Subgroups — #11987539

Order 1280.

434 conjugacy classes of subgroups.

The subgroup lattice has 322 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	#10600184	C₂	1	1	yes	maximal, characteristic
3	#10600182	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	C₂×#3935	—	2	1	no	—
19	C₂×#3935	—	2	1	no	—
20	#64744	—	2	1	no	—
21	#64744	C₂²	1	2	yes	—
22	#64744	C₂²	1	2	yes	—
23	#7630	—	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	—	2	4	no	—
52	#3935	—	2	4	no	—
53	#3935	—	4	2	no	—
54	#2295	D₅	1	1	yes	characteristic
55	#1767	—	5	1	no	—
56	#1745	—	5	1	no	—
57	#1272	—	5	2	no	—
58	#1271	—	5	2	no	—
59	C₂×C₅×Q₈	—	2	1	no	—
60	C₂×C₅×Q₈	—	4	1	no	—
61	C₂×C₅×Q₈	—	4	1	no	—
62	C₂×C₅×Q₈	M₁₆	1	2	yes	—
63	C₂²×C₄×C₅	—	2	1	no	—
64	C₂²×C₄×C₅	—	2	1	no	—
65	C₂²×C₄×C₅	—	2	1	no	—
66	C₂²×C₄×C₅	—	2	1	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₅×C₄⋊C₄	—	2	4	no	—
75	C₅×C₄⋊C₄	—	2	1	no	—
76	C₅×C₄⋊C₄	—	4	1	no	—
77	C₅×C₄⋊C₄	—	4	1	no	—
78	C₅×C₄⋊C₄	—	4	2	no	—
79	C₅×C₄⋊C₄	—	4	1	no	—
80	C₅×C₄⋊C₄	—	4	1	no	—
81	C₅×C₄⋊C₄	M₁₆	1	2	yes	—
82	C₄²×C₅	—	2	4	no	—
83	C₄²×C₅	—	2	2	no	—
84	C₄²×C₅	—	2	2	no	—
85	C₄²×C₅	—	2	1	no	—
86	C₄²×C₅	—	2	1	no	—
87	C₄²×C₅	—	2	1	no	—
88	C₄²×C₅	—	2	1	no	—
89	C₄²×C₅	—	2	2	no	—
90	C₄²×C₅	—	4	1	no	—
91	C₄²×C₅	—	4	1	no	—
92	C₄²×C₅	—	4	2	no	—
93	C₂×#168	—	2	4	no	—
94	C₂×#168	—	2	4	no	—
95	C₂×#168	—	4	2	no	—
96	C₂×#168	—	4	2	no	—
97	C₂×#168	—	4	2	no	—
98	C₂×#168	—	4	2	no	—
99	C₂×C₄×Q₈	Dic₅	1	1	yes	characteristic
100	C₂×C₄⋊C₈	—	10	1	no	—
101	C₂×#97	—	10	1	no	—
102	C₂×#97	—	10	1	no	—
103	#391	Dic₅	1	1	yes	characteristic
104	#391	C₂×D₅	1	1	yes	characteristic
105	#389	Dic₅	1	1	yes	characteristic
106	#389	C₂×D₅	1	1	yes	characteristic
107	#387	—	2	1	no	—
108	#387	—	2	1	no	—
109	C₄×C₄⋊C₄	—	2	1	no	—
110	C₄×C₄⋊C₄	Dic₅	1	1	yes	characteristic
111	#376	—	2	1	no	—
112	C₄³	C₂×D₅	1	1	yes	characteristic
113	#336	—	5	2	no	—
114	#336	—	5	2	no	—
115	#336	—	10	1	no	—
116	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₅×Q₈	—	4	2	no	—
118	C₅×Q₈	—	4	2	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	1	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	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	—	4	1	no	—
145	C₂×C₄×C₅	—	4	1	no	—
146	C₂×C₄×C₅	—	4	1	no	—
147	C₂×C₄×C₅	#104	1	2	yes	—
148	C₂×C₄×C₅	#104	1	2	yes	—
149	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
151	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄×C₅	#137	1	1	yes	characteristic
153	C₂×C₄×C₅	#137	1	2	yes	—
154	#168	—	4	4	no	—
155	#168	—	4	4	no	—
156	C₂²×Q₈	C₂×Dic₅	1	1	yes	characteristic
157	C₂²×C₈	—	10	2	no	—
158	C₂²×C₈	—	10	2	no	—
159	C₄×Q₈	—	2	2	no	—
160	C₄×Q₈	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
166	C₂×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	—	2	1	no	—
169	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
170	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₅	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	C₄⋊C₈	—	10	4	no	—
177	#97	—	10	4	no	—
178	#97	—	20	2	no	—
179	#95	—	2	1	no	—
180	#95	—	2	1	no	—
181	#95	—	2	1	no	—
