Subgroups — #12117784

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	#10610909	C₂	1	2	yes	maximal, complemented
2	#10610293	C₂	1	2	yes	maximal
3	C₂×#64989	C₂	1	1	yes	maximal, characteristic
4	C₅×#1772	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601513	C₂	1	2	yes	maximal, complemented
6	#10600687	C₂	1	2	yes	maximal, complemented
7	#10600559	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599565	C₂	1	4	yes	maximal, complemented
9	C₂×#4050	C₂²	1	1	yes	characteristic
10	C₂×#4047	C₂²	1	2	yes	—
11	#65857	C₂²	1	4	yes	—
12	C₂×C₅×#108	C₂²	1	1	yes	characteristic
13	C₅×#427	C₂²	1	1	yes	characteristic
14	C₅×#405	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	#65422	C₂²	1	4	yes	—
17	C₂×#3951	C₂²	1	1	yes	characteristic
18	C₂×#3950	C₂²	1	2	yes	—
19	C₂×#3949	C₂²	1	2	yes	—
20	#65284	C₂²	1	4	yes	—
21	#65225	C₂²	1	2	yes	—
22	C₂×#3937	C₂²	1	2	yes	—
23	#65170	—	2	4	no	—
24	#65149	—	2	2	no	—
25	D₅×#106	—	2	2	no	—
26	#64999	—	2	4	no	—
27	#64989	C₄	1	8	yes	complemented
28	C₄×Dic₂₀	—	2	2	no	—
29	C₄×D₄₀	—	2	2	no	—
30	C₅×#339	C₂²	1	2	yes	—
31	#64774	C₂²	1	4	yes	—
32	#22690	—	5	1	no	maximal, Sylow
33	C₂×#709	C₂³	1	2	yes	—
34	C₂×C₅×M₁₆	C₂³	1	2	yes	—
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₅×#117	C₂³	1	2	yes	—
38	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊Dic₅	C₂³	1	4	yes	—
40	C₂×#681	C₂³	1	2	yes	—
41	C₂×Dic₂₀	D₄	1	1	yes	characteristic
42	#4050	C₂×C₄	1	4	yes	—
43	#4050	Q₈	1	4	yes	—
44	C₂×D₄₀	D₄	1	1	yes	characteristic
45	C₂×#677	D₄	1	2	yes	—
46	#4047	—	2	4	no	—
47	#4047	C₂×C₄	1	8	yes	complemented
48	C₂×#676	—	2	2	no	—
49	C₂×C₈×D₅	—	2	2	no	—
50	#4038	—	2	4	no	—
51	D₅×C₄⋊C₄	—	2	4	no	—
52	#4027	—	4	4	no	—
53	C₄×D₂₀	—	2	4	no	—
54	C₄×Dic₁₀	—	2	4	no	—
55	C₅×#108	—	2	2	no	—
56	C₅×#107	D₄	1	2	yes	—
57	C₅×#106	D₄	1	1	yes	characteristic
58	C₅×#106	D₄	1	1	yes	characteristic
59	C₅×#97	—	2	2	no	—
60	C₄×C₅×C₈	—	2	2	no	—
61	C₄×C₅×C₈	C₂×C₄	1	4	yes	complemented
62	#3963	—	4	2	no	—
63	#3953	—	2	4	no	—
64	#3951	C₂×C₄	1	4	yes	complemented
65	#3950	—	2	2	no	—
66	#3950	—	2	2	no	—
67	#3949	—	2	2	no	—
68	#3949	—	2	2	no	—
69	#3948	—	2	4	no	—
70	#3940	—	2	4	no	—
71	#3939	—	2	4	no	—
72	#3937	C₂×C₄	1	8	yes	complemented
73	C₂×#443	—	5	1	no	—
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	D₅	1	1	yes	characteristic, complemented
81	C₂²×C₄×C₅	—	2	1	no	—
82	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
83	C₂×C₂²⋊D₅	—	2	2	no	—
84	C₂²×Dic₅	—	2	4	no	—
85	#709	—	2	2	no	—
86	#709	—	2	2	no	—
87	#709	C₂²×C₄	1	4	yes	—
88	#709	C₂×Q₈	1	4	yes	—
89	C₂×D₂₀	C₂×D₄	1	2	yes	—
90	C₂×C₄×D₅	—	2	2	no	—
91	C₂×C₄×D₅	—	4	4	no	—
92	C₂×Dic₁₀	C₂×D₄	1	2	yes	—
93	C₅×M₁₆	—	2	2	no	—
94	C₅×M₁₆	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
103	C₂×C₅×C₈	#41	1	2	yes	—
104	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
105	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
106	C₅×C₂²⋊C₄	—	2	2	no	—
107	C₄²×C₅	—	2	2	no	—
108	C₄²×C₅	—	2	1	no	—
109	C₄²×C₅	C₂²×C₄	1	2	yes	—
110	#685	—	4	4	no	—
111	C₄⋊Dic₅	—	2	4	no	—
