Subgroups — #11993142

Order 1280.

500 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65122	C₂	1	1	yes	maximal, characteristic
2	#10601268	C₂	1	1	yes	maximal, characteristic
3	C₂×#64700	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#3952	C₂	1	1	yes	maximal, characteristic
5	C₂×#64678	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1277	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598194	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄²×D₅	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×#3952	—	2	1	no	—
13	C₂×#3952	—	2	1	no	—
14	C₂×#3952	C₄	1	1	yes	characteristic
15	C₂×#3952	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₅	—	2	1	no	—
17	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
19	C₂×C₄×#168	C₂²	1	1	yes	characteristic
20	#65122	—	2	2	no	—
21	#65122	—	2	2	no	—
22	#64770	—	2	1	no	—
23	#64770	C₄	1	1	yes	characteristic
24	#64770	C₄	1	1	yes	characteristic
25	#64700	—	2	2	no	—
26	#64678	—	2	2	no	—
27	#8831	—	5	1	no	maximal, Sylow
28	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₅	D₄	1	1	yes	characteristic
30	C₂²×C₄×D₅	Q₈	1	1	yes	characteristic
31	C₂²×C₅×C₈	—	2	1	no	—
32	C₂²×C₅×C₈	—	2	1	no	—
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
38	C₂×C₄×Dic₅	Q₈	1	1	yes	characteristic
39	C₂²×#168	—	2	1	no	—
40	C₂²×#168	—	2	1	no	—
41	C₂²×#168	C₂×C₄	1	1	yes	characteristic
42	C₂²×#168	C₂×C₄	1	1	yes	characteristic
43	C₄²×D₅	—	2	1	no	—
44	C₄²×D₅	—	2	1	no	—
45	C₄²×D₅	—	2	1	no	—
46	C₄²×D₅	—	2	1	no	—
47	C₅×C₄⋊C₈	—	2	1	no	—
48	C₅×C₄⋊C₈	—	2	1	no	—
49	C₅×C₄⋊C₈	—	2	1	no	—
50	C₅×C₄⋊C₈	D₄	1	2	yes	—
51	C₄×C₅×C₈	—	2	1	no	—
52	C₄×C₅×C₈	—	2	1	no	—
53	#3952	—	2	4	no	—
54	#3952	—	2	2	no	—
55	#3952	—	4	2	no	—
56	#3952	C₈	1	4	yes	complemented
57	C₈×Dic₅	—	2	4	no	—
58	C₄⋊#168	—	2	1	no	—
59	C₄⋊#168	—	2	1	no	—
60	C₄⋊#168	—	2	1	no	—
61	C₄⋊#168	D₄	1	2	yes	—
62	C₄×#168	—	2	1	no	—
63	C₄×#168	—	2	1	no	—
64	C₂×#431	—	5	1	no	—
65	#1797	—	5	1	no	—
66	C₄×C₂²⋊C₈	—	5	1	no	—
67	C₂×#335	—	5	1	no	—
68	C₂×#334	—	5	1	no	—
69	#1277	—	5	1	no	—
70	#1277	D₅	1	1	yes	characteristic, complemented
71	C₂³×D₅	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
75	C₂²×Dic₅	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×Dic₅	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×Dic₅	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×D₅	—	2	1	no	—
79	C₂×C₄×D₅	—	2	1	no	—
80	C₂×C₄×D₅	—	2	1	no	—
