Subgroups — #12071051

Order 1280.

652 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#135	C₂	1	1	yes	maximal, characteristic
2	#10614253	C₂	1	2	yes	maximal, complemented
3	#10602434	C₂	1	2	yes	maximal, complemented
4	#10601699	C₂	1	2	yes	maximal, complemented
5	#10601688	C₂	1	4	yes	maximal, complemented
6	#10601671	C₂	1	2	yes	maximal, complemented
7	#10600523	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600518	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#135	C₄	1	1	yes	characteristic
10	C₄×C₅×#41	C₂²	1	2	yes	—
11	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic
12	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₅×SD₁₆	C₄	1	2	yes	—
14	C₄×C₅×SD₁₆	C₂²	1	2	yes	—
15	C₄×C₅×D₈	C₄	1	1	yes	characteristic
16	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
17	C₅×#427	C₄	1	1	yes	characteristic
18	C₅×#416	C₄	1	2	yes	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	#65313	C₂²	1	4	yes	—
21	#65300	—	2	4	no	—
22	D₄×#168	C₂²	1	4	yes	—
23	#65215	—	2	2	no	—
24	#65214	C₂²	1	1	yes	characteristic
25	#65213	C₂²	1	2	yes	—
26	C₂×C₄⋊#168	—	2	2	no	—
27	C₄×#681	C₂²	1	1	yes	characteristic
28	Q₈⋊#168	C₂²	1	4	yes	—
29	D₄⋊#168	C₂²	1	4	yes	—
30	#64677	C₂²	1	2	yes	—
31	#64677	C₂²	1	2	yes	—
32	#64675	C₂²	1	2	yes	—
33	#64674	C₂²	1	2	yes	—
34	#18065	—	5	1	no	maximal, Sylow
35	C₂×C₅×#41	C₂×C₄	1	2	yes	—
36	C₅×#135	—	2	4	no	—
37	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅×SD₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
43	C₄×C₅×Q₈	C₂³	1	2	yes	—
44	C₄×C₅×D₄	—	2	2	no	—
45	C₄×C₅×D₄	—	2	2	no	—
46	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
47	C₄×C₅×D₄	C₂³	1	2	yes	—
48	C₅×#117	—	2	2	no	—
49	C₅×#117	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅	—	2	2	no	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₂×#681	—	2	1	no	—
53	C₂×#681	—	2	2	no	—
54	C₂²×#168	—	2	4	no	—
55	C₅×#107	C₂×C₄	1	1	yes	characteristic
56	C₅×#107	C₂×C₄	1	1	yes	characteristic
57	C₅×#106	C₂×C₄	1	2	yes	—
58	C₅×#103	C₂×C₄	1	2	yes	—
59	C₅×#103	C₂×C₄	1	2	yes	—
60	C₅×#102	C₂×C₄	1	2	yes	—
61	C₅×#102	C₂×C₄	1	2	yes	—
62	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
64	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
65	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
66	C₂²⋊#168	—	2	4	no	—
67	C₂²⋊#168	D₄	1	2	yes	—
68	C₄⋊#168	—	2	2	no	—
69	C₄⋊#168	—	2	4	no	—
70	C₄⋊#168	D₄	1	2	yes	—
71	C₄⋊#168	C₂³	1	4	yes	—
72	#3935	—	2	1	no	—
73	#3935	—	2	4	no	—
74	C₄×#168	C₂³	1	2	yes	—
75	C₄×#168	C₂³	1	2	yes	—
76	#2990	—	5	2	no	—
77	C₄×#135	D₅	1	1	yes	characteristic
78	#1593	—	5	2	no	—
79	#1582	—	5	4	no	—
80	#1580	—	5	2	no	—
81	#1566	—	5	1	no	—
82	#1489	—	5	2	no	—
83	#1451	—	5	1	no	—
84	C₅×#41	—	2	4	no	—
85	C₅×#41	—	4	2	no	—
86	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
87	C₂×C₅×D₄	—	2	2	no	—
88	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅	—	2	2	no	—
90	C₂²×C₄×C₅	—	2	2	no	—
91	C₂²×C₄×C₅	—	2	2	no	—
92	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅	#41	1	1	yes	characteristic
95	C₅×Q₁₆	—	2	2	no	—
96	C₅×SD₁₆	—	2	4	no	—
97	C₅×D₈	M₁₆	1	4	yes	complemented
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	4	1	no	—
