Subgroups — #11990029

Order 1280.

482 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#709	C₂	1	1	yes	maximal, characteristic
2	C₅×#1744	C₂	1	1	yes	maximal, characteristic, complemented
3	#10600527	C₂	1	1	yes	maximal, characteristic
4	#10600518	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#3937	C₂	1	1	yes	maximal, characteristic
6	#10598147	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598102	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#709	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₄³×C₅	C₂²	1	1	yes	characteristic
12	C₈×C₂²⋊D₅	—	2	2	no	—
13	C₂×#3937	C₄	1	2	yes	—
14	#65214	C₂²	1	1	yes	characteristic
15	#65214	C₂²	1	1	yes	characteristic
16	C₄×#681	C₂²	1	1	yes	characteristic
17	C₈×D₂₀	C₄	1	2	yes	complemented
18	#64974	—	2	1	no	—
19	C₄×C₈×D₅	—	2	1	no	—
20	C₈×Dic₁₀	C₄	1	2	yes	complemented
21	C₅×#336	—	2	2	no	—
22	#64743	C₄	1	2	yes	—
23	#64679	C₄	1	2	yes	complemented
24	#64676	C₄	1	2	yes	complemented
25	#64675	—	2	1	no	—
26	#64674	—	2	1	no	—
27	#8193	—	5	1	no	maximal, Sylow
28	C₂×#709	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	—	2	1	no	—
30	C₂²×C₅×C₈	D₄	1	2	yes	—
31	C₂×C₄²×C₅	—	2	1	no	—
32	C₂×C₄²×C₅	—	2	1	no	—
33	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₄×C₂²⋊D₅	—	2	1	no	—
36	C₄×C₂²⋊D₅	—	2	1	no	—
37	#4072	C₂×C₄	1	1	yes	characteristic
38	C₂×#681	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₅	—	2	2	no	—
40	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
41	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
42	#4018	—	2	1	no	—
43	C₄²×D₅	—	2	1	no	—
44	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
45	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
46	C₄×C₅×C₈	—	2	1	no	—
47	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
48	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
49	C₄×C₅×C₈	D₄	1	1	yes	characteristic
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₂²⋊#168	—	2	1	no	—
52	C₂²⋊#168	—	2	1	no	—
53	#3952	—	2	2	no	—
54	#3946	—	2	2	no	—
55	C₈×Dic₅	—	2	2	no	—
56	#3937	—	2	4	no	—
57	C₄⋊#168	C₂×C₄	1	2	yes	—
58	C₄⋊#168	C₂×C₄	1	2	yes	—
59	#3935	—	2	1	no	—
60	C₄×#168	—	2	1	no	—
61	C₄×#168	—	2	1	no	—
62	C₄×#168	—	2	1	no	—
63	C₈×#41	—	5	1	no	—
64	C₄×#104	—	5	1	no	—
65	#1744	D₅	1	1	yes	characteristic, complemented
66	#1476	—	5	1	no	—
67	#1451	—	5	1	no	—
68	#1278	—	5	1	no	—
69	#1276	—	5	1	no	—
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	#41	1	1	yes	characteristic
74	C₂×C₂²⋊D₅	—	2	1	no	—
75	#709	—	2	2	no	—
76	#709	M₁₆	1	4	yes	—
77	C₂×D₂₀	C₄²	1	1	yes	characteristic
78	C₂×C₄×D₅	—	2	1	no	—
79	C₂×C₄×D₅	—	2	1	no	—
80	C₂×C₄×D₅	—	2	1	no	—
81	C₂×Dic₁₀	C₄²	1	1	yes	characteristic
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	4	2	no	—
84	C₂×C₅×C₈	—	4	2	no	—
85	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
86	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₅	—	2	4	no	—
88	C₄²×C₅	—	2	2	no	—
89	C₄²×C₅	—	2	2	no	—
90	C₄²×C₅	—	4	2	no	—
91	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅	#41	1	1	yes	characteristic
95	C₂²⋊Dic₅	—	2	1	no	—
96	#685	—	2	1	no	—
97	#685	—	2	1	no	—
