Subgroups — #12069147

Order 1280.

646 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#4058	C₂	1	1	yes	maximal, characteristic
2	#10611254	C₂	1	2	yes	maximal, complemented
3	C₅×#1589	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601674	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601653	C₂	1	2	yes	maximal, complemented
6	D₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
7	#10599357	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598959	C₂	1	2	yes	maximal, complemented
9	#10598955	C₂	1	2	yes	maximal, complemented
10	#10598849	C₂	1	1	yes	maximal, characteristic, complemented
11	#10598840	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#4058	C₄	1	1	yes	characteristic
13	C₄×#711	C₂²	1	2	yes	—
14	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
15	C₅×#436	C₂²	1	2	yes	—
16	D₈×Dic₅	C₄	1	1	yes	characteristic
17	#65416	—	2	2	no	—
18	#65407	—	2	2	no	—
19	C₄×#687	C₂²	1	2	yes	—
20	#65301	C₂²	1	2	yes	—
21	#65169	C₄	1	1	yes	characteristic
22	#65162	C₄	1	1	yes	characteristic
23	D₅×C₄⋊C₈	C₂²	1	2	yes	—
24	#65120	C₂²	1	2	yes	—
25	#65061	C₄	1	2	yes	—
26	#65047	C₄	1	2	yes	—
27	#65017	—	2	2	no	—
28	#65008	—	2	2	no	—
29	C₄×Dic₂₀	C₂²	1	1	yes	characteristic
30	#64974	C₂²	1	1	yes	characteristic
31	#64974	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
33	#64968	C₂²	1	2	yes	—
34	C₅×#335	C₂²	1	1	yes	characteristic
35	C₅×D₄⋊C₈	C₂²	1	2	yes	—
36	C₅×C₈²	C₂²	1	1	yes	characteristic
37	D₄⋊#168	C₂²	1	2	yes	—
38	#64704	C₂²	1	2	yes	—
39	#64699	C₂²	1	2	yes	—
40	#64678	C₂²	1	1	yes	characteristic
41	#64676	C₂²	1	2	yes	—
42	#64674	C₂²	1	1	yes	characteristic
43	#64674	C₂²	1	1	yes	characteristic
44	C₈×#168	C₂²	1	1	yes	characteristic
45	#17898	—	5	1	no	maximal, Sylow
46	C₂×#711	C₂×C₄	1	2	yes	—
47	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₅×M₁₆	—	2	2	no	—
49	C₄×C₅×D₄	C₂³	1	2	yes	—
50	D₄×Dic₅	C₂×C₄	1	2	yes	—
51	C₂×#687	C₂×C₄	1	2	yes	—
52	C₄×C₂²⋊D₅	—	2	2	no	—
53	C₂×C₄×Dic₅	—	2	2	no	—
54	C₂×#681	—	2	2	no	—
55	#4058	—	2	4	no	—
56	C₂×Dic₂₀	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₅	—	2	2	no	—
58	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
60	#4038	C₂×C₄	1	2	yes	—
61	#4033	C₂×C₄	1	2	yes	—
62	#4027	—	2	2	no	—
63	#4023	—	2	2	no	—
64	C₄²×D₅	C₂³	1	1	yes	characteristic
65	C₄×Dic₁₀	C₂³	1	2	yes	—
66	C₅×#107	C₂×C₄	1	1	yes	characteristic
67	C₅×C₄⋊C₈	C₂³	1	2	yes	—
68	C₅×#102	C₂×C₄	1	2	yes	—
