Subgroups — #12014347

Order 1280.

490 conjugacy classes of subgroups.

The subgroup lattice has 338 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	C₂×#64735	C₂	1	1	yes	maximal, characteristic
3	Dic₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#64728	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1379	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#4053	C₄	1	1	yes	characteristic
9	C₂×#4047	C₂²	1	1	yes	characteristic
10	#65856	C₄	1	1	yes	characteristic
11	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₅×#405	C₂²	1	1	yes	characteristic
14	M₁₆×Dic₅	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×#3944	C₂²	1	1	yes	characteristic
17	C₂²×#672	C₂²	1	1	yes	characteristic
18	#65286	C₄	1	1	yes	characteristic
19	#64997	—	2	1	no	—
20	#64996	—	2	1	no	—
21	#64978	—	2	1	no	—
22	C₄×C₈×D₅	—	2	1	no	—
23	#64735	C₄	1	4	yes	complemented
24	#64728	C₄	1	4	yes	complemented
25	#64722	—	2	1	no	—
26	C₁₆×Dic₅	—	2	1	no	—
27	C₁₆×Dic₅	—	2	1	no	—
28	C₁₆×Dic₅	—	2	1	no	—
29	C₄×#350	—	5	1	no	maximal, Sylow
30	C₂×#709	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₅×#117	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₂×#681	C₂×C₄	1	1	yes	characteristic
36	C₂²×#168	C₂×C₄	1	1	yes	characteristic
37	#4053	—	2	2	no	—
38	#4053	C₈	1	4	yes	—
39	#4047	—	2	1	no	—
40	#4047	—	2	1	no	—
41	#4047	C₂×C₄	1	2	yes	—
42	#4047	C₂×C₄	1	2	yes	—
43	C₂×#676	—	2	1	no	—
44	C₂×#676	—	2	1	no	—
45	C₂×C₈×D₅	—	2	1	no	—
46	C₂×C₈×D₅	—	2	1	no	—
47	#4039	C₈	1	2	yes	—
48	#4033	C₈	1	2	yes	—
49	#4027	—	2	2	no	—
50	#4018	—	2	1	no	—
51	C₄²×D₅	—	2	1	no	—
52	C₅×M₃₂	—	2	1	no	—
53	C₅×M₃₂	C₂×C₄	1	2	yes	complemented
54	C₂×C₅×C₁₆	—	2	1	no	—
55	C₂×C₅×C₁₆	—	2	1	no	—
56	C₂×C₅×C₁₆	—	2	1	no	—
57	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
58	C₅×#97	—	2	1	no	—
59	C₄×C₅×C₈	—	2	1	no	—
60	#3947	—	2	1	no	—
61	C₈×Dic₅	—	2	1	no	—
62	C₈×Dic₅	D₄	1	1	yes	characteristic
63	C₈×Dic₅	D₄	1	1	yes	characteristic
64	C₈×Dic₅	D₄	1	1	yes	characteristic
65	C₈×Dic₅	D₄	1	1	yes	characteristic
66	#3944	—	2	1	no	—
67	#3944	C₂×C₄	1	2	yes	complemented
68	C₂×#672	—	2	1	no	—
69	C₂×#672	—	2	1	no	—
70	C₂×#672	—	2	1	no	—
71	C₂×#672	C₂×C₄	1	2	yes	complemented
72	#3935	—	2	1	no	—
73	C₄×#168	—	2	1	no	—
74	C₄×#131	—	5	1	no	—
75	C₂×#350	—	5	1	no	—
76	C₂×#350	—	5	1	no	—
77	C₄×M₃₂	—	5	1	no	—
78	C₂×C₄×C₁₆	—	5	1	no	—
79	#1379	—	5	1	no	—
80	#1379	D₅	1	1	yes	characteristic, complemented
81	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₂²⋊D₅	—	2	1	no	—
83	C₂²×Dic₅	M₁₆	1	2	yes	—
84	#709	—	2	1	no	—
85	#709	—	2	1	no	—
86	#709	C₄²	1	2	yes	—
