Subgroups — #12070522

Order 1280.

598 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10615083	C₂	1	1	yes	maximal, characteristic
2	#10615063	C₂	1	1	yes	maximal, characteristic, complemented
3	#10610783	C₂	1	1	yes	maximal, characteristic
4	C₅×#1493	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601288	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600527	C₂	1	1	yes	maximal, characteristic
7	#10600517	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599017	C₂	1	2	yes	maximal, complemented
9	#10599016	C₂	1	2	yes	maximal, complemented
10	#10599015	C₂	1	2	yes	maximal, complemented
11	#10599006	C₂	1	2	yes	maximal, complemented
12	C₅×#533	C₂²	1	1	yes	characteristic
13	C₄×#709	C₂²	1	1	yes	characteristic
14	C₅×#423	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	#65417	—	2	2	no	—
17	#65383	C₄	1	1	yes	characteristic
18	#65379	C₄	1	1	yes	characteristic
19	#65376	C₄	1	1	yes	characteristic
20	#65372	C₄	1	1	yes	characteristic
21	#65336	C₄	1	2	yes	—
22	#65334	C₄	1	2	yes	—
23	C₄×#689	C₂²	1	2	yes	—
24	C₄×#688	C₂²	1	2	yes	—
25	#65313	—	2	1	no	—
26	#65313	C₂²	1	2	yes	—
27	Q₈×#168	—	2	1	no	—
28	Q₈×#168	C₂²	1	2	yes	—
29	#65289	—	2	2	no	—
30	#65214	C₂²	1	1	yes	characteristic
31	#65214	C₂²	1	1	yes	characteristic
32	#65214	C₂²	1	1	yes	characteristic
33	#65127	C₂²	1	2	yes	—
34	#65126	—	2	1	no	—
35	#65122	—	2	1	no	—
36	#65119	C₂²	1	2	yes	—
37	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
39	#64700	C₂²	1	2	yes	—
40	#64699	C₂²	1	2	yes	—
41	#64679	C₂²	1	1	yes	characteristic
42	#64679	C₂²	1	1	yes	characteristic
43	#64676	C₂²	1	1	yes	characteristic
44	#64676	C₂²	1	1	yes	characteristic
45	#64675	C₂²	1	2	yes	—
46	C₈×#168	C₂²	1	2	yes	—
47	#18019	—	5	1	no	maximal, Sylow
48	C₂×#709	C₂×C₄	1	1	yes	characteristic
49	C₂×C₅×M₁₆	—	2	1	no	—
50	C₂²×C₅×C₈	—	2	1	no	—
51	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
52	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
53	C₅×#125	—	2	1	no	—
54	C₅×C₂²⋊Q₈	—	2	1	no	—
55	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
56	C₄×C₅×Q₈	—	2	1	no	—
57	C₄×C₅×Q₈	C₂³	1	2	yes	—
58	C₅×#117	—	2	1	no	—
59	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
60	C₂×#689	C₂×C₄	1	2	yes	—
61	C₂×#688	C₂×C₄	1	2	yes	—
62	C₄×C₂²⋊D₅	—	2	1	no	—
63	C₄×C₂²⋊D₅	—	2	1	no	—
64	#4072	C₂×C₄	1	1	yes	characteristic
65	C₂×#676	—	2	2	no	—
66	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
67	C₄×D₂₀	C₂³	1	1	yes	characteristic
68	#4018	—	2	1	no	—
69	C₄²×D₅	—	2	1	no	—
