Subgroups — #27484665

Order 13568.

810 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283113	C₂	1	1	yes	maximal, characteristic
2	#20276048	C₂	1	4	yes	maximal
3	#20275677	C₂	1	2	yes	maximal
4	#20275478	C₂	1	4	yes	maximal
5	#20274871	C₂	1	2	yes	maximal
6	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133217	—	2	1	no	—
9	C₂×#13133217	—	2	1	no	—
10	C₂×#13133217	C₂²	1	2	yes	—
11	C₂×#13133148	C₂²	1	8	yes	—
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×C₄×#88322	C₂²	1	2	yes	—
14	#16810493	—	2	1	no	—
15	#16810493	—	2	1	no	—
16	#16810493	C₂²	1	2	yes	—
17	#16810493	C₂²	1	2	yes	—
18	#16810385	C₂²	1	8	yes	—
19	#16810378	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
22	C₅₃×#391	—	2	1	no	—
23	C₅₃×#389	—	2	1	no	—
24	C₅₃×#389	C₂²	1	2	yes	—
25	C₅₃×#384	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₄³×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×Dic₁₀₆	—	2	2	no	—
30	C₂²×#88322	C₂³	1	4	yes	—
31	#13133217	—	4	4	no	—
32	#13133217	—	4	4	no	—
33	#13133217	—	4	4	no	—
34	C₂×C₄⋊Dic₅₃	—	2	2	no	—
35	C₂×C₄⋊Dic₅₃	—	2	2	no	—
36	C₂×C₄⋊Dic₅₃	—	2	2	no	—
37	C₂×C₄⋊Dic₅₃	—	2	2	no	—
38	C₂×C₄⋊Dic₅₃	—	2	2	no	—
39	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
40	#13133148	—	4	8	no	—
41	C₄⋊#88322	Q₈	1	8	yes	complemented
42	C₄×#88322	—	2	4	no	—
43	C₂²×Q₈×C₅₃	—	2	1	no	—
44	C₄⋊Q₈×C₅₃	—	2	2	no	—
45	C₄⋊Q₈×C₅₃	Q₈	1	4	yes	complemented
46	C₄×Q₈×C₅₃	—	4	4	no	—
47	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
48	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
49	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
50	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₅₃	—	2	2	no	—
54	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	#95×C₅₃	—	2	2	no	—
62	#95×C₅₃	—	2	2	no	—
63	C₂×Q₈×C₅₃	—	2	4	no	—
64	C₂×Q₈×C₅₃	—	4	2	no	—
65	C₂×Q₈×C₅₃	—	4	2	no	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	2	2	no	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₅₃	—	2	4	no	—
76	C₂×Dic₁₀₆	—	4	4	no	—
77	C₂×Dic₁₀₆	—	4	4	no	—
78	C₂×Dic₁₀₆	—	4	4	no	—
79	C₄⋊C₄×C₅₃	—	2	4	no	—
80	C₄⋊C₄×C₅₃	—	2	2	no	—
81	C₄⋊C₄×C₅₃	—	2	4	no	—
82	C₄⋊C₄×C₅₃	—	4	4	no	—
83	C₄⋊C₄×C₅₃	—	4	2	no	—
84	C₄⋊C₄×C₅₃	—	4	2	no	—
85	C₄⋊C₄×C₅₃	C₂×Q₈	1	4	yes	—
86	C₄²×C₅₃	—	2	2	no	—
87	C₄²×C₅₃	—	2	2	no	—
88	C₄²×C₅₃	—	2	2	no	—
89	C₄²×C₅₃	—	4	4	no	—
90	C₄²×C₅₃	SD₁₆	1	4	yes	—
91	C₄²×C₅₃	SD₁₆	1	4	yes	—
92	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
93	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
94	C₄²×C₅₃	#41	1	2	yes	—
95	C₄⋊Dic₅₃	—	2	8	no	—
96	C₄⋊Dic₅₃	—	4	4	no	—
97	C₄⋊Dic₅₃	—	4	4	no	—
98	C₄⋊Dic₅₃	—	4	4	no	—
99	C₄⋊Dic₅₃	—	4	4	no	—
100	C₄⋊Dic₅₃	—	4	4	no	—
101	C₂×#88322	—	2	4	no	—
102	C₂×#88322	—	2	4	no	—
103	C₂×#88322	C₂×Q₈	1	8	yes	—
104	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
105	Q₈×C₅₃	—	4	8	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	4	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
124	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
