Subgroups — #11987593

Order 1280.

402 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	C₅×#2295	C₂	1	1	yes	maximal, characteristic
2	#10600186	C₂	1	1	yes	maximal, characteristic
3	#10600185	C₂	1	1	yes	maximal, characteristic
4	#10598133	C₂	1	2	yes	maximal
5	#10598129	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×Q₈	C₄	1	1	yes	characteristic
7	C₅×#391	C₄	1	1	yes	characteristic
8	C₅×#391	C₂²	1	1	yes	characteristic
9	C₅×#389	C₄	1	1	yes	characteristic
10	C₅×#389	C₂²	1	1	yes	characteristic
11	C₅×#387	—	2	1	no	—
12	C₅×#387	—	2	1	no	—
13	C₄×C₅×C₄⋊C₄	—	2	1	no	—
14	C₄×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
15	C₅×#376	—	2	1	no	—
16	C₄³×C₅	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#168	—	2	1	no	—
18	#64744	—	2	1	no	—
19	#64744	—	2	1	no	—
20	#64744	—	2	1	no	—
21	#64744	C₂²	1	2	yes	—
22	#64744	C₂²	1	2	yes	—
23	#7644	—	5	1	no	maximal, Sylow
24	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₅×Q₈	—	2	2	no	—
26	C₄×C₅×Q₈	—	2	2	no	—
27	C₂×C₅×C₄⋊C₄	—	2	1	no	—
28	C₂×C₅×C₄⋊C₄	—	2	1	no	—
29	C₂×C₅×C₄⋊C₄	—	2	1	no	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	—	2	1	no	—
34	C₂×C₄²×C₅	—	2	1	no	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₂²×#168	—	2	2	no	—
43	C₂²×#168	—	2	2	no	—
44	C₅×#95	—	2	1	no	—
45	C₅×#95	—	2	1	no	—
46	C₅×#95	—	2	1	no	—
47	C₅×#95	—	2	1	no	—
48	C₅×#95	—	2	1	no	—
49	C₅×#95	—	2	1	no	—
50	C₅×#95	—	2	1	no	—
51	C₄⋊#168	—	4	2	no	—
52	#2295	D₅	1	1	yes	characteristic
53	#1839	—	5	1	no	—
54	#1831	—	5	1	no	—
55	#1272	—	5	2	no	—
56	#1271	—	5	2	no	—
57	C₂×C₅×Q₈	—	2	1	no	—
58	C₂×C₅×Q₈	—	4	1	no	—
59	C₂×C₅×Q₈	—	4	1	no	—
60	C₂×C₅×Q₈	M₁₆	1	2	yes	—
61	C₂²×C₄×C₅	—	2	1	no	—
62	C₂²×C₄×C₅	—	2	1	no	—
63	C₂²×C₄×C₅	—	2	1	no	—
64	C₂²×C₄×C₅	—	2	1	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₅×C₄⋊C₄	—	2	4	no	—
73	C₅×C₄⋊C₄	—	2	1	no	—
74	C₅×C₄⋊C₄	—	4	1	no	—
75	C₅×C₄⋊C₄	—	4	1	no	—
76	C₅×C₄⋊C₄	—	4	2	no	—
77	C₅×C₄⋊C₄	—	4	2	no	—
78	C₅×C₄⋊C₄	M₁₆	1	2	yes	—
79	C₄²×C₅	—	2	4	no	—
80	C₄²×C₅	—	2	1	no	—
81	C₄²×C₅	—	2	1	no	—
82	C₄²×C₅	—	2	2	no	—
83	C₄²×C₅	—	4	1	no	—
84	C₄²×C₅	—	4	1	no	—
85	C₄²×C₅	—	4	1	no	—
86	C₄²×C₅	—	4	1	no	—
87	C₄²×C₅	—	4	2	no	—
88	C₄²×C₅	SD₁₆	1	2	yes	—
89	C₄²×C₅	Q₁₆	1	2	yes	—
90	C₂×#168	—	4	2	no	—
91	C₂×#168	—	4	2	no	—
92	C₂×#168	—	4	2	no	—
93	C₂×#168	—	4	2	no	—
94	C₂×#168	—	4	2	no	—
95	C₂×#168	—	4	2	no	—
96	C₂×C₄×Q₈	Dic₅	1	1	yes	characteristic
97	C₂×C₄⋊C₈	—	10	1	no	—
98	#391	Dic₅	1	1	yes	characteristic
99	#391	C₂×D₅	1	1	yes	characteristic
100	#389	Dic₅	1	1	yes	characteristic
101	#389	C₂×D₅	1	1	yes	characteristic
102	#387	—	2	1	no	—
103	#387	—	2	1	no	—
104	C₄×C₄⋊C₄	—	2	1	no	—
105	C₄×C₄⋊C₄	Dic₅	1	1	yes	characteristic
106	#376	—	2	1	no	—
107	C₄³	C₂×D₅	1	1	yes	characteristic
108	#336	—	5	2	no	—
109	#336	—	5	2	no	—
110	#336	—	10	1	no	—
111	#336	—	10	1	no	—
112	#336	—	10	1	no	—
113	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₅×Q₈	—	4	2	no	—
115	C₅×Q₈	—	4	2	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	2	1	no	—
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	2	no	—
128	C₂×C₄×C₅	—	2	2	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	—	4	1	no	—
135	C₂×C₄×C₅	—	4	1	no	—
136	C₂×C₄×C₅	—	4	1	no	—
137	C₂×C₄×C₅	—	4	1	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	—	4	1	no	—
140	C₂×C₄×C₅	—	4	1	no	—
141	C₂×C₄×C₅	—	4	1	no	—
142	C₂×C₄×C₅	—	4	1	no	—
143	C₂×C₄×C₅	—	4	1	no	—
144	C₂×C₄×C₅	#103	1	2	yes	—
145	C₂×C₄×C₅	#103	1	2	yes	—
146	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
150	C₂×C₄×C₅	#135	1	2	yes	—
151	#168	—	8	2	no	—
152	#168	—	8	2	no	—
153	C₂²×Q₈	C₂×Dic₅	1	1	yes	characteristic
154	C₂²×C₈	—	10	2	no	—
155	C₂²×C₈	—	10	2	no	—
156	C₄×Q₈	—	2	2	no	—
157	C₄×Q₈	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