98	C₄⋊Dic₅	C₄²	1	1	yes	characteristic
99	C₄⋊Dic₅	C₄²	1	1	yes	characteristic
100	#683	—	2	1	no	—
101	#683	—	2	1	no	—
102	C₄×Dic₅	—	2	1	no	—
103	C₄×Dic₅	—	2	1	no	—
104	C₄×Dic₅	—	2	1	no	—
105	#681	—	2	2	no	—
106	#681	—	2	2	no	—
107	C₂×#168	—	2	2	no	—
108	C₂×#168	—	2	2	no	—
109	C₂×#168	—	2	2	no	—
110	C₈×D₅	—	4	4	no	—
111	C₄×#41	—	5	1	no	—
112	C₈×Q₈	—	5	2	no	—
113	C₈×D₄	—	5	2	no	—
114	C₈×D₄	—	10	2	no	—
115	#431	—	5	1	no	—
116	#431	—	5	1	no	—
117	#431	—	10	1	no	—
118	C₂×#104	—	5	2	no	—
119	C₄×M₁₆	—	5	1	no	—
120	C₂×C₄×C₈	—	10	1	no	—
121	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
123	C₄³	C₂×D₅	1	1	yes	characteristic
124	#339	—	5	2	no	—
125	#336	—	2	2	no	—
126	Q₈⋊C₈	—	5	2	no	—
127	D₄⋊C₈	—	5	2	no	—
128	#322	—	10	1	no	—
129	#322	—	10	1	no	—
130	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
131	C₂²×D₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	4	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	#117	1	1	yes	characteristic
140	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
144	C₂×C₄×C₅	#131	1	2	yes	—
145	C₂²⋊D₅	—	2	4	no	—
146	C₂×Dic₅	—	2	1	no	—
147	C₂×Dic₅	—	2	1	no	—
148	C₂×Dic₅	—	2	1	no	—
149	D₂₀	—	2	1	no	—
150	D₂₀	#97	1	2	yes	—
151	C₄×D₅	—	2	4	no	—
152	C₄×D₅	—	4	2	no	—
153	C₄×D₅	—	4	2	no	—
154	Dic₁₀	—	2	1	no	—
155	Dic₁₀	#97	1	2	yes	—
156	C₅×C₈	—	2	2	no	—
157	C₅×C₈	—	2	2	no	—
158	C₅×C₈	—	4	2	no	—
159	#168	—	4	2	no	—
160	#168	—	4	2	no	—
161	#168	—	4	2	no	—
162	C₂×#41	—	5	1	no	—
163	C₂×M₁₆	—	5	2	no	—
164	C₂²×C₈	—	2	1	no	—
165	C₂²×C₈	—	10	2	no	—
166	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
167	C₄×Q₈	—	5	1	no	—
168	C₄×Q₈	—	5	1	no	—
169	C₄×D₄	—	5	1	no	—
170	C₄×D₄	—	5	1	no	—
171	C₄×D₄	—	10	1	no	—
172	C₄×D₄	—	10	1	no	—
173	#117	—	5	1	no	—
174	#117	—	10	1	no	—
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	—	2	1	no	—
177	C₂×C₄²	—	10	1	no	—
178	C₂×C₄²	C₄×D₅	1	2	yes	—
179	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
180	C₄⋊C₈	—	5	2	no	—
181	C₄⋊C₈	—	5	2	no	—
182	C₄⋊C₈	—	10	2	no	—
183	#104	—	10	4	no	—
184	C₂²⋊C₈	—	10	1	no	—
185	C₂²⋊C₈	—	10	1	no	—
186	C₂²⋊C₈	—	10	2	no	—
187	#97	—	10	1	no	—
188	C₄×C₈	—	2	1	no	—
189	C₄×C₈	—	10	2	no	—
190	C₄×C₈	—	10	1	no	—
191	C₄×C₈	—	10	1	no	—
192	C₄×C₈	—	10	1	no	—
193	C₄×C₈	C₄×D₅	1	2	yes	—
194	C₄×C₈	C₄×D₅	1	2	yes	—
195	C₄×C₈	D₂₀	1	1	yes	characteristic
196	C₄×C₈	D₂₀	1	1	yes	characteristic
197	C₂²×C₅	—	2	1	no	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
200	C₂²×C₅	#435	1	2	yes	—
201	C₂×D₅	—	2	2	no	—
202	C₂×D₅	—	4	1	no	—
203	C₂×D₅	—	4	1	no	—
204	C₄×C₅	—	2	1	no	—
205	C₄×C₅	—	2	1	no	—
206	C₄×C₅	—	4	4	no	—
207	C₄×C₅	C₂×#97	1	1	yes	characteristic
208	C₄×C₅	#405	1	1	yes	characteristic
209	C₄×C₅	#421	1	2	yes	—
210	C₄×C₅	#435	1	2	yes	—
211	C₄×C₅	#443	1	2	yes	—
212	Dic₅	—	2	2	no	—
213	Dic₅	—	4	1	no	—
214	Dic₅	—	4	1	no	—
215	#41	—	5	4	no	—
216	#41	—	10	2	no	—
217	C₂×Q₈	—	5	1	no	—
218	C₂×D₄	—	5	1	no	—
219	C₂×D₄	—	10	1	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	10	1	no	—
222	C₂²×C₄	—	10	1	no	—
223	C₂²×C₄	—	10	1	no	—
224	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