112	C₄⋊Dic₅	—	2	4	no	—
113	#683	—	4	4	no	—
114	C₄×Dic₅	—	4	4	no	—
115	#681	—	2	2	no	—
116	#681	C₂²×C₄	1	4	yes	—
117	C₂×#168	—	2	2	no	—
118	Dic₂₀	C₄⋊C₄	1	4	yes	—
119	D₄₀	C₄⋊C₄	1	4	yes	—
120	#677	C₄⋊C₄	1	8	yes	—
121	#676	—	2	8	no	—
122	C₈×D₅	—	2	8	no	—
123	C₂×#135	—	5	1	no	—
124	C₂×#131	—	5	2	no	—
125	#520	—	5	4	no	—
126	#443	—	5	8	no	—
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	—	10	2	no	—
135	#427	C₂×D₅	1	1	yes	characteristic
136	C₂×#107	—	5	2	no	—
137	C₂×#106	—	5	2	no	—
138	C₂×#106	—	10	2	no	—
139	C₂×#104	—	5	2	no	—
140	#417	—	5	4	no	—
141	#405	C₂×D₅	1	2	yes	—
142	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
143	#382	—	5	2	no	—
144	#343	—	5	4	no	—
145	#339	C₂×D₅	1	2	yes	—
146	C₂³×C₅	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂²×D₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	4	2	no	—
152	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
158	C₂²⋊D₅	—	2	4	no	—
159	C₂²⋊D₅	—	2	4	no	—
160	C₂×Dic₅	—	2	2	no	—
161	C₂×Dic₅	—	4	4	no	—
162	C₂×Dic₅	—	8	4	no	—
163	D₂₀	—	2	2	no	—
164	D₂₀	C₂×C₄⋊C₄	1	4	yes	—
165	C₄×D₅	—	2	4	no	—
166	C₄×D₅	—	2	4	no	—
167	C₄×D₅	—	16	4	no	—
168	Dic₁₀	—	2	2	no	—
169	Dic₁₀	C₂×C₄⋊C₄	1	4	yes	—
170	C₅×C₈	—	4	2	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	—	2	4	no	—
176	C₂×#41	—	5	2	no	—
177	#135	—	5	4	no	—
178	#135	—	5	4	no	—
179	C₂×Q₁₆	—	5	1	no	—
180	C₂×SD₁₆	—	5	2	no	—
181	C₂×D₈	—	5	1	no	—
182	#131	—	5	8	no	—
183	#131	—	10	4	no	—
184	C₂×M₁₆	—	5	2	no	—
185	C₂×M₁₆	—	10	2	no	—
186	C₂×M₁₆	C₂²×D₅	1	2	yes	—
187	C₂²×C₈	—	10	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	—	10	4	no	—
194	#117	C₂²×D₅	1	2	yes	—
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	—	2	2	no	—
199	#108	—	5	4	no	—
200	#107	—	10	2	no	—
201	#107	—	10	2	no	—
202	#107	—	10	4	no	—
203	#107	D₂₀	1	2	yes	—
204	#106	—	10	2	no	—
205	#106	—	10	2	no	—
206	#106	—	10	4	no	—
207	#106	D₂₀	1	1	yes	characteristic
208	#106	D₂₀	1	1	yes	characteristic
209	#104	—	5	8	no	—
210	#103	—	10	4	no	—
211	#102	—	10	4	no	—
212	#97	—	2	2	no	—
213	C₄×C₈	—	2	2	no	—
214	C₄×C₈	C₄×D₅	1	4	yes	complemented
215	#95	—	20	2	no	—
216	C₂²×C₅	—	2	2	no	—
217	C₂²×C₅	#520	1	1	yes	characteristic
218	C₂²×C₅	D₄²	1	1	yes	characteristic
219	C₂²×C₅	#554	1	1	yes	characteristic
220	C₂×D₅	—	2	4	no	—
221	C₂×D₅	—	4	4	no	—
222	C₄×C₅	—	4	2	no	—
223	C₄×C₅	—	4	2	no	—
224	C₄×C₅	—	4	2	no	—
225	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
226	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
227	C₄×C₅	#547	1	1	yes	characteristic
228	C₄×C₅	D₄×Q₈	1	1	yes	characteristic
229	Dic₅	—	2	4	no	—
230	Dic₅	—	8	4	no	—
231	#41	—	5	4	no	—
232	#41	—	5	4	no	—
233	#41	—	10	2	no	—
234	#41	—	10	2	no	—
235	C₂×Q₈	—	5	2	no	—
236	C₂×D₄	—	5	2	no	—
237	C₂×D₄	—	10	2	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	10	2	no	—
240	C₂²×C₄	—	10	4	no	—
241	C₂²×C₄	—	20	4	no	—
242	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
243	Q₁₆	—	5	4	no	—
244	SD₁₆	—	5	8	no	—
245	D₈	—	5	4	no	—
246	M₁₆	—	2	2	no	—
247	M₁₆	—	2	2	no	—
