Subgroups — #12068439

Order 1280.

602 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611023	C₂	1	1	yes	maximal, characteristic
2	#10611022	C₂	1	1	yes	maximal, characteristic
3	#10610186	C₂	1	1	yes	maximal, characteristic
4	C₅×#1564	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601539	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601539	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#64678	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599366	C₂	1	2	yes	maximal, complemented
9	#10599365	C₂	1	2	yes	maximal, complemented
10	#10599328	C₂	1	2	yes	maximal, complemented
11	#10599327	C₂	1	2	yes	maximal, complemented
12	C₄×#709	C₂²	1	1	yes	characteristic
13	C₅×#431	C₂²	1	1	yes	characteristic
14	C₅×#431	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	#65399	C₄	1	1	yes	characteristic
17	#65399	C₄	1	1	yes	characteristic
18	#65391	C₄	1	1	yes	characteristic
19	#65391	C₄	1	1	yes	characteristic
20	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
22	C₄×#681	C₂²	1	1	yes	characteristic
23	#65126	C₂²	1	2	yes	—
24	#65124	C₂²	1	2	yes	—
25	#65122	C₂²	1	2	yes	—
26	D₅×C₄⋊C₈	C₂²	1	2	yes	—
27	#65019	—	2	2	no	—
28	#65014	—	2	2	no	—
29	#64979	C₄	1	2	yes	—
30	#64979	C₄	1	2	yes	—
31	#64977	—	2	1	no	—
32	C₈×D₂₀	—	2	1	no	—
33	#64974	C₂²	1	2	yes	—
34	#64974	C₂²	1	2	yes	—
35	#64968	—	2	1	no	—
36	C₈×Dic₁₀	—	2	1	no	—
37	C₅×#335	C₂²	1	1	yes	characteristic
38	C₅×#335	C₂²	1	1	yes	characteristic
39	C₅×#334	C₂²	1	1	yes	characteristic
40	C₅×#334	C₂²	1	1	yes	characteristic
41	#64700	C₂²	1	2	yes	—
42	#64700	C₂²	1	2	yes	—
43	#64699	C₂²	1	2	yes	—
44	#64699	C₂²	1	2	yes	—
45	#64678	C₂²	1	2	yes	—
46	#64678	C₂²	1	2	yes	—
47	#17812	—	5	1	no	maximal, Sylow
48	C₂×#709	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₄×C₂²⋊D₅	D₄	1	2	yes	—
53	C₄×C₂²⋊D₅	Q₈	1	2	yes	—
54	#4072	C₂×C₄	1	1	yes	characteristic
55	C₂×#681	C₂×C₄	1	1	yes	characteristic
56	C₂×#681	C₂×C₄	1	1	yes	characteristic
57	C₂²×#168	—	2	1	no	—
58	C₂²×#168	—	2	1	no	—
59	C₂×#676	—	2	2	no	—
60	C₂×C₈×D₅	—	2	2	no	—
61	C₄×D₂₀	D₄	1	1	yes	characteristic
62	C₄×D₂₀	Q₈	1	1	yes	characteristic
63	#4018	C₂×C₄	1	2	yes	—
64	C₄²×D₅	C₂³	1	2	yes	—
65	C₄×Dic₁₀	D₄	1	1	yes	characteristic
66	C₄×Dic₁₀	Q₈	1	1	yes	characteristic
67	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
68	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
69	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
70	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
71	C₅×C₂²⋊C₈	—	2	1	no	—
72	C₅×C₂²⋊C₈	—	2	1	no	—
73	C₄×C₅×C₈	—	2	1	no	—
74	C₄×C₅×C₈	—	2	1	no	—
75	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
76	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
77	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
78	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
79	#3952	—	2	2	no	—
80	#3952	—	2	2	no	—
81	#3952	C₂×C₄	1	2	yes	—
82	#3952	C₂×C₄	1	2	yes	—
83	#3952	C₂×C₄	1	2	yes	—
84	#3952	C₂×C₄	1	2	yes	—
85	#3947	—	2	2	no	—
86	#3946	—	2	2	no	—
87	#3946	—	2	2	no	—
88	#3946	C₂×C₄	1	2	yes	—
89	#3946	C₂×C₄	1	2	yes	—