225	C₂²×C₄	#709	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
227	M₁₆	—	10	2	no	—
228	M₁₆	—	10	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	10	2	no	—
233	C₂×C₈	—	10	2	no	—
234	C₂×C₈	—	10	2	no	—
235	C₂×C₈	—	20	4	no	—
236	C₂×C₈	#685	1	2	yes	—
237	C₂×C₈	#685	1	2	yes	—
238	C₄⋊C₄	—	5	1	no	—
239	C₄⋊C₄	—	5	1	no	—
240	C₄⋊C₄	—	10	1	no	—
241	C₄⋊C₄	—	10	1	no	—
242	C₂²⋊C₄	—	10	1	no	—
243	C₂²⋊C₄	—	10	1	no	—
244	C₂²⋊C₄	—	10	1	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	4	2	no	—
249	C₄²	—	10	1	no	—
250	C₄²	—	10	1	no	—
251	C₄²	—	10	1	no	—
252	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
253	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
254	C₄²	C₂×D₂₀	1	1	yes	characteristic
255	C₄²	#709	1	1	yes	characteristic
256	C₂×C₅	—	2	2	no	—
257	C₂×C₅	#1752	1	1	yes	characteristic
258	C₂×C₅	#1759	1	1	yes	characteristic
259	C₂×C₅	#1764	1	1	yes	characteristic
260	D₅	—	4	2	no	—
261	C₂³	—	10	1	no	—
262	C₂³	C₄×C₂²⋊D₅	1	1	yes	characteristic
263	Q₈	—	5	2	no	—
264	Q₈	—	10	1	no	—
265	D₄	—	5	2	no	—
266	D₄	—	10	4	no	—
267	D₄	—	10	1	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	10	1	no	—
274	C₂×C₄	—	10	4	no	—
275	C₂×C₄	—	10	1	no	—
276	C₂×C₄	—	10	1	no	—
277	C₂×C₄	—	20	2	no	—
278	C₂×C₄	—	20	2	no	—
279	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
280	C₂×C₄	#4018	1	1	yes	characteristic
281	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
282	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
283	C₂×C₄	D₅×M₁₆	1	2	yes	—
284	C₂×C₄	#4053	1	2	yes	—
285	C₂×C₄	C₂×#685	1	1	yes	characteristic
286	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	—	2	2	no	—
289	C₈	—	4	2	no	—
290	C₈	—	20	2	no	—
291	C₈	—	20	2	no	—
292	C₈	—	20	2	no	—
293	C₅	#8193	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	10	2	no	—
297	C₂²	—	20	1	no	—
298	C₂²	—	20	1	no	—
299	C₂²	C₄×#685	1	1	yes	characteristic
300	C₂²	#65416	1	2	yes	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	4	no	—
304	C₄	—	10	2	no	—
305	C₄	—	20	1	no	—
306	C₄	—	20	1	no	—
307	C₄	#64989	1	2	yes	—
308	C₄	D₅×#97	1	1	yes	characteristic
309	C₄	#64995	1	2	yes	—
310	C₄	#64997	1	1	yes	characteristic
311	C₄	#65294	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	20	2	no	—
314	C₂	#10600223	1	1	yes	characteristic
315	C₂	#10600582	1	1	yes	characteristic
316	C₂	#10600624	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1744	C₂
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	C₂×#709	C₂×C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1744	D₅
hypocenter	C₅	#8193
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1744

Complements

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

N	H ≅ G/N	type
2. C₅×#1744	C₂	semidirect
4. #10600518	C₂	semidirect
6. #10598147	C₂	semidirect
7. #10598102	C₂	semidirect
17. C₈×D₂₀	C₄	semidirect
20. C₈×Dic₁₀	C₄	semidirect
23. #64679	C₄	semidirect
24. #64676	C₄	semidirect
65. #1744	D₅	semidirect
293. C₅	#8193	semidirect

Maximal subgroups

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

subgroup	index
C₈×#709	2
C₅×#1744	2
#10600527	2
#10600518	2
C₄×#3937	2
#10598147	2
#10598102	2
#8193	5