Subgroups — #11990236

Order 1280.

488 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#4016	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×#64685	C₂	1	2	yes	maximal
4	#10600182	C₂	1	1	yes	maximal, characteristic
5	#10598150	C₂	1	2	yes	maximal
6	C₂×#4016	C₂²	1	1	yes	characteristic
7	C₂×C₅×#97	—	2	1	no	—
8	C₂×C₅×#97	C₂²	1	2	yes	—
9	C₂×C₄×C₅×C₈	—	2	1	no	—
10	C₄³×C₅	C₂²	1	1	yes	characteristic
11	#65225	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊Dic₅	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Dic₅	C₄	1	1	yes	characteristic
14	#65222	—	2	1	no	—
15	#65222	—	2	1	no	—
16	C₄×#683	—	2	1	no	—
17	C₄×#683	—	2	1	no	—
18	#65217	C₄	1	1	yes	characteristic
19	#65217	C₄	1	1	yes	characteristic
20	C₂×#3935	C₂²	1	2	yes	—
21	#64744	—	2	1	no	—
22	#64744	—	2	1	no	—
23	#64685	C₄	1	8	yes	complemented
24	#8227	—	5	1	no	maximal, Sylow
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
37	C₂×#683	—	2	1	no	—
38	C₂×#683	—	2	1	no	—
39	C₂×#683	—	2	1	no	—
40	C₂×#683	—	2	1	no	—
41	C₂×C₄×Dic₅	—	2	1	no	—
42	C₂×C₄×Dic₅	—	2	1	no	—
43	C₂²×#168	—	2	2	no	—
44	#4016	—	2	2	no	—
45	#4016	C₂×C₄	1	4	yes	—
46	C₅×#97	—	2	4	no	—
47	C₅×#97	—	2	2	no	—
48	C₅×#97	C₂×C₄	1	4	yes	—
49	C₄×C₅×C₈	—	2	4	no	—
50	#3963	—	2	1	no	—
51	#3963	—	2	1	no	—
52	#3963	—	2	1	no	—
53	#3963	—	2	1	no	—
54	#3963	—	2	1	no	—
55	#3963	—	2	1	no	—
56	#3935	—	2	2	no	—
57	#3935	C₂×C₄	1	4	yes	—
58	C₄×#125	—	5	1	no	—
59	#1745	—	5	1	no	—
60	C₄×#97	D₅	1	1	yes	characteristic
61	C₂×#330	—	5	2	no	—
62	#1295	—	5	2	no	—
63	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₅	—	2	1	no	—
71	C₂²×Dic₅	—	2	1	no	—
72	C₂²×Dic₅	—	2	1	no	—
73	C₂²×Dic₅	—	2	1	no	—
74	C₂×C₅×C₈	—	2	4	no	—
75	C₂×C₅×C₈	—	2	4	no	—
76	C₂×C₅×C₈	—	2	4	no	—
77	C₄²×C₅	—	2	2	no	—
78	C₄²×C₅	—	2	2	no	—
79	C₄²×C₅	—	2	2	no	—
80	C₄²×C₅	—	4	1	no	—
81	C₄²×C₅	—	4	1	no	—
82	C₄²×C₅	—	4	1	no	—
83	C₄²×C₅	C₂²×C₄	1	2	yes	—
84	C₄²×C₅	#41	1	2	yes	—
85	C₄⋊Dic₅	—	2	1	no	—
86	C₄⋊Dic₅	—	2	1	no	—
87	C₄⋊Dic₅	—	4	1	no	—
88	C₄⋊Dic₅	—	4	1	no	—
89	C₄⋊Dic₅	C₄²	1	2	yes	—
90	C₄⋊Dic₅	C₄²	1	2	yes	—
91	#683	—	2	4	no	—
92	#683	—	2	4	no	—
93	#683	—	4	2	no	—
94	#683	—	4	2	no	—
95	C₄×Dic₅	—	4	2	no	—
96	C₄×Dic₅	—	4	2	no	—
97	C₂×#168	—	2	4	no	—
98	C₂×#168	—	4	2	no	—
99	C₂×#168	—	4	2	no	—
100	C₂×#125	—	5	1	no	—
101	C₂×#97	—	2	1	no	—
102	C₂×#97	—	5	2	no	—
103	C₂×#97	C₂×D₅	1	2	yes	—
104	C₂×C₄×C₈	—	2	1	no	—
105	#389	—	5	1	no	—
106	#389	—	5	1	no	—
107	#387	—	10	1	no	—
108	#387	—	10	1	no	—
109	#382	—	5	1	no	—
110	C₄×C₄⋊C₄	—	5	1	no	—
