Subgroups — #28319418

Order 13568.

616 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290645	C₂	1	2	yes	maximal, complemented
2	#20290634	C₂	1	2	yes	maximal, complemented
3	#20290624	C₂	1	2	yes	maximal
4	#20290599	C₂	1	1	yes	maximal, characteristic
5	#20290597	C₂	1	1	yes	maximal, characteristic
6	#20290517	C₂	1	2	yes	maximal, complemented
7	#20290470	C₂	1	2	yes	maximal
8	#20290436	C₂	1	2	yes	maximal, complemented
9	C₄×#13133223	C₂	1	1	yes	maximal, characteristic
10	D₅₃⋊#108	C₂²	1	2	yes	—
11	#16811633	C₂²	1	2	yes	—
12	#16811609	C₂²	1	2	yes	—
13	#16811602	C₂²	1	2	yes	—
14	#16811586	C₂²	1	4	yes	—
15	#16811505	C₂²	1	4	yes	—
16	#16811494	C₂²	1	4	yes	—
17	#16811492	C₂²	1	2	yes	—
18	#16811491	C₂²	1	2	yes	—
19	#16811490	C₂²	1	4	yes	—
20	C₂×#13133223	C₂²	1	1	yes	characteristic
21	C₂×#13133223	C₂²	1	1	yes	characteristic
22	C₄×#11719945	C₂²	1	1	yes	characteristic
23	#16810578	C₄	1	1	yes	characteristic, complemented
24	#16810578	C₄	1	1	yes	characteristic, complemented
25	#16810577	C₄	1	1	yes	characteristic, complemented
26	#16810577	C₄	1	1	yes	characteristic, complemented
27	#16810577	C₂²	1	1	yes	characteristic
28	#16810577	C₂²	1	1	yes	characteristic
29	C₄×#11719917	C₄	1	1	yes	characteristic, complemented
30	C₄×#11719912	C₄	1	1	yes	characteristic, complemented
31	C₄×#11719912	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic, complemented
33	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
35	D₅₃⋊M₁₆	C₂³	1	4	yes	—
36	#13133296	C₂³	1	4	yes	—
37	#13133288	—	2	2	no	—
38	#13133287	—	2	2	no	—
39	#13133286	—	2	2	no	—
40	#13133285	—	2	2	no	—
41	#13133278	—	2	2	no	—
42	#13133277	—	2	2	no	—
43	#13133276	—	2	2	no	—
44	#13133275	—	2	2	no	—
45	#13133271	—	2	4	no	—
46	C₈×#6157	—	2	4	no	—
47	C₂×#11719945	C₂³	1	1	yes	characteristic
48	#13133223	—	2	2	no	—
49	#13133223	—	2	2	no	—
50	#13133223	—	2	2	no	—
51	#13133223	—	2	2	no	—
52	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
53	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
54	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
55	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
56	C₂×#11719912	C₂³	1	1	yes	characteristic
57	C₂×#11719912	C₂³	1	1	yes	characteristic
58	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
62	C₄×Dic₁₀₆	—	2	1	no	—
63	C₄×D₂₁₂	—	2	1	no	—
64	C₄×C₂²⋊D₅₃	—	2	1	no	—
65	C₄×C₂²⋊D₅₃	—	2	1	no	—
66	#13133176	C₂×C₄	1	1	yes	characteristic
67	#13133174	C₂×C₄	1	1	yes	characteristic
68	#13133174	C₂³	1	1	yes	characteristic
69	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄²×D₅₃	C₂³	1	1	yes	characteristic
71	#13133128	C₂×C₄	1	1	yes	characteristic
72	#13133128	C₂×C₄	1	1	yes	characteristic
73	#13133127	C₂×C₄	1	1	yes	characteristic
74	#13133127	C₂×C₄	1	1	yes	characteristic
75	#13133127	C₂×C₄	1	1	yes	characteristic
76	#13133127	C₂×C₄	1	1	yes	characteristic
77	C₄×#88322	C₂×C₄	1	1	yes	characteristic
78	C₄×#88322	C₂×C₄	1	1	yes	characteristic
79	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
89	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	—	2	1	no	—
