Subgroups — #12252396

Order 1280.

582 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₅×#131	C₂	1	1	yes	maximal, characteristic
2	C₅×#2113	C₂	1	1	yes	maximal, characteristic
3	#10602797	C₂	1	1	yes	maximal, characteristic
4	#10602214	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#10602194	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#64722	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599887	C₂	1	4	yes	maximal, complemented
9	#10599886	C₂	1	4	yes	maximal, complemented
10	C₂×C₅×#131	C₂²	1	1	yes	characteristic
11	#41×Dic₅	C₄	1	1	yes	characteristic
12	C₂×#4094	C₄	1	1	yes	characteristic
13	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	#65411	C₄	1	2	yes	—
16	#65411	C₄	1	1	yes	characteristic
17	M₁₆×Dic₅	C₄	1	1	yes	characteristic
18	M₁₆×Dic₅	C₄	1	2	yes	—
19	#65390	C₂²	1	2	yes	—
20	#65390	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₅	C₂²	1	2	yes	—
22	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
23	C₂×#3944	C₂²	1	1	yes	characteristic
24	C₂²×#672	C₂²	1	1	yes	characteristic
25	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	—
26	C₂²⋊#672	C₂²	1	4	yes	—
27	#64722	C₂²	1	2	yes	—
28	#64722	C₂²	1	2	yes	—
29	#64722	C₂²	1	2	yes	—
30	#64721	C₂²	1	4	yes	—
31	#64721	C₂²	1	4	yes	—
32	C₁₆×Dic₅	C₂²	1	2	yes	—
33	#33421	—	5	1	no	maximal, Sylow
34	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
35	C₅×#131	—	2	2	no	—
36	C₅×#131	—	2	2	no	—
37	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
39	C₂²×C₅×C₈	C₂³	1	2	yes	—
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	#4094	—	2	4	no	—
42	Q₈×Dic₅	C₈	1	2	yes	—
43	D₄×Dic₅	C₈	1	4	yes	—
44	D₄×Dic₅	C₈	1	2	yes	—
45	#4072	C₂×C₄	1	2	yes	—
46	#4072	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
49	C₂×#681	C₂×C₄	1	1	yes	characteristic
50	C₂×#681	C₂×C₄	1	2	yes	—
51	C₂²×#168	C₂×C₄	1	2	yes	—
52	C₂²×#168	C₂×C₄	1	1	yes	characteristic
53	C₅×M₃₂	—	2	2	no	—
54	C₂×C₅×C₁₆	—	2	2	no	—
55	C₂×C₅×C₁₆	C₂³	1	2	yes	—
56	C₂×C₅×C₁₆	C₂³	1	2	yes	—
57	#3947	C₂×C₄	1	2	yes	—
58	#3947	C₂×C₄	1	2	yes	—
59	#3947	C₂×C₄	1	2	yes	—
60	C₈×Dic₅	C₂×C₄	1	2	yes	—
61	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
62	C₈×Dic₅	C₂×C₄	1	2	yes	—
63	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
64	C₈×Dic₅	C₂³	1	1	yes	characteristic
65	C₈×Dic₅	C₂³	1	1	yes	characteristic
66	C₈×Dic₅	C₂³	1	2	yes	—
67	#3944	—	2	2	no	—
68	C₂×#672	—	2	2	no	—
69	C₂×#672	C₂³	1	2	yes	—
70	C₂×#672	C₂³	1	2	yes	—
71	C₄×#131	—	5	1	no	—
72	#2164	—	5	4	no	—
73	#2163	—	5	4	no	—
74	#2151	—	5	1	no	—
75	#2151	—	5	1	no	—
76	#2113	—	5	1	no	—
77	#2113	D₅	1	1	yes	characteristic
78	C₄×M₃₂	—	5	1	no	—
79	C₂×#346	—	5	1	no	—
80	C₅×#41	—	2	2	no	—
81	C₅×#41	—	2	2	no	—
82	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
83	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
84	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂²×Dic₅	C₂×C₈	1	4	yes	—
88	C₂²×Dic₅	C₂×C₈	1	2	yes	—
89	C₅×M₁₆	—	2	2	no	—
90	C₅×M₁₆	—	2	4	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₅×C₈	#41	1	2	yes	—
96	C₂×C₅×C₈	#41	1	2	yes	—
97	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
