Subgroups — #12068519

Order 1280.

572 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611070	C₂	1	2	yes	maximal
2	#10610186	C₂	1	1	yes	maximal, characteristic
3	C₅×#1568	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601588	C₂	1	2	yes	maximal, complemented
5	#10600524	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599366	C₂	1	2	yes	maximal, complemented
7	#10599365	C₂	1	2	yes	maximal, complemented
8	#10599328	C₂	1	2	yes	maximal, complemented
9	#10599327	C₂	1	2	yes	maximal, complemented
10	C₄×#709	C₂²	1	1	yes	characteristic
11	C₅×#432	C₂²	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	#65399	C₄	1	2	yes	—
14	#65391	C₄	1	2	yes	—
15	#65213	C₂²	1	2	yes	—
16	C₄×#681	C₂²	1	1	yes	characteristic
17	#65126	C₂²	1	2	yes	—
18	#65124	C₂²	1	2	yes	—
19	#65122	C₂²	1	2	yes	—
20	D₅×C₄⋊C₈	C₂²	1	2	yes	—
21	#65018	—	2	2	no	—
22	#65017	—	2	2	no	—
23	#64995	—	2	2	no	—
24	#64990	—	2	2	no	—
25	#64979	C₄	1	4	yes	—
26	#64974	C₂²	1	2	yes	—
27	#64974	C₂²	1	2	yes	—
28	C₅×#335	C₂²	1	1	yes	characteristic
29	C₅×#335	C₂²	1	1	yes	characteristic
30	C₅×#334	C₂²	1	2	yes	—
31	#64700	C₂²	1	2	yes	—
32	#64700	C₂²	1	2	yes	—
33	#64699	C₂²	1	2	yes	—
34	#64699	C₂²	1	2	yes	—
35	#64678	C₂²	1	1	yes	characteristic
36	#64678	C₂²	1	1	yes	characteristic
37	#64678	C₂²	1	2	yes	—
38	#17825	—	5	1	no	maximal, Sylow
39	C₂×#709	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₄×C₂²⋊D₅	D₄	1	2	yes	—
43	C₄×C₂²⋊D₅	Q₈	1	2	yes	—
44	#4072	C₂×C₄	1	1	yes	characteristic
45	C₂×#681	—	2	2	no	—
46	C₂×#681	C₂×C₄	1	2	yes	—
47	C₂×#676	—	2	2	no	—
48	C₂×C₈×D₅	—	2	2	no	—
49	C₄×D₂₀	D₄	1	1	yes	characteristic
50	C₄×D₂₀	Q₈	1	1	yes	characteristic
51	#4018	C₂×C₄	1	2	yes	—
52	C₄²×D₅	C₂³	1	2	yes	—
53	C₄×Dic₁₀	D₄	1	1	yes	characteristic
54	C₄×Dic₁₀	Q₈	1	1	yes	characteristic
55	C₅×C₄⋊C₈	C₂³	1	2	yes	—
56	C₅×C₄⋊C₈	C₂³	1	2	yes	—
57	C₅×C₂²⋊C₈	—	2	2	no	—
58	C₅×#97	—	2	2	no	—
59	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
60	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
61	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
62	#3952	—	2	2	no	—
63	#3952	—	2	2	no	—
64	#3952	C₂×C₄	1	4	yes	—
65	#3952	C₂×C₄	1	4	yes	—
66	#3947	—	2	2	no	—
67	#3946	—	2	2	no	—
68	#3946	—	2	2	no	—
69	#3946	C₂×C₄	1	4	yes	—
70	#3946	C₂×C₄	1	4	yes	—
71	C₈×Dic₅	—	2	2	no	—
72	C₄⋊#168	—	2	2	no	—
73	C₄⋊#168	C₂³	1	2	yes	—
74	C₄⋊#168	C₂³	1	2	yes	—
75	#3935	C₂×C₄	1	2	yes	—
76	C₄×#168	C₂³	1	2	yes	—
77	#2989	—	5	2	no	—
78	#2922	—	5	1	no	—
79	#1574	—	5	2	no	—
80	#1573	—	5	2	no	—
81	#1572	—	5	2	no	—
82	#1571	—	5	2	no	—
83	#1569	—	5	2	no	—
84	#1568	D₅	1	1	yes	characteristic, complemented
85	#1567	—	5	1	no	—
86	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₂²⋊D₅	C₄⋊C₄	1	2	yes	—
90	#709	—	4	2	no	—
91	C₂×D₂₀	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₅	C₂×D₄	1	2	yes	—
94	C₂×C₄×D₅	C₂×Q₈	1	2	yes	—
95	C₂×Dic₁₀	C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	4	2	no	—
99	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
