Subgroups — #12252482

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×#65190	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#3930	C₂	1	1	yes	maximal, characteristic, complemented
3	#10610284	C₂	1	1	yes	maximal, characteristic
4	C₅×#2108	C₂	1	1	yes	maximal, characteristic, complemented
5	#10602225	C₂	1	1	yes	maximal, characteristic, complemented
6	#10602224	C₂	1	1	yes	maximal, characteristic, complemented
7	#10600675	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599868	C₂	1	4	yes	maximal, complemented
9	#10599848	C₂	1	4	yes	maximal, complemented
10	C₂×#4047	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
12	#65858	C₄	1	1	yes	characteristic
13	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₅×#405	C₂²	1	1	yes	characteristic
16	#65418	C₄	1	1	yes	characteristic
17	#65415	C₄	1	1	yes	characteristic
18	C₂×#3944	C₂²	1	1	yes	characteristic
19	C₂²×#672	C₂²	1	1	yes	characteristic
20	#65288	C₄	1	1	yes	characteristic
21	#65190	—	2	4	no	—
22	C₂×#3930	—	2	4	no	—
23	C₂×#3930	C₂²	1	2	yes	—
24	C₂×#3930	C₂²	1	2	yes	—
25	C₂×#3930	C₂²	1	2	yes	complemented
26	C₂×D₅×C₁₆	C₂²	1	2	yes	complemented
27	#64995	C₄	1	4	yes	—
28	C₈×D₂₀	C₂²	1	4	yes	—
29	C₅×#346	C₂²	1	2	yes	complemented
30	C₄×C₅×C₁₆	C₂²	1	2	yes	complemented
31	#64727	C₂²	1	4	yes	—
32	#64727	C₂²	1	4	yes	complemented
33	C₄⋊#672	C₂²	1	4	yes	complemented
34	#33433	—	5	1	no	maximal, Sylow
35	C₂×#709	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₅×#117	C₂×C₄	1	1	yes	characteristic
40	C₂×#681	C₂×C₄	1	1	yes	characteristic
41	C₂²×#168	C₂×C₄	1	1	yes	characteristic
42	#4047	—	2	2	no	—
43	#4047	—	2	2	no	—
44	C₂×#676	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₅	—	2	2	no	—
46	C₂×C₈×D₅	—	2	2	no	—
47	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₅	C₂³	1	2	yes	—
49	C₂×C₈×D₅	C₂³	1	2	yes	—
50	#4043	C₈	1	2	yes	—
51	#4041	C₈	1	2	yes	—
52	#4032	C₈	1	2	yes	—
53	#4028	C₈	1	2	yes	—
54	C₄×D₂₀	C₂×C₄	1	4	yes	—
55	C₅×M₃₂	—	2	2	no	—
56	C₂×C₅×C₁₆	D₄	1	4	yes	complemented
57	C₂×C₅×C₁₆	C₂³	1	2	yes	—
58	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
60	C₅×#97	C₂×C₄	1	2	yes	—
61	C₄×C₅×C₈	C₂³	1	2	yes	—
62	#3952	C₂×C₄	1	4	yes	—
63	#3952	C₂×C₄	1	4	yes	—
64	#3944	—	2	2	no	—
65	C₂×#672	—	2	2	no	—
66	C₂×#672	C₂³	1	2	yes	—
67	C₂×#672	C₂³	1	2	yes	—
68	C₄⋊#168	C₂×C₄	1	4	yes	—
69	#3930	—	2	4	no	—
70	#3930	—	2	4	no	—
71	#3930	—	2	4	no	—
72	D₅×C₁₆	—	2	4	no	—
73	C₂×#504	—	5	1	no	—
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	D₅	1	1	yes	characteristic, complemented
82	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
85	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
86	#709	—	2	2	no	—
87	#709	—	2	2	no	—
88	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
89	C₂×D₂₀	C₂×C₈	1	4	yes	—
90	C₂×C₄×D₅	—	2	2	no	—
91	C₂×C₄×D₅	—	2	2	no	—
92	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
93	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
95	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
96	C₅×M₁₆	—	2	2	no	—
97	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