90	#3946	C₂×C₄	1	2	yes	—
91	#3946	C₂×C₄	1	2	yes	—
92	C₈×Dic₅	—	2	2	no	—
93	C₄⋊#168	—	2	1	no	—
94	C₄⋊#168	—	2	1	no	—
95	C₄⋊#168	C₂³	1	2	yes	—
96	C₄⋊#168	C₂³	1	2	yes	—
97	#3935	C₂×C₄	1	2	yes	—
98	C₄×#168	C₂³	1	2	yes	—
99	#2989	—	5	1	no	—
100	#2988	—	5	1	no	—
101	#2922	—	5	1	no	—
102	#1574	—	5	2	no	—
103	#1573	—	5	2	no	—
104	#1572	—	5	2	no	—
105	#1571	—	5	2	no	—
106	#1565	—	5	1	no	—
107	#1565	—	5	1	no	—
108	#1564	D₅	1	1	yes	characteristic, complemented
109	C₂×#335	—	5	1	no	—
110	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
113	C₂×C₂²⋊D₅	C₄⋊C₄	1	2	yes	—
114	#709	—	2	2	no	—
115	#709	—	2	2	no	—
116	C₂×D₂₀	C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
118	C₂×C₄×D₅	C₂×D₄	1	2	yes	—
119	C₂×C₄×D₅	C₂×Q₈	1	2	yes	—
120	C₂×Dic₁₀	C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₅×C₈	—	2	2	no	—
122	C₂×C₅×C₈	—	2	2	no	—
123	C₂×C₅×C₈	—	2	1	no	—
124	C₂×C₅×C₈	—	2	1	no	—
125	C₂×C₅×C₈	—	2	1	no	—
126	C₂×C₅×C₈	—	2	1	no	—
127	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
132	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
134	C₄²×C₅	C₂⁴	1	1	yes	characteristic
135	C₂²⋊Dic₅	C₄⋊C₄	1	2	yes	—
136	#685	C₄⋊C₄	1	2	yes	—
137	#685	C₄⋊C₄	1	2	yes	—
138	C₄⋊Dic₅	C₄⋊C₄	1	1	yes	characteristic
139	C₄⋊Dic₅	C₄⋊C₄	1	1	yes	characteristic
140	#683	C₄⋊C₄	1	2	yes	—
141	#683	C₄⋊C₄	1	2	yes	—
142	C₄×Dic₅	C₂²×C₄	1	2	yes	—
143	C₄×Dic₅	C₂×D₄	1	2	yes	—
144	C₄×Dic₅	C₂×Q₈	1	2	yes	—
145	#681	—	2	2	no	—
146	#681	—	2	2	no	—
147	C₂×#168	—	2	2	no	—
148	C₂×#168	—	2	2	no	—
149	C₂×#168	—	8	1	no	—
150	C₂×#168	—	8	1	no	—
151	C₂×#168	C₂²×C₄	1	2	yes	—
152	C₂×#168	C₂²×C₄	1	2	yes	—
153	#676	—	8	2	no	—
154	C₈×D₅	—	8	2	no	—
155	C₄×#41	—	5	1	no	—
156	#446	—	10	1	no	—
157	C₈×Q₈	—	10	1	no	—
158	#436	—	10	1	no	—
159	#435	—	10	2	no	—
160	C₈×D₄	—	10	1	no	—
161	C₈×D₄	—	10	2	no	—
162	#433	—	5	2	no	—
163	#433	—	5	2	no	—
164	#432	—	5	2	no	—
165	#431	—	5	2	no	—
166	#431	—	5	2	no	—
167	#431	—	5	2	no	—
168	#431	C₂×D₅	1	1	yes	characteristic
169	#431	C₂×D₅	1	1	yes	characteristic
170	#424	—	5	1	no	—
171	#424	—	5	1	no	—
172	#423	—	5	2	no	—
173	C₂×C₄⋊C₈	—	5	1	no	—
174	C₂×C₄⋊C₈	—	5	1	no	—
175	C₂×C₄⋊C₈	—	5	2	no	—
176	#408	—	5	1	no	—
177	#408	—	5	1	no	—
178	C₄×M₁₆	—	5	1	no	—
179	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
180	#335	—	5	2	no	—
181	#335	—	5	2	no	—
182	#335	—	5	2	no	—
183	#335	—	5	2	no	—
184	#335	C₂×D₅	1	1	yes	characteristic
185	#335	C₂×D₅	1	1	yes	characteristic
186	#334	—	5	2	no	—
187	#334	—	5	2	no	—
188	#334	C₂×D₅	1	1	yes	characteristic
189	#334	C₂×D₅	1	1	yes	characteristic
190	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂²×D₅	C₂×C₄⋊C₄	1	2	yes	—
192	C₂×C₄×C₅	—	2	1	no	—
193	C₂×C₄×C₅	—	2	1	no	—
194	C₂×C₄×C₅	—	4	1	no	—
195	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
196	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₄×C₅	#131	1	2	yes	—
199	C₂×C₄×C₅	#131	1	2	yes	—
200	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
201	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
203	C₂²⋊D₅	—	2	4	no	—
204	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
