Subgroups — #12014499

Order 1280.

452 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2932	C₂	1	1	yes	maximal, characteristic
2	#10602804	C₂	1	2	yes	maximal, complemented
3	#10602613	C₂	1	1	yes	maximal, characteristic
4	#10602612	C₂	1	1	yes	maximal, characteristic
5	#10598183	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#131	C₄	1	1	yes	characteristic, complemented
7	C₂×C₅×#131	C₂²	1	1	yes	characteristic
8	C₅×#532	C₂²	1	1	yes	characteristic
9	C₅×#435	—	2	2	no	—
10	C₅×#435	—	2	2	no	—
11	C₅×#423	C₄	1	1	yes	characteristic, complemented
12	C₅×#408	C₄	1	1	yes	characteristic, complemented
13	C₅×#408	C₂²	1	1	yes	characteristic
14	C₂×C₅×#97	C₄	1	1	yes	characteristic, complemented
15	#65529	C₂²	1	2	yes	—
16	#65526	C₂²	1	2	yes	—
17	#65390	—	2	2	no	—
18	#64776	—	2	2	no	—
19	#64768	C₄	1	2	yes	complemented
20	#64765	C₄	1	2	yes	—
21	#64762	C₄	1	2	yes	complemented
22	#64761	C₄	1	2	yes	—
23	#12634	—	5	1	no	maximal, Sylow
24	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅×#41	C₂³	1	1	yes	characteristic
26	C₅×#131	—	4	2	no	—
27	C₅×#131	—	4	2	no	—
28	C₂×C₅×M₁₆	—	2	1	no	—
29	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×M₁₆	D₄	1	2	yes	—
32	C₂²×C₅×C₈	—	2	1	no	—
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	D₄	1	2	yes	—
36	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
37	C₅×#127	C₂×C₄	1	1	yes	characteristic
38	C₅×#124	C₂×C₄	1	1	yes	characteristic
39	C₅×#124	C₂×C₄	1	1	yes	characteristic
40	C₅×#121	—	2	2	no	—
41	C₄×C₅×D₄	—	2	2	no	—
42	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	#4072	—	2	2	no	—
44	C₂×#681	—	2	2	no	—
45	C₅×C₄⋊C₈	—	2	2	no	—
46	C₅×C₂²⋊C₈	—	2	2	no	—
47	C₅×C₂²⋊C₈	—	2	2	no	—
48	C₅×#97	—	2	1	no	—
49	C₅×#97	—	2	1	no	—
50	#3968	C₂×C₄	1	4	yes	—
51	#3966	—	2	2	no	—
52	#3966	C₂×C₄	1	4	yes	—
53	#3965	—	2	2	no	—
54	#3947	—	4	2	no	—
55	C₈×Dic₅	—	4	2	no	—
56	#2932	D₅	1	1	yes	characteristic
57	#2132	—	5	1	no	—
58	#2126	—	5	1	no	—
59	#1757	—	5	2	no	—
60	#1395	—	5	2	no	—
61	C₅×#41	—	4	1	no	—
62	C₅×#41	—	4	1	no	—
63	C₅×#41	—	4	1	no	—
64	C₅×#41	—	4	1	no	—
65	C₂×C₅×Q₈	C₄²	1	1	yes	characteristic
66	C₂×C₅×Q₈	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₅×D₄	—	2	1	no	—
68	C₂×C₅×D₄	—	2	2	no	—
69	C₂×C₅×D₄	C₄²	1	1	yes	characteristic
70	C₂×C₅×D₄	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₅×D₄	#41	1	2	yes	—
72	C₂²×C₄×C₅	—	2	1	no	—
73	C₂²×C₄×C₅	—	2	1	no	—
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
76	C₅×M₁₆	—	4	1	no	—
77	C₅×M₁₆	—	4	2	no	—
78	C₅×M₁₆	—	4	2	no	—
79	C₅×M₁₆	—	4	1	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	2	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	—	4	2	no	—
86	C₂×C₅×C₈	—	4	1	no	—
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	C₅×C₄⋊C₄	—	2	2	no	—