98	C₂²⋊Dic₅	C₂×C₈	1	4	yes	—
99	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
100	C₄⋊Dic₅	C₂×C₈	1	4	yes	—
101	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
102	C₄×Dic₅	C₂×C₈	1	4	yes	—
103	C₄×Dic₅	C₂×C₈	1	2	yes	—
104	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
105	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
106	C₄×Dic₅	C₂²×C₄	1	2	yes	—
107	#681	—	2	2	no	—
108	#681	—	2	4	no	—
109	C₂×#168	—	2	4	no	—
110	C₂×#168	M₁₆	1	4	yes	—
111	C₂×#168	C₂²×C₄	1	2	yes	—
112	C₂×#168	C₂²×C₄	1	1	yes	characteristic
113	C₂×#168	C₂²×C₄	1	1	yes	characteristic
114	C₅×C₁₆	—	2	2	no	—
115	C₅×C₁₆	—	2	2	no	—
116	#672	—	2	2	no	—
117	#672	—	2	2	no	—
118	C₂×#131	—	5	1	no	—
119	C₂×#131	C₂×D₅	1	1	yes	characteristic
120	C₄×#41	—	5	1	no	—
121	C₂×M₃₂	—	5	1	no	—
122	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
123	C₂²×C₁₆	—	5	1	no	—
124	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
125	#405	—	5	1	no	—
126	#405	—	5	2	no	—
127	#405	—	5	1	no	—
128	#405	—	5	2	no	—
129	C₄×M₁₆	—	5	2	no	—
130	C₄×M₁₆	—	5	1	no	—
131	C₂×C₄×C₈	—	5	1	no	—
132	C₂×C₄×C₈	—	5	2	no	—
133	C₄⋊C₁₆	—	5	4	no	—
134	C₄⋊C₁₆	—	5	4	no	—
135	C₂²⋊C₁₆	—	5	4	no	—
136	C₂²⋊C₁₆	C₂×D₅	1	4	yes	—
137	#346	—	5	2	no	—
138	#346	—	5	2	no	—
139	#346	—	5	2	no	—
140	C₄×C₁₆	—	5	2	no	—
141	C₂³×C₅	C₂²×C₈	1	2	yes	—
142	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
143	C₅×Q₈	—	2	2	no	—
144	C₅×D₄	—	2	4	no	—
145	C₅×D₄	—	2	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	#117	1	2	yes	—
149	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
150	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
152	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
153	C₂×Dic₅	—	4	2	no	—
154	C₂×Dic₅	—	4	4	no	—
155	C₂×Dic₅	C₂²×C₈	1	2	yes	—
156	C₂×Dic₅	C₂²×C₈	1	4	yes	—
157	C₂×Dic₅	C₂²×C₈	1	2	yes	—
158	C₅×C₈	—	2	2	no	—
159	C₅×C₈	#117	1	2	yes	—
160	C₅×C₈	C₂×#41	1	1	yes	characteristic
161	C₅×C₈	C₂×#41	1	1	yes	characteristic
162	#168	—	2	2	no	—
163	#168	C₂×M₁₆	1	2	yes	—
164	#168	C₂×M₁₆	1	2	yes	—
165	C₂×#41	C₄×D₅	1	1	yes	characteristic
166	#131	—	2	2	no	—
167	#131	—	2	2	no	—
168	#131	—	10	4	no	—
169	C₂×M₁₆	—	5	2	no	—
170	C₂×M₁₆	—	5	1	no	—
171	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
172	C₂×M₁₆	C₄×D₅	1	2	yes	—
173	C₂²×C₈	—	5	1	no	—
174	C₂²×C₈	—	5	2	no	—
175	C₂²×C₈	C₂²×D₅	1	2	yes	—
176	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
177	C₄×Q₈	—	5	2	no	—
178	C₄×D₄	—	5	4	no	—
179	C₄×D₄	—	5	2	no	—
180	#117	—	5	2	no	—
181	#117	—	5	1	no	—
182	C₂×C₄²	—	5	2	no	—
183	C₂×C₄²	—	5	1	no	—
184	M₃₂	—	2	2	no	—
185	M₃₂	—	10	2	no	—
186	C₂×C₁₆	—	2	2	no	—
187	C₂×C₁₆	—	5	2	no	—
188	C₂×C₁₆	—	5	2	no	—
189	C₂×C₁₆	—	10	2	no	—
190	C₂×C₁₆	C₂²×D₅	1	2	yes	—
191	C₂×C₁₆	C₂²×D₅	1	2	yes	—
192	#97	—	5	2	no	—
193	#97	—	5	2	no	—
194	#97	—	5	2	no	—
195	C₄×C₈	—	5	1	no	—
196	C₄×C₈	—	5	2	no	—
197	C₄×C₈	—	5	2	no	—
198	C₄×C₈	—	5	1	no	—
199	C₄×C₈	—	5	1	no	—
200	C₄×C₈	—	5	1	no	—
201	C₄×C₈	—	5	2	no	—
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	—	2	4	no	—
204	C₂²×C₅	—	2	2	no	—
205	C₂²×C₅	#431	1	2	yes	—
206	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
207	C₄×C₅	—	2	2	no	—