81	C₂×C₄×D₅	—	2	1	no	—
82	C₂×C₄×D₅	—	2	1	no	—
83	C₂×C₄×D₅	—	2	1	no	—
84	C₂×C₄×D₅	—	2	1	no	—
85	C₂×C₄×D₅	—	2	1	no	—
86	C₂×C₄×D₅	—	2	1	no	—
87	C₂×C₄×D₅	—	2	1	no	—
88	C₂×C₄×D₅	—	2	1	no	—
89	C₂×C₄×D₅	—	2	1	no	—
90	C₂×C₄×D₅	—	2	2	no	—
91	C₂×C₄×D₅	—	4	1	no	—
92	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
93	C₂×C₄×D₅	M₁₆	1	2	yes	—
94	C₂×C₅×C₈	—	2	1	no	—
95	C₂×C₅×C₈	—	2	1	no	—
96	C₂×C₅×C₈	—	2	1	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	2	1	no	—
99	C₂×C₅×C₈	—	2	1	no	—
100	C₂×C₅×C₈	—	4	1	no	—
101	C₂×C₅×C₈	—	4	1	no	—
102	C₂×C₅×C₈	—	4	1	no	—
103	C₂×C₅×C₈	—	4	1	no	—
104	C₂×C₅×C₈	—	4	1	no	—
105	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
106	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅	#41	1	1	yes	characteristic
109	C₄²×C₅	#41	1	1	yes	characteristic
110	C₄×Dic₅	—	2	1	no	—
111	C₄×Dic₅	—	2	1	no	—
112	C₄×Dic₅	—	2	1	no	—
113	C₄×Dic₅	—	2	1	no	—
114	C₄×Dic₅	—	2	1	no	—
115	C₄×Dic₅	—	2	1	no	—
116	C₂×#168	—	2	1	no	—
117	C₂×#168	—	2	1	no	—
118	C₂×#168	—	2	1	no	—
119	C₂×#168	—	2	2	no	—
120	C₂×#168	—	2	1	no	—
121	C₂×#168	—	2	1	no	—
122	C₂×#168	—	4	1	no	—
123	C₂×#168	—	4	1	no	—
124	C₂×#168	—	4	1	no	—
125	C₂×#168	—	4	1	no	—
126	C₂×#168	—	4	1	no	—
127	C₂×#168	C₂×C₈	1	2	yes	—
128	C₂²×C₄²	—	5	1	no	—
129	#431	—	10	2	no	—
130	#431	—	10	2	no	—
131	C₂×C₄⋊C₈	—	5	1	no	—
132	C₂×C₄⋊C₈	—	5	1	no	—
133	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
134	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
135	C₂×C₂²⋊C₈	—	5	1	no	—
136	C₂×C₂²⋊C₈	—	5	1	no	—
137	C₂×C₂²⋊C₈	—	10	1	no	—
138	C₂×C₂²⋊C₈	—	10	1	no	—
139	C₂×C₄×C₈	—	5	1	no	—
140	C₂×C₄×C₈	—	10	1	no	—
141	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
142	#336	—	5	1	no	—
143	#336	—	5	1	no	—
144	#336	—	10	1	no	—
145	#335	—	10	2	no	—
146	#334	—	10	2	no	—
147	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×D₅	—	2	1	no	—
149	C₂²×D₅	—	2	1	no	—
150	C₂²×D₅	—	2	1	no	—
151	C₂²×D₅	—	2	1	no	—
152	C₂²×D₅	—	2	1	no	—
153	C₂²×D₅	—	2	1	no	—
154	C₂²×D₅	C₄⋊C₈	1	2	yes	—
155	C₂×C₄×C₅	—	2	1	no	—
156	C₂×C₄×C₅	—	2	1	no	—
157	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅	#120	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₄×C₅	#123	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
165	C₂×C₄×C₅	#131	1	1	yes	characteristic
166	C₂×C₄×C₅	#131	1	1	yes	characteristic
167	C₂×C₄×C₅	#135	1	1	yes	characteristic
168	C₂×C₄×C₅	#135	1	1	yes	characteristic
