Subgroups — #11987883

Order 1280.

372 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2863	C₂	1	1	yes	maximal, characteristic
2	#10600185	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	C₅×#391	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	C₄	1	1	yes	characteristic
11	C₅×#389	C₂²	1	1	yes	characteristic
12	C₅×#387	—	2	1	no	—
13	C₅×#387	—	2	1	no	—
14	C₅×#387	—	2	1	no	—
15	C₅×#383	—	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	#64744	—	2	1	no	—
20	#64744	—	2	1	no	—
21	#64744	C₂²	1	2	yes	—
22	#64744	C₂²	1	2	yes	—
23	#7709	—	5	1	no	maximal, Sylow
24	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅×C₄⋊C₄	—	2	1	no	—
26	C₂×C₅×C₄⋊C₄	—	2	1	no	—
27	C₂×C₅×C₄⋊C₄	—	2	1	no	—
28	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₂²×#168	—	2	2	no	—
39	C₂²×#168	—	2	2	no	—
40	C₅×#95	—	2	1	no	—
41	C₅×#95	—	2	1	no	—
42	C₅×#95	—	2	1	no	—
43	C₅×#95	—	2	1	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	C₄⋊#168	—	4	2	no	—
50	#3935	—	2	4	no	—
51	#2863	D₅	1	1	yes	characteristic
52	#1831	—	5	1	no	—
53	#1745	—	5	1	no	—
54	#1272	—	5	2	no	—
55	#1271	—	5	2	no	—
56	C₂×C₅×Q₈	—	2	2	no	—
57	C₂×C₅×Q₈	—	4	1	no	—
58	C₂×C₅×Q₈	—	4	1	no	—
59	C₂²×C₄×C₅	—	2	1	no	—
60	C₂²×C₄×C₅	—	2	1	no	—
61	C₂²×C₄×C₅	—	2	1	no	—
62	C₂²×C₄×C₅	—	2	1	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	C₅×C₄⋊C₄	—	2	2	no	—
71	C₅×C₄⋊C₄	—	4	1	no	—
72	C₅×C₄⋊C₄	—	4	1	no	—
73	C₅×C₄⋊C₄	—	4	2	no	—
74	C₅×C₄⋊C₄	—	4	2	no	—
75	C₅×C₄⋊C₄	—	4	1	no	—
76	C₅×C₄⋊C₄	—	4	1	no	—
77	C₄²×C₅	—	2	2	no	—
78	C₄²×C₅	—	2	2	no	—
79	C₄²×C₅	—	2	1	no	—
80	C₄²×C₅	—	2	1	no	—
81	C₄²×C₅	—	2	2	no	—
82	C₄²×C₅	—	4	1	no	—
83	C₄²×C₅	—	4	1	no	—
84	C₄²×C₅	—	4	1	no	—
85	C₂×#168	—	2	4	no	—
86	C₂×#168	—	4	2	no	—
87	C₂×#168	—	4	2	no	—
88	C₂×#168	—	4	2	no	—
89	C₂×#168	—	4	2	no	—
90	C₂×#168	—	4	2	no	—
91	C₂×C₄⋊C₈	—	10	1	no	—
92	C₂×#97	—	10	1	no	—
93	#391	Dic₅	1	1	yes	characteristic
94	#391	Dic₅	1	1	yes	characteristic
95	#391	C₂×D₅	1	1	yes	characteristic
96	#389	Dic₅	1	1	yes	characteristic
97	#389	Dic₅	1	1	yes	characteristic
98	#389	C₂×D₅	1	1	yes	characteristic
99	#387	—	2	1	no	—
100	#387	—	2	1	no	—
101	#387	—	2	1	no	—
102	#383	—	2	1	no	—
103	C₄³	C₂×D₅	1	1	yes	characteristic
104	#336	—	5	2	no	—
105	#336	—	5	2	no	—
106	#336	—	10	1	no	—
107	#336	—	10	1	no	—
108	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₅×Q₈	—	8	2	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	1	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	—	4	1	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	—	4	1	no	—
131	C₂×C₄×C₅	—	4	1	no	—
132	C₂×C₄×C₅	—	4	1	no	—
133	C₂×C₄×C₅	—	4	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	#104	1	2	yes	—
140	C₂×C₄×C₅	#131	1	2	yes	—
141	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₅	#137	1	1	yes	characteristic
143	C₂×C₄×C₅	#137	1	2	yes	—
144	#168	—	4	4	no	—
145	#168	—	8	2	no	—
146	C₂²×Q₈	C₂×Dic₅	1	1	yes	characteristic
147	C₂²×C₈	—	10	2	no	—
148	C₂²×C₈	—	10	2	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
155	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
156	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
157	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
158	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