69	C₅×C₂²⋊C₈	—	2	2	no	—
70	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
71	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
73	#3964	C₂×C₄	1	2	yes	—
74	C₂²⋊#168	—	2	2	no	—
75	#3952	—	2	1	no	—
76	#3952	D₄	1	1	yes	characteristic
77	#3952	D₄	1	1	yes	characteristic
78	#3950	C₂×C₄	1	1	yes	characteristic
79	#3948	C₂×C₄	1	2	yes	—
80	#3947	—	2	2	no	—
81	#3947	—	2	2	no	—
82	#3946	—	2	2	no	—
83	#3946	—	2	2	no	—
84	#3946	—	2	1	no	—
85	#3946	—	2	2	no	—
86	#3946	D₄	1	1	yes	characteristic
87	#3946	D₄	1	1	yes	characteristic
88	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
89	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
90	#3942	C₂×C₄	1	2	yes	—
91	#3940	C₂×C₄	1	2	yes	—
92	C₄⋊#168	C₂³	1	2	yes	—
93	C₄×#168	C₂³	1	1	yes	characteristic
94	C₄×#168	C₂³	1	1	yes	characteristic
95	C₄×#168	C₂³	1	1	yes	characteristic
96	#2989	—	5	2	no	—
97	C₄×#135	—	5	1	no	—
98	#1592	—	5	2	no	—
99	#1592	—	5	2	no	—
100	#1591	—	5	1	no	—
101	#1591	—	5	1	no	—
102	#1589	—	5	1	no	—
103	#1589	D₅	1	1	yes	characteristic, complemented
104	C₂×#335	—	5	1	no	—
105	#1477	—	5	2	no	—
106	#1450	—	5	1	no	—
107	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₅	—	2	2	no	—
109	C₂×C₂²⋊D₅	—	2	2	no	—
110	C₂²×Dic₅	—	2	2	no	—
111	C₂²×Dic₅	—	2	2	no	—
112	#711	—	2	4	no	—
113	#711	—	2	4	no	—
114	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×D₅	#41	1	1	yes	characteristic
117	C₂×Dic₁₀	C₂²×C₄	1	2	yes	—
118	C₅×D₈	—	2	2	no	—
119	C₅×M₁₆	—	8	2	no	—
120	C₂×C₅×C₈	—	2	2	no	—
121	C₂×C₅×C₈	—	2	1	no	—
122	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
126	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
127	C₅×C₂²⋊C₄	—	2	2	no	—
128	C₄²×C₅	C₂⁴	1	1	yes	characteristic
129	C₂²⋊Dic₅	—	2	2	no	—
130	#687	—	2	4	no	—
131	#685	—	2	2	no	—
132	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
133	#683	—	2	2	no	—
134	#683	—	2	2	no	—
135	C₄×Dic₅	—	2	2	no	—
136	C₄×Dic₅	—	2	2	no	—
137	C₄×Dic₅	—	2	2	no	—
138	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
139	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
140	C₄×Dic₅	#41	1	1	yes	characteristic
141	#681	—	8	2	no	—
142	C₂×#168	—	2	2	no	—
143	C₂×#168	—	2	1	no	—
144	C₂×#168	C₂²×C₄	1	1	yes	characteristic
145	C₂×#168	C₂²×C₄	1	1	yes	characteristic
146	C₂×#168	C₂×D₄	1	1	yes	characteristic
147	C₂×#168	C₂×D₄	1	1	yes	characteristic
148	Dic₂₀	—	2	2	no	—
149	C₈×D₅	—	2	2	no	—
