Subgroups — #12014564

Order 1280.

428 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65184	C₂	1	1	yes	maximal, characteristic
2	#10602235	C₂	1	1	yes	maximal, characteristic
3	Dic₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601452	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601449	C₂	1	1	yes	maximal, characteristic
6	C₅×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598224	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#4053	—	2	1	no	—
9	C₂×#4050	C₂²	1	1	yes	characteristic
10	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₅×#108	C₂²	1	1	yes	characteristic
12	C₅×#427	C₂²	1	1	yes	characteristic
13	C₂×#3957	—	2	1	no	—
14	C₂×#3951	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×#3944	C₂²	1	1	yes	characteristic
17	#65286	—	2	1	no	—
18	#65184	—	2	1	no	—
19	#65184	—	2	1	no	—
20	#65184	C₄	1	2	yes	complemented
21	#65184	C₄	1	2	yes	complemented
22	#64752	—	2	1	no	—
23	#64737	C₄	1	2	yes	complemented
24	#64736	C₄	1	2	yes	complemented
25	#64722	—	2	1	no	—
26	C₁₆×Dic₅	—	2	1	no	—
27	#64708	C₄	1	4	yes	complemented
28	#12646	—	5	1	no	maximal, Sylow
29	C₂×#709	—	2	1	no	—
30	C₂×C₅×M₁₆	—	2	1	no	—
31	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₅×#117	—	2	1	no	—
33	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
34	C₂×#681	—	2	1	no	—
35	C₂²×#168	D₄	1	1	yes	characteristic
36	#4053	—	2	2	no	—
37	#4053	—	2	2	no	—
38	#4053	—	4	1	no	—
39	#4053	—	4	1	no	—
40	C₂×Dic₂₀	C₂×C₄	1	1	yes	characteristic
41	#4050	—	2	1	no	—
42	#4050	—	2	1	no	—
43	#4050	C₂×C₄	1	2	yes	—
44	#4050	C₂×C₄	1	2	yes	—
45	C₂×D₄₀	C₂×C₄	1	1	yes	characteristic
46	C₂×#677	—	2	1	no	—
47	C₂×#676	—	4	1	no	—
48	C₂×C₈×D₅	—	4	1	no	—
49	C₅×M₃₂	—	2	1	no	—
50	C₅×M₃₂	C₂×C₄	1	2	yes	—
51	C₂×C₅×C₁₆	—	2	1	no	—
52	C₅×#108	—	2	1	no	—
53	C₅×#108	C₂×C₄	1	2	yes	—
54	C₅×#107	—	2	1	no	—
55	C₅×#106	C₂×C₄	1	2	yes	—
56	#3957	—	2	2	no	—
57	#3957	—	2	2	no	—
58	#3951	—	2	1	no	—
59	#3951	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	D₄	1	1	yes	characteristic
63	C₈×Dic₅	D₄	1	1	yes	characteristic
64	#3944	—	2	1	no	—
65	#3944	C₂×C₄	1	2	yes	—
66	C₂×#672	—	2	1	no	—
67	#3935	—	4	1	no	—
68	C₄×#168	—	4	1	no	—
69	C₂×#443	—	5	1	no	—
70	#2148	—	5	1	no	—
71	#2145	—	5	1	no	—
72	C₄×M₃₂	—	5	1	no	—
73	#1772	—	5	1	no	—
74	#1389	D₅	1	1	yes	characteristic, complemented
75	#1388	—	5	1	no	—
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₅	—	4	1	no	—
78	C₂²×Dic₅	—	2	1	no	—
79	#709	—	2	2	no	—
80	#709	—	2	2	no	—
81	#709	—	4	1	no	—
82	#709	—	4	1	no	—
83	C₂×D₂₀	—	2	1	no	—
84	C₂×C₄×D₅	—	4	1	no	—
85	C₂×Dic₁₀	—	2	1	no	—
86	C₅×M₁₆	—	2	2	no	—
87	C₅×M₁₆	—	4	1	no	—
88	C₂×C₅×C₈	—	4	1	no	—
89	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₅×C₈	#41	1	1	yes	characteristic
94	C₂×C₅×C₈	#41	1	1	yes	characteristic
95	C₅×C₄⋊C₄	—	2	2	no	—
96	C₅×C₂²⋊C₄	—	4	1	no	—
97	C₄²×C₅	—	4	1	no	—
98	C₄×Dic₅	—	2	1	no	—
99	C₄×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
100	C₄×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
101	#681	—	2	2	no	—
102	#681	—	4	1	no	—
103	C₂×#168	—	4	1	no	—
104	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×#168	D₈	1	1	yes	characteristic
107	C₂×#168	D₈	1	1	yes	characteristic
108	C₂×#168	SD₁₆	1	1	yes	characteristic
109	C₂×#168	SD₁₆	1	1	yes	characteristic
