Subgroups — #28318038

Order 13568.

908 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291278	C₂	1	2	yes	maximal
2	#20290437	C₂	1	4	yes	maximal, complemented
3	#20290436	C₂	1	4	yes	maximal, complemented
4	C₂×#16811492	C₂	1	1	yes	maximal, characteristic
5	C₂×#16811491	C₂	1	1	yes	maximal, characteristic
6	#20290385	C₂	1	2	yes	maximal
7	#20283131	C₂	1	1	yes	maximal, characteristic
8	C₂×D₅₃⋊M₁₆	C₂²	1	2	yes	—
9	C₂×#13133296	C₂²	1	2	yes	—
10	#16811589	C₄	1	8	yes	complemented
11	#16811586	C₂²	1	8	yes	—
12	M₁₆×#6157	C₄	1	8	yes	complemented
13	#16811492	C₂²	1	2	yes	—
14	#16811492	C₂²	1	2	yes	—
15	#16811491	C₂²	1	2	yes	—
16	#16811491	C₂²	1	2	yes	—
17	#16811490	C₂²	1	8	yes	—
18	C₂²×#11719912	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
20	#117×D₅₃	C₂²	1	1	yes	characteristic
21	#16810581	C₄	1	1	yes	characteristic, complemented
22	#16810580	C₂²	1	2	yes	—
23	#16810579	C₄	1	1	yes	characteristic, complemented
24	#16810578	C₄	1	1	yes	characteristic, complemented
25	#16810577	C₂²	1	2	yes	—
26	#16810575	C₄	1	2	yes	complemented
27	C₄×#11719912	C₄	1	2	yes	complemented
28	C₅₃×#405	C₄	1	1	yes	characteristic, complemented
29	C₂×#11719936	—	2	2	no	—
30	D₅₃⋊M₁₆	—	2	2	no	—
31	D₅₃⋊M₁₆	—	2	2	no	—
32	D₅₃⋊M₁₆	C₂×C₄	1	4	yes	—
33	D₅₃⋊M₁₆	C₂³	1	4	yes	—
34	C₂×#11719940	—	2	2	no	—
35	C₂×D₅₃⋊C₈	C₂×C₄	1	4	yes	—
36	#13133296	—	2	2	no	—
37	#13133296	—	2	2	no	—
38	#13133296	C₂×C₄	1	4	yes	—
39	#13133296	C₂³	1	4	yes	—
40	C₂×C₄⋊#6157	—	2	2	no	—
41	C₂×C₂²⋊#6157	—	2	2	no	—
42	C₂×C₄×#6157	C₂×C₄	1	4	yes	—
43	#13133271	C₂×C₄	1	16	yes	complemented
44	C₈×#6157	C₂×C₄	1	16	yes	complemented
45	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
46	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
47	C₂×#11719912	—	2	2	no	—
48	C₂×#11719912	—	2	2	no	—
49	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
50	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
51	C₂×#11719912	C₂×C₄	1	2	yes	—
52	C₂×#11719912	C₂³	1	1	yes	characteristic
53	C₂×#11719912	C₂³	1	1	yes	characteristic
54	M₁₆×D₅₃	C₂×C₄	1	4	yes	—
55	M₁₆×D₅₃	C₂×C₄	1	4	yes	—
56	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₅₃	C₂³	1	2	yes	—
58	#13133176	C₂×C₄	1	1	yes	characteristic
59	#13133175	—	2	1	no	—
60	#13133174	C₂³	1	2	yes	—
61	#13133173	—	2	1	no	—
62	C₄⋊C₄×D₅₃	—	2	1	no	—
63	C₂²⋊C₄×D₅₃	—	2	1	no	—
64	C₄²×D₅₃	C₂×C₄	1	2	yes	—
65	#13133128	C₂×C₄	1	2	yes	complemented
66	#13133127	C₂×C₄	1	2	yes	complemented
67	#13133127	C₂×C₄	1	2	yes	complemented