100	C₂×C₅×C₈	#41	1	2	yes	—
101	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
102	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
103	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
104	C₄²×C₅	C₂²×C₄	1	2	yes	—
105	#685	C₂×C₈	1	4	yes	—
106	#685	C₂×C₈	1	4	yes	—
107	C₄⋊Dic₅	C₂×C₈	1	4	yes	—
108	#681	—	2	2	no	—
109	C₂×#168	—	2	2	no	—
110	C₂×#168	C₂²×C₄	1	2	yes	—
111	C₂×#168	C₂²×C₄	1	2	yes	—
112	#676	—	2	4	no	—
113	C₈×D₅	—	2	4	no	—
114	C₈×D₅	—	2	2	no	—
115	C₈×D₅	—	2	2	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	—	5	1	no	—
124	C₂³×C₈	—	5	1	no	—
125	#525	—	5	1	no	—
126	#504	—	10	4	no	—
127	C₂×M₃₂	—	5	1	no	—
128	C₂×M₃₂	—	5	2	no	—
129	C₂×M₃₂	—	5	2	no	—
130	C₂×M₃₂	—	5	2	no	—
131	C₂×M₃₂	—	10	4	no	—
132	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
133	C₂²×C₁₆	—	5	1	no	—
134	C₂²×C₁₆	—	5	2	no	—
135	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
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	C₂×D₅	1	1	yes	characteristic
142	C₄⋊C₁₆	—	5	4	no	—
143	C₂²⋊C₁₆	—	5	4	no	—
144	C₂²⋊C₁₆	—	5	4	no	—
145	#346	C₂×D₅	1	2	yes	complemented
146	C₄×C₁₆	C₂×D₅	1	2	yes	complemented
147	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
148	C₂²×D₅	—	2	2	no	—
149	C₂²×D₅	—	4	2	no	—
150	C₂²×D₅	C₂²×C₈	1	2	yes	—
151	C₂²×D₅	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
153	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
154	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
155	C₂×C₄×C₅	#131	1	2	yes	—
156	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
157	C₂²⋊D₅	—	2	4	no	—
158	C₂×Dic₅	—	2	2	no	—
159	C₂×Dic₅	C₂²×C₈	1	2	yes	—
160	C₂×Dic₅	C₂²×C₈	1	2	yes	—
161	D₂₀	—	2	2	no	—
162	D₂₀	—	4	4	no	—
163	C₄×D₅	—	2	2	no	—
164	C₄×D₅	—	2	2	no	—
165	C₄×D₅	—	2	2	no	—
166	C₄×D₅	—	2	2	no	—
167	C₄×D₅	—	2	4	no	—
168	Dic₁₀	—	2	2	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	—	5	1	no	—
176	C₂³×C₄	—	5	1	no	—
177	#131	—	10	2	no	—
178	#131	—	10	2	no	—
179	C₂×M₁₆	—	5	1	no	—
180	C₂×M₁₆	—	5	2	no	—
181	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
182	C₂²×C₈	—	5	1	no	—
183	C₂²×C₈	—	5	2	no	—
184	C₂²×C₈	—	5	2	no	—
185	C₂²×C₈	—	5	2	no	—
186	C₂²×C₈	—	10	2	no	—
187	C₂²×C₈	—	10	2	no	—
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	C₄×D₅	1	1	yes	characteristic
195	M₃₂	—	2	2	no	—
196	M₃₂	—	10	2	no	—
197	M₃₂	—	10	4	no	—
198	M₃₂	—	10	4	no	—
199	M₃₂	—	10	4	no	—
200	C₂×C₁₆	—	5	2	no	—
201	C₂×C₁₆	—	5	2	no	—
202	C₂×C₁₆	—	10	2	no	—
203	C₂×C₁₆	—	10	4	no	—
204	C₂×C₁₆	D₂₀	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	C₄×D₅	1	2	yes	—
212	C₄×C₈	C₂²×D₅	1	2	yes	—
213	C₂²×C₅	—	2	2	no	—
214	C₂²×C₅	C₈×D₄	1	2	yes	—
215	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
216	C₂×D₅	—	2	2	no	—
217	C₂×D₅	—	2	4	no	—
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	4	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	2	no	—
227	Dic₅	—	2	2	no	—
228	C₂⁴	—	5	1	no	—
229	#41	—	10	2	no	—
230	#41	—	10	2	no	—
231	C₂×Q₈	—	5	1	no	—
232	C₂×D₄	—	5	2	no	—
233	C₂×D₄	—	5	1	no	—
234	C₂×D₄	—	5	4	no	—
235	C₂²×C₄	—	5	1	no	—