159	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
160	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
161	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
162	C₄⋊C₈	—	20	2	no	—
163	#97	—	10	4	no	—
164	#95	—	2	1	no	—
165	#95	—	2	1	no	—
166	#95	—	2	1	no	—
167	#95	—	2	1	no	—
168	#95	—	2	1	no	—
169	#95	—	2	1	no	—
170	#95	—	2	1	no	—
171	#95	—	2	1	no	—
172	#95	—	2	1	no	—
173	C₂²×C₅	#408	1	1	yes	characteristic
174	C₂²×C₅	#410	1	1	yes	characteristic
175	C₂²×C₅	#413	1	1	yes	characteristic
176	C₂²×C₅	#417	1	1	yes	characteristic
177	C₂²×C₅	#419	1	1	yes	characteristic
178	C₂²×C₅	C₂×#104	1	1	yes	characteristic
179	C₂²×C₅	#421	1	1	yes	characteristic
180	C₄×C₅	—	2	2	no	—
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	2	1	no	—
184	C₄×C₅	—	2	1	no	—
185	C₄×C₅	—	4	1	no	—
186	C₄×C₅	—	4	1	no	—
187	C₄×C₅	—	4	1	no	—
188	C₄×C₅	—	8	1	no	—
189	C₄×C₅	—	8	1	no	—
190	C₄×C₅	—	8	1	no	—
191	C₄×C₅	—	8	1	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	4	1	no	—
194	C₂×Q₈	—	4	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
201	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
202	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
206	C₂×C₈	—	10	4	no	—
207	C₂×C₈	—	20	2	no	—
208	C₂×C₈	—	20	2	no	—
209	C₂×C₈	—	20	2	no	—
210	C₂×C₈	—	20	2	no	—
211	C₂×C₈	—	20	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	4	1	no	—
225	C₄²	—	4	1	no	—
226	C₄²	—	4	1	no	—
227	C₂×C₅	#1473	1	1	yes	characteristic
228	C₂×C₅	#1483	1	1	yes	characteristic
229	C₂×C₅	#1497	1	1	yes	characteristic
230	C₂×C₅	#1506	1	1	yes	characteristic
231	C₂×C₅	#1512	1	1	yes	characteristic
232	C₂×C₅	#1529	1	1	yes	characteristic
233	C₂×C₅	#1559	1	1	yes	characteristic
234	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
235	Q₈	—	8	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	#3969	1	2	yes	—
266	C₂×C₄	#4088	1	2	yes	—
267	C₂×C₄	#4094	1	2	yes	—
268	C₂×C₄	#4095	1	1	yes	characteristic
269	C₂×C₄	#4097	1	1	yes	characteristic
270	C₈	—	20	4	no	—
271	C₈	—	40	2	no	—
272	C₅	#7709	1	1	yes	characteristic, complemented, Sylow
273	C₂²	#65514	1	1	yes	characteristic
274	C₂²	#65521	1	1	yes	characteristic
275	C₂²	#65523	1	1	yes	characteristic
276	C₂²	C₂×#3969	1	1	yes	characteristic
277	C₂²	#65525	1	1	yes	characteristic
278	C₂²	#65526	1	1	yes	characteristic
279	C₂²	#65529	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	2	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	1	no	—
288	C₄	—	8	1	no	—
289	C₄	—	8	1	no	—
290	C₄	—	8	1	no	—
291	C₄	—	8	1	no	—
292	C₂	#10602538	1	1	yes	characteristic
293	C₂	#10602540	1	1	yes	characteristic
294	C₂	#10602544	1	1	yes	characteristic
295	C₂	#10602549	1	1	yes	characteristic
296	C₂	#10602620	1	1	yes	characteristic
297	C₂	#10602626	1	1	yes	characteristic
298	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×#2863	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#2863	D₅
hypocenter	C₅	#7709
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2863

Complements

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

N	H ≅ G/N	type
272. C₅	#7709	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2863	2
#10600185	2
#10600182	2
#10598133	2
#10598129	2
#7709	5