208	C₄×C₅	#431	1	2	yes	—
209	C₄×C₅	C₂×#117	1	1	yes	characteristic
210	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
211	Dic₅	—	2	2	no	—
212	Dic₅	—	2	2	no	—
213	Dic₅	—	2	4	no	—
214	#41	—	2	2	no	—
215	#41	—	2	2	no	—
216	C₂×Q₈	C₈×D₅	1	1	yes	characteristic
217	C₂×D₄	C₈×D₅	1	2	yes	—
218	C₂×D₄	C₈×D₅	1	1	yes	characteristic
219	C₂²×C₄	—	5	4	no	—
220	C₂²×C₄	—	5	2	no	—
221	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
222	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
223	M₁₆	—	2	2	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	10	4	no	—
226	M₁₆	—	10	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	5	1	no	—
230	C₂×C₈	—	5	1	no	—
231	C₂×C₈	—	5	4	no	—
232	C₂×C₈	—	5	2	no	—
233	C₂×C₈	—	10	4	no	—
234	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
236	C₂×C₈	#711	1	2	yes	—
237	C₂×C₈	#711	1	2	yes	—
238	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
239	C₄⋊C₄	—	5	4	no	—
240	C₄⋊C₄	—	5	2	no	—
241	C₂²⋊C₄	—	5	4	no	—
242	C₂²⋊C₄	—	5	2	no	—
243	C₄²	—	5	1	no	—
244	C₄²	—	5	1	no	—
245	C₄²	—	5	2	no	—
246	C₄²	—	5	4	no	—
247	C₄²	—	5	2	no	—
248	C₁₆	—	2	2	no	—
249	C₁₆	—	2	2	no	—
250	C₁₆	—	10	2	no	—
251	C₁₆	—	10	2	no	—
252	C₂×C₅	—	2	2	no	—
253	C₂×C₅	—	4	2	no	—
254	C₂×C₅	C₂×#431	1	1	yes	characteristic
255	C₂×C₅	#3407	1	2	yes	—
256	C₂³	C₂×C₈×D₅	1	2	yes	—
257	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
258	Q₈	—	2	2	no	—
259	D₄	—	2	4	no	—
260	D₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	5	2	no	—
264	C₂×C₄	—	5	2	no	—
265	C₂×C₄	—	5	4	no	—
266	C₂×C₄	—	20	2	no	—
267	C₂×C₄	—	20	4	no	—
268	C₂×C₄	#4023	1	2	yes	—
269	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
270	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
271	C₂×C₄	D₅×M₁₆	1	2	yes	—
272	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	—	5	2	no	—
275	C₈	—	5	2	no	—
276	C₈	—	10	2	no	—
277	C₈	#4023	1	2	yes	—
278	C₈	C₂×#711	1	1	yes	characteristic
279	C₈	C₂×#711	1	1	yes	characteristic
280	C₅	#33421	1	1	yes	characteristic, complemented, Sylow
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	#65007	1	2	yes	—
285	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	10	2	no	—
288	C₄	—	10	2	no	—
289	C₄	—	10	4	no	—
290	C₄	#65007	1	2	yes	—
291	C₄	C₂×#4023	1	1	yes	characteristic
292	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	4	2	no	—
295	C₂	C₂×#65007	1	1	yes	characteristic
296	C₂	#10613743	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₅×#2113	C₂
prime radical	C₂×C₅×#41	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2113	D₅
hypocenter	C₅	#33421
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ #2113

Complements

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

N	H ≅ G/N	type
4. #10602214	C₂	semidirect
5. Dic₅×M₃₂	C₂	semidirect
6. #10602194	C₂	semidirect
7. C₂×#64722	C₂	semidirect
8. #10599887	C₂	semidirect
9. #10599886	C₂	semidirect
280. C₅	#33421	semidirect

Maximal subgroups

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

subgroup	index
Dic₅×#131	2
C₅×#2113	2
#10602797	2
#10602214	2
Dic₅×M₃₂	2
#10602194	2
C₂×#64722	2
#10599887	2
#10599886	2
#33421	5