Subgroups — #27403016

Order 13568.

812 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	C₂×#16810754	C₂	1	2	yes	maximal
2	#20275439	C₂	1	4	yes	maximal
3	#20275434	C₂	1	2	yes	maximal
4	#20274832	C₂	1	2	yes	maximal
5	#20274832	C₂	1	1	yes	maximal, characteristic
6	#20274568	C₂	1	2	yes	maximal
7	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
9	#16810754	—	2	8	no	—
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₄⋊#88322	C₂²	1	4	yes	—
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×C₄⋊#88322	C₂²	1	4	yes	—
14	C₂×#13133128	C₂²	1	4	yes	—
15	C₂×C₄×#88322	C₂²	1	2	yes	—
16	#16810378	C₂²	1	1	yes	characteristic
17	#16810378	C₂²	1	1	yes	characteristic
18	#16810378	C₂²	1	2	yes	—
19	#16810378	C₂²	1	4	yes	—
20	#16810378	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
27	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
30	C₄³×C₅₃	C₂²	1	1	yes	characteristic
31	C₄³×C₅₃	C₂²	1	2	yes	—
32	C₂²×#88322	C₂³	1	2	yes	—
33	C₂²×#88322	C₂³	1	4	yes	—
34	C₂²×#88322	C₂³	1	2	yes	—
35	C₄⋊#88322	—	2	8	no	—
36	C₄⋊#88322	—	2	4	no	—
37	C₄⋊#88322	—	2	4	no	—
38	C₄⋊#88322	—	2	8	no	—
39	#13133128	—	2	8	no	—
40	C₄×#88322	—	2	4	no	—
41	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×Q₈×C₅₃	—	2	4	no	—
43	C₄×Q₈×C₅₃	—	2	4	no	—
44	C₄×Q₈×C₅₃	D₄	1	4	yes	—
45	C₄×Q₈×C₅₃	Q₈	1	4	yes	—
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
65	#95×C₅₃	C₂×C₄	1	2	yes	—
66	#95×C₅₃	C₂×C₄	1	2	yes	—
67	#95×C₅₃	C₂×C₄	1	2	yes	—
68	C₂×Q₈×C₅₃	—	2	2	no	—
69	C₂×Q₈×C₅₃	—	2	2	no	—
70	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₅₃	—	2	4	no	—
83	C₄⋊C₄×C₅₃	—	2	4	no	—
84	C₄⋊C₄×C₅₃	—	2	2	no	—
85	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
86	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
87	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
88	C₄²×C₅₃	—	2	4	no	—
89	C₄²×C₅₃	—	2	4	no	—
90	C₄²×C₅₃	—	2	2	no	—
91	C₄²×C₅₃	—	2	2	no	—
92	C₄²×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	—	2	4	no	—
95	C₄²×C₅₃	—	2	2	no	—
96	C₄²×C₅₃	—	4	2	no	—
97	C₄²×C₅₃	—	4	1	no	—
98	C₄²×C₅₃	M₁₆	1	4	yes	—
99	C₄²×C₅₃	C₂×D₄	1	2	yes	—
100	C₄²×C₅₃	C₂×D₄	1	4	yes	—
101	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
102	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
103	C₂×#88322	—	2	2	no	—
104	C₂×#88322	—	2	2	no	—
105	C₂×#88322	—	2	2	no	—
106	C₂×#88322	—	2	4	no	—
107	C₂×#88322	—	2	2	no	—
108	C₂×#88322	—	2	4	no	—
109	C₂×#88322	—	2	2	no	—
110	C₂×#88322	—	2	4	no	—
111	C₂×#88322	#41	1	4	yes	—
112	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	Q₈×C₅₃	—	2	8	no	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	1	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
135	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
137	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
139	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
140	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
141	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
142	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
143	C₂×C₄×C₅₃	#131	1	2	yes	—
144	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
147	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
148	#88322	—	2	4	no	—
149	#88322	—	4	2	no	—
150	#88322	—	4	4	no	—
151	#22736	—	53	1	no	maximal, Sylow
152	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
154	C₂²×C₅₃	#519	1	1	yes	characteristic
155	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
156	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
157	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
158	C₂²×C₅₃	#568	1	1	yes	characteristic
159	C₄×C₅₃	—	2	4	no	—
160	C₄×C₅₃	—	2	2	no	—
161	C₄×C₅₃	—	2	4	no	—
162	C₄×C₅₃	—	2	2	no	—
163	C₄×C₅₃	—	2	4	no	—
164	C₄×C₅₃	—	2	2	no	—
165	C₄×C₅₃	—	2	2	no	—
166	C₄×C₅₃	—	4	2	no	—
167	C₄×C₅₃	—	4	1	no	—
168	C₄×C₅₃	#423	1	4	yes	—
169	C₄×C₅₃	#548	1	2	yes	—
170	C₄×C₅₃	#557	1	2	yes	—
171	C₂×#446	—	53	2	no	—
172	C₄²×Q₈	D₅₃	1	1	yes	characteristic
173	#1940	—	53	4	no	—
174	#1939	—	53	2	no	—
175	#1831	—	53	2	no	—
176	#1831	—	53	1	no	—
177	#1767	—	53	2	no	—
178	C₄×C₄⋊C₈	—	53	1	no	—
179	C₂×C₅₃	#2351	1	1	yes	characteristic
180	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
181	C₂×C₅₃	#2906	1	1	yes	characteristic
182	C₂×C₅₃	#2965	1	1	yes	characteristic
183	C₂×C₅₃	#2970	1	1	yes	characteristic
184	C₂×C₅₃	#2983	1	1	yes	characteristic
185	C₂×C₅₃	#2984	1	1	yes	characteristic
186	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
187	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
188	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
189	#446	—	106	8	no	—
190	C₂×C₄⋊C₈	—	53	2	no	—
191	C₂×C₄⋊C₈	—	53	4	no	—
192	C₂×C₄⋊C₈	—	53	4	no	—
193	C₂×C₄⋊C₈	—	53	2	no	—
194	C₂×#97	—	53	4	no	—
195	C₂×C₄×C₈	—	53	2	no	—
196	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
197	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
198	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
199	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
