Subgroups — #12005111

Order 1280.

1110 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
2	#10613793	C₂	1	1	yes	maximal, characteristic
3	C₂×#65213	C₂	1	1	yes	maximal, characteristic
4	C₂×#64743	C₂	1	4	yes	maximal
5	C₂²×C₄²×C₅	—	2	2	no	—
6	C₂²×C₄²×C₅	C₂²	1	2	yes	—
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×#4072	C₂²	1	2	yes	—
9	C₂²×#681	C₂²	1	2	yes	—
10	C₄³×C₅	—	2	2	no	—
11	C₄³×C₅	C₄	1	4	yes	complemented
12	#65501	—	2	2	no	—
13	#65225	C₄	1	2	yes	complemented
14	#65224	C₄	1	2	yes	complemented
15	C₄×C₄⋊Dic₅	—	2	2	no	—
16	#65213	—	2	2	no	—
17	#65213	C₄	1	4	yes	complemented
18	C₂×C₄⋊#168	—	2	2	no	—
19	#64743	—	2	8	no	—
20	#10839	—	5	1	no	maximal, Sylow
21	C₂³×C₄×C₅	—	2	2	no	—
22	C₂³×C₄×C₅	D₄	1	2	yes	—
23	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₅	—	2	16	no	—
25	C₂×C₄²×C₅	—	2	8	no	—
26	C₂×C₄²×C₅	—	2	2	no	—
27	C₂×C₄²×C₅	—	2	2	no	—
28	C₂×C₄²×C₅	—	2	2	no	—
29	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅	D₄	1	4	yes	—
35	C₂×C₄²×C₅	D₄	1	2	yes	—
36	C₂×C₄²×C₅	D₄	1	2	yes	—
37	C₂×C₄²×C₅	Q₈	1	4	yes	—
38	C₂×C₄²×C₅	Q₈	1	2	yes	—
39	C₂×C₂²⋊Dic₅	—	2	2	no	—
40	#4072	—	2	4	no	—
41	#4072	—	2	4	no	—
42	C₂×C₄⋊Dic₅	—	2	4	no	—
43	C₂×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
44	C₂×C₄×Dic₅	—	2	2	no	—
45	C₂×#681	—	2	2	no	—
46	C₂×#681	—	2	4	no	—
47	C₂×#681	—	2	4	no	—
48	C₂×#681	C₂×C₄	1	4	yes	—
49	C₂²×#168	—	2	2	no	—
50	#3963	—	2	4	no	—
51	C₄⋊#168	—	2	8	no	—
52	C₂×#423	—	5	1	no	—
53	#2285	—	5	1	no	—
54	C₂×C₄³	D₅	1	1	yes	characteristic
55	C₂×#339	—	5	4	no	—
56	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	—	2	16	no	—
58	C₂²×C₄×C₅	—	2	4	no	—
59	C₂²×C₄×C₅	—	2	4	no	—
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	—	2	8	no	—
62	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
65	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₅	#41	1	2	yes	—
75	C₂²×C₄×C₅	#41	1	2	yes	—
76	C₂²×Dic₅	—	2	4	no	—
77	C₄²×C₅	—	2	32	no	—
78	C₄²×C₅	—	2	4	no	—
79	C₄²×C₅	—	2	4	no	—
80	C₄²×C₅	—	2	4	no	—
81	C₄²×C₅	C₄²	1	2	yes	—
82	C₄²×C₅	C₄²	1	2	yes	—
83	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
84	C₄²×C₅	C₄⋊C₄	1	2	yes	—
85	C₄²×C₅	C₄⋊C₄	1	2	yes	—
86	C₄²×C₅	C₄⋊C₄	1	4	yes	—
87	C₄²×C₅	C₄⋊C₄	1	8	yes	complemented
88	C₂²⋊Dic₅	—	4	4	no	—
89	C₄⋊Dic₅	—	2	8	no	—
90	C₄⋊Dic₅	—	4	8	no	—
91	C₄×Dic₅	—	4	2	no	—
92	C₄×Dic₅	—	4	2	no	—
93	#681	—	2	8	no	—
94	#681	—	4	4	no	—
95	C₂×#168	—	2	4	no	—
96	C₂×#168	—	4	2	no	—
97	C₂×#168	—	4	2	no	—
98	C₂²×M₁₆	—	5	2	no	—
99	C₂×#117	—	5	2	no	—
100	C₂²×C₄²	—	2	2	no	—
101	C₂²×C₄²	C₂×D₅	1	2	yes	—
102	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
103	#423	—	5	4	no	—
104	#423	—	10	2	no	—
105	C₂×C₄⋊C₈	—	10	2	no	—
106	#384	—	5	2	no	—
107	#382	—	5	2	no	—
108	#380	—	10	2	no	—
109	C₄×C₄⋊C₄	—	10	2	no	—
110	C₄³	—	2	2	no	—
111	C₄³	Dic₅	1	4	yes	complemented
112	#339	—	10	8	no	—
113	C₂³×C₅	—	2	4	no	—
114	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂³×C₅	#121	1	2	yes	—
118	C₂³×C₅	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₅	—	2	16	no	—