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	—	4	2	no	—
116	#681	—	8	2	no	—
117	C₂×#168	—	2	2	no	—
118	C₂×#168	—	2	2	no	—
119	C₂×#168	C₂²×C₄	1	4	yes	—
120	#676	—	8	2	no	—
121	C₈×D₅	—	8	2	no	—
122	C₄×#41	—	5	1	no	—
123	#446	—	10	2	no	—
124	#436	—	10	2	no	—
125	#435	—	10	2	no	—
126	#435	—	10	2	no	—
127	#433	—	5	4	no	—
128	#432	—	5	2	no	—
129	#432	C₂×D₅	1	2	yes	—
130	#431	—	5	2	no	—
131	#431	—	5	2	no	—
132	#431	—	5	2	no	—
133	#424	—	5	2	no	—
134	#423	—	5	2	no	—
135	#423	—	5	2	no	—
136	C₂×C₄⋊C₈	—	5	2	no	—
137	#408	—	5	2	no	—
138	C₄×M₁₆	—	5	1	no	—
139	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
140	#335	—	5	2	no	—
141	#335	—	5	2	no	—
142	#335	—	5	2	no	—
143	#335	—	5	1	no	—
144	#335	—	5	1	no	—
145	#335	C₂×D₅	1	1	yes	characteristic
146	#335	C₂×D₅	1	1	yes	characteristic
147	#334	—	5	2	no	—
148	#334	—	5	2	no	—
149	#334	C₂×D₅	1	2	yes	—
150	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×D₅	C₂×C₄⋊C₄	1	2	yes	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	2	1	no	—
155	C₂×C₄×C₅	—	4	1	no	—
156	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
162	C₂²⋊D₅	—	4	2	no	—
163	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
165	C₂×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
166	D₂₀	—	4	1	no	—
167	C₄×D₅	—	2	4	no	—
168	C₄×D₅	—	2	2	no	—
169	C₄×D₅	—	2	2	no	—
170	C₄×D₅	—	4	2	no	—
171	Dic₁₀	—	4	1	no	—
172	C₅×C₈	—	2	2	no	—
173	C₅×C₈	—	2	2	no	—
174	C₅×C₈	—	4	2	no	—
175	C₅×C₈	—	4	2	no	—
176	#168	—	2	4	no	—
177	#168	—	4	2	no	—
178	#168	—	4	2	no	—
179	C₂×#41	—	5	1	no	—
180	C₂×M₁₆	—	5	2	no	—
181	C₂×M₁₆	—	10	2	no	—
182	C₂×M₁₆	—	10	2	no	—
183	C₂²×C₈	—	10	2	no	—
184	C₂²×C₈	C₄×D₅	1	2	yes	—
185	C₄×Q₈	—	5	1	no	—
186	C₄×Q₈	—	5	1	no	—
187	C₄×D₄	—	5	2	no	—
188	C₄×D₄	—	5	2	no	—
189	C₄×D₄	—	5	1	no	—
190	C₄×D₄	—	5	1	no	—
191	#117	—	5	1	no	—
192	#117	—	5	2	no	—
193	C₂×C₄²	—	5	2	no	—
194	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
195	C₄⋊C₈	—	5	4	no	—
196	C₄⋊C₈	—	5	4	no	—
197	C₄⋊C₈	—	5	2	no	—
198	C₄⋊C₈	—	5	2	no	—
199	C₄⋊C₈	—	10	2	no	—
200	C₄⋊C₈	—	10	2	no	—
201	C₄⋊C₈	—	10	2	no	—
202	C₄⋊C₈	C₂²×D₅	1	2	yes	—
203	C₄⋊C₈	C₂²×D₅	1	2	yes	—
204	C₂²⋊C₈	—	2	2	no	—
205	C₂²⋊C₈	—	5	4	no	—
206	C₂²⋊C₈	—	5	4	no	—
207	C₂²⋊C₈	—	10	2	no	—
208	C₂²⋊C₈	—	10	2	no	—
209	#97	—	2	2	no	—
210	#97	—	5	2	no	—
211	#97	—	10	2	no	—
212	C₄×C₈	—	5	2	no	—
213	C₄×C₈	—	10	2	no	—
214	C₄×C₈	C₄×D₅	1	2	yes	—
215	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
217	C₂²×C₅	—	2	1	no	—
218	C₂²×C₅	—	2	1	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
221	C₂×D₅	—	2	2	no	—
222	C₂×D₅	—	2	2	no	—
223	C₂×D₅	—	2	2	no	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	#568	1	2	yes	—
229	C₄×C₅	#576	1	1	yes	characteristic
230	C₄×C₅	#578	1	1	yes	characteristic
231	Dic₅	—	2	2	no	—
232	Dic₅	—	2	2	no	—
233	Dic₅	—	2	2	no	—
234	#41	—	20	2	no	—
235	C₂×Q₈	—	5	1	no	—
236	C₂×D₄	—	5	2	no	—
237	C₂×D₄	—	5	1	no	—
238	C₂²×C₄	—	5	2	no	—
