Subgroups — #12004514

Order 1280.

842 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×#65390	C₂	1	2	yes	maximal
3	C₂×#64776	C₂	1	2	yes	maximal
4	#10602385	C₂	1	1	yes	maximal, characteristic
5	#10602380	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#117	C₂²	1	1	yes	characteristic
9	C₂×#4072	C₂²	1	2	yes	—
10	C₂²×#681	C₂²	1	2	yes	—
11	C₅×#405	—	2	2	no	—
12	C₅×#405	—	2	2	no	—
13	C₂×C₅×#97	C₄	1	2	yes	complemented
14	C₂×C₄×C₅×C₈	C₄	1	2	yes	complemented
15	#65390	—	2	4	no	—
16	#65390	C₄	1	8	yes	complemented
17	C₂×#3947	—	2	2	no	—
18	C₂×C₈×Dic₅	—	2	2	no	—
19	#64776	C₄	1	8	yes	complemented
20	#64750	C₄	1	4	yes	complemented
21	#64746	C₄	1	4	yes	—
22	#10711	—	5	1	no	maximal, Sylow
23	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	—	2	1	no	—
26	C₂×C₅×M₁₆	—	2	1	no	—
27	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
28	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	2	yes	—
33	C₂²×C₅×C₈	D₄	1	2	yes	—
34	C₂²×C₅×C₈	D₄	1	2	yes	—
35	C₂²×C₅×C₈	Q₈	1	2	yes	—
36	C₅×#117	—	2	2	no	—
37	C₅×#117	—	2	2	no	—
38	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊Dic₅	C₂×C₄	1	4	yes	—
42	#4072	—	2	2	no	—
43	#4072	—	2	2	no	—
44	#4072	C₂×C₄	1	4	yes	—
45	#4072	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊Dic₅	—	2	2	no	—
47	C₂×C₄×Dic₅	—	2	2	no	—
48	C₂×#681	—	2	2	no	—
49	C₂×#681	—	2	2	no	—
50	C₂×#681	—	2	2	no	—
51	C₂×#681	C₂×C₄	1	4	yes	—
52	C₂×#681	C₂×C₄	1	4	yes	—
53	C₂×#681	C₂×C₄	1	4	yes	—
54	C₂²×#168	—	2	2	no	—
55	C₅×#97	—	2	4	no	—
56	C₄×C₅×C₈	—	2	2	no	—
57	C₄×C₅×C₈	—	2	2	no	—
58	#3947	—	2	8	no	—
59	C₈×Dic₅	—	2	8	no	—
60	C₂×#405	—	5	2	no	—
61	C₂×#405	D₅	1	1	yes	characteristic
62	C₂×#344	—	5	2	no	—
63	#1742	—	5	1	no	—
64	#1740	—	5	1	no	—
65	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₄²	1	4	yes	—
67	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₅	#41	1	1	yes	characteristic
75	C₂²×C₄×C₅	#41	1	2	yes	—
76	C₂²×C₄×C₅	#41	1	1	yes	characteristic
77	C₂²×Dic₅	—	2	4	no	—
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	—	2	4	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	C₄²	1	4	yes	—
83	C₂×C₅×C₈	C₄²	1	2	yes	—
84	C₂×C₅×C₈	C₄²	1	2	yes	—
85	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
86	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
87	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
89	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
91	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
92	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
93	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
94	C₅×C₄⋊C₄	—	2	4	no	—
95	C₅×C₂²⋊C₄	—	2	4	no	—
96	C₄²×C₅	—	2	2	no	—
97	C₄²×C₅	—	2	2	no	—
98	C₂²⋊Dic₅	—	2	4	no	—
99	C₂²⋊Dic₅	C₄²	1	8	yes	—
100	C₄⋊Dic₅	—	2	4	no	—
101	C₄⋊Dic₅	—	2	4	no	—
102	C₄×Dic₅	—	2	4	no	—
103	C₄×Dic₅	—	2	4	no	—
104	#681	—	2	4	no	—
105	#681	—	2	4	no	—
106	#681	—	2	4	no	—
107	#681	C₄²	1	8	yes	—
108	C₂×#168	—	2	4	no	—
109	C₂×#168	—	2	4	no	—
110	C₂×#168	—	2	4	no	—
111	C₂²×M₁₆	—	5	2	no	—
112	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
113	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
114	C₂×#117	—	5	2	no	—
115	C₂×#117	C₂×D₅	1	1	yes	characteristic
116	#405	—	2	2	no	—
117	#405	—	2	2	no	—
118	#405	—	5	8	no	—
119	#405	—	10	4	no	—
120	C₂×#97	—	10	2	no	—
121	C₂×#97	Dic₅	1	2	yes	complemented
122	C₂×C₄×C₈	—	10	2	no	—
