Subgroups — #28316998

Order 13568.

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	#20291503	C₂	1	2	yes	maximal
2	#20290355	C₂	1	1	yes	maximal, characteristic, complemented
3	#20290354	C₂	1	1	yes	maximal, characteristic, complemented
4	#20290353	C₂	1	2	yes	maximal, complemented
5	#20290303	C₂	1	1	yes	maximal, characteristic, complemented
6	#20290302	C₂	1	2	yes	maximal, complemented
7	Q₁₆×#88324	C₂	1	1	yes	maximal, characteristic, complemented
8	#20290272	C₂	1	1	yes	maximal, characteristic
9	#20290063	C₂	1	2	yes	maximal
10	C₂×C₈×#88324	C₂	1	1	yes	maximal, characteristic
11	#20285296	C₂	1	1	yes	maximal, characteristic
12	#16811675	—	2	2	no	—
13	#16811674	C₂²	1	2	yes	—
14	#16811673	C₂²	1	2	yes	—
15	#16811672	—	2	2	no	—
16	#16811672	C₂²	1	2	yes	—
17	Q₈×#88324	C₂²	1	2	yes	—
18	#16811629	—	2	2	no	—
19	#16811626	C₂²	1	2	yes	—
20	#16811587	—	2	2	no	—
21	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
22	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
23	#16811487	C₂²	1	2	yes	—
24	#16811486	C₂²	1	1	yes	characteristic
25	#16811486	C₂²	1	1	yes	characteristic
26	#16811473	C₂²	1	2	yes	—
27	#16811473	C₂²	1	2	yes	—
28	Q₈⋊#88324	C₂²	1	2	yes	—
29	Q₈⋊#88324	C₂²	1	2	yes	—
30	C₈×#88324	C₂²	1	1	yes	characteristic
31	C₈×#88324	C₂²	1	1	yes	characteristic
32	C₈×#88324	C₂²	1	1	yes	characteristic
33	C₈×#88324	C₂²	1	1	yes	characteristic
34	#16811175	C₂²	1	2	yes	—
35	#16811027	C₄	1	1	yes	characteristic
36	#16811001	C₄	1	1	yes	characteristic
37	#16810972	C₄	1	2	yes	—
38	#16810864	C₄	1	1	yes	characteristic
39	#16810862	C₄	1	2	yes	—
40	Q₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
41	#16810717	C₂²	1	1	yes	characteristic
42	#16810715	C₂²	1	2	yes	—
43	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
44	C₅₃×#467	C₄	1	1	yes	characteristic
45	C₂×#11719940	—	2	2	no	—
46	C₂²×#88324	—	2	1	no	—
47	C₂²×#88324	D₄	1	1	yes	characteristic
48	C₂²×#88324	D₄	1	1	yes	characteristic
49	#13133284	—	2	2	no	—
50	#13133284	—	2	2	no	—
51	#13133284	—	2	2	no	—
52	C₄⋊#88324	—	2	2	no	—
53	C₄⋊#88324	—	2	2	no	—
54	C₄⋊#88324	C₂³	1	2	yes	—
55	C₄⋊#88324	C₂³	1	2	yes	—
56	#13133272	—	2	2	no	—
57	#13133272	—	2	2	no	—
58	C₄×#88324	—	2	1	no	—
59	C₄×#88324	—	2	2	no	—
60	C₄×#88324	—	2	2	no	—
61	C₄×#88324	D₄	1	1	yes	characteristic
62	C₄×#88324	D₄	1	1	yes	characteristic
63	C₄×#88324	C₂³	1	1	yes	characteristic
64	C₄×#88324	C₂³	1	1	yes	characteristic
65	C₄×#88324	C₂³	1	1	yes	characteristic
66	C₄×#88324	C₂³	1	1	yes	characteristic
67	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
68	C₂×#11719925	C₂×C₄	1	2	yes	—
69	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
70	#13133216	C₂×C₄	1	2	yes	—
71	#13133215	C₂×C₄	1	2	yes	—
72	#13133207	—	2	2	no	—
73	#13133196	C₂×C₄	1	2	yes	—
74	#13133194	—	2	2	no	—
75	Q₈×Dic₅₃	C₂³	1	2	yes	—
76	#13133182	—	2	2	no	—
77	#13133182	C₂³	1	2	yes	—
78	#13133173	—	2	2	no	—
79	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