205	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
206	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
207	D₂₀	—	2	2	no	—
208	C₄×D₅	—	2	2	no	—
209	C₄×D₅	—	2	2	no	—
210	C₄×D₅	—	2	2	no	—
211	C₄×D₅	—	2	2	no	—
212	C₄×D₅	—	4	2	no	—
213	Dic₁₀	—	2	2	no	—
214	C₅×C₈	—	2	1	no	—
215	C₅×C₈	—	2	1	no	—
216	C₅×C₈	—	2	1	no	—
217	C₅×C₈	—	2	1	no	—
218	C₅×C₈	—	4	1	no	—
219	C₅×C₈	—	4	1	no	—
220	C₅×C₈	—	4	1	no	—
221	C₅×C₈	—	4	1	no	—
222	#168	—	2	2	no	—
223	#168	—	2	2	no	—
224	#168	—	4	2	no	—
225	#168	—	4	2	no	—
226	C₂×#41	—	5	1	no	—
227	C₂×M₁₆	—	5	1	no	—
228	C₂×M₁₆	—	5	1	no	—
229	C₂×M₁₆	—	10	2	no	—
230	C₂²×C₈	—	10	2	no	—
231	C₂²×C₈	—	10	1	no	—
232	C₂²×C₈	—	10	1	no	—
233	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
234	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
235	C₄×Q₈	—	5	1	no	—
236	C₄×Q₈	—	5	1	no	—
237	C₄×D₄	—	5	2	no	—
238	C₄×D₄	—	5	2	no	—
239	C₄×D₄	—	5	1	no	—
240	C₄×D₄	—	5	1	no	—
241	#117	—	5	1	no	—
242	#117	—	5	2	no	—
243	C₂×C₄²	—	5	2	no	—
244	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
245	C₄⋊C₈	—	5	2	no	—
246	C₄⋊C₈	—	5	2	no	—
247	C₄⋊C₈	—	5	2	no	—
248	C₄⋊C₈	—	5	2	no	—
249	C₄⋊C₈	—	5	2	no	—
250	C₄⋊C₈	—	5	2	no	—
251	C₄⋊C₈	—	10	2	no	—
252	C₄⋊C₈	—	10	2	no	—
253	C₄⋊C₈	—	10	1	no	—
254	C₄⋊C₈	—	10	1	no	—
255	C₄⋊C₈	C₂²×D₅	1	1	yes	characteristic
256	C₄⋊C₈	C₂²×D₅	1	1	yes	characteristic
257	C₄⋊C₈	C₂²×D₅	1	1	yes	characteristic
258	C₄⋊C₈	C₂²×D₅	1	1	yes	characteristic
259	C₂²⋊C₈	—	2	1	no	—
260	C₂²⋊C₈	—	2	1	no	—
261	C₂²⋊C₈	—	5	2	no	—
262	C₂²⋊C₈	—	5	2	no	—
263	C₂²⋊C₈	—	5	2	no	—
264	C₂²⋊C₈	—	5	2	no	—
265	C₂²⋊C₈	—	10	2	no	—
266	C₂²⋊C₈	—	10	2	no	—
267	#97	—	5	2	no	—
268	#97	—	10	2	no	—
269	C₄×C₈	—	2	1	no	—
270	C₄×C₈	—	2	1	no	—
271	C₄×C₈	—	5	2	no	—
272	C₄×C₈	—	10	2	no	—
273	C₄×C₈	C₄×D₅	1	1	yes	characteristic
274	C₄×C₈	C₄×D₅	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
277	C₂²×C₅	—	2	1	no	—
278	C₂²×C₅	—	2	1	no	—
279	C₂²×C₅	#424	1	2	yes	—
280	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
281	C₂×D₅	—	2	2	no	—
282	C₂×D₅	—	2	2	no	—
283	C₂×D₅	—	2	2	no	—
284	C₄×C₅	—	2	1	no	—
285	C₄×C₅	—	2	1	no	—
286	C₄×C₅	—	2	1	no	—
287	C₄×C₅	#424	1	2	yes	—
288	C₄×C₅	C₂×#131	1	1	yes	characteristic
289	C₄×C₅	C₂×#131	1	1	yes	characteristic
290	C₄×C₅	#576	1	1	yes	characteristic
291	C₄×C₅	#578	1	1	yes	characteristic
292	Dic₅	—	2	2	no	—
293	Dic₅	—	2	2	no	—
294	Dic₅	—	2	2	no	—
295	#41	—	10	2	no	—
296	#41	—	10	2	no	—
297	C₂×Q₈	—	5	1	no	—
298	C₂×D₄	—	5	2	no	—
299	C₂×D₄	—	5	1	no	—
300	C₂²×C₄	—	5	2	no	—
301	C₂²×C₄	—	5	2	no	—
302	C₂²×C₄	—	5	2	no	—
303	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
304	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
305	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
306	M₁₆	—	10	2	no	—
307	M₁₆	—	10	2	no	—
308	M₁₆	—	40	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	5	2	no	—
316	C₂×C₈	—	5	2	no	—
317	C₂×C₈	—	10	2	no	—
318	C₂×C₈	—	10	2	no	—
319	C₂×C₈	—	40	2	no	—
320	C₂×C₈	—	40	1	no	—
321	C₂×C₈	—	40	1	no	—
322	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
