Subgroups — #12090880

Order 1280.

602 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₄₀	C₂	1	1	yes	maximal, characteristic
2	#10611643	C₂	1	1	yes	maximal, characteristic
3	#10611470	C₂	1	2	yes	maximal, complemented
4	C₅×C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601737	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601735	C₂	1	1	yes	maximal, characteristic, complemented
7	#10599378	C₂	1	2	yes	maximal, complemented
8	#10599067	C₂	1	2	yes	maximal, complemented
9	#10599044	C₂	1	2	yes	maximal, complemented
10	#10598856	C₂	1	1	yes	maximal, characteristic, complemented
11	#10598845	C₂	1	1	yes	maximal, characteristic
12	#65918	C₂²	1	2	yes	—
13	Q₈×D₂₀	C₂²	1	2	yes	—
14	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
15	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
16	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
17	#65475	—	2	2	no	—
18	#65320	C₂²	1	2	yes	—
19	#65309	C₂²	1	2	yes	—
20	#65177	—	2	2	no	—
21	#65135	—	2	2	no	—
22	#65135	C₂²	1	2	yes	—
23	#65134	—	2	2	no	—
24	#65130	C₂²	1	2	yes	—
25	#65105	—	2	2	no	—
26	#65091	—	2	2	no	—
27	#64988	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₀	—	2	1	no	—
29	#64985	—	2	1	no	—
30	C₄×D₄₀	—	2	1	no	—
31	C₄×D₄₀	C₂²	1	1	yes	characteristic
32	#64977	C₂²	1	2	yes	—
33	#64971	—	2	1	no	—
34	#64971	C₂²	1	1	yes	characteristic
35	#64970	C₂²	1	2	yes	—
36	#64969	C₂²	1	2	yes	—
37	C₅×#335	C₂²	1	1	yes	characteristic
38	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic
39	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic
40	C₅×C₈²	C₂²	1	1	yes	characteristic
41	#64706	C₂²	1	2	yes	—
42	#64702	C₂²	1	2	yes	—
43	#64679	C₂²	1	2	yes	—
44	#64678	C₂²	1	1	yes	characteristic
45	#19907	—	5	1	no	maximal, Sylow
46	C₂×Q₈×D₅	—	2	2	no	—
47	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
48	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
49	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
50	#4092	—	2	2	no	—
51	C₂×#689	—	2	2	no	—
52	C₂×Dic₂₀	—	2	1	no	—
53	C₂×Dic₂₀	—	2	1	no	—
54	C₂×D₄₀	D₄	1	1	yes	characteristic
55	C₂×#677	—	2	2	no	—
56	C₂×#676	—	2	2	no	—
57	#4044	—	4	2	no	—
58	#4043	—	2	2	no	—
59	#4043	—	4	2	no	—
60	#4038	—	2	2	no	—
61	#4021	—	2	2	no	—
62	C₄×D₂₀	—	2	2	no	—
63	C₄×D₂₀	C₂³	1	2	yes	—
64	#4015	—	2	2	no	—
65	#4015	C₂³	1	2	yes	—
66	#4015	C₂³	1	2	yes	—
67	C₄×Dic₁₀	—	2	2	no	—
68	C₅×#107	D₄	1	1	yes	characteristic
69	C₅×C₄⋊C₈	—	2	1	no	—
70	C₅×C₄⋊C₈	—	2	1	no	—
71	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
72	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
73	C₅×#103	D₄	1	1	yes	characteristic
74	C₅×#103	D₄	1	1	yes	characteristic
75	C₄×C₅×C₈	—	2	1	no	—
76	C₄×C₅×C₈	—	2	1	no	—
77	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
78	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
