Subgroups — #12068601

Order 1280.

616 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611024	C₂	1	2	yes	maximal
2	C₈×#709	C₂	1	1	yes	maximal, characteristic
3	C₅×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601541	C₂	1	2	yes	maximal, complemented
5	#10600524	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599363	C₂	1	2	yes	maximal, complemented
7	D₅×#335	C₂	1	2	yes	maximal, complemented
8	#10599327	C₂	1	4	yes	maximal, complemented
9	C₄×#709	C₂²	1	1	yes	characteristic
10	C₅×#431	C₂²	1	2	yes	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₈×C₂²⋊D₅	C₄	1	2	yes	—
13	C₈×C₂²⋊D₅	C₄	1	2	yes	—
14	#65214	C₂²	1	1	yes	characteristic
15	#65213	C₂²	1	2	yes	—
16	#65124	C₂²	1	4	yes	—
17	D₅×C₄⋊C₈	C₂²	1	4	yes	—
18	#65017	—	2	4	no	—
19	C₈×D₂₀	—	2	2	no	—
20	C₈×D₂₀	C₄	1	1	yes	characteristic
21	C₈×D₂₀	C₄	1	1	yes	characteristic
22	#64974	C₂²	1	2	yes	—
23	C₄×C₈×D₅	C₂²	1	2	yes	—
24	#64968	—	2	2	no	—
25	C₈×Dic₁₀	C₄	1	1	yes	characteristic
26	C₈×Dic₁₀	C₄	1	1	yes	characteristic
27	C₅×#335	C₂²	1	1	yes	characteristic
28	C₅×#335	C₂²	1	1	yes	characteristic
29	C₅×#334	C₂²	1	2	yes	—
30	#64700	C₂²	1	4	yes	—
31	#64699	C₂²	1	4	yes	—
32	#64678	C₂²	1	2	yes	—
33	#64678	C₂²	1	2	yes	—
34	#17836	—	5	1	no	maximal, Sylow
35	C₂×#709	Q₈	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₄×C₂²⋊D₅	C₂×C₄	1	2	yes	—
40	C₄×C₂²⋊D₅	D₄	1	2	yes	—
41	#4072	Q₈	1	1	yes	characteristic
42	C₂×#681	—	2	2	no	—
43	C₂×C₈×D₅	—	2	4	no	—
44	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
46	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
47	C₄×D₂₀	D₄	1	1	yes	characteristic
48	#4018	Q₈	1	2	yes	—
49	C₄²×D₅	C₂³	1	2	yes	—
50	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₁₀	D₄	1	1	yes	characteristic
52	C₅×C₄⋊C₈	C₂³	1	2	yes	—
53	C₅×C₄⋊C₈	C₂³	1	2	yes	—
54	C₅×C₂²⋊C₈	—	2	2	no	—
55	C₄×C₅×C₈	—	2	2	no	—
56	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
57	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
58	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
59	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
60	C₂²⋊#168	C₂×C₄	1	2	yes	—
61	#3952	—	2	4	no	—
62	#3952	C₂×C₄	1	2	yes	—
63	#3952	C₂×C₄	1	2	yes	—
64	#3947	—	2	4	no	—
65	#3946	—	2	4	no	—
66	#3946	C₂×C₄	1	2	yes	—
67	#3946	C₂×C₄	1	2	yes	—
68	C₈×Dic₅	C₂×C₄	1	2	yes	—
69	C₈×Dic₅	C₂×C₄	1	2	yes	—
70	C₄⋊#168	—	2	2	no	—
71	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
72	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
73	C₄⋊#168	C₂³	1	4	yes	—
74	C₄×#168	C₂³	1	2	yes	—
75	#2990	—	5	2	no	—
76	C₈×#41	—	5	1	no	—
77	#1574	—	5	2	no	—
78	#1573	—	5	4	no	—
79	#1568	—	5	2	no	—
80	#1567	—	5	1	no	—
81	#1564	D₅	1	1	yes	characteristic, complemented
82	C₂×#335	—	5	2	no	—
83	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₂²⋊D₅	C₄⋊C₄	1	2	yes	—
87	#709	—	2	4	no	—
88	C₂×D₂₀	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
90	C₂×C₄×D₅	C₂×D₄	1	2	yes	—
91	C₂×C₄×D₅	C₂×Q₈	1	2	yes	—
92	C₂×Dic₁₀	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	4	1	no	—
97	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
104	C₄²×C₅	C₂⁴	1	1	yes	characteristic
105	C₂²⋊Dic₅	C₄⋊C₄	1	2	yes	—
106	#685	C₄⋊C₄	1	2	yes	—