123	C₂×C₄×C₈	Dic₅	1	2	yes	complemented
124	#344	—	5	8	no	—
125	#343	—	5	4	no	—
126	#342	—	5	4	no	—
127	C₂³×C₅	—	4	2	no	—
128	C₂³×C₅	C₂×C₄²	1	2	yes	—
129	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
130	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
131	C₂³×C₅	C₄×D₄	1	2	yes	—
132	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	4	4	no	—
135	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
136	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
140	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	#117	1	2	yes	—
143	C₂×C₄×C₅	#117	1	2	yes	—
144	C₂×C₄×C₅	#117	1	1	yes	characteristic
145	C₂×C₄×C₅	#117	1	1	yes	characteristic
146	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
148	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
149	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
150	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
151	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
152	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
153	C₂×Dic₅	—	2	8	no	—
154	C₂×Dic₅	—	4	8	no	—
155	C₅×C₈	—	2	4	no	—
156	C₅×C₈	#95	1	4	yes	—
157	C₅×C₈	#95	1	2	yes	—
158	C₅×C₈	#95	1	2	yes	—
159	#168	—	2	8	no	—
160	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
161	C₂×M₁₆	—	2	1	no	—
162	C₂×M₁₆	—	2	1	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	5	4	no	—
165	C₂×M₁₆	—	5	4	no	—
166	C₂×M₁₆	—	5	4	no	—
167	C₂×M₁₆	—	10	2	no	—
168	C₂×M₁₆	—	10	2	no	—
169	C₂×M₁₆	—	10	2	no	—
170	C₂×M₁₆	C₄×D₅	1	4	yes	—
171	C₂²×C₈	—	10	2	no	—
172	C₂²×C₈	Dic₁₀	1	2	yes	—
173	C₂²×C₈	C₄×D₅	1	4	yes	—
174	C₂²×C₈	D₂₀	1	2	yes	—
175	C₂²×C₈	C₂×Dic₅	1	2	yes	—
176	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
177	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
178	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
179	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
180	#117	—	2	2	no	—
181	#117	—	2	2	no	—
182	#117	—	5	4	no	—
183	#117	—	5	4	no	—
184	#117	—	10	2	no	—
185	#117	—	10	2	no	—
186	C₂×C₄⋊C₄	—	10	2	no	—
187	C₂×C₄⋊C₄	C₄×D₅	1	2	yes	—
188	C₂×C₂²⋊C₄	—	5	4	no	—
189	C₂×C₂²⋊C₄	C₄×D₅	1	2	yes	—
190	C₂×C₄²	—	10	2	no	—
191	C₂×C₄²	C₂×Dic₅	1	2	yes	—
192	#97	—	2	4	no	—
193	#97	—	10	8	no	—
194	C₄×C₈	—	2	2	no	—
195	C₄×C₈	—	2	2	no	—
196	C₄×C₈	—	10	8	no	—
197	C₂²×C₅	—	2	4	no	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	4	2	no	—
200	C₂²×C₅	—	4	1	no	—
201	C₂²×C₅	—	4	1	no	—
202	C₂²×C₅	C₄³	1	1	yes	characteristic
203	C₂²×C₅	C₄×C₂²⋊C₄	1	2	yes	—
204	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
205	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
206	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
207	C₄×C₅	—	4	4	no	—
208	C₄×C₅	C₂×#95	1	1	yes	characteristic
209	C₄×C₅	#376	1	1	yes	characteristic
210	C₄×C₅	C₄×C₂²⋊C₄	1	2	yes	—
211	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
212	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
213	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
214	Dic₅	—	4	8	no	—
215	C₂⁴	C₂×C₄×D₅	1	1	yes	characteristic
216	C₂²×C₄	—	10	4	no	—
217	C₂²×C₄	C₄×Dic₅	1	4	yes	—
218	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
221	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
222	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
223	C₂²×C₄	#709	1	2	yes	—
224	C₂²×C₄	#709	1	1	yes	characteristic