68	C₄×#88322	C₂×C₄	1	2	yes	complemented
69	C₈×Dic₅₃	C₂×C₄	1	2	yes	complemented
70	C₈×Dic₅₃	C₂×C₄	1	2	yes	complemented
71	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
74	#97×C₅₃	C₂×C₄	1	2	yes	complemented
75	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	complemented
76	C₂³×D₅₃	#41	1	1	yes	characteristic
77	C₂²×#6157	—	2	4	no	—
78	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×Dic₅₃	#41	1	1	yes	characteristic
80	C₂×C₄×D₅₃	—	2	2	no	—
81	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
85	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
86	C₂×C₄×D₅₃	#41	1	2	yes	—
87	C₂×C₄×D₅₃	#41	1	2	yes	—
88	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
89	C₂²⋊#6157	—	2	4	no	—
90	C₂²⋊#6157	—	2	4	no	—
91	#11719940	—	2	4	no	—
92	#11719940	—	2	4	no	—
93	C₂×#88324	—	2	4	no	—
94	C₄⋊#6157	—	2	4	no	—
95	C₄⋊#6157	—	2	4	no	—
96	C₄×#6157	—	2	4	no	—
97	C₄×#6157	C₂²×C₄	1	8	yes	—
98	#11719936	—	2	4	no	—
99	#11719936	—	2	4	no	—
100	D₅₃⋊C₈	—	2	4	no	—
101	D₅₃⋊C₈	C₂²×C₄	1	8	yes	—
102	M₁₆×C₅₃	C₄²	1	2	yes	—
103	M₁₆×C₅₃	C₄²	1	2	yes	—
104	C₂×C₈×C₅₃	—	2	1	no	—
105	C₂×C₈×C₅₃	—	2	1	no	—
106	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
109	C₄⋊C₄×C₅₃	—	2	1	no	—
110	C₂²⋊C₄×C₅₃	—	2	1	no	—
111	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
112	C₂²⋊Dic₅₃	—	2	1	no	—
113	#11719921	—	2	2	no	—
114	C₄⋊Dic₅₃	—	2	1	no	—
115	#11719919	—	2	2	no	—
116	C₄×Dic₅₃	—	2	2	no	—
117	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
118	#11719917	C₄²	1	2	yes	—
119	#11719917	C₄²	1	2	yes	—
120	C₂×#88322	—	2	1	no	—
121	C₂×#88322	—	2	1	no	—
122	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
123	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
124	C₂×#88322	C₂²×C₄	1	2	yes	—
125	#11719912	—	2	2	no	—
126	#11719912	—	2	2	no	—
127	#11719912	—	2	2	no	—
128	#11719912	—	2	2	no	—
129	#11719912	C₄²	1	8	yes	—
130	C₈×D₅₃	C₂²×C₄	1	8	yes	—
131	C₂³×C₅₃	#117	1	1	yes	characteristic
132	C₂²×D₅₃	—	8	1	no	—
133	C₂²×D₅₃	#117	1	2	yes	—
134	C₂²×D₅₃	#117	1	2	yes	—
135	C₂²×D₅₃	#117	1	1	yes	characteristic
136	C₂²×D₅₃	C₂×#41	1	1	yes	characteristic
137	C₂×#6157	—	2	8	no	—
138	C₂×#6157	—	8	4	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
146	C₂×Dic₅₃	—	2	2	no	—
147	C₂×Dic₅₃	#117	1	2	yes	—
148	C₂×Dic₅₃	#117	1	2	yes	—
149	C₂×Dic₅₃	#117	1	1	yes	characteristic
150	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
151	C₄×D₅₃	—	8	2	no	—
152	C₄×D₅₃	C₂×C₄²	1	2	yes	—
153	C₄×D₅₃	#117	1	2	yes	—
