Subgroups — #12252493

Order 1280.

674 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	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	#10615517	C₂	1	1	yes	maximal, characteristic
3	C₂×#65190	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602797	C₂	1	1	yes	maximal, characteristic, complemented
5	#10602226	C₂	1	4	yes	maximal, complemented
6	#10602225	C₂	1	1	yes	maximal, characteristic, complemented
7	#10602213	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602200	C₂	1	4	yes	maximal, complemented
9	C₂×#64722	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#131	C₂²	1	1	yes	characteristic
11	#66166	C₄	1	1	yes	characteristic
12	C₂×#4047	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	—
16	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
17	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
18	#65523	C₄	1	1	yes	characteristic
19	#65418	C₄	1	1	yes	characteristic
20	#65417	C₄	1	4	yes	—
21	#65407	C₄	1	1	yes	characteristic
22	C₈×C₂²⋊D₅	C₂²	1	4	yes	—
23	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
24	C₂×#3944	C₂²	1	1	yes	characteristic
25	C₂²×#672	C₂²	1	1	yes	characteristic
26	#65190	—	2	4	no	—
27	C₂×#3930	C₂²	1	2	yes	—
28	C₂×D₅×C₁₆	C₂²	1	2	yes	—
29	C₂²⋊#672	C₂²	1	4	yes	—
30	#64727	C₂²	1	4	yes	—
31	#64722	C₂²	1	2	yes	—
32	#64722	C₂²	1	2	yes	—
33	#64721	C₂²	1	4	yes	—
34	#33435	—	5	1	no	maximal, Sylow
35	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
36	C₂×#709	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	#4093	C₈	1	1	yes	characteristic
44	#4090	C₈	1	1	yes	characteristic
45	#4086	C₈	1	1	yes	characteristic
46	#4085	C₈	1	4	yes	—
47	#4082	C₈	1	1	yes	characteristic
48	C₄×C₂²⋊D₅	C₂×C₄	1	4	yes	—
49	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
50	C₂×#681	C₂×C₄	1	1	yes	characteristic
51	C₂²×#168	C₂×C₄	1	1	yes	characteristic
52	#4047	—	2	2	no	—
53	#4047	—	2	2	no	—
54	C₂×#676	C₂×C₄	1	2	yes	—
55	C₂×C₈×D₅	C₂³	1	2	yes	—
56	C₅×M₃₂	—	2	2	no	—
57	C₅×M₃₂	—	2	4	no	—
58	C₂×C₅×C₁₆	—	2	4	no	—
59	C₂×C₅×C₁₆	D₄	1	4	yes	complemented
60	C₂×C₅×C₁₆	C₂³	1	2	yes	—
61	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
63	C₂²⋊#168	C₂×C₄	1	4	yes	—
64	#3952	C₂×C₄	1	4	yes	—
65	#3946	C₂×C₄	1	4	yes	—
66	C₈×Dic₅	C₂×C₄	1	2	yes	—
67	C₈×Dic₅	C₂³	1	2	yes	—
68	#3944	—	2	2	no	—
69	C₂×#672	—	2	2	no	—
70	C₂×#672	C₂³	1	2	yes	—
71	C₂×#672	C₂³	1	2	yes	—
72	#3930	—	2	4	no	—
73	D₅×C₁₆	—	2	4	no	—
74	C₂×#504	—	5	1	no	—
75	C₂×#504	D₅	1	1	yes	characteristic, complemented
76	#2931	—	5	1	no	—
77	#2168	—	5	4	no	—
78	#2168	—	5	4	no	—
79	#2150	—	5	1	no	—
80	#2113	—	5	1	no	—
81	#2113	—	5	1	no	—
82	C₂×#346	—	5	1	no	—
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	1	yes	characteristic
87	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
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	2	yes	—
91	C₂×C₂²⋊D₅	C₂×C₈	1	4	yes	—
92	C₂²×Dic₅	C₂×C₈	1	2	yes	—
93	#709	—	2	2	no	—
94	#709	—	2	2	no	—
95	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
96	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
97	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
98	C₅×M₁₆	—	2	2	no	—
99	C₅×M₁₆	—	2	4	no	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
103	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