200	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
201	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
202	C₄³	C₂×D₅₃	1	1	yes	characteristic
203	C₄³	C₂×D₅₃	1	2	yes	—
204	#336	—	53	1	no	—
205	#336	—	53	2	no	—
206	#336	—	53	4	no	—
207	#336	—	53	2	no	—
208	#336	—	53	1	no	—
209	C₅₃	#22736	1	1	yes	characteristic, complemented, Sylow
210	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
211	C₂²×C₈	—	53	2	no	—
212	C₂²×C₈	—	53	2	no	—
213	C₂²×C₈	—	53	4	no	—
214	C₄×Q₈	—	2	4	no	—
215	C₄×Q₈	—	2	4	no	—
216	C₄×Q₈	Dic₁₀₆	1	4	yes	—
217	C₄×Q₈	D₂₁₂	1	4	yes	—
218	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
220	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
221	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
222	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
223	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
224	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
225	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
226	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
227	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
228	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
229	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
230	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
231	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
233	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
234	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
237	C₄⋊C₈	—	106	8	no	—
238	C₄⋊C₈	—	106	4	no	—
239	C₄⋊C₈	—	106	4	no	—
240	C₄⋊C₈	—	106	8	no	—
241	#97	—	106	8	no	—
242	C₄×C₈	—	106	4	no	—
243	#95	C₂×Dic₅₃	1	2	yes	—
244	#95	C₂×Dic₅₃	1	2	yes	—
245	#95	C₂×Dic₅₃	1	2	yes	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
249	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
250	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
253	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
254	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
255	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
256	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
258	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
259	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
260	C₂×C₈	—	53	4	no	—
261	C₂×C₈	—	106	2	no	—
262	C₂×C₈	—	106	2	no	—
263	C₂×C₈	—	106	2	no	—
264	C₂×C₈	—	106	4	no	—
265	C₂×C₈	—	106	2	no	—
266	C₂×C₈	—	106	4	no	—
267	C₂×C₈	—	106	2	no	—
268	C₂×C₈	—	106	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	#11719917	1	4	yes	—
273	C₄⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
274	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
275	C₄²	—	2	4	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	4	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	4	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	4	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	—	4	1	no	—
285	C₄²	#11719917	1	4	yes	—
286	C₄²	C₂×Dic₁₀₆	1	4	yes	—
287	C₄²	C₂×Dic₁₀₆	1	2	yes	—
288	C₄²	C₂×D₂₁₂	1	2	yes	—
289	C₄²	C₂×D₂₁₂	1	4	yes	—
290	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
291	Q₈	—	2	8	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
313	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
314	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
315	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
316	C₂×C₄	C₂×#11719917	1	2	yes	—
317	C₂×C₄	C₂×#11719917	1	4	yes	—
318	C₂×C₄	C₂×#11719917	1	2	yes	—
319	C₂×C₄	C₂×#11719917	1	2	yes	—
320	C₂×C₄	C₂×#11719917	1	2	yes	—
321	C₂×C₄	#13133224	1	2	yes	—
322	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
323	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
324	C₂×C₄	#41×D₅₃	1	2	yes	—
325	C₂×C₄	#13133265	1	2	yes	—
326	C₈	—	106	4	no	—
327	C₈	—	212	2	no	—
328	C₈	—	212	4	no	—
329	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
330	C₂²	#41×Dic₅₃	1	1	yes	characteristic
331	C₂²	#16811101	1	1	yes	characteristic
332	C₂²	C₂²×#11719917	1	1	yes	characteristic
333	C₂²	C₂²×#11719917	1	1	yes	characteristic
334	C₂²	C₂×#13133224	1	1	yes	characteristic
335	C₂²	#16811390	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	4	no	—
338	C₄	—	2	4	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	4	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	4	2	no	—
344	C₄	—	4	1	no	—
345	C₄	#16810647	1	4	yes	—
346	C₄	#16811285	1	2	yes	—
347	C₄	#16811349	1	2	yes	—
348	C₂	#20277636	1	1	yes	characteristic
349	C₂	C₂×#16810647	1	1	yes	characteristic
350	C₂	#20283283	1	1	yes	characteristic
351	C₂	#20283682	1	1	yes	characteristic
352	C₂	#20283729	1	1	yes	characteristic
353	C₂	#20283800	1	1	yes	characteristic
354	C₂	#20283804	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂³×C₄
Fitting	C₄²×Q₈×C₅₃	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄²×Q₈	D₅₃
hypocenter	C₅₃	#22736
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
209. C₅₃	#22736	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810754	2
#20275439	2
#20275434	2
#20274832	2
#20274568	2
C₄×C₄⋊#88322	2
C₄²×Q₈×C₅₃	2
#22736	53