80	#13133151	C₂×C₄	1	1	yes	characteristic
81	#13133150	C₂×C₄	1	2	yes	—
82	#13133142	C₂×C₄	1	2	yes	—
83	#13133141	C₂×C₄	1	2	yes	—
84	#13133140	C₂×C₄	1	2	yes	—
85	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
86	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
87	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
88	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
89	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
92	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
93	#103×C₅₃	C₂×C₄	1	2	yes	—
94	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×Dic₅₃	#41	1	1	yes	characteristic
98	C₂×Dic₁₀₆	—	2	2	no	—
99	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
100	#11719940	—	4	4	no	—
101	C₂×#88324	—	2	4	no	—
102	C₂×#88324	—	2	2	no	—
103	C₂×#88324	—	2	1	no	—
104	C₂×#88324	—	2	1	no	—
105	C₂×#88324	—	2	2	no	—
106	C₂×#88324	—	2	2	no	—
107	C₂×#88324	Q₁₆	1	4	yes	complemented
108	C₂×#88324	C₂×D₄	1	1	yes	characteristic
109	C₂×#88324	C₂×D₄	1	1	yes	characteristic
110	C₂×#88324	C₂×D₄	1	1	yes	characteristic
111	C₂×#88324	C₂×D₄	1	1	yes	characteristic
112	Q₁₆×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₄⋊C₄×C₅₃	—	2	2	no	—
117	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
118	C₂²⋊C₄×C₅₃	—	2	2	no	—
119	C₂²⋊Dic₅₃	—	2	2	no	—
120	#11719925	—	2	4	no	—
121	C₄⋊Dic₅₃	—	2	2	no	—
122	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
123	#11719919	—	2	2	no	—
124	#11719919	—	2	2	no	—
125	#11719919	—	2	2	no	—
126	#11719919	—	2	2	no	—
127	#11719919	—	2	2	no	—
128	C₄×Dic₅₃	—	2	2	no	—
129	C₄×Dic₅₃	—	2	2	no	—
130	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
132	C₄×Dic₅₃	#41	1	1	yes	characteristic
133	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
134	C₂×#88322	—	2	2	no	—
135	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
136	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
137	Dic₂₁₂	—	2	2	no	—
138	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
139	Q₈×C₅₃	—	2	2	no	—
140	Q₈×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
147	C₂×Dic₅₃	—	2	2	no	—
148	C₂×Dic₅₃	—	2	2	no	—
149	C₂×Dic₅₃	—	2	2	no	—
150	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
151	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
152	C₂×Dic₅₃	C₂×Q₁₆	1	2	yes	—
153	C₂×Dic₅₃	#135	1	2	yes	—
154	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
156	Dic₁₀₆	—	2	2	no	—
157	Dic₁₀₆	—	2	2	no	—
158	Dic₁₀₆	—	4	4	no	—
159	#88324	—	2	2	no	—
160	#88324	—	2	2	no	—
161	#88324	—	2	1	no	—
162	#88324	—	2	1	no	—
163	#88324	—	4	2	no	—
164	#88324	—	4	2	no	—
165	#88324	C₂×Q₁₆	1	2	yes	—
166	#88324	C₂×Q₁₆	1	2	yes	—
167	C₈×C₅₃	—	2	1	no	—
168	C₈×C₅₃	#131	1	2	yes	—
169	#88322	—	2	1	no	—
170	#88322	#131	1	2	yes	—
171	#17958	—	53	1	no	maximal, Sylow
172	C₂²×C₅₃	—	2	1	no	—
173	C₂²×C₅₃	—	2	1	no	—
174	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
175	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	4	2	no	—
178	C₄×C₅₃	—	4	2	no	—
