Subgroups — #12252399

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	#10610280	C₂	1	1	yes	maximal, characteristic
2	C₅×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602225	C₂	1	1	yes	maximal, characteristic
4	Dic₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	#10602202	C₂	1	1	yes	maximal, characteristic
6	C₂×#64722	C₂	1	1	yes	maximal, characteristic, complemented
7	#10600675	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600069	C₂	1	4	yes	maximal, complemented
9	#10600068	C₂	1	4	yes	maximal, complemented
10	C₂×#4053	C₄	1	1	yes	characteristic
11	C₂×#4047	C₂²	1	1	yes	characteristic
12	#65856	C₄	1	1	yes	characteristic
13	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₅×#405	C₂²	1	1	yes	characteristic
16	M₁₆×Dic₅	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
18	C₂×#3944	C₂²	1	1	yes	characteristic
19	C₂²×#672	C₂²	1	1	yes	characteristic
20	#65286	C₄	1	1	yes	characteristic
21	#64997	C₄	1	2	yes	—
22	#64996	C₄	1	2	yes	—
23	#64978	C₂²	1	2	yes	—
24	C₄×C₈×D₅	C₂²	1	2	yes	—
25	C₅×C₄⋊C₁₆	C₂²	1	4	yes	—
26	#64727	C₂²	1	4	yes	—
27	#64727	C₂²	1	4	yes	—
28	#64722	C₂²	1	2	yes	—
29	#64722	C₂²	1	2	yes	—
30	#64722	C₂²	1	2	yes	—
31	C₁₆×Dic₅	C₂²	1	2	yes	—
32	C₄⋊#672	C₂²	1	4	yes	—
33	#33421	—	5	1	no	maximal, Sylow
34	C₂×#709	C₂×C₄	1	1	yes	characteristic
35	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₅×#117	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
39	C₂×#681	C₂×C₄	1	1	yes	characteristic
40	C₂²×#168	C₂×C₄	1	1	yes	characteristic
41	#4053	—	2	4	no	—
42	#4047	—	2	2	no	—
43	#4047	—	2	2	no	—
44	C₂×#676	C₂×C₄	1	2	yes	—
45	C₂×#676	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₅	C₂³	1	2	yes	—
48	#4039	C₈	1	2	yes	—
49	#4033	C₈	1	2	yes	—
50	#4027	C₈	1	4	yes	—
51	#4018	C₂×C₄	1	2	yes	—
52	C₄²×D₅	C₂×C₄	1	2	yes	—
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	C₅×#97	C₂×C₄	1	2	yes	—
58	C₄×C₅×C₈	C₂³	1	2	yes	—
59	#3947	C₂×C₄	1	2	yes	—
60	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
61	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
62	C₈×Dic₅	C₂×C₄	1	2	yes	—
63	C₈×Dic₅	C₂³	1	1	yes	characteristic
64	C₈×Dic₅	C₂³	1	1	yes	characteristic
65	#3944	—	2	2	no	—
66	C₂×#672	—	2	2	no	—
67	C₂×#672	C₂³	1	2	yes	—
68	C₂×#672	C₂³	1	2	yes	—
69	#3935	C₂×C₄	1	2	yes	—
70	C₄×#168	C₂×C₄	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	D₅	1	1	yes	characteristic, complemented
76	#2113	—	5	1	no	—
77	#2113	—	5	1	no	—
78	C₄×M₃₂	—	5	1	no	—
79	C₂×#346	—	5	1	no	—
80	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
82	C₂²×Dic₅	C₂×C₈	1	2	yes	—
83	#709	—	2	2	no	—
84	#709	—	2	2	no	—
85	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₅	C₂×C₈	1	4	yes	—
87	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
88	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
89	C₅×M₁₆	—	2	2	no	—
90	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	#41	1	2	yes	—
93	C₂×C₅×C₈	#41	1	2	yes	—
94	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
95	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
96	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
97	C₄²×C₅	C₂²×C₄	1	2	yes	—
98	#685	C₂×C₈	1	4	yes	—
99	#683	C₂×C₈	1	4	yes	—
100	C₄×Dic₅	C₂×C₈	1	4	yes	—
101	C₄×Dic₅	C₂×C₈	1	2	yes	—
102	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
103	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
104	#681	—	2	2	no	—
105	C₂×#168	M₁₆	1	4	yes	—
106	C₂×#168	C₂²×C₄	1	2	yes	—
107	C₂×#168	C₂²×C₄	1	1	yes	characteristic
108	C₂×#168	C₂²×C₄	1	1	yes	characteristic
109	#676	—	2	4	no	—
110	#676	—	2	4	no	—
111	C₈×D₅	—	2	2	no	—
112	C₈×D₅	—	2	2	no	—
113	C₈×D₅	—	2	4	no	—
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	—	5	1	no	—
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	2	no	—
128	#405	C₂×D₅	1	1	yes	characteristic
129	C₄×M₁₆	—	5	1	no	—
130	C₄×M₁₆	—	5	2	no	—
131	C₂×C₄×C₈	—	5	1	no	—
132	C₂×C₄×C₈	—	5	2	no	—
133	C₄⋊C₁₆	—	5	4	no	—
134	C₄⋊C₁₆	C₂×D₅	1	4	yes	—
135	C₂²⋊C₁₆	—	5	4	no	—
136	C₂²⋊C₁₆	—	5	4	no	—
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	1	yes	characteristic
