Subgroups — #12252659

Order 1280.

718 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅×D₁₆	C₂	1	1	yes	maximal, characteristic
2	#10611319	C₂	1	2	yes	maximal, complemented
3	C₅×#2177	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602254	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#3958	C₂	1	2	yes	maximal, complemented
6	#10600138	C₂	1	1	yes	maximal, characteristic, complemented
7	#10600132	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599983	C₂	1	2	yes	maximal, complemented
9	#10599967	C₂	1	2	yes	maximal, complemented
10	#10599871	C₂	1	1	yes	maximal, characteristic, complemented
11	#10599865	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₅×D₈	C₂²	1	2	yes	—
13	#65879	—	2	2	no	—
14	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₅×D₈	C₂²	1	2	yes	complemented
16	D₈×Dic₅	C₂²	1	2	yes	complemented
17	C₂×#3958	C₂²	1	2	yes	—
18	#65303	—	2	2	no	—
19	D₅×D₁₆	C₄	1	8	yes	complemented
20	C₂×D₈₀	C₂²	1	1	yes	characteristic
21	C₂×#3930	C₂²	1	1	yes	characteristic
22	C₂×D₅×C₁₆	C₂²	1	1	yes	characteristic
23	#65169	C₂²	1	2	yes	—
24	#65161	C₂²	1	2	yes	complemented
25	#65074	—	2	2	no	—
26	#65059	—	2	2	no	—
27	C₄×D₄₀	C₂²	1	2	yes	complemented
28	#64978	C₂²	1	1	yes	characteristic
29	C₅×#367	C₂²	1	1	yes	characteristic, complemented
30	C₅×#357	C₂²	1	2	yes	complemented
31	C₅×#346	C₂²	1	1	yes	characteristic, complemented
32	#64782	C₂²	1	2	yes	complemented
33	#64729	C₂²	1	2	yes	complemented
34	#64725	C₂²	1	1	yes	characteristic, complemented
35	#64722	C₂²	1	1	yes	characteristic, complemented
36	C₁₆×Dic₅	C₂²	1	1	yes	characteristic, complemented
37	#64711	C₂²	1	2	yes	complemented
38	#64709	C₂²	1	2	yes	complemented
39	C₄×#672	C₂²	1	1	yes	characteristic, complemented
40	#33456	—	5	1	no	maximal, Sylow
41	C₂×D₄×D₅	—	2	2	no	—
42	C₂×C₅×D₈	C₂³	1	2	yes	—
43	C₄×C₅×D₄	—	2	2	no	—
44	D₄×Dic₅	—	2	2	no	—
45	C₂×#686	—	2	2	no	—
46	C₄×C₂²⋊D₅	—	4	2	no	—
47	C₂×#685	—	4	2	no	—
48	D₅×D₈	—	2	4	no	—
49	D₅×D₈	C₂×C₄	1	8	yes	complemented
50	C₂×D₄₀	C₂³	1	2	yes	—
51	C₂×#676	D₄	1	1	yes	characteristic
52	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
53	#4039	—	2	2	no	—
54	#4038	—	2	2	no	—
55	#4027	—	4	2	no	—
56	D₅×C₂²⋊C₄	—	4	2	no	—
57	C₄×D₂₀	—	2	2	no	—
58	#4018	D₄	1	1	yes	characteristic
59	C₅×D₁₆	C₂×C₄	1	4	yes	complemented
60	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
62	C₅×#107	C₂³	1	2	yes	—
63	C₅×#102	—	2	2	no	—
64	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
65	#3964	—	2	2	no	—
66	#3958	C₂×C₄	1	8	yes	complemented
67	#3953	—	2	2	no	—
68	#3950	C₂³	1	2	yes	—
69	#3947	D₄	1	1	yes	characteristic
70	C₈×Dic₅	C₂³	1	1	yes	characteristic
71	C₂×#672	C₂³	1	1	yes	characteristic
72	C₂×#672	C₂³	1	1	yes	characteristic
73	#3941	—	2	2	no	—
74	#3940	—	2	2	no	—
