Subgroups — #27537812

Order 13568.

502 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276624	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276547	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276541	C₂	1	2	yes	maximal, complemented
5	D₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×#11719908	C₂	1	2	yes	maximal, complemented
7	#20276522	C₂	1	2	yes	maximal, complemented
8	#20276514	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276474	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2966	C₂	1	1	yes	maximal, characteristic
12	C₂×#13133157	C₂²	1	2	yes	—
13	C₂²×#11719908	C₂²	1	2	yes	—
14	C₂²×#11719908	C₂²	1	2	yes	—
15	C₄⋊#11719908	C₂²	1	1	yes	characteristic
16	C₄⋊#11719908	C₂²	1	1	yes	characteristic
17	C₄⋊#11719908	C₂²	1	2	yes	—
18	C₄⋊#11719908	C₂²	1	1	yes	characteristic
19	C₄⋊#11719908	C₂²	1	2	yes	—
20	C₄⋊#11719908	C₂²	1	2	yes	—
21	C₄⋊#11719908	C₂²	1	1	yes	characteristic
22	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
23	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
24	C₂²⋊#11719908	C₂²	1	2	yes	—
25	C₂²⋊#11719908	C₂²	1	2	yes	—
26	#16810399	C₂²	1	2	yes	—
27	C₄×#11719908	C₂²	1	1	yes	characteristic
28	C₄×#11719908	C₂²	1	1	yes	characteristic
29	C₄×#11719908	C₂²	1	2	yes	—
30	C₅₃×#529	C₄	1	1	yes	characteristic
31	C₅₃×#446	C₄	1	1	yes	characteristic
32	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#436	C₄	1	1	yes	characteristic
34	C₅₃×#435	C₄	1	2	yes	—
35	C₈×D₄×C₅₃	C₂²	1	2	yes	—
36	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#433	C₄	1	2	yes	—
38	C₅₃×#432	C₄	1	1	yes	characteristic
39	C₅₃×#405	C₂²	1	2	yes	—
40	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
41	#13133157	—	4	2	no	—
42	C₂×#11719908	—	4	2	no	—
43	C₂×#11719908	—	4	2	no	—
44	C₂×#11719908	C₂³	1	1	yes	characteristic
45	C₂×#11719908	C₂³	1	2	yes	—
46	C₂×#11719908	C₂³	1	2	yes	—
47	C₂×#11719908	C₂³	1	1	yes	characteristic
48	C₂×#11719908	C₂³	1	1	yes	characteristic
49	C₂×#11719908	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
54	#126×C₅₃	C₈	1	2	yes	—
55	#125×C₅₃	C₈	1	1	yes	characteristic
56	#124×C₅₃	C₈	1	1	yes	characteristic
57	#123×C₅₃	C₈	1	2	yes	—
58	C₂²⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
59	#121×C₅₃	C₈	1	1	yes	characteristic
60	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
62	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
63	#117×C₅₃	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
68	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
69	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
70	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
71	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
72	#97×C₅₃	C₂×C₄	1	2	yes	—
73	#97×C₅₃	C₂×C₄	1	2	yes	—
74	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₅₃	C₂³	1	2	yes	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
80	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
81	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
86	M₁₆×C₅₃	—	4	2	no	—
87	C₂×C₈×C₅₃	—	2	1	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	1	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	4	1	no	—
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
99	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
100	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
101	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
102	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
103	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
104	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
105	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
106	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
107	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
108	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
109	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
110	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
111	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
114	#11719908	—	2	1	no	—
115	#11719908	—	2	2	no	—
116	#11719908	—	2	2	no	—
117	#11719908	—	2	1	no	—
118	#11719908	—	2	1	no	—
119	#11719908	—	2	1	no	—
120	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
121	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
122	Q₈×C₅₃	—	2	2	no	—
123	D₄×C₅₃	—	2	4	no	—
124	D₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	1	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	—	2	2	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	—	2	2	no	—
145	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
146	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
147	#33508	—	53	1	no	maximal, Sylow
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	2	no	—
151	C₂²×C₅₃	—	2	2	no	—
152	C₂²×C₅₃	—	2	2	no	—
153	C₂²×C₅₃	#504	1	2	yes	—
154	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	2	2	no	—
162	C₄×C₅₃	#504	1	2	yes	—
163	C₄×C₅₃	#520	1	1	yes	characteristic
164	C₄×C₅₃	#568	1	1	yes	characteristic
165	#2966	D₅₃	1	1	yes	characteristic
166	#2183	—	53	1	no	—
167	Q₈×C₁₆	—	53	1	no	—
168	#2169	—	53	1	no	—
169	#2168	—	53	2	no	—
170	D₄×C₁₆	—	53	1	no	—
171	D₄×C₁₆	—	53	2	no	—
172	#2164	—	53	2	no	—
173	#2162	—	53	1	no	—
174	#2151	—	53	2	no	—
175	C₂×C₄⋊C₁₆	—	53	1	no	—
176	C₂×C₅₃	—	2	2	no	—
177	C₂×C₅₃	—	4	2	no	—
178	C₂×C₅₃	#2988	1	1	yes	characteristic
179	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
180	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
181	#529	Dic₅₃	1	1	yes	characteristic
182	C₂×M₃₂	—	53	2	no	—
183	C₂²×C₁₆	—	53	2	no	—
184	C₂²×C₁₆	—	53	2	no	—
185	#446	Dic₅₃	1	1	yes	characteristic