79	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
80	#3967	—	2	2	no	—
81	#3953	—	2	2	no	—
82	#3953	D₄	1	2	yes	—
83	#3952	—	2	2	no	—
84	#3950	—	2	1	no	—
85	#3950	—	2	1	no	—
86	#3950	—	2	2	no	—
87	#3950	D₄	1	1	yes	characteristic
88	#3949	—	4	2	no	—
89	#3948	—	2	2	no	—
90	#3948	—	2	2	no	—
91	#3948	—	2	2	no	—
92	#3942	—	2	2	no	—
93	#3940	—	2	2	no	—
94	C₄⋊#168	C₂³	1	2	yes	—
95	#3393	—	5	1	no	—
96	Q₈×D₈	—	5	1	no	—
97	#3328	—	5	2	no	—
98	#1707	—	5	1	no	—
99	#1699	—	5	2	no	—
100	#1674	—	5	1	no	—
101	#1670	—	5	1	no	—
102	#1652	—	5	2	no	—
103	#1651	—	5	2	no	—
104	#1589	—	5	1	no	—
105	C₈×Q₁₆	D₅	1	1	yes	characteristic, complemented
106	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
108	Q₈×D₅	—	4	4	no	—
109	Q₈×D₅	—	8	2	no	—
110	C₂×D₂₀	C₂×D₄	1	2	yes	—
111	C₂×C₄×D₅	—	2	2	no	—
112	C₂×C₄×D₅	—	4	2	no	—
113	C₂×Dic₁₀	—	2	2	no	—
114	C₂×Dic₁₀	—	2	2	no	—
115	C₂×Dic₁₀	—	4	2	no	—
116	C₅×Q₁₆	—	2	2	no	—
117	C₂×C₅×C₈	—	2	1	no	—
118	C₂×C₅×C₈	—	2	1	no	—
119	C₂×C₅×C₈	—	2	1	no	—
120	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
124	C₅×C₄⋊C₄	—	2	1	no	—
125	C₅×C₄⋊C₄	—	2	1	no	—
126	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
127	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₅	—	2	1	no	—
129	C₄²×C₅	—	2	1	no	—
130	C₄²×C₅	C₂⁴	1	1	yes	characteristic
131	#689	—	8	2	no	—
132	#685	—	2	2	no	—
133	#685	—	4	2	no	—
134	#685	—	4	2	no	—
135	#685	—	4	2	no	—
136	C₄⋊Dic₅	—	2	2	no	—
137	C₄⋊Dic₅	—	2	2	no	—
138	C₄⋊Dic₅	—	2	2	no	—
139	C₄⋊Dic₅	—	2	2	no	—
140	C₄⋊Dic₅	—	4	2	no	—
141	C₄⋊Dic₅	C₂×D₄	1	2	yes	—
142	#683	—	4	2	no	—
143	C₄×Dic₅	—	4	2	no	—
144	C₂×#168	—	2	2	no	—
145	Dic₂₀	—	4	2	no	—
146	Dic₂₀	—	8	1	no	—
147	D₄₀	—	2	2	no	—
148	#677	—	8	2	no	—
149	#676	—	8	2	no	—
150	#556	—	5	2	no	—
151	D₄×Q₈	—	5	2	no	—
152	#500	—	5	1	no	—
153	#500	—	10	1	no	—
154	#498	—	5	2	no	—
155	#495	—	10	1	no	—
156	#494	—	5	1	no	—
157	#487	—	5	2	no	—
158	#485	—	10	2	no	—
159	#475	—	5	2	no	—
160	#474	—	5	2	no	—
161	#474	—	10	2	no	—
162	#470	—	10	2	no	—
163	#464	—	10	2	no	—
164	#462	—	5	2	no	—
165	#461	—	5	2	no	—
166	#451	—	10	2	no	—
167	C₈×Q₈	C₂×D₅	1	1	yes	characteristic
168	C₈×Q₈	C₂×D₅	1	1	yes	characteristic
169	C₄×Q₁₆	—	10	1	no	—
170	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
171	C₄×D₈	—	5	1	no	—
172	C₄×D₈	—	10	1	no	—
173	#436	—	5	2	no	—
174	#419	—	10	2	no	—
175	#335	—	5	1	no	—
176	#335	C₂×D₅	1	1	yes	characteristic
177	#332	—	5	2	no	—
178	#332	—	5	2	no	—
179	Q₈⋊C₈	C₂×D₅	1	1	yes	characteristic
180	Q₈⋊C₈	C₂×D₅	1	1	yes	characteristic
181	D₄⋊C₈	—	5	2	no	—
182	C₈²	C₂×D₅	1	1	yes	characteristic
183	C₂²×D₅	—	2	2	no	—
184	C₅×Q₈	—	2	1	no	—
185	C₅×Q₈	—	2	1	no	—
186	C₅×Q₈	—	2	1	no	—
187	C₅×Q₈	#135	1	2	yes	—
188	C₂×C₄×C₅	—	2	1	no	—
189	C₂×C₄×C₅	—	2	1	no	—
190	C₂×C₄×C₅	—	2	1	no	—
