Subgroups — #12117783

Order 1280.

846 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#443	C₂	1	1	yes	maximal, characteristic
2	#10615629	C₂	1	2	yes	maximal
3	#10615515	C₂	1	2	yes	maximal, complemented
4	#10602808	C₂	1	2	yes	maximal, complemented
5	#10601494	C₂	1	2	yes	maximal, complemented
6	#10601157	C₂	1	4	yes	maximal, complemented
7	#10600572	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600559	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#135	C₂²	1	1	yes	characteristic
10	C₂×C₅×#131	C₂²	1	2	yes	—
11	#66161	C₂²	1	4	yes	—
12	C₅×#443	C₄	1	8	yes	complemented
13	C₂×C₅×#108	C₂²	1	1	yes	characteristic
14	C₂×C₅×#104	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	#65521	C₂²	1	4	yes	—
17	Q₁₆×Dic₅	—	2	2	no	—
18	#65453	—	2	4	no	—
19	D₈×Dic₅	—	2	2	no	—
20	#65410	—	2	4	no	—
21	#65410	C₂²	1	4	yes	—
22	C₂×#3951	C₂²	1	1	yes	characteristic
23	#65395	—	2	2	no	—
24	#65395	C₂²	1	1	yes	characteristic
25	C₂×#3950	C₂²	1	2	yes	—
26	C₂×#3949	—	2	2	no	—
27	C₂×#3949	C₂²	1	2	yes	—
28	#65390	C₂²	1	2	yes	—
29	#65225	C₂²	1	2	yes	—
30	#64774	C₂²	1	4	yes	—
31	#64743	C₂²	1	2	yes	—
32	#22690	—	5	1	no	maximal, Sylow
33	C₂×C₅×#41	C₂³	1	2	yes	—
34	C₅×#135	C₂×C₄	1	4	yes	—
35	C₅×#135	Q₈	1	4	yes	—
36	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
37	C₂×C₅×SD₁₆	D₄	1	2	yes	—
38	C₂×C₅×D₈	D₄	1	1	yes	characteristic
39	C₅×#131	—	2	4	no	—
40	C₅×#131	C₂×C₄	1	8	yes	complemented
41	C₂×C₅×M₁₆	—	2	2	no	—
42	C₂×C₅×M₁₆	C₂³	1	2	yes	—
43	C₂²×C₅×C₈	—	2	2	no	—
44	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	Q₈×Dic₅	—	2	4	no	—
48	D₄×Dic₅	—	2	4	no	—
49	D₄×Dic₅	—	4	4	no	—
50	#4072	—	2	4	no	—
51	#4072	C₂³	1	2	yes	—
52	C₂×C₄⋊Dic₅	—	2	4	no	—
53	C₂×C₄⋊Dic₅	C₂³	1	4	yes	—
54	C₂×#681	C₂³	1	2	yes	—
55	C₅×#108	C₂×C₄	1	4	yes	complemented
56	C₅×#104	C₂×C₄	1	8	yes	complemented
57	C₄×C₅×C₈	C₂×C₄	1	4	yes	complemented
58	#3967	—	2	4	no	—
59	#3964	—	2	4	no	—
60	#3963	—	4	2	no	—
61	#3951	—	2	2	no	—
62	#3950	—	2	2	no	—
63	#3950	—	2	2	no	—
64	#3950	—	2	4	no	—
65	#3950	D₄	1	2	yes	—
66	#3949	—	2	2	no	—
67	#3949	—	2	2	no	—
68	#3949	—	2	4	no	—
69	#3949	D₄	1	1	yes	characteristic
70	#3949	D₄	1	1	yes	characteristic
71	#3947	—	2	2	no	—
72	C₈×Dic₅	—	2	2	no	—
73	C₂×#443	D₅	1	1	yes	characteristic
74	#2951	—	5	2	no	—
75	#2912	—	5	2	no	—
76	#1952	—	5	2	no	—
77	#1860	—	5	4	no	—
78	#1846	—	5	1	no	—
79	#1815	—	5	2	no	—
80	#1772	—	5	1	no	—
81	C₅×#41	—	2	2	no	—
82	C₅×#41	—	2	2	no	—
83	C₅×#41	C₂²×C₄	1	4	yes	—
84	C₅×#41	C₂×Q₈	1	4	yes	—
