Subgroups — #12252415

Order 1280.

590 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	C₄×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	#10602797	C₂	1	2	yes	maximal
3	#10600676	C₂	1	4	yes	maximal, complemented
4	#10600674	C₂	1	2	yes	maximal, complemented
5	#10600538	C₂	1	2	yes	maximal, complemented
6	#10600537	C₂	1	2	yes	maximal, complemented
7	#10600536	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#131	C₄	1	1	yes	characteristic
10	C₂×C₅×#131	C₂²	1	1	yes	characteristic
11	C₄×C₅×#41	C₄	1	1	yes	characteristic
12	C₅×#405	C₄	1	2	yes	—
13	C₅×#405	C₄	1	1	yes	characteristic
14	C₅×#405	C₂²	1	1	yes	characteristic
15	C₅×#405	C₂²	1	2	yes	—
16	C₄×C₅×M₁₆	C₄	1	2	yes	—
17	C₄×C₅×M₁₆	C₄	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
20	C₂×#3944	C₂²	1	2	yes	—
21	C₂²×#672	C₂²	1	2	yes	—
22	C₂²⋊#672	C₂²	1	4	yes	—
23	C₂²⋊#672	C₂²	1	4	yes	—
24	C₄⋊#672	C₂²	1	1	yes	characteristic
25	C₄⋊#672	C₂²	1	1	yes	characteristic
26	C₄⋊#672	C₂²	1	2	yes	—
27	C₄⋊#672	C₂²	1	2	yes	—
28	C₄⋊#672	C₂²	1	2	yes	—
29	#64681	C₂²	1	4	yes	—
30	C₄×#672	C₂²	1	4	yes	—
31	#33423	—	5	1	no	maximal, Sylow
32	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
33	C₅×#131	—	2	4	no	—
34	C₅×#131	—	2	2	no	—
35	C₅×#131	—	2	2	no	—
36	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
42	C₂²×C₅×C₈	C₂³	1	2	yes	—
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₄×C₅×Q₈	C₈	1	1	yes	characteristic
45	C₄×C₅×Q₈	C₈	1	1	yes	characteristic
46	C₄×C₅×D₄	C₈	1	2	yes	—
47	C₄×C₅×D₄	C₈	1	1	yes	characteristic
48	C₄×C₅×D₄	C₈	1	2	yes	—
49	C₄×C₅×D₄	C₈	1	1	yes	characteristic
50	C₅×#117	C₂×C₄	1	2	yes	—
51	C₅×#117	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
54	C₅×#97	C₂×C₄	1	2	yes	—
55	C₅×#97	C₂×C₄	1	2	yes	—
56	C₅×#97	C₂×C₄	1	2	yes	—
57	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
58	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
59	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
60	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂³	1	2	yes	—
63	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
64	#3944	—	2	4	no	—
65	C₂×#672	—	2	4	no	—
66	C₂×#672	C₂³	1	2	yes	—
67	C₂×#672	C₂³	1	2	yes	—
68	C₂×#672	C₂³	1	4	yes	—
69	C₄×#131	D₅	1	1	yes	characteristic
70	#2164	—	5	4	no	—
71	#2163	—	5	2	no	—
72	#2162	—	5	2	no	—
73	#2150	—	5	1	no	—
74	C₂×C₄⋊C₁₆	—	5	1	no	—
75	#2113	—	5	2	no	—
76	#2108	—	5	2	no	—
77	C₅×#41	—	2	2	no	—
78	C₅×#41	—	2	2	no	—
79	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
80	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
81	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
83	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
85	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₅×M₁₆	—	2	2	no	—
89	C₅×M₁₆	—	2	4	no	—
90	C₅×M₁₆	—	2	4	no	—
91	C₅×M₁₆	M₁₆	1	4	yes	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	4	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
101	C₂×C₅×C₈	#41	1	2	yes	—
102	C₂×C₅×C₈	#41	1	2	yes	—
103	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
104	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
105	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
106	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
107	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
108	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
109	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