191	C₂×C₄×C₅	—	2	1	no	—
192	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
195	C₂×Dic₅	—	2	2	no	—
196	C₂×Dic₅	—	4	2	no	—
197	C₂×Dic₅	—	4	2	no	—
198	D₂₀	—	2	2	no	—
199	D₂₀	—	2	2	no	—
200	C₄×D₅	—	4	2	no	—
201	C₄×D₅	—	4	2	no	—
202	C₄×D₅	—	8	4	no	—
203	Dic₁₀	—	4	2	no	—
204	Dic₁₀	—	4	2	no	—
205	Dic₁₀	—	4	2	no	—
206	Dic₁₀	—	4	2	no	—
207	Dic₁₀	—	8	4	no	—
208	C₅×C₈	—	2	2	no	—
209	C₅×C₈	—	2	1	no	—
210	C₅×C₈	—	2	1	no	—
211	C₅×C₈	—	2	1	no	—
212	C₅×C₈	—	4	1	no	—
213	C₅×C₈	—	4	1	no	—
214	C₅×C₈	#137	1	2	yes	—
215	#168	—	4	2	no	—
216	C₂²×Q₈	—	10	2	no	—
217	C₂×Q₁₆	—	10	2	no	—
218	C₂×Q₁₆	—	10	1	no	—
219	C₂×Q₁₆	—	10	1	no	—
220	C₂×Q₁₆	D₂₀	1	1	yes	characteristic
221	C₂×SD₁₆	—	10	2	no	—
222	C₂×D₈	—	5	1	no	—
223	C₂×M₁₆	—	10	2	no	—
224	C₄⋊Q₈	—	5	2	no	—
225	C₄⋊Q₈	—	5	2	no	—
226	C₄⋊Q₈	—	10	2	no	—
227	#126	—	20	2	no	—
228	#124	—	10	2	no	—
229	C₂²⋊Q₈	—	10	2	no	—
230	C₂²⋊Q₈	—	10	2	no	—
231	C₂²⋊Q₈	—	20	2	no	—
232	C₄×Q₈	—	10	2	no	—
233	C₄×Q₈	C₂²×D₅	1	1	yes	characteristic
234	C₄×Q₈	C₂²×D₅	1	1	yes	characteristic
235	C₄×D₄	—	5	2	no	—
236	C₄×D₄	—	10	2	no	—
237	#117	—	10	2	no	—
238	#107	—	5	1	no	—
239	#107	—	10	2	no	—
240	#107	—	10	1	no	—
241	#107	—	10	1	no	—
242	#107	D₂₀	1	1	yes	characteristic
243	#106	—	10	2	no	—
244	#106	—	20	2	no	—
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	5	2	no	—
248	C₄⋊C₈	C₂²×D₅	1	1	yes	characteristic
249	C₄⋊C₈	C₂²×D₅	1	1	yes	characteristic
250	#103	—	10	2	no	—
251	#103	—	10	2	no	—
252	#103	—	10	2	no	—
253	#103	—	10	2	no	—
254	#103	—	10	2	no	—
255	#103	D₂₀	1	1	yes	characteristic
256	#103	D₂₀	1	1	yes	characteristic
257	#102	—	5	2	no	—
258	#102	—	10	2	no	—
259	C₂²⋊C₈	—	10	2	no	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	—	2	1	no	—
262	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
265	C₂²×C₅	D₄²	1	1	yes	characteristic
266	C₂×D₅	—	4	2	no	—
267	C₂×D₅	—	4	2	no	—
268	C₂×D₅	—	4	2	no	—
269	C₄×C₅	—	2	2	no	—
270	C₄×C₅	—	2	1	no	—
271	C₄×C₅	—	4	1	no	—
272	C₄×C₅	—	4	1	no	—
273	C₄×C₅	—	4	1	no	—
274	C₄×C₅	C₂×#135	1	1	yes	characteristic
275	C₄×C₅	C₂×#137	1	1	yes	characteristic
276	C₄×C₅	#576	1	1	yes	characteristic
277	C₄×C₅	#577	1	1	yes	characteristic
278	Dic₅	—	4	2	no	—
279	Dic₅	—	8	2	no	—
280	Dic₅	—	8	2	no	—
281	C₂×Q₈	—	10	2	no	—
282	C₂×Q₈	—	10	2	no	—
283	C₂×Q₈	—	20	2	no	—
284	C₂×Q₈	—	20	4	no	—
285	C₂×Q₈	—	40	2	no	—
286	C₂×Q₈	C₂×D₂₀	1	1	yes	characteristic
287	C₂×Q₈	C₂×D₂₀	1	1	yes	characteristic
288	C₂×D₄	—	5	2	no	—
289	C₂²×C₄	—	10	2	no	—
290	C₂²×C₄	—	20	2	no	—
291	Q₁₆	—	2	2	no	—
292	Q₁₆	—	20	2	no	—
293	Q₁₆	—	40	1	no	—
294	Q₁₆	—	40	2	no	—
295	SD₁₆	—	40	2	no	—
296	D₈	—	10	2	no	—
297	M₁₆	—	40	2	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	10	2	no	—
302	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
303	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
