	2	no	—
198	C₂×Dic₅	—	4	2	no	—
199	D₂₀	—	2	2	no	—
200	D₂₀	—	2	2	no	—
201	D₂₀	—	4	2	no	—
202	D₂₀	—	4	2	no	—
203	D₂₀	—	4	2	no	—
204	D₂₀	—	4	2	no	—
205	D₂₀	—	8	4	no	—
206	D₂₀	—	8	2	no	—
207	D₂₀	—	16	2	no	—
208	C₄×D₅	—	4	2	no	—
209	C₄×D₅	—	4	2	no	—
210	C₄×D₅	—	8	4	no	—
211	C₄×D₅	—	16	2	no	—
212	Dic₁₀	—	8	2	no	—
213	C₅×C₈	—	2	2	no	—
214	C₅×C₈	—	2	1	no	—
215	C₅×C₈	—	2	1	no	—
216	C₅×C₈	—	2	1	no	—
217	C₅×C₈	—	4	1	no	—
218	C₅×C₈	—	4	1	no	—
219	C₅×C₈	#137	1	2	yes	—
220	#168	—	4	2	no	—
221	C₂×#41	—	10	2	no	—
222	C₂×Q₁₆	D₂₀	1	1	yes	characteristic
223	C₂×SD₁₆	—	10	2	no	—
224	C₂×SD₁₆	—	20	2	no	—
225	C₂×D₈	—	5	1	no	—
226	C₂×D₈	—	10	2	no	—
227	C₂×D₈	—	10	1	no	—
228	C₂×D₈	—	10	1	no	—
229	C₂×M₁₆	—	10	2	no	—
230	C₄⋊Q₈	—	5	2	no	—
231	#127	—	5	2	no	—
232	#127	—	10	2	no	—
233	#125	—	10	2	no	—
234	C₂²⋊Q₈	—	10	2	no	—
235	C₂²⋊Q₈	—	20	2	no	—
236	#121	—	10	2	no	—
237	#121	—	20	2	no	—
238	C₄×Q₈	C₂²×D₅	1	1	yes	characteristic
239	C₄×Q₈	C₂²×D₅	1	1	yes	characteristic
240	C₄×D₄	—	5	2	no	—
241	C₄×D₄	—	10	2	no	—
242	C₄×D₄	—	10	2	no	—
243	C₂×C₄⋊C₄	—	10	2	no	—
244	#107	—	5	1	no	—
245	#107	—	10	1	no	—
246	#107	—	10	1	no	—
247	#107	—	10	2	no	—
248	#107	D₂₀	1	1	yes	characteristic
249	#106	—	10	2	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	2	1	no	—
252	C₄⋊C₈	—	5	2	no	—
253	C₄⋊C₈	C₂²×D₅	1	1	yes	characteristic
254	C₄⋊C₈	C₂²×D₅	1	1	yes	characteristic
255	#103	—	10	2	no	—
256	#103	D₂₀	1	1	yes	characteristic
257	#103	D₂₀	1	1	yes	characteristic
258	#102	—	5	2	no	—
259	#102	—	10	2	no	—
260	#102	—	10	2	no	—
261	#102	—	10	2	no	—
262	#102	—	10	2	no	—
263	#102	—	10	2	no	—
264	C₂²⋊C₈	—	10	2	no	—
265	C₄×C₈	—	2	1	no	—
266	C₄×C₈	—	2	1	no	—
267	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
270	C₂²×C₅	D₄²	1	1	yes	characteristic
271	C₂×D₅	—	4	2	no	—
272	C₂×D₅	—	4	2	no	—
273	C₂×D₅	—	4	2	no	—
274	C₂×D₅	—	8	2	no	—
275	C₂×D₅	—	8	2	no	—
276	C₂×D₅	—	8	2	no	—
277	C₄×C₅	—	2	2	no	—
278	C₄×C₅	—	2	1	no	—
279	C₄×C₅	—	4	1	no	—
280	C₄×C₅	—	4	1	no	—
281	C₄×C₅	—	4	1	no	—
282	C₄×C₅	C₂×#135	1	1	yes	characteristic
283	C₄×C₅	C₂×#137	1	1	yes	characteristic
284	C₄×C₅	#576	1	1	yes	characteristic
285	C₄×C₅	#577	1	1	yes	characteristic
286	Dic₅	—	4	2	no	—
287	Dic₅	—	8	2	no	—
288	#41	—	20	4	no	—
289	#41	—	40	2	no	—
290	C₂×Q₈	—	10	2	no	—
291	C₂×Q₈	C₂×D₂₀	1	1	yes	characteristic
292	C₂×Q₈	C₂×D₂₀	1	1	yes	characteristic
293	C₂×D₄	—	5	2	no	—
294	C₂×D₄	—	10	2	no	—
295	C₂×D₄	—	10	2	no	—
296	C₂×D₄	—	10	2	no	—
297	C₂×D₄	—	20	2	no	—
298	C₂×D₄	—	20	2	no	—
299	C₂²×C₄	—	10	2	no	—
300	C₂²×C₄	—	20	2	no	—
301	C₂²×C₄	—	20	2	no	—
302	Q₁₆	—	2	2	no	—
303	SD₁₆	—	40	4	no	—
304	SD₁₆	—	40	2	no	—
305	D₈	—	10	2	no	—
306	D₈	—	20	2	no	—
307	D₈	—	40	1	no	—
308	D₈	—	40	2	no	—
309	M₁₆	—	40	2	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	10	2	no	—
314	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
315	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