154	C₄×D₅₃	#117	1	2	yes	—
155	C₄×D₅₃	#117	1	2	yes	—
156	C₄×D₅₃	#117	1	2	yes	—
157	C₄×D₅₃	#117	1	2	yes	—
158	C₄×D₅₃	C₂³×C₄	1	2	yes	—
159	C₄×D₅₃	C₂×#41	1	2	yes	—
160	#88324	—	2	8	no	—
161	C₈×C₅₃	—	2	1	no	—
162	C₈×C₅₃	—	2	1	no	—
163	C₈×C₅₃	C₂×C₄²	1	4	yes	—
164	#88322	—	2	1	no	—
165	#88322	—	2	1	no	—
166	#88322	C₂×C₄²	1	4	yes	—
167	#20370	—	53	1	no	maximal, Sylow
168	C₂²×C₅₃	—	4	1	no	—
169	C₂²×C₅₃	#376	1	1	yes	characteristic
170	C₂²×C₅₃	#376	1	1	yes	characteristic
171	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
172	C₂×D₅₃	—	4	1	no	—
173	C₂×D₅₃	—	4	2	no	—
174	C₂×D₅₃	—	4	2	no	—
175	C₂×D₅₃	—	4	1	no	—
176	C₂×D₅₃	#376	1	2	yes	—
177	C₂×D₅₃	C₂×#117	1	2	yes	—
178	#6157	—	4	8	no	—
179	C₄×C₅₃	—	4	2	no	—
180	C₄×C₅₃	#376	1	1	yes	characteristic
181	C₄×C₅₃	#376	1	1	yes	characteristic
182	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
183	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
184	Dic₅₃	—	4	2	no	—
185	Dic₅₃	#376	1	2	yes	—
186	Dic₅₃	C₂×#117	1	2	yes	—
187	#2873	—	53	2	no	—
188	#2873	—	53	1	no	—
189	#1747	—	53	4	no	—
190	#1747	—	53	4	no	—
191	C₂×#338	—	53	1	no	—
192	C₂×H(3,ℤ/4)	—	53	2	no	—
193	C₂×H(3,ℤ/4)	—	53	1	no	—
194	C₂×C₅₃	—	2	1	no	—
195	C₂×C₅₃	—	2	1	no	—
196	C₂×C₅₃	—	2	1	no	—
197	C₂×C₅₃	C₂×#376	1	1	yes	characteristic
198	D₅₃	—	2	2	no	—
199	D₅₃	—	2	2	no	—
200	C₂²×M₁₆	—	53	1	no	—
201	C₂²×M₁₆	—	53	2	no	—
202	C₂²×M₁₆	—	53	1	no	—
203	C₂×#117	—	53	2	no	—
204	C₂×#117	—	53	1	no	—
205	#405	—	53	8	no	—
206	#405	—	53	8	no	—
207	#405	—	53	1	no	—
208	#405	—	53	1	no	—
209	#405	—	53	1	no	—
210	#405	—	53	2	no	—
211	#405	—	53	2	no	—
212	#405	#6157	1	1	yes	characteristic, complemented
213	C₄×M₁₆	—	53	8	no	—
214	C₄×M₁₆	—	53	2	no	—
215	C₄×M₁₆	—	53	2	no	—
216	#338	—	53	2	no	—
217	#338	—	53	2	no	—
218	H(3,ℤ/4)	—	53	8	no	—
219	H(3,ℤ/4)	—	53	2	no	—
220	H(3,ℤ/4)	—	53	2	no	—
221	C₅₃	#20370	1	1	yes	characteristic, complemented, Sylow
222	C₂³×C₄	—	53	1	no	—
223	C₂×M₁₆	—	53	1	no	—
224	C₂×M₁₆	—	53	4	no	—
225	C₂×M₁₆	—	53	4	no	—
226	C₂×M₁₆	—	53	4	no	—
227	C₂×M₁₆	—	53	4	no	—
228	C₂×M₁₆	—	53	2	no	—
229	C₂×M₁₆	—	53	1	no	—
230	C₂×M₁₆	—	53	1	no	—
231	C₂×M₁₆	—	53	1	no	—
232	C₂×M₁₆	—	53	1	no	—
233	C₂×M₁₆	—	106	2	no	—
234	C₂×M₁₆	—	106	2	no	—
235	C₂×M₁₆	—	106	2	no	—
236	C₂×M₁₆	—	106	2	no	—
237	C₂×M₁₆	—	106	2	no	—
238	C₂×M₁₆	—	106	2	no	—
239	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
240	C₂²×C₈	—	53	2	no	—
241	C₂²×C₈	—	53	4	no	—
242	C₂²×C₈	—	53	1	no	—
