Subgroups — #12252478

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	#10615525	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3930	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602797	C₂	1	1	yes	maximal, characteristic, complemented
5	#10602224	C₂	1	1	yes	maximal, characteristic, complemented
6	#10602223	C₂	1	4	yes	maximal, complemented
7	#10602214	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602200	C₂	1	4	yes	maximal, complemented
9	#10602192	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#131	C₂²	1	1	yes	characteristic
11	#66162	C₄	1	1	yes	characteristic
12	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
13	C₅×#504	—	2	4	no	—
14	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₅×M₃₂	C₂²	1	2	yes	complemented
16	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₅×C₁₆	C₂²	1	2	yes	complemented
18	#65523	C₄	1	1	yes	characteristic
19	#65416	C₄	1	4	yes	—
20	#65415	C₄	1	1	yes	characteristic
21	#65408	C₄	1	1	yes	characteristic
22	C₈×C₂²⋊D₅	C₂²	1	4	yes	—
23	#65390	C₂²	1	1	yes	characteristic
24	C₂×#3944	C₂²	1	1	yes	characteristic
25	C₂²×#672	C₂²	1	1	yes	characteristic
26	C₂×#3930	—	2	4	no	—
27	C₂×#3930	C₂²	1	2	yes	—
28	C₂×#3930	C₂²	1	2	yes	—
29	C₂²⋊#672	C₂²	1	4	yes	complemented
30	#64727	C₂²	1	4	yes	—
31	#64722	C₂²	1	2	yes	complemented
32	#64721	C₂²	1	4	yes	complemented
33	C₁₆×Dic₅	C₂²	1	2	yes	complemented
34	#33433	—	5	1	no	maximal, Sylow
35	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
37	C₅×#131	—	2	2	no	—
38	C₅×#131	—	2	2	no	—
39	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
41	C₂²×C₅×C₈	C₂³	1	2	yes	—
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	#4092	C₈	1	2	yes	—
44	#4084	C₈	1	2	yes	—
45	#4083	C₈	1	2	yes	—
46	#4081	C₈	1	2	yes	—
47	C₄×C₂²⋊D₅	C₂×C₄	1	4	yes	—
48	#4072	C₂×C₄	1	1	yes	characteristic
49	C₂×#681	C₂×C₄	1	1	yes	characteristic
50	C₂²×#168	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₅	—	2	2	no	—
52	C₂×C₈×D₅	—	2	2	no	—
53	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
54	C₂×C₈×D₅	C₂³	1	2	yes	—
55	C₅×M₃₂	—	2	2	no	—
56	C₅×M₃₂	—	2	4	no	—
57	C₂×C₅×C₁₆	—	2	4	no	—
58	C₂×C₅×C₁₆	D₄	1	4	yes	complemented
59	C₂×C₅×C₁₆	C₂³	1	2	yes	—
60	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
62	C₂²⋊#168	C₂×C₄	1	4	yes	—
63	#3952	C₂×C₄	1	4	yes	—
64	#3947	C₂×C₄	1	2	yes	—
65	#3946	C₂×C₄	1	4	yes	—
66	C₈×Dic₅	C₂³	1	2	yes	—
67	#3944	—	2	2	no	—
68	C₂×#672	—	2	2	no	—
69	C₂×#672	C₂³	1	2	yes	—
70	C₂×#672	C₂³	1	2	yes	—
71	#3930	—	2	4	no	—
72	#3930	—	2	4	no	—
73	C₂×#504	D₅	1	1	yes	characteristic, complemented
74	C₂²×M₃₂	—	5	1	no	—
75	#2929	—	5	1	no	—
76	#2169	—	5	4	no	—
77	#2168	—	5	4	no	—
78	#2151	—	5	1	no	—
79	#2113	—	5	1	no	—
80	#2112	—	5	1	no	—
81	#2108	—	5	1	no	—
82	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
83	C₅×#41	—	2	2	no	—
84	C₅×#41	—	2	2	no	—
85	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
86	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
87	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₂²⋊D₅	C₂×C₈	1	4	yes	—
91	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
92	C₂×C₄×D₅	—	2	2	no	—
93	C₂×C₄×D₅	—	2	2	no	—
94	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
95	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
96	C₅×M₁₆	—	2	2	no	—
97	C₅×M₁₆	—	2	4	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
101	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
103	C₂×C₅×C₈	#41	1	2	yes	—
104	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
