Subgroups — #12069825

Order 1280.

682 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	#10610199	C₂	1	1	yes	maximal, characteristic
2	#10610189	C₂	1	1	yes	maximal, characteristic
3	#10610176	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
5	#10600531	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600528	C₂	1	1	yes	maximal, characteristic
7	#10600523	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598848	C₂	1	4	yes	maximal, complemented
9	C₈×#677	C₂	1	4	yes	maximal, complemented
10	C₂×#4047	—	2	1	no	—
11	#65811	C₂²	1	1	yes	characteristic
12	#65804	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	#65403	C₄	1	4	yes	—
17	#65399	—	2	1	no	—
18	#65391	—	2	1	no	—
19	#65390	—	2	1	no	—
20	#65213	C₂²	1	1	yes	characteristic
21	#65213	C₂²	1	1	yes	characteristic
22	#64983	C₄	1	1	yes	characteristic
23	#64982	C₄	1	1	yes	characteristic
24	C₄×#677	C₂²	1	4	yes	—
25	#64977	—	2	1	no	—
26	#64977	C₂²	1	2	yes	—
27	C₈×D₂₀	—	2	1	no	—
28	C₈×D₂₀	C₂²	1	2	yes	—
29	#64970	C₄	1	1	yes	characteristic
30	#64969	C₄	1	1	yes	characteristic
31	#64968	—	2	1	no	—
32	#64968	C₂²	1	2	yes	—
33	C₈×Dic₁₀	—	2	1	no	—
34	C₈×Dic₁₀	C₂²	1	2	yes	—
35	C₅×C₈²	C₂²	1	2	yes	—
36	C₅×C₈²	C₂²	1	2	yes	—
37	#64679	C₂²	1	4	yes	—
38	#64677	C₂²	1	4	yes	—
39	#64676	C₂²	1	4	yes	—
40	#17957	—	5	1	no	maximal, Sylow
41	C₂×#709	—	2	1	no	—
42	C₂²×C₅×C₈	—	2	1	no	—
43	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
44	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	#4076	—	2	1	no	—
48	#4076	C₂×C₄	1	2	yes	—
49	#4072	—	2	1	no	—
50	#4070	—	2	1	no	—
51	#4070	C₂×C₄	1	2	yes	—
52	C₂×#681	—	2	1	no	—
53	C₂×#681	—	2	1	no	—
54	C₂×#677	C₂×C₄	1	4	yes	—
55	#4047	—	4	4	no	—
56	C₂×#676	—	2	2	no	—
57	C₂×C₈×D₅	—	2	2	no	—
58	#4021	—	2	1	no	—
59	#4020	C₂×C₄	1	1	yes	characteristic
60	C₄×D₂₀	—	2	1	no	—
61	C₄×D₂₀	C₂³	1	2	yes	—
62	#4016	—	2	1	no	—
63	#4015	C₂×C₄	1	1	yes	characteristic
64	#4015	C₂×C₄	1	1	yes	characteristic
65	#4015	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₀	—	2	1	no	—
67	C₄×Dic₁₀	C₂³	1	2	yes	—
68	C₄×C₅×C₈	—	2	1	no	—
69	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₅×C₈	D₄	1	1	yes	characteristic
72	C₄×C₅×C₈	D₄	1	1	yes	characteristic
73	C₄×C₅×C₈	C₂³	1	2	yes	—
74	C₄×C₅×C₈	C₂³	1	2	yes	—
75	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
76	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
77	#3953	C₂×C₄	1	4	yes	—
78	#3952	—	2	2	no	—
79	#3952	—	2	2	no	—
80	#3949	C₂×C₄	1	4	yes	—
81	#3948	C₂×C₄	1	4	yes	—
82	#3947	—	2	2	no	—
83	#3946	—	2	2	no	—
84	#3946	—	2	2	no	—
85	C₈×Dic₅	—	2	2	no	—
86	C₄⋊#168	—	2	1	no	—
87	C₄⋊#168	—	2	1	no	—
88	C₄⋊#168	C₂³	1	2	yes	—
89	C₄⋊#168	C₂³	1	2	yes	—
90	#3167	—	5	1	no	—
91	#2960	—	5	1	no	—
92	#2931	—	5	1	no	—
93	#1590	—	5	4	no	—
94	C₈×SD₁₆	—	5	4	no	—
95	#1566	—	5	1	no	—
96	#1493	—	5	1	no	—
