Subgroups — #11989389

Order 1280.

706 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2285	C₂	1	1	yes	maximal, characteristic
2	C₂×#65214	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
4	#10598125	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#117	—	2	1	no	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂³×#168	—	2	1	no	—
8	C₅×#384	C₂²	1	2	yes	—
9	C₅×#382	—	2	1	no	—
10	C₅×#380	—	2	1	no	—
11	C₅×#380	C₄	1	2	yes	—
12	C₄×C₅×C₄⋊C₄	—	2	1	no	—
13	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
14	C₂×C₂²⋊#168	—	2	1	no	—
15	#65214	—	2	2	no	—
16	#65214	C₄	1	4	yes	complemented
17	C₂×C₄⋊#168	—	2	2	no	—
18	C₂×C₄⋊#168	—	2	1	no	—
19	C₂×C₄⋊#168	C₄	1	4	yes	complemented
20	C₂×C₄⋊#168	C₂²	1	2	yes	—
21	C₂×C₄⋊#168	C₂²	1	2	yes	—
22	C₂×C₄×#168	—	2	1	no	—
23	#8039	—	5	1	no	maximal, Sylow
24	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
27	C₅×#117	—	4	2	no	—
28	C₅×#117	—	4	2	no	—
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₅×C₄⋊C₄	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
32	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
33	C₂×C₄²×C₅	—	2	2	no	—
34	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	2	yes	—
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	Q₈	1	2	yes	—
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₂²×#168	—	2	4	no	—
43	C₂²×#168	—	2	2	no	—
44	C₂²×#168	—	2	2	no	—
45	C₂²×#168	—	4	2	no	—
46	C₂²×#168	—	4	2	no	—
47	C₅×#95	—	2	2	no	—
48	C₅×#95	—	2	2	no	—
49	C₂²⋊#168	—	2	4	no	—
50	C₄⋊#168	—	2	4	no	—
51	C₄⋊#168	—	2	2	no	—
52	C₄⋊#168	—	2	2	no	—
53	C₄⋊#168	C₂×C₄	1	4	yes	—
54	C₄⋊#168	C₂×C₄	1	4	yes	—
55	C₄×#168	—	2	4	no	—
56	C₂×#431	—	5	1	no	—
57	C₂²×C₄⋊C₈	—	5	1	no	—
58	#2285	D₅	1	1	yes	characteristic
59	#1296	—	5	4	no	—
60	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	—	2	4	no	—
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	—	4	2	no	—
64	C₂²×C₄×C₅	C₄²	1	2	yes	—
65	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
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	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₅	D₈	1	2	yes	—
74	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
75	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
77	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	4	4	no	—
81	C₅×C₄⋊C₄	—	4	2	no	—
82	C₅×C₄⋊C₄	—	4	2	no	—
83	C₅×C₄⋊C₄	—	4	4	no	—
84	C₅×C₂²⋊C₄	—	4	4	no	—
85	C₄²×C₅	—	4	4	no	—
86	C₄²×C₅	C₄²	1	2	yes	—
87	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₅	C₄⋊C₄	1	2	yes	—
90	C₄²×C₅	C₄⋊C₄	1	2	yes	—
91	C₄²×C₅	C₄⋊C₄	1	2	yes	—
92	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
96	C₂×#168	—	2	4	no	—
97	C₂×#168	—	2	4	no	—
98	C₂×#168	—	4	8	no	—
99	C₂×#168	—	4	2	no	—
100	C₂×#168	—	4	2	no	—
101	C₂×#168	—	4	4	no	—
102	C₂³×C₈	—	10	1	no	—
103	C₂×#117	—	2	1	no	—
104	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
105	#431	—	5	4	no	—
106	#431	—	10	2	no	—
107	C₂×C₄⋊C₈	—	5	4	no	—