169	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₂×C₄×C₅	#137	1	1	yes	characteristic
171	C₂×Dic₅	—	2	1	no	—
172	C₂×Dic₅	—	2	1	no	—
173	C₂×Dic₅	—	2	1	no	—
174	C₂×Dic₅	—	2	1	no	—
175	C₂×Dic₅	—	2	1	no	—
176	C₂×Dic₅	—	2	1	no	—
177	C₂×Dic₅	—	2	1	no	—
178	C₂×Dic₅	C₄⋊C₈	1	2	yes	—
179	C₂×Dic₅	#108	1	2	yes	—
180	C₄×D₅	—	2	2	no	—
181	C₄×D₅	—	2	2	no	—
182	C₄×D₅	—	2	2	no	—
183	C₄×D₅	—	2	2	no	—
184	C₄×D₅	—	4	2	no	—
185	C₄×D₅	—	4	2	no	—
186	C₄×D₅	—	4	1	no	—
187	C₄×D₅	—	4	1	no	—
188	C₄×D₅	—	4	1	no	—
189	C₄×D₅	—	4	1	no	—
190	C₅×C₈	—	2	2	no	—
191	C₅×C₈	—	4	1	no	—
192	C₅×C₈	—	4	2	no	—
193	C₅×C₈	—	8	1	no	—
194	#168	—	2	2	no	—
195	#168	—	4	1	no	—
196	#168	—	4	2	no	—
197	#168	—	8	1	no	—
198	C₂³×C₄	—	5	1	no	—
199	C₂³×C₄	—	5	1	no	—
200	C₂³×C₄	—	5	1	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	5	1	no	—
204	C₂²×C₈	—	5	1	no	—
205	C₂²×C₈	—	10	1	no	—
206	C₂²×C₈	—	10	1	no	—
207	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
208	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
209	C₂×C₄²	—	5	1	no	—
210	C₂×C₄²	—	5	1	no	—
211	C₂×C₄²	—	5	1	no	—
212	C₂×C₄²	—	10	1	no	—
213	C₂×C₄²	—	10	1	no	—
214	C₂×C₄²	—	10	1	no	—
215	C₂×C₄²	—	10	1	no	—
216	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	5	2	no	—
221	C₄⋊C₈	—	10	1	no	—
222	C₄⋊C₈	—	10	1	no	—
223	C₄⋊C₈	—	10	1	no	—
224	C₄⋊C₈	D₂₀	1	2	yes	—
225	C₂²⋊C₈	—	5	4	no	—
226	C₂²⋊C₈	—	10	2	no	—
227	C₂²⋊C₈	—	10	4	no	—
228	C₂²⋊C₈	—	20	2	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	10	4	no	—
232	C₄×C₈	—	10	1	no	—
233	C₄×C₈	—	10	1	no	—
234	C₂²×C₅	#385	1	1	yes	characteristic
235	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
236	C₂²×C₅	#424	1	1	yes	characteristic
237	C₂²×C₅	#427	1	1	yes	characteristic
238	C₂²×C₅	#428	1	1	yes	characteristic
239	C₂²×C₅	C₂×#108	1	1	yes	characteristic
240	C₂²×C₅	#430	1	1	yes	characteristic
241	C₂×D₅	—	2	2	no	—
242	C₂×D₅	—	2	2	no	—
243	C₂×D₅	—	2	2	no	—
244	C₂×D₅	—	4	1	no	—
245	C₂×D₅	—	4	1	no	—
246	C₂×D₅	—	4	1	no	—
247	C₂×D₅	—	4	1	no	—
248	C₄×C₅	—	2	1	no	—
249	C₄×C₅	—	2	1	no	—
250	C₄×C₅	C₈×D₄	1	1	yes	characteristic
251	C₄×C₅	C₈×D₄	1	1	yes	characteristic
252	C₄×C₅	#435	1	1	yes	characteristic
253	C₄×C₅	#435	1	1	yes	characteristic
254	C₄×C₅	#449	1	1	yes	characteristic
255	C₄×C₅	#452	1	1	yes	characteristic
256	C₄×C₅	#482	1	1	yes	characteristic
257	C₄×C₅	#485	1	1	yes	characteristic
258	Dic₅	—	2	2	no	—
259	Dic₅	—	2	2	no	—
