Subgroups — #12070721

Order 1280.

794 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610774	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₄₀	C₂	1	1	yes	maximal, characteristic
3	C₅×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601289	C₂	1	2	yes	maximal, complemented
5	C₈×#681	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599356	C₂	1	2	yes	maximal, complemented
7	#10599352	C₂	1	4	yes	maximal, complemented
8	#10599349	C₂	1	2	yes	maximal, complemented
9	C₂×D₅×M₁₆	—	2	2	no	—
10	C₂²×D₄₀	C₄	1	1	yes	characteristic
11	C₂×C₄×D₂₀	C₂²	1	2	yes	—
12	C₅×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	#65415	—	2	2	no	—
15	C₂×#3953	C₄	1	2	yes	—
16	C₂×#3950	C₄	1	1	yes	characteristic
17	#65214	C₂²	1	1	yes	characteristic
18	C₄×#681	C₂²	1	1	yes	characteristic
19	#65127	C₂²	1	4	yes	complemented
20	#65125	—	2	4	no	—
21	#65123	C₂²	1	4	yes	complemented
22	C₄×D₄₀	C₄	1	2	yes	—
23	C₄×D₄₀	C₄	1	1	yes	characteristic
24	C₄×D₄₀	C₄	1	1	yes	characteristic
25	C₄×D₄₀	C₂²	1	1	yes	characteristic
26	C₄×D₄₀	C₂²	1	2	yes	—
27	C₄×D₄₀	C₂²	1	1	yes	characteristic
28	C₅×#335	C₂²	1	2	yes	complemented
29	C₅×#335	C₂²	1	2	yes	complemented
30	#64703	C₂²	1	4	yes	complemented
31	#64703	C₂²	1	4	yes	complemented
32	#64674	C₂²	1	2	yes	complemented
33	C₈×#168	C₂²	1	2	yes	complemented
34	M₁₆×D₈	—	5	1	no	maximal, Sylow
35	C₂²×D₂₀	C₂×C₄	1	2	yes	—
36	C₂²×C₄×D₅	—	2	2	no	—
37	C₂×C₅×M₁₆	—	2	2	no	—
38	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₂×#685	—	2	2	no	—
42	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
43	C₂×#681	D₄	1	1	yes	characteristic
44	C₂×#681	D₄	1	1	yes	characteristic
45	D₅×M₁₆	—	4	8	no	—
46	C₂×D₄₀	—	2	4	no	—
47	C₂×D₄₀	C₂×C₄	1	2	yes	—
48	C₂×D₄₀	C₂×C₄	1	1	yes	characteristic
49	C₂×D₄₀	C₂×C₄	1	1	yes	characteristic
50	C₂×#676	—	2	4	no	—
51	C₂×C₈×D₅	—	2	4	no	—
52	C₄×D₂₀	—	2	2	no	—
53	C₄×D₂₀	—	2	2	no	—
54	C₄×D₂₀	C₂×C₄	1	2	yes	—
55	C₄×D₂₀	C₂×C₄	1	2	yes	—
56	C₄×D₂₀	C₂³	1	2	yes	—
57	C₄×D₂₀	C₂³	1	2	yes	—
58	C₅×C₄⋊C₈	—	2	2	no	—
59	C₅×C₄⋊C₈	C₂³	1	4	yes	—
60	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
63	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
64	C₂²⋊#168	—	2	1	no	—
65	#3953	C₂×C₄	1	2	yes	—
66	#3953	C₂×C₄	1	2	yes	—
67	#3953	C₂×C₄	1	2	yes	—
68	#3953	C₂×C₄	1	2	yes	—
69	#3952	—	2	4	no	—
70	#3952	—	2	4	no	—
71	#3950	C₂×C₄	1	2	yes	—
72	#3950	C₂×C₄	1	1	yes	characteristic
73	#3950	C₂×C₄	1	1	yes	characteristic
74	C₄⋊#168	—	2	1	no	—
75	#3935	D₄	1	2	yes	—
76	C₄×#168	C₂³	1	2	yes	—
77	C₄×#168	C₂³	1	2	yes	—
78	C₂×C₄×D₈	—	5	1	no	—
79	D₄×M₁₆	—	5	2	no	—
80	#1589	—	5	2	no	—
81	#1581	—	5	4	no	—
82	C₈×D₈	—	5	2	no	—
83	#1567	D₅	1	1	yes	characteristic, complemented
84	#1486	—	5	2	no	—
85	C₈×M₁₆	—	5	1	no	—
86	C₂³×D₅	—	2	2	no	—
87	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	#41	1	1	yes	characteristic
90	C₂²×Dic₅	—	2	2	no	—