150	C₈×D₅	—	8	2	no	—
151	C₈×D₅	M₁₆	1	4	yes	complemented
152	C₂×#135	—	5	1	no	—
153	C₄×#41	—	5	2	no	—
154	#446	—	5	2	no	—
155	#446	—	5	2	no	—
156	C₄×Q₁₆	—	5	1	no	—
157	C₄×Q₁₆	—	5	1	no	—
158	C₄×SD₁₆	—	5	2	no	—
159	C₄×SD₁₆	—	5	2	no	—
160	C₄×D₈	—	5	1	no	—
161	C₄×D₈	C₂×D₅	1	1	yes	characteristic
162	#436	—	5	2	no	—
163	#436	C₂×D₅	1	2	yes	—
164	#435	—	10	2	no	—
165	#435	—	10	2	no	—
166	#432	—	10	2	no	—
167	#431	—	5	1	no	—
168	#431	—	5	1	no	—
169	#427	—	5	1	no	—
170	#423	—	10	2	no	—
171	C₂×C₄⋊C₈	—	5	2	no	—
172	#416	—	5	2	no	—
173	C₂×C₄×C₈	—	5	1	no	—
174	#335	—	5	2	no	—
175	#335	—	5	1	no	—
176	#335	C₂×D₅	1	1	yes	characteristic
177	Q₈⋊C₈	—	5	2	no	—
178	Q₈⋊C₈	—	5	2	no	—
179	D₄⋊C₈	—	5	2	no	—
180	D₄⋊C₈	C₂×D₅	1	2	yes	—
181	#322	—	5	1	no	—
182	#322	—	5	1	no	—
183	C₈²	—	5	1	no	—
184	C₈²	C₂×D₅	1	1	yes	characteristic
185	C₂³×C₅	—	2	2	no	—
186	C₂²×D₅	C₄×D₄	1	1	yes	characteristic
187	C₅×D₄	—	2	2	no	—
188	C₅×D₄	—	2	2	no	—
189	C₂×C₄×C₅	—	2	2	no	—
190	C₂×C₄×C₅	—	4	2	no	—
191	C₂×C₄×C₅	—	4	2	no	—
192	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
195	C₂²⋊D₅	—	4	4	no	—
196	C₂×Dic₅	—	2	2	no	—
197	C₂×Dic₅	—	2	2	no	—
198	C₂×Dic₅	—	4	4	no	—
199	C₂×Dic₅	—	8	2	no	—
200	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
201	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
202	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
203	C₄×D₅	—	2	1	no	—
204	C₄×D₅	—	2	1	no	—
205	C₄×D₅	—	4	1	no	—
206	C₄×D₅	C₂×M₁₆	1	2	yes	—
207	C₄×D₅	#131	1	2	yes	—
208	Dic₁₀	—	2	2	no	—
209	Dic₁₀	—	2	2	no	—
210	C₅×C₈	—	2	1	no	—
211	C₅×C₈	—	2	1	no	—
212	C₅×C₈	—	2	1	no	—
213	C₅×C₈	—	4	1	no	—
214	C₅×C₈	—	4	2	no	—
215	C₅×C₈	C₂×M₁₆	1	2	yes	—
216	#168	—	2	1	no	—
217	#168	—	2	1	no	—
218	#168	—	2	1	no	—
219	#168	—	4	1	no	—
220	#168	—	4	2	no	—
221	#168	C₂×M₁₆	1	2	yes	—
222	C₂×#41	—	5	2	no	—
223	#135	—	10	4	no	—
224	C₂×Q₁₆	—	5	1	no	—
225	C₂×SD₁₆	—	5	2	no	—
226	C₂×D₈	C₄×D₅	1	1	yes	characteristic
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	—	10	2	no	—
229	C₂²×C₈	—	5	1	no	—
230	C₂²×C₈	—	5	1	no	—
231	C₂²×C₈	—	10	2	no	—
232	C₄×Q₈	—	5	2	no	—
233	C₄×Q₈	—	5	2	no	—
234	C₄×D₄	—	5	2	no	—
235	C₄×D₄	—	10	2	no	—
236	C₄×D₄	—	10	2	no	—
237	C₄×D₄	C₂²×D₅	1	2	yes	—
238	#117	—	5	2	no	—
239	#117	—	10	2	no	—
240	C₂×C₄²	—	5	1	no	—
241	C₂×C₄²	—	10	2	no	—
242	#107	—	5	1	no	—
243	#107	C₄×D₅	1	1	yes	characteristic
244	#106	—	5	2	no	—
245	C₄⋊C₈	—	5	1	no	—
246	C₄⋊C₈	—	5	1	no	—