91	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₅₃	#41	1	1	yes	characteristic
93	C₂×C₂²⋊D₅₃	#41	1	1	yes	characteristic
94	#11719945	—	2	2	no	—
95	#11719945	—	2	2	no	—
96	C₂×D₂₁₂	#41	1	1	yes	characteristic
97	C₂×C₄×D₅₃	—	2	1	no	—
98	C₂×C₄×D₅₃	—	2	1	no	—
99	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
101	C₂×Dic₁₀₆	#41	1	1	yes	characteristic
102	C₂²⋊#6157	—	2	4	no	—
103	#11719940	—	2	4	no	—
104	C₄⋊#6157	—	2	4	no	—
105	C₄×#6157	—	2	4	no	—
106	#11719936	—	2	4	no	—
107	D₅₃⋊C₈	—	2	4	no	—
108	C₂×C₈×C₅₃	—	2	1	no	—
109	C₂×C₈×C₅₃	—	2	1	no	—
110	C₂×C₈×C₅₃	—	2	1	no	—
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₅₃	—	2	1	no	—
117	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂²⋊Dic₅₃	—	2	1	no	—
120	#11719921	—	2	1	no	—
121	#11719921	—	2	1	no	—
122	C₄⋊Dic₅₃	—	2	1	no	—
123	#11719919	—	2	1	no	—
124	#11719919	—	2	1	no	—
125	C₄×Dic₅₃	—	2	1	no	—
126	C₄×Dic₅₃	—	2	1	no	—
127	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
129	#11719917	—	2	2	no	—
130	#11719917	—	2	2	no	—
131	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
132	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
133	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
134	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
135	#11719912	—	2	2	no	—
136	#11719912	—	2	2	no	—
137	#11719912	—	2	2	no	—
138	#11719912	—	2	2	no	—
139	C₈×D₅₃	—	2	1	no	—
140	C₈×D₅₃	—	2	1	no	—
141	C₈×D₅₃	—	2	1	no	—
142	C₈×D₅₃	—	2	1	no	—
143	C₈×D₅₃	#41	1	2	yes	—
144	C₈×D₅₃	#41	1	2	yes	—
145	C₈×D₅₃	#41	1	2	yes	—
146	C₈×D₅₃	#41	1	2	yes	—
147	C₂³×C₅₃	#117	1	1	yes	characteristic
148	C₂²×D₅₃	#117	1	1	yes	characteristic
149	C₂²×D₅₃	C₂×#41	1	1	yes	characteristic
150	C₂×#6157	—	4	4	no	—
151	C₂×C₄×C₅₃	—	2	1	no	—
152	C₂×C₄×C₅₃	—	2	1	no	—
153	C₂×C₄×C₅₃	—	2	1	no	—
154	C₂×C₄×C₅₃	—	2	1	no	—
155	C₂×C₄×C₅₃	—	4	2	no	—
156	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
158	C₂²⋊D₅₃	—	2	2	no	—
159	C₂²⋊D₅₃	—	2	2	no	—
160	C₂×Dic₅₃	—	2	1	no	—
161	C₂×Dic₅₃	—	2	1	no	—
162	C₂×Dic₅₃	#117	1	1	yes	characteristic
163	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
164	D₂₁₂	—	2	2	no	—
165	C₄×D₅₃	—	2	1	no	—
166	C₄×D₅₃	—	2	1	no	—
167	C₄×D₅₃	—	4	2	no	—
168	C₄×D₅₃	—	4	2	no	—
169	C₄×D₅₃	C₂×#41	1	2	yes	—
170	C₄×D₅₃	C₂×#41	1	2	yes	—
171	Dic₁₀₆	—	2	2	no	—
172	#88324	—	4	4	no	—
173	C₈×C₅₃	—	2	1	no	—
174	C₈×C₅₃	—	2	1	no	—
175	C₈×C₅₃	#117	1	1	yes	characteristic
176	C₈×C₅₃	#117	1	1	yes	characteristic
177	C₈×C₅₃	#117	1	1	yes	characteristic
178	C₈×C₅₃	#117	1	1	yes	characteristic
179	#88322	—	2	1	no	—
180	#88322	—	2	1	no	—
181	#88322	#117	1	2	yes	—
182	#88322	#117	1	2	yes	—
183	#20787	—	53	1	no	maximal, Sylow
184	C₂²×C₅₃	—	2	1	no	—
185	C₂²×C₅₃	—	4	1	no	—
186	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
187	C₂×D₅₃	—	2	1	no	—
188	C₂×D₅₃	—	4	1	no	—
189	C₂×D₅₃	—	4	1	no	—
190	C₂×D₅₃	#529	1	2	yes	—
191	#6157	—	8	4	no	—
192	C₄×C₅₃	—	2	2	no	—
193	C₄×C₅₃	—	2	1	no	—
194	C₄×C₅₃	—	4	1	no	—
195	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
196	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
197	Dic₅₃	—	2	1	no	—
198	Dic₅₃	—	4	1	no	—
199	Dic₅₃	—	4	1	no	—
200	Dic₅₃	#529	1	2	yes	—
