Subgroups — #12004347

Order 1280.

510 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	C₅×#1834	C₂	1	1	yes	maximal, characteristic, complemented
2	#10600565	C₂	1	1	yes	maximal, characteristic
3	#10600373	C₂	1	1	yes	maximal, characteristic
4	#10600373	C₂	1	1	yes	maximal, characteristic
5	#10600370	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600369	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598144	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#4020	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₅×#384	C₂²	1	1	yes	characteristic
11	C₅×#384	C₂²	1	1	yes	characteristic
12	C₂×#3953	—	2	2	no	—
13	C₂×#3941	—	2	2	no	—
14	C₂×#3941	—	2	2	no	—
15	C₂×#3940	—	2	2	no	—
16	C₂×#3939	—	2	2	no	—
17	#65224	C₂²	1	1	yes	characteristic
18	C₂×C₄×#168	C₂²	1	1	yes	characteristic
19	C₂×C₄×#168	C₂²	1	1	yes	characteristic
20	C₅×#340	—	2	2	no	—
21	#10672	—	5	1	no	maximal, Sylow
22	C₂²×D₂₀	—	2	1	no	—
23	C₂²×D₂₀	—	2	1	no	—
24	C₂²×D₂₀	—	2	1	no	—
25	C₂²×C₅×C₈	D₄	1	2	yes	—
26	C₂×C₅×C₄⋊C₄	—	2	1	no	—
27	C₂×C₅×C₄⋊C₄	—	2	1	no	—
28	C₂×C₅×C₄⋊C₄	—	2	1	no	—
29	C₂×C₅×C₄⋊C₄	—	2	1	no	—
30	C₂×C₅×C₄⋊C₄	—	2	1	no	—
31	C₂×C₅×C₄⋊C₄	—	2	1	no	—
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊Dic₅	—	2	1	no	—
34	C₂×C₄⋊Dic₅	—	2	1	no	—
35	C₂×C₄⋊Dic₅	—	2	1	no	—
36	C₂²×#168	D₄	1	2	yes	—
37	C₂²×#168	D₄	1	2	yes	—
38	#4020	—	2	1	no	—
39	#4020	—	2	1	no	—
40	#4020	—	2	1	no	—
41	#4020	Q₈	1	2	yes	—
42	C₄×C₅×C₈	—	2	1	no	—
43	C₄×C₅×C₈	—	2	1	no	—
44	#3953	—	4	4	no	—
45	#3941	—	4	4	no	—
46	#3941	—	4	4	no	—
47	#3940	—	4	4	no	—
48	#3939	—	4	4	no	—
49	C₄×#168	—	2	1	no	—
50	C₄×#168	—	2	1	no	—
51	C₄×#168	—	2	1	no	—
52	C₄×#168	—	2	1	no	—
53	#1957	—	5	1	no	—
54	#1957	—	5	1	no	—
55	#1957	—	5	1	no	—
56	#1845	—	5	1	no	—
57	#1844	—	5	1	no	—
58	#1834	D₅	1	1	yes	characteristic, complemented
59	#1273	—	5	1	no	—
60	C₂³×D₅	—	4	1	no	—
61	C₂²×C₄×C₅	—	4	1	no	—
62	C₂²×C₄×C₅	—	4	1	no	—
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₅	—	4	1	no	—
67	C₂×D₂₀	—	2	2	no	—
68	C₂×D₂₀	—	2	2	no	—
69	C₂×D₂₀	—	2	2	no	—
70	C₂×D₂₀	—	4	2	no	—
71	C₂×D₂₀	—	4	2	no	—
72	C₂×D₂₀	—	4	2	no	—
73	C₂×D₂₀	—	4	2	no	—
74	C₂×D₂₀	—	4	2	no	—
75	C₂×D₂₀	—	4	2	no	—
76	C₂×D₂₀	—	4	1	no	—
77	C₂×D₂₀	—	4	1	no	—
78	C₂×D₂₀	—	4	1	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₅×C₄⋊C₄	—	4	2	no	—
83	C₅×C₄⋊C₄	—	4	2	no	—
84	C₅×C₄⋊C₄	—	4	2	no	—
85	C₅×C₄⋊C₄	—	4	2	no	—
86	C₅×C₄⋊C₄	—	4	2	no	—
87	C₅×C₄⋊C₄	—	4	2	no	—
88	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
89	C₄²×C₅	#41	1	1	yes	characteristic
90	C₄²×C₅	#41	1	1	yes	characteristic
91	C₄²×C₅	#41	1	1	yes	characteristic
92	C₄⋊Dic₅	—	4	2	no	—
93	C₄⋊Dic₅	—	4	2	no	—
94	C₄⋊Dic₅	—	4	2	no	—
95	C₂×#168	—	2	2	no	—
96	C₂×#168	—	2	2	no	—
97	C₂×#168	—	2	2	no	—
98	C₂×#168	—	2	2	no	—
99	C₂×#168	D₈	1	4	yes	—
100	C₂×#168	SD₁₆	1	4	yes	—
101	C₂×#127	—	5	1	no	—
102	C₂×#107	—	10	2	no	—
103	C₂×#106	—	10	2	no	—
104	C₂×#102	—	10	2	no	—
105	C₂×#102	—	10	2	no	—
106	C₂×#102	—	10	2	no	—
107	C₂×C₄×C₈	—	5	1	no	—