70	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₁₀	C₂³	1	1	yes	characteristic
72	C₅×C₄⋊C₈	—	2	1	no	—
73	C₅×C₄⋊C₈	C₂³	1	2	yes	—
74	C₄×C₅×C₈	—	2	1	no	—
75	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
76	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
77	#3967	C₂×C₄	1	2	yes	—
78	#3967	C₂×C₄	1	2	yes	—
79	C₂²⋊#168	—	2	1	no	—
80	C₂²⋊#168	C₂×C₄	1	2	yes	—
81	C₂²⋊#168	D₄	1	2	yes	—
82	#3952	—	2	2	no	—
83	#3946	—	2	2	no	—
84	C₈×Dic₅	—	2	2	no	—
85	#3942	C₂×C₄	1	2	yes	—
86	#3941	C₂×C₄	1	2	yes	—
87	#3940	C₂×C₄	1	2	yes	—
88	#3939	C₂×C₄	1	2	yes	—
89	C₄⋊#168	—	2	2	no	—
90	C₄⋊#168	—	2	2	no	—
91	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
92	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
93	C₄⋊#168	D₄	1	1	yes	characteristic
94	C₄⋊#168	D₄	1	1	yes	characteristic
95	C₄⋊#168	C₂³	1	1	yes	characteristic
96	C₄⋊#168	C₂³	1	1	yes	characteristic
97	#3935	—	2	2	no	—
98	C₄×#168	—	2	1	no	—
99	C₄×#168	—	2	2	no	—
100	C₄×#168	C₂³	1	2	yes	—
101	C₄×#168	C₂³	1	2	yes	—
102	#3107	—	5	1	no	—
103	#2991	—	5	1	no	—
104	#2986	—	5	1	no	—
105	#1590	—	5	2	no	—
106	#1588	—	5	2	no	—
107	#1587	—	5	2	no	—
108	C₈×Q₁₆	—	5	2	no	—
109	#1574	—	5	1	no	—
110	#1493	D₅	1	1	yes	characteristic, complemented
111	#1476	—	5	1	no	—
112	#1450	—	5	1	no	—
113	C₂×C₅×Q₈	—	2	1	no	—
114	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₅	#41	1	1	yes	characteristic
118	C₂×C₂²⋊D₅	—	2	1	no	—
119	#709	—	2	2	no	—
120	#709	—	4	1	no	—
121	C₂×D₂₀	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄×D₅	—	2	1	no	—
123	C₂×C₄×D₅	—	2	1	no	—
124	C₂×C₄×D₅	—	2	1	no	—
125	C₂×Dic₁₀	C₂²×C₄	1	1	yes	characteristic
126	C₅×M₁₆	—	8	1	no	—
127	C₂×C₅×C₈	—	2	4	no	—
128	C₂×C₅×C₈	—	2	1	no	—
129	C₂×C₅×C₈	—	2	1	no	—
130	C₂×C₅×C₈	—	2	2	no	—
131	C₅×C₄⋊C₄	—	2	1	no	—
132	C₅×C₄⋊C₄	—	2	1	no	—
133	C₅×C₄⋊C₄	—	2	1	no	—
134	C₅×C₄⋊C₄	—	2	2	no	—
135	C₅×C₄⋊C₄	—	2	2	no	—
136	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
137	C₅×C₂²⋊C₄	—	2	2	no	—
138	C₄²×C₅	—	2	2	no	—
139	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
140	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₅	#41	1	1	yes	characteristic
142	C₄²×C₅	C₂⁴	1	1	yes	characteristic
143	C₂²⋊Dic₅	—	2	1	no	—
144	#689	—	2	4	no	—
145	#688	—	4	2	no	—
146	#685	—	2	1	no	—
147	#685	—	2	1	no	—
148	C₄⋊Dic₅	C₂²×C₄	1	1	yes	characteristic
149	C₄⋊Dic₅	C₂²×C₄	1	1	yes	characteristic
150	#683	—	2	1	no	—
151	#683	—	2	1	no	—
152	C₄×Dic₅	—	2	1	no	—
153	C₄×Dic₅	—	2	1	no	—
154	C₄×Dic₅	—	2	1	no	—
155	C₂×#168	—	2	1	no	—
156	C₂×#168	—	2	1	no	—