179	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
180	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
181	Dic₅₃	—	2	1	no	—
182	Dic₅₃	—	4	2	no	—
183	Dic₅₃	—	4	2	no	—
184	Dic₅₃	C₄×Q₁₆	1	2	yes	—
185	Dic₅₃	#443	1	2	yes	—
186	Dic₅₃	C₂²×Q₁₆	1	1	yes	characteristic
187	Dic₅₃	C₂×#135	1	1	yes	characteristic
188	#3170	—	53	1	no	—
189	#2960	—	53	2	no	—
190	#1591	—	53	2	no	—
191	#1591	—	53	1	no	—
192	#1591	—	53	1	no	—
193	C₈×Q₁₆	—	53	1	no	—
194	C₈×Q₁₆	—	53	2	no	—
195	C₈×Q₁₆	—	53	1	no	—
196	#1567	—	53	1	no	—
197	#1493	—	53	2	no	—
198	C₂×C₈²	—	53	1	no	—
199	C₂×C₅₃	—	2	2	no	—
200	C₂×C₅₃	#2931	1	1	yes	characteristic
201	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
202	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
203	#533	—	53	2	no	—
204	#500	—	53	1	no	—
205	#494	—	53	1	no	—
206	#487	—	53	2	no	—
207	#467	—	53	1	no	—
208	#467	—	53	2	no	—
209	#467	#6157	1	1	yes	characteristic
210	#446	—	53	2	no	—
211	#446	—	53	2	no	—
212	#446	—	106	2	no	—
213	C₈×Q₈	—	53	2	no	—
214	C₈×Q₈	—	53	2	no	—
215	C₈×Q₈	—	106	2	no	—
216	C₄×Q₁₆	—	53	1	no	—
217	C₄×Q₁₆	—	53	1	no	—
218	C₄×Q₁₆	—	53	2	no	—
219	#424	—	106	2	no	—
220	#423	—	53	2	no	—
221	#405	—	106	2	no	—
222	C₂×C₄×C₈	—	53	1	no	—
223	C₂×C₄×C₈	—	53	1	no	—
224	C₂×C₄×C₈	—	53	1	no	—
225	#335	—	53	1	no	—
226	#335	—	53	1	no	—
227	#335	—	53	2	no	—
228	Q₈⋊C₈	—	53	2	no	—
229	Q₈⋊C₈	—	53	2	no	—
230	Q₈⋊C₈	—	53	2	no	—
231	Q₈⋊C₈	—	53	2	no	—
232	C₈²	—	53	1	no	—
233	C₈²	—	53	1	no	—
234	C₈²	—	53	1	no	—
235	C₈²	—	53	1	no	—
236	C₅₃	#17958	1	1	yes	characteristic, complemented, Sylow
237	C₂×Q₁₆	—	53	2	no	—
238	C₂×Q₁₆	—	53	1	no	—
239	C₂×Q₁₆	C₂×#6157	1	1	yes	characteristic
240	C₂×M₁₆	—	106	2	no	—
241	C₂²×C₈	—	53	1	no	—
242	C₂²×C₈	—	53	1	no	—
243	C₂²×C₈	—	53	1	no	—
244	C₂²×C₈	—	106	1	no	—
245	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
246	C₄⋊Q₈	—	53	2	no	—
247	C₄⋊Q₈	—	53	2	no	—
248	#125	—	106	2	no	—
249	C₂²⋊Q₈	—	53	2	no	—
250	C₂²⋊Q₈	—	106	2	no	—
251	C₂²⋊Q₈	C₂×#6157	1	2	yes	—
252	C₄×Q₈	—	53	2	no	—
253	C₄×Q₈	—	53	2	no	—
254	C₄×Q₈	—	106	2	no	—
255	#117	—	106	2	no	—
256	C₂×C₄²	—	53	1	no	—
257	#107	—	53	1	no	—
258	#107	—	53	2	no	—
259	#107	C₂×#6157	1	1	yes	characteristic
260	C₄⋊C₈	—	53	2	no	—
261	C₄⋊C₈	—	53	2	no	—
262	C₄⋊C₈	—	106	2	no	—
263	C₄⋊C₈	—	106	2	no	—
264	C₄⋊C₈	—	106	2	no	—
265	C₄⋊C₈	—	106	2	no	—
266	C₄⋊C₈	—	106	2	no	—
267	#103	—	53	2	no	—
268	#103	—	53	2	no	—
269	#103	—	53	2	no	—
270	#103	C₂×#6157	1	2	yes	—
271	#97	—	106	2	no	—
272	#97	—	106	2	no	—
273	C₄×C₈	—	53	1	no	—
274	C₄×C₈	—	53	1	no	—
275	C₄×C₈	—	53	1	no	—
276	C₄×C₈	—	53	1	no	—
277	C₄×C₈	—	53	1	no	—
278	C₄×C₈	—	53	1	no	—
279	C₄×C₈	—	53	1	no	—
280	C₄×C₈	—	53	1	no	—
281	C₄×C₈	—	53	1	no	—
282	C₄×C₈	—	53	1	no	—
283	C₄×C₈	—	106	1	no	—
284	C₄×C₈	—	106	2	no	—
285	C₄×C₈	—	106	2	no	—
286	C₂×Q₈	—	53	2	no	—
287	C₂×Q₈	—	106	2	no	—
288	C₂×Q₈	C₂²×#6157	1	2	yes	—
289	C₂²×C₄	—	53	1	no	—
290	C₂²×C₄	—	53	1	no	—
291	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