108	C₂×C₄×C₈	—	5	1	no	—
109	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
110	#384	—	5	1	no	—
111	#384	C₂×D₅	1	1	yes	characteristic
112	#384	C₂×D₅	1	1	yes	characteristic
113	#340	—	2	2	no	—
114	C₂³×C₅	#127	1	1	yes	characteristic
115	C₂²×D₅	—	4	2	no	—
116	C₂²×D₅	—	8	1	no	—
117	C₂²×D₅	—	8	1	no	—
118	C₂²×D₅	—	8	1	no	—
119	C₂²×D₅	—	8	1	no	—
120	C₂²×D₅	—	8	1	no	—
121	C₂²×D₅	—	8	1	no	—
122	C₂×C₄×C₅	—	8	1	no	—
123	C₂×C₄×C₅	—	8	1	no	—
124	C₂×C₄×C₅	—	8	1	no	—
125	C₂×C₄×C₅	—	8	1	no	—
126	C₂×C₄×C₅	—	8	1	no	—
127	C₂×C₄×C₅	—	8	1	no	—
128	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₅	#123	1	1	yes	characteristic
132	C₂×C₄×C₅	#123	1	1	yes	characteristic
133	C₂×C₄×C₅	#123	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
136	C₂×C₄×C₅	#135	1	2	yes	—
137	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
139	C₂×C₄×C₅	#137	1	2	yes	—
140	C₂×Dic₅	—	8	1	no	—
141	C₂×Dic₅	—	8	1	no	—
142	C₂×Dic₅	—	8	1	no	—
143	D₂₀	—	4	4	no	—
144	D₂₀	—	4	4	no	—
145	D₂₀	—	4	4	no	—
146	D₂₀	—	8	2	no	—
147	D₂₀	—	8	2	no	—
148	D₂₀	—	8	2	no	—
149	C₅×C₈	—	4	2	no	—
150	#168	—	2	4	no	—
151	#168	—	2	4	no	—
152	C₂²×D₄	—	10	1	no	—
153	C₂²×D₄	—	10	1	no	—
154	C₂²×D₄	—	10	1	no	—
155	C₂²×C₈	—	5	2	no	—
156	C₂²×C₈	—	5	2	no	—
157	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
158	#127	—	5	2	no	—
159	#127	—	10	1	no	—
160	#127	—	10	1	no	—
161	#127	—	10	1	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	—	10	1	no	—
169	C₂×C₄⋊C₄	—	10	1	no	—
170	C₂×C₄⋊C₄	—	10	1	no	—
171	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
172	#107	—	20	4	no	—
173	#106	—	20	4	no	—
174	#102	—	20	4	no	—
175	#102	—	20	4	no	—
176	#102	—	20	4	no	—
177	C₄×C₈	—	2	1	no	—
178	C₄×C₈	—	2	1	no	—
179	C₄×C₈	—	10	1	no	—
180	C₄×C₈	—	10	1	no	—
181	C₄×C₈	—	10	1	no	—
182	C₄×C₈	—	10	1	no	—
183	C₂²×C₅	#399	1	1	yes	characteristic
184	C₂²×C₅	#492	1	1	yes	characteristic
185	C₂²×C₅	#493	1	1	yes	characteristic
186	C₂²×C₅	#495	1	1	yes	characteristic
187	C₂²×C₅	#496	1	1	yes	characteristic
188	C₂²×C₅	#496	1	1	yes	characteristic
189	C₂²×C₅	#497	1	1	yes	characteristic
190	C₂×D₅	—	8	2	no	—
191	C₂×D₅	—	8	2	no	—
192	C₂×D₅	—	8	2	no	—
193	C₂×D₅	—	16	1	no	—
194	C₂×D₅	—	16	1	no	—
195	C₂×D₅	—	16	1	no	—
196	C₂×D₅	—	16	1	no	—
197	C₄×C₅	—	16	1	no	—
198	C₄×C₅	—	16	1	no	—
199	C₄×C₅	#474	1	2	yes	—
200	C₄×C₅	#476	1	2	yes	—
201	C₄×C₅	#478	1	2	yes	—
202	C₄×C₅	#480	1	2	yes	—
203	C₄×C₅	#481	1	2	yes	—
204	C₄×C₅	#482	1	2	yes	—
205	Dic₅	—	16	1	no	—
206	C₂⁴	—	20	1	no	—
207	C₂×D₄	—	10	2	no	—
208	C₂×D₄	—	10	2	no	—
209	C₂×D₄	—	10	2	no	—
210	C₂×D₄	—	20	2	no	—
211	C₂×D₄	—	20	1	no	—
212	C₂×D₄	—	20	1	no	—
213	C₂×D₄	—	20	2	no	—
214	C₂×D₄	—	20	2	no	—
215	C₂×D₄	—	20	2	no	—
216	C₂×D₄	—	20	2	no	—
217	C₂×D₄	—	20	1	no	—
218	C₂×D₄	—	20	2	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	20	1	no	—
222	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
223	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	5	4	no	—
229	C₂×C₈	—	5	4	no	—
230	C₂×C₈	—	10	2	no	—