111	C₄×C₄⋊C₄	—	5	1	no	—
112	C₄×C₄⋊C₄	—	10	1	no	—
113	C₄×C₄⋊C₄	—	10	1	no	—
114	C₄³	C₂×D₅	1	1	yes	characteristic
115	#336	—	10	1	no	—
116	#336	—	10	1	no	—
117	#330	—	5	8	no	—
118	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	#101	1	2	yes	—
135	C₂×C₄×C₅	#104	1	2	yes	—
136	C₂×C₄×C₅	#104	1	2	yes	—
137	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₅	#117	1	1	yes	characteristic
139	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
141	C₂×Dic₅	—	2	2	no	—
142	C₂×Dic₅	—	2	2	no	—
143	C₂×Dic₅	—	4	1	no	—
144	C₂×Dic₅	—	4	1	no	—
145	C₂×Dic₅	—	4	1	no	—
146	C₂×Dic₅	—	4	1	no	—
147	C₂×Dic₅	—	4	1	no	—
148	C₂×Dic₅	—	4	1	no	—
149	C₂×Dic₅	—	4	1	no	—
150	C₂×Dic₅	—	4	1	no	—
151	C₂×Dic₅	—	4	1	no	—
152	C₂×Dic₅	—	4	1	no	—
153	C₅×C₈	—	2	8	no	—
154	#168	—	4	4	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	10	2	no	—
157	#125	—	5	4	no	—
158	#125	—	10	2	no	—
159	C₂×C₄⋊C₄	—	5	1	no	—
160	C₂×C₄⋊C₄	—	5	1	no	—
161	C₂×C₄⋊C₄	—	5	1	no	—
162	C₂×C₄⋊C₄	—	5	1	no	—
163	C₂×C₄⋊C₄	—	10	1	no	—
164	C₂×C₄⋊C₄	—	10	1	no	—
165	C₂×C₄⋊C₄	—	10	1	no	—
166	C₂×C₄⋊C₄	—	10	1	no	—
167	C₂×C₄²	—	10	1	no	—
168	C₂×C₄²	—	10	1	no	—
169	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
171	C₂×C₄²	D₂₀	1	1	yes	characteristic
172	C₂×C₄²	D₂₀	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
174	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
176	#97	—	2	4	no	—
177	#97	—	2	2	no	—
178	#97	—	5	4	no	—
179	#97	—	10	2	no	—
180	#97	C₄×D₅	1	4	yes	—
181	C₄×C₈	—	2	4	no	—
182	#95	—	10	1	no	—
183	#95	—	10	1	no	—
184	#95	—	10	1	no	—
185	#95	—	10	1	no	—
186	#95	—	10	1	no	—
187	#95	—	10	1	no	—
188	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
189	C₂²×C₅	C₂×#101	1	1	yes	characteristic
190	C₂²×C₅	#413	1	1	yes	characteristic
191	C₂²×C₅	C₂×#104	1	1	yes	characteristic
192	C₂²×C₅	C₂×#104	1	1	yes	characteristic
193	C₂²×C₅	#421	1	1	yes	characteristic
194	C₂²×C₅	#421	1	1	yes	characteristic
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	2	2	no	—
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	2	2	no	—
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	—	4	1	no	—
203	Dic₅	—	4	2	no	—
204	Dic₅	—	4	2	no	—
205	Dic₅	—	8	1	no	—
206	Dic₅	—	8	1	no	—
207	C₂²×C₄	—	10	1	no	—
208	C₂²×C₄	—	10	1	no	—
209	C₂²×C₄	—	10	1	no	—
210	C₂²×C₄	—	10	1	no	—
211	C₂²×C₄	#685	1	1	yes	characteristic
212	C₂²×C₄	#685	1	1	yes	characteristic
213	C₂²×C₄	#685	1	1	yes	characteristic
214	C₂²×C₄	#685	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	10	4	no	—
