Subgroups — #27388779

Order 13568.

814 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275652	C₂	1	2	yes	maximal
2	#20275434	C₂	1	2	yes	maximal
3	#20275274	C₂	1	4	yes	maximal
4	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊#88322	C₂	1	2	yes	maximal
6	#20274461	C₂	1	2	yes	maximal
7	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊#88322	C₂²	1	4	yes	—
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₄×#88322	C₂²	1	2	yes	—
12	C₂×C₄×#88322	C₂²	1	2	yes	—
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	C₂×C₄×#88322	C₂²	1	2	yes	—
15	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
16	#16810378	C₂²	1	4	yes	—
17	#16810378	C₂²	1	4	yes	—
18	#16810378	C₂²	1	2	yes	—
19	#16810378	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
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	1	yes	characteristic
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₄³×C₅₃	C₂²	1	2	yes	—
30	C₄³×C₅₃	C₂²	1	1	yes	characteristic
31	C₂²×#88322	C₂³	1	4	yes	—
32	C₂²×#88322	C₂³	1	2	yes	—
33	C₂²×#88322	C₂³	1	2	yes	—
34	C₄⋊#88322	—	2	4	no	—
35	C₄⋊#88322	—	2	8	no	—
36	C₄×#88322	—	2	4	no	—
37	C₄×#88322	—	2	2	no	—
38	C₄×#88322	—	2	4	no	—
39	C₄×#88322	—	2	4	no	—
40	C₄×#88322	Q₈	1	4	yes	complemented
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	C₈	1	8	yes	complemented
45	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
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×C₄²×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
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	#95×C₅₃	C₂×C₄	1	2	yes	—
65	#95×C₅₃	C₂×C₄	1	2	yes	—
66	#95×C₅₃	C₂×C₄	1	2	yes	—
67	C₂×Q₈×C₅₃	—	2	2	no	—
68	C₂×Q₈×C₅₃	—	2	2	no	—
69	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	2	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	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₅₃	—	2	4	no	—
82	C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₄⋊C₄×C₅₃	—	2	4	no	—
84	C₄⋊C₄×C₅₃	—	2	2	no	—
85	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
86	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
87	C₄²×C₅₃	—	2	4	no	—
88	C₄²×C₅₃	—	2	2	no	—
89	C₄²×C₅₃	—	2	2	no	—
90	C₄²×C₅₃	—	2	4	no	—
91	C₄²×C₅₃	—	2	4	no	—
92	C₄²×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	—	2	4	no	—
95	C₄²×C₅₃	C₂×C₈	1	8	yes	—
96	C₄²×C₅₃	C₂×C₈	1	4	yes	—
97	C₄²×C₅₃	M₁₆	1	4	yes	—
98	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
99	C₄²×C₅₃	#41	1	4	yes	—
100	C₄²×C₅₃	#41	1	2	yes	—
101	C₂×#88322	—	2	2	no	—
102	C₂×#88322	—	2	2	no	—
103	C₂×#88322	—	2	2	no	—
104	C₂×#88322	—	2	2	no	—
105	C₂×#88322	—	2	4	no	—
106	C₂×#88322	—	2	4	no	—
107	C₂×#88322	—	2	4	no	—
108	C₂×#88322	C₂×Q₈	1	4	yes	—
109	C₂×#88322	#41	1	4	yes	—
110	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
111	Q₈×C₅₃	—	2	8	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	#117	1	2	yes	—
129	C₂×C₄×C₅₃	#117	1	2	yes	—
130	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
133	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
135	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
138	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
139	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
140	C₂×C₄×C₅₃	#131	1	2	yes	—
141	C₂×C₄×C₅₃	#131	1	2	yes	—
142	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
144	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
145	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
146	#88322	—	2	4	no	—
147	#88322	—	2	4	no	—
148	#88322	—	4	4	no	—
149	#21339	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
151	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
153	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
154	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
155	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
156	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
157	C₄×C₅₃	—	2	4	no	—
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	2	4	no	—
161	C₄×C₅₃	—	2	2	no	—
162	C₄×C₅₃	—	4	2	no	—
163	C₄×C₅₃	—	4	2	no	—
164	C₄×C₅₃	#431	1	4	yes	—
165	C₄×C₅₃	C₈×Q₈	1	4	yes	—
166	C₄×C₅₃	#446	1	4	yes	—
167	C₄×C₅₃	#529	1	2	yes	—
168	C₄×C₅₃	#533	1	2	yes	—
169	C₄²×Q₈	D₅₃	1	1	yes	characteristic
170	#1984	—	53	2	no	—
171	#1939	—	53	2	no	—
172	#1916	—	53	4	no	—
173	C₄×C₄⋊C₈	—	53	1	no	—
174	C₄×C₄⋊C₈	—	53	2	no	—
175	#1744	—	53	2	no	—
176	C₄²×C₈	—	53	1	no	—
177	C₂×C₅₃	#2295	1	1	yes	characteristic
178	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
179	C₂×C₅₃	#2922	1	1	yes	characteristic
180	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
181	C₂×C₅₃	C₂×#446	1	1	yes	characteristic
182	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
183	C₂×C₅₃	#2965	1	1	yes	characteristic
184	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
185	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
186	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
187	C₂×C₄⋊C₈	—	53	4	no	—
188	C₂×C₄⋊C₈	—	53	2	no	—
189	C₂×C₄×C₈	—	53	1	no	—
190	C₂×C₄×C₈	—	53	2	no	—
191	C₂×C₄×C₈	—	53	2	no	—
192	C₂×C₄×C₈	—	53	1	no	—
193	C₂×C₄×C₈	—	53	2	no	—
194	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
195	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