247	C₄⋊C₈	—	5	2	no	—
248	C₄⋊C₈	—	10	2	no	—
249	C₄⋊C₈	—	10	1	no	—
250	C₄⋊C₈	—	10	2	no	—
251	C₄⋊C₈	—	10	2	no	—
252	C₄⋊C₈	C₂²×D₅	1	2	yes	—
253	#103	—	5	2	no	—
254	#103	—	5	2	no	—
255	#102	—	5	2	no	—
256	#102	C₄×D₅	1	2	yes	—
257	C₂²⋊C₈	—	2	2	no	—
258	C₂²⋊C₈	—	5	1	no	—
259	C₂²⋊C₈	—	5	1	no	—
260	C₂²⋊C₈	—	10	1	no	—
261	C₂²⋊C₈	—	10	2	no	—
262	#97	—	10	2	no	—
263	#97	—	10	2	no	—
264	C₄×C₈	—	5	1	no	—
265	C₄×C₈	—	5	1	no	—
266	C₄×C₈	—	5	1	no	—
267	C₄×C₈	—	5	1	no	—
268	C₄×C₈	—	5	1	no	—
269	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
272	C₂²×C₅	—	4	2	no	—
273	C₂²×C₅	—	4	2	no	—
274	C₂²×C₅	—	4	2	no	—
275	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
276	C₂×D₅	—	2	1	no	—
277	C₂×D₅	—	2	1	no	—
278	C₂×D₅	#436	1	2	yes	—
279	C₄×C₅	—	2	1	no	—
280	C₄×C₅	—	4	2	no	—
281	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
282	C₄×C₅	C₂×#131	1	1	yes	characteristic
283	C₄×C₅	#576	1	1	yes	characteristic
284	C₄×C₅	#578	1	1	yes	characteristic
285	Dic₅	—	2	1	no	—
286	Dic₅	—	2	1	no	—
287	Dic₅	—	4	2	no	—
288	Dic₅	—	4	2	no	—
289	Dic₅	#436	1	2	yes	—
290	#41	—	10	4	no	—
291	#41	—	10	4	no	—
292	C₂×Q₈	—	5	2	no	—
293	C₂×D₄	—	10	2	no	—
294	C₂×D₄	C₂×C₄×D₅	1	2	yes	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	5	1	no	—
297	C₂²×C₄	—	5	1	no	—
298	C₂²×C₄	—	5	1	no	—
299	C₂²×C₄	—	10	2	no	—
300	C₂²×C₄	—	10	2	no	—
301	Q₁₆	—	10	2	no	—
302	SD₁₆	—	10	4	no	—
303	D₈	—	2	2	no	—
304	M₁₆	—	8	2	no	—
305	M₁₆	—	40	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	5	1	no	—
309	C₂×C₈	—	5	4	no	—
310	C₂×C₈	—	5	1	no	—
311	C₂×C₈	—	5	1	no	—
312	C₂×C₈	—	5	1	no	—
313	C₂×C₈	—	10	2	no	—
314	C₂×C₈	—	10	2	no	—
315	C₂×C₈	—	10	1	no	—
316	C₂×C₈	—	40	2	no	—
317	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
318	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
319	C₂×C₈	D₄×D₅	1	1	yes	characteristic
320	C₂×C₈	D₄×D₅	1	1	yes	characteristic
321	C₄⋊C₄	—	5	2	no	—
322	C₄⋊C₄	—	10	2	no	—
323	C₄⋊C₄	—	10	2	no	—
324	C₄⋊C₄	C₂×C₄×D₅	1	2	yes	—
325	C₂²⋊C₄	—	2	2	no	—
326	C₂²⋊C₄	—	10	2	no	—
327	C₂²⋊C₄	—	10	2	no	—
328	C₄²	—	5	1	no	—
329	C₄²	—	5	1	no	—
330	C₄²	—	5	1	no	—
331	C₄²	—	10	2	no	—
332	C₄²	—	10	2	no	—
333	C₄²	—	10	2	no	—
334	C₄²	C₂³×D₅	1	1	yes	characteristic
335	C₂×C₅	—	8	2	no	—
336	C₂×C₅	C₂×#436	1	1	yes	characteristic
337	C₂×C₅	#2951	1	1	yes	characteristic
338	C₂×C₅	#2957	1	1	yes	characteristic
339	D₅	—	2	2	no	—
340	C₂³	—	2	2	no	—
341	C₂³	—	5	1	no	—