162	C₂×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
163	C₂×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
167	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
173	C₄⋊C₈	—	20	2	no	—
174	#95	—	2	1	no	—
175	#95	—	2	1	no	—
176	#95	—	2	1	no	—
177	#95	—	2	1	no	—
178	#95	—	2	1	no	—
179	#95	—	2	1	no	—
180	#95	—	2	1	no	—
181	C₂²×C₅	#407	1	1	yes	characteristic
182	C₂²×C₅	#410	1	1	yes	characteristic
183	C₂²×C₅	#413	1	1	yes	characteristic
184	C₂²×C₅	C₂×#103	1	1	yes	characteristic
185	C₂²×C₅	#416	1	1	yes	characteristic
186	C₂²×C₅	#421	1	1	yes	characteristic
187	C₂²×C₅	#421	1	1	yes	characteristic
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	2	1	no	—
190	C₄×C₅	—	2	1	no	—
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	—	2	1	no	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	2	2	no	—
195	C₄×C₅	—	4	2	no	—
196	C₄×C₅	—	4	1	no	—
197	C₄×C₅	—	4	2	no	—
198	C₄×C₅	—	4	1	no	—
199	C₄×C₅	—	8	1	no	—
200	C₄×C₅	—	8	1	no	—
201	C₂×Q₈	—	2	1	no	—
202	C₂×Q₈	—	4	1	no	—
203	C₂×Q₈	—	4	1	no	—
204	C₂×Q₈	#681	1	2	yes	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
211	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
213	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
216	C₂×C₈	—	20	2	no	—
217	C₂×C₈	—	20	2	no	—
218	C₂×C₈	—	20	2	no	—
219	C₂×C₈	—	20	2	no	—
220	C₂×C₈	—	20	2	no	—
221	C₂×C₈	—	20	2	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	#681	1	2	yes	—
229	C₄²	—	2	4	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	4	1	no	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	1	no	—
236	C₄²	—	4	1	no	—
237	C₄²	—	4	2	no	—
238	C₄²	#688	1	2	yes	—
239	C₄²	#689	1	2	yes	—
240	C₂×C₅	#1464	1	1	yes	characteristic
241	C₂×C₅	#1487	1	1	yes	characteristic
242	C₂×C₅	#1488	1	1	yes	characteristic
243	C₂×C₅	#1518	1	1	yes	characteristic
244	C₂×C₅	#1521	1	1	yes	characteristic
245	C₂×C₅	#1535	1	1	yes	characteristic
246	C₂×C₅	#1542	1	1	yes	characteristic
247	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
248	Q₈	—	4	2	no	—
249	Q₈	—	4	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
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	2	no	—
262	C₂×C₄	—	2	2	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	—	2	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	#3967	1	2	yes	—
279	C₂×C₄	#3967	1	2	yes	—
280	C₂×C₄	C₂×#681	1	1	yes	characteristic
281	C₂×C₄	C₂×#681	1	1	yes	characteristic
282	C₂×C₄	C₂×#688	1	1	yes	characteristic
283	C₂×C₄	C₂×#689	1	1	yes	characteristic
284	C₂×C₄	#4096	1	2	yes	—
285	C₈	—	40	2	no	—
286	C₈	—	40	2	no	—
287	C₅	#7644	1	1	yes	characteristic, complemented, Sylow
288	C₂²	#65496	1	1	yes	characteristic
289	C₂²	C₂×#3967	1	1	yes	characteristic
290	C₂²	#65522	1	1	yes	characteristic
291	C₂²	#65525	1	1	yes	characteristic
292	C₂²	#65525	1	1	yes	characteristic
293	C₂²	#65526	1	1	yes	characteristic
294	C₂²	#65529	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	2	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	2	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	2	no	—
306	C₄	—	8	1	no	—
307	C₄	—	8	1	no	—
308	C₂	#10602365	1	1	yes	characteristic
309	C₂	#10602415	1	1	yes	characteristic
310	C₂	#10602433	1	1	yes	characteristic
311	C₂	#10602542	1	1	yes	characteristic
312	C₂	#10602550	1	1	yes	characteristic
313	C₂	#10602621	1	1	yes	characteristic
314	C₂	#10602631	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2295	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#2295	D₅
hypocenter	C₅	#7644
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2295

Complements

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

N	H ≅ G/N	type
287. C₅	#7644	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2295	2
#10600186	2
#10600185	2
#10598133	2
#10598129	2
#7644	5