105	C₂×C₅×C₈	#41	1	2	yes	—
106	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
107	C₂²⋊Dic₅	C₂×C₈	1	4	yes	—
108	#685	C₂×C₈	1	4	yes	—
109	#683	C₂×C₈	1	4	yes	—
110	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
111	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
112	C₄×Dic₅	C₂²×C₄	1	2	yes	—
113	#681	—	2	2	no	—
114	C₂×#168	—	2	2	no	—
115	C₂×#168	C₂²×C₄	1	2	yes	—
116	C₂×#168	C₂²×C₄	1	2	yes	—
117	#676	—	2	4	no	—
118	C₈×D₅	—	2	2	no	—
119	C₈×D₅	—	2	2	no	—
120	C₅×C₁₆	—	2	2	no	—
121	C₅×C₁₆	C₂×D₄	1	2	yes	—
122	C₅×C₁₆	C₂×D₄	1	2	yes	—
123	#672	—	2	2	no	—
124	#672	—	2	2	no	—
125	C₂×#131	—	5	1	no	—
126	C₂×#131	C₂×D₅	1	1	yes	characteristic
127	#532	—	5	1	no	—
128	#504	—	2	4	no	—
129	#504	—	10	4	no	—
130	C₂×M₃₂	—	5	1	no	—
131	C₂×M₃₂	—	5	2	no	—
132	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
133	C₂×M₃₂	C₂×D₅	1	2	yes	—
134	C₂²×C₁₆	—	5	1	no	—
135	C₂²×C₁₆	—	5	2	no	—
136	C₂²×C₁₆	C₂×D₅	1	2	yes	—
137	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
138	#436	—	5	4	no	—
139	C₈×D₄	—	5	4	no	—
140	#423	—	5	1	no	—
141	#408	—	5	1	no	—
142	#408	—	5	1	no	—
143	C₂×C₄×C₈	—	5	1	no	—
144	C₄⋊C₁₆	—	5	4	no	—
145	C₂²⋊C₁₆	—	5	4	no	—
146	C₂²⋊C₁₆	—	5	4	no	—
147	#346	—	5	2	no	—
148	#346	—	5	2	no	—
149	C₂³×C₅	C₂²×C₈	1	2	yes	—
150	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
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	—	2	4	no	—
163	C₂²⋊D₅	—	4	4	no	—
164	C₂×Dic₅	—	4	2	no	—
165	C₂×Dic₅	C₂²×C₈	1	2	yes	—
166	C₂×Dic₅	C₂²×C₈	1	2	yes	—
167	C₂×Dic₅	C₂²×C₈	1	2	yes	—
168	D₂₀	—	2	2	no	—
169	C₄×D₅	—	2	2	no	—
170	C₄×D₅	—	2	2	no	—
171	Dic₁₀	—	2	2	no	—
172	C₅×C₈	—	2	2	no	—
173	C₅×C₈	C₄×D₄	1	2	yes	—
174	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
175	C₅×C₈	C₂×#41	1	1	yes	characteristic
176	#168	—	2	2	no	—
177	#168	—	2	2	no	—
178	C₂×#41	—	5	1	no	—
179	C₂×#41	C₄×D₅	1	1	yes	characteristic
180	#131	—	2	2	no	—
181	#131	—	2	2	no	—
182	#131	—	10	2	no	—
183	#131	—	10	2	no	—
184	C₂×M₁₆	—	5	1	no	—
185	C₂×M₁₆	—	5	2	no	—
186	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
187	C₂×M₁₆	C₄×D₅	1	2	yes	—
188	C₂²×C₈	—	5	1	no	—
189	C₂²×C₈	—	5	2	no	—
190	C₂²×C₈	C₂²×D₅	1	2	yes	—
191	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
192	C₄⋊Q₈	—	5	1	no	—
193	#127	—	5	1	no	—
194	#124	—	5	1	no	—
195	#124	—	5	1	no	—
196	#121	—	5	4	no	—
197	C₄×D₄	—	5	4	no	—
198	C₂×C₄²	—	5	1	no	—
199	M₃₂	—	2	2	no	—
200	M₃₂	—	2	4	no	—
201	M₃₂	—	10	2	no	—
202	M₃₂	—	10	4	no	—
203	C₂×C₁₆	—	2	4	no	—
204	C₂×C₁₆	—	5	2	no	—
205	C₂×C₁₆	—	5	2	no	—
206	C₂×C₁₆	—	10	2	no	—
207	C₂×C₁₆	—	10	4	no	—
208	C₂×C₁₆	C₂²⋊D₅	1	4	yes	complemented
209	C₂×C₁₆	C₂²×D₅	1	2	yes	—
210	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
211	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
212	C₄⋊C₈	—	5	4	no	—
213	C₂²⋊C₈	—	5	4	no	—
214	C₂²⋊C₈	—	5	4	no	—
215	C₄×C₈	—	5	2	no	—
216	C₄×C₈	—	5	2	no	—
217	C₂²×C₅	—	2	2	no	—
218	C₂²×C₅	—	2	4	no	—
219	C₂²×C₅	—	2	2	no	—
220	C₂²×C₅	C₈×D₄	1	2	yes	—
221	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
222	C₂×D₅	—	2	4	no	—
223	C₂×D₅	—	2	2	no	—
224	C₄×C₅	—	2	2	no	—
225	C₄×C₅	C₈×D₄	1	2	yes	—
226	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
227	C₄×C₅	C₂×#131	1	1	yes	characteristic
228	Dic₅	—	2	2	no	—
229	Dic₅	—	2	2	no	—
230	Dic₅	—	2	2	no	—