125	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
126	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
127	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
128	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
129	C₂×C₄×C₅₃	#137	1	2	yes	—
130	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
133	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
134	C₂×Dic₅₃	—	4	4	no	—
135	C₂×Dic₅₃	—	4	4	no	—
136	C₂×Dic₅₃	—	4	4	no	—
137	Dic₁₀₆	—	8	8	no	—
138	#88322	—	2	8	no	—
139	#29206	—	53	1	no	maximal, Sylow
140	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
141	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
142	C₂²×C₅₃	#535	1	1	yes	characteristic
143	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
144	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
145	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂²×C₅₃	C₂×#137	1	1	yes	characteristic
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	2	2	no	—
150	C₄×C₅₃	—	4	4	no	—
151	C₄×C₅₃	—	8	2	no	—
152	C₄×C₅₃	#475	1	4	yes	—
153	C₄×C₅₃	#476	1	4	yes	—
154	C₄×C₅₃	#498	1	4	yes	—
155	C₄×C₅₃	#554	1	2	yes	—
156	C₄×C₅₃	Q₈²	1	2	yes	—
157	Dic₅₃	—	8	4	no	—
158	#2868	—	53	1	no	—
159	C₄×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
160	#2058	—	53	4	no	—
161	#1990	—	53	2	no	—
162	#1948	—	53	4	no	—
163	#1844	—	53	2	no	—
164	C₄×C₄⋊C₈	—	53	1	no	—
165	C₂×C₅₃	#2567	1	1	yes	characteristic
166	C₂×C₅₃	C₂×#476	1	1	yes	characteristic
167	C₂×C₅₃	C₂×#475	1	1	yes	characteristic
168	C₂×C₅₃	C₂×#498	1	1	yes	characteristic
169	C₂×C₅₃	#3165	1	1	yes	characteristic
170	C₂×C₅₃	#3206	1	1	yes	characteristic
171	C₂×C₅₃	#3208	1	1	yes	characteristic
172	C₂×C₄⋊Q₈	—	53	2	no	—
173	C₂×C₄⋊Q₈	—	106	1	no	—
174	C₂×C₄⋊Q₈	—	106	1	no	—
175	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
176	C₂×C₄×Q₈	—	2	1	no	—
177	C₂×#106	—	53	8	no	—
178	C₂×C₄⋊C₈	—	53	2	no	—
179	C₂×C₄×C₈	—	53	2	no	—
180	#391	—	2	1	no	—
181	#389	—	2	1	no	—
182	#389	C₂×D₅₃	1	2	yes	—
183	#384	—	53	2	no	—
184	#384	—	53	2	no	—
185	#384	—	106	1	no	—
186	#384	—	106	1	no	—
187	#384	C₂×D₅₃	1	1	yes	characteristic
188	C₄×C₄⋊C₄	—	2	1	no	—
189	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
190	C₄³	C₂×D₅₃	1	1	yes	characteristic
191	#340	—	53	8	no	—
192	#336	—	53	2	no	—
193	C₅₃	#29206	1	1	yes	characteristic, complemented, Sylow
194	C₂²×Q₈	—	2	1	no	—
195	C₂²×Q₈	—	106	2	no	—
196	C₂²×C₈	—	53	4	no	—
197	C₄⋊Q₈	—	2	2	no	—
198	C₄⋊Q₈	—	212	4	no	—
199	C₄⋊Q₈	—	212	4	no	—
200	C₄⋊Q₈	—	212	4	no	—
201	C₄⋊Q₈	Dic₁₀₆	1	4	yes	complemented
202	C₄×Q₈	—	4	4	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	53	4	no	—
208	C₂×C₄⋊C₄	—	106	2	no	—
209	C₂×C₄⋊C₄	—	106	2	no	—
210	C₂×C₄⋊C₄	—	106	2	no	—
211	C₂×C₄⋊C₄	—	106	2	no	—
212	C₂×C₄⋊C₄	—	106	2	no	—
213	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
214	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
219	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
220	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
223	#106	—	212	8	no	—