304	C₂×C₈	D₄×D₅	1	1	yes	characteristic
305	C₂×C₈	D₄×D₅	1	1	yes	characteristic
306	C₄⋊C₄	—	2	1	no	—
307	C₄⋊C₄	—	2	1	no	—
308	C₄⋊C₄	—	5	2	no	—
309	C₄⋊C₄	—	10	2	no	—
310	C₄⋊C₄	—	10	2	no	—
311	C₄⋊C₄	—	10	2	no	—
312	C₄⋊C₄	—	10	2	no	—
313	C₄⋊C₄	—	20	2	no	—
314	C₄⋊C₄	—	20	2	no	—
315	C₄⋊C₄	C₂×D₂₀	1	1	yes	characteristic
316	C₄⋊C₄	C₂×D₂₀	1	1	yes	characteristic
317	C₂²⋊C₄	—	10	2	no	—
318	C₂²⋊C₄	—	20	2	no	—
319	C₂²⋊C₄	—	20	2	no	—
320	C₂²⋊C₄	—	20	2	no	—
321	C₄²	—	2	1	no	—
322	C₄²	—	2	1	no	—
323	C₄²	—	20	2	no	—
324	C₄²	C₂³×D₅	1	1	yes	characteristic
325	C₂×C₅	#3278	1	1	yes	characteristic
326	C₂×C₅	#3280	1	1	yes	characteristic
327	C₂×C₅	#3293	1	1	yes	characteristic
328	D₅	—	8	2	no	—
329	C₂³	—	10	2	no	—
330	Q₈	—	2	1	no	—
331	Q₈	—	2	1	no	—
332	Q₈	—	2	1	no	—
333	Q₈	—	20	2	no	—
334	Q₈	—	20	2	no	—
335	Q₈	—	20	2	no	—
336	Q₈	—	20	2	no	—
337	Q₈	—	40	4	no	—
338	Q₈	#4050	1	2	yes	—
339	D₄	—	10	2	no	—
340	D₄	—	10	2	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	10	2	no	—
346	C₂×C₄	—	20	2	no	—
347	C₂×C₄	—	20	2	no	—
348	C₂×C₄	—	20	2	no	—
349	C₂×C₄	—	20	2	no	—
350	C₂×C₄	—	40	4	no	—
351	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
352	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
353	C₂×C₄	#4149	1	1	yes	characteristic
354	C₈	—	2	2	no	—
355	C₈	—	2	1	no	—
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	4	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	20	2	no	—
361	C₈	#4064	1	2	yes	—
362	C₅	#19907	1	1	yes	characteristic, complemented, Sylow
363	C₂²	—	20	2	no	—
364	C₂²	—	20	2	no	—
365	C₂²	—	20	2	no	—
366	C₂²	D₄×D₂₀	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	2	2	no	—
369	C₄	—	4	1	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	20	2	no	—
373	C₄	—	40	2	no	—
374	C₄	—	40	2	no	—
375	C₄	C₂×#4050	1	1	yes	characteristic
376	C₄	#66086	1	1	yes	characteristic
377	C₄	C₂×#4064	1	1	yes	characteristic
378	C₄	#66104	1	1	yes	characteristic
379	C₂	—	40	2	no	—
380	C₂	#10611164	1	1	yes	characteristic
381	C₂	#10611462	1	1	yes	characteristic
382	C₂	#10611626	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₀
commutator, cosocle	C₄²×C₅	C₂⁴
Frattini	C₄²	C₂³×D₅
Fitting	C₅×C₈×Q₁₆	C₂
socle	C₂²×C₅	D₄²
prime radical	C₂×D₄₀	D₄
hypercenter	C₈×Q₁₆	D₅
hypocenter	C₅	#19907
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #10611470	C₂	semidirect
4. C₅×C₈×Q₁₆	C₂	semidirect
5. #10601737	C₂	semidirect
6. #10601735	C₂	semidirect
7. #10599378	C₂	semidirect
8. #10599067	C₂	semidirect
9. #10599044	C₂	semidirect
10. #10598856	C₂	semidirect
105. C₈×Q₁₆	D₅	semidirect
362. C₅	#19907	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₄₀	2
#10611643	2
#10611470	2
C₅×C₈×Q₁₆	2
#10601737	2
#10601735	2
#10599378	2
#10599067	2
#10599044	2
#10598856	2
#10598845	2
#19907	5