110	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
111	C₄²×C₅	C₂×C₈	1	2	yes	—
112	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
113	C₄²×C₅	C₂×C₈	1	2	yes	—
114	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
115	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₅	C₂²×C₄	1	2	yes	—
117	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
118	#672	—	2	2	no	—
119	#672	—	2	2	no	—
120	#672	—	2	4	no	—
121	C₂×#131	Dic₅	1	1	yes	characteristic
122	C₂×#131	C₂×D₅	1	1	yes	characteristic
123	C₄×#41	Dic₅	1	1	yes	characteristic
124	C₂×M₃₂	—	5	2	no	—
125	C₂²×C₁₆	—	5	2	no	—
126	#405	Dic₅	1	1	yes	characteristic
127	#405	Dic₅	1	2	yes	—
128	#405	C₂×D₅	1	1	yes	characteristic
129	#405	C₂×D₅	1	2	yes	—
130	C₄×M₁₆	Dic₅	1	2	yes	—
131	C₄×M₁₆	Dic₅	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
134	C₄⋊C₁₆	—	5	1	no	—
135	C₄⋊C₁₆	—	5	1	no	—
136	C₄⋊C₁₆	—	5	2	no	—
137	C₄⋊C₁₆	—	5	2	no	—
138	C₄⋊C₁₆	—	5	2	no	—
139	C₂²⋊C₁₆	—	5	4	no	—
140	C₂²⋊C₁₆	—	5	4	no	—
141	#346	—	5	4	no	—
142	C₄×C₁₆	—	5	4	no	—
143	C₂³×C₅	C₂²×C₈	1	2	yes	—
144	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
145	C₅×Q₈	—	2	2	no	—
146	C₅×D₄	—	2	4	no	—
147	C₅×D₄	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	—	4	1	no	—
153	C₂×C₄×C₅	—	4	2	no	—
154	C₂×C₄×C₅	#117	1	2	yes	—
155	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
157	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
158	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
160	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
164	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
165	C₅×C₈	—	2	2	no	—
166	C₅×C₈	—	2	1	no	—
167	C₅×C₈	—	2	1	no	—
168	C₅×C₈	#117	1	2	yes	—
169	C₅×C₈	C₂×M₁₆	1	4	yes	—
170	C₅×C₈	C₂×#41	1	1	yes	characteristic
171	C₅×C₈	C₂×#41	1	1	yes	characteristic
172	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
173	#131	—	2	4	no	—
174	#131	—	2	2	no	—
175	#131	—	2	2	no	—
176	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
177	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
178	C₂×M₁₆	C₂×Dic₅	1	2	yes	—
179	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
180	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
181	C₂²×C₈	C₂×Dic₅	1	2	yes	—
182	C₂²×C₈	C₂²×D₅	1	2	yes	—
183	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
184	C₄×Q₈	#168	1	1	yes	characteristic
185	C₄×Q₈	#168	1	1	yes	characteristic
186	C₄×D₄	#168	1	2	yes	—
187	C₄×D₄	#168	1	1	yes	characteristic
188	C₄×D₄	#168	1	2	yes	—
189	C₄×D₄	#168	1	1	yes	characteristic
190	#117	C₂×Dic₅	1	2	yes	—
191	#117	C₂×Dic₅	1	1	yes	characteristic
192	C₂×C₄²	C₂×Dic₅	1	2	yes	—
193	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
194	M₃₂	—	10	4	no	—
195	C₂×C₁₆	—	5	2	no	—
196	C₂×C₁₆	—	5	4	no	—
197	C₂×C₁₆	—	5	2	no	—
198	C₂×C₁₆	—	10	4	no	—
199	#97	C₂×Dic₅	1	2	yes	—
200	#97	C₂×Dic₅	1	2	yes	—
201	#97	C₂×Dic₅	1	2	yes	—
202	C₄×C₈	C₂×Dic₅	1	2	yes	—
203	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
204	C₄×C₈	C₂×Dic₅	1	2	yes	—
205	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₅	1	2	yes	—
207	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	—	2	2	no	—
213	C₂²×C₅	—	2	2	no	—
214	C₂²×C₅	#431	1	2	yes	—
215	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
216	C₄×C₅	—	2	1	no	—
217	C₄×C₅	—	2	2	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	2	2	no	—
221	C₄×C₅	—	2	2	no	—
222	C₄×C₅	#431	1	2	yes	—
223	C₄×C₅	#504	1	2	yes	—