97	#1492	—	5	1	no	—
98	C₂×C₈²	D₅	1	1	yes	characteristic, complemented
99	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅	#41	1	1	yes	characteristic
102	C₂×C₂²⋊D₅	—	2	2	no	—
103	#709	—	4	2	no	—
104	#709	—	4	2	no	—
105	C₂×D₂₀	—	2	1	no	—
106	C₂×D₂₀	—	2	1	no	—
107	C₂×D₂₀	C₂²×C₄	1	2	yes	—
108	C₂×C₄×D₅	—	2	2	no	—
109	C₂×Dic₁₀	—	2	1	no	—
110	C₂×Dic₁₀	—	2	1	no	—
111	C₂×Dic₁₀	C₂²×C₄	1	2	yes	—
112	C₂×C₅×C₈	—	2	4	no	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	—	2	4	no	—
116	C₂×C₅×C₈	SD₁₆	1	4	yes	complemented
117	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
118	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
119	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
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	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₅	#41	1	1	yes	characteristic
126	C₄²×C₅	C₂⁴	1	1	yes	characteristic
127	C₂²⋊Dic₅	—	2	2	no	—
128	#685	—	2	2	no	—
129	#685	—	2	2	no	—
130	C₄⋊Dic₅	—	2	1	no	—
131	C₄⋊Dic₅	—	2	1	no	—
132	C₄⋊Dic₅	—	2	1	no	—
133	C₄⋊Dic₅	—	2	1	no	—
134	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
135	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
136	#683	—	2	2	no	—
137	#683	—	2	2	no	—
138	C₄×Dic₅	—	2	2	no	—
139	#681	—	4	2	no	—
140	#681	—	4	2	no	—
141	C₂×#168	—	2	2	no	—
142	C₂×#168	—	2	2	no	—
143	#677	—	2	8	no	—
144	#676	—	4	4	no	—
145	C₈×D₅	—	4	4	no	—
146	C₂×#131	—	10	1	no	—
147	#533	—	5	1	no	—
148	#532	—	5	1	no	—
149	#498	—	5	1	no	—
150	#492	—	5	1	no	—
151	#487	—	5	1	no	—
152	#486	—	5	1	no	—
153	#465	—	5	4	no	—
154	#446	—	5	2	no	—
155	#446	—	10	1	no	—
156	C₈×Q₈	—	5	2	no	—
157	C₈×Q₈	—	10	1	no	—
158	C₄×SD₁₆	—	5	4	no	—
159	#436	—	5	2	no	—
160	#436	—	10	1	no	—
161	C₈×D₄	—	5	2	no	—
162	C₈×D₄	—	10	1	no	—
163	#424	—	10	1	no	—
164	#423	—	5	1	no	—
165	#423	—	5	1	no	—
166	#408	—	10	1	no	—
167	#405	—	10	1	no	—
168	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
171	#334	—	5	4	no	—
172	Q₈⋊C₈	—	5	4	no	—
173	D₄⋊C₈	—	5	4	no	—
174	C₈²	C₂×D₅	1	2	yes	—
175	C₈²	C₂×D₅	1	2	yes	—
176	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
177	C₂²×D₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
184	C₂×C₄×C₅	#135	1	2	yes	—
185	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
188	C₂²⋊D₅	—	4	4	no	—
189	C₂×Dic₅	—	2	2	no	—
190	C₂×Dic₅	—	2	2	no	—
191	C₂×Dic₅	—	2	2	no	—
192	D₂₀	—	2	4	no	—
193	D₂₀	—	4	2	no	—
194	D₂₀	—	4	2	no	—
195	C₄×D₅	—	4	2	no	—
196	C₄×D₅	—	4	2	no	—
197	Dic₁₀	—	2	4	no	—
198	Dic₁₀	—	4	2	no	—
199	Dic₁₀	—	4	2	no	—
200	C₅×C₈	—	2	4	no	—
201	C₅×C₈	—	2	2	no	—
202	C₅×C₈	—	2	1	no	—
203	C₅×C₈	—	2	1	no	—
204	C₅×C₈	#131	1	4	yes	complemented
205	C₅×C₈	C₂×SD₁₆	1	2	yes	—
206	C₅×C₈	C₂×SD₁₆	1	2	yes	—
207	#168	—	4	2	no	—
208	#168	—	4	2	no	—
209	C₂×#41	—	10	1	no	—
210	C₂×SD₁₆	—	5	4	no	—
211	#131	—	20	4	no	—
212	C₂×M₁₆	—	10	1	no	—
