Subgroups — #12069998

Order 1280.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610784	C₂	1	2	yes	maximal, complemented
2	#10610200	C₂	1	1	yes	maximal, characteristic
3	C₅×#1566	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601292	C₂	1	2	yes	maximal, complemented
5	#10600517	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599318	C₂	1	1	yes	maximal, characteristic, complemented
7	#10599317	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599316	C₂	1	2	yes	maximal, complemented
9	#10599311	C₂	1	1	yes	maximal, characteristic, complemented
10	#10599310	C₂	1	1	yes	maximal, characteristic, complemented
11	#10599309	C₂	1	2	yes	maximal, complemented
12	#65813	C₂²	1	2	yes	—
13	C₅×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	#65417	—	2	2	no	—
16	#65416	—	2	2	no	—
17	#65405	C₄	1	1	yes	characteristic
18	#65404	C₄	1	1	yes	characteristic
19	#65403	C₄	1	2	yes	—
20	#65214	C₂²	1	1	yes	characteristic
21	#65214	C₂²	1	1	yes	characteristic
22	#65129	—	2	2	no	—
23	#65128	—	2	2	no	—
24	#65127	C₂²	1	2	yes	—
25	#65123	C₂²	1	2	yes	—
26	#65120	C₂²	1	2	yes	—
27	#65119	C₂²	1	2	yes	—
28	C₄×Dic₂₀	C₂²	1	1	yes	characteristic
29	#64986	C₄	1	2	yes	—
30	#64985	C₄	1	1	yes	characteristic
31	C₄×D₄₀	C₂²	1	1	yes	characteristic
32	C₄×#677	C₂²	1	2	yes	—
33	#64972	C₄	1	1	yes	characteristic
34	C₅×#334	C₂²	1	2	yes	—
35	C₅×#334	C₂²	1	1	yes	characteristic
36	C₅×#334	C₂²	1	1	yes	characteristic
37	#64704	C₂²	1	2	yes	—
38	#64704	C₂²	1	2	yes	—
39	#64703	C₂²	1	2	yes	—
40	#64703	C₂²	1	2	yes	—
41	#64675	C₂²	1	1	yes	characteristic
42	#64675	C₂²	1	1	yes	characteristic
43	C₈×#168	C₂²	1	1	yes	characteristic
44	C₈×#168	C₂²	1	1	yes	characteristic
45	#17974	—	5	1	no	maximal, Sylow
46	C₂×C₅×M₁₆	—	2	2	no	—
47	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
50	#4076	C₂×C₄	1	2	yes	—
51	#4075	—	2	2	no	—
52	C₄×C₂²⋊D₅	—	2	2	no	—
53	#4070	C₂×C₄	1	2	yes	—
54	C₂×Dic₂₀	C₂×C₄	1	1	yes	characteristic
55	C₂×D₄₀	C₂×C₄	1	1	yes	characteristic
56	C₂×#677	C₂×C₄	1	2	yes	—
57	C₂×#676	—	2	2	no	—
58	C₂×C₈×D₅	—	2	2	no	—
59	#4022	—	2	2	no	—
60	#4021	C₂×C₄	1	2	yes	—
61	C₄×D₂₀	C₂³	1	2	yes	—
62	#4018	—	2	2	no	—
63	#4016	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₀	C₂³	1	2	yes	—
65	C₅×C₄⋊C₈	—	2	2	no	—
66	C₅×C₄⋊C₈	C₂³	1	2	yes	—
67	C₅×C₄⋊C₈	C₂³	1	2	yes	—
68	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
69	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
71	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₂²⋊#168	—	2	1	no	—
73	C₂²⋊#168	D₄	1	2	yes	—
74	#3953	C₂×C₄	1	2	yes	—
75	#3953	C₂×C₄	1	2	yes	—
76	#3952	—	2	2	no	—
77	#3952	—	2	2	no	—
78	#3950	C₂×C₄	1	1	yes	characteristic
79	#3950	C₂×C₄	1	1	yes	characteristic