239	C₂²×C₄	—	5	2	no	—
240	C₂²×C₄	—	5	2	no	—
241	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
242	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
243	C₂²×C₄	Q₈×D₅	1	1	yes	characteristic
244	M₁₆	—	20	2	no	—
245	M₁₆	—	40	2	no	—
246	M₁₆	—	40	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	5	4	no	—
251	C₂×C₈	—	10	2	no	—
252	C₂×C₈	—	10	2	no	—
253	C₂×C₈	—	40	2	no	—
254	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
255	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
256	C₄⋊C₄	—	5	2	no	—
257	C₄⋊C₄	—	5	2	no	—
258	C₄⋊C₄	—	5	1	no	—
259	C₄⋊C₄	—	5	1	no	—
260	C₂²⋊C₄	—	5	2	no	—
261	C₂²⋊C₄	—	5	2	no	—
262	C₂²⋊C₄	—	5	2	no	—
263	C₄²	—	5	2	no	—
264	C₄²	—	5	2	no	—
265	C₄²	—	5	2	no	—
266	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
267	C₄²	D₄×D₅	1	1	yes	characteristic
268	C₄²	Q₈×D₅	1	1	yes	characteristic
269	C₄²	C₂³×D₅	1	1	yes	characteristic
270	C₂×C₅	—	4	1	no	—
271	C₂×C₅	#2905	1	1	yes	characteristic
272	C₂×C₅	#2913	1	1	yes	characteristic
273	C₂×C₅	#2916	1	1	yes	characteristic
274	D₅	—	4	2	no	—
275	C₂³	—	5	2	no	—
276	C₂³	D₅×C₄⋊C₄	1	1	yes	characteristic
277	Q₈	—	20	1	no	—
278	D₄	—	20	1	no	—
279	D₄	—	20	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	5	2	no	—
285	C₂×C₄	—	5	2	no	—
286	C₂×C₄	—	5	2	no	—
287	C₂×C₄	—	10	4	no	—
288	C₂×C₄	—	10	2	no	—
289	C₂×C₄	—	10	2	no	—
290	C₂×C₄	—	20	2	no	—
291	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
292	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
293	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
296	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	2	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	4	2	no	—
301	C₈	—	10	4	no	—
302	C₈	—	20	2	no	—
303	C₈	—	20	2	no	—
304	C₅	#17825	1	1	yes	characteristic, complemented, Sylow
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	10	2	no	—
309	C₂²	—	10	2	no	—
310	C₂²	—	10	2	no	—
311	C₂²	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	10	2	no	—
317	C₄	—	10	2	no	—
318	C₄	—	10	2	no	—
319	C₄	#66079	1	2	yes	—
320	C₄	#66103	1	1	yes	characteristic
321	C₄	#66105	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	20	2	no	—
324	C₂	#10610779	1	1	yes	characteristic
325	C₂	#10610910	1	1	yes	characteristic
326	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×#1568	C₂
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂×#709	C₂×C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1568	D₅
hypocenter	C₅	#17825
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. C₅×#1568	C₂	semidirect
4. #10601588	C₂	semidirect
5. #10600524	C₂	semidirect
6. #10599366	C₂	semidirect
7. #10599365	C₂	semidirect
8. #10599328	C₂	semidirect
9. #10599327	C₂	semidirect
84. #1568	D₅	semidirect
304. C₅	#17825	semidirect

Maximal subgroups

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

subgroup	index
#10611070	2
#10610186	2
C₅×#1568	2
#10601588	2
#10600524	2
#10599366	2
#10599365	2
#10599328	2
#10599327	2
#17825	5