85	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
86	C₂×C₅×D₄	—	2	2	no	—
87	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₅	—	2	2	no	—
89	C₂²×C₄×C₅	—	2	1	no	—
90	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	C₂²×Dic₅	—	2	4	no	—
92	C₂²×Dic₅	—	4	4	no	—
93	C₅×Q₁₆	C₄⋊C₄	1	4	yes	—
94	C₅×SD₁₆	C₄⋊C₄	1	8	yes	—
95	C₅×D₈	C₄⋊C₄	1	4	yes	—
96	C₅×M₁₆	—	2	2	no	—
97	C₅×M₁₆	—	2	8	no	—
98	C₅×M₁₆	C₂²×C₄	1	4	yes	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	—	2	8	no	—
101	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
102	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
103	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
108	C₂×C₅×C₈	#41	1	2	yes	—
109	C₄²×C₅	—	2	1	no	—
110	C₄²×C₅	C₂²×C₄	1	2	yes	—
111	C₂²⋊Dic₅	—	2	2	no	—
112	C₂²⋊Dic₅	—	4	4	no	—
113	C₄⋊Dic₅	—	2	4	no	—
114	C₄⋊Dic₅	—	2	4	no	—
115	C₄⋊Dic₅	—	4	4	no	—
116	C₄⋊Dic₅	C₂×D₄	1	2	yes	—
117	C₄⋊Dic₅	C₂×D₄	1	2	yes	—
118	C₄×Dic₅	—	2	2	no	—
119	C₄×Dic₅	—	4	4	no	—
120	#681	—	2	2	no	—
121	#681	—	2	2	no	—
122	C₂×#168	—	2	2	no	—
123	C₂×#135	C₂×D₅	1	1	yes	characteristic
124	C₂×#131	C₂×D₅	1	2	yes	—
125	#520	—	5	4	no	—
126	#443	Dic₅	1	8	yes	complemented
127	#440	—	10	4	no	—
128	C₄×Q₁₆	—	10	2	no	—
129	C₄×D₈	—	10	2	no	—
130	C₂×#108	—	5	1	no	—
131	C₂×#108	C₂×D₅	1	1	yes	characteristic
132	#428	—	5	4	no	—
133	#428	—	10	4	no	—
134	#427	—	5	1	no	—
135	#427	—	10	2	no	—
136	C₂×#107	—	5	2	no	—
137	C₂×#106	—	5	2	no	—
138	C₂×#106	—	10	2	no	—
139	C₂×#104	C₂×D₅	1	2	yes	—
140	#417	—	5	4	no	—
141	#405	—	5	2	no	—
142	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
143	#382	—	5	2	no	—
144	#343	—	5	4	no	—
145	#339	—	5	2	no	—
146	C₂³×C₅	—	2	2	no	—
147	C₂³×C₅	H(5,𝔽₂)	1	1	yes	characteristic
148	C₅×Q₈	—	2	2	no	—
149	C₅×Q₈	C₂×C₄⋊C₄	1	4	yes	—
150	C₅×D₄	—	2	4	no	—
151	C₅×D₄	—	2	4	no	—
152	C₅×D₄	—	2	2	no	—
153	C₅×D₄	C₂×C₄⋊C₄	1	4	yes	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	4	no	—
156	C₂×C₄×C₅	—	2	4	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	4	2	no	—
159	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
164	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
165	C₂×Dic₅	—	2	2	no	—
166	C₂×Dic₅	—	2	2	no	—
167	C₂×Dic₅	—	4	4	no	—
168	C₂×Dic₅	—	8	4	no	—
169	C₂×Dic₅	—	16	4	no	—
170	C₅×C₈	—	2	4	no	—
171	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
172	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
173	C₅×C₈	C₄×D₄	1	2	yes	—
174	C₅×C₈	C₄×D₄	1	2	yes	—
175	#168	—	4	2	no	—
176	C₂×#41	C₂²×D₅	1	2	yes	—