80	#3949	C₂×C₄	1	2	yes	—
81	#3948	C₂×C₄	1	2	yes	—
82	#3948	C₂×C₄	1	2	yes	—
83	#3947	—	2	2	no	—
84	#3946	—	2	2	no	—
85	#3946	—	2	2	no	—
86	C₈×Dic₅	—	2	2	no	—
87	C₄⋊#168	—	2	1	no	—
88	C₄⋊#168	D₄	1	2	yes	—
89	C₄×#168	C₂³	1	1	yes	characteristic
90	C₄×#168	C₂³	1	1	yes	characteristic
91	C₄×#168	C₂³	1	1	yes	characteristic
92	C₄×#168	C₂³	1	1	yes	characteristic
93	#3168	—	5	1	no	—
94	#2993	—	5	2	no	—
95	#1594	—	5	1	no	—
96	#1593	—	5	1	no	—
97	#1590	—	5	2	no	—
98	#1580	—	5	1	no	—
99	#1579	—	5	1	no	—
100	C₈×SD₁₆	—	5	2	no	—
101	#1566	D₅	1	1	yes	characteristic, complemented
102	#1495	—	5	2	no	—
103	#1450	—	5	1	no	—
104	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅	#41	1	1	yes	characteristic
107	C₂×C₂²⋊D₅	—	2	2	no	—
108	C₂×D₂₀	C₂²×C₄	1	2	yes	—
109	C₂×C₄×D₅	—	2	2	no	—
110	C₂×Dic₁₀	C₂²×C₄	1	2	yes	—
111	C₅×M₁₆	—	8	2	no	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	—	2	2	no	—
116	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₅	#41	1	1	yes	characteristic
123	C₄²×C₅	C₂⁴	1	1	yes	characteristic
124	C₂²⋊Dic₅	—	2	2	no	—
125	#685	—	2	2	no	—
126	#685	—	2	2	no	—
127	#685	—	2	2	no	—
128	#685	—	2	2	no	—
129	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
130	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
131	#683	—	2	2	no	—
132	#683	—	2	2	no	—
133	#683	—	2	2	no	—
134	#683	—	2	2	no	—
135	C₄×Dic₅	—	2	2	no	—
136	C₂×#168	—	2	1	no	—
137	C₂×#168	—	2	1	no	—
138	C₂×#168	C₂×D₄	1	2	yes	—
139	C₂×#168	C₂×D₄	1	2	yes	—
140	Dic₂₀	—	2	2	no	—
141	D₄₀	—	2	2	no	—
142	#677	—	2	4	no	—
143	#676	—	8	2	no	—
144	C₈×D₅	—	8	2	no	—
145	#529	—	5	2	no	—
146	#499	—	5	1	no	—
147	#495	—	5	1	no	—
148	#488	—	5	2	no	—
149	#467	—	5	1	no	—
150	#466	—	5	1	no	—
151	#465	—	5	2	no	—
152	#446	—	5	2	no	—
153	C₈×Q₈	—	5	2	no	—
154	C₄×Q₁₆	—	5	1	no	—
155	C₄×SD₁₆	—	5	2	no	—
156	C₄×D₈	—	5	1	no	—
157	#436	—	5	2	no	—
158	#436	—	10	2	no	—
159	#435	—	10	2	no	—
160	C₈×D₄	—	5	2	no	—
161	#433	—	10	2	no	—
162	#431	—	5	1	no	—
163	#431	—	5	1	no	—
164	#424	—	10	2	no	—
165	#423	C₂×D₅	1	2	yes	—
166	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
167	#334	C₂×D₅	1	2	yes	—
168	#334	C₂×D₅	1	1	yes	characteristic
169	#334	C₂×D₅	1	1	yes	characteristic
170	Q₈⋊C₈	—	5	2	no	—
171	Q₈⋊C₈	—	5	2	no	—
172	D₄⋊C₈	—	5	2	no	—
173	D₄⋊C₈	—	5	2	no	—
174	#322	—	5	1	no	—
175	#322	—	5	1	no	—
176	C₈²	—	5	1	no	—
177	C₈²	—	5	1	no	—
178	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
179	C₂²×D₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	1	no	—
181	C₂×C₄×C₅	—	2	1	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	—	4	1	no	—
184	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
186	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