157	C₂×#168	C₂²×C₄	1	2	yes	—
158	C₂×#168	C₂²×C₄	1	2	yes	—
159	C₂×#168	C₂×D₄	1	2	yes	—
160	C₂×#168	C₂×D₄	1	2	yes	—
161	#676	—	8	2	no	—
162	#533	C₂×D₅	1	1	yes	characteristic
163	C₄×#41	—	5	1	no	—
164	#485	—	5	2	no	—
165	#475	—	5	1	no	—
166	#474	—	5	1	no	—
167	#462	—	5	1	no	—
168	#461	—	5	1	no	—
169	#452	—	5	2	no	—
170	#446	—	5	2	no	—
171	#446	—	10	1	no	—
172	C₈×Q₈	—	5	2	no	—
173	C₈×Q₈	—	5	2	no	—
174	C₈×Q₈	—	10	1	no	—
175	C₄×Q₁₆	—	5	2	no	—
176	C₄×SD₁₆	—	5	2	no	—
177	#436	—	5	2	no	—
178	#436	—	10	2	no	—
179	#433	—	10	2	no	—
180	#431	—	5	1	no	—
181	#431	—	5	1	no	—
182	#431	—	5	1	no	—
183	#431	—	10	1	no	—
184	#423	—	10	1	no	—
185	#423	C₂×D₅	1	1	yes	characteristic
186	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
187	#335	—	5	2	no	—
188	#334	—	5	2	no	—
189	Q₈⋊C₈	—	5	1	no	—
190	Q₈⋊C₈	—	5	1	no	—
191	Q₈⋊C₈	C₂×D₅	1	2	yes	—
192	Q₈⋊C₈	C₂×D₅	1	2	yes	—
193	D₄⋊C₈	—	5	1	no	—
194	D₄⋊C₈	—	5	1	no	—
195	#322	—	5	2	no	—
196	C₈²	—	5	2	no	—
197	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
198	C₂²×D₅	—	2	1	no	—
199	C₅×Q₈	—	2	2	no	—
200	C₅×Q₈	—	2	2	no	—
201	C₅×Q₈	—	4	2	no	—
202	C₂×C₄×C₅	—	2	1	no	—
203	C₂×C₄×C₅	—	2	1	no	—
204	C₂×C₄×C₅	—	2	1	no	—
205	C₂×C₄×C₅	—	2	1	no	—
206	C₂×C₄×C₅	—	2	2	no	—
207	C₂×C₄×C₅	—	2	2	no	—
208	C₂×C₄×C₅	—	2	2	no	—
209	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
210	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
211	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
212	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
215	C₂²⋊D₅	—	4	2	no	—
216	C₂×Dic₅	—	2	1	no	—
217	C₂×Dic₅	—	2	1	no	—
218	C₂×Dic₅	—	2	1	no	—
219	D₂₀	—	2	1	no	—
220	D₂₀	—	2	1	no	—
221	C₄×D₅	—	4	2	no	—
222	C₄×D₅	—	4	1	no	—
223	C₄×D₅	—	4	1	no	—
224	C₄×D₅	—	4	2	no	—
225	Dic₁₀	—	2	1	no	—
226	Dic₁₀	#131	1	2	yes	—
227	C₅×C₈	—	2	2	no	—
228	C₅×C₈	—	2	2	no	—
229	C₅×C₈	—	4	2	no	—
230	#168	—	2	2	no	—
231	#168	—	2	2	no	—
232	#168	—	2	2	no	—
233	#168	—	2	2	no	—
234	#168	—	4	1	no	—
235	#168	—	4	1	no	—
236	C₂×#41	—	5	1	no	—
237	C₂×Q₁₆	—	5	2	no	—
238	C₂×SD₁₆	—	5	2	no	—
239	C₂×M₁₆	—	2	1	no	—
240	C₂×M₁₆	—	10	2	no	—
241	C₂²×C₈	—	2	1	no	—
242	C₄⋊Q₈	C₄×D₅	1	1	yes	characteristic
243	C₄⋊Q₈	C₄×D₅	1	1	yes	characteristic
244	#125	—	2	1	no	—
245	C₂²⋊Q₈	—	2	1	no	—
246	C₂²⋊Q₈	C₄×D₅	1	2	yes	—
247	C₄×Q₈	—	2	1	no	—
248	C₄×Q₈	—	5	1	no	—
249	C₄×Q₈	—	5	1	no	—
250	C₄×Q₈	C₂²×D₅	1	2	yes	—
251	C₄×D₄	—	5	1	no	—
252	C₄×D₄	—	5	1	no	—
253	C₄×D₄	—	10	1	no	—
254	C₄×D₄	—	10	1	no	—
255	#117	—	2	1	no	—
256	#117	—	5	1	no	—
257	#117	—	10	1	no	—
258	C₂×C₄²	—	10	1	no	—