110	Dic₂₀	—	2	1	no	—
111	Dic₂₀	C₄²	1	2	yes	—
112	D₄₀	—	2	1	no	—
113	D₄₀	C₄²	1	2	yes	—
114	#677	—	2	2	no	—
115	#677	—	2	2	no	—
116	#676	—	4	2	no	—
117	#676	—	4	2	no	—
118	C₈×D₅	—	4	2	no	—
119	C₈×D₅	—	4	2	no	—
120	C₅×C₁₆	—	2	2	no	—
121	#672	—	2	2	no	—
122	C₂×#135	—	5	1	no	—
123	C₂×#131	—	10	1	no	—
124	C₂×M₃₂	—	5	1	no	—
125	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
126	#443	—	5	2	no	—
127	#443	—	5	2	no	—
128	#443	—	10	1	no	—
129	#443	—	10	1	no	—
130	C₂×#108	—	5	1	no	—
131	C₂×#108	C₂×D₅	1	1	yes	characteristic
132	#427	C₂×D₅	1	1	yes	characteristic
133	C₂×#104	—	10	1	no	—
134	#405	—	10	1	no	—
135	C₂×C₄×C₈	—	5	1	no	—
136	#362	—	5	2	no	—
137	#361	—	5	2	no	—
138	#360	—	5	4	no	—
139	#346	—	10	1	no	—
140	C₄×C₁₆	—	10	1	no	—
141	#339	—	10	1	no	—
142	C₂³×C₅	#135	1	1	yes	characteristic
143	C₂²×D₅	—	4	1	no	—
144	C₂×C₄×C₅	—	4	2	no	—
145	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
150	C₂×C₄×C₅	#135	1	1	yes	characteristic
151	C₂²⋊D₅	—	4	2	no	—
152	C₂²⋊D₅	—	4	2	no	—
153	C₂×Dic₅	—	2	1	no	—
154	C₂×Dic₅	—	2	1	no	—
155	C₂×Dic₅	—	4	2	no	—
156	C₂×Dic₅	—	4	1	no	—
157	D₂₀	—	2	2	no	—
158	D₂₀	—	4	1	no	—
159	C₄×D₅	—	4	2	no	—
160	C₄×D₅	—	4	2	no	—
161	Dic₁₀	—	2	2	no	—
162	Dic₁₀	—	4	1	no	—
163	C₅×C₈	—	4	2	no	—
164	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
165	C₅×C₈	#117	1	1	yes	characteristic
166	C₅×C₈	C₄×D₄	1	1	yes	characteristic
167	C₅×C₈	C₄×D₄	1	1	yes	characteristic
168	#168	—	4	2	no	—
169	#168	#102	1	1	yes	characteristic
170	#168	#102	1	1	yes	characteristic
171	#168	#102	1	1	yes	characteristic
172	#168	#102	1	1	yes	characteristic
173	C₂×#41	—	10	1	no	—
174	#135	—	5	2	no	—
175	#135	—	5	2	no	—
176	#135	—	10	1	no	—
177	#135	—	10	1	no	—
178	C₂×Q₁₆	—	5	1	no	—
179	C₂×SD₁₆	—	10	1	no	—
180	C₂×D₈	—	5	1	no	—
181	#131	—	10	2	no	—
182	#131	—	10	2	no	—
183	#131	—	20	1	no	—
184	#131	—	20	1	no	—
185	C₂×M₁₆	—	2	1	no	—
186	C₂×M₁₆	—	10	1	no	—
187	C₂×M₁₆	—	20	1	no	—
188	C₂²×C₈	—	5	1	no	—
189	C₂²×C₈	—	20	1	no	—
190	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
191	#117	—	2	1	no	—
192	C₂×C₄²	—	5	1	no	—
193	M₃₂	—	2	1	no	—
194	M₃₂	—	5	2	no	—
195	M₃₂	—	10	1	no	—
196	M₃₂	C₄×D₅	1	2	yes	—
197	C₂×C₁₆	—	2	1	no	—
198	C₂×C₁₆	—	10	1	no	—
199	#108	—	2	1	no	—
200	#108	—	5	2	no	—
201	#108	—	10	1	no	—
202	#108	C₄×D₅	1	2	yes	—
203	#107	—	2	1	no	—
204	#106	C₄×D₅	1	2	yes	—
205	#104	—	10	2	no	—
206	#104	—	10	2	no	—
207	#97	—	20	1	no	—
208	C₄×C₈	—	5	1	no	—
209	C₄×C₈	—	5	1	no	—
210	C₄×C₈	—	5	1	no	—
211	C₄×C₈	—	5	1	no	—
212	C₄×C₈	—	20	1	no	—
213	C₂²×C₅	—	4	1	no	—
214	C₂²×C₅	#416	1	1	yes	characteristic
215	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
216	C₂²×C₅	C₄×SD₁₆	1	1	yes	characteristic
217	C₂×D₅	—	4	2	no	—
218	C₂×D₅	—	8	2	no	—
219	C₄×C₅	—	8	2	no	—
220	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
221	C₄×C₅	C₂×#102	1	1	yes	characteristic
222	C₄×C₅	C₄×D₈	1	1	yes	characteristic
223	C₄×C₅	C₄×SD₁₆	1	1	yes	characteristic
224	Dic₅	—	2	2	no	—
225	Dic₅	—	4	1	no	—
226	Dic₅	—	4	2	no	—
227	#41	—	10	2	no	—
228	#41	—	10	2	no	—
229	#41	—	20	1	no	—
230	#41	—	20	1	no	—
231	C₂×Q₈	—	10	1	no	—
232	C₂×D₄	—	10	1	no	—
233	C₂×D₄	—	20	1	no	—
234	C₂²×C₄	—	10	1	no	—
235	C₂²×C₄	—	20	1	no	—
236	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
237	Q₁₆	—	5	2	no	—
