Subgroups — #12252301

Order 1280.

684 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	#10613738	C₂	1	2	yes	maximal
2	C₄×#4047	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602216	C₂	1	2	yes	maximal, complemented
5	C₂×#64683	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600096	C₂	1	4	yes	maximal, complemented
7	D₅×#364	C₂	1	4	yes	maximal, complemented
8	C₂×#4047	C₄	1	1	yes	characteristic
9	C₂×#4047	C₂²	1	1	yes	characteristic
10	C₄×#709	C₄	1	1	yes	characteristic
11	C₂×C₅×M₃₂	C₂²	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	#65390	C₄	1	1	yes	characteristic
14	#65390	C₂²	1	1	yes	characteristic
15	C₂×#3944	C₂²	1	2	yes	—
16	C₄×#681	C₄	1	1	yes	characteristic
17	#65196	—	2	4	no	—
18	D₅×M₃₂	C₂²	1	8	yes	—
19	#65190	—	2	4	no	—
20	#64978	C₄	1	2	yes	—
21	#64978	C₂²	1	2	yes	—
22	C₄×#676	C₄	1	2	yes	—
23	C₄×C₈×D₅	C₂²	1	2	yes	—
24	C₅×#364	C₂²	1	2	yes	—
25	C₅×#364	C₂²	1	2	yes	—
26	#64731	C₂²	1	8	yes	—
27	#64683	C₂²	1	2	yes	—
28	#64683	C₂²	1	2	yes	—
29	#33401	—	5	1	no	maximal, Sylow
30	C₂×#709	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₅	—	2	2	no	—
35	#4072	C₂×C₄	1	1	yes	characteristic
36	C₂×#681	C₂×C₄	1	1	yes	characteristic
37	C₂×#681	C₂×C₄	1	1	yes	characteristic
38	#4047	C₈	1	4	yes	—
39	#4047	C₈	1	4	yes	—
40	#4047	D₄	1	2	yes	—
41	#4047	D₄	1	2	yes	—
42	#4047	Q₈	1	2	yes	—
43	#4047	Q₈	1	2	yes	—
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	C₄×D₂₀	—	2	1	no	—
49	#4018	C₂×C₄	1	2	yes	—
50	C₄²×D₅	C₂×C₄	1	2	yes	—
51	C₄×Dic₁₀	—	2	1	no	—
52	C₅×M₃₂	—	2	2	no	—
53	C₅×M₃₂	C₂³	1	4	yes	—
54	C₂×C₅×C₁₆	—	2	2	no	—
55	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
56	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
57	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
58	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
59	#3947	C₂×C₄	1	2	yes	—
60	#3947	C₂×C₄	1	2	yes	—
61	C₈×Dic₅	C₂×C₄	1	2	yes	—
62	C₈×Dic₅	C₂³	1	2	yes	—
63	#3944	—	2	2	no	—
64	#3944	C₂³	1	4	yes	—
65	C₂×#672	—	2	2	no	—
66	#3935	C₂×C₄	1	2	yes	—
67	C₄×#168	C₂×C₄	1	2	yes	—
68	#3930	—	2	8	no	—
69	D₅×C₁₆	—	2	8	no	—
70	#3407	—	5	2	no	—
71	C₄×#131	—	5	1	no	—
72	#2153	—	5	2	no	—
73	#2153	—	5	4	no	—
74	C₂×#364	—	5	1	no	—
75	C₂×#364	—	5	4	no	—
76	C₂×#364	D₅	1	1	yes	characteristic, complemented
77	C₂²×C₄×C₅	—	2	1	no	—
78	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₂²⋊D₅	M₁₆	1	2	yes	—
80	#709	C₄⋊C₄	1	2	yes	—
81	#709	C₄⋊C₄	1	2	yes	—
82	#709	C₂×C₈	1	2	yes	—
83	#709	C₂×C₈	1	2	yes	—
84	C₂×D₂₀	M₁₆	1	1	yes	characteristic
85	C₂×C₄×D₅	—	2	2	no	—
86	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
87	C₂×Dic₁₀	M₁₆	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
89	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
90	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
98	C₄²×C₅	—	2	1	no	—
99	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
101	C₂²⋊Dic₅	—	2	1	no	—
102	#685	—	2	2	no	—
103	C₄⋊Dic₅	—	2	1	no	—
104	#683	—	2	2	no	—
105	C₄×Dic₅	—	2	2	no	—
106	C₄×Dic₅	C₂²×C₄	1	2	yes	—
107	#681	C₄⋊C₄	1	2	yes	—
108	#681	C₄⋊C₄	1	2	yes	—
109	#681	C₂×C₈	1	2	yes	—
110	#681	C₂×C₈	1	2	yes	—
111	C₂×#168	C₂²×C₄	1	2	yes	—
112	C₂×#168	C₂²×C₄	1	2	yes	—
113	#676	C₄⋊C₄	1	4	yes	—
114	#676	C₄⋊C₄	1	4	yes	—
115	#676	C₂×C₈	1	8	yes	—
116	C₈×D₅	C₂×C₈	1	8	yes	—
117	C₈×D₅	C₂×D₄	1	4	yes	—