201	C₄×#131	—	53	1	no	—
202	#1841	—	53	2	no	—
203	#1833	—	53	2	no	—
204	#1827	—	53	2	no	—
205	#1813	—	53	1	no	—
206	#1812	—	53	1	no	—
207	#1772	—	53	2	no	—
208	#1757	—	53	2	no	—
209	#1747	—	53	2	no	—
210	C₂×C₅₃	—	2	1	no	—
211	C₂×C₅₃	—	4	1	no	—
212	C₂×C₅₃	#2283	1	1	yes	characteristic
213	D₅₃	—	2	2	no	—
214	D₅₃	—	4	1	no	—
215	C₂×#131	—	53	1	no	—
216	C₂×#131	—	53	1	no	—
217	C₄×#41	—	53	1	no	—
218	C₂×#108	—	53	2	no	—
219	#427	—	53	2	no	—
220	#413	—	53	2	no	—
221	#410	—	53	2	no	—
222	#405	—	53	1	no	—
223	#405	—	53	1	no	—
224	#405	—	53	1	no	—
225	#405	—	53	1	no	—
226	#405	—	53	1	no	—
227	#405	—	53	1	no	—
228	#405	—	53	4	no	—
229	C₄×M₁₆	—	53	1	no	—
230	C₄×M₁₆	—	53	1	no	—
231	C₄×M₁₆	—	53	1	no	—
232	C₂×C₄×C₈	—	53	1	no	—
233	C₂×C₄×C₈	—	53	1	no	—
234	C₂×C₄×C₈	#6157	1	1	yes	characteristic, complemented
235	#344	—	53	4	no	—
236	#339	—	53	4	no	—
237	#338	—	53	2	no	—
238	H(3,ℤ/4)	—	53	2	no	—
239	H(3,ℤ/4)	—	53	4	no	—
240	C₅₃	#20787	1	1	yes	characteristic, complemented, Sylow
241	C₂×#41	—	53	1	no	—
242	#131	—	106	2	no	—
243	#131	—	106	2	no	—
244	#131	—	106	2	no	—
245	#131	—	106	2	no	—
246	C₂×M₁₆	—	53	1	no	—
247	C₂×M₁₆	—	53	1	no	—
248	C₂×M₁₆	—	53	1	no	—
249	C₂×M₁₆	—	53	1	no	—
250	C₂×M₁₆	—	53	1	no	—
251	C₂×M₁₆	—	53	1	no	—
252	C₂×M₁₆	—	53	4	no	—
253	C₂²×C₈	—	53	1	no	—
254	C₂²×C₈	—	53	1	no	—
255	C₂²×C₈	—	53	1	no	—
256	C₂²×C₈	—	53	1	no	—
257	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
258	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
259	C₄×Q₈	—	106	1	no	—
260	C₄×D₄	—	106	1	no	—
261	C₄×D₄	—	106	1	no	—
262	C₄×D₄	—	106	1	no	—
263	#117	—	53	1	no	—
264	#117	—	53	1	no	—
265	#117	—	53	1	no	—
266	#117	—	53	4	no	—
267	C₂×C₄²	—	53	1	no	—
268	C₂×C₄²	—	53	1	no	—
269	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
270	#108	—	106	2	no	—
271	#108	—	106	2	no	—
272	#107	—	106	2	no	—
273	#106	—	106	2	no	—
274	#101	—	106	2	no	—
275	#100	—	106	2	no	—
276	#99	—	106	2	no	—
277	#99	—	106	2	no	—
278	#97	—	53	1	no	—
279	#97	—	53	1	no	—
280	#97	—	53	1	no	—
281	#97	—	53	1	no	—
282	#97	—	53	1	no	—
283	#97	—	53	1	no	—
284	#97	—	106	4	no	—
285	C₄×C₈	—	53	1	no	—
286	C₄×C₈	—	53	1	no	—
287	C₄×C₈	—	53	1	no	—
288	C₄×C₈	—	53	1	no	—
289	C₄×C₈	—	53	1	no	—
290	C₄×C₈	—	53	1	no	—
291	C₄×C₈	—	106	4	no	—
292	C₄×C₈	C₂×#6157	1	1	yes	characteristic
293	C₄×C₈	C₂×#6157	1	1	yes	characteristic
294	C₄×C₈	C₂×#6157	1	1	yes	characteristic
295	C₄×C₈	C₂×#6157	1	1	yes	characteristic
296	#41	—	106	2	no	—
297	#41	—	106	2	no	—
298	C₂×Q₈	—	53	1	no	—
299	C₂×D₄	—	53	1	no	—
300	C₂×D₄	—	53	1	no	—
301	C₂×D₄	—	53	1	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	53	1	no	—
304	C₂²×C₄	—	53	1	no	—
305	C₂²×C₄	—	106	1	no	—
306	C₂²×C₄	—	106	1	no	—
307	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
308	M₁₆	—	106	2	no	—
309	M₁₆	—	106	2	no	—
310	M₁₆	—	106	2	no	—
311	M₁₆	—	106	2	no	—
312	M₁₆	—	106	2	no	—
313	M₁₆	—	106	2	no	—
314	M₁₆	—	106	4	no	—
315	M₁₆	—	106	4	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	53	1	no	—
321	C₂×C₈	—	53	1	no	—
322	C₂×C₈	—	53	2	no	—
323	C₂×C₈	—	53	2	no	—
324	C₂×C₈	—	53	2	no	—
325	C₂×C₈	—	53	2	no	—