224	C₄×C₅	C₂×#117	1	1	yes	characteristic
225	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	C₂×Q₈	C₂×#168	1	1	yes	characteristic
229	C₂×D₄	C₂×#168	1	2	yes	—
230	C₂×D₄	C₂×#168	1	1	yes	characteristic
231	C₂²×C₄	C₂×#168	1	2	yes	—
232	C₂²×C₄	C₂×#168	1	1	yes	characteristic
233	C₂²×C₄	C₂×#168	1	2	yes	—
234	C₂²×C₄	C₂×#168	1	1	yes	characteristic
235	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
236	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
237	M₁₆	—	2	2	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	#681	1	4	yes	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	#709	1	2	yes	—
246	C₂×C₈	#709	1	2	yes	—
247	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
248	C₂×C₈	C₂²×Dic₅	1	2	yes	—
249	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
250	C₂×C₈	C₂²×Dic₅	1	2	yes	—
251	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
252	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
253	C₄⋊C₄	C₂×#168	1	2	yes	—
254	C₄⋊C₄	C₂×#168	1	2	yes	—
255	C₄⋊C₄	C₂×#168	1	1	yes	characteristic
256	C₄⋊C₄	C₂×#168	1	1	yes	characteristic
257	C₂²⋊C₄	C₂×#168	1	2	yes	—
258	C₂²⋊C₄	C₂×#168	1	2	yes	—
259	C₂²⋊C₄	C₂×#168	1	2	yes	—
260	C₄²	C₂×#168	1	2	yes	—
261	C₄²	C₂×#168	1	1	yes	characteristic
262	C₄²	C₂×#168	1	2	yes	—
263	C₄²	C₂×#168	1	1	yes	characteristic
264	C₄²	C₂²×Dic₅	1	1	yes	characteristic
265	C₄²	C₂²×Dic₅	1	2	yes	—
266	C₄²	C₂²×Dic₅	1	1	yes	characteristic
267	C₁₆	—	10	2	no	—
268	C₁₆	—	10	2	no	—
269	C₁₆	—	10	4	no	—
270	C₂×C₅	—	2	2	no	—
271	C₂×C₅	—	4	2	no	—
272	C₂×C₅	C₂×#431	1	1	yes	characteristic
273	C₂×C₅	C₂×#504	1	1	yes	characteristic
274	C₂×C₅	#3407	1	1	yes	characteristic
275	C₂³	C₂²×#168	1	2	yes	—
276	C₂³	C₂²×#168	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	D₄	—	2	4	no	—
279	D₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₂²×#168	1	1	yes	characteristic
287	C₂×C₄	C₂²×#168	1	2	yes	—
288	C₂×C₄	C₂²×#168	1	2	yes	—
289	C₂×C₄	C₂²×#168	1	1	yes	characteristic
290	C₂×C₄	C₂²×#168	1	2	yes	—
291	C₂×C₄	C₂²×#168	1	1	yes	characteristic
292	C₂×C₄	C₂²×#168	1	1	yes	characteristic
293	C₂×C₄	C₂²×#168	1	1	yes	characteristic
294	C₂×C₄	C₂×#681	1	2	yes	—
295	C₂×C₄	#4072	1	2	yes	—
296	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	2	no	—
300	C₈	C₂×#681	1	4	yes	—
301	C₈	#4072	1	2	yes	—
302	C₈	C₂×#709	1	1	yes	characteristic
303	C₈	C₂×#709	1	1	yes	characteristic
304	C₅	#33423	1	1	yes	characteristic, complemented, Sylow
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	#65214	1	2	yes	—
311	C₂²	C₂³×#168	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	2	no	—
318	C₄	#65214	1	2	yes	—
319	C₄	#65429	1	2	yes	—
320	C₄	C₂²×#681	1	1	yes	characteristic
321	C₄	C₂×#4072	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	C₂×#65214	1	1	yes	characteristic
325	C₂	C₂×#65429	1	1	yes	characteristic
326	C₂	#10615511	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₄×C₅×#131	C₂
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₄×#131	D₅
hypocenter	C₅	#33423
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
3. #10600676	C₂	semidirect
4. #10600674	C₂	semidirect
5. #10600538	C₂	semidirect
6. #10600537	C₂	semidirect
7. #10600536	C₂	semidirect
8. C₂×C₄⋊#672	C₂	semidirect
304. C₅	#33423	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#131	2
#10602797	2
#10600676	2
#10600674	2
#10600538	2
#10600537	2
#10600536	2
C₂×C₄⋊#672	2
#33423	5