118	C₈×D₅	C₂×Q₈	1	4	yes	—
119	C₅×C₁₆	—	2	4	no	—
120	#672	—	2	4	no	—
121	C₂×#131	—	5	1	no	—
122	C₂×#131	—	5	1	no	—
123	C₄×#41	—	5	1	no	—
124	#504	—	10	4	no	—
125	#504	—	10	4	no	—
126	C₂×M₃₂	—	5	2	no	—
127	C₂×M₃₂	—	5	8	no	—
128	C₂×M₃₂	C₂×D₅	1	2	yes	—
129	#405	—	5	1	no	—
130	#405	—	5	1	no	—
131	#405	—	5	2	no	—
132	#405	—	5	2	no	—
133	C₄×M₁₆	—	5	1	no	—
134	C₄×M₁₆	—	5	2	no	—
135	C₂×C₄×C₈	—	5	2	no	—
136	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
137	#364	—	5	8	no	—
138	#364	—	5	2	no	—
139	#364	—	5	2	no	—
140	#364	C₂×D₅	1	2	yes	—
141	#364	C₂×D₅	1	2	yes	—
142	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
143	C₂²×D₅	C₂×M₁₆	1	2	yes	—
144	C₂×C₄×C₅	—	2	1	no	—
145	C₂×C₄×C₅	—	2	1	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
153	C₂²⋊D₅	C₄⋊C₈	1	4	yes	—
154	C₂²⋊D₅	C₄⋊C₈	1	4	yes	—
155	C₂×Dic₅	—	2	2	no	—
156	C₂×Dic₅	C₂×M₁₆	1	2	yes	—
157	D₂₀	C₄⋊C₈	1	2	yes	—
158	D₂₀	C₄⋊C₈	1	2	yes	—
159	C₄×D₅	—	4	4	no	—
160	C₄×D₅	C₂×C₄⋊C₄	1	4	yes	—
161	C₄×D₅	C₂²×C₈	1	4	yes	—
162	Dic₁₀	C₄⋊C₈	1	2	yes	—
163	Dic₁₀	C₄⋊C₈	1	2	yes	—
164	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₅×C₈	C₂²×C₈	1	2	yes	—
167	C₅×C₈	C₂²×C₈	1	2	yes	—
168	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
169	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
170	#168	C₂×C₄⋊C₄	1	4	yes	—
171	#168	C₂²×C₈	1	4	yes	—
172	C₂×#41	—	5	1	no	—
173	#131	—	5	4	no	—
174	#131	—	5	4	no	—
175	#131	—	5	2	no	—
176	#131	—	5	2	no	—
177	#131	—	5	2	no	—
178	#131	—	5	2	no	—
179	C₂×M₁₆	—	5	1	no	—
180	C₂×M₁₆	—	5	1	no	—
181	C₂×M₁₆	—	5	2	no	—
182	C₂×M₁₆	—	5	2	no	—
183	C₂²×C₈	—	5	2	no	—
184	C₂²×C₈	—	5	2	no	—
185	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
187	C₄×Q₈	—	10	1	no	—
188	C₄×D₄	—	10	1	no	—
189	C₄×D₄	—	10	2	no	—
190	#117	—	5	1	no	—
191	#117	—	5	2	no	—
192	C₂×C₄²	—	5	2	no	—
193	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
194	M₃₂	—	2	2	no	—
195	M₃₂	—	5	4	no	—
196	M₃₂	—	10	8	no	—
197	M₃₂	—	10	2	no	—
198	M₃₂	C₂²×D₅	1	4	yes	—
199	C₂×C₁₆	—	2	2	no	—
200	C₂×C₁₆	—	10	8	no	—
201	C₂×C₁₆	—	10	2	no	—
202	#97	—	5	2	no	—
203	#97	—	5	2	no	—
204	#97	—	5	2	no	—
205	C₄×C₈	—	5	2	no	—
206	C₄×C₈	—	5	2	no	—
207	C₄×C₈	—	5	2	no	—
208	C₄×C₈	C₄×D₅	1	1	yes	characteristic
209	C₄×C₈	C₄×D₅	1	1	yes	characteristic
210	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
211	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
212	C₂²×C₅	—	4	1	no	—
213	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
214	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
215	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
216	C₂×D₅	—	4	2	no	—
217	C₂×D₅	C₂×C₄⋊C₈	1	4	yes	—
218	C₄×C₅	—	2	2	no	—
219	C₄×C₅	—	4	1	no	—
220	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
221	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
222	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
223	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
224	Dic₅	—	4	2	no	—
225	Dic₅	C₂×C₄⋊C₈	1	4	yes	—
226	#41	—	5	2	no	—
227	#41	—	5	2	no	—
228	#41	—	5	2	no	—
229	#41	—	5	2	no	—
230	C₂×Q₈	—	5	1	no	—
231	C₂×D₄	—	5	2	no	—
232	C₂×D₄	—	5	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	5	2	no	—
235	C₂²×C₄	—	10	2	no	—
236	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
237	M₁₆	—	5	2	no	—
238	M₁₆	—	5	2	no	—
239	M₁₆	—	5	2	no	—
240	M₁₆	—	5	2	no	—