105	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
106	C₂²⋊Dic₅	C₂×C₈	1	4	yes	—
107	#685	C₂×C₈	1	4	yes	—
108	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
109	#683	C₂×C₈	1	4	yes	—
110	C₄×Dic₅	C₂²×C₄	1	2	yes	—
111	#681	—	2	2	no	—
112	C₂×#168	—	2	2	no	—
113	C₂×#168	C₂²×C₄	1	2	yes	—
114	C₂×#168	C₂²×C₄	1	2	yes	—
115	C₈×D₅	—	2	4	no	—
116	C₈×D₅	—	2	2	no	—
117	C₈×D₅	—	2	2	no	—
118	C₅×C₁₆	—	2	2	no	—
119	C₅×C₁₆	C₂×D₄	1	2	yes	—
120	C₅×C₁₆	C₂×D₄	1	2	yes	—
121	#672	—	2	2	no	—
122	#672	—	2	2	no	—
123	C₂×#131	C₂×D₅	1	1	yes	characteristic
124	C₂³×C₈	—	5	1	no	—
125	#525	—	5	1	no	—
126	#504	—	2	4	no	—
127	C₂×M₃₂	—	5	1	no	—
128	C₂×M₃₂	—	5	2	no	—
129	C₂×M₃₂	—	5	2	no	—
130	C₂×M₃₂	—	10	4	no	—
131	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
132	C₂×M₃₂	C₂×D₅	1	2	yes	complemented
133	C₂²×C₁₆	—	5	1	no	—
134	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
135	C₂²×C₁₆	C₂×D₅	1	2	yes	complemented
136	#435	—	5	4	no	—
137	C₈×D₄	—	5	4	no	—
138	#424	—	5	1	no	—
139	#408	—	5	1	no	—
140	#407	—	5	1	no	—
141	#405	—	5	1	no	—
142	C₄⋊C₁₆	—	5	4	no	—
143	C₂²⋊C₁₆	—	5	4	no	—
144	C₂²⋊C₁₆	—	5	4	no	—
145	#346	—	5	2	no	—
146	C₄×C₁₆	—	5	2	no	—
147	C₂³×C₅	C₂²×C₈	1	2	yes	—
148	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
149	C₂²×D₅	—	2	2	no	—
150	C₂²×D₅	—	4	2	no	—
151	C₂²×D₅	C₂²×C₈	1	2	yes	—
152	C₅×Q₈	—	2	2	no	—
153	C₅×D₄	—	2	4	no	—
154	C₅×D₄	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
158	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
159	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
160	C₂×C₄×C₅	#131	1	2	yes	—
161	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
162	C₂²⋊D₅	—	4	4	no	—
163	C₂×Dic₅	—	2	2	no	—
164	C₂×Dic₅	C₂²×C₈	1	4	yes	—
165	C₂×Dic₅	C₂²×C₈	1	2	yes	—
166	C₄×D₅	—	2	2	no	—
167	C₄×D₅	—	2	2	no	—
168	C₄×D₅	—	2	4	no	—
169	C₅×C₈	—	2	2	no	—
170	C₅×C₈	C₄×D₄	1	2	yes	—
171	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
172	C₅×C₈	C₂×#41	1	1	yes	characteristic
173	#168	—	2	2	no	—
174	#168	—	2	2	no	—
175	C₂×#41	C₄×D₅	1	1	yes	characteristic
176	C₂³×C₄	—	5	1	no	—
177	#131	—	2	2	no	—
178	#131	—	2	2	no	—
179	C₂×M₁₆	—	5	1	no	—
180	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
181	C₂×M₁₆	C₄×D₅	1	2	yes	—
182	C₂²×C₈	—	5	1	no	—
183	C₂²×C₈	—	5	2	no	—
184	C₂²×C₈	—	5	2	no	—
185	C₂²×C₈	—	10	2	no	—
186	C₂²×C₈	—	10	2	no	—
187	C₂²×C₈	C₂²×D₅	1	2	yes	—
188	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
189	#123	—	5	2	no	—
190	C₂²⋊Q₈	—	5	2	no	—
191	#121	—	5	2	no	—
192	#120	—	5	2	no	—
193	C₄×D₄	—	5	4	no	—
194	#117	—	5	1	no	—
195	M₃₂	—	2	2	no	—
196	M₃₂	—	2	4	no	—
197	M₃₂	—	10	2	no	—
198	M₃₂	—	10	4	no	—
199	M₃₂	—	10	4	no	—
200	C₂×C₁₆	—	2	4	no	—
201	C₂×C₁₆	—	5	2	no	—
202	C₂×C₁₆	—	5	2	no	—
203	C₂×C₁₆	—	10	2	no	—
204	C₂×C₁₆	C₂²⋊D₅	1	4	yes	complemented
205	C₂×C₁₆	C₂²×D₅	1	2	yes	—
206	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
207	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
208	C₄⋊C₈	—	5	4	no	—
209	C₂²⋊C₈	—	5	4	no	—
210	C₂²⋊C₈	—	5	4	no	—
211	#97	—	5	2	no	—
212	C₄×C₈	—	5	2	no	—
213	C₂²×C₅	—	2	2	no	—
214	C₂²×C₅	—	2	4	no	—
215	C₂²×C₅	—	2	2	no	—
216	C₂²×C₅	C₈×D₄	1	2	yes	—
217	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
218	C₂×D₅	—	2	2	no	—
219	C₂×D₅	—	2	2	no	—
220	C₂×D₅	—	2	2	no	—
221	C₂×D₅	—	4	4	no	—
222	C₄×C₅	—	2	2	no	—
223	C₄×C₅	C₈×D₄	1	2	yes	—
224	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
225	C₄×C₅	C₂×#131	1	1	yes	characteristic
226	Dic₅	—	2	4	no	—
227	Dic₅	—	2	2	no	—
228	C₂⁴	—	5	1	no	—
229	#41	—	2	2	no	—
