Subgroups — #11987271

Order 1280.

444 conjugacy classes of subgroups.

The subgroup lattice has 306 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	C₅×#1744	C₂	1	1	yes	maximal, characteristic
3	#10600181	C₂	1	1	yes	maximal, characteristic
4	#10597969	C₂	1	2	yes	maximal
5	#10597968	C₂	1	2	yes	maximal
6	C₂×#4016	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	—	2	1	no	—
8	C₄³×C₅	C₂²	1	1	yes	characteristic
9	#65225	C₄	1	1	yes	characteristic
10	C₄×C₄⋊Dic₅	C₄	1	1	yes	characteristic
11	C₄×C₄⋊Dic₅	C₄	1	1	yes	characteristic
12	#65222	—	2	1	no	—
13	#65222	—	2	1	no	—
14	C₄×#683	—	2	1	no	—
15	C₄×#683	—	2	1	no	—
16	#65217	C₂²	1	1	yes	characteristic
17	#65217	C₂²	1	1	yes	characteristic
18	C₂×C₄×#168	—	2	1	no	—
19	C₅×#336	—	2	1	no	—
20	C₅×#336	C₂²	1	2	yes	—
21	#64744	—	2	1	no	—
22	#64744	C₂²	1	2	yes	—
23	#7568	—	5	1	no	maximal, Sylow
24	C₂²×C₅×C₈	—	2	2	no	—
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	D₄	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	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₅	C₂×C₄	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×#683	—	2	1	no	—
37	C₂×#683	—	2	1	no	—
38	C₂×#683	—	2	1	no	—
39	C₂×#683	—	2	1	no	—
40	C₂×C₄×Dic₅	—	2	1	no	—
41	C₂×C₄×Dic₅	—	2	1	no	—
42	C₂²×#168	—	2	2	no	—
43	#4016	—	2	2	no	—
44	#4016	C₈	1	4	yes	complemented
45	C₄×C₅×C₈	—	2	4	no	—
46	#3963	—	2	1	no	—
47	#3963	—	2	1	no	—
48	#3963	—	2	1	no	—
49	#3963	—	2	1	no	—
50	#3963	—	2	1	no	—
51	#3963	—	2	1	no	—
52	C₄×#168	—	2	4	no	—
53	C₄×#125	—	5	1	no	—
54	#1744	—	5	1	no	—
55	#1744	D₅	1	1	yes	characteristic
56	#1271	—	5	2	no	—
57	#1271	—	5	2	no	—
58	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₅	—	2	1	no	—
66	C₂²×Dic₅	—	2	1	no	—
67	C₂²×Dic₅	—	2	1	no	—
68	C₂²×Dic₅	—	2	1	no	—
69	C₂×C₅×C₈	—	2	4	no	—
70	C₂×C₅×C₈	—	4	2	no	—
71	C₂×C₅×C₈	—	4	2	no	—
72	C₄²×C₅	—	2	2	no	—
73	C₄²×C₅	—	2	2	no	—
74	C₄²×C₅	—	2	2	no	—
75	C₄²×C₅	—	2	2	no	—
76	C₄²×C₅	—	2	2	no	—
77	C₄²×C₅	—	4	1	no	—
78	C₄²×C₅	C₂×C₈	1	2	yes	—
79	C₄²×C₅	M₁₆	1	2	yes	—
80	C₄⋊Dic₅	—	2	1	no	—
81	C₄⋊Dic₅	—	2	1	no	—
82	C₄⋊Dic₅	—	4	1	no	—
83	C₄⋊Dic₅	—	4	1	no	—
84	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
85	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
86	#683	—	2	4	no	—
87	#683	—	2	4	no	—
88	#683	—	4	2	no	—
89	#683	—	4	2	no	—
90	C₄×Dic₅	—	4	2	no	—
91	C₄×Dic₅	—	4	2	no	—
92	C₂×#168	—	2	4	no	—
93	C₂×#168	—	4	2	no	—
94	C₂×#168	—	4	2	no	—
95	C₂×#125	—	5	1	no	—
96	C₂×C₄×C₈	—	2	1	no	—
97	C₂×C₄×C₈	—	10	1	no	—
98	#389	—	5	1	no	—
99	#389	—	5	1	no	—