120	C₂×C₄×C₅	—	2	16	no	—
121	C₂×C₄×C₅	—	2	16	no	—
122	C₂×C₄×C₅	—	2	8	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	8	no	—
125	C₂×C₄×C₅	—	2	8	no	—
126	C₂×C₄×C₅	#95	1	4	yes	—
127	C₂×C₄×C₅	#95	1	4	yes	—
128	C₂×C₄×C₅	#104	1	8	yes	—
129	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	#117	1	2	yes	—
135	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
138	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
139	C₂×C₄×C₅	#121	1	2	yes	—
140	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
141	C₂×C₄×C₅	#124	1	1	yes	characteristic
142	C₂×C₄×C₅	#124	1	1	yes	characteristic
143	C₂×C₄×C₅	#125	1	2	yes	—
144	C₂×C₄×C₅	#127	1	1	yes	characteristic
145	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
146	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
147	C₂×Dic₅	—	4	4	no	—
148	C₂×Dic₅	—	4	8	no	—
149	#168	—	4	4	no	—
150	C₂³×C₄	—	2	2	no	—
151	C₂³×C₄	C₂²⋊D₅	1	2	yes	—
152	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
153	C₂×M₁₆	—	5	4	no	—
154	C₂×M₁₆	—	10	2	no	—
155	C₂×M₁₆	—	10	4	no	—
156	C₂×M₁₆	—	10	4	no	—
157	C₂²×C₈	—	10	2	no	—
158	#117	—	10	4	no	—
159	#117	—	10	4	no	—
160	C₂×C₄⋊C₄	—	5	4	no	—
161	C₂×C₄⋊C₄	—	10	4	no	—
162	C₂×C₂²⋊C₄	—	10	2	no	—
163	C₂×C₄²	—	2	16	no	—
164	C₂×C₄²	—	2	8	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	—	10	2	no	—
169	C₂×C₄²	Dic₁₀	1	4	yes	—
170	C₂×C₄²	Dic₁₀	1	2	yes	—
171	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
172	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₅	1	2	yes	—
174	C₂×C₄²	D₂₀	1	4	yes	—
175	C₂×C₄²	D₂₀	1	2	yes	—
176	C₂×C₄²	C₂×Dic₅	1	4	yes	—
177	C₂×C₄²	C₂×Dic₅	1	2	yes	—
178	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
179	C₄⋊C₈	—	10	8	no	—
180	#95	—	10	4	no	—
181	C₂²×C₅	—	2	4	no	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	—	2	4	no	—
185	C₂²×C₅	C₂×#95	1	1	yes	characteristic
186	C₂²×C₅	#380	1	2	yes	—
187	C₂²×C₅	#389	1	2	yes	—
188	C₂²×C₅	#390	1	1	yes	characteristic
189	C₂²×C₅	#391	1	1	yes	characteristic
190	C₂²×C₅	C₂×#104	1	4	yes	—
191	C₄×C₅	—	2	16	no	—
192	C₄×C₅	—	2	4	no	—
193	C₄×C₅	#339	1	8	yes	complemented
194	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
195	C₄×C₅	#382	1	1	yes	characteristic
196	C₄×C₅	#384	1	1	yes	characteristic
197	C₄×C₅	#389	1	2	yes	—
198	C₄×C₅	#390	1	1	yes	characteristic
199	C₄×C₅	#391	1	1	yes	characteristic
200	Dic₅	—	8	4	no	—
201	C₂⁴	C₂×C₂²⋊D₅	1	1	yes	characteristic
202	C₂²×C₄	—	2	16	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	10	4	no	—
208	C₂²×C₄	#683	1	4	yes	—
209	C₂²×C₄	#685	1	2	yes	—
210	C₂²×C₄	#685	1	2	yes	—
211	C₂²×C₄	C₂²⋊Dic₅	1	4	yes	—
212	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
213	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₂₀	1	2	yes	—
217	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
218	C₂²×C₄	#709	1	2	yes	—
219	C₂²×C₄	#709	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
222	M₁₆	—	10	8	no	—
223	M₁₆	—	20	4	no	—
224	C₂×C₈	—	10	4	no	—
225	C₂×C₈	—	20	2	no	—