107	#685	C₄⋊C₄	1	2	yes	—
108	C₄⋊Dic₅	C₄⋊C₄	1	1	yes	characteristic
109	C₄⋊Dic₅	C₄⋊C₄	1	1	yes	characteristic
110	#683	C₄⋊C₄	1	2	yes	—
111	#683	C₄⋊C₄	1	2	yes	—
112	C₄×Dic₅	C₂²×C₄	1	2	yes	—
113	C₄×Dic₅	C₂×D₄	1	2	yes	—
114	C₄×Dic₅	C₂×Q₈	1	2	yes	—
115	#681	—	8	2	no	—
116	C₂×#168	—	2	4	no	—
117	C₂×#168	C₂²×C₄	1	2	yes	—
118	C₂×#168	C₂²×C₄	1	2	yes	—
119	C₈×D₅	—	2	4	no	—
120	C₈×D₅	—	4	2	no	—
121	C₈×D₅	—	8	4	no	—
122	C₄×#41	—	5	1	no	—
123	#446	—	10	2	no	—
124	C₈×Q₈	—	5	1	no	—
125	C₈×Q₈	—	5	1	no	—
126	#435	—	10	4	no	—
127	C₈×D₄	—	5	2	no	—
128	C₈×D₄	—	5	2	no	—
129	C₈×D₄	—	5	1	no	—
130	C₈×D₄	—	5	1	no	—
131	C₈×D₄	—	10	2	no	—
132	#432	—	5	4	no	—
133	#431	—	5	1	no	—
134	#431	—	5	2	no	—
135	#431	C₂×D₅	1	2	yes	—
136	#423	—	5	2	no	—
137	C₂×C₄⋊C₈	—	5	4	no	—
138	C₂×C₄×C₈	—	5	2	no	—
139	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
140	#335	—	5	4	no	—
141	#335	—	5	2	no	—
142	#335	—	5	2	no	—
143	#335	C₂×D₅	1	1	yes	characteristic
144	#335	C₂×D₅	1	1	yes	characteristic
145	#334	—	5	4	no	—
146	#334	C₂×D₅	1	2	yes	—
147	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×D₅	C₂×C₄⋊C₄	1	2	yes	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
160	C₂²⋊D₅	—	2	4	no	—
161	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
162	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
163	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
164	D₂₀	#108	1	4	yes	—
165	C₄×D₅	—	2	4	no	—
166	C₄×D₅	—	2	2	no	—
167	C₄×D₅	—	2	2	no	—
168	C₄×D₅	—	2	2	no	—
169	C₄×D₅	—	2	2	no	—
170	Dic₁₀	—	2	2	no	—
171	C₅×C₈	—	2	1	no	—
172	C₅×C₈	—	2	1	no	—
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	—	4	2	no	—
175	C₅×C₈	—	4	2	no	—
176	C₅×C₈	#131	1	2	yes	—
177	#168	—	2	2	no	—
178	#168	—	2	2	no	—
179	#168	—	4	4	no	—
180	C₂×#41	—	5	1	no	—
181	C₂×M₁₆	—	10	2	no	—
182	C₂²×C₈	—	5	2	no	—
183	C₂²×C₈	—	5	2	no	—
184	C₂²×C₈	—	10	4	no	—
185	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
187	C₄×Q₈	—	5	1	no	—
188	C₄×Q₈	—	5	1	no	—
189	C₄×D₄	—	5	2	no	—
190	C₄×D₄	—	5	2	no	—
191	C₄×D₄	—	5	1	no	—
192	C₄×D₄	—	5	1	no	—
193	#117	—	5	1	no	—
194	#117	—	5	2	no	—
195	C₂×C₄²	—	5	2	no	—
196	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
197	C₄⋊C₈	—	5	2	no	—
198	C₄⋊C₈	—	5	2	no	—
199	C₄⋊C₈	—	5	1	no	—
200	C₄⋊C₈	—	5	1	no	—
201	C₄⋊C₈	—	5	4	no	—
202	C₄⋊C₈	—	10	4	no	—
203	C₄⋊C₈	—	10	2	no	—
204	C₄⋊C₈	C₂²×D₅	1	2	yes	—
205	C₄⋊C₈	C₂²×D₅	1	2	yes	—
206	C₂²⋊C₈	—	2	2	no	—
207	C₂²⋊C₈	—	5	2	no	—
208	C₂²⋊C₈	—	5	2	no	—
209	C₂²⋊C₈	—	5	2	no	—
210	C₂²⋊C₈	—	10	4	no	—
211	#97	—	10	4	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	5	2	no	—
214	C₄×C₈	—	5	2	no	—
215	C₄×C₈	—	5	2	no	—
216	C₄×C₈	C₄×D₅	1	1	yes	characteristic
217	C₄×C₈	C₄×D₅	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
220	C₂²×C₅	—	2	1	no	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
224	C₂×D₅	—	2	2	no	—
225	C₂×D₅	—	2	2	no	—
226	C₂×D₅	C₂×#108	1	4	yes	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	C₂×#108	1	2	yes	—
231	C₄×C₅	C₂×#131	1	1	yes	characteristic
232	C₄×C₅	#568	1	1	yes	characteristic
233	C₄×C₅	#576	1	1	yes	characteristic
234	C₄×C₅	#578	1	1	yes	characteristic
235	Dic₅	—	2	2	no	—
236	Dic₅	—	2	2	no	—
237	Dic₅	—	2	2	no	—