316	C₂×C₈	D₄×D₅	1	1	yes	characteristic
317	C₂×C₈	D₄×D₅	1	1	yes	characteristic
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	2	1	no	—
320	C₄⋊C₄	—	5	2	no	—
321	C₄⋊C₄	—	10	2	no	—
322	C₄⋊C₄	—	10	2	no	—
323	C₄⋊C₄	—	20	2	no	—
324	C₄⋊C₄	—	20	2	no	—
325	C₄⋊C₄	C₂×D₂₀	1	1	yes	characteristic
326	C₄⋊C₄	C₂×D₂₀	1	1	yes	characteristic
327	C₂²⋊C₄	—	10	2	no	—
328	C₂²⋊C₄	—	20	2	no	—
329	C₂²⋊C₄	—	20	2	no	—
330	C₄²	—	2	1	no	—
331	C₄²	—	2	1	no	—
332	C₄²	C₂³×D₅	1	1	yes	characteristic
333	C₂×C₅	#3278	1	1	yes	characteristic
334	C₂×C₅	#3280	1	1	yes	characteristic
335	C₂×C₅	#3292	1	1	yes	characteristic
336	D₅	—	8	2	no	—
337	D₅	—	16	2	no	—
338	C₂³	—	10	2	no	—
339	C₂³	—	20	2	no	—
340	Q₈	—	2	1	no	—
341	Q₈	—	2	1	no	—
342	Q₈	—	2	1	no	—
343	Q₈	—	40	2	no	—
344	Q₈	#4050	1	2	yes	—
345	D₄	—	10	2	no	—
346	D₄	—	10	2	no	—
347	D₄	—	20	2	no	—
348	D₄	—	20	2	no	—
349	D₄	—	20	2	no	—
350	D₄	—	20	2	no	—
351	D₄	—	40	4	no	—
352	D₄	—	40	2	no	—
353	D₄	—	80	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	10	2	no	—
359	C₂×C₄	—	20	2	no	—
360	C₂×C₄	—	20	2	no	—
361	C₂×C₄	—	20	2	no	—
362	C₂×C₄	—	40	4	no	—
363	C₂×C₄	—	80	2	no	—
364	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
365	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
366	C₂×C₄	#4149	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	4	1	no	—
372	C₈	—	4	1	no	—
373	C₈	—	20	2	no	—
374	C₈	#4064	1	2	yes	—
375	C₅	#19909	1	1	yes	characteristic, complemented, Sylow
376	C₂²	—	20	2	no	—
377	C₂²	—	20	2	no	—
378	C₂²	—	20	2	no	—
379	C₂²	—	40	2	no	—
380	C₂²	—	40	2	no	—
381	C₂²	—	40	2	no	—
382	C₂²	D₄×D₂₀	1	1	yes	characteristic
383	C₄	—	2	1	no	—
384	C₄	—	2	2	no	—
385	C₄	—	4	1	no	—
386	C₄	—	4	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	20	2	no	—
389	C₄	—	40	2	no	—
390	C₄	C₂×#4050	1	1	yes	characteristic
391	C₄	#66086	1	1	yes	characteristic
392	C₄	C₂×#4064	1	1	yes	characteristic
393	C₄	#66104	1	1	yes	characteristic
394	C₂	—	40	2	no	—
395	C₂	—	80	2	no	—
396	C₂	#10611164	1	1	yes	characteristic
397	C₂	#10611462	1	1	yes	characteristic
398	C₂	#10611625	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₀
commutator, cosocle	C₄²×C₅	C₂⁴
Frattini	C₄²	C₂³×D₅
Fitting	C₅×C₈×Q₁₆	C₂
socle	C₂²×C₅	D₄²
prime radical	#10611641	C₂
hypercenter	C₈×Q₁₆	D₅
hypocenter	C₅	#19909
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #10611661	C₂	semidirect
3. #10611471	C₂	semidirect
4. C₅×C₈×Q₁₆	C₂	semidirect
5. #10601738	C₂	semidirect
6. #10601732	C₂	semidirect
7. #10599377	C₂	semidirect
8. #10599054	C₂	semidirect
9. #10599037	C₂	semidirect
10. #10598851	C₂	semidirect
11. #10598845	C₂	semidirect
105. C₈×Q₁₆	D₅	semidirect
375. C₅	#19909	semidirect

Maximal subgroups

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

subgroup	index
#10611661	2
#10611641	2
#10611471	2
C₅×C₈×Q₁₆	2
#10601738	2
#10601732	2
#10599377	2
#10599054	2
#10599037	2
#10598851	2
#10598845	2
#19909	5