Subgroups — #12252409

Order 1280.

600 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	C₂×#65390	C₂	1	1	yes	maximal, characteristic
2	C₅×#2112	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊#672	C₂	1	1	yes	maximal, characteristic
4	#10602214	C₂	1	2	yes	maximal, complemented
5	#10602212	C₂	1	2	yes	maximal, complemented
6	#10599887	C₂	1	4	yes	maximal, complemented
7	#10599886	C₂	1	2	yes	maximal, complemented
8	#10599885	C₂	1	2	yes	maximal, complemented
9	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×#4072	C₄	1	1	yes	characteristic
11	C₂²×#681	C₄	1	1	yes	characteristic
12	C₂×C₅×M₃₂	C₂²	1	2	yes	—
13	#65390	C₄	1	1	yes	characteristic
14	#65390	C₄	1	2	yes	—
15	#65390	C₄	1	1	yes	characteristic
16	#65390	C₂²	1	2	yes	—
17	#65390	C₂²	1	1	yes	characteristic
18	#65390	C₂²	1	1	yes	characteristic
19	C₂×#3947	C₄	1	2	yes	—
20	C₂×C₈×Dic₅	C₂²	1	2	yes	—
21	C₂²×#672	C₂²	1	2	yes	—
22	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
23	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
24	C₂²⋊#672	C₂²	1	1	yes	characteristic
25	C₂²⋊#672	C₂²	1	1	yes	characteristic
26	C₂²⋊#672	C₂²	1	2	yes	—
27	#64722	C₂²	1	4	yes	complemented
28	#64721	C₂²	1	4	yes	complemented
29	#64721	C₂²	1	4	yes	complemented
30	C₁₆×Dic₅	C₂²	1	4	yes	complemented
31	#33422	—	5	1	no	maximal, Sylow
32	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	—	2	4	no	—
34	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊Dic₅	C₈	1	2	yes	—
41	#4072	C₈	1	2	yes	—
42	#4072	C₈	1	1	yes	characteristic
43	#4072	C₈	1	1	yes	characteristic
44	#4072	C₂×C₄	1	2	yes	—
45	#4072	C₂×C₄	1	1	yes	characteristic
46	#4072	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊Dic₅	C₈	1	1	yes	characteristic
48	C₂×C₄⋊Dic₅	C₈	1	1	yes	characteristic
49	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
50	C₂×#681	—	2	4	no	—
51	C₂×#681	C₂×C₄	1	1	yes	characteristic
52	C₂×#681	C₂×C₄	1	2	yes	—
53	C₂×#681	C₂×C₄	1	1	yes	characteristic
54	C₂²×#168	C₂×C₄	1	2	yes	—
55	C₅×M₃₂	—	2	4	no	—
56	C₂×C₅×C₁₆	C₂³	1	4	yes	—
57	#3947	C₂×C₄	1	2	yes	—
58	#3947	C₂×C₄	1	2	yes	—
59	#3947	C₂×C₄	1	2	yes	—
60	#3947	C₂×C₄	1	2	yes	—
61	C₈×Dic₅	C₂×C₄	1	2	yes	—
62	C₈×Dic₅	C₂×C₄	1	2	yes	—
63	C₈×Dic₅	C₂³	1	2	yes	—
64	C₈×Dic₅	C₂³	1	2	yes	—
65	C₂×#672	—	2	4	no	—
66	C₂×#672	C₂³	1	2	yes	—
67	C₂×#672	C₂³	1	2	yes	—
68	C₂×#405	—	5	1	no	—
69	#2164	—	5	4	no	—
70	#2163	—	5	2	no	—
71	#2162	—	5	2	no	—
72	#2151	—	5	2	no	—
73	#2112	D₅	1	1	yes	characteristic
74	C₂×C₂²⋊C₁₆	—	5	1	no	—
75	#2108	—	5	2	no	—
76	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₅	—	2	2	no	—
78	C₂²×C₄×C₅	—	2	2	no	—
79	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂²×Dic₅	C₂×C₈	1	2	yes	—
86	C₂²×Dic₅	C₂×C₈	1	2	yes	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	—	2	2	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₅×C₈	#41	1	4	yes	—
96	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
97	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
98	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
99	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
100	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
101	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
102	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