213	C₂×M₁₆	—	10	1	no	—
214	C₂×M₁₆	—	10	2	no	—
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	—	10	2	no	—
217	C₂²×C₈	C₄×D₅	1	2	yes	—
218	C₂²×C₈	D₂₀	1	1	yes	characteristic
219	C₂²×C₈	D₂₀	1	1	yes	characteristic
220	C₄⋊Q₈	—	5	1	no	—
221	C₄⋊Q₈	—	5	1	no	—
222	C₄⋊Q₈	—	5	1	no	—
223	#127	—	5	1	no	—
224	#125	—	10	1	no	—
225	#124	—	10	1	no	—
226	C₂²⋊Q₈	—	5	2	no	—
227	C₂²⋊Q₈	—	10	1	no	—
228	#121	—	5	2	no	—
229	#121	—	10	1	no	—
230	C₄×Q₈	—	5	2	no	—
231	C₄×Q₈	—	10	1	no	—
232	C₄×D₄	—	5	2	no	—
233	C₄×D₄	—	10	1	no	—
234	#117	—	10	1	no	—
235	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
236	#106	—	5	4	no	—
237	C₄⋊C₈	—	5	2	no	—
238	C₄⋊C₈	—	5	2	no	—
239	C₄⋊C₈	—	10	2	no	—
240	C₄⋊C₈	—	10	2	no	—
241	C₄⋊C₈	—	10	1	no	—
242	C₄⋊C₈	—	10	1	no	—
243	#103	—	5	4	no	—
244	#102	—	5	4	no	—
245	C₂²⋊C₈	—	10	2	no	—
246	C₂²⋊C₈	—	10	2	no	—
247	#97	—	10	2	no	—
248	C₄×C₈	—	2	1	no	—
249	C₄×C₈	—	10	2	no	—
250	C₄×C₈	C₄×D₅	1	1	yes	characteristic
251	C₄×C₈	C₄×D₅	1	1	yes	characteristic
252	C₄×C₈	D₂₀	1	1	yes	characteristic
253	C₄×C₈	D₂₀	1	1	yes	characteristic
254	C₄×C₈	C₂²×D₅	1	2	yes	—
255	C₄×C₈	C₂²×D₅	1	2	yes	—
256	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
257	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
258	C₂²×C₅	—	2	1	no	—
259	C₂²×C₅	—	2	1	no	—
260	C₂²×C₅	C₄×SD₁₆	1	2	yes	—
261	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
262	C₂×D₅	—	4	2	no	—
263	C₂×D₅	—	4	2	no	—
264	C₂×D₅	—	4	2	no	—
265	C₄×C₅	—	2	1	no	—
266	C₄×C₅	—	2	1	no	—
267	C₄×C₅	C₄×SD₁₆	1	2	yes	—
268	C₄×C₅	#443	1	2	yes	—
269	C₄×C₅	C₂×#131	1	2	yes	—
270	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
271	C₄×C₅	C₂×#135	1	1	yes	characteristic
272	Dic₅	—	4	2	no	—
273	Dic₅	—	4	2	no	—
274	Dic₅	—	4	2	no	—
275	#41	—	20	2	no	—
276	#41	—	20	2	no	—
277	C₂×Q₈	—	5	2	no	—
278	C₂×Q₈	—	10	1	no	—
279	C₂×Q₈	—	10	1	no	—
280	C₂×D₄	—	5	2	no	—
281	C₂×D₄	—	10	2	no	—
282	C₂×D₄	—	10	1	no	—
283	C₂×D₄	—	10	1	no	—
284	C₂²×C₄	—	10	2	no	—
285	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
287	C₂²×C₄	#709	1	1	yes	characteristic
288	SD₁₆	—	10	8	no	—
289	M₁₆	—	20	2	no	—
290	M₁₆	—	20	2	no	—
291	M₁₆	—	20	4	no	—
292	C₂×C₈	—	2	4	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	4	no	—
296	C₂×C₈	—	10	2	no	—
297	C₂×C₈	—	10	2	no	—
298	C₂×C₈	—	20	4	no	—
299	C₂×C₈	#677	1	4	yes	complemented
300	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
301	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
302	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
303	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
304	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
305	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
306	C₄⋊C₄	—	5	2	no	—
307	C₄⋊C₄	—	5	2	no	—
308	C₄⋊C₄	—	10	2	no	—
309	C₄⋊C₄	—	10	1	no	—
310	C₄⋊C₄	—	10	1	no	—
311	C₄⋊C₄	—	10	1	no	—
312	C₄⋊C₄	—	10	2	no	—