226	C₂×C₈	—	20	2	no	—
227	C₄⋊C₄	—	10	8	no	—
228	C₄⋊C₄	—	20	8	no	—
229	C₂²⋊C₄	—	20	4	no	—
230	C₄²	—	2	32	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	20	2	no	—
235	C₄²	—	20	2	no	—
236	C₄²	C₄×Dic₅	1	2	yes	—
237	C₄²	C₄×Dic₅	1	2	yes	—
238	C₄²	#683	1	2	yes	—
239	C₄²	#683	1	2	yes	—
240	C₄²	C₄⋊Dic₅	1	4	yes	—
241	C₄²	C₄⋊Dic₅	1	8	yes	complemented
242	C₄²	#685	1	4	yes	—
243	C₂×C₅	—	2	2	no	—
244	C₂×C₅	—	2	2	no	—
245	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
246	C₂×C₅	C₂×#339	1	2	yes	—
247	C₂×C₅	#1962	1	2	yes	—
248	C₂×C₅	#1994	1	2	yes	—
249	C₂³	—	2	4	no	—
250	C₂³	C₂×#683	1	1	yes	characteristic
251	C₂³	#4070	1	2	yes	—
252	C₂³	C₂×#685	1	1	yes	characteristic
253	C₂³	#4076	1	2	yes	—
254	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
255	C₂×C₄	—	2	16	no	—
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	20	8	no	—
263	C₂×C₄	—	20	4	no	—
264	C₂×C₄	#3937	1	8	yes	—
265	C₂×C₄	#3963	1	4	yes	—
266	C₂×C₄	#3963	1	4	yes	—
267	C₂×C₄	C₄×Dic₁₀	1	2	yes	—
268	C₂×C₄	C₄×Dic₁₀	1	2	yes	—
269	C₂×C₄	#4015	1	2	yes	—
270	C₂×C₄	#4015	1	1	yes	characteristic
271	C₂×C₄	#4016	1	2	yes	—
272	C₂×C₄	C₄×D₂₀	1	2	yes	—
273	C₂×C₄	C₄×D₂₀	1	2	yes	—
274	C₂×C₄	#4020	1	1	yes	characteristic
275	C₂×C₄	#4021	1	1	yes	characteristic
276	C₂×C₄	#4021	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
278	C₂×C₄	C₂×#683	1	1	yes	characteristic
279	C₂×C₄	#4070	1	2	yes	—
280	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
281	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
282	C₂×C₄	#4072	1	2	yes	—
283	C₂×C₄	C₂×#685	1	1	yes	characteristic
284	C₂×C₄	#4076	1	2	yes	—
285	C₈	—	20	4	no	—
286	C₅	#10839	1	1	yes	characteristic, complemented, Sylow
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	C₂×#3937	1	4	yes	—
292	C₂²	#65217	1	2	yes	—
293	C₂²	#65218	1	1	yes	characteristic
294	C₂²	#65228	1	1	yes	characteristic
295	C₂²	C₂×#3963	1	1	yes	characteristic
296	C₂²	#65501	1	2	yes	—
297	C₄	—	2	16	no	—
298	C₄	—	2	4	no	—
299	C₄	—	40	4	no	—
300	C₄	#64743	1	8	yes	complemented
301	C₄	#65217	1	2	yes	—
302	C₄	#65218	1	1	yes	characteristic
303	C₄	C₄×C₄⋊Dic₅	1	2	yes	—
304	C₄	#65224	1	1	yes	characteristic
305	C₄	#65225	1	1	yes	characteristic
306	C₄	#65228	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	#10600188	1	2	yes	—
310	C₂	#10600189	1	2	yes	—
311	C₂	C₂×#64743	1	2	yes	—
312	C₂	#10602328	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂×C₄³×C₅	C₂
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅	C₂×D₄
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂×C₄³	D₅
hypocenter	C₅	#10839
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
11. C₄³×C₅	C₄	semidirect
13. #65225	C₄	semidirect
14. #65224	C₄	semidirect
17. #65213	C₄	semidirect
87. C₄²×C₅	C₄⋊C₄	semidirect
111. C₄³	Dic₅	semidirect
193. C₄×C₅	#339	semidirect
241. C₄²	C₄⋊Dic₅	semidirect
286. C₅	#10839	semidirect
300. C₄	#64743	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄³×C₅	2
#10613793	2
C₂×#65213	2
C₂×#64743	2
#10839	5