190	C₂²⋊D₅	—	8	2	no	—
191	C₂×Dic₅	—	2	2	no	—
192	C₂×Dic₅	—	2	2	no	—
193	C₂×Dic₅	—	2	2	no	—
194	D₂₀	—	2	2	no	—
195	D₂₀	—	2	2	no	—
196	C₄×D₅	—	4	2	no	—
197	C₄×D₅	—	4	2	no	—
198	Dic₁₀	—	2	2	no	—
199	Dic₁₀	—	2	2	no	—
200	C₅×C₈	—	2	1	no	—
201	C₅×C₈	—	2	1	no	—
202	C₅×C₈	—	4	2	no	—
203	C₅×C₈	—	4	2	no	—
204	C₅×C₈	#131	1	2	yes	—
205	C₅×C₈	#131	1	2	yes	—
206	#168	—	2	2	no	—
207	#168	—	2	2	no	—
208	#168	—	4	1	no	—
209	#168	—	4	1	no	—
210	C₂×Q₁₆	—	5	1	no	—
211	C₂×SD₁₆	—	5	2	no	—
212	C₂×D₈	—	5	1	no	—
213	C₂×M₁₆	—	2	2	no	—
214	C₂×M₁₆	—	10	2	no	—
215	C₂²×C₈	—	10	2	no	—
216	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
217	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
218	#126	—	10	2	no	—
219	#125	—	5	2	no	—
220	#124	—	5	2	no	—
221	#123	—	10	2	no	—
222	C₂²⋊Q₈	—	5	2	no	—
223	#121	—	5	2	no	—
224	C₄×Q₈	—	5	2	no	—
225	C₄×D₄	—	5	2	no	—
226	C₄×D₄	—	10	2	no	—
227	#117	—	10	2	no	—
228	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
229	#107	—	5	1	no	—
230	#107	—	5	1	no	—
231	#106	—	5	2	no	—
232	C₄⋊C₈	—	2	2	no	—
233	C₄⋊C₈	—	5	2	no	—
234	C₄⋊C₈	—	10	2	no	—
235	C₄⋊C₈	—	10	2	no	—
236	C₄⋊C₈	—	10	1	no	—
237	C₄⋊C₈	C₂²×D₅	1	2	yes	—
238	C₄⋊C₈	C₂²×D₅	1	2	yes	—
239	#103	—	5	2	no	—
240	#103	—	5	2	no	—
241	#102	—	5	2	no	—
242	#102	—	5	2	no	—
243	C₂²⋊C₈	—	5	2	no	—
244	C₂²⋊C₈	—	10	2	no	—
245	C₂²⋊C₈	—	10	2	no	—
246	C₂²⋊C₈	—	10	1	no	—
247	#97	—	10	2	no	—
248	C₄×C₈	—	5	1	no	—
249	C₄×C₈	—	5	1	no	—
250	C₄×C₈	—	5	1	no	—
251	C₄×C₈	—	5	1	no	—
252	C₄×C₈	—	10	2	no	—
253	C₄×C₈	C₄×D₅	1	1	yes	characteristic
254	C₄×C₈	C₄×D₅	1	1	yes	characteristic
255	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
256	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
257	C₂²×C₅	—	2	1	no	—
258	C₂²×C₅	—	2	1	no	—
259	C₂²×C₅	—	2	1	no	—
260	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
261	C₂×D₅	—	4	2	no	—
262	C₂×D₅	—	4	2	no	—
263	C₂×D₅	—	4	2	no	—
264	C₄×C₅	—	2	1	no	—
265	C₄×C₅	—	2	1	no	—
266	C₄×C₅	—	2	1	no	—
267	C₄×C₅	—	2	1	no	—
268	C₄×C₅	C₂×#131	1	1	yes	characteristic
269	C₄×C₅	C₂×#131	1	1	yes	characteristic
270	C₄×C₅	#577	1	2	yes	—
271	Dic₅	—	4	2	no	—
272	Dic₅	—	4	2	no	—
273	Dic₅	—	4	2	no	—
274	C₂×Q₈	—	5	2	no	—
275	C₂×D₄	—	5	2	no	—
276	C₂×D₄	—	10	2	no	—
277	C₂²×C₄	—	10	2	no	—
278	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
279	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
280	C₂²×C₄	#713	1	1	yes	characteristic
281	Q₁₆	—	10	2	no	—
282	SD₁₆	—	10	4	no	—
283	D₈	—	10	2	no	—
284	M₁₆	—	8	2	no	—
285	M₁₆	—	40	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	5	2	no	—
291	C₂×C₈	—	5	2	no	—
292	C₂×C₈	—	10	1	no	—
293	C₂×C₈	—	10	1	no	—
294	C₂×C₈	—	40	2	no	—
295	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
