Subgroups — #12069829

Order 1280.

614 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	#10611689	C₂	1	2	yes	maximal
2	#10611456	C₂	1	1	yes	maximal, characteristic
3	C₅×#1591	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601744	C₂	1	2	yes	maximal, complemented
5	Q₁₆×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599362	C₂	1	1	yes	maximal, characteristic
7	#10599350	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599033	C₂	1	2	yes	maximal
9	#10599006	C₂	1	2	yes	maximal, complemented
10	#10598849	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈²×D₅	C₂	1	1	yes	maximal, characteristic
12	#65912	C₂²	1	2	yes	—
13	C₅×#446	C₂²	1	2	yes	—
14	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
15	#65477	C₄	1	1	yes	characteristic
16	#65470	C₄	1	1	yes	characteristic
17	C₄×#689	C₂²	1	2	yes	—
18	Q₈×#168	C₂²	1	2	yes	—
19	#65176	C₄	1	1	yes	characteristic
20	#65163	C₄	1	1	yes	characteristic
21	#65128	—	2	2	no	—
22	#65126	C₂²	1	2	yes	—
23	#65119	—	2	2	no	—
24	#65119	C₂²	1	2	yes	—
25	#65102	C₄	1	2	yes	—
26	#65081	C₄	1	2	yes	—
27	#64997	—	2	2	no	—
28	#64990	—	2	2	no	—
29	C₄×Dic₂₀	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
33	#64968	C₂²	1	2	yes	—
34	C₅×#335	C₂²	1	1	yes	characteristic
35	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₅×C₈²	C₂²	1	1	yes	characteristic
37	Q₈⋊#168	C₂²	1	2	yes	—
38	#64704	C₂²	1	2	yes	—
39	#64699	C₂²	1	2	yes	—
40	#64678	C₂²	1	1	yes	characteristic
41	#64676	C₂²	1	2	yes	—
42	C₈×#168	C₂²	1	1	yes	characteristic
43	C₈×#168	C₂²	1	1	yes	characteristic
44	C₈×#168	C₂²	1	1	yes	characteristic
45	#17958	—	5	1	no	maximal, Sylow
46	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
47	C₄×C₅×Q₈	C₂³	1	2	yes	—
48	#4092	C₂×C₄	1	2	yes	—
49	#4090	C₂×C₄	1	2	yes	—
50	C₂×#689	C₂×C₄	1	2	yes	—
51	C₂×Dic₂₀	C₂×C₄	1	1	yes	characteristic
52	C₂×#676	—	2	2	no	—
53	C₂×C₈×D₅	—	2	1	no	—
54	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
56	C₂×C₈×D₅	D₄	1	1	yes	characteristic
57	C₂×C₈×D₅	D₄	1	1	yes	characteristic
58	#4043	C₂×C₄	1	2	yes	—
59	#4042	—	2	2	no	—
60	#4035	—	2	2	no	—
61	#4034	C₂×C₄	1	2	yes	—
62	#4018	—	2	2	no	—
63	C₄²×D₅	C₂³	1	1	yes	characteristic
64	C₄×Dic₁₀	—	2	2	no	—
65	C₄×Dic₁₀	C₂³	1	2	yes	—
66	C₅×#107	C₂×C₄	1	1	yes	characteristic
67	C₅×C₄⋊C₈	—	2	2	no	—
68	C₅×C₄⋊C₈	C₂³	1	2	yes	—
69	C₅×#103	C₂×C₄	1	2	yes	—
70	C₅×#97	—	2	2	no	—
71	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
74	#3967	C₂×C₄	1	2	yes	—
75	#3950	C₂×C₄	1	1	yes	characteristic
76	#3948	C₂×C₄	1	2	yes	—
77	#3947	—	2	2	no	—
78	#3946	—	2	2	no	—
79	#3946	—	2	2	no	—
80	#3946	—	2	2	no	—
81	C₈×Dic₅	—	2	1	no	—
82	C₈×Dic₅	—	2	2	no	—
83	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
85	C₈×Dic₅	D₄	1	1	yes	characteristic
86	C₈×Dic₅	D₄	1	1	yes	characteristic