91	C₂×D₂₀	—	2	4	no	—
92	C₂×D₂₀	—	2	2	no	—
93	C₂×D₂₀	—	4	2	no	—
94	C₂×D₂₀	C₂²×C₄	1	2	yes	—
95	C₂×D₂₀	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₅	—	2	2	no	—
97	C₂×C₄×D₅	—	2	2	no	—
98	C₂×C₄×D₅	—	2	2	no	—
99	C₂×C₄×D₅	—	2	2	no	—
100	C₂×C₄×D₅	—	4	4	no	—
101	C₂×C₄×D₅	—	4	4	no	—
102	C₅×M₁₆	—	4	4	no	—
103	C₂×C₅×C₈	—	2	2	no	—
104	C₂×C₅×C₈	—	2	4	no	—
105	C₂×C₅×C₈	—	4	1	no	—
106	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅	#41	1	1	yes	characteristic
113	C₄²×C₅	C₂⁴	1	1	yes	characteristic
114	#685	—	2	2	no	—
115	#685	—	2	2	no	—
116	#685	—	2	2	no	—
117	#685	—	2	2	no	—
118	C₄⋊Dic₅	—	2	2	no	—
119	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
120	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
121	#681	—	2	2	no	—
122	#681	D₈	1	4	yes	complemented
123	C₂×#168	—	2	2	no	—
124	C₂×#168	C₂×D₄	1	2	yes	—
125	C₂×#168	C₂×D₄	1	2	yes	—
126	D₄₀	—	2	4	no	—
127	D₄₀	—	2	2	no	—
128	D₄₀	M₁₆	1	4	yes	complemented
129	#676	—	4	8	no	—
130	C₈×D₅	—	4	8	no	—
131	C₂²×D₈	—	5	1	no	—
132	C₂²×M₁₆	—	10	2	no	—
133	C₂×C₄×D₄	—	5	2	no	—
134	C₄×D₈	—	5	1	no	—
135	C₄×D₈	—	5	2	no	—
136	C₄×D₈	—	5	1	no	—
137	C₄×D₈	—	5	1	no	—
138	C₄×D₈	—	5	2	no	—
139	C₄×D₈	—	5	1	no	—
140	#436	—	5	4	no	—
141	#435	—	10	4	no	—
142	C₈×D₄	—	5	4	no	—
143	#431	—	5	1	no	—
144	C₂×#107	—	5	1	no	—
145	#423	C₂×D₅	1	2	yes	—
146	C₂×#102	—	5	2	no	—
147	#407	—	10	2	no	—
148	C₄×M₁₆	—	5	1	no	—
149	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
150	#335	C₂×D₅	1	2	yes	complemented
151	#335	C₂×D₅	1	2	yes	complemented
152	D₄⋊C₈	—	5	4	no	—
153	D₄⋊C₈	—	5	4	no	—
154	#322	—	5	2	no	—
155	C₈²	—	5	2	no	—
156	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
157	C₂²×D₅	—	2	2	no	—
158	C₂²×D₅	—	2	2	no	—
159	C₂²×D₅	—	4	4	no	—
160	C₂²×D₅	—	4	4	no	—
161	C₂²×D₅	—	8	2	no	—
162	C₂×C₄×C₅	—	2	1	no	—
163	C₂×C₄×C₅	—	2	1	no	—
164	C₂×C₄×C₅	—	4	1	no	—
165	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
167	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
169	C₂×C₄×C₅	#135	1	2	yes	—
170	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
173	C₂×Dic₅	—	2	2	no	—
174	C₂×Dic₅	—	2	2	no	—
175	C₂×Dic₅	—	4	4	no	—
176	D₂₀	—	2	2	no	—
177	D₂₀	—	2	2	no	—
178	D₂₀	—	2	2	no	—
179	D₂₀	—	4	4	no	—
180	D₂₀	C₂×M₁₆	1	4	yes	complemented
181	C₄×D₅	—	4	2	no	—
182	C₄×D₅	—	4	2	no	—
183	C₄×D₅	—	4	2	no	—
184	C₄×D₅	—	4	2	no	—
185	C₄×D₅	—	4	4	no	—
186	C₄×D₅	—	4	4	no	—
187	C₅×C₈	—	2	1	no	—
188	C₅×C₈	—	2	1	no	—
189	C₅×C₈	—	2	1	no	—
190	C₅×C₈	—	4	4	no	—
191	C₅×C₈	C₂×M₁₆	1	2	yes	complemented
192	#168	—	2	2	no	—
193	#168	—	4	2	no	—
194	#168	C₂×D₈	1	4	yes	complemented
195	C₂²×D₄	—	5	2	no	—
196	C₂³×C₄	—	10	2	no	—
197	C₂×D₈	—	5	1	no	—
198	C₂×D₈	—	5	2	no	—
199	C₂×D₈	—	5	1	no	—
200	C₂×D₈	—	10	4	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	5	1	no	—
203	C₂×M₁₆	—	5	1	no	—
204	C₂×M₁₆	—	10	4	no	—