87	#709	C₂×C₈	1	2	yes	—
88	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
89	C₂×C₄×D₅	—	2	1	no	—
90	C₂×C₄×D₅	—	2	2	no	—
91	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
92	C₅×M₁₆	—	2	1	no	—
93	C₅×M₁₆	C₂×C₈	1	2	yes	—
94	C₂×C₅×C₈	—	2	1	no	—
95	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₅×C₈	#41	1	1	yes	characteristic
100	C₂×C₅×C₈	#41	1	1	yes	characteristic
101	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
102	C₅×C₂²⋊C₄	—	2	1	no	—
103	C₄²×C₅	—	2	1	no	—
104	#685	—	2	2	no	—
105	#683	—	2	2	no	—
106	C₄×Dic₅	—	2	2	no	—
107	C₄×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
108	C₄×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
109	C₄×Dic₅	C₂×C₈	1	2	yes	—
110	#681	—	2	1	no	—
111	#681	C₄²	1	2	yes	—
112	C₂×#168	—	2	1	no	—
113	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
114	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
115	C₂×#168	C₂×C₈	1	2	yes	—
116	C₂×#168	M₁₆	1	2	yes	—
117	#676	—	2	2	no	—
118	#676	—	2	2	no	—
119	#676	—	2	4	no	—
120	C₈×D₅	—	2	2	no	—
121	C₈×D₅	—	2	2	no	—
122	C₈×D₅	—	2	4	no	—
123	C₅×C₁₆	—	2	2	no	—
124	C₅×C₁₆	—	2	2	no	—
125	#672	—	2	2	no	—
126	#672	—	2	2	no	—
127	C₂×#131	—	5	1	no	—
128	C₂×#131	—	5	1	no	—
129	C₄×#41	—	5	1	no	—
130	C₂×M₃₂	—	5	1	no	—
131	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
132	C₂²×C₁₆	—	5	1	no	—
133	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
134	#405	—	5	1	no	—
135	#405	—	10	1	no	—
136	#405	—	10	1	no	—
137	#405	C₂×D₅	1	1	yes	characteristic
138	C₄×M₁₆	—	5	1	no	—
139	C₄×M₁₆	—	10	1	no	—
140	C₂×C₄×C₈	—	5	1	no	—
141	C₂×C₄×C₈	—	10	1	no	—
142	#350	—	5	4	no	—
143	#350	—	5	4	no	—
144	#346	—	10	1	no	—
145	C₄×C₁₆	—	10	1	no	—
146	C₄×C₁₆	—	10	1	no	—
147	C₄×C₁₆	—	10	1	no	—
148	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
149	C₂²×D₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅	#117	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
157	C₂²⋊D₅	—	2	2	no	—
158	C₂²⋊D₅	—	2	2	no	—
159	C₂×Dic₅	—	2	1	no	—
160	C₂×Dic₅	—	2	4	no	—
161	C₂×Dic₅	C₂²⋊C₈	1	2	yes	—
162	C₂×Dic₅	C₂²⋊C₈	1	2	yes	—
163	D₂₀	—	2	1	no	—
164	D₂₀	C₄×C₈	1	2	yes	—
165	C₄×D₅	—	2	2	no	—
166	C₄×D₅	—	2	2	no	—
167	C₄×D₅	—	4	4	no	—
168	Dic₁₀	—	2	1	no	—
169	Dic₁₀	C₄×C₈	1	2	yes	—
170	C₅×C₈	—	2	2	no	—
171	C₅×C₈	C₄×D₄	1	1	yes	characteristic
172	C₅×C₈	C₄×D₄	1	1	yes	characteristic
173	C₅×C₈	C₄×D₄	1	1	yes	characteristic
174	C₅×C₈	C₄×D₄	1	1	yes	characteristic
175	#168	—	2	2	no	—
176	#168	C₂²⋊C₈	1	2	yes	—
177	#168	C₂²⋊C₈	1	2	yes	—
178	C₂×#41	—	5	1	no	—
179	#131	—	5	4	no	—
180	#131	—	5	2	no	—
181	#131	—	5	2	no	—