224	C₄⋊C₈	—	53	8	no	—
225	C₄×C₈	—	106	4	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	C₂×Q₈	—	2	4	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂×Q₈	—	212	4	no	—
232	C₂×Q₈	—	212	4	no	—
233	C₂×Q₈	—	212	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	106	4	no	—
237	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
243	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
244	C₂×C₈	—	53	8	no	—
245	C₂×C₈	—	106	4	no	—
246	C₂×C₈	—	106	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	106	8	no	—
254	C₄⋊C₄	—	212	4	no	—
255	C₄⋊C₄	—	212	4	no	—
256	C₄⋊C₄	—	212	4	no	—
257	C₄⋊C₄	—	212	4	no	—
258	C₄⋊C₄	—	212	4	no	—
259	C₄⋊C₄	C₂×Dic₁₀₆	1	4	yes	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	4	4	no	—
264	C₄²	#11719923	1	4	yes	—
265	C₄²	#11719924	1	4	yes	—
266	C₄²	C₂×Dic₁₀₆	1	2	yes	—
267	C₄²	#11719945	1	2	yes	—
268	C₄²	Q₈×D₅₃	1	4	yes	—
269	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
270	Q₈	—	4	8	no	—
271	Q₈	—	424	8	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	212	4	no	—
285	C₂×C₄	—	212	4	no	—
286	C₂×C₄	—	212	4	no	—
287	C₂×C₄	#13133196	1	2	yes	—
288	C₂×C₄	#13133196	1	2	yes	—
289	C₂×C₄	#13133215	1	2	yes	—
290	C₂×C₄	#13133215	1	2	yes	—
291	C₂×C₄	C₂×#11719924	1	2	yes	—
292	C₂×C₄	C₂×#11719924	1	2	yes	—
293	C₂×C₄	C₂×#11719924	1	2	yes	—
294	C₂×C₄	C₂×#11719923	1	2	yes	—
295	C₂×C₄	C₂×#11719923	1	2	yes	—
296	C₂×C₄	C₂×#11719923	1	2	yes	—
297	C₂×C₄	#13133242	1	2	yes	—
298	C₂×C₄	#13133247	1	2	yes	—
299	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
300	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
301	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
302	C₂×C₄	#13133263	1	2	yes	—
303	C₈	—	106	8	no	—
304	C₂²	C₂×#13133196	1	1	yes	characteristic
305	C₂²	C₂×#13133215	1	1	yes	characteristic
306	C₂²	#16811187	1	1	yes	characteristic
307	C₂²	C₂²×#11719924	1	1	yes	characteristic
308	C₂²	C₂²×#11719923	1	1	yes	characteristic
309	C₂²	C₂×#13133242	1	1	yes	characteristic
310	C₂²	C₂×#13133247	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	4	4	no	—
315	C₄	—	8	2	no	—
316	C₄	—	424	4	no	—
317	C₄	#16810908	1	4	yes	—
318	C₄	#16810913	1	4	yes	—
319	C₄	#16811019	1	4	yes	—
320	C₄	#16811333	1	2	yes	—
321	C₄	Q₈×Dic₁₀₆	1	2	yes	—
322	C₂	#20279920	1	1	yes	characteristic
323	C₂	C₂×#16810913	1	1	yes	characteristic
324	C₂	C₂×#16810908	1	1	yes	characteristic
325	C₂	C₂×#16811019	1	1	yes	characteristic
326	C₂	#20285262	1	1	yes	characteristic
327	C₂	#20285637	1	1	yes	characteristic
328	C₂	#20285653	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×C₄⋊Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	C₄×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#29206
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
41. C₄⋊#88322	Q₈	semidirect
45. C₄⋊Q₈×C₅₃	Q₈	semidirect
193. C₅₃	#29206	semidirect
201. C₄⋊Q₈	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20283113	2
#20276048	2
#20275677	2
#20275478	2
#20274871	2
C₄×C₄⋊#88322	2
C₄×C₄⋊Q₈×C₅₃	2
#29206	53