238	Q₁₆	—	10	1	no	—
239	SD₁₆	—	10	2	no	—
240	SD₁₆	—	10	2	no	—
241	D₈	—	5	2	no	—
242	D₈	—	10	1	no	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	4	1	no	—
245	M₁₆	—	10	2	no	—
246	M₁₆	—	20	2	no	—
247	M₁₆	—	20	2	no	—
248	M₁₆	—	20	1	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	5	1	no	—
251	C₂×C₈	—	5	1	no	—
252	C₂×C₈	—	5	1	no	—
253	C₂×C₈	—	5	1	no	—
254	C₂×C₈	—	5	1	no	—
255	C₂×C₈	—	5	1	no	—
256	C₂×C₈	—	20	2	no	—
257	C₂×C₈	—	20	2	no	—
258	C₂×C₈	—	20	1	no	—
259	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
260	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
261	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
262	C₂×C₈	D₄×D₅	1	1	yes	characteristic
263	C₂×C₈	#711	1	1	yes	characteristic
264	C₂×C₈	#713	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₂²⋊C₄	—	4	1	no	—
267	C₄²	—	4	1	no	—
268	C₄²	—	5	1	no	—
269	C₄²	—	5	1	no	—
270	C₄²	—	10	1	no	—
271	C₁₆	—	2	2	no	—
272	C₁₆	—	10	2	no	—
273	C₂×C₅	—	2	1	no	—
274	C₂×C₅	—	2	1	no	—
275	C₂×C₅	—	2	1	no	—
276	C₂×C₅	C₄×#102	1	1	yes	characteristic
277	D₅	—	8	2	no	—
278	C₂³	—	20	1	no	—
279	C₂³	#4062	1	1	yes	characteristic
280	Q₈	—	10	2	no	—
281	Q₈	—	20	1	no	—
282	D₄	—	10	2	no	—
283	D₄	—	20	2	no	—
284	D₄	—	20	2	no	—
285	D₄	—	20	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	10	1	no	—
288	C₂×C₄	—	10	1	no	—
289	C₂×C₄	—	20	2	no	—
290	C₂×C₄	—	20	1	no	—
291	C₂×C₄	—	20	2	no	—
292	C₂×C₄	—	20	2	no	—
293	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
294	C₂×C₄	#4027	1	1	yes	characteristic
295	C₂×C₄	#4039	1	1	yes	characteristic
296	C₂×C₄	D₅×D₈	1	1	yes	characteristic
297	C₂×C₄	D₅×SD₁₆	1	1	yes	characteristic
298	C₂×C₄	#4065	1	1	yes	characteristic
299	C₈	—	4	2	no	—
300	C₈	—	5	1	no	—
301	C₈	—	5	1	no	—
302	C₈	—	5	1	no	—
303	C₈	—	5	1	no	—
304	C₈	—	20	2	no	—
305	C₈	C₄²×D₅	1	1	yes	characteristic
306	C₈	#4027	1	1	yes	characteristic
307	C₈	#4038	1	1	yes	characteristic
308	C₈	#4039	1	1	yes	characteristic
309	C₅	#12646	1	1	yes	characteristic, complemented, Sylow
310	C₂²	—	4	1	no	—
311	C₂²	—	20	2	no	—
312	C₂²	—	40	2	no	—
313	C₂²	#65045	1	1	yes	characteristic
314	C₂²	#65093	1	1	yes	characteristic
315	C₂²	#65141	1	1	yes	characteristic
316	C₄	—	8	2	no	—
317	C₄	—	10	2	no	—
318	C₄	—	20	1	no	—
319	C₄	—	20	2	no	—
320	C₄	#64955	1	1	yes	characteristic
321	C₄	D₅×#102	1	1	yes	characteristic
322	C₄	#65077	1	1	yes	characteristic
323	C₄	#65161	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	40	2	no	—
328	C₂	#10599573	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₅×#102
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1389	C₂
socle	C₂×C₅	C₄×#102
prime radical	C₂×#4050	C₂²
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1389	D₅
hypocenter	C₅	#12646
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1389

Complements

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

N	H ≅ G/N	type
3. Dic₅×M₃₂	C₂	semidirect
4. #10601452	C₂	semidirect
6. C₅×#1389	C₂	semidirect
7. #10598224	C₂	semidirect
20. #65184	C₄	semidirect
21. #65184	C₄	semidirect
23. #64737	C₄	semidirect
24. #64736	C₄	semidirect
27. #64708	C₄	semidirect
74. #1389	D₅	semidirect
309. C₅	#12646	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65184	2
#10602235	2
Dic₅×M₃₂	2
#10601452	2
#10601449	2
C₅×#1389	2
#10598224	2
#12646	5