260	Dic₅	—	4	1	no	—
261	C₂⁴	—	5	1	no	—
262	C₂²×C₄	—	5	1	no	—
263	C₂²×C₄	—	5	1	no	—
264	C₂²×C₄	—	5	2	no	—
265	C₂²×C₄	—	5	2	no	—
266	C₂²×C₄	—	5	1	no	—
267	C₂²×C₄	—	10	2	no	—
268	C₂²×C₄	—	10	1	no	—
269	C₂²×C₄	—	10	1	no	—
270	C₂²×C₄	—	10	1	no	—
271	C₂²×C₄	—	10	1	no	—
272	C₂²×C₄	—	10	1	no	—
273	C₂²×C₄	—	10	1	no	—
274	C₂²×C₄	—	10	1	no	—
275	C₂²×C₄	—	10	1	no	—
276	C₂²×C₄	—	10	1	no	—
277	C₂²×C₄	—	10	1	no	—
278	C₂²×C₄	—	10	1	no	—
279	C₂²×C₄	—	10	1	no	—
280	C₂²×C₄	—	20	1	no	—
281	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
282	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
283	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	4	1	no	—
291	C₂×C₈	—	4	1	no	—
292	C₂×C₈	—	4	1	no	—
293	C₂×C₈	—	4	1	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	—	5	2	no	—
296	C₂×C₈	—	10	1	no	—
297	C₂×C₈	—	10	1	no	—
298	C₂×C₈	—	10	1	no	—
299	C₂×C₈	—	10	2	no	—
300	C₂×C₈	—	10	1	no	—
301	C₂×C₈	—	10	1	no	—
302	C₂×C₈	—	20	1	no	—
303	C₂×C₈	—	20	1	no	—
304	C₂×C₈	—	20	1	no	—
305	C₂×C₈	—	20	1	no	—
306	C₂×C₈	—	20	1	no	—
307	C₂×C₈	C₈×D₅	1	2	yes	—
308	C₄²	—	10	1	no	—
309	C₄²	—	10	1	no	—
310	C₄²	—	10	1	no	—
311	C₄²	—	10	1	no	—
312	C₄²	—	10	1	no	—
313	C₄²	—	10	1	no	—
314	C₄²	C₂×D₂₀	1	1	yes	characteristic
315	C₄²	#709	1	1	yes	characteristic
316	C₄²	D₄×D₅	1	1	yes	characteristic
317	C₄²	#711	1	1	yes	characteristic
318	C₂×C₅	#1564	1	1	yes	characteristic
319	C₂×C₅	#1565	1	1	yes	characteristic
320	C₂×C₅	#1572	1	1	yes	characteristic
321	C₂×C₅	#1574	1	1	yes	characteristic
322	C₂×C₅	#1850	1	1	yes	characteristic
323	C₂×C₅	#1854	1	1	yes	characteristic
324	C₂×C₅	#1855	1	1	yes	characteristic
325	D₅	—	4	2	no	—
326	C₂³	—	5	2	no	—
327	C₂³	—	10	1	no	—
328	C₂³	—	10	1	no	—
329	C₂³	—	10	1	no	—
330	C₂³	—	10	1	no	—
331	C₂³	—	10	1	no	—
332	C₂³	—	10	1	no	—
333	C₂³	D₅×C₄⋊C₄	1	1	yes	characteristic
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	5	2	no	—
337	C₂×C₄	—	5	2	no	—
338	C₂×C₄	—	10	1	no	—
339	C₂×C₄	—	10	1	no	—
340	C₂×C₄	—	10	1	no	—
341	C₂×C₄	—	10	1	no	—
342	C₂×C₄	—	10	2	no	—
343	C₂×C₄	—	10	2	no	—
344	C₂×C₄	—	10	2	no	—
345	C₂×C₄	—	10	2	no	—
346	C₂×C₄	—	10	1	no	—
347	C₂×C₄	—	10	1	no	—
348	C₂×C₄	—	10	1	no	—
349	C₂×C₄	—	20	2	no	—
350	C₂×C₄	—	20	2	no	—
351	C₂×C₄	—	20	1	no	—
352	C₂×C₄	—	20	1	no	—
353	C₂×C₄	—	20	1	no	—
354	C₂×C₄	—	20	1	no	—
355	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
356	C₂×C₄	#4027	1	1	yes	characteristic