196	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
197	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
198	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
199	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
200	C₄³	C₂×D₅₃	1	1	yes	characteristic
201	C₄³	C₂×D₅₃	1	2	yes	—
202	#336	—	53	2	no	—
203	#336	—	53	4	no	—
204	#336	—	53	4	no	—
205	#336	—	53	4	no	—
206	C₅₃	#21339	1	1	yes	characteristic, complemented, Sylow
207	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
208	C₂²×C₈	—	53	2	no	—
209	C₂²×C₈	—	53	2	no	—
210	C₂²×C₈	—	53	4	no	—
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	—	2	4	no	—
213	C₄×Q₈	#88322	1	8	yes	complemented
214	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
216	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
217	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
218	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
219	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
220	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
221	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
222	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
223	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
224	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
225	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
226	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
227	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
229	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
231	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
233	C₄⋊C₈	—	106	8	no	—
234	C₄⋊C₈	—	106	4	no	—
235	C₄×C₈	—	53	4	no	—
236	C₄×C₈	—	106	4	no	—
237	C₄×C₈	—	106	4	no	—
238	C₄×C₈	—	106	2	no	—
239	C₄×C₈	—	106	4	no	—
240	#95	C₂×Dic₅₃	1	2	yes	—
241	#95	C₂×Dic₅₃	1	2	yes	—
242	#95	C₂×Dic₅₃	1	2	yes	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	C₂×#88322	1	4	yes	—
246	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
247	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
248	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
251	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
252	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
253	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
255	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
257	C₂×C₈	—	53	4	no	—
258	C₂×C₈	—	53	4	no	—
259	C₂×C₈	—	106	2	no	—
260	C₂×C₈	—	106	2	no	—
261	C₂×C₈	—	106	2	no	—
262	C₂×C₈	—	106	2	no	—
263	C₂×C₈	—	106	4	no	—
264	C₂×C₈	—	106	4	no	—
265	C₂×C₈	—	106	4	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	C₂×#88322	1	8	yes	—
271	C₄⋊C₄	C₂×#88322	1	4	yes	—
272	C₄²	—	2	4	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	4	no	—
276	C₄²	—	2	4	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	4	no	—
279	C₄²	—	2	4	no	—
280	C₄²	C₂×#88322	1	8	yes	—
281	C₄²	C₂×#88322	1	4	yes	—
282	C₄²	#11719917	1	4	yes	—
283	C₄²	#11719945	1	4	yes	—
284	C₄²	Q₈×D₅₃	1	2	yes	—
285	C₄²	#11719949	1	2	yes	—
286	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
287	Q₈	—	2	8	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	#13133176	1	2	yes	—
305	C₂×C₄	#13133176	1	2	yes	—
306	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
307	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
308	C₂×C₄	C₂²×#88322	1	4	yes	—
309	C₂×C₄	C₂²×#88322	1	2	yes	—
310	C₂×C₄	C₂²×#88322	1	4	yes	—
311	C₂×C₄	C₂²×#88322	1	2	yes	—
312	C₂×C₄	C₂²×#88322	1	2	yes	—
313	C₂×C₄	C₂×#11719917	1	2	yes	—
314	C₂×C₄	C₂×#11719917	1	2	yes	—
315	C₂×C₄	C₂×#11719917	1	2	yes	—
316	C₂×C₄	#13133224	1	2	yes	—
317	C₂×C₄	#13133224	1	2	yes	—
318	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
319	C₂×C₄	#41×D₅₃	1	2	yes	—
320	C₂×C₄	C₂×#11719945	1	2	yes	—
321	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
322	C₈	—	106	4	no	—
323	C₈	—	106	4	no	—
324	C₈	—	212	4	no	—
325	C₂²	C₂×#13133176	1	1	yes	characteristic
326	C₂²	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
327	C₂²	#41×Dic₅₃	1	1	yes	characteristic
328	C₂²	C₂³×#88322	1	1	yes	characteristic
329	C₂²	C₂²×#11719917	1	1	yes	characteristic
330	C₂²	C₂×#13133224	1	1	yes	characteristic
331	C₂²	C₂×#13133224	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	4	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	4	no	—
336	C₄	—	2	2	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	2	no	—
339	C₄	#16810676	1	4	yes	—
340	C₄	Q₈×#88322	1	4	yes	—
341	C₄	#16810754	1	4	yes	—
342	C₄	#16811160	1	2	yes	—
343	C₄	#16811177	1	2	yes	—
344	C₂	#20277193	1	1	yes	characteristic
345	C₂	C₂×#16810676	1	1	yes	characteristic
346	C₂	#20283359	1	1	yes	characteristic
347	C₂	C₂×Q₈×#88322	1	1	yes	characteristic
348	C₂	C₂×#16810754	1	1	yes	characteristic
349	C₂	#41×#88322	1	1	yes	characteristic
350	C₂	#20283682	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄²×Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄²×Q₈	D₅₃
hypocenter	C₅₃	#21339
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
40. C₄×#88322	Q₈	semidirect
44. C₄×Q₈×C₅₃	C₈	semidirect
206. C₅₃	#21339	semidirect
213. C₄×Q₈	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20275652	2
#20275434	2
#20275274	2
C₄×C₄⋊#88322	2
#20274461	2
C₄²×#88322	2
C₄²×Q₈×C₅₃	2
#21339	53