87	#3942	C₂×C₄	1	2	yes	—
88	#3939	C₂×C₄	1	2	yes	—
89	C₄⋊#168	—	2	2	no	—
90	C₄⋊#168	C₂³	1	2	yes	—
91	C₄×#168	—	2	2	no	—
92	C₄×#168	C₂³	1	1	yes	characteristic
93	C₄×#168	C₂³	1	1	yes	characteristic
94	C₄×#168	C₂³	1	1	yes	characteristic
95	#3170	—	5	1	no	—
96	#2960	—	5	2	no	—
97	#1591	—	5	2	no	—
98	#1591	—	5	1	no	—
99	#1591	D₅	1	1	yes	characteristic, complemented
100	C₈×Q₁₆	—	5	1	no	—
101	C₈×Q₁₆	—	5	1	no	—
102	C₈×Q₁₆	—	5	2	no	—
103	#1567	—	5	1	no	—
104	#1493	—	5	2	no	—
105	C₂×C₈²	—	5	1	no	—
106	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
107	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×D₅	#41	1	1	yes	characteristic
110	C₂×Dic₁₀	—	2	2	no	—
111	C₂×Dic₁₀	C₂²×C₄	1	2	yes	—
112	C₅×Q₁₆	—	2	2	no	—
113	C₂×C₅×C₈	—	2	1	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
119	C₅×C₄⋊C₄	—	2	2	no	—
120	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
121	C₄²×C₅	—	2	2	no	—
122	C₄²×C₅	C₂⁴	1	1	yes	characteristic
123	#689	—	2	4	no	—
124	#685	—	2	2	no	—
125	#685	—	2	2	no	—
126	C₄⋊Dic₅	—	2	2	no	—
127	C₄⋊Dic₅	C₂²×C₄	1	2	yes	—
128	#683	—	2	2	no	—
129	#683	—	2	2	no	—
130	#683	—	2	2	no	—
131	#683	—	2	2	no	—
132	#683	—	2	2	no	—
133	C₄×Dic₅	—	2	2	no	—
134	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
135	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
136	C₄×Dic₅	#41	1	1	yes	characteristic
137	C₂×#168	—	2	1	no	—
138	C₂×#168	—	2	2	no	—
139	C₂×#168	C₂²×C₄	1	1	yes	characteristic
140	C₂×#168	C₂²×C₄	1	1	yes	characteristic
141	C₂×#168	C₂×D₄	1	1	yes	characteristic
142	C₂×#168	C₂×D₄	1	1	yes	characteristic
143	Dic₂₀	—	2	2	no	—
144	#676	—	4	4	no	—
145	C₈×D₅	—	2	4	no	—
146	C₈×D₅	—	2	2	no	—
147	C₈×D₅	—	2	2	no	—
148	C₈×D₅	—	2	2	no	—
149	C₈×D₅	Q₁₆	1	4	yes	complemented
150	#533	—	5	2	no	—
151	#500	—	5	1	no	—
152	#494	—	5	1	no	—
153	#487	—	5	2	no	—
154	#467	—	5	1	no	—
155	#467	—	5	1	no	—
156	#467	—	5	2	no	—
157	#446	—	5	2	no	—
158	#446	—	10	2	no	—
159	#446	C₂×D₅	1	2	yes	—
160	C₈×Q₈	—	5	2	no	—
161	C₈×Q₈	—	5	2	no	—
162	C₈×Q₈	—	10	2	no	—
163	C₄×Q₁₆	—	5	2	no	—
164	C₄×Q₁₆	—	5	1	no	—
165	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
166	#424	—	10	2	no	—
167	#423	—	5	2	no	—
168	#405	—	10	2	no	—
169	C₂×C₄×C₈	—	5	1	no	—
170	C₂×C₄×C₈	—	5	1	no	—
171	C₂×C₄×C₈	—	5	1	no	—
172	#335	—	5	2	no	—
173	#335	—	5	1	no	—
174	#335	C₂×D₅	1	1	yes	characteristic
175	Q₈⋊C₈	—	5	2	no	—
176	Q₈⋊C₈	—	5	2	no	—
177	Q₈⋊C₈	—	5	2	no	—
178	Q₈⋊C₈	C₂×D₅	1	2	yes	—
179	C₈²	—	5	1	no	—
180	C₈²	—	5	1	no	—
181	C₈²	—	5	1	no	—
182	C₈²	C₂×D₅	1	1	yes	characteristic
183	C₂²×D₅	C₄×D₄	1	1	yes	characteristic
184	C₅×Q₈	—	2	2	no	—
185	C₅×Q₈	—	2	2	no	—
186	C₂×C₄×C₅	—	2	2	no	—
187	C₂×C₄×C₅	—	2	2	no	—
188	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
191	C₂×Dic₅	—	2	2	no	—
192	C₂×Dic₅	—	2	2	no	—