108	C₂×C₄⋊C₈	—	5	2	no	—
109	C₂×C₄⋊C₈	—	5	2	no	—
110	C₂×C₄⋊C₈	—	10	2	no	—
111	C₂×C₄⋊C₈	—	10	1	no	—
112	C₂×C₂²⋊C₈	—	10	1	no	—
113	C₂×C₄×C₈	—	10	1	no	—
114	#384	C₂×D₅	1	2	yes	—
115	#382	—	2	1	no	—
116	#380	—	2	1	no	—
117	#380	Dic₅	1	2	yes	—
118	C₄×C₄⋊C₄	—	2	1	no	—
119	C₄×C₄⋊C₄	Dic₅	1	2	yes	—
120	C₂³×C₅	—	2	2	no	—
121	C₂³×C₅	—	2	2	no	—
122	C₂³×C₅	#95	1	2	yes	—
123	C₂³×C₅	#102	1	2	yes	—
124	C₂³×C₅	#103	1	2	yes	—
125	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	—	4	4	no	—
135	C₂×C₄×C₅	—	4	4	no	—
136	C₂×C₄×C₅	#95	1	2	yes	—
137	C₂×C₄×C₅	#95	1	2	yes	—
138	C₂×C₄×C₅	#95	1	2	yes	—
139	C₂×C₄×C₅	#102	1	4	yes	—
140	C₂×C₄×C₅	#102	1	2	yes	—
141	C₂×C₄×C₅	#103	1	4	yes	—
142	C₂×C₄×C₅	#103	1	2	yes	—
143	C₂×C₄×C₅	#106	1	4	yes	—
144	C₂×C₄×C₅	#107	1	4	yes	—
145	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
155	C₂×C₄×C₅	#135	1	4	yes	—
156	#168	—	4	4	no	—
157	#168	—	4	4	no	—
158	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
159	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
160	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
161	C₂²×C₈	—	10	4	no	—
162	C₂²×C₈	—	10	2	no	—
163	C₂²×C₈	—	10	2	no	—
164	C₂²×C₈	—	20	2	no	—
165	C₂²×C₈	—	20	2	no	—
166	#117	—	4	2	no	—
167	#117	—	4	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	Dic₁₀	1	2	yes	—
174	C₂×C₄²	C₄×D₅	1	2	yes	—
175	C₂×C₄²	C₄×D₅	1	2	yes	—
176	C₂×C₄²	D₂₀	1	2	yes	—
177	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
178	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
181	C₄⋊C₈	—	5	4	no	—
182	C₄⋊C₈	—	5	4	no	—
183	C₄⋊C₈	—	10	4	no	—
184	C₄⋊C₈	—	10	2	no	—
185	C₄⋊C₈	—	10	2	no	—
186	C₂²⋊C₈	—	10	4	no	—
187	C₄×C₈	—	10	4	no	—
188	#95	—	2	2	no	—
189	#95	—	2	2	no	—
190	C₂²×C₅	—	2	4	no	—
191	C₂²×C₅	—	2	4	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	#340	1	4	yes	—
195	C₂²×C₅	C₂×#95	1	1	yes	characteristic
196	C₂²×C₅	#413	1	2	yes	—
197	C₂²×C₅	C₂×#102	1	1	yes	characteristic
198	C₂²×C₅	C₂×#103	1	1	yes	characteristic
199	C₂²×C₅	#416	1	2	yes	—
200	C₄×C₅	—	2	2	no	—
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	2	1	no	—
204	C₄×C₅	—	8	4	no	—
205	C₄×C₅	#340	1	4	yes	—
206	C₄×C₅	C₂×#102	1	1	yes	characteristic
207	C₄×C₅	C₂×#103	1	1	yes	characteristic
208	C₄×C₅	#416	1	2	yes	—
209	C₄×C₅	C₂×#106	1	1	yes	characteristic
210	C₄×C₅	C₂×#107	1	1	yes	characteristic
211	C₄×C₅	#427	1	2	yes	—
212	C₂⁴	C₂²⋊Dic₅	1	1	yes	characteristic
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	C₄×Dic₅	1	2	yes	—
217	C₂²×C₄	#683	1	2	yes	—
218	C₂²×C₄	#683	1	2	yes	—
219	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
220	C₂²×C₄	#685	1	2	yes	—
221	C₂²×C₄	#685	1	2	yes	—
222	C₂²×C₄	#686	1	2	yes	—
223	C₂²×C₄	#687	1	2	yes	—
224	C₂²×C₄	#688	1	2	yes	—
225	C₂²×C₄	#689	1	2	yes	—
226	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
229	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
232	C₂×C₈	—	10	4	no	—
233	C₂×C₈	—	10	4	no	—