326	C₂×C₈	—	53	1	no	—
327	C₂×C₈	—	53	1	no	—
328	C₂×C₈	—	106	1	no	—
329	C₂×C₈	—	106	1	no	—
330	C₂×C₈	—	106	1	no	—
331	C₂×C₈	—	106	1	no	—
332	C₂×C₈	—	106	4	no	—
333	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
334	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
335	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
336	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
337	C₄⋊C₄	—	106	1	no	—
338	C₄⋊C₄	—	106	1	no	—
339	C₄⋊C₄	—	106	4	no	—
340	C₄⋊C₄	—	106	1	no	—
341	C₂²⋊C₄	—	106	1	no	—
342	C₂²⋊C₄	—	106	1	no	—
343	C₂²⋊C₄	—	106	1	no	—
344	C₂²⋊C₄	—	106	4	no	—
345	C₄²	—	2	1	no	—
346	C₄²	—	53	1	no	—
347	C₄²	—	53	1	no	—
348	C₄²	—	106	1	no	—
349	C₄²	—	106	1	no	—
350	C₄²	—	106	4	no	—
351	C₄²	C₂²×#6157	1	1	yes	characteristic
352	C₄²	C₂²×#6157	1	1	yes	characteristic
353	C₂³	—	53	1	no	—
354	C₂³	—	53	1	no	—
355	C₂³	#13133296	1	1	yes	characteristic
356	Q₈	—	106	2	no	—
357	D₄	—	106	2	no	—
358	D₄	—	106	2	no	—
359	D₄	—	106	2	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	53	1	no	—
366	C₂×C₄	—	53	2	no	—
367	C₂×C₄	—	53	2	no	—
368	C₂×C₄	—	53	1	no	—
369	C₂×C₄	—	106	1	no	—
370	C₂×C₄	—	106	1	no	—
371	C₂×C₄	—	106	1	no	—
372	C₂×C₄	—	106	1	no	—
373	C₂×C₄	—	212	2	no	—
374	C₂×C₄	—	212	2	no	—
375	C₂×C₄	—	212	4	no	—
376	C₂×C₄	#13133296	1	1	yes	characteristic
377	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
378	C₈	—	2	1	no	—
379	C₈	—	2	1	no	—
380	C₈	—	53	2	no	—
381	C₈	—	53	2	no	—
382	C₈	—	106	1	no	—
383	C₈	—	106	1	no	—
384	C₈	—	212	4	no	—
385	C₈	#13133296	1	1	yes	characteristic
386	C₈	#13133296	1	1	yes	characteristic
387	C₈	#13133296	1	1	yes	characteristic
388	C₈	#13133296	1	1	yes	characteristic
389	C₂²	—	2	1	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	53	2	no	—
392	C₂²	—	106	1	no	—
393	C₂²	—	212	1	no	—
394	C₂²	—	212	1	no	—
395	C₂²	C₂×#13133296	1	1	yes	characteristic
396	C₄	—	2	2	no	—
397	C₄	—	2	1	no	—
398	C₄	—	4	1	no	—
399	C₄	—	53	2	no	—
400	C₄	—	106	1	no	—
401	C₄	—	424	4	no	—
402	C₄	—	212	1	no	—
403	C₄	—	212	1	no	—
404	C₄	C₂×#13133296	1	1	yes	characteristic
405	C₄	C₂×#13133296	1	1	yes	characteristic
406	C₂	—	2	1	no	—
407	C₂	—	4	1	no	—
408	C₂	—	106	2	no	—
409	C₂	—	212	1	no	—
410	C₂	#20291162	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133296
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂×C₅₃	#2283
prime radical	C₂×#11719945	C₂³
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#20787
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #20290645	C₂	semidirect
2. #20290634	C₂	semidirect
6. #20290517	C₂	semidirect
8. #20290436	C₂	semidirect
23. #16810578	C₄	semidirect
24. #16810578	C₄	semidirect
25. #16810577	C₄	semidirect
26. #16810577	C₄	semidirect
29. C₄×#11719917	C₄	semidirect
30. C₄×#11719912	C₄	semidirect
32. C₄×C₈×D₅₃	C₄	semidirect
34. C₂×C₄×C₈×C₅₃	C₄	semidirect
234. C₂×C₄×C₈	#6157	semidirect
240. C₅₃	#20787	semidirect

Maximal subgroups

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

subgroup	index
#20290645	2
#20290634	2
#20290624	2
#20290599	2
#20290597	2
#20290517	2
#20290470	2
#20290436	2
C₄×#13133223	2
#20787	53