193	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
194	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
195	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
196	C₄×D₅	—	2	1	no	—
197	C₄×D₅	—	2	1	no	—
198	C₄×D₅	—	2	2	no	—
199	C₄×D₅	C₂×Q₁₆	1	2	yes	—
200	C₄×D₅	#135	1	2	yes	—
201	Dic₁₀	—	2	2	no	—
202	Dic₁₀	—	2	2	no	—
203	Dic₁₀	—	4	4	no	—
204	C₅×C₈	—	2	2	no	—
205	C₅×C₈	—	2	1	no	—
206	C₅×C₈	—	2	1	no	—
207	C₅×C₈	—	4	2	no	—
208	C₅×C₈	#131	1	2	yes	—
209	C₅×C₈	C₂×Q₁₆	1	2	yes	—
210	#168	—	2	2	no	—
211	#168	—	2	1	no	—
212	#168	—	2	1	no	—
213	#168	—	4	2	no	—
214	#168	#131	1	2	yes	—
215	#168	C₂×Q₁₆	1	2	yes	—
216	C₂×Q₁₆	—	5	2	no	—
217	C₂×Q₁₆	—	5	1	no	—
218	C₂×Q₁₆	C₄×D₅	1	1	yes	characteristic
219	C₂×M₁₆	—	10	2	no	—
220	C₂²×C₈	—	5	1	no	—
221	C₂²×C₈	—	5	1	no	—
222	C₂²×C₈	—	5	1	no	—
223	C₂²×C₈	—	5	1	no	—
224	C₂²×C₈	—	10	1	no	—
225	C₄⋊Q₈	—	5	2	no	—
226	C₄⋊Q₈	—	5	2	no	—
227	#125	—	10	2	no	—
228	C₂²⋊Q₈	—	5	2	no	—
229	C₂²⋊Q₈	—	5	2	no	—
230	C₂²⋊Q₈	—	10	2	no	—
231	C₄×Q₈	—	5	2	no	—
232	C₄×Q₈	—	10	2	no	—
233	C₄×Q₈	C₂²×D₅	1	2	yes	—
234	#117	—	10	2	no	—
235	C₂×C₄²	—	5	1	no	—
236	#107	—	5	1	no	—
237	#107	—	5	2	no	—
238	#107	C₄×D₅	1	1	yes	characteristic
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	5	2	no	—
241	C₄⋊C₈	—	10	2	no	—
242	C₄⋊C₈	—	10	2	no	—
243	C₄⋊C₈	—	10	2	no	—
244	C₄⋊C₈	—	10	2	no	—
245	C₄⋊C₈	C₂²×D₅	1	2	yes	—
246	#103	—	5	2	no	—
247	#103	—	5	2	no	—
248	#103	—	5	2	no	—
249	#103	C₄×D₅	1	2	yes	—
250	#97	—	2	2	no	—
251	#97	—	10	2	no	—
252	C₄×C₈	—	5	1	no	—
253	C₄×C₈	—	5	1	no	—
254	C₄×C₈	—	5	1	no	—
255	C₄×C₈	—	5	1	no	—
256	C₄×C₈	—	5	1	no	—
257	C₄×C₈	—	5	1	no	—
258	C₄×C₈	—	5	1	no	—
259	C₄×C₈	—	10	2	no	—
260	C₄×C₈	—	10	1	no	—
261	C₄×C₈	—	10	2	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	C₂×C₄×D₄	1	1	yes	characteristic
266	C₂×D₅	—	2	1	no	—
267	C₂×D₅	—	2	1	no	—
268	C₂×D₅	C₄×Q₁₆	1	2	yes	—
269	C₄×C₅	—	2	1	no	—
270	C₄×C₅	—	4	2	no	—
271	C₄×C₅	—	4	2	no	—
272	C₄×C₅	C₂×#131	1	1	yes	characteristic
273	C₄×C₅	C₂×#131	1	1	yes	characteristic
274	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
275	C₄×C₅	C₂×#135	1	1	yes	characteristic
276	Dic₅	—	2	1	no	—
277	Dic₅	—	4	2	no	—
278	Dic₅	—	4	2	no	—
279	Dic₅	C₄×Q₁₆	1	2	yes	—
280	Dic₅	#443	1	2	yes	—
281	C₂×Q₈	—	5	2	no	—
282	C₂×Q₈	—	10	2	no	—
283	C₂×Q₈	C₂×C₄×D₅	1	2	yes	—
284	C₂²×C₄	—	5	1	no	—
285	C₂²×C₄	—	5	1	no	—
286	C₂²×C₄	—	5	1	no	—
287	Q₁₆	—	2	2	no	—
288	Q₁₆	—	10	4	no	—
289	Q₁₆	—	10	2	no	—
290	M₁₆	—	20	4	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	5	1	no	—
294	C₂×C₈	—	5	1	no	—
295	C₂×C₈	—	5	4	no	—
296	C₂×C₈	—	5	1	no	—
297	C₂×C₈	—	5	1	no	—
298	C₂×C₈	—	10	2	no	—
299	C₂×C₈	—	10	2	no	—
300	C₂×C₈	—	10	4	no	—
301	C₂×C₈	—	10	1	no	—
302	C₂×C₈	—	10	2	no	—
303	C₂×C₈	—	10	2	no	—
304	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