259	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
260	#107	—	5	2	no	—
261	#106	—	5	2	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	5	1	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	—	10	2	no	—
270	C₄⋊C₈	—	10	2	no	—
271	C₄⋊C₈	—	10	2	no	—
272	C₄⋊C₈	C₂²×D₅	1	2	yes	—
273	#103	—	5	2	no	—
274	#103	—	5	2	no	—
275	#103	—	5	2	no	—
276	#102	—	5	2	no	—
277	C₂²⋊C₈	—	5	2	no	—
278	C₂²⋊C₈	—	5	2	no	—
279	C₂²⋊C₈	—	10	2	no	—
280	C₂²⋊C₈	—	10	1	no	—
281	#97	—	10	2	no	—
282	C₄×C₈	—	2	1	no	—
283	C₄×C₈	—	5	2	no	—
284	C₄×C₈	—	5	2	no	—
285	C₄×C₈	—	10	2	no	—
286	C₄×C₈	—	10	2	no	—
287	C₄×C₈	—	10	1	no	—
288	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
290	C₂²×C₅	—	2	1	no	—
291	C₂²×C₅	—	2	1	no	—
292	C₂²×C₅	—	2	1	no	—
293	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
294	C₂×D₅	—	4	1	no	—
295	C₂×D₅	—	4	1	no	—
296	C₂×D₅	—	4	1	no	—
297	C₄×C₅	—	2	1	no	—
298	C₄×C₅	—	2	1	no	—
299	C₄×C₅	—	2	1	no	—
300	C₄×C₅	—	4	2	no	—
301	C₄×C₅	—	4	2	no	—
302	C₄×C₅	#443	1	2	yes	—
303	C₄×C₅	C₂×#131	1	1	yes	characteristic
304	C₄×C₅	#568	1	1	yes	characteristic
305	C₄×C₅	#577	1	1	yes	characteristic
306	C₄×C₅	#578	1	1	yes	characteristic
307	Dic₅	—	2	2	no	—
308	Dic₅	—	4	1	no	—
309	Dic₅	—	4	1	no	—
310	#41	—	10	2	no	—
311	#41	—	20	1	no	—
312	C₂×Q₈	—	2	1	no	—
313	C₂×Q₈	—	5	1	no	—
314	C₂×Q₈	C₂×C₄×D₅	1	2	yes	—
315	C₂×D₄	—	5	1	no	—
316	C₂×D₄	—	10	1	no	—
317	C₂²×C₄	—	10	1	no	—
318	C₂²×C₄	—	10	1	no	—
319	C₂²×C₄	—	10	1	no	—
320	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
321	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
322	C₂²×C₄	#711	1	1	yes	characteristic
323	Q₁₆	—	10	4	no	—
324	SD₁₆	—	20	2	no	—
325	M₁₆	—	8	1	no	—
326	M₁₆	—	40	2	no	—
327	C₂×C₈	—	2	4	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	5	2	no	—
332	C₂×C₈	—	5	2	no	—
333	C₂×C₈	—	5	2	no	—
334	C₂×C₈	—	5	2	no	—
335	C₂×C₈	—	10	1	no	—
336	C₂×C₈	—	10	1	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	2	1	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	5	1	no	—
343	C₄⋊C₄	—	5	1	no	—
344	C₄⋊C₄	—	10	1	no	—
345	C₄⋊C₄	—	10	1	no	—
346	C₄⋊C₄	C₂×C₄×D₅	1	2	yes	—
347	C₂²⋊C₄	—	2	2	no	—
348	C₂²⋊C₄	—	10	1	no	—
349	C₂²⋊C₄	—	10	1	no	—
350	C₂²⋊C₄	—	10	1	no	—
351	C₄²	—	2	2	no	—
352	C₄²	—	10	1	no	—
353	C₄²	—	10	1	no	—
354	C₄²	—	10	1	no	—
355	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
356	C₄²	D₄×D₅	1	1	yes	characteristic
357	C₄²	#711	1	1	yes	characteristic
358	C₄²	C₂³×D₅	1	1	yes	characteristic
359	C₂×C₅	—	4	1	no	—
360	C₂×C₅	#2933	1	1	yes	characteristic