186	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
187	#436	Dic₅₃	1	1	yes	characteristic
188	#435	Dic₅₃	1	2	yes	—
189	C₈×D₄	C₂×D₅₃	1	2	yes	—
190	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
191	#433	Dic₅₃	1	2	yes	—
192	#432	Dic₅₃	1	1	yes	characteristic
193	#405	C₂×D₅₃	1	2	yes	—
194	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
195	C₄⋊C₁₆	—	53	2	no	—
196	C₄⋊C₁₆	—	53	1	no	—
197	C₄⋊C₁₆	—	53	1	no	—
198	C₄⋊C₁₆	—	53	1	no	—
199	C₄⋊C₁₆	—	53	2	no	—
200	C₄⋊C₁₆	—	53	2	no	—
201	C₄⋊C₁₆	—	53	1	no	—
202	C₂²⋊C₁₆	—	53	1	no	—
203	C₂²⋊C₁₆	—	53	1	no	—
204	C₂²⋊C₁₆	—	53	2	no	—
205	C₂²⋊C₁₆	—	53	2	no	—
206	#346	—	53	2	no	—
207	C₄×C₁₆	—	53	1	no	—
208	C₄×C₁₆	—	53	1	no	—
209	C₄×C₁₆	—	53	2	no	—
210	C₅₃	#33508	1	1	yes	characteristic, complemented, Sylow
211	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
212	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
215	#126	#88322	1	2	yes	—
216	#125	#88322	1	1	yes	characteristic
217	#124	#88322	1	1	yes	characteristic
218	#123	#88322	1	2	yes	—
219	C₂²⋊Q₈	#88322	1	1	yes	characteristic
220	#121	#88322	1	1	yes	characteristic
221	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
222	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
223	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
224	#117	C₂×Dic₅₃	1	2	yes	—
225	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
226	M₃₂	—	212	2	no	—
227	C₂×C₁₆	—	53	1	no	—
228	C₂×C₁₆	—	53	1	no	—
229	C₂×C₁₆	—	53	1	no	—
230	C₂×C₁₆	—	53	2	no	—
231	C₂×C₁₆	—	53	1	no	—
232	C₂×C₁₆	—	53	2	no	—
233	C₂×C₁₆	—	212	2	no	—
234	C₂×C₁₆	—	212	2	no	—
235	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
236	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
237	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
238	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
239	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
240	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
241	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
242	#97	C₂×Dic₅₃	1	2	yes	—
243	#97	C₂×Dic₅₃	1	2	yes	—
244	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
245	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₅₃	1	2	yes	—
248	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
249	C₂×Q₈	C₂×#88322	1	1	yes	characteristic
250	C₂×D₄	C₂×#88322	1	2	yes	—
251	C₂×D₄	C₂×#88322	1	1	yes	characteristic
252	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
253	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
255	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
256	M₁₆	—	4	2	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
263	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
264	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
265	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
266	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
267	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
268	C₄⋊C₄	C₂×#88322	1	2	yes	—
269	C₄⋊C₄	C₂×#88322	1	2	yes	—
270	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
271	C₄⋊C₄	C₂×#88322	1	2	yes	—
272	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
273	C₄⋊C₄	C₂×#88322	1	2	yes	—
274	C₂²⋊C₄	C₂×#88322	1	2	yes	—
275	C₂²⋊C₄	C₂×#88322	1	2	yes	—
276	C₂²⋊C₄	C₂×#88322	1	2	yes	—
277	C₂²⋊C₄	C₂×#88322	1	2	yes	—
278	C₂²⋊C₄	C₂×#88322	1	2	yes	—
279	C₄²	C₂×#88322	1	1	yes	characteristic
280	C₄²	C₂×#88322	1	1	yes	characteristic
281	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
282	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
283	C₄²	C₂²×Dic₅₃	1	2	yes	—
284	C₁₆	—	106	1	no	—
285	C₁₆	—	106	2	no	—
286	C₁₆	—	106	2	no	—
287	C₁₆	—	106	1	no	—
288	C₁₆	—	106	1	no	—
289	C₁₆	—	106	1	no	—
290	C₂³	C₂²×#88322	1	2	yes	—
291	C₂³	C₂²×#88322	1	1	yes	characteristic
292	Q₈	—	2	2	no	—
293	D₄	—	2	4	no	—
294	D₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
302	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
303	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
304	C₂×C₄	C₂²×#88322	1	2	yes	—
305	C₂×C₄	C₂²×#88322	1	2	yes	—
306	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
307	C₂×C₄	C₂²×#88322	1	2	yes	—
308	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
309	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	#13133263	1	1	yes	characteristic
316	C₈	#13133264	1	1	yes	characteristic
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	#16811040	1	2	yes	—
323	C₂²	C₂³×#88322	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	#16811040	1	2	yes	—
332	C₄	#16811106	1	1	yes	characteristic
333	C₄	#16811390	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	4	2	no	—
336	C₂	#20283826	1	1	yes	characteristic
337	C₂	C₂×#16811040	1	1	yes	characteristic
338	C₂	C₂×#16811040	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2966	C₂
prime radical	#121×C₅₃	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2966	D₅₃
hypocenter	C₅₃	#33508
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2966

Complements

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

N	H ≅ G/N	type
1. #20276624	C₂	semidirect
2. Q₈×#11719908	C₂	semidirect
3. #20276547	C₂	semidirect
4. #20276541	C₂	semidirect
5. D₄×#11719908	C₂	semidirect
6. D₄×#11719908	C₂	semidirect
7. #20276522	C₂	semidirect
8. #20276514	C₂	semidirect
9. #20276474	C₂	semidirect
10. C₂×C₄⋊#11719908	C₂	semidirect
210. C₅₃	#33508	semidirect

Maximal subgroups

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

subgroup	index
#20276624	2
Q₈×#11719908	2
#20276547	2
#20276541	2
D₄×#11719908	2
#20276522	2
#20276514	2
#20276474	2
C₂×C₄⋊#11719908	2
C₅₃×#2966	2
#33508	53