91	C₅×C₂²⋊C₄	—	2	2	no	—
92	C₅×C₂²⋊C₄	—	2	2	no	—
93	C₄²×C₅	—	2	1	no	—
94	C₄²×C₅	C₄²	1	1	yes	characteristic
95	C₄²×C₅	C₄²	1	1	yes	characteristic
96	C₂²⋊Dic₅	—	2	4	no	—
97	C₄⋊Dic₅	—	4	2	no	—
98	C₄×Dic₅	—	4	2	no	—
99	#681	—	2	4	no	—
100	#681	—	4	2	no	—
101	C₂×#168	—	4	2	no	—
102	C₂×#131	Dic₅	1	1	yes	characteristic, complemented
103	C₂×#131	C₂×D₅	1	1	yes	characteristic
104	#532	C₂×D₅	1	1	yes	characteristic
105	#435	—	2	2	no	—
106	#435	—	2	2	no	—
107	#423	Dic₅	1	1	yes	characteristic, complemented
108	#413	—	5	2	no	—
109	#410	—	5	2	no	—
110	#408	Dic₅	1	1	yes	characteristic, complemented
111	#408	C₂×D₅	1	1	yes	characteristic
112	#405	—	10	2	no	—
113	C₂×#97	Dic₅	1	1	yes	characteristic, complemented
114	#356	—	5	2	no	—
115	#355	—	5	2	no	—
116	#354	—	5	2	no	—
117	#353	—	5	2	no	—
118	#344	—	10	2	no	—
119	C₂³×C₅	—	2	1	no	—
120	C₂³×C₅	#117	1	1	yes	characteristic
121	C₂³×C₅	C₄×D₄	1	2	yes	—
122	C₅×Q₈	—	4	1	no	—
123	C₅×Q₈	—	4	1	no	—
124	C₅×D₄	—	4	1	no	—
125	C₅×D₄	—	4	2	no	—
126	C₅×D₄	—	4	1	no	—
127	C₅×D₄	—	4	2	no	—
128	C₅×D₄	—	8	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	4	1	no	—
137	C₂×C₄×C₅	—	4	1	no	—
138	C₂×C₄×C₅	—	8	1	no	—
139	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
142	C₂×Dic₅	—	4	4	no	—
143	C₅×C₈	—	2	1	no	—
144	C₅×C₈	—	2	2	no	—
145	C₅×C₈	—	4	1	no	—
146	C₅×C₈	—	4	1	no	—
147	C₅×C₈	#99	1	2	yes	—
148	#168	—	4	4	no	—
149	C₂×#41	C₂×Dic₅	1	1	yes	characteristic
150	C₂×#41	C₂²×D₅	1	1	yes	characteristic
151	#131	—	4	2	no	—
152	#131	—	4	2	no	—
153	C₂×M₁₆	—	2	1	no	—
154	C₂×M₁₆	—	10	2	no	—
155	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
156	C₂×M₁₆	C₂×Dic₅	1	1	yes	characteristic
157	C₂×M₁₆	C₂²⋊D₅	1	2	yes	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
160	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
162	C₄⋊Q₈	C₄×D₅	1	1	yes	characteristic
163	#127	C₄×D₅	1	1	yes	characteristic
164	#124	C₄×D₅	1	1	yes	characteristic
165	#124	C₄×D₅	1	1	yes	characteristic
166	#121	—	2	2	no	—
167	C₄×D₄	—	2	2	no	—
168	#117	—	10	2	no	—
169	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
170	C₄⋊C₈	—	2	2	no	—
171	#101	—	5	4	no	—
172	#100	—	10	2	no	—
173	#99	—	5	4	no	—
174	#99	—	10	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	#97	—	2	1	no	—
178	#97	—	2	1	no	—
179	#97	—	20	2	no	—
180	C₄×C₈	—	20	2	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	2	1	no	—
183	C₂²×C₅	—	4	1	no	—
184	C₂²×C₅	—	4	2	no	—
185	C₂²×C₅	—	4	2	no	—
186	C₂²×C₅	—	4	1	no	—
187	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	2	1	no	—
190	C₄×C₅	—	4	1	no	—
191	C₄×C₅	—	4	1	no	—
192	C₄×C₅	—	4	1	no	—
193	C₄×C₅	C₂×#99	1	1	yes	characteristic
194	C₄×C₅	#410	1	1	yes	characteristic
195	Dic₅	—	8	4	no	—
196	#41	—	4	1	no	—
197	#41	—	4	1	no	—
198	#41	—	4	1	no	—