361	C₂×C₅	#2947	1	1	yes	characteristic
362	C₂×C₅	C₂×#443	1	1	yes	characteristic
363	D₅	—	8	1	no	—
364	C₂³	—	10	1	no	—
365	C₂³	#4027	1	1	yes	characteristic
366	Q₈	—	2	2	no	—
367	Q₈	—	2	2	no	—
368	Q₈	—	4	2	no	—
369	Q₈	—	5	2	no	—
370	Q₈	—	10	1	no	—
371	D₄	—	10	1	no	—
372	D₄	—	10	1	no	—
373	D₄	—	20	2	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	10	1	no	—
382	C₂×C₄	—	10	1	no	—
383	C₂×C₄	—	10	1	no	—
384	C₂×C₄	—	20	2	no	—
385	C₂×C₄	—	20	1	no	—
386	C₂×C₄	—	20	1	no	—
387	C₂×C₄	—	20	2	no	—
388	C₂×C₄	#4027	1	1	yes	characteristic
389	C₂×C₄	#4027	1	1	yes	characteristic
390	C₂×C₄	#4027	1	1	yes	characteristic
391	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
392	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
393	C₂×C₄	C₂×#711	1	1	yes	characteristic
394	C₈	—	2	2	no	—
395	C₈	—	2	2	no	—
396	C₈	—	4	2	no	—
397	C₈	—	10	2	no	—
398	C₈	—	10	2	no	—
399	C₈	—	10	2	no	—
400	C₈	—	10	2	no	—
401	C₈	—	20	1	no	—
402	C₈	—	20	1	no	—
403	C₅	#18019	1	1	yes	characteristic, complemented, Sylow
404	C₂²	—	2	1	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	2	1	no	—
407	C₂²	—	20	1	no	—
408	C₂²	—	20	1	no	—
409	C₂²	—	20	1	no	—
410	C₂²	C₂×#4027	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	4	2	no	—
415	C₄	—	4	2	no	—
416	C₄	—	10	2	no	—
417	C₄	—	20	1	no	—
418	C₄	—	20	1	no	—
419	C₄	#65113	1	2	yes	—
420	C₄	D₅×#131	1	1	yes	characteristic
421	C₄	#66084	1	1	yes	characteristic
422	C₄	#66096	1	1	yes	characteristic
423	C₄	#66100	1	1	yes	characteristic
424	C₂	—	4	1	no	—
425	C₂	—	40	1	no	—
426	C₂	#10610390	1	1	yes	characteristic
427	C₂	#10610505	1	1	yes	characteristic
428	C₂	C₂×#65113	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×#1493	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂×#709	C₂×C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1493	D₅
hypocenter	C₅	#18019
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1493

Complements

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

N	H ≅ G/N	type
2. #10615063	C₂	semidirect
4. C₅×#1493	C₂	semidirect
5. #10601288	C₂	semidirect
7. #10600517	C₂	semidirect
8. #10599017	C₂	semidirect
9. #10599016	C₂	semidirect
10. #10599015	C₂	semidirect
11. #10599006	C₂	semidirect
110. #1493	D₅	semidirect
403. C₅	#18019	semidirect

Maximal subgroups

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

subgroup	index
#10615083	2
#10615063	2
#10610783	2
C₅×#1493	2
#10601288	2
#10600527	2
#10600517	2
#10599017	2
#10599016	2
#10599015	2
#10599006	2
#18019	5