231	#41	—	2	2	no	—
232	#41	—	2	2	no	—
233	#41	—	10	2	no	—
234	#41	—	10	2	no	—
235	C₂×Q₈	—	5	1	no	—
236	C₂×Q₈	C₈×D₅	1	1	yes	characteristic
237	C₂×D₄	—	5	2	no	—
238	C₂×D₄	—	5	4	no	—
239	C₂×D₄	—	5	1	no	—
240	C₂×D₄	C₈×D₅	1	2	yes	—
241	C₂×D₄	C₈×D₅	1	1	yes	characteristic
242	C₂²×C₄	—	5	2	no	—
243	C₂²×C₄	—	5	2	no	—
244	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
245	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
246	M₁₆	—	2	2	no	—
247	M₁₆	—	2	4	no	—
248	M₁₆	—	10	2	no	—
249	M₁₆	—	10	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	5	2	no	—
253	C₂×C₈	—	5	2	no	—
254	C₂×C₈	—	10	2	no	—
255	C₂×C₈	—	10	2	no	—
256	C₂×C₈	—	10	2	no	—
257	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
258	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
259	C₂×C₈	#709	1	2	yes	—
260	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
261	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
262	C₄⋊C₄	—	5	4	no	—
263	C₂²⋊C₄	—	5	4	no	—
264	C₂²⋊C₄	—	5	4	no	—
265	C₄²	—	5	2	no	—
266	C₄²	—	5	1	no	—
267	C₄²	—	5	1	no	—
268	C₁₆	—	2	2	no	—
269	C₁₆	—	10	2	no	—
270	C₁₆	—	10	2	no	—
271	C₁₆	C₂×C₂²⋊D₅	1	2	yes	—
272	C₁₆	C₂×C₂²⋊D₅	1	2	yes	—
273	C₂×C₅	—	2	2	no	—
274	C₂×C₅	—	4	2	no	—
275	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
276	C₂×C₅	#3407	1	2	yes	—
277	D₅	—	4	2	no	—
278	C₂³	—	5	2	no	—
279	C₂³	C₂×C₈×D₅	1	2	yes	—
280	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
281	Q₈	—	2	2	no	—
282	Q₈	—	10	2	no	—
283	D₄	—	2	4	no	—
284	D₄	—	2	2	no	—
285	D₄	—	10	4	no	—
286	D₄	—	10	2	no	—
287	D₄	—	20	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	5	2	no	—
291	C₂×C₄	—	5	2	no	—
292	C₂×C₄	—	5	2	no	—
293	C₂×C₄	—	10	2	no	—
294	C₂×C₄	—	10	2	no	—
295	C₂×C₄	—	20	2	no	—
296	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
297	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
298	C₂×C₄	#4047	1	2	yes	—
299	C₂×C₄	C₄×C₂²⋊D₅	1	2	yes	—
300	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
301	C₈	—	2	2	no	—
302	C₈	—	10	2	no	—
303	C₈	—	10	2	no	—
304	C₈	C₄×C₂²⋊D₅	1	2	yes	—
305	C₈	C₂×#709	1	1	yes	characteristic
306	C₈	C₂²×C₂²⋊D₅	1	1	yes	characteristic
307	C₅	#33435	1	1	yes	characteristic, complemented, Sylow
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	10	2	no	—
312	C₂²	—	10	4	no	—
313	C₂²	C₈×C₂²⋊D₅	1	2	yes	—
314	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	10	2	no	—
317	C₄	—	10	2	no	—
318	C₄	—	10	2	no	—
319	C₄	C₈×C₂²⋊D₅	1	2	yes	—
320	C₄	C₂×#4047	1	1	yes	characteristic
321	C₄	C₂×C₄×C₂²⋊D₅	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	—	20	2	no	—
325	C₂	#10613743	1	2	yes	—
326	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	#66166	C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×#504	D₅
hypocenter	C₅	#33435
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. C₂×#65190	C₂	semidirect
4. #10602797	C₂	semidirect
5. #10602226	C₂	semidirect
6. #10602225	C₂	semidirect
7. #10602213	C₂	semidirect
8. #10602200	C₂	semidirect
9. C₂×#64722	C₂	semidirect
59. C₂×C₅×C₁₆	D₄	semidirect
75. C₂×#504	D₅	semidirect
208. C₂×C₁₆	C₂²⋊D₅	semidirect
307. C₅	#33435	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
#10615517	2
C₂×#65190	2
#10602797	2
#10602226	2
#10602225	2
#10602213	2
#10602200	2
C₂×#64722	2
#33435	5