222	C₂×C₈	—	20	2	no	—
223	C₂×C₈	—	20	2	no	—
224	C₄⋊C₄	—	5	2	no	—
225	C₄⋊C₄	—	5	2	no	—
226	C₄⋊C₄	—	10	1	no	—
227	C₄⋊C₄	—	10	1	no	—
228	C₄⋊C₄	—	10	4	no	—
229	C₄⋊C₄	—	10	4	no	—
230	C₄⋊C₄	—	20	1	no	—
231	C₄⋊C₄	—	20	1	no	—
232	C₄⋊C₄	—	20	2	no	—
233	C₄⋊C₄	—	20	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	20	2	no	—
241	C₄²	—	20	2	no	—
242	C₄²	C₂×C₄×D₅	1	2	yes	—
243	C₄²	#709	1	2	yes	—
244	C₂×C₅	#1528	1	1	yes	characteristic
245	C₂×C₅	#1529	1	1	yes	characteristic
246	C₂×C₅	#1531	1	1	yes	characteristic
247	C₂×C₅	#1532	1	1	yes	characteristic
248	C₂×C₅	C₄×#101	1	1	yes	characteristic
249	C₂×C₅	C₄×#104	1	1	yes	characteristic
250	C₂×C₅	#1759	1	1	yes	characteristic
251	C₂³	C₂×#685	1	1	yes	characteristic
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	10	2	no	—
268	C₂×C₄	—	10	2	no	—
269	C₂×C₄	—	20	1	no	—
270	C₂×C₄	—	20	1	no	—
271	C₂×C₄	—	20	1	no	—
272	C₂×C₄	—	20	1	no	—
273	C₂×C₄	—	20	1	no	—
274	C₂×C₄	—	20	1	no	—
275	C₂×C₄	—	20	1	no	—
276	C₂×C₄	—	20	1	no	—
277	C₂×C₄	—	20	1	no	—
278	C₂×C₄	—	20	1	no	—
279	C₂×C₄	#3937	1	2	yes	—
280	C₂×C₄	#3937	1	2	yes	—
281	C₂×C₄	#3956	1	2	yes	—
282	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
283	C₂×C₄	#4018	1	1	yes	characteristic
284	C₂×C₄	C₄×D₂₀	1	2	yes	—
285	C₂×C₄	C₄×C₂²⋊D₅	1	2	yes	—
286	C₈	—	2	8	no	—
287	C₈	—	20	4	no	—
288	C₅	#8227	1	1	yes	characteristic, complemented, Sylow
289	C₂²	C₂×#3937	1	1	yes	characteristic
290	C₂²	C₂×#3937	1	1	yes	characteristic
291	C₂²	C₄×#685	1	1	yes	characteristic
292	C₂²	#65421	1	1	yes	characteristic
293	C₂²	C₂×#3956	1	1	yes	characteristic
294	C₂²	#65428	1	1	yes	characteristic
295	C₂²	#65428	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	20	2	no	—
305	C₄	—	20	2	no	—
306	C₄	—	40	1	no	—
307	C₄	—	40	1	no	—
308	C₂	C₄×#3937	1	1	yes	characteristic
309	C₂	#10600592	1	1	yes	characteristic
310	C₂	#10600594	1	1	yes	characteristic
311	C₂	#10600654	1	1	yes	characteristic
312	C₂	#10600656	1	1	yes	characteristic
313	C₂	#10600659	1	1	yes	characteristic
314	C₂	C₄×#3956	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#685
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄×C₅×#97	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×#97	D₅
hypocenter	C₅	#8227
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
23. #64685	C₄	semidirect
288. C₅	#8227	semidirect

Maximal subgroups

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

subgroup	index
C₄×#4016	2
C₄×C₅×#97	2
C₂×#64685	2
#10600182	2
#10598150	2
#8227	5