103	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
104	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
105	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
106	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
107	C₄×Dic₅	C₂×C₈	1	2	yes	—
108	C₄×Dic₅	C₂×C₈	1	2	yes	—
109	C₄×Dic₅	C₂²×C₄	1	2	yes	—
110	C₄×Dic₅	C₂²×C₄	1	2	yes	—
111	#681	—	2	2	no	—
112	#681	—	2	4	no	—
113	#681	M₁₆	1	4	yes	—
114	C₂×#168	—	2	4	no	—
115	C₂×#168	C₂²×C₄	1	2	yes	—
116	C₂×#168	C₂²×C₄	1	2	yes	—
117	C₅×C₁₆	—	2	4	no	—
118	#672	—	2	2	no	—
119	#672	—	2	2	no	—
120	C₂²×M₁₆	—	5	1	no	—
121	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
122	C₂×#117	—	5	1	no	—
123	C₂×M₃₂	C₂×D₅	1	2	yes	—
124	C₂²×C₁₆	—	5	2	no	—
125	#405	—	5	1	no	—
126	#405	—	5	2	no	—
127	#405	—	5	1	no	—
128	#405	—	5	1	no	—
129	#405	—	5	1	no	—
130	#405	—	5	2	no	—
131	C₂×#97	—	5	2	no	—
132	C₂×C₄×C₈	—	5	2	no	—
133	C₄⋊C₁₆	—	5	4	no	—
134	C₄⋊C₁₆	—	5	4	no	—
135	C₂²⋊C₁₆	—	5	1	no	—
136	C₂²⋊C₁₆	—	5	1	no	—
137	C₂²⋊C₁₆	—	5	2	no	—
138	C₂²⋊C₁₆	C₂×D₅	1	2	yes	—
139	C₂²⋊C₁₆	C₂×D₅	1	2	yes	—
140	#346	—	5	4	no	—
141	C₄×C₁₆	—	5	4	no	—
142	C₂³×C₅	—	2	2	no	—
143	C₂³×C₅	—	2	2	no	—
144	C₂³×C₅	—	4	1	no	—
145	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
146	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
147	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
148	C₂×C₄×C₅	—	2	2	no	—
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	—	2	2	no	—
155	C₂×C₄×C₅	#117	1	2	yes	—
156	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
160	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
161	C₂×Dic₅	—	2	4	no	—
162	C₂×Dic₅	—	4	2	no	—
163	C₂×Dic₅	C₂²×C₈	1	2	yes	—
164	C₂×Dic₅	C₂²×C₈	1	2	yes	—
165	C₂×Dic₅	C₂²×C₈	1	2	yes	—
166	C₂×Dic₅	C₂²×C₈	1	2	yes	—
167	C₅×C₈	—	2	1	no	—
168	C₅×C₈	—	2	1	no	—
169	C₅×C₈	#117	1	2	yes	—
170	C₅×C₈	C₂×#41	1	2	yes	—
171	#168	—	2	2	no	—
172	#168	C₂×M₁₆	1	4	yes	—
173	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
174	C₂×M₁₆	—	5	1	no	—
175	C₂×M₁₆	—	5	2	no	—
176	C₂×M₁₆	—	5	1	no	—
177	C₂×M₁₆	—	10	4	no	—
178	C₂²×C₈	—	2	4	no	—
179	C₂²×C₈	—	5	2	no	—
180	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
181	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
182	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
186	#117	—	5	1	no	—
187	#117	—	5	1	no	—
188	#117	—	5	2	no	—
189	#117	—	5	2	no	—
190	#117	—	5	1	no	—
191	#117	—	5	1	no	—
192	C₂×C₄⋊C₄	—	5	1	no	—
193	C₂×C₄⋊C₄	—	5	1	no	—
194	C₂×C₂²⋊C₄	—	5	2	no	—
195	C₂×C₄²	—	5	2	no	—
196	M₃₂	—	2	4	no	—
197	C₂×C₁₆	—	5	2	no	—
198	C₂×C₁₆	—	5	2	no	—
199	C₂×C₁₆	—	10	4	no	—
200	C₂×C₁₆	C₂²×D₅	1	4	yes	—
201	#97	—	5	2	no	—
202	#97	—	5	2	no	—
203	#97	—	5	2	no	—
204	#97	—	5	2	no	—
205	C₄×C₈	—	5	2	no	—
206	C₄×C₈	—	5	2	no	—
207	C₄×C₈	—	5	2	no	—
208	C₄×C₈	—	5	2	no	—
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	1	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	—	2	1	no	—
214	C₂²×C₅	—	2	1	no	—
215	C₂²×C₅	—	2	1	no	—
216	C₂²×C₅	—	4	4	no	—
217	C₂²×C₅	#431	1	2	yes	—
218	C₂²×C₅	#504	1	2	yes	—
219	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	—	2	1	no	—
222	C₄×C₅	#431	1	2	yes	—
223	C₄×C₅	C₂×#117	1	1	yes	characteristic
224	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
225	Dic₅	—	2	2	no	—
226	Dic₅	—	2	2	no	—