182	#131	—	10	2	no	—
183	#131	—	10	1	no	—
184	#131	—	10	1	no	—
185	C₂×M₁₆	—	5	1	no	—
186	C₂×M₁₆	—	10	1	no	—
187	C₂×M₁₆	—	10	1	no	—
188	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
189	C₂²×C₈	—	5	1	no	—
190	C₂²×C₈	—	10	1	no	—
191	C₂²×C₈	—	10	1	no	—
192	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
193	C₄×Q₈	—	5	2	no	—
194	C₄×D₄	—	5	2	no	—
195	C₄×D₄	—	10	2	no	—
196	#117	—	10	1	no	—
197	#117	C₄×D₅	1	1	yes	characteristic
198	C₂×C₄²	—	5	1	no	—
199	C₂×C₄²	—	10	1	no	—
200	M₃₂	—	2	1	no	—
201	M₃₂	—	5	2	no	—
202	M₃₂	—	10	1	no	—
203	M₃₂	C₄×D₅	1	2	yes	complemented
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	5	2	no	—
208	C₂×C₁₆	—	10	1	no	—
209	C₂×C₁₆	—	10	1	no	—
210	C₂×C₁₆	—	10	1	no	—
211	C₂×C₁₆	C₄×D₅	1	2	yes	complemented
212	#97	—	2	1	no	—
213	#97	—	10	1	no	—
214	#97	—	10	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	—	5	1	no	—
217	C₄×C₈	—	5	1	no	—
218	C₄×C₈	—	5	1	no	—
219	C₄×C₈	—	5	1	no	—
220	C₄×C₈	—	10	1	no	—
221	C₄×C₈	—	10	1	no	—
222	C₂²×C₅	—	2	2	no	—
223	C₂²×C₅	C₄×M₁₆	1	1	yes	characteristic
224	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
225	C₂²×C₅	#431	1	1	yes	characteristic
226	C₂×D₅	—	2	2	no	—
227	C₂×D₅	—	4	2	no	—
228	C₄×C₅	—	4	2	no	—
229	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
230	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
231	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
232	C₄×C₅	#431	1	1	yes	characteristic
233	Dic₅	—	2	2	no	—
234	Dic₅	—	2	2	no	—
235	Dic₅	#350	1	4	yes	complemented
236	#41	—	5	2	no	—
237	#41	—	5	2	no	—
238	#41	—	10	1	no	—
239	#41	—	10	1	no	—
240	C₂×Q₈	—	5	1	no	—
241	C₂×D₄	—	5	1	no	—
242	C₂×D₄	—	10	1	no	—
243	C₂²×C₄	—	5	2	no	—
244	C₂²×C₄	—	10	2	no	—
245	C₂²×C₄	—	10	1	no	—
246	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
247	M₁₆	—	2	1	no	—
248	M₁₆	—	5	2	no	—
249	M₁₆	—	10	4	no	—
250	M₁₆	—	10	1	no	—
251	M₁₆	—	10	2	no	—
252	M₁₆	—	10	2	no	—
253	M₁₆	C₈×D₅	1	2	yes	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	5	1	no	—
256	C₂×C₈	—	5	1	no	—
257	C₂×C₈	—	5	2	no	—
258	C₂×C₈	—	5	2	no	—
259	C₂×C₈	—	10	4	no	—
260	C₂×C₈	—	10	1	no	—
261	C₂×C₈	—	10	2	no	—
262	C₂×C₈	—	10	2	no	—
263	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
264	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
265	C₂×C₈	D₄×D₅	1	1	yes	characteristic
266	C₂×C₈	D₄×D₅	1	1	yes	characteristic
267	C₂×C₈	#711	1	1	yes	characteristic
268	C₂×C₈	#713	1	1	yes	characteristic
269	C₄⋊C₄	—	10	2	no	—
270	C₄⋊C₄	C₈×D₅	1	2	yes	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	10	2	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	5	1	no	—