142	C₂²×D₅	C₂²×C₈	1	2	yes	—
143	C₂×C₄×C₅	#117	1	2	yes	—
144	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
145	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
147	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
148	C₂²⋊D₅	—	2	4	no	—
149	C₂×Dic₅	—	4	2	no	—
150	C₂×Dic₅	C₂²×C₈	1	2	yes	—
151	C₂×Dic₅	C₂²×C₈	1	2	yes	—
152	C₂×Dic₅	C₂²×C₈	1	2	yes	—
153	D₂₀	—	2	2	no	—
154	C₄×D₅	—	2	2	no	—
155	C₄×D₅	—	2	2	no	—
156	C₄×D₅	—	4	4	no	—
157	Dic₁₀	—	2	2	no	—
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	—	5	1	no	—
166	#131	—	10	4	no	—
167	#131	—	10	2	no	—
168	#131	—	10	2	no	—
169	C₂×M₁₆	—	5	1	no	—
170	C₂×M₁₆	—	5	2	no	—
171	C₂×M₁₆	—	5	2	no	—
172	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
173	C₂²×C₈	—	5	1	no	—
174	C₂²×C₈	—	5	2	no	—
175	C₂²×C₈	—	5	2	no	—
176	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
177	C₄×Q₈	—	5	2	no	—
178	C₄×D₄	—	5	2	no	—
179	C₄×D₄	—	5	4	no	—
180	#117	—	5	2	no	—
181	#117	C₄×D₅	1	1	yes	characteristic
182	C₂×C₄²	—	5	1	no	—
183	C₂×C₄²	—	5	2	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	C₄×D₅	1	2	yes	—
195	C₄×C₈	—	5	1	no	—
196	C₄×C₈	—	5	2	no	—
197	C₄×C₈	—	5	1	no	—
198	C₄×C₈	—	5	1	no	—
199	C₄×C₈	—	5	1	no	—
200	C₄×C₈	—	5	2	no	—
201	C₄×C₈	C₂²×D₅	1	2	yes	—
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	#431	1	2	yes	—
204	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
205	C₂×D₅	—	2	4	no	—
206	C₂×D₅	—	2	2	no	—
207	C₄×C₅	—	2	4	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	2	no	—
214	#41	—	10	2	no	—
215	#41	—	10	2	no	—
216	C₂×Q₈	—	5	1	no	—
217	C₂×D₄	—	5	2	no	—
218	C₂×D₄	—	5	1	no	—
219	C₂²×C₄	—	5	2	no	—
220	C₂²×C₄	—	5	2	no	—
221	C₂²×C₄	—	5	4	no	—
222	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
223	M₁₆	—	2	2	no	—
224	M₁₆	—	10	4	no	—
225	M₁₆	—	10	2	no	—
226	M₁₆	—	10	4	no	—
227	C₂×C₈	—	5	1	no	—
228	C₂×C₈	—	5	1	no	—
229	C₂×C₈	—	5	4	no	—
230	C₂×C₈	—	5	2	no	—
231	C₂×C₈	—	10	4	no	—
232	C₂×C₈	—	10	2	no	—
233	C₂×C₈	—	10	2	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	#713	1	2	yes	—
238	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
239	C₄⋊C₄	—	5	4	no	—
240	C₄⋊C₄	C₈×D₅	1	2	yes	—
241	C₂²⋊C₄	—	5	4	no	—
242	C₂²⋊C₄	C₈×D₅	1	2	yes	—
243	C₄²	—	5	1	no	—
244	C₄²	—	5	1	no	—
245	C₄²	—	5	2	no	—
246	C₄²	—	5	4	no	—
247	C₄²	C₂×C₄×D₅	1	2	yes	—
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	C₂×#431	1	1	yes	characteristic
254	C₂×C₅	#3407	1	2	yes	—
255	D₅	—	4	2	no	—
256	C₂³	—	5	2	no	—
257	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
258	Q₈	—	10	2	no	—
259	D₄	—	10	4	no	—
260	D₄	—	10	2	no	—
261	C₂×C₄	—	5	2	no	—
262	C₂×C₄	—	5	2	no	—
263	C₂×C₄	—	5	2	no	—
264	C₂×C₄	—	10	2	no	—
265	C₂×C₄	—	10	2	no	—
266	C₂×C₄	—	20	4	no	—
267	C₂×C₄	—	20	2	no	—
268	C₂×C₄	#4038	1	2	yes	—
269	C₂×C₄	C₂×C₈×D₅	1	4	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	#4038	1	2	yes	—
278	C₈	C₂×#711	1	1	yes	characteristic
279	C₈	C₂×#713	1	1	yes	characteristic
280	C₅	#33421	1	1	yes	characteristic, complemented, Sylow
281	C₂²	—	2	2	no	—
282	C₂²	—	10	2	no	—
283	C₂²	—	10	4	no	—
284	C₂²	#65122	1	2	yes	—
285	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	10	2	no	—
288	C₄	—	10	2	no	—
289	C₄	—	10	2	no	—
290	C₄	#65122	1	2	yes	—
291	C₄	C₂×#4038	1	1	yes	characteristic
292	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	20	2	no	—
295	C₂	C₂×#65122	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×#2151	C₂
prime radical	C₂×#709	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2151	D₅
hypocenter	C₅	#33421
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ #2151

Complements

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

N	H ≅ G/N	type
2. C₅×#2151	C₂	semidirect
4. Dic₅×M₃₂	C₂	semidirect
6. C₂×#64722	C₂	semidirect
7. #10600675	C₂	semidirect
8. #10600069	C₂	semidirect
9. #10600068	C₂	semidirect
75. #2151	D₅	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
#10610280	2
C₅×#2151	2
#10602225	2
Dic₅×M₃₂	2
#10602202	2
C₂×#64722	2
#10600675	2
#10600069	2
#10600068	2
#33421	5