313	C₄⋊C₄	—	10	1	no	—
314	C₂²⋊C₄	—	10	2	no	—
315	C₂²⋊C₄	—	10	2	no	—
316	C₂²⋊C₄	—	10	2	no	—
317	C₄²	—	10	2	no	—
318	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
319	C₄²	C₂×D₂₀	1	1	yes	characteristic
320	C₄²	#709	1	1	yes	characteristic
321	C₄²	C₂³×D₅	1	1	yes	characteristic
322	C₂×C₅	—	2	2	no	—
323	C₂×C₅	#2931	1	1	yes	characteristic
324	C₂×C₅	C₂×C₄×SD₁₆	1	1	yes	characteristic
325	C₂×C₅	C₂×#443	1	1	yes	characteristic
326	D₅	—	8	2	no	—
327	C₂³	—	10	2	no	—
328	C₂³	C₄×D₂₀	1	1	yes	characteristic
329	Q₈	—	10	4	no	—
330	Q₈	—	20	2	no	—
331	Q₈	—	20	2	no	—
332	D₄	—	10	4	no	—
333	D₄	—	20	4	no	—
334	D₄	—	20	2	no	—
335	D₄	—	20	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	10	2	no	—
339	C₂×C₄	—	10	2	no	—
340	C₂×C₄	—	10	2	no	—
341	C₂×C₄	—	20	2	no	—
342	C₂×C₄	—	20	2	no	—
343	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
344	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
345	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
346	C₂×C₄	C₂×#677	1	2	yes	—
347	C₂×C₄	#4050	1	2	yes	—
348	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
349	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
350	C₂×C₄	C₂×#709	1	1	yes	characteristic
351	C₈	—	2	4	no	—
352	C₈	—	2	2	no	—
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	20	2	no	—
356	C₈	—	20	2	no	—
357	C₈	#4047	1	4	yes	complemented
358	C₈	C₂×#677	1	2	yes	—
359	C₈	C₂×#677	1	2	yes	—
360	C₅	#17957	1	1	yes	characteristic, complemented, Sylow
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	20	2	no	—
364	C₂²	—	20	2	no	—
365	C₂²	—	20	2	no	—
366	C₂²	C₄×#677	1	2	yes	—
367	C₂²	C₂×C₄×D₂₀	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	20	2	no	—
371	C₄	—	20	2	no	—
372	C₄	—	20	2	no	—
373	C₄	C₄×#677	1	2	yes	—
374	C₄	#64989	1	2	yes	—
375	C₄	C₂×#4047	1	2	yes	—
376	C₄	C₂²×#677	1	1	yes	characteristic
377	C₄	C₂×#4050	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	40	2	no	—
380	C₂	#10610189	1	1	yes	characteristic
381	C₂	C₂×C₄×#677	1	1	yes	characteristic
382	C₂	C₂×#64989	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₅×C₈²	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	#65811	C₂²
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂×C₈²	D₅
hypocenter	C₅	#17957
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #10610176	C₂	semidirect
4. C₂×C₅×C₈²	C₂	semidirect
5. #10600531	C₂	semidirect
7. #10600523	C₂	semidirect
8. #10598848	C₂	semidirect
9. C₈×#677	C₂	semidirect
98. C₂×C₈²	D₅	semidirect
116. C₂×C₅×C₈	SD₁₆	semidirect
204. C₅×C₈	#131	semidirect
299. C₂×C₈	#677	semidirect
357. C₈	#4047	semidirect
360. C₅	#17957	semidirect

Maximal subgroups

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

subgroup	index
#10610199	2
#10610189	2
#10610176	2
C₂×C₅×C₈²	2
#10600531	2
#10600528	2
#10600523	2
#10598848	2
C₈×#677	2
#17957	5