227	Dic₅	—	2	2	no	—
228	Dic₅	—	2	2	no	—
229	C₂⁴	C₈×D₅	1	1	yes	characteristic
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	5	2	no	—
233	C₂²×C₄	—	5	2	no	—
234	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
235	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
236	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
240	M₁₆	—	5	4	no	—
241	M₁₆	—	10	4	no	—
242	M₁₆	—	10	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	5	2	no	—
249	C₂×C₈	—	5	2	no	—
250	C₂×C₈	—	10	4	no	—
251	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
252	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
253	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
254	C₂×C₈	#711	1	4	yes	—
255	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
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	C₄⋊C₄	—	5	1	no	—
261	C₄⋊C₄	—	5	1	no	—
262	C₂²⋊C₄	—	5	2	no	—
263	C₂²⋊C₄	—	5	2	no	—
264	C₂²⋊C₄	—	5	2	no	—
265	C₂²⋊C₄	—	5	2	no	—
266	C₄²	—	5	2	no	—
267	C₄²	—	5	2	no	—
268	C₄²	—	5	2	no	—
269	C₄²	—	5	2	no	—
270	C₁₆	—	2	4	no	—
271	C₁₆	—	10	2	no	—
272	C₁₆	—	10	2	no	—
273	C₂×C₅	—	2	2	no	—
274	C₂×C₅	—	2	2	no	—
275	C₂×C₅	—	4	1	no	—
276	C₂×C₅	C₂×#431	1	1	yes	characteristic
277	C₂×C₅	C₂×#504	1	1	yes	characteristic
278	C₂×C₅	#3407	1	1	yes	characteristic
279	C₂³	—	2	2	no	—
280	C₂³	—	2	2	no	—
281	C₂³	—	4	1	no	—
282	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
283	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
284	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	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	—	5	2	no	—
296	C₂×C₄	—	10	4	no	—
297	C₂×C₄	—	20	2	no	—
298	C₂×C₄	#4023	1	2	yes	—
299	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
300	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
301	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
302	C₂×C₄	D₅×M₁₆	1	2	yes	—
303	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	5	4	no	—
307	C₈	—	10	2	no	—
308	C₈	#4023	1	2	yes	—
309	C₈	C₂×#711	1	2	yes	—
310	C₅	#33422	1	1	yes	characteristic, complemented, Sylow
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	4	no	—
319	C₂²	#65007	1	2	yes	—
320	C₂²	#65196	1	2	yes	—
321	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	10	2	no	—
325	C₄	—	10	2	no	—
326	C₄	—	10	2	no	—
327	C₄	—	10	2	no	—
328	C₄	#65007	1	2	yes	—
329	C₄	C₂×#4023	1	1	yes	characteristic
330	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	—	2	2	no	—
333	C₂	—	4	1	no	—
334	C₂	C₂×#65007	1	1	yes	characteristic
335	C₂	C₂×#65196	1	1	yes	characteristic
336	C₂	#10613738	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#2112	C₂
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2112	D₅
hypocenter	C₅	#33422
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ #2112

Complements

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

N	H ≅ G/N	type
4. #10602214	C₂	semidirect
5. #10602212	C₂	semidirect
6. #10599887	C₂	semidirect
7. #10599886	C₂	semidirect
8. #10599885	C₂	semidirect
27. #64722	C₂²	semidirect
28. #64721	C₂²	semidirect
29. #64721	C₂²	semidirect
30. C₁₆×Dic₅	C₂²	semidirect
310. C₅	#33422	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65390	2
C₅×#2112	2
C₂×C₂²⋊#672	2
#10602214	2
#10602212	2
#10599887	2
#10599886	2
#10599885	2
#33422	5