75	#3935	D₄	1	1	yes	characteristic
76	D₈₀	C₂×C₄	1	4	yes	complemented
77	#3930	—	2	2	no	—
78	D₅×C₁₆	C₂×C₄	1	4	yes	complemented
79	C₂²×D₁₆	—	5	1	no	—
80	#2948	—	5	2	no	—
81	#2177	—	5	1	no	—
82	#2177	—	5	1	no	—
83	#2177	—	5	1	no	—
84	#2177	D₅	1	1	yes	characteristic, complemented
85	C₄×D₁₆	—	5	2	no	—
86	C₄×D₁₆	—	5	2	no	—
87	#2158	—	5	1	no	—
88	#2140	—	5	2	no	—
89	#2108	—	5	1	no	—
90	C₂³×D₅	—	4	2	no	—
91	C₂×C₅×D₄	—	2	2	no	—
92	C₂²×C₄×C₅	—	4	2	no	—
93	C₂×C₂²⋊D₅	—	4	2	no	—
94	C₂²×Dic₅	—	4	2	no	—
95	D₄×D₅	—	2	8	no	—
96	D₄×D₅	—	4	4	no	—
97	C₂×D₂₀	—	2	2	no	—
98	C₂×C₄×D₅	—	4	2	no	—
99	C₂×C₄×D₅	—	4	2	no	—
100	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
101	C₅×D₈	—	2	2	no	—
102	C₅×D₈	C₂²×C₄	1	4	yes	—
103	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₂²⋊C₄	—	4	2	no	—
107	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
108	C₂²⋊Dic₅	—	4	2	no	—
109	#686	—	2	8	no	—
110	#685	—	4	2	no	—
111	#685	—	4	2	no	—
112	#685	—	4	2	no	—
113	#685	D₈	1	2	yes	—
114	C₄⋊Dic₅	—	2	2	no	—
115	#683	D₈	1	2	yes	—
116	C₄×Dic₅	—	4	2	no	—
117	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
118	C₂×#168	C₂×D₄	1	1	yes	characteristic
119	C₂×#168	C₂×D₄	1	1	yes	characteristic
120	D₄₀	—	2	2	no	—
121	D₄₀	C₂²×C₄	1	4	yes	—
122	#676	—	2	2	no	—
123	C₈×D₅	C₂²×C₄	1	2	yes	—
124	C₈×D₅	#41	1	2	yes	—
125	C₅×C₁₆	—	2	1	no	—
126	C₅×C₁₆	C₂²×C₄	1	2	yes	—
127	#672	—	2	1	no	—
128	#672	C₂²×C₄	1	2	yes	—
129	C₂²×D₈	—	5	2	no	—
130	#518	—	10	2	no	—
131	C₂×D₁₆	—	5	2	no	—
132	C₂×D₁₆	—	5	8	no	—
133	C₂×D₁₆	—	5	1	no	—
134	C₂×D₁₆	C₂×D₅	1	1	yes	characteristic
135	C₂×M₃₂	—	5	1	no	—
136	C₂²×C₁₆	—	5	1	no	—
137	#442	—	10	2	no	—
138	#442	—	10	2	no	—
139	C₄×D₈	—	5	2	no	—
140	C₄×D₈	—	5	2	no	—
141	C₄×D₈	—	5	2	no	—
142	C₄×D₈	C₂×D₅	1	2	yes	complemented
143	#427	—	5	2	no	—
144	#418	—	10	2	no	—
145	#405	—	5	1	no	—
146	#367	—	5	1	no	—
147	#367	C₂×D₅	1	1	yes	characteristic, complemented
148	#366	—	5	2	no	—
149	#357	—	5	2	no	—
150	#357	—	5	2	no	—
151	#357	—	5	2	no	—
152	#357	C₂×D₅	1	2	yes	complemented
153	#346	—	5	1	no	—
154	#346	C₂×D₅	1	1	yes	characteristic, complemented
155	C₄×C₁₆	—	5	1	no	—
156	C₄×C₁₆	—	5	1	no	—
157	C₂³×C₅	—	4	2	no	—
158	C₂²×D₅	—	4	2	no	—
159	C₂²×D₅	—	4	8	no	—
160	C₂²×D₅	—	16	2	no	—
161	C₂²×D₅	—	16	2	no	—
162	C₂²×D₅	C₂×D₈	1	1	yes	characteristic
163	C₅×D₄	—	2	4	no	—
164	C₅×D₄	—	4	2	no	—
165	C₂×C₄×C₅	—	4	2	no	—
166	C₂×C₄×C₅	—	16	2	no	—
167	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
168	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
169	C₂²⋊D₅	—	4	8	no	—
170	C₂×Dic₅	—	4	2	no	—
171	C₂×Dic₅	—	16	2	no	—
172	C₂×Dic₅	C₂×D₈	1	2	yes	—
173	C₂×Dic₅	C₂×D₈	1	1	yes	characteristic
174	D₂₀	—	2	4	no	—
175	D₂₀	—	4	2	no	—
176	C₄×D₅	—	16	2	no	—
177	C₄×D₅	—	16	2	no	—
178	C₄×D₅	C₄×D₄	1	2	yes	—
179	C₄×D₅	#135	1	2	yes	—