100	#387	—	10	1	no	—
101	#387	—	10	1	no	—
102	#382	—	5	1	no	—
103	C₄×C₄⋊C₄	—	5	1	no	—
104	C₄×C₄⋊C₄	—	5	1	no	—
105	C₄×C₄⋊C₄	—	10	1	no	—
106	C₄×C₄⋊C₄	—	10	1	no	—
107	C₄³	C₂×D₅	1	1	yes	characteristic
108	#336	—	2	1	no	—
109	#336	—	5	2	no	—
110	#336	—	10	1	no	—
111	#336	C₂×D₅	1	2	yes	—
112	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	C₂²⋊C₈	1	2	yes	—
127	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
128	C₂×C₄×C₅	#101	1	2	yes	—
129	C₂×C₄×C₅	#104	1	2	yes	—
130	C₂×C₄×C₅	#104	1	2	yes	—
131	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
133	C₂×C₄×C₅	#135	1	2	yes	—
134	C₂×C₄×C₅	#135	1	2	yes	—
135	C₂×Dic₅	—	2	2	no	—
136	C₂×Dic₅	—	2	2	no	—
137	C₂×Dic₅	—	4	1	no	—
138	C₂×Dic₅	—	4	1	no	—
139	C₂×Dic₅	—	4	1	no	—
140	C₂×Dic₅	—	4	1	no	—
141	C₂×Dic₅	—	4	1	no	—
142	C₂×Dic₅	—	4	1	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	C₅×C₈	—	4	4	no	—
148	#168	—	4	4	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	10	2	no	—
151	#125	—	5	4	no	—
152	#125	—	10	2	no	—
153	C₂×C₄⋊C₄	—	5	1	no	—
154	C₂×C₄⋊C₄	—	5	1	no	—
155	C₂×C₄⋊C₄	—	5	1	no	—
156	C₂×C₄⋊C₄	—	5	1	no	—
157	C₂×C₄⋊C₄	—	10	1	no	—
158	C₂×C₄⋊C₄	—	10	1	no	—
159	C₂×C₄⋊C₄	—	10	1	no	—
160	C₂×C₄⋊C₄	—	10	1	no	—
161	C₂×C₄²	—	10	1	no	—
162	C₂×C₄²	—	10	1	no	—
163	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
164	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
165	C₂×C₄²	D₂₀	1	1	yes	characteristic
166	C₂×C₄²	D₂₀	1	1	yes	characteristic
167	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
170	C₄×C₈	—	2	4	no	—
171	C₄×C₈	—	10	4	no	—
172	#95	—	10	1	no	—
173	#95	—	10	1	no	—
174	#95	—	10	1	no	—
175	#95	—	10	1	no	—
176	#95	—	10	1	no	—
177	#95	—	10	1	no	—
178	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
179	C₂²×C₅	#410	1	1	yes	characteristic
180	C₂²×C₅	C₂×#101	1	1	yes	characteristic
181	C₂²×C₅	#416	1	1	yes	characteristic
182	C₂²×C₅	#416	1	1	yes	characteristic
183	C₂²×C₅	C₂×#104	1	1	yes	characteristic
184	C₂²×C₅	C₂×#104	1	1	yes	characteristic
185	C₄×C₅	—	2	1	no	—
186	C₄×C₅	—	2	1	no	—
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	4	1	no	—
192	C₄×C₅	#330	1	4	yes	—
193	Dic₅	—	4	2	no	—
194	Dic₅	—	4	2	no	—
195	Dic₅	—	8	1	no	—
196	Dic₅	—	8	1	no	—
197	C₂²×C₄	—	10	1	no	—
198	C₂²×C₄	—	10	1	no	—
199	C₂²×C₄	—	10	1	no	—
200	C₂²×C₄	—	10	1	no	—
201	C₂²×C₄	#685	1	1	yes	characteristic
202	C₂²×C₄	#685	1	1	yes	characteristic
203	C₂²×C₄	#685	1	1	yes	characteristic
204	C₂²×C₄	#685	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	10	4	no	—
212	C₂×C₈	—	20	2	no	—
213	C₂×C₈	—	20	2	no	—