177	#135	Dic₁₀	1	4	yes	—
178	#135	C₂×Dic₅	1	4	yes	—
179	C₂×Q₁₆	D₂₀	1	1	yes	characteristic
180	C₂×SD₁₆	D₂₀	1	2	yes	—
181	C₂×D₈	D₂₀	1	1	yes	characteristic
182	#131	—	2	4	no	—
183	#131	C₂×Dic₅	1	8	yes	complemented
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	5	2	no	—
186	C₂×M₁₆	C₂²×D₅	1	2	yes	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
190	C₄×Q₈	—	10	4	no	—
191	C₄×D₄	—	10	4	no	—
192	C₄×D₄	—	20	4	no	—
193	#117	—	5	2	no	—
194	#117	—	10	4	no	—
195	C₂×C₄⋊C₄	—	5	4	no	—
196	C₂×C₄⋊C₄	—	10	4	no	—
197	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
198	#108	—	10	2	no	—
199	#108	C₂×Dic₅	1	4	yes	complemented
200	#107	—	5	2	no	—
201	#107	—	10	4	no	—
202	#107	—	10	2	no	—
203	#107	—	10	2	no	—
204	#106	—	5	1	no	—
205	#106	—	5	1	no	—
206	#106	—	10	4	no	—
207	#106	—	10	2	no	—
208	#106	—	10	2	no	—
209	#104	C₂×Dic₅	1	8	yes	complemented
210	#103	—	10	4	no	—
211	#102	—	10	4	no	—
212	#97	—	10	2	no	—
213	C₄×C₈	—	10	2	no	—
214	C₄×C₈	C₂×Dic₅	1	4	yes	complemented
215	#95	—	20	2	no	—
216	C₂²×C₅	—	2	2	no	—
217	C₂²×C₅	—	2	4	no	—
218	C₂²×C₅	—	4	4	no	—
219	C₂²×C₅	#520	1	1	yes	characteristic
220	C₂²×C₅	D₄²	1	1	yes	characteristic
221	C₂²×C₅	#554	1	1	yes	characteristic
222	C₄×C₅	—	2	4	no	—
223	C₄×C₅	—	4	2	no	—
224	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
225	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
226	C₄×C₅	#547	1	1	yes	characteristic
227	C₄×C₅	D₄×Q₈	1	1	yes	characteristic
228	Dic₅	—	4	2	no	—
229	Dic₅	—	4	2	no	—
230	Dic₅	—	8	4	no	—
231	#41	—	2	2	no	—
232	#41	—	2	2	no	—
233	#41	C₂×Dic₁₀	1	4	yes	—
234	#41	C₂²×Dic₅	1	4	yes	—
235	C₂×Q₈	C₂×D₂₀	1	2	yes	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	C₂×D₂₀	1	2	yes	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	10	4	no	—
241	C₂²×C₄	—	20	4	no	—
242	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
243	Q₁₆	C₄⋊Dic₅	1	4	yes	—
244	SD₁₆	C₄⋊Dic₅	1	8	yes	—
245	D₈	C₄⋊Dic₅	1	4	yes	—
246	M₁₆	—	2	2	no	—
247	M₁₆	—	2	8	no	—
248	M₁₆	—	10	2	no	—
249	M₁₆	—	10	2	no	—
250	M₁₆	C₂²×Dic₅	1	4	yes	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	8	no	—
253	C₂×C₈	—	10	2	no	—
254	C₂×C₈	C₂×Dic₁₀	1	2	yes	—
255	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
256	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
257	C₂×C₈	D₄×D₅	1	1	yes	characteristic
258	C₂×C₈	D₄×D₅	1	1	yes	characteristic
259	C₂×C₈	#711	1	2	yes	—
260	C₂×C₈	C₂²×Dic₅	1	2	yes	—
261	C₂×C₈	C₂²×Dic₅	1	2	yes	—
262	C₄⋊C₄	—	5	2	no	—
263	C₄⋊C₄	—	5	2	no	—
264	C₄⋊C₄	—	10	4	no	—