241	M₁₆	—	5	4	no	—
242	M₁₆	—	5	4	no	—
243	M₁₆	—	5	8	no	—
244	C₂×C₈	—	5	2	no	—
245	C₂×C₈	—	5	2	no	—
246	C₂×C₈	—	5	4	no	—
247	C₂×C₈	—	5	4	no	—
248	C₂×C₈	—	5	8	no	—
249	C₂×C₈	C₈×D₅	1	2	yes	—
250	C₂×C₈	C₈×D₅	1	2	yes	—
251	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
252	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
253	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
254	C₂×C₈	D₄×D₅	1	1	yes	characteristic
255	C₂×C₈	D₄×D₅	1	1	yes	characteristic
256	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
257	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
258	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
259	C₄⋊C₄	—	10	2	no	—
260	C₄⋊C₄	—	10	1	no	—
261	C₂²⋊C₄	—	10	2	no	—
262	C₂²⋊C₄	—	10	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	5	2	no	—
265	C₄²	—	10	2	no	—
266	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
267	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
268	C₁₆	—	2	4	no	—
269	C₁₆	—	10	4	no	—
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	—	2	1	no	—
272	C₂×C₅	—	2	1	no	—
273	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
274	D₅	—	2	4	no	—
275	C₂³	—	5	2	no	—
276	C₂³	D₅×M₁₆	1	1	yes	characteristic
277	Q₈	—	5	2	no	—
278	Q₈	—	5	2	no	—
279	D₄	—	5	4	no	—
280	D₄	—	5	4	no	—
281	D₄	—	5	2	no	—
282	D₄	—	5	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	5	2	no	—
287	C₂×C₄	—	5	4	no	—
288	C₂×C₄	—	5	4	no	—
289	C₂×C₄	—	10	2	no	—
290	C₂×C₄	—	20	4	no	—
291	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
292	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
293	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
294	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
295	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
296	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
297	C₈	—	5	4	no	—
298	C₈	—	5	4	no	—
299	C₈	D₅×C₄⋊C₄	1	1	yes	characteristic
300	C₈	D₅×C₄⋊C₄	1	1	yes	characteristic
301	C₈	C₂×C₈×D₅	1	2	yes	—
302	C₈	C₂×C₈×D₅	1	2	yes	—
303	C₈	C₂×D₄×D₅	1	1	yes	characteristic
304	C₈	C₂×Q₈×D₅	1	1	yes	characteristic
305	C₅	#33401	1	1	yes	characteristic, complemented, Sylow
306	C₂²	—	4	1	no	—
307	C₂²	—	5	4	no	—
308	C₂²	—	20	2	no	—
309	C₂²	D₅×C₄⋊C₈	1	1	yes	characteristic
310	C₂²	D₅×C₄⋊C₈	1	1	yes	characteristic
311	C₂²	C₂×D₅×M₁₆	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	5	4	no	—
315	C₄	—	20	2	no	—
316	C₄	D₅×C₄⋊C₈	1	1	yes	characteristic
317	C₄	D₅×C₄⋊C₈	1	1	yes	characteristic
318	C₄	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
319	C₄	C₂²×C₈×D₅	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	10	4	no	—
324	C₂	C₂×D₅×C₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₅
commutator	C₄×C₅	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#364	C₂
socle	C₂×C₅	C₂²×C₄⋊C₈
prime radical	C₂×#709	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×#364	D₅
hypocenter	C₅	#33401
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#364	C₂	semidirect
4. #10602216	C₂	semidirect
5. C₂×#64683	C₂	semidirect
6. #10600096	C₂	semidirect
7. D₅×#364	C₂	semidirect
76. C₂×#364	D₅	semidirect
305. C₅	#33401	semidirect

Maximal subgroups

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

subgroup	index
#10613738	2
C₄×#4047	2
C₂×C₅×#364	2
#10602216	2
C₂×#64683	2
#10600096	2
D₅×#364	2
#33401	5