357	C₂×C₄	#4028	1	1	yes	characteristic
358	C₂×C₄	#4029	1	1	yes	characteristic
359	C₂×C₄	#4042	1	1	yes	characteristic
360	C₂×C₄	#4043	1	1	yes	characteristic
361	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
362	C₂×C₄	#4047	1	1	yes	characteristic
363	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
364	C₂×C₄	#4053	1	1	yes	characteristic
365	C₂×C₄	#4058	1	1	yes	characteristic
366	C₂×C₄	#4060	1	1	yes	characteristic
367	C₂×C₄	#4061	1	1	yes	characteristic
368	C₂×C₄	#4065	1	1	yes	characteristic
369	C₈	—	2	2	no	—
370	C₈	—	4	2	no	—
371	C₈	—	4	1	no	—
372	C₈	—	8	1	no	—
373	C₈	—	10	2	no	—
374	C₈	—	20	2	no	—
375	C₈	—	20	1	no	—
376	C₈	—	40	1	no	—
377	C₅	#8831	1	1	yes	characteristic, complemented, Sylow
378	C₂²	—	10	2	no	—
379	C₂²	—	10	2	no	—
380	C₂²	—	10	2	no	—
381	C₂²	—	20	1	no	—
382	C₂²	—	20	1	no	—
383	C₂²	—	20	1	no	—
384	C₂²	—	20	1	no	—
385	C₂²	#64957	1	1	yes	characteristic
386	C₂²	D₅×C₄⋊C₈	1	1	yes	characteristic
387	C₂²	#65128	1	1	yes	characteristic
388	C₂²	#65150	1	1	yes	characteristic
389	C₂²	#65169	1	1	yes	characteristic
390	C₂²	D₅×#108	1	1	yes	characteristic
391	C₂²	#65182	1	1	yes	characteristic
392	C₄	—	2	1	no	—
393	C₄	—	2	1	no	—
394	C₄	—	10	2	no	—
395	C₄	—	10	2	no	—
396	C₄	—	20	1	no	—
397	C₄	C₈×D₂₀	1	1	yes	characteristic
398	C₄	#64995	1	1	yes	characteristic
399	C₄	#65014	1	1	yes	characteristic
400	C₄	#65017	1	1	yes	characteristic
401	C₄	#65072	1	1	yes	characteristic
402	C₄	#65073	1	1	yes	characteristic
403	C₄	#65100	1	1	yes	characteristic
404	C₄	#65103	1	1	yes	characteristic
405	C₂	—	20	2	no	—
406	C₂	#10599321	1	1	yes	characteristic
407	C₂	#10599328	1	1	yes	characteristic
408	C₂	#10599360	1	1	yes	characteristic
409	C₂	#10599366	1	1	yes	characteristic
410	C₂	#10599459	1	1	yes	characteristic
411	C₂	#10599670	1	1	yes	characteristic
412	C₂	#10599734	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1277	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂³×D₅	C₄⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1277	D₅
hypocenter	C₅	#8831
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
3. C₂×#64700	C₂	semidirect
5. C₂×#64678	C₂	semidirect
6. C₅×#1277	C₂	semidirect
7. #10598194	C₂	semidirect
56. #3952	C₈	semidirect
70. #1277	D₅	semidirect
377. C₅	#8831	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65122	2
#10601268	2
C₂×#64700	2
C₄×#3952	2
C₂×#64678	2
C₅×#1277	2
#10598194	2
#8831	5