Subgroups — #27355835

Order 13568.

718 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283750	C₂	1	2	yes	maximal, complemented
2	#20273629	C₂	1	2	yes	maximal, complemented
3	#20273545	C₂	1	4	yes	maximal, complemented
4	#20273539	C₂	1	2	yes	maximal, complemented
5	#20273478	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273158	C₂	1	2	yes	maximal, complemented
7	#20273004	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
9	#16810701	C₂²	1	4	yes	complemented
10	#16810695	—	2	4	no	—
11	D₄×#88322	C₂²	1	4	yes	complemented
12	#16810680	C₂²	1	2	yes	—
13	#16810676	C₂²	1	1	yes	characteristic
14	#16810587	—	2	2	no	—
15	C₂×C₂²⋊#88322	—	2	2	no	—
16	C₄×#11719917	C₂²	1	1	yes	characteristic
17	#16810376	C₂²	1	2	yes	complemented
18	#16810374	C₂²	1	2	yes	complemented
19	D₄⋊#88322	C₂²	1	4	yes	complemented
20	D₄⋊#88322	C₂²	1	4	yes	complemented
21	#16810345	C₂²	1	2	yes	complemented
22	#16810344	C₂²	1	2	yes	complemented
23	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic
24	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
25	C₄×D₈×C₅₃	C₄	1	2	yes	—
26	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic
27	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic
28	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₄×D₈×C₅₃	C₂²	1	2	yes	—
30	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
32	C₂×#102×C₅₃	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#11719917	—	2	4	no	—
35	C₂×#11719917	—	2	1	no	—
36	C₂²×#88322	—	2	4	no	—
37	C₄⋊#88322	D₄	1	2	yes	—
38	C₄⋊#88322	C₂³	1	4	yes	—
39	C₂²⋊#88322	—	2	4	no	—
40	C₂²⋊#88322	—	2	4	no	—
41	C₂²⋊#88322	—	2	2	no	—
42	C₂²⋊#88322	D₄	1	2	yes	—
43	#13133128	—	2	2	no	—
44	#13133128	—	2	1	no	—
45	C₄×#88322	C₂³	1	2	yes	—
46	C₄×#88322	C₂³	1	2	yes	—
47	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₂³×C₄×C₅₃	—	2	2	no	—
49	C₂×D₈×C₅₃	—	2	4	no	—
50	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄×D₄×C₅₃	—	2	2	no	—
56	C₄×D₄×C₅₃	—	2	2	no	—
57	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
58	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₄×D₄×C₅₃	C₂³	1	2	yes	—
60	C₄×D₄×C₅₃	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	#107×C₅₃	C₂×C₄	1	2	yes	—
65	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
66	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
67	#102×C₅₃	C₂×C₄	1	2	yes	—
68	#102×C₅₃	C₂×C₄	1	2	yes	—
69	#102×C₅₃	C₂×C₄	1	2	yes	—
70	#102×C₅₃	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂⁴×C₅₃	—	2	2	no	—
76	C₂×D₄×C₅₃	—	2	4	no	—
77	C₂×D₄×C₅₃	—	2	2	no	—
78	C₂×D₄×C₅₃	—	4	2	no	—
79	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₃	—	2	2	no	—
82	C₂²×C₄×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	—	2	2	no	—
84	C₂²×C₄×C₅₃	—	2	2	no	—
85	C₂²×C₄×C₅₃	—	2	2	no	—
86	C₂²×C₄×C₅₃	—	8	4	no	—
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
90	D₈×C₅₃	—	2	4	no	—
91	D₈×C₅₃	—	2	2	no	—
92	D₈×C₅₃	M₁₆	1	4	yes	complemented
93	C₂×C₈×C₅₃	—	2	2	no	—
94	C₂×C₈×C₅₃	—	4	1	no	—
95	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
101	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
102	C₂²⋊C₄×C₅₃	—	2	2	no	—
103	C₂²⋊C₄×C₅₃	—	2	2	no	—
104	C₂²⋊C₄×C₅₃	—	2	2	no	—
105	C₂²⋊C₄×C₅₃	—	2	2	no	—
106	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅₃	#41	1	1	yes	characteristic
109	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
110	#11719917	—	4	2	no	—
111	#11719917	—	8	4	no	—
112	C₂×#88322	—	2	4	no	—
113	C₂×#88322	—	2	2	no	—
114	C₂×#88322	—	8	4	no	—
115	C₂×#88322	C₂×D₄	1	4	yes	—
116	C₂³×C₅₃	—	2	2	no	—
117	C₂³×C₅₃	—	2	2	no	—