182	#95	—	2	1	no	—
183	#95	—	2	1	no	—
184	#95	—	2	1	no	—
185	#95	—	2	1	no	—
186	C₂²×C₅	#407	1	1	yes	characteristic
187	C₂²×C₅	#410	1	1	yes	characteristic
188	C₂²×C₅	#413	1	1	yes	characteristic
189	C₂²×C₅	#417	1	1	yes	characteristic
190	C₂²×C₅	#419	1	1	yes	characteristic
191	C₂²×C₅	C₂×#104	1	1	yes	characteristic
192	C₂²×C₅	C₂×#104	1	1	yes	characteristic
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	2	1	no	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	—	2	2	no	—
197	C₄×C₅	—	2	1	no	—
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	4	1	no	—
201	C₄×C₅	—	4	2	no	—
202	C₄×C₅	—	4	1	no	—
203	C₄×C₅	—	4	2	no	—
204	C₄×C₅	—	8	1	no	—
205	C₄×C₅	—	8	1	no	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	4	1	no	—
208	C₂×Q₈	—	4	1	no	—
209	C₂×Q₈	#681	1	2	yes	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
215	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
217	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
221	C₂×C₈	—	10	4	no	—
222	C₂×C₈	—	10	4	no	—
223	C₂×C₈	—	20	2	no	—
224	C₂×C₈	—	20	2	no	—
225	C₂×C₈	—	20	2	no	—
226	C₂×C₈	—	20	2	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	#681	1	2	yes	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	2	no	—
246	C₂×C₅	#1464	1	1	yes	characteristic
247	C₂×C₅	#1489	1	1	yes	characteristic
248	C₂×C₅	#1491	1	1	yes	characteristic
249	C₂×C₅	#1506	1	1	yes	characteristic
250	C₂×C₅	#1509	1	1	yes	characteristic
251	C₂×C₅	#1524	1	1	yes	characteristic
252	C₂×C₅	#1559	1	1	yes	characteristic
253	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
254	Q₈	—	4	2	no	—
255	Q₈	—	4	2	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	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	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	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	#3969	1	2	yes	—
285	C₂×C₄	#3969	1	2	yes	—
286	C₂×C₄	C₂×#681	1	1	yes	characteristic
287	C₂×C₄	C₂×#681	1	1	yes	characteristic
288	C₂×C₄	#4088	1	2	yes	—
289	C₂×C₄	#4095	1	1	yes	characteristic
290	C₂×C₄	#4097	1	1	yes	characteristic
291	C₈	—	20	4	no	—
292	C₈	—	20	4	no	—
293	C₅	#7630	1	1	yes	characteristic, complemented, Sylow
294	C₂²	#65496	1	1	yes	characteristic
295	C₂²	#65514	1	1	yes	characteristic
296	C₂²	#65521	1	1	yes	characteristic
297	C₂²	C₂×#3969	1	1	yes	characteristic
298	C₂²	C₂×#3969	1	1	yes	characteristic
299	C₂²	#65526	1	1	yes	characteristic
300	C₂²	#65529	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	2	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	1	no	—
313	C₄	—	8	1	no	—
314	C₂	#10602365	1	1	yes	characteristic
315	C₂	#10602416	1	1	yes	characteristic
316	C₂	#10602434	1	1	yes	characteristic
317	C₂	#10602539	1	1	yes	characteristic
318	C₂	#10602548	1	1	yes	characteristic
319	C₂	#10602626	1	1	yes	characteristic
320	C₂	#10602629	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	#7630
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
293. C₅	#7630	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
#10600184	2
#10600182	2
#10598133	2
#10598129	2
#7630	5