248	M₁₆	—	5	4	no	—
249	M₁₆	—	10	2	no	—
250	M₁₆	—	10	8	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	10	2	no	—
253	C₂×C₈	—	10	8	no	—
254	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
255	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
256	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
257	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
258	C₂×C₈	#709	1	2	yes	—
259	C₂×C₈	D₄×D₅	1	1	yes	characteristic
260	C₂×C₈	D₄×D₅	1	1	yes	characteristic
261	C₂×C₈	Q₈×D₅	1	2	yes	—
262	C₄⋊C₄	—	10	4	no	—
263	C₄⋊C₄	—	10	4	no	—
264	C₄⋊C₄	—	20	4	no	—
265	C₄⋊C₄	C₂×D₂₀	1	2	yes	—
266	C₄⋊C₄	C₂×D₂₀	1	2	yes	—
267	C₂²⋊C₄	—	2	2	no	—
268	C₂²⋊C₄	—	20	4	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	20	4	no	—
272	C₄²	C₂×C₄×D₅	1	2	yes	—
273	C₂×C₅	—	2	1	no	—
274	C₂×C₅	—	2	1	no	—
275	C₂×C₅	D₄×C₄⋊C₄	1	1	yes	characteristic
276	C₂×C₅	#3290	1	2	yes	—
277	D₅	—	4	4	no	—
278	C₂³	—	10	2	no	—
279	C₂³	#4149	1	1	yes	characteristic
280	Q₈	—	5	4	no	—
281	Q₈	—	10	2	no	—
282	D₄	—	5	4	no	—
283	D₄	—	10	4	no	—
284	D₄	—	10	4	no	—
285	D₄	—	10	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	10	2	no	—
291	C₂×C₄	—	10	4	no	—
292	C₂×C₄	—	10	4	no	—
293	C₂×C₄	—	20	4	no	—
294	C₂×C₄	—	40	4	no	—
295	C₂×C₄	—	80	4	no	—
296	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
298	C₂×C₄	C₂×#709	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
301	C₂×C₄	#4150	1	1	yes	characteristic
302	C₈	—	4	2	no	—
303	C₈	—	10	4	no	—
304	C₈	C₄×D₂₀	1	2	yes	—
305	C₈	C₄×D₂₀	1	2	yes	—
306	C₈	D₅×C₄⋊C₄	1	2	yes	—
307	C₈	D₅×C₄⋊C₄	1	2	yes	—
308	C₅	#22690	1	1	yes	characteristic, complemented, Sylow
309	C₂²	—	2	2	no	—
310	C₂²	—	10	4	no	—
311	C₂²	—	20	4	no	—
312	C₂²	#65857	1	1	yes	characteristic
313	C₂²	D₄×D₂₀	1	1	yes	characteristic
314	C₂²	#65913	1	1	yes	characteristic
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	10	4	no	—
319	C₄	—	40	4	no	—
320	C₄	C₂×C₄×D₂₀	1	1	yes	characteristic
321	C₄	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
322	C₄	#65885	1	1	yes	characteristic
323	C₄	Q₈×D₂₀	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	20	4	no	—
327	C₂	C₄⋊C₄×D₂₀	1	1	yes	characteristic
328	C₂	#10611622	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₅×#1772	C₂
socle	C₂²×C₅	D₄²
prime radical	C₂×#4050	C₂²
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	#1772	D₅
hypocenter	C₅	#22690
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772

Complements

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

N	H ≅ G/N	type
1. #10610909	C₂	semidirect
4. C₅×#1772	C₂	semidirect
5. #10601513	C₂	semidirect
6. #10600687	C₂	semidirect
7. #10600559	C₂	semidirect
8. #10599565	C₂	semidirect
27. #64989	C₄	semidirect
47. #4047	C₂×C₄	semidirect
61. C₄×C₅×C₈	C₂×C₄	semidirect
64. #3951	C₂×C₄	semidirect
72. #3937	C₂×C₄	semidirect
80. #1772	D₅	semidirect
214. C₄×C₈	C₄×D₅	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
#10610909	2
#10610293	2
C₂×#64989	2
C₅×#1772	2
#10601513	2
#10600687	2
#10600559	2
#10599565	2
#22690	5