118	C₂³×C₅₃	—	4	4	no	—
119	C₂³×C₅₃	—	4	4	no	—
120	C₂³×C₅₃	—	8	2	no	—
121	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
122	D₄×C₅₃	—	2	2	no	—
123	D₄×C₅₃	—	2	2	no	—
124	D₄×C₅₃	—	2	2	no	—
125	D₄×C₅₃	—	4	4	no	—
126	D₄×C₅₃	C₂×M₁₆	1	4	yes	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	4	4	no	—
134	C₂×C₄×C₅₃	—	4	4	no	—
135	C₂×C₄×C₅₃	—	8	2	no	—
136	C₂×C₄×C₅₃	—	8	2	no	—
137	C₂×C₄×C₅₃	—	8	2	no	—
138	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	—	2	1	no	—
146	C₈×C₅₃	—	2	1	no	—
147	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
148	#88322	—	2	4	no	—
149	#88322	—	4	2	no	—
150	#88322	—	4	4	no	—
151	#18064	—	53	1	no	maximal, Sylow
152	C₂²×C₅₃	—	2	1	no	—
153	C₂²×C₅₃	—	2	1	no	—
154	C₂²×C₅₃	—	2	4	no	—
155	C₂²×C₅₃	—	4	2	no	—
156	C₂²×C₅₃	—	4	2	no	—
157	C₂²×C₅₃	—	4	2	no	—
158	C₂²×C₅₃	—	4	2	no	—
159	C₂²×C₅₃	—	4	2	no	—
160	C₂²×C₅₃	—	8	8	no	—
161	C₂²×C₅₃	#444	1	2	yes	—
162	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
163	C₄×C₅₃	—	2	1	no	—
164	C₄×C₅₃	—	2	1	no	—
165	C₄×C₅₃	—	2	1	no	—
166	C₄×C₅₃	—	2	1	no	—
167	C₄×C₅₃	—	4	2	no	—
168	C₄×C₅₃	—	4	2	no	—
169	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
170	C₄×C₅₃	#568	1	1	yes	characteristic
171	C₄×C₅₃	#576	1	2	yes	—
172	#2974	—	53	2	no	—
173	C₂×C₄×D₈	D₅₃	1	1	yes	characteristic
174	#1594	—	53	2	no	—
175	#1581	—	53	4	no	—
176	#1580	—	53	2	no	—
177	#1568	—	53	1	no	—
178	#1490	—	53	2	no	—
179	#1451	—	53	1	no	—
180	C₂×C₅₃	—	2	2	no	—
181	C₂×C₅₃	—	4	4	no	—
182	C₂×C₅₃	—	8	2	no	—
183	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
184	C₂×C₅₃	#2948	1	1	yes	characteristic
185	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
186	C₂²×D₈	Dic₅₃	1	1	yes	characteristic
187	C₂×C₄×D₄	C₂×D₅₃	1	2	yes	—
188	C₄×D₈	Dic₅₃	1	2	yes	—
189	C₄×D₈	Dic₅₃	1	1	yes	characteristic
190	C₄×D₈	Dic₅₃	1	1	yes	characteristic
191	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
192	C₄×D₈	C₂×D₅₃	1	2	yes	—
193	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
194	#436	—	53	4	no	—
195	#435	—	106	4	no	—
196	C₈×D₄	—	53	4	no	—
197	#432	—	53	2	no	—
198	#431	—	53	1	no	—
199	C₂×#107	Dic₅₃	1	1	yes	characteristic
200	C₂×#102	Dic₅₃	1	2	yes	—
201	#407	—	106	2	no	—
202	C₂×C₂²⋊C₈	—	106	2	no	—
203	C₄×M₁₆	—	53	1	no	—
204	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
205	#335	—	53	2	no	—
206	#334	—	53	2	no	—
207	D₄⋊C₈	—	53	4	no	—
208	D₄⋊C₈	—	53	4	no	—
209	#322	—	53	2	no	—
210	#322	—	53	2	no	—
211	C₅₃	#18064	1	1	yes	characteristic, complemented, Sylow
212	C₂²×D₄	C₂×Dic₅₃	1	2	yes	—
213	C₂³×C₄	—	2	2	no	—
214	C₂×D₈	—	2	4	no	—
215	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
216	C₂×D₈	C₂×Dic₅₃	1	1	yes	characteristic
217	C₂×D₈	C₂×Dic₅₃	1	1	yes	characteristic
218	C₂×M₁₆	—	106	4	no	—
219	C₂×M₁₆	—	106	1	no	—
220	C₂²×C₈	—	106	4	no	—
221	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
222	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
223	C₄×D₄	—	2	2	no	—
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
226	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
227	C₄×D₄	C₂²×D₅₃	1	2	yes	—
228	C₄×D₄	C₂²×D₅₃	1	2	yes	—
229	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
232	#107	C₂×Dic₅₃	1	2	yes	—
233	#107	C₂×Dic₅₃	1	1	yes	characteristic
234	#107	C₂×Dic₅₃	1	1	yes	characteristic
235	C₄⋊C₈	—	53	2	no	—
236	C₄⋊C₈	—	53	4	no	—
237	#102	C₂×Dic₅₃	1	2	yes	—
238	#102	C₂×Dic₅₃	1	2	yes	—
239	#102	C₂×Dic₅₃	1	2	yes	—
240	#102	C₂×Dic₅₃	1	2	yes	—
241	C₂²⋊C₈	—	53	2	no	—
242	C₂²⋊C₈	—	106	4	no	—
243	C₂²⋊C₈	—	106	4	no	—
244	C₂²⋊C₈	—	106	2	no	—
245	#97	—	106	2	no	—
246	#97	—	106	1	no	—