231	C₂×C₈	—	10	2	no	—
232	C₂×C₈	—	10	2	no	—
233	C₂×C₈	—	10	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	20	2	no	—
241	C₄⋊C₄	—	20	2	no	—
242	C₄⋊C₄	—	20	2	no	—
243	C₄²	#709	1	1	yes	characteristic
244	C₄²	#709	1	1	yes	characteristic
245	C₄²	Q₈×D₅	1	1	yes	characteristic
246	C₄²	#713	1	1	yes	characteristic
247	C₂×C₅	#1706	1	1	yes	characteristic
248	C₂×C₅	#1714	1	1	yes	characteristic
249	C₂×C₅	#1720	1	1	yes	characteristic
250	C₂×C₅	#1721	1	1	yes	characteristic
251	C₂×C₅	#2092	1	1	yes	characteristic
252	C₂×C₅	#2093	1	1	yes	characteristic
253	C₂×C₅	#2094	1	1	yes	characteristic
254	D₅	—	16	2	no	—
255	C₂³	—	20	2	no	—
256	C₂³	—	40	1	no	—
257	C₂³	—	40	1	no	—
258	C₂³	—	40	1	no	—
259	C₂³	—	40	1	no	—
260	C₂³	—	40	1	no	—
261	C₂³	—	40	1	no	—
262	C₂³	#4086	1	1	yes	characteristic
263	D₄	—	20	4	no	—
264	D₄	—	20	4	no	—
265	D₄	—	20	4	no	—
266	D₄	—	40	2	no	—
267	D₄	—	40	2	no	—
268	D₄	—	40	2	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	40	1	no	—
276	C₂×C₄	—	40	1	no	—
277	C₂×C₄	—	40	1	no	—
278	C₂×C₄	#4040	1	1	yes	characteristic
279	C₂×C₄	#4040	1	1	yes	characteristic
280	C₂×C₄	#4042	1	1	yes	characteristic
281	C₂×C₄	#4042	1	1	yes	characteristic
282	C₂×C₄	D₅×D₈	1	2	yes	—
283	C₂×C₄	D₅×SD₁₆	1	2	yes	—
284	C₂×C₄	#4060	1	2	yes	—
285	C₂×C₄	#4064	1	2	yes	—
286	C₂×C₄	#4075	1	1	yes	characteristic
287	C₂×C₄	#4092	1	1	yes	characteristic
288	C₂×C₄	#4095	1	2	yes	—
289	C₂×C₄	#4096	1	2	yes	—
290	C₈	—	4	2	no	—
291	C₈	—	10	4	no	—
292	C₈	—	10	4	no	—
293	C₅	#10672	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	40	2	no	—
295	C₂²	—	40	2	no	—
296	C₂²	—	40	2	no	—
297	C₂²	—	80	1	no	—
298	C₂²	—	80	1	no	—
299	C₂²	—	80	1	no	—
300	C₂²	—	80	1	no	—
301	C₂²	#65281	1	1	yes	characteristic
302	C₂²	#65347	1	1	yes	characteristic
303	C₂²	#65348	1	1	yes	characteristic
304	C₂²	#65440	1	1	yes	characteristic
305	C₂²	#65464	1	1	yes	characteristic
306	C₂²	#65465	1	1	yes	characteristic
307	C₂²	#65479	1	1	yes	characteristic
308	C₄	—	16	1	no	—
309	C₄	—	16	1	no	—
310	C₄	—	80	1	no	—
311	C₄	#65152	1	2	yes	—
312	C₄	#65159	1	2	yes	—
313	C₄	#65171	1	2	yes	—
314	C₄	#65179	1	2	yes	—
315	C₄	#65292	1	2	yes	—
316	C₄	#65355	1	2	yes	—
317	C₂	—	80	2	no	—
318	C₂	#10600351	1	1	yes	characteristic
319	C₂	#10601380	1	1	yes	characteristic
320	C₂	#10601434	1	1	yes	characteristic
321	C₂	#10601973	1	1	yes	characteristic
322	C₂	#10601987	1	1	yes	characteristic
323	C₂	#10602009	1	1	yes	characteristic
324	C₂	#10602018	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4086
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1834	C₂
socle	C₂³×C₅	#127
prime radical	C₂×#4020	C₂²
hypercenter	#1834	D₅
hypocenter	C₅	#10672
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
1. C₅×#1834	C₂	semidirect
5. #10600370	C₂	semidirect
6. #10600369	C₂	semidirect
7. #10598144	C₂	semidirect
58. #1834	D₅	semidirect
293. C₅	#10672	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1834	2
#10600565	2
#10600373	2
#10600370	2
#10600369	2
#10598144	2
#10672	5