100	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₅×C₄⋊C₄	—	2	2	no	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
107	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
108	C₅×C₂²⋊C₄	—	2	2	no	—
109	C₅×C₂²⋊C₄	—	2	2	no	—
110	C₅×C₂²⋊C₄	—	2	2	no	—
111	C₄²×C₅	—	2	2	no	—
112	C₄²×C₅	—	2	2	no	—
113	C₄²×C₅	—	2	2	no	—
114	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅	#41	1	1	yes	characteristic
117	C₄²×C₅	C₂⁴	1	1	yes	characteristic
118	#681	—	4	2	no	—
119	#681	—	8	2	no	—
120	C₂×#168	—	2	4	no	—
121	C₂×#168	—	2	2	no	—
122	C₂×#168	—	8	4	no	—
123	C₂×#168	C₂×D₄	1	4	yes	—
124	C₂×#135	Dic₅	1	1	yes	characteristic
125	C₄×#41	C₂×D₅	1	2	yes	—
126	#446	—	5	4	no	—
127	C₄×Q₁₆	Dic₅	1	1	yes	characteristic
128	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
129	C₄×SD₁₆	Dic₅	1	2	yes	—
130	C₄×SD₁₆	C₂×D₅	1	2	yes	—
131	C₄×D₈	Dic₅	1	1	yes	characteristic
132	C₄×D₈	C₂×D₅	1	1	yes	characteristic
133	#435	—	10	4	no	—
134	C₈×D₄	—	5	4	no	—
135	#432	—	10	2	no	—
136	#431	—	5	1	no	—
137	#427	Dic₅	1	1	yes	characteristic
138	#423	—	5	2	no	—
139	C₂×C₄⋊C₈	—	10	2	no	—
140	#416	Dic₅	1	2	yes	—
141	C₄×M₁₆	—	5	1	no	—
142	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
143	#334	—	5	2	no	—
144	#334	—	5	2	no	—
145	Q₈⋊C₈	—	5	4	no	—
146	D₄⋊C₈	—	5	4	no	—
147	#322	—	5	2	no	—
148	#322	—	5	2	no	—
149	C₂³×C₅	—	2	2	no	—
150	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
151	C₅×Q₈	—	2	2	no	—
152	C₅×Q₈	—	2	2	no	—
153	C₅×D₄	—	2	2	no	—
154	C₅×D₄	—	4	4	no	—
155	C₅×D₄	C₂×M₁₆	1	4	yes	—
156	C₂×C₄×C₅	—	2	1	no	—
157	C₂×C₄×C₅	—	2	1	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	4	4	no	—
164	C₂×C₄×C₅	—	4	2	no	—
165	C₂×C₄×C₅	—	4	2	no	—
166	C₂×C₄×C₅	—	4	4	no	—
167	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	—	2	1	no	—
175	C₅×C₈	—	2	1	no	—
176	C₅×C₈	C₂×M₁₆	1	2	yes	—
177	#168	—	2	4	no	—
178	#168	—	4	2	no	—
179	#168	—	4	4	no	—
180	C₂×#41	C₂×Dic₅	1	2	yes	—
181	#135	—	2	4	no	—
182	C₂×Q₁₆	C₂×Dic₅	1	1	yes	characteristic
183	C₂×SD₁₆	C₂×Dic₅	1	2	yes	—
184	C₂×D₈	C₂×Dic₅	1	1	yes	characteristic
185	C₂×M₁₆	—	10	1	no	—
186	C₂×M₁₆	—	10	2	no	—
187	C₂²×C₈	—	10	4	no	—
188	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
189	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
190	C₄×Q₈	C₂×Dic₅	1	2	yes	—
191	C₄×Q₈	C₂²×D₅	1	2	yes	—
192	C₄×D₄	—	2	2	no	—
193	C₄×D₄	—	2	2	no	—
194	C₄×D₄	C₂×Dic₅	1	2	yes	—
195	C₄×D₄	C₂²×D₅	1	2	yes	—
196	#117	—	2	2	no	—
197	#117	C₂×Dic₅	1	2	yes	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
200	#107	C₂×Dic₅	1	1	yes	characteristic
201	#107	C₂×Dic₅	1	1	yes	characteristic
202	#106	C₂×Dic₅	1	2	yes	—
203	C₄⋊C₈	—	5	2	no	—
204	C₄⋊C₈	—	5	4	no	—
205	C₄⋊C₈	—	10	4	no	—
206	C₄⋊C₈	—	10	2	no	—
207	#103	C₂×Dic₅	1	2	yes	—
208	#103	C₂×Dic₅	1	2	yes	—
209	#102	C₂×Dic₅	1	2	yes	—
210	#102	C₂×Dic₅	1	2	yes	—
211	C₂²⋊C₈	—	5	2	no	—
212	C₂²⋊C₈	—	10	4	no	—
213	#97	—	10	4	no	—
214	#97	—	10	1	no	—
215	C₄×C₈	—	5	2	no	—
216	C₄×C₈	—	5	2	no	—
217	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
218	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
220	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	2	4	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	2	1	no	—