214	C₄⋊C₄	—	5	2	no	—
215	C₄⋊C₄	—	5	2	no	—
216	C₄⋊C₄	—	10	4	no	—
217	C₄⋊C₄	—	10	4	no	—
218	C₄⋊C₄	—	10	1	no	—
219	C₄⋊C₄	—	10	1	no	—
220	C₄⋊C₄	—	20	1	no	—
221	C₄⋊C₄	—	20	1	no	—
222	C₄⋊C₄	—	20	2	no	—
223	C₄⋊C₄	—	20	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	4	1	no	—
230	C₄²	—	20	2	no	—
231	C₄²	—	20	2	no	—
232	C₄²	C₈×D₅	1	2	yes	—
233	C₄²	#676	1	2	yes	—
234	C₂×C₅	#1461	1	1	yes	characteristic
235	C₂×C₅	#1476	1	1	yes	characteristic
236	C₂×C₅	#1476	1	1	yes	characteristic
237	C₂×C₅	#1500	1	1	yes	characteristic
238	C₂×C₅	#1500	1	1	yes	characteristic
239	C₂×C₅	C₂×#330	1	1	yes	characteristic
240	C₂×C₅	#1544	1	1	yes	characteristic
241	C₂³	C₂×#685	1	1	yes	characteristic
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	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	10	2	no	—
256	C₂×C₄	—	10	2	no	—
257	C₂×C₄	—	20	1	no	—
258	C₂×C₄	—	20	1	no	—
259	C₂×C₄	—	20	1	no	—
260	C₂×C₄	—	20	1	no	—
261	C₂×C₄	—	20	1	no	—
262	C₂×C₄	—	20	1	no	—
263	C₂×C₄	—	20	1	no	—
264	C₂×C₄	—	20	1	no	—
265	C₂×C₄	—	20	1	no	—
266	C₂×C₄	—	20	1	no	—
267	C₂×C₄	#3937	1	2	yes	—
268	C₂×C₄	#3952	1	2	yes	—
269	C₂×C₄	#3952	1	2	yes	—
270	C₂×C₄	#3956	1	2	yes	—
271	C₂×C₄	#3957	1	2	yes	—
272	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
273	C₂×C₄	C₂×#676	1	1	yes	characteristic
274	C₂×C₄	#4050	1	2	yes	—
275	C₂×C₄	#4096	1	2	yes	—
276	C₈	—	4	4	no	—
277	C₈	—	20	4	no	—
278	C₅	#7568	1	1	yes	characteristic, complemented, Sylow
279	C₂²	C₂×#3937	1	1	yes	characteristic
280	C₂²	#65291	1	1	yes	characteristic
281	C₂²	#65295	1	1	yes	characteristic
282	C₂²	C₂×#3952	1	1	yes	characteristic
283	C₂²	#65402	1	1	yes	characteristic
284	C₂²	C₂×#3956	1	1	yes	characteristic
285	C₂²	C₂×#3957	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	1	no	—
293	C₄	—	20	2	no	—
294	C₄	—	20	2	no	—
295	C₄	—	40	1	no	—
296	C₄	—	40	1	no	—
297	C₄	#64685	1	4	yes	—
298	C₂	#10600224	1	1	yes	characteristic
299	C₂	#10600527	1	1	yes	characteristic
300	C₂	C₂×#64685	1	1	yes	characteristic
301	C₂	#10601547	1	1	yes	characteristic
302	C₂	#10601570	1	1	yes	characteristic
303	C₂	#10601579	1	1	yes	characteristic
304	C₂	#10601581	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#685
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1744	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1744	D₅
hypocenter	C₅	#7568
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
44. #4016	C₈	semidirect
278. C₅	#7568	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
C₅×#1744	2
#10600181	2
#10597969	2
#10597968	2
#7568	5