243	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
244	#117	—	53	4	no	—
245	#117	—	53	4	no	—
246	#117	—	53	1	no	—
247	#117	—	53	2	no	—
248	#117	—	106	1	no	—
249	#117	—	106	1	no	—
250	#117	—	106	2	no	—
251	#117	—	106	2	no	—
252	#117	C₂×#6157	1	1	yes	characteristic
253	C₂×C₄⋊C₄	—	106	1	no	—
254	C₂×C₄⋊C₄	—	106	2	no	—
255	C₂×C₂²⋊C₄	—	106	1	no	—
256	C₂×C₂²⋊C₄	—	106	2	no	—
257	C₂×C₄²	—	53	4	no	—
258	C₂×C₄²	—	53	2	no	—
259	#97	—	53	16	no	—
260	#97	—	53	2	no	—
261	#97	—	53	2	no	—
262	#97	—	53	2	no	—
263	#97	C₂×#6157	1	2	yes	complemented
264	C₄×C₈	—	53	16	no	—
265	C₄×C₈	—	53	2	no	—
266	C₄×C₈	—	53	2	no	—
267	C₄×C₈	—	53	2	no	—
268	C₄×C₈	C₂×#6157	1	2	yes	complemented
269	C₂⁴	—	53	1	no	—
270	C₂²×C₄	—	53	1	no	—
271	C₂²×C₄	—	53	2	no	—
272	C₂²×C₄	—	53	2	no	—
273	C₂²×C₄	—	53	2	no	—
274	C₂²×C₄	—	53	1	no	—
275	C₂²×C₄	—	53	1	no	—
276	C₂²×C₄	—	53	2	no	—
277	C₂²×C₄	—	53	1	no	—
278	C₂²×C₄	—	53	1	no	—
279	C₂²×C₄	—	106	2	no	—
280	C₂²×C₄	—	106	4	no	—
281	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
282	M₁₆	—	53	2	no	—
283	M₁₆	—	53	2	no	—
284	M₁₆	—	53	8	no	—
285	M₁₆	—	106	2	no	—
286	M₁₆	—	106	2	no	—
287	M₁₆	—	106	2	no	—
288	M₁₆	—	106	2	no	—
289	M₁₆	—	106	4	no	—
290	M₁₆	—	106	4	no	—
291	M₁₆	—	106	4	no	—
292	M₁₆	—	106	4	no	—
293	M₁₆	C₄×#6157	1	2	yes	—
294	M₁₆	C₄×#6157	1	2	yes	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	53	2	no	—
298	C₂×C₈	—	53	8	no	—
299	C₂×C₈	—	53	8	no	—
300	C₂×C₈	—	53	1	no	—
301	C₂×C₈	—	53	1	no	—
302	C₂×C₈	—	106	4	no	—
303	C₂×C₈	—	106	4	no	—
304	C₂×C₈	—	106	1	no	—
305	C₂×C₈	—	106	1	no	—
306	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
307	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
308	C₂×C₈	C₂²×#6157	1	2	yes	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	106	2	no	—
311	C₄⋊C₄	—	106	4	no	—
312	C₄⋊C₄	—	106	4	no	—
313	C₄⋊C₄	—	106	1	no	—
314	C₂²⋊C₄	—	2	1	no	—
315	C₂²⋊C₄	—	106	2	no	—
316	C₂²⋊C₄	—	106	4	no	—
317	C₂²⋊C₄	—	106	4	no	—
318	C₂²⋊C₄	—	106	1	no	—
319	C₄²	—	53	8	no	—
320	C₄²	—	53	2	no	—
321	C₄²	—	106	2	no	—
322	C₄²	—	106	4	no	—
323	C₄²	C₂²×#6157	1	2	yes	—
324	C₂³	—	53	2	no	—
325	C₂³	—	53	2	no	—
326	C₂³	—	53	1	no	—
327	C₂³	—	53	1	no	—
328	C₂³	—	424	1	no	—
329	C₂³	#13133296	1	1	yes	characteristic
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	53	2	no	—
332	C₂×C₄	—	53	2	no	—
333	C₂×C₄	—	53	1	no	—
334	C₂×C₄	—	53	1	no	—
335	C₂×C₄	—	53	2	no	—
336	C₂×C₄	—	53	2	no	—
337	C₂×C₄	—	53	2	no	—
338	C₂×C₄	—	53	2	no	—
339	C₂×C₄	—	53	2	no	—
340	C₂×C₄	—	53	2	no	—