230	#41	—	2	2	no	—
231	C₂×Q₈	C₈×D₅	1	1	yes	characteristic
232	C₂×D₄	—	5	4	no	—
233	C₂×D₄	C₈×D₅	1	2	yes	—
234	C₂×D₄	C₈×D₅	1	1	yes	characteristic
235	C₂²×C₄	—	5	1	no	—
236	C₂²×C₄	—	5	2	no	—
237	C₂²×C₄	—	5	2	no	—
238	C₂²×C₄	—	10	2	no	—
239	C₂²×C₄	—	10	2	no	—
240	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
241	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
242	M₁₆	—	2	2	no	—
243	M₁₆	—	2	4	no	—
244	M₁₆	—	10	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	5	2	no	—
248	C₂×C₈	—	5	2	no	—
249	C₂×C₈	—	10	2	no	—
250	C₂×C₈	—	10	2	no	—
251	C₂×C₈	—	10	2	no	—
252	C₂×C₈	—	10	4	no	—
253	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
254	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
255	C₂×C₈	#709	1	2	yes	—
256	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
257	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
258	C₄⋊C₄	—	5	2	no	—
259	C₄⋊C₄	—	5	4	no	—
260	C₂²⋊C₄	—	5	2	no	—
261	C₂²⋊C₄	—	5	4	no	—
262	C₂²⋊C₄	—	5	4	no	—
263	C₄²	—	5	2	no	—
264	C₁₆	—	2	2	no	—
265	C₁₆	—	10	2	no	—
266	C₁₆	—	10	2	no	—
267	C₁₆	C₂×C₂²⋊D₅	1	2	yes	—
268	C₁₆	C₂×C₂²⋊D₅	1	2	yes	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	—	4	2	no	—
271	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
272	C₂×C₅	#3407	1	2	yes	—
273	D₅	—	4	2	no	—
274	C₂³	—	5	2	no	—
275	C₂³	—	10	2	no	—
276	C₂³	—	20	2	no	—
277	C₂³	C₂×C₈×D₅	1	2	yes	—
278	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
279	Q₈	—	2	2	no	—
280	D₄	—	2	4	no	—
281	D₄	—	2	2	no	—
282	D₄	—	20	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	5	4	no	—
286	C₂×C₄	—	5	2	no	—
287	C₂×C₄	—	10	2	no	—
288	C₂×C₄	—	10	2	no	—
289	C₂×C₄	—	10	2	no	—
290	C₂×C₄	—	10	4	no	—
291	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
292	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
293	C₂×C₄	#4047	1	2	yes	—
294	C₂×C₄	C₄×C₂²⋊D₅	1	2	yes	—
295	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	10	2	no	—
298	C₈	—	10	2	no	—
299	C₈	C₄×C₂²⋊D₅	1	2	yes	—
300	C₈	C₂×#709	1	1	yes	characteristic
301	C₈	C₂²×C₂²⋊D₅	1	1	yes	characteristic
302	C₅	#33433	1	1	yes	characteristic, complemented, Sylow
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	10	2	no	—
307	C₂²	—	10	2	no	—
308	C₂²	—	10	2	no	—
309	C₂²	—	20	4	no	—
310	C₂²	C₈×C₂²⋊D₅	1	2	yes	—
311	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	10	2	no	—
314	C₄	—	10	4	no	—
315	C₄	C₈×C₂²⋊D₅	1	2	yes	—
316	C₄	C₂×#4047	1	1	yes	characteristic
317	C₄	C₂×C₄×C₂²⋊D₅	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	2	no	—
320	C₂	—	20	2	no	—
321	C₂	#10613743	1	2	yes	—
322	C₂	C₂×C₈×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₂×C₅×#504	C₂
prime radical	#66162	C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×#504	D₅
hypocenter	C₅	#33433
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#504	C₂	semidirect
3. C₂²×#3930	C₂	semidirect
4. #10602797	C₂	semidirect
5. #10602224	C₂	semidirect
6. #10602223	C₂	semidirect
7. #10602214	C₂	semidirect
8. #10602200	C₂	semidirect
9. #10602192	C₂	semidirect
15. C₂×C₅×M₃₂	C₂²	semidirect
17. C₂²×C₅×C₁₆	C₂²	semidirect
29. C₂²⋊#672	C₂²	semidirect
31. #64722	C₂²	semidirect
32. #64721	C₂²	semidirect
33. C₁₆×Dic₅	C₂²	semidirect
58. C₂×C₅×C₁₆	D₄	semidirect
73. C₂×#504	D₅	semidirect
132. C₂×M₃₂	C₂×D₅	semidirect
135. C₂²×C₁₆	C₂×D₅	semidirect
204. C₂×C₁₆	C₂²⋊D₅	semidirect
302. C₅	#33433	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#504	2
#10615525	2
C₂²×#3930	2
#10602797	2
#10602224	2
#10602223	2
#10602214	2
#10602200	2
#10602192	2
#33433	5