265	C₄⋊C₄	—	10	4	no	—
266	C₄⋊C₄	—	20	4	no	—
267	C₂²⋊C₄	—	10	2	no	—
268	C₂²⋊C₄	—	20	4	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	10	2	no	—
271	C₄²	—	20	4	no	—
272	C₄²	C₂²×Dic₅	1	2	yes	—
273	C₂×C₅	—	2	1	no	—
274	C₂×C₅	—	2	1	no	—
275	C₂×C₅	—	4	4	no	—
276	C₂×C₅	D₄×C₄⋊C₄	1	1	yes	characteristic
277	C₂×C₅	#3290	1	2	yes	—
278	C₂³	—	2	2	no	—
279	C₂³	#4149	1	1	yes	characteristic
280	Q₈	—	2	2	no	—
281	Q₈	C₂×C₄⋊Dic₅	1	4	yes	—
282	D₄	—	2	4	no	—
283	D₄	—	2	4	no	—
284	D₄	—	2	2	no	—
285	D₄	C₂×C₄⋊Dic₅	1	4	yes	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	10	2	no	—
292	C₂×C₄	—	10	2	no	—
293	C₂×C₄	—	20	4	no	—
294	C₂×C₄	—	40	4	no	—
295	C₂×C₄	—	80	4	no	—
296	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
299	C₂×C₄	C₂×#711	1	1	yes	characteristic
300	C₂×C₄	#4150	1	1	yes	characteristic
301	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
302	C₈	—	2	4	no	—
303	C₈	—	20	2	no	—
304	C₈	C₂×C₄⋊Dic₅	1	2	yes	—
305	C₈	C₂×C₄⋊Dic₅	1	2	yes	—
306	C₈	D₄×Dic₅	1	2	yes	—
307	C₈	D₄×Dic₅	1	2	yes	—
308	C₅	#22690	1	1	yes	characteristic, complemented, Sylow
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	#65877	1	1	yes	characteristic
313	C₂²	D₄×D₂₀	1	1	yes	characteristic
314	C₂²	#66161	1	1	yes	characteristic
315	C₄	—	2	4	no	—
316	C₄	—	4	2	no	—
317	C₄	—	20	2	no	—
318	C₄	—	20	2	no	—
319	C₄	—	40	4	no	—
320	C₄	D₄×Dic₁₀	1	1	yes	characteristic
321	C₄	#65889	1	1	yes	characteristic
322	C₄	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
323	C₄	C₂×D₄×Dic₅	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	4	4	no	—
327	C₂	#10611623	1	2	yes	—
328	C₂	D₄×C₄⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂×C₅×#443	C₂
socle	C₂²×C₅	D₄²
prime radical	C₂×C₅×#135	C₂²
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂×#443	D₅
hypocenter	C₅	#22690
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
3. #10615515	C₂	semidirect
4. #10602808	C₂	semidirect
5. #10601494	C₂	semidirect
6. #10601157	C₂	semidirect
7. #10600572	C₂	semidirect
8. #10600559	C₂	semidirect
12. C₅×#443	C₄	semidirect
40. C₅×#131	C₂×C₄	semidirect
55. C₅×#108	C₂×C₄	semidirect
56. C₅×#104	C₂×C₄	semidirect
57. C₄×C₅×C₈	C₂×C₄	semidirect
126. #443	Dic₅	semidirect
183. #131	C₂×Dic₅	semidirect
199. #108	C₂×Dic₅	semidirect
209. #104	C₂×Dic₅	semidirect
214. C₄×C₈	C₂×Dic₅	semidirect
308. C₅	#22690	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#443	2
#10615629	2
#10615515	2
#10602808	2
#10601494	2
#10601157	2
#10600572	2
#10600559	2
#22690	5