205	C₂×M₁₆	—	20	8	no	—
206	C₂²×C₈	—	10	4	no	—
207	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
208	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
209	C₄×D₄	—	5	2	no	—
210	C₄×D₄	—	5	2	no	—
211	C₄×D₄	—	5	2	no	—
212	C₄×D₄	—	5	2	no	—
213	C₄×D₄	—	10	2	no	—
214	C₄×D₄	—	10	2	no	—
215	C₂×C₄⋊C₄	—	5	2	no	—
216	C₂×C₂²⋊C₄	—	10	2	no	—
217	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
218	#107	—	5	2	no	—
219	#107	—	5	1	no	—
220	#107	—	5	1	no	—
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	—	10	1	no	—
223	C₄⋊C₈	C₂²×D₅	1	4	yes	—
224	#102	—	5	2	no	—
225	#102	—	5	2	no	—
226	#102	—	5	2	no	—
227	#102	—	5	2	no	—
228	C₂²⋊C₈	—	10	4	no	—
229	C₂²⋊C₈	—	10	4	no	—
230	C₂²⋊C₈	—	10	1	no	—
231	#97	—	5	2	no	—
232	C₄×C₈	—	5	2	no	—
233	C₄×C₈	—	5	2	no	—
234	C₄×C₈	C₄×D₅	1	1	yes	characteristic
235	C₄×C₈	C₄×D₅	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
238	C₂²×C₅	—	2	1	no	—
239	C₂²×C₅	—	2	1	no	—
240	C₂²×C₅	C₄×D₈	1	2	yes	—
241	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
242	C₂×D₅	—	2	4	no	—
243	C₂×D₅	—	4	2	no	—
244	C₂×D₅	—	4	2	no	—
245	C₂×D₅	—	4	2	no	—
246	C₂×D₅	—	4	2	no	—
247	C₂×D₅	—	4	2	no	—
248	C₂×D₅	—	8	8	no	—
249	C₄×C₅	—	2	1	no	—
250	C₄×C₅	—	2	1	no	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	C₄×D₈	1	2	yes	—
253	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
254	C₄×C₅	#568	1	1	yes	characteristic
255	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
256	C₄×C₅	C₂×#135	1	1	yes	characteristic
257	Dic₅	—	4	2	no	—
258	Dic₅	—	4	2	no	—
259	C₂⁴	—	10	2	no	—
260	C₂×D₄	—	5	2	no	—
261	C₂×D₄	—	5	2	no	—
262	C₂×D₄	—	10	4	no	—
263	C₂×D₄	—	10	2	no	—
264	C₂×D₄	—	20	2	no	—
265	C₂²×C₄	—	10	2	no	—
266	C₂²×C₄	—	10	2	no	—
267	C₂²×C₄	—	10	2	no	—
268	C₂²×C₄	—	10	2	no	—
269	C₂²×C₄	—	10	2	no	—
270	C₂²×C₄	—	20	4	no	—
271	C₂²×C₄	—	20	4	no	—
272	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
273	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
274	C₂²×C₄	#713	1	1	yes	characteristic
275	D₈	—	5	4	no	—
276	D₈	—	10	4	no	—
277	D₈	—	10	2	no	—
278	M₁₆	—	4	4	no	—
279	M₁₆	—	5	4	no	—
280	M₁₆	—	10	2	no	—
281	M₁₆	—	20	8	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	5	2	no	—
286	C₂×C₈	—	5	2	no	—
287	C₂×C₈	—	10	2	no	—
288	C₂×C₈	—	20	8	no	—
289	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
293	C₄⋊C₄	—	5	2	no	—
294	C₄⋊C₄	—	5	2	no	—
295	C₄⋊C₄	—	10	2	no	—
296	C₂²⋊C₄	—	10	2	no	—
297	C₂²⋊C₄	—	10	2	no	—
298	C₂²⋊C₄	—	10	2	no	—
299	C₂²⋊C₄	—	10	2	no	—
300	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
301	C₄²	D₄×D₅	1	1	yes	characteristic
302	C₄²	#713	1	1	yes	characteristic
303	C₄²	C₂³×D₅	1	1	yes	characteristic
304	C₂×C₅	—	2	2	no	—
305	C₂×C₅	D₄×M₁₆	1	1	yes	characteristic
306	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
307	C₂×C₅	#2957	1	1	yes	characteristic
308	D₅	—	4	4	no	—
309	D₅	—	8	2	no	—