180	C₅×C₈	—	2	1	no	—
181	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
182	C₅×C₈	C₂×#41	1	1	yes	characteristic
183	#168	—	2	1	no	—
184	#168	C₄×D₄	1	2	yes	—
185	C₂²×D₄	—	10	2	no	—
186	C₂×D₈	—	5	8	no	—
187	C₂×D₈	—	5	2	no	—
188	C₂×D₈	—	10	4	no	—
189	C₂×D₈	—	10	2	no	—
190	C₂×D₈	C₂²×D₅	1	2	yes	—
191	C₂×M₁₆	—	5	1	no	—
192	C₂²×C₈	—	5	1	no	—
193	C₄×D₄	—	2	2	no	—
194	C₄×D₄	—	10	2	no	—
195	C₄×D₄	—	10	2	no	—
196	C₄×D₄	—	10	2	no	—
197	C₄×D₄	—	20	2	no	—
198	C₄×D₄	—	20	2	no	—
199	#117	—	5	1	no	—
200	#117	—	10	2	no	—
201	C₂×C₂²⋊C₄	—	20	2	no	—
202	C₂×C₂²⋊C₄	—	20	2	no	—
203	D₁₆	—	5	8	no	—
204	D₁₆	—	5	4	no	—
205	D₁₆	C₄×D₅	1	4	yes	complemented
206	M₃₂	—	10	2	no	—
207	C₂×C₁₆	—	5	1	no	—
208	C₂×C₁₆	—	5	4	no	—
209	C₂×C₁₆	—	5	1	no	—
210	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
211	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
212	#107	—	5	2	no	—
213	#107	C₂²×D₅	1	2	yes	—
214	#106	—	10	2	no	—
215	#102	—	2	2	no	—
216	#102	—	10	2	no	—
217	#102	—	10	2	no	—
218	#102	—	10	2	no	—
219	#97	—	5	1	no	—
220	#97	—	5	1	no	—
221	C₄×C₈	—	5	1	no	—
222	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
223	C₂²×C₅	—	4	4	no	—
224	C₂²×C₅	—	8	2	no	—
225	C₂²×C₅	—	8	2	no	—
226	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
227	C₂×D₅	—	2	2	no	—
228	C₂×D₅	—	4	4	no	—
229	C₂×D₅	—	8	2	no	—
230	C₂×D₅	—	8	2	no	—
231	C₂×D₅	—	16	8	no	—
232	C₂×D₅	C₄×D₈	1	2	yes	—
233	C₄×C₅	—	2	2	no	—
234	C₄×C₅	—	8	2	no	—
235	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
236	C₄×C₅	C₂×#135	1	1	yes	characteristic
237	Dic₅	—	2	2	no	—
238	Dic₅	—	8	2	no	—
239	Dic₅	C₄×D₈	1	2	yes	—
240	C₂⁴	—	20	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	10	8	no	—
243	C₂×D₄	—	10	2	no	—
244	C₂×D₄	—	20	4	no	—
245	C₂×D₄	—	20	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	5	1	no	—
248	C₂²×C₄	—	20	2	no	—
249	C₂²×C₄	—	20	2	no	—
250	C₂²×C₄	—	20	2	no	—
251	D₈	—	2	2	no	—
252	D₈	—	5	4	no	—
253	D₈	—	10	8	no	—
254	D₈	—	10	2	no	—
255	D₈	C₂×C₄×D₅	1	4	yes	—
256	M₁₆	—	10	2	no	—
257	C₂×C₈	—	5	1	no	—
258	C₂×C₈	—	5	2	no	—
259	C₂×C₈	—	5	2	no	—
260	C₂×C₈	—	5	1	no	—
261	C₂×C₈	D₄×D₅	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	5	2	no	—
265	C₄⋊C₄	—	10	2	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	5	2	no	—
268	C₂²⋊C₄	—	20	2	no	—
269	C₂²⋊C₄	—	20	2	no	—
270	C₂²⋊C₄	—	20	2	no	—
271	C₂²⋊C₄	—	20	2	no	—
272	C₄²	—	5	1	no	—
273	C₄²	—	20	2	no	—
274	C₄²	D₄×D₅	1	1	yes	characteristic
275	C₁₆	—	2	1	no	—
276	C₁₆	—	5	2	no	—
277	C₁₆	—	10	1	no	—
278	C₁₆	C₂×C₄×D₅	1	2	yes	—
279	C₂×C₅	—	8	4	no	—
280	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
281	C₂×C₅	#3415	1	2	yes	—
282	D₅	—	2	2	no	—
283	D₅	—	8	4	no	—
284	C₂³	—	4	2	no	—
285	C₂³	—	5	1	no	—
286	C₂³	—	20	8	no	—
287	C₂³	—	20	2	no	—
288	C₂³	—	80	2	no	—