225	C₂²×C₅	—	4	2	no	—
226	C₂²×C₅	—	4	2	no	—
227	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	—	4	2	no	—
232	C₄×C₅	—	4	2	no	—
233	C₄×C₅	—	4	2	no	—
234	C₄×C₅	#444	1	2	yes	—
235	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
236	C₄×C₅	#568	1	1	yes	characteristic
237	C₄×C₅	#577	1	2	yes	—
238	#41	—	2	4	no	—
239	#41	—	4	2	no	—
240	C₂×Q₈	C₂²×Dic₅	1	2	yes	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	C₂²×Dic₅	1	2	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
247	C₂²×C₄	#711	1	1	yes	characteristic
248	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
249	Q₁₆	—	2	2	no	—
250	SD₁₆	—	2	4	no	—
251	D₈	#681	1	4	yes	complemented
252	M₁₆	—	20	2	no	—
253	M₁₆	—	40	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	5	4	no	—
257	C₂×C₈	—	10	4	no	—
258	C₂×C₈	—	10	2	no	—
259	C₂×C₈	—	40	4	no	—
260	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
261	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
262	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
263	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	C₂²×Dic₅	1	2	yes	—
267	C₄⋊C₄	C₂²×Dic₅	1	2	yes	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	D₄×D₅	1	1	yes	characteristic
275	C₄²	#711	1	1	yes	characteristic
276	C₄²	C₂²×Dic₅	1	1	yes	characteristic
277	C₄²	C₂³×D₅	1	1	yes	characteristic
278	C₂×C₅	—	4	1	no	—
279	C₂×C₅	—	4	4	no	—
280	C₂×C₅	D₄×M₁₆	1	1	yes	characteristic
281	C₂×C₅	#2952	1	1	yes	characteristic
282	C₂×C₅	C₂×#444	1	1	yes	characteristic
283	C₂³	—	2	2	no	—
284	C₂³	D₄×Dic₅	1	1	yes	characteristic
285	Q₈	—	2	2	no	—
286	Q₈	—	2	2	no	—
287	D₄	—	2	2	no	—
288	D₄	—	4	4	no	—
289	D₄	C₂×#681	1	4	yes	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
302	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
303	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
305	C₂×C₄	C₂×#711	1	1	yes	characteristic
306	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	10	4	no	—
311	C₈	—	20	2	no	—
312	C₈	—	20	4	no	—
313	C₈	C₂×#681	1	2	yes	—
314	C₅	#18065	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	2	1	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	C₂×D₄×Dic₅	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	#65486	1	2	yes	—
329	C₄	#66104	1	2	yes	—
330	C₄	C₂²×#681	1	1	yes	characteristic
331	C₄	#66153	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	4	4	no	—
334	C₂	D₄×#681	1	1	yes	characteristic
335	C₂	#10615630	1	1	yes	characteristic
336	C₂	C₂×#65486	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₄×C₅×#135	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂×C₅×#135	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₄×#135	D₅
hypocenter	C₅	#18065
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. #10614253	C₂	semidirect
3. #10602434	C₂	semidirect
4. #10601699	C₂	semidirect
5. #10601688	C₂	semidirect
6. #10601671	C₂	semidirect
7. #10600523	C₂	semidirect
8. #10600518	C₂	semidirect
97. C₅×D₈	M₁₆	semidirect
251. D₈	#681	semidirect
314. C₅	#18065	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#135	2
#10614253	2
#10602434	2
#10601699	2
#10601688	2
#10601671	2
#10600523	2
#10600518	2
#18065	5