310	C₂³	—	10	2	no	—
311	C₂³	—	10	2	no	—
312	C₂³	—	20	4	no	—
313	C₂³	—	20	4	no	—
314	C₂³	—	40	2	no	—
315	C₂³	#4039	1	1	yes	characteristic
316	D₄	—	5	4	no	—
317	D₄	—	10	2	no	—
318	D₄	—	10	2	no	—
319	D₄	—	10	2	no	—
320	D₄	—	20	4	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	10	2	no	—
325	C₂×C₄	—	10	2	no	—
326	C₂×C₄	—	20	4	no	—
327	C₂×C₄	—	20	2	no	—
328	C₂×C₄	—	20	2	no	—
329	C₂×C₄	—	20	2	no	—
330	C₂×C₄	—	20	2	no	—
331	C₂×C₄	—	20	4	no	—
332	C₂×C₄	—	20	4	no	—
333	C₂×C₄	#4039	1	1	yes	characteristic
334	C₂×C₄	#4039	1	1	yes	characteristic
335	C₂×C₄	#4039	1	1	yes	characteristic
336	C₂×C₄	D₅×D₈	1	2	yes	—
337	C₂×C₄	#4065	1	2	yes	—
338	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
339	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
340	C₂×C₄	C₂×#713	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	4	4	no	—
345	C₈	—	5	4	no	—
346	C₈	—	10	2	no	—
347	C₈	—	20	2	no	—
348	C₈	D₅×M₁₆	1	2	yes	complemented
349	C₅	M₁₆×D₈	1	1	yes	characteristic, complemented, Sylow
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	10	4	no	—
353	C₂²	—	20	2	no	—
354	C₂²	—	20	2	no	—
355	C₂²	—	20	2	no	—
356	C₂²	—	20	2	no	—
357	C₂²	—	20	2	no	—
358	C₂²	—	40	8	no	—
359	C₂²	#65161	1	2	yes	—
360	C₂²	C₂×#4039	1	1	yes	characteristic
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	20	2	no	—
365	C₄	—	20	2	no	—
366	C₄	#65161	1	2	yes	—
367	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
368	C₄	#66079	1	1	yes	characteristic
369	C₄	C₂×D₅×D₈	1	1	yes	characteristic
370	C₄	C₂×#4065	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	20	4	no	—
373	C₂	—	40	2	no	—
374	C₂	#10610774	1	1	yes	characteristic
375	C₂	C₂×#65161	1	1	yes	characteristic
376	C₂	#10610980	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#1567	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂²×D₄₀	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1567	D₅
hypocenter	C₅	M₁₆×D₈
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
1. #10610774	C₂	semidirect
3. C₅×#1567	C₂	semidirect
4. #10601289	C₂	semidirect
5. C₈×#681	C₂	semidirect
6. #10599356	C₂	semidirect
7. #10599352	C₂	semidirect
8. #10599349	C₂	semidirect
19. #65127	C₂²	semidirect
21. #65123	C₂²	semidirect
28. C₅×#335	C₂²	semidirect
29. C₅×#335	C₂²	semidirect
30. #64703	C₂²	semidirect
31. #64703	C₂²	semidirect
32. #64674	C₂²	semidirect
33. C₈×#168	C₂²	semidirect
83. #1567	D₅	semidirect
122. #681	D₈	semidirect
128. D₄₀	M₁₆	semidirect
150. #335	C₂×D₅	semidirect
151. #335	C₂×D₅	semidirect
180. D₂₀	C₂×M₁₆	semidirect
191. C₅×C₈	C₂×M₁₆	semidirect
194. #168	C₂×D₈	semidirect
348. C₈	D₅×M₁₆	semidirect
349. C₅	M₁₆×D₈	semidirect

Maximal subgroups

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

subgroup	index
#10610774	2
C₂×C₄×D₄₀	2
C₅×#1567	2
#10601289	2
C₈×#681	2
#10599356	2
#10599352	2
#10599349	2
M₁₆×D₈	5