234	C₂×C₈	—	20	4	no	—
235	C₂×C₈	—	20	8	no	—
236	C₂×C₈	—	20	2	no	—
237	C₂×C₈	—	20	2	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	4	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₄²	—	4	4	no	—
244	C₄²	C₄×Dic₅	1	2	yes	—
245	C₄²	#683	1	2	yes	—
246	C₄²	#683	1	2	yes	—
247	C₄²	C₄⋊Dic₅	1	2	yes	—
248	C₄²	#685	1	2	yes	—
249	C₄²	#685	1	2	yes	—
250	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
251	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
252	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
253	C₄²	C₂×D₂₀	1	1	yes	characteristic
254	C₂×C₅	—	2	4	no	—
255	C₂×C₅	#1540	1	2	yes	—
256	C₂×C₅	#1542	1	2	yes	—
257	C₂×C₅	C₂×#340	1	1	yes	characteristic
258	C₂×C₅	#1735	1	1	yes	characteristic
259	C₂×C₅	#1742	1	1	yes	characteristic
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	#3963	1	2	yes	—
263	C₂³	#3964	1	2	yes	—
264	C₂³	#3967	1	2	yes	—
265	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	#3939	1	4	yes	—
277	C₂×C₄	#3940	1	4	yes	—
278	C₂×C₄	#3941	1	4	yes	—
279	C₂×C₄	#3942	1	4	yes	—
280	C₂×C₄	#3963	1	2	yes	—
281	C₂×C₄	#3963	1	2	yes	—
282	C₂×C₄	#3963	1	2	yes	—
283	C₂×C₄	#3964	1	2	yes	—
284	C₂×C₄	#3967	1	2	yes	—
285	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
286	C₂×C₄	C₂×#683	1	1	yes	characteristic
287	C₂×C₄	C₂×#683	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
289	C₂×C₄	C₂×#685	1	1	yes	characteristic
290	C₂×C₄	C₂×#685	1	1	yes	characteristic
291	C₂×C₄	C₂×#686	1	1	yes	characteristic
292	C₂×C₄	C₂×#687	1	1	yes	characteristic
293	C₂×C₄	C₂×#688	1	1	yes	characteristic
294	C₂×C₄	C₂×#689	1	1	yes	characteristic
295	C₂×C₄	#4096	1	4	yes	—
296	C₈	—	20	4	no	—
297	C₈	—	20	4	no	—
298	C₅	#8039	1	1	yes	characteristic, complemented, Sylow
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	#64749	1	4	yes	—
304	C₂²	C₂×#3963	1	1	yes	characteristic
305	C₂²	C₂×#3964	1	1	yes	characteristic
306	C₂²	C₂×#3967	1	1	yes	characteristic
307	C₂²	#65522	1	2	yes	—
308	C₂²	#65526	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	8	4	no	—
314	C₄	#64749	1	4	yes	—
315	C₄	C₂×#3939	1	1	yes	characteristic
316	C₄	C₂×#3940	1	1	yes	characteristic
317	C₄	C₂×#3941	1	1	yes	characteristic
318	C₄	C₂×#3942	1	1	yes	characteristic
319	C₄	#65287	1	2	yes	—
320	C₄	#65291	1	2	yes	—
321	C₂	—	2	4	no	—
322	C₂	C₂×#64749	1	1	yes	characteristic
323	C₂	#10602373	1	1	yes	characteristic
324	C₂	#10602380	1	1	yes	characteristic
325	C₂	#10602597	1	2	yes	—
326	C₂	#10602621	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2285	C₂
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅	C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#2285	D₅
hypocenter	C₅	#8039
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
4. #10598125	C₂	semidirect
16. #65214	C₄	semidirect
19. C₂×C₄⋊#168	C₄	semidirect
298. C₅	#8039	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2285	2
C₂×#65214	2
C₂²×C₄⋊#168	2
#10598125	2
#8039	5