225	C₂²×C₄	#709	1	1	yes	characteristic
226	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	5	8	no	—
231	M₁₆	—	10	4	no	—
232	M₁₆	—	10	4	no	—
233	M₁₆	—	10	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	10	4	no	—
237	C₂×C₈	—	10	4	no	—
238	C₂×C₈	—	10	4	no	—
239	C₂×C₈	C₄×Dic₅	1	4	yes	—
240	C₂×C₈	C₄×Dic₅	1	2	yes	—
241	C₂×C₈	C₄×Dic₅	1	2	yes	—
242	C₂×C₈	#683	1	4	yes	—
243	C₂×C₈	#683	1	4	yes	—
244	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
245	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
246	C₂×C₈	#685	1	4	yes	—
247	C₂×C₈	#685	1	4	yes	—
248	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
250	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	10	4	no	—
253	C₄⋊C₄	—	10	4	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	—	5	8	no	—
256	C₂²⋊C₄	—	10	4	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	10	4	no	—
260	C₄²	—	10	4	no	—
261	C₂×C₅	—	2	2	no	—
262	C₂×C₅	—	2	1	no	—
263	C₂×C₅	—	2	2	no	—
264	C₂×C₅	—	2	1	no	—
265	C₂×C₅	C₄×#95	1	1	yes	characteristic
266	C₂×C₅	#1757	1	2	yes	—
267	C₂³	—	4	2	no	—
268	C₂³	C₄×Dic₁₀	1	1	yes	characteristic
269	C₂³	C₄²×D₅	1	2	yes	—
270	C₂³	C₄×D₂₀	1	1	yes	characteristic
271	C₂³	C₂×C₄×Dic₅	1	1	yes	characteristic
272	C₂³	C₄×C₂²⋊D₅	1	2	yes	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	10	8	no	—
276	C₂×C₄	—	20	8	no	—
277	C₂×C₄	C₄×Dic₁₀	1	1	yes	characteristic
278	C₂×C₄	C₄×Dic₁₀	1	2	yes	—
279	C₂×C₄	C₄²×D₅	1	2	yes	—
280	C₂×C₄	#4018	1	2	yes	—
281	C₂×C₄	#4018	1	2	yes	—
282	C₂×C₄	C₄×D₂₀	1	2	yes	—
283	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
286	C₂×C₄	C₂×#683	1	2	yes	—
287	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
288	C₂×C₄	#4072	1	1	yes	characteristic
289	C₂×C₄	#4072	1	1	yes	characteristic
290	C₂×C₄	C₂×#685	1	2	yes	—
291	C₂×C₄	C₄×C₂²⋊D₅	1	2	yes	—
292	C₂×C₄	C₄×C₂²⋊D₅	1	2	yes	—
293	C₂×C₄	C₄×C₂²⋊D₅	1	2	yes	—
294	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	10	8	no	—
297	C₈	#3963	1	4	yes	—
298	C₈	#3963	1	2	yes	—
299	C₈	#3963	1	2	yes	—
300	C₅	#10711	1	1	yes	characteristic, complemented, Sylow
301	C₂²	—	2	2	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	C₄²×Dic₅	1	1	yes	characteristic
307	C₂²	C₄×#683	1	2	yes	—
308	C₂²	C₄×C₄⋊Dic₅	1	1	yes	characteristic
309	C₂²	C₄×#685	1	2	yes	—
310	C₂²	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
311	C₄	—	4	4	no	—
312	C₄	—	20	8	no	—
313	C₄	C₄×#683	1	2	yes	—
314	C₄	#65221	1	1	yes	characteristic
315	C₄	C₄×C₄⋊Dic₅	1	1	yes	characteristic
316	C₄	C₄×#685	1	2	yes	—
317	C₄	C₂×#3963	1	1	yes	characteristic
318	C₄	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
319	C₂	—	2	2	no	—
320	C₂	—	2	1	no	—
321	C₂	—	2	2	no	—
322	C₂	—	2	1	no	—
323	C₂	C₄×#3963	1	1	yes	characteristic
324	C₂	#10602804	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₅
commutator	C₂²×C₅	C₄³
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂×C₅×#405	C₂
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂×#405	D₅
hypocenter	C₅	#10711
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
13. C₂×C₅×#97	C₄	semidirect
14. C₂×C₄×C₅×C₈	C₄	semidirect
16. #65390	C₄	semidirect
19. #64776	C₄	semidirect
20. #64750	C₄	semidirect
121. C₂×#97	Dic₅	semidirect
123. C₂×C₄×C₈	Dic₅	semidirect
300. C₅	#10711	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#405	2
C₂×#65390	2
C₂×#64776	2
#10602385	2
#10602380	2
#10711	5