238	#41	—	10	4	no	—
239	C₂×Q₈	—	5	1	no	—
240	C₂×D₄	—	5	2	no	—
241	C₂×D₄	—	5	1	no	—
242	C₂²×C₄	—	5	2	no	—
243	C₂²×C₄	—	5	2	no	—
244	C₂²×C₄	—	5	2	no	—
245	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
246	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
247	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
248	M₁₆	—	40	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	5	2	no	—
254	C₂×C₈	—	5	2	no	—
255	C₂×C₈	—	10	4	no	—
256	C₂×C₈	—	10	4	no	—
257	C₂×C₈	—	20	2	no	—
258	C₂×C₈	—	40	4	no	—
259	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
260	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
261	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
262	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
263	C₄⋊C₄	—	5	2	no	—
264	C₄⋊C₄	—	5	2	no	—
265	C₄⋊C₄	—	5	1	no	—
266	C₄⋊C₄	—	5	1	no	—
267	C₂²⋊C₄	—	5	2	no	—
268	C₂²⋊C₄	—	5	2	no	—
269	C₂²⋊C₄	—	5	2	no	—
270	C₄²	—	5	2	no	—
271	C₄²	—	5	2	no	—
272	C₄²	—	5	2	no	—
273	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
274	C₄²	D₄×D₅	1	1	yes	characteristic
275	C₄²	Q₈×D₅	1	1	yes	characteristic
276	C₄²	C₂³×D₅	1	1	yes	characteristic
277	C₂×C₅	—	4	1	no	—
278	C₂×C₅	#2906	1	1	yes	characteristic
279	C₂×C₅	#2913	1	1	yes	characteristic
280	C₂×C₅	C₂²×#108	1	1	yes	characteristic
281	D₅	—	2	4	no	—
282	C₂³	—	5	2	no	—
283	C₂³	D₅×C₄⋊C₄	1	1	yes	characteristic
284	Q₈	—	10	2	no	—
285	D₄	—	5	4	no	—
286	D₄	—	10	4	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	5	2	no	—
293	C₂×C₄	—	5	2	no	—
294	C₂×C₄	—	5	2	no	—
295	C₂×C₄	—	10	4	no	—
296	C₂×C₄	—	10	2	no	—
297	C₂×C₄	—	10	2	no	—
298	C₂×C₄	—	10	2	no	—
299	C₂×C₄	—	10	2	no	—
300	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
301	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
302	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
305	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	10	2	no	—
312	C₈	—	10	2	no	—
313	C₈	—	20	4	no	—
314	C₈	#4047	1	2	yes	—
315	C₅	#17836	1	1	yes	characteristic, complemented, Sylow
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	5	4	no	—
320	C₂²	—	10	2	no	—
321	C₂²	—	10	2	no	—
322	C₂²	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	10	2	no	—
327	C₄	—	10	2	no	—
328	C₄	—	10	2	no	—
329	C₄	D₅×#108	1	2	yes	—
330	C₄	C₂×#4047	1	1	yes	characteristic
331	C₄	#66079	1	1	yes	characteristic
332	C₄	#66103	1	1	yes	characteristic
333	C₄	#66105	1	1	yes	characteristic
334	C₂	—	4	1	no	—
335	C₂	—	10	4	no	—
336	C₂	#10610743	1	1	yes	characteristic
337	C₂	#10610910	1	1	yes	characteristic
338	C₂	C₂×D₅×#108	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	Q₈
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1564	D₅
hypocenter	C₅	#17836
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
3. C₅×#1564	C₂	semidirect
4. #10601541	C₂	semidirect
5. #10600524	C₂	semidirect
6. #10599363	C₂	semidirect
7. D₅×#335	C₂	semidirect
8. #10599327	C₂	semidirect
81. #1564	D₅	semidirect
315. C₅	#17836	semidirect

Maximal subgroups

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

subgroup	index
#10611024	2
C₈×#709	2
C₅×#1564	2
#10601541	2
#10600524	2
#10599363	2
D₅×#335	2
#10599327	2
#17836	5