292	Q₁₆	—	2	2	no	—
293	Q₁₆	—	106	4	no	—
294	Q₁₆	—	106	2	no	—
295	M₁₆	—	212	4	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	53	4	no	—
298	C₂×C₈	—	53	1	no	—
299	C₂×C₈	—	53	1	no	—
300	C₂×C₈	—	53	1	no	—
301	C₂×C₈	—	53	1	no	—
302	C₂×C₈	—	53	1	no	—
303	C₂×C₈	—	53	1	no	—
304	C₂×C₈	—	106	2	no	—
305	C₂×C₈	—	106	1	no	—
306	C₂×C₈	—	106	1	no	—
307	C₂×C₈	—	106	2	no	—
308	C₂×C₈	—	106	4	no	—
309	C₂×C₈	—	106	2	no	—
310	C₂×C₈	—	106	2	no	—
311	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
312	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	53	2	no	—
315	C₄⋊C₄	—	106	2	no	—
316	C₄⋊C₄	—	106	2	no	—
317	C₄⋊C₄	—	106	2	no	—
318	C₄⋊C₄	—	106	2	no	—
319	C₄⋊C₄	—	106	2	no	—
320	C₄⋊C₄	—	106	2	no	—
321	C₄⋊C₄	C₂²×#6157	1	2	yes	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₂²⋊C₄	—	106	2	no	—
324	C₄²	—	53	1	no	—
325	C₄²	—	53	1	no	—
326	C₄²	—	53	1	no	—
327	C₄²	—	53	1	no	—
328	C₄²	—	106	2	no	—
329	C₄²	—	106	2	no	—
330	C₂³	D₄×#6157	1	1	yes	characteristic
331	Q₈	—	2	2	no	—
332	Q₈	—	2	2	no	—
333	Q₈	—	106	2	no	—
334	Q₈	—	106	2	no	—
335	Q₈	—	212	4	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	53	2	no	—
341	C₂×C₄	—	53	2	no	—
342	C₂×C₄	—	53	1	no	—
343	C₂×C₄	—	53	1	no	—
344	C₂×C₄	—	53	1	no	—
345	C₂×C₄	—	53	1	no	—
346	C₂×C₄	—	106	2	no	—
347	C₂×C₄	—	106	2	no	—
348	C₂×C₄	—	106	2	no	—
349	C₂×C₄	D₄×#6157	1	1	yes	characteristic
350	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
351	C₈	—	2	1	no	—
352	C₈	—	53	2	no	—
353	C₈	—	53	2	no	—
354	C₈	—	53	2	no	—
355	C₈	—	106	1	no	—
356	C₈	—	106	1	no	—
357	C₈	—	106	2	no	—
358	C₈	—	106	2	no	—
359	C₈	—	106	1	no	—
360	C₈	—	212	2	no	—
361	C₈	—	212	2	no	—
362	C₈	#13133304	1	2	yes	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	Q₁₆×#6157	1	2	yes	—
366	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	4	2	no	—
369	C₄	—	4	2	no	—
370	C₄	—	53	1	no	—
371	C₄	—	53	1	no	—
372	C₄	—	53	2	no	—
373	C₄	—	53	2	no	—
374	C₄	—	106	1	no	—
375	C₄	—	212	2	no	—
376	C₄	—	212	2	no	—
377	C₄	C₂×#13133304	1	1	yes	characteristic
378	C₄	D₅₃⋊#131	1	1	yes	characteristic
379	C₂	—	2	2	no	—
380	C₂	#20291438	1	1	yes	characteristic
381	C₂	C₂×Q₁₆×#6157	1	1	yes	characteristic
382	C₂	D₅₃⋊#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#467	C₄
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	#467	#6157
hypocenter	C₅₃	#17958
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #467

Complements

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

N	H ≅ G/N	type
2. #20290355	C₂	semidirect
3. #20290354	C₂	semidirect
4. #20290353	C₂	semidirect
5. #20290303	C₂	semidirect
6. #20290302	C₂	semidirect
7. Q₁₆×#88324	C₂	semidirect
107. C₂×#88324	Q₁₆	semidirect
236. C₅₃	#17958	semidirect

Maximal subgroups

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

subgroup	index
#20291503	2
#20290355	2
#20290354	2
#20290353	2
#20290303	2
#20290302	2
Q₁₆×#88324	2
#20290272	2
#20290063	2
C₂×C₈×#88324	2
#20285296	2
#17958	53