299	C₄⋊C₄	—	5	2	no	—
300	C₄⋊C₄	—	5	2	no	—
301	C₄⋊C₄	—	10	2	no	—
302	C₄⋊C₄	—	10	2	no	—
303	C₄⋊C₄	—	10	2	no	—
304	C₄⋊C₄	—	10	2	no	—
305	C₂²⋊C₄	—	10	2	no	—
306	C₂²⋊C₄	—	10	2	no	—
307	C₂²⋊C₄	—	10	2	no	—
308	C₂²⋊C₄	—	10	2	no	—
309	C₂²⋊C₄	—	10	2	no	—
310	C₄²	—	10	2	no	—
311	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
312	C₄²	D₄×D₅	1	1	yes	characteristic
313	C₄²	#713	1	1	yes	characteristic
314	C₄²	C₂³×D₅	1	1	yes	characteristic
315	C₂×C₅	—	4	1	no	—
316	C₂×C₅	#2931	1	1	yes	characteristic
317	C₂×C₅	#2952	1	1	yes	characteristic
318	C₂×C₅	#2957	1	1	yes	characteristic
319	D₅	—	8	2	no	—
320	C₂³	—	10	2	no	—
321	C₂³	#4039	1	1	yes	characteristic
322	Q₈	—	10	2	no	—
323	Q₈	—	10	2	no	—
324	D₄	—	10	2	no	—
325	D₄	—	10	2	no	—
326	D₄	—	40	2	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	10	2	no	—
332	C₂×C₄	—	10	2	no	—
333	C₂×C₄	—	10	2	no	—
334	C₂×C₄	—	20	2	no	—
335	C₂×C₄	—	20	2	no	—
336	C₂×C₄	#4039	1	1	yes	characteristic
337	C₂×C₄	#4039	1	1	yes	characteristic
338	C₂×C₄	#4039	1	1	yes	characteristic
339	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
341	C₂×C₄	C₂×#713	1	1	yes	characteristic
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	4	2	no	—
345	C₈	—	4	2	no	—
346	C₈	—	10	2	no	—
347	C₈	—	10	2	no	—
348	C₈	—	20	1	no	—
349	C₈	—	20	1	no	—
350	C₈	#4053	1	2	yes	—
351	C₈	#4053	1	2	yes	—
352	C₅	#17974	1	1	yes	characteristic, complemented, Sylow
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	20	2	no	—
357	C₂²	—	20	2	no	—
358	C₂²	—	20	2	no	—
359	C₂²	C₂×#4039	1	1	yes	characteristic
360	C₄	—	2	1	no	—
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	—	20	2	no	—
367	C₄	C₂×#4053	1	1	yes	characteristic
368	C₄	C₂×#4053	1	1	yes	characteristic
369	C₄	#66096	1	2	yes	—
370	C₂	—	4	1	no	—
371	C₂	—	40	2	no	—
372	C₂	#10610772	1	1	yes	characteristic
373	C₂	#10610837	1	1	yes	characteristic
374	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×#1566	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	#65404	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1566	D₅
hypocenter	C₅	#17974
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
1. #10610784	C₂	semidirect
3. C₅×#1566	C₂	semidirect
4. #10601292	C₂	semidirect
5. #10600517	C₂	semidirect
6. #10599318	C₂	semidirect
7. #10599317	C₂	semidirect
8. #10599316	C₂	semidirect
9. #10599311	C₂	semidirect
10. #10599310	C₂	semidirect
11. #10599309	C₂	semidirect
101. #1566	D₅	semidirect
352. C₅	#17974	semidirect

Maximal subgroups

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

subgroup	index
#10610784	2
#10610200	2
C₅×#1566	2
#10601292	2
#10600517	2
#10599318	2
#10599317	2
#10599316	2
#10599311	2
#10599310	2
#10599309	2
#17974	5