236	C₂²×C₄	—	5	2	no	—
237	C₂²×C₄	—	5	2	no	—
238	C₂²×C₄	—	5	2	no	—
239	C₂²×C₄	—	10	2	no	—
240	C₂²×C₄	—	10	2	no	—
241	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
242	M₁₆	—	2	2	no	—
243	M₁₆	—	10	2	no	—
244	M₁₆	—	10	4	no	—
245	C₂×C₈	—	5	2	no	—
246	C₂×C₈	—	5	2	no	—
247	C₂×C₈	—	10	2	no	—
248	C₂×C₈	—	10	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	C₂×D₂₀	1	2	yes	—
256	C₂×C₈	#709	1	2	yes	—
257	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
258	C₄⋊C₄	—	5	4	no	—
259	C₄⋊C₄	C₈×D₅	1	2	yes	—
260	C₂²⋊C₄	—	5	4	no	—
261	C₂²⋊C₄	—	5	4	no	—
262	C₂²⋊C₄	C₈×D₅	1	2	yes	—
263	C₄²	C₂×C₄×D₅	1	2	yes	—
264	C₁₆	—	2	2	no	—
265	C₁₆	—	10	2	no	—
266	C₁₆	—	10	2	no	—
267	C₁₆	C₂×D₂₀	1	2	yes	—
268	C₁₆	C₂×D₂₀	1	2	yes	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
271	C₂×C₅	#3407	1	2	yes	—
272	D₅	—	4	2	no	—
273	D₅	—	4	2	no	—
274	C₂³	—	5	2	no	—
275	C₂³	—	5	2	no	—
276	C₂³	—	10	2	no	—
277	C₂³	—	20	2	no	—
278	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
279	Q₈	—	10	2	no	—
280	D₄	—	10	4	no	—
281	D₄	—	10	2	no	—
282	D₄	—	20	4	no	—
283	C₂×C₄	—	5	2	no	—
284	C₂×C₄	—	5	2	no	—
285	C₂×C₄	—	10	2	no	—
286	C₂×C₄	—	10	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	C₄×D₂₀	1	2	yes	—
292	C₂×C₄	C₂×C₈×D₅	1	4	yes	—
293	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
294	C₂×C₄	#4047	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×D₂₀	1	2	yes	—
300	C₈	C₂²×D₂₀	1	1	yes	characteristic
301	C₈	C₂×#709	1	1	yes	characteristic
302	C₅	#33433	1	1	yes	characteristic, complemented, Sylow
303	C₂²	—	2	2	no	—
304	C₂²	—	10	2	no	—
305	C₂²	—	10	2	no	—
306	C₂²	—	10	2	no	—
307	C₂²	—	10	4	no	—
308	C₂²	—	10	2	no	—
309	C₂²	—	20	4	no	—
310	C₂²	C₈×D₂₀	1	2	yes	—
311	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
312	C₄	—	2	4	no	—
313	C₄	—	10	2	no	—
314	C₄	—	10	2	no	—
315	C₄	C₈×D₂₀	1	2	yes	—
316	C₄	C₂×C₄×D₂₀	1	1	yes	characteristic
317	C₄	C₂×#4047	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	20	2	no	—
320	C₂	—	20	2	no	—
321	C₂	C₂×C₈×D₂₀	1	1	yes	characteristic
322	C₂	#10613743	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#2108	C₂
prime radical	#65858	C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2108	D₅
hypocenter	C₅	#33433
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ #2108

Complements

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

N	H ≅ G/N	type
1. C₂×#65190	C₂	semidirect
2. C₂²×#3930	C₂	semidirect
4. C₅×#2108	C₂	semidirect
5. #10602225	C₂	semidirect
6. #10602224	C₂	semidirect
7. #10600675	C₂	semidirect
8. #10599868	C₂	semidirect
9. #10599848	C₂	semidirect
25. C₂×#3930	C₂²	semidirect
26. C₂×D₅×C₁₆	C₂²	semidirect
29. C₅×#346	C₂²	semidirect
30. C₄×C₅×C₁₆	C₂²	semidirect
32. #64727	C₂²	semidirect
33. C₄⋊#672	C₂²	semidirect
56. C₂×C₅×C₁₆	D₄	semidirect
81. #2108	D₅	semidirect
145. #346	C₂×D₅	semidirect
146. C₄×C₁₆	C₂×D₅	semidirect
204. C₂×C₁₆	D₂₀	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×#65190	2
C₂²×#3930	2
#10610284	2
C₅×#2108	2
#10602225	2
#10602224	2
#10600675	2
#10599868	2
#10599848	2
#33433	5