341	C₂×C₄	—	53	2	no	—
342	C₂×C₄	—	53	2	no	—
343	C₂×C₄	—	106	8	no	—
344	C₂×C₄	—	106	2	no	—
345	C₂×C₄	—	424	2	no	—
346	C₂×C₄	—	424	4	no	—
347	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
348	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
349	C₂×C₄	#13133296	1	1	yes	characteristic
350	C₂×C₄	#13133296	1	1	yes	characteristic
351	C₂×C₄	#13133296	1	1	yes	characteristic
352	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	53	4	no	—
356	C₈	—	106	8	no	—
357	C₈	—	106	1	no	—
358	C₈	—	106	1	no	—
359	C₈	C₂×C₄×#6157	1	4	yes	—
360	C₂²	—	4	1	no	—
361	C₂²	—	53	2	no	—
362	C₂²	—	53	2	no	—
363	C₂²	—	212	1	no	—
364	C₂²	—	212	2	no	—
365	C₂²	—	212	1	no	—
366	C₂²	—	212	2	no	—
367	C₂²	#16811559	1	1	yes	characteristic
368	C₂²	#16811559	1	1	yes	characteristic
369	C₂²	C₂×#13133296	1	1	yes	characteristic
370	C₄	—	4	2	no	—
371	C₄	—	53	2	no	—
372	C₄	—	53	2	no	—
373	C₄	—	212	2	no	—
374	C₄	—	212	8	no	—
375	C₄	#16811559	1	1	yes	characteristic
376	C₄	#16811559	1	1	yes	characteristic
377	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
378	C₄	C₂×#13133296	1	1	yes	characteristic
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	2	1	no	—
382	C₂	—	106	2	no	—
383	C₂	—	106	2	no	—
384	C₂	C₂×#16811559	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×#6157
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#405	C₄
socle	C₂×C₅₃	C₂×#376
prime radical	C₂³×D₅₃	#41
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	#405	#6157
hypocenter	C₅₃	#20370
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #20290437	C₂	semidirect
3. #20290436	C₂	semidirect
10. #16811589	C₄	semidirect
12. M₁₆×#6157	C₄	semidirect
21. #16810581	C₄	semidirect
23. #16810579	C₄	semidirect
24. #16810578	C₄	semidirect
26. #16810575	C₄	semidirect
27. C₄×#11719912	C₄	semidirect
28. C₅₃×#405	C₄	semidirect
43. #13133271	C₂×C₄	semidirect
44. C₈×#6157	C₂×C₄	semidirect
65. #13133128	C₂×C₄	semidirect
66. #13133127	C₂×C₄	semidirect
67. #13133127	C₂×C₄	semidirect
68. C₄×#88322	C₂×C₄	semidirect
69. C₈×Dic₅₃	C₂×C₄	semidirect
70. C₈×Dic₅₃	C₂×C₄	semidirect
74. #97×C₅₃	C₂×C₄	semidirect
75. C₄×C₈×C₅₃	C₂×C₄	semidirect
212. #405	#6157	semidirect
221. C₅₃	#20370	semidirect
263. #97	C₂×#6157	semidirect
268. C₄×C₈	C₂×#6157	semidirect

Maximal subgroups

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

subgroup	index
#20291278	2
#20290437	2
#20290436	2
C₂×#16811492	2
C₂×#16811491	2
#20290385	2
#20283131	2
#20370	53