323	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
324	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
325	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
326	C₄⋊C₄	—	5	2	no	—
327	C₄⋊C₄	—	5	2	no	—
328	C₄⋊C₄	—	5	1	no	—
329	C₄⋊C₄	—	5	1	no	—
330	C₂²⋊C₄	—	5	2	no	—
331	C₂²⋊C₄	—	5	2	no	—
332	C₂²⋊C₄	—	5	2	no	—
333	C₄²	—	5	2	no	—
334	C₄²	—	5	2	no	—
335	C₄²	—	5	2	no	—
336	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
337	C₄²	D₄×D₅	1	1	yes	characteristic
338	C₄²	Q₈×D₅	1	1	yes	characteristic
339	C₄²	C₂³×D₅	1	1	yes	characteristic
340	C₂×C₅	—	2	2	no	—
341	C₂×C₅	C₂×#424	1	1	yes	characteristic
342	C₂×C₅	#2913	1	1	yes	characteristic
343	C₂×C₅	#2916	1	1	yes	characteristic
344	D₅	—	4	2	no	—
345	C₂³	—	5	2	no	—
346	C₂³	D₅×C₄⋊C₄	1	1	yes	characteristic
347	Q₈	—	10	2	no	—
348	D₄	—	10	4	no	—
349	D₄	—	10	2	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	5	2	no	—
354	C₂×C₄	—	5	2	no	—
355	C₂×C₄	—	5	2	no	—
356	C₂×C₄	—	10	2	no	—
357	C₂×C₄	—	10	2	no	—
358	C₂×C₄	—	10	2	no	—
359	C₂×C₄	—	10	2	no	—
360	C₂×C₄	—	20	2	no	—
361	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
362	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
363	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
364	C₂×C₄	#4047	1	2	yes	—
365	C₂×C₄	#4053	1	2	yes	—
366	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
367	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
368	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	4	1	no	—
374	C₈	—	4	1	no	—
375	C₈	—	4	1	no	—
376	C₈	—	4	1	no	—
377	C₈	—	10	2	no	—
378	C₈	—	10	2	no	—
379	C₈	—	20	2	no	—
380	C₈	—	20	2	no	—
381	C₅	#17812	1	1	yes	characteristic, complemented, Sylow
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	10	2	no	—
385	C₂²	—	10	2	no	—
386	C₂²	—	10	2	no	—
387	C₂²	#65128	1	2	yes	—
388	C₂²	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
389	C₄	—	2	1	no	—
390	C₄	—	2	1	no	—
391	C₄	—	2	1	no	—
392	C₄	—	10	2	no	—
393	C₄	—	10	2	no	—
394	C₄	—	10	2	no	—
395	C₄	#65128	1	2	yes	—
396	C₄	C₂×#4047	1	1	yes	characteristic
397	C₄	C₂×#4053	1	1	yes	characteristic
398	C₄	#66103	1	1	yes	characteristic
399	C₄	#66105	1	1	yes	characteristic
400	C₂	—	2	2	no	—
401	C₂	—	20	2	no	—
402	C₂	C₂×#65128	1	1	yes	characteristic
403	C₂	#10610910	1	1	yes	characteristic
404	C₂	#10610983	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×#1564	C₂
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂×#709	C₂×C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1564	D₅
hypocenter	C₅	#17812
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
4. C₅×#1564	C₂	semidirect
5. #10601539	C₂	semidirect
6. #10601539	C₂	semidirect
7. C₂×#64678	C₂	semidirect
8. #10599366	C₂	semidirect
9. #10599365	C₂	semidirect
10. #10599328	C₂	semidirect
11. #10599327	C₂	semidirect
108. #1564	D₅	semidirect
381. C₅	#17812	semidirect

Maximal subgroups

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

subgroup	index
#10611023	2
#10611022	2
#10610186	2
C₅×#1564	2
#10601539	2
C₂×#64678	2
#10599366	2
#10599365	2
#10599328	2
#10599327	2
#17812	5