305	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
306	C₂×C₈	D₄×D₅	1	1	yes	characteristic
307	C₂×C₈	D₄×D₅	1	1	yes	characteristic
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	5	2	no	—
310	C₄⋊C₄	—	10	2	no	—
311	C₄⋊C₄	—	10	2	no	—
312	C₄⋊C₄	—	10	2	no	—
313	C₄⋊C₄	—	10	2	no	—
314	C₄⋊C₄	—	10	2	no	—
315	C₄⋊C₄	—	10	2	no	—
316	C₄⋊C₄	C₂×C₄×D₅	1	2	yes	—
317	C₂²⋊C₄	—	10	2	no	—
318	C₂²⋊C₄	—	10	2	no	—
319	C₄²	—	2	2	no	—
320	C₄²	—	5	1	no	—
321	C₄²	—	5	1	no	—
322	C₄²	—	5	1	no	—
323	C₄²	—	10	2	no	—
324	C₄²	C₂³×D₅	1	1	yes	characteristic
325	C₂×C₅	#2931	1	1	yes	characteristic
326	C₂×C₅	C₂×C₄×Q₁₆	1	1	yes	characteristic
327	C₂×C₅	C₂×#443	1	1	yes	characteristic
328	D₅	—	2	2	no	—
329	C₂³	—	5	1	no	—
330	Q₈	—	2	2	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	10	2	no	—
333	Q₈	—	10	2	no	—
334	Q₈	—	20	4	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	5	1	no	—
338	C₂×C₄	—	5	1	no	—
339	C₂×C₄	—	5	2	no	—
340	C₂×C₄	—	5	2	no	—
341	C₂×C₄	—	5	1	no	—
342	C₂×C₄	—	10	2	no	—
343	C₂×C₄	—	10	1	no	—
344	C₂×C₄	—	10	1	no	—
345	C₂×C₄	—	10	2	no	—
346	C₂×C₄	—	10	2	no	—
347	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
349	C₂×C₄	D₅×#41	1	1	yes	characteristic
350	C₈	—	2	2	no	—
351	C₈	—	2	1	no	—
352	C₈	—	2	1	no	—
353	C₈	—	4	2	no	—
354	C₈	—	5	2	no	—
355	C₈	—	5	2	no	—
356	C₈	—	10	2	no	—
357	C₈	—	10	1	no	—
358	C₈	—	10	1	no	—
359	C₈	—	20	2	no	—
360	C₈	#4053	1	2	yes	—
361	C₈	D₅×Q₁₆	1	2	yes	—
362	C₅	#17958	1	1	yes	characteristic, complemented, Sylow
363	C₂²	—	5	2	no	—
364	C₂²	—	10	1	no	—
365	C₂²	—	10	1	no	—
366	C₂²	C₄×D₄×D₅	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	4	2	no	—
369	C₄	—	4	2	no	—
370	C₄	—	5	2	no	—
371	C₄	—	5	2	no	—
372	C₄	—	10	1	no	—
373	C₄	—	20	2	no	—
374	C₄	—	20	2	no	—
375	C₄	C₂×#4053	1	1	yes	characteristic
376	C₄	D₅×#131	1	1	yes	characteristic
377	C₄	C₂×D₅×Q₁₆	1	1	yes	characteristic
378	C₄	D₅×#135	1	1	yes	characteristic
379	C₂	—	10	2	no	—
380	C₂	#10611269	1	1	yes	characteristic
381	C₂	C₄×D₅×Q₁₆	1	1	yes	characteristic
382	C₂	D₅×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#1591	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂²×D₅	C₄×D₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	#1591	D₅
hypocenter	C₅	#17958
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
3. C₅×#1591	C₂	semidirect
4. #10601744	C₂	semidirect
5. Q₁₆×#168	C₂	semidirect
7. #10599350	C₂	semidirect
9. #10599006	C₂	semidirect
10. #10598849	C₂	semidirect
99. #1591	D₅	semidirect
149. C₈×D₅	Q₁₆	semidirect
362. C₅	#17958	semidirect

Maximal subgroups

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

subgroup	index
#10611689	2
#10611456	2
C₅×#1591	2
#10601744	2
Q₁₆×#168	2
#10599362	2
#10599350	2
#10599033	2
#10599006	2
#10598849	2
C₈²×D₅	2
#17958	5