342	Q₈	—	10	2	no	—
343	Q₈	—	10	2	no	—
344	D₄	—	2	2	no	—
345	D₄	—	2	2	no	—
346	D₄	—	20	4	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	5	1	no	—
351	C₂×C₄	—	5	2	no	—
352	C₂×C₄	—	5	2	no	—
353	C₂×C₄	—	5	1	no	—
354	C₂×C₄	—	5	1	no	—
355	C₂×C₄	—	10	1	no	—
356	C₂×C₄	—	10	1	no	—
357	C₂×C₄	—	10	2	no	—
358	C₂×C₄	—	10	2	no	—
359	C₂×C₄	—	20	1	no	—
360	C₂×C₄	—	20	4	no	—
361	C₂×C₄	—	40	2	no	—
362	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
363	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
364	C₂×C₄	D₅×#41	1	1	yes	characteristic
365	C₈	—	2	1	no	—
366	C₈	—	2	1	no	—
367	C₈	—	2	1	no	—
368	C₈	—	4	1	no	—
369	C₈	—	4	2	no	—
370	C₈	—	5	2	no	—
371	C₈	—	10	1	no	—
372	C₈	—	10	1	no	—
373	C₈	—	10	1	no	—
374	C₈	—	20	1	no	—
375	C₈	—	20	2	no	—
376	C₈	D₅×M₁₆	1	2	yes	—
377	C₅	#17898	1	1	yes	characteristic, complemented, Sylow
378	C₂²	—	4	2	no	—
379	C₂²	—	4	2	no	—
380	C₂²	—	4	2	no	—
381	C₂²	—	5	2	no	—
382	C₂²	—	10	1	no	—
383	C₂²	—	10	1	no	—
384	C₂²	C₄×D₄×D₅	1	1	yes	characteristic
385	C₄	—	2	1	no	—
386	C₄	—	4	2	no	—
387	C₄	—	5	2	no	—
388	C₄	—	10	1	no	—
389	C₄	—	10	1	no	—
390	C₄	—	20	2	no	—
391	C₄	—	20	2	no	—
392	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
393	C₄	D₅×#131	1	1	yes	characteristic
394	C₄	#66091	1	1	yes	characteristic
395	C₄	#66105	1	1	yes	characteristic
396	C₂	—	8	2	no	—
397	C₂	—	10	2	no	—
398	C₂	D₅×#436	1	1	yes	characteristic
399	C₂	#10611320	1	1	yes	characteristic
400	C₂	#10611621	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×#1589	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂×#4058	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1589	D₅
hypocenter	C₅	#17898
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1589

Complements

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

N	H ≅ G/N	type
2. #10611254	C₂	semidirect
3. C₅×#1589	C₂	semidirect
4. #10601674	C₂	semidirect
5. #10601653	C₂	semidirect
6. D₅×#335	C₂	semidirect
7. #10599357	C₂	semidirect
8. #10598959	C₂	semidirect
9. #10598955	C₂	semidirect
10. #10598849	C₂	semidirect
11. #10598840	C₂	semidirect
103. #1589	D₅	semidirect
151. C₈×D₅	M₁₆	semidirect
377. C₅	#17898	semidirect

Maximal subgroups

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

subgroup	index
C₄×#4058	2
#10611254	2
C₅×#1589	2
#10601674	2
#10601653	2
D₅×#335	2
#10599357	2
#10598959	2
#10598955	2
#10598849	2
#10598840	2
#17898	5