275	C₄²	—	5	1	no	—
276	C₄²	—	5	2	no	—
277	C₄²	—	10	2	no	—
278	C₁₆	—	2	2	no	—
279	C₁₆	—	2	2	no	—
280	C₁₆	—	10	2	no	—
281	C₁₆	—	10	2	no	—
282	C₂×C₅	—	2	1	no	—
283	C₂×C₅	—	2	1	no	—
284	C₂×C₅	C₄×C₂²⋊C₈	1	1	yes	characteristic
285	C₂×C₅	C₂×#350	1	2	yes	—
286	D₅	—	4	2	no	—
287	C₂³	—	10	1	no	—
288	C₂³	D₅×M₁₆	1	1	yes	characteristic
289	Q₈	—	5	2	no	—
290	Q₈	—	10	1	no	—
291	D₄	—	5	2	no	—
292	D₄	—	10	2	no	—
293	D₄	—	10	2	no	—
294	D₄	—	10	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	5	2	no	—
297	C₂×C₄	—	5	2	no	—
298	C₂×C₄	—	10	4	no	—
299	C₂×C₄	—	10	1	no	—
300	C₂×C₄	—	10	2	no	—
301	C₂×C₄	—	10	2	no	—
302	C₂×C₄	—	20	4	no	—
303	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
304	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
305	C₂×C₄	#4038	1	1	yes	characteristic
306	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
308	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	5	2	no	—
311	C₈	—	5	2	no	—
312	C₈	—	10	2	no	—
313	C₈	#4027	1	1	yes	characteristic
314	C₈	#4027	1	1	yes	characteristic
315	C₈	#4039	1	1	yes	characteristic
316	C₈	#4039	1	1	yes	characteristic
317	C₅	C₄×#350	1	1	yes	characteristic, complemented, Sylow
318	C₂²	—	2	2	no	—
319	C₂²	—	10	2	no	—
320	C₂²	—	20	2	no	—
321	C₂²	#64996	1	1	yes	characteristic
322	C₂²	D₅×C₂²⋊C₈	1	1	yes	characteristic
323	C₂²	#65122	1	1	yes	characteristic
324	C₄	—	4	2	no	—
325	C₄	—	5	4	no	—
326	C₄	—	10	2	no	—
327	C₄	—	10	2	no	—
328	C₄	#64955	1	1	yes	characteristic
329	C₄	C₄×C₈×D₅	1	1	yes	characteristic
330	C₄	D₅×C₂²⋊C₈	1	1	yes	characteristic
331	C₄	#65122	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	20	2	no	—
335	C₂	#10599392	1	1	yes	characteristic
336	C₂	D₅×#350	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1379	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂×#709	C₂×C₄
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1379	D₅
hypocenter	C₅	C₄×#350
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
3. Dic₅×M₃₂	C₂	semidirect
5. C₂×C₁₆×Dic₅	C₂	semidirect
6. C₅×#1379	C₂	semidirect
7. #10598162	C₂	semidirect
23. #64735	C₄	semidirect
24. #64728	C₄	semidirect
53. C₅×M₃₂	C₂×C₄	semidirect
57. C₂×C₅×C₁₆	C₂×C₄	semidirect
67. #3944	C₂×C₄	semidirect
71. C₂×#672	C₂×C₄	semidirect
80. #1379	D₅	semidirect
203. M₃₂	C₄×D₅	semidirect
211. C₂×C₁₆	C₄×D₅	semidirect
235. Dic₅	#350	semidirect
317. C₅	C₄×#350	semidirect

Maximal subgroups

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

subgroup	index
#10610280	2
C₂×#64735	2
Dic₅×M₃₂	2
C₂×#64728	2
C₂×C₁₆×Dic₅	2
C₅×#1379	2
#10598162	2
C₄×#350	5