289	C₂³	—	80	2	no	—
290	D₄	—	2	4	no	—
291	D₄	—	4	2	no	—
292	D₄	—	10	4	no	—
293	D₄	—	20	8	no	—
294	D₄	—	20	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	5	1	no	—
297	C₂×C₄	—	5	2	no	—
298	C₂×C₄	—	5	2	no	—
299	C₂×C₄	—	5	2	no	—
300	C₂×C₄	—	16	2	no	—
301	C₂×C₄	—	20	2	no	—
302	C₂×C₄	—	80	2	no	—
303	C₂×C₄	—	80	2	no	—
304	C₂×C₄	—	80	2	no	—
305	C₂×C₄	D₅×D₈	1	2	yes	—
306	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	5	2	no	—
309	C₈	—	10	1	no	—
310	C₈	C₂²×C₄×D₅	1	1	yes	characteristic
311	C₈	D₅×#41	1	1	yes	characteristic
312	C₅	#33456	1	1	yes	characteristic, complemented, Sylow
313	C₂²	—	4	4	no	—
314	C₂²	—	5	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	10	2	no	—
318	C₂²	—	20	4	no	—
319	C₂²	—	40	2	no	—
320	C₂²	—	40	2	no	—
321	C₂²	—	80	8	no	—
322	C₂²	C₂×D₅×D₈	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	5	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	10	2	no	—
327	C₄	—	40	2	no	—
328	C₄	C₄×D₄×D₅	1	1	yes	characteristic
329	C₄	D₅×#135	1	1	yes	characteristic
330	C₂	—	8	4	no	—
331	C₂	—	10	2	no	—
332	C₂	—	40	4	no	—
333	C₂	C₄×D₅×D₈	1	1	yes	characteristic
334	C₂	#10613767	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₅×D₈
commutator	C₅×C₈	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₅×#2177	C₂
socle	C₂²×C₅	C₂²×D₈
prime radical	C₂×D₅×D₁₆	C₂
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2177	D₅
hypocenter	C₅	#33456
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

Complements

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

N	H ≅ G/N	type
2. #10611319	C₂	semidirect
3. C₅×#2177	C₂	semidirect
4. #10602254	C₂	semidirect
5. C₄×#3958	C₂	semidirect
6. #10600138	C₂	semidirect
7. #10600132	C₂	semidirect
8. #10599983	C₂	semidirect
9. #10599967	C₂	semidirect
10. #10599871	C₂	semidirect
11. #10599865	C₂	semidirect
15. C₄×C₅×D₈	C₂²	semidirect
16. D₈×Dic₅	C₂²	semidirect
19. D₅×D₁₆	C₄	semidirect
24. #65161	C₂²	semidirect
27. C₄×D₄₀	C₂²	semidirect
29. C₅×#367	C₂²	semidirect
30. C₅×#357	C₂²	semidirect
31. C₅×#346	C₂²	semidirect
32. #64782	C₂²	semidirect
33. #64729	C₂²	semidirect
34. #64725	C₂²	semidirect
35. #64722	C₂²	semidirect
36. C₁₆×Dic₅	C₂²	semidirect
37. #64711	C₂²	semidirect
38. #64709	C₂²	semidirect
39. C₄×#672	C₂²	semidirect
49. D₅×D₈	C₂×C₄	semidirect
59. C₅×D₁₆	C₂×C₄	semidirect
66. #3958	C₂×C₄	semidirect
76. D₈₀	C₂×C₄	semidirect
78. D₅×C₁₆	C₂×C₄	semidirect
84. #2177	D₅	semidirect
142. C₄×D₈	C₂×D₅	semidirect
147. #367	C₂×D₅	semidirect
152. #357	C₂×D₅	semidirect
154. #346	C₂×D₅	semidirect
205. D₁₆	C₄×D₅	semidirect
312. C₅	#33456	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅×D₁₆	2
#10611319	2
C₅×#2177	2
#10602254	2
C₄×#3958	2
#10600138	2
#10600132	2
#10599983	2
#10599967	2
#10599871	2
#10599865	2
#33456	5