247	C₄×C₈	—	53	2	no	—
248	C₄×C₈	—	53	2	no	—
249	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
250	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
252	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
253	C₂⁴	—	2	2	no	—
254	C₂×D₄	—	2	4	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
258	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	8	4	no	—
265	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
266	C₂²×C₄	#11719947	1	1	yes	characteristic
267	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
268	D₈	—	2	4	no	—
269	D₈	—	2	2	no	—
270	D₈	#11719917	1	4	yes	complemented
271	M₁₆	—	424	4	no	—
272	M₁₆	—	212	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	—	53	4	no	—
276	C₂×C₈	—	106	4	no	—
277	C₂×C₈	—	106	2	no	—
278	C₂×C₈	—	424	4	no	—
279	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
280	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
281	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
282	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
285	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₂²⋊C₄	—	2	2	no	—
288	C₂²⋊C₄	—	2	2	no	—
289	C₂²⋊C₄	—	2	2	no	—
290	C₄²	D₄×D₅₃	1	1	yes	characteristic
291	C₄²	#11719947	1	1	yes	characteristic
292	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
293	C₄²	C₂³×D₅₃	1	1	yes	characteristic
294	C₂³	—	2	2	no	—
295	C₂³	—	2	2	no	—
296	C₂³	—	4	4	no	—
297	C₂³	—	4	4	no	—
298	C₂³	—	8	2	no	—
299	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
300	D₄	—	2	2	no	—
301	D₄	—	2	2	no	—
302	D₄	—	2	2	no	—
303	D₄	—	4	4	no	—
304	D₄	C₂×#11719917	1	4	yes	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
317	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
318	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
319	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
320	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
321	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	106	4	no	—
326	C₈	—	212	2	no	—
327	C₈	—	212	4	no	—
328	C₈	C₂×#11719917	1	2	yes	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	4	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	2	no	—
335	C₂²	—	4	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	8	8	no	—
338	C₂²	#16810745	1	2	yes	—
339	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	2	no	—
346	C₄	C₂²×#11719917	1	1	yes	characteristic
347	C₄	#16811390	1	1	yes	characteristic
348	C₄	#16811424	1	2	yes	—
349	C₂	—	2	2	no	—
350	C₂	—	4	4	no	—
351	C₂	—	8	2	no	—
352	C₂	D₄×#11719917	1	1	yes	characteristic
353	C₂	#20283567	1	1	yes	characteristic
354	C₂	C₂×#16810745	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₄×D₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₈×C₅₃	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₄×D₈	D₅₃
hypocenter	C₅₃	#18064
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #20283750	C₂	semidirect
2. #20273629	C₂	semidirect
3. #20273545	C₂	semidirect
4. #20273539	C₂	semidirect
5. #20273478	C₂	semidirect
6. #20273158	C₂	semidirect
7. #20273004	C₂	semidirect
9. #16810701	C₂²	semidirect
11. D₄×#88322	C₂²	semidirect
17. #16810376	C₂²	semidirect
18. #16810374	C₂²	semidirect
19. D₄⋊#88322	C₂²	semidirect
20. D₄⋊#88322	C₂²	semidirect
21. #16810345	C₂²	semidirect
22. #16810344	C₂²	semidirect
92. D₈×C₅₃	M₁₆	semidirect
211. C₅₃	#18064	semidirect
270. D₈	#11719917	semidirect

Maximal subgroups

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

subgroup	index
#20283750	2
#20273629	2
#20273545	2
#20273539	2
#20273478	2
#20273158	2
#20273004	2
C₂×C₄×D₈×C₅₃	2
#18064	53