199	#41	—	4	1	no	—
200	C₂×Q₈	C₄×Dic₅	1	1	yes	characteristic
201	C₂×Q₈	C₂×C₄×D₅	1	1	yes	characteristic
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	C₄×Dic₅	1	1	yes	characteristic
205	C₂×D₄	C₂×C₄×D₅	1	1	yes	characteristic
206	C₂×D₄	#709	1	2	yes	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	1	no	—
215	M₁₆	—	10	4	no	—
216	M₁₆	—	20	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	20	2	no	—
225	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
226	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
227	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	20	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	10	4	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	20	2	no	—
235	C₄²	C₄×Dic₅	1	1	yes	characteristic
236	C₄²	C₄×Dic₅	1	1	yes	characteristic
237	C₂×C₅	—	2	1	no	—
238	C₂×C₅	—	4	2	no	—
239	C₂×C₅	—	4	1	no	—
240	C₂×C₅	—	8	1	no	—
241	C₂×C₅	C₄×#99	1	1	yes	characteristic
242	C₂³	—	2	1	no	—
243	C₂³	#4072	1	1	yes	characteristic
244	C₂³	C₄×C₂²⋊D₅	1	2	yes	—
245	Q₈	—	4	1	no	—
246	Q₈	—	4	1	no	—
247	D₄	—	4	1	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	1	no	—
250	D₄	—	4	2	no	—
251	D₄	—	8	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	8	1	no	—
262	C₂×C₄	—	20	4	no	—
263	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
264	C₂×C₄	C₄×C₂²⋊D₅	1	2	yes	—
265	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	2	no	—
268	C₈	—	4	1	no	—
269	C₈	—	4	1	no	—
270	C₈	—	20	4	no	—
271	C₈	#3966	1	2	yes	—
272	C₅	#12634	1	1	yes	characteristic, complemented, Sylow
273	C₂²	—	2	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	1	no	—
279	C₂²	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	1	no	—
285	C₄	—	40	4	no	—
286	C₄	C₂×#3966	1	1	yes	characteristic
287	C₄	#65529	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	4	2	no	—
290	C₂	—	4	1	no	—
291	C₂	—	8	1	no	—
292	C₂	C₄×#3966	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#3966
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×#41	C₂²×D₅
Fitting	C₅×#2932	C₂
socle	C₂×C₅	C₄×#99
prime radical	C₅×#532	C₂²
cosocle	C₂×C₅×#41	C₂³
hypercenter	#2932	D₅
hypocenter	C₅	#12634
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×#131 ◁ #2932

Complements

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

N	H ≅ G/N	type
2. #10602804	C₂	semidirect
5. #10598183	C₂	semidirect
6. C₂×C₅×#131	C₄	semidirect
11. C₅×#423	C₄	semidirect
12. C₅×#408	C₄	semidirect
14. C₂×C₅×#97	C₄	semidirect
19. #64768	C₄	semidirect
21. #64762	C₄	semidirect
102. C₂×#131	Dic₅	semidirect
107. #423	Dic₅	semidirect
110. #408	Dic₅	semidirect
113. C₂×#97	Dic₅	semidirect
272. C₅	#12634	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2932	2
#10602804	2
#10602613	2
#10602612	2
#10598183	2
#12634	5