Subgroups — #27293921

Order 13568.

378 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11719947	C₂	1	1	yes	maximal, characteristic
2	#20276513	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272695	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272693	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11719947	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₅₃	—	2	1	no	—
10	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
11	#16810747	C₄	1	1	yes	characteristic
12	D₄×#88322	C₄	1	1	yes	characteristic
13	C₈×C₂²⋊D₅₃	—	2	1	no	—
14	#16810686	—	2	1	no	—
15	#16810675	C₄	1	1	yes	characteristic
16	#16810673	—	2	1	no	—
17	C₂×C₈×Dic₅₃	—	2	1	no	—
18	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊#11719908	C₂²	1	1	yes	characteristic
20	#16810402	—	2	1	no	—
21	C₄×#11719908	C₂²	1	1	yes	characteristic
22	C₁₆×Dic₅₃	—	2	1	no	—
23	C₁₆×Dic₅₃	—	2	1	no	—
24	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#11719947	C₈	1	1	yes	characteristic
28	C₂²×#88322	—	2	1	no	—
29	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
30	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
31	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
32	#13133182	C₈	1	1	yes	characteristic
33	D₄×Dic₅₃	C₈	1	1	yes	characteristic
34	C₄×C₂²⋊D₅₃	—	2	1	no	—
35	#13133177	—	2	1	no	—
36	#13133175	C₈	1	1	yes	characteristic
37	#13133173	—	2	1	no	—
38	C₂×C₄×Dic₅₃	—	2	1	no	—
39	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×#11719908	—	2	1	no	—
41	C₂×#11719908	—	2	1	no	—
42	C₁₆×D₅₃	—	2	4	no	—
43	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
44	#13133146	—	2	1	no	—
45	#13133146	—	2	1	no	—
46	C₂²⋊#88322	—	2	1	no	—
47	#13133129	—	2	1	no	—
48	C₄×#88322	C₂×C₄	1	1	yes	characteristic
49	C₈×Dic₅₃	—	2	1	no	—
50	C₈×Dic₅₃	—	2	1	no	—
51	C₈×Dic₅₃	—	2	1	no	—
52	C₈×Dic₅₃	D₄	1	1	yes	characteristic
53	C₈×Dic₅₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	—	2	1	no	—
55	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₆×C₅₃	—	2	1	no	—
57	C₂×C₁₆×C₅₃	—	2	1	no	—
58	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂²⋊C₈×C₅₃	—	2	1	no	—
60	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	—	2	1	no	—
63	C₂×C₂²⋊D₅₃	—	2	1	no	—
64	C₂²×Dic₅₃	—	2	1	no	—
65	C₂²×Dic₅₃	—	2	1	no	—
66	#11719947	—	2	2	no	—
67	#11719947	C₁₆	1	4	yes	complemented
68	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
71	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	—	4	1	no	—
74	C₂×C₈×C₅₃	—	4	1	no	—
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
78	C₂²⋊C₄×C₅₃	—	2	1	no	—
79	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²⋊Dic₅₃	—	2	1	no	—
81	#11719921	—	2	1	no	—
82	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
83	#11719919	—	2	1	no	—
84	#11719919	—	2	1	no	—
85	C₄×Dic₅₃	—	2	1	no	—
86	C₄×Dic₅₃	—	2	1	no	—
87	C₄×Dic₅₃	—	2	1	no	—
88	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
90	C₄×Dic₅₃	M₁₆	1	1	yes	characteristic
91	C₂×#88322	—	2	1	no	—
92	C₂×#88322	—	4	2	no	—
93	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
95	C₈×D₅₃	—	2	1	no	—
96	C₈×D₅₃	—	2	1	no	—
97	C₈×D₅₃	—	2	2	no	—
98	C₁₆×C₅₃	—	2	2	no	—
99	C₁₆×C₅₃	—	4	1	no	—
100	#11719908	—	2	2	no	—
101	#11719908	—	4	1	no	—
102	C₂³×C₅₃	—	2	1	no	—
103	C₂²×D₅₃	C₂²⋊C₈	1	1	yes	characteristic
104	D₄×C₅₃	—	2	1	no	—
105	D₄×C₅₃	C₂×C₁₆	1	2	yes	complemented
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	4	1	no	—
108	C₂×C₄×C₅₃	—	4	1	no	—
109	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
112	C₂²⋊D₅₃	—	2	4	no	—
113	C₂×Dic₅₃	—	2	1	no	—
114	C₂×Dic₅₃	—	2	4	no	—
115	C₂×Dic₅₃	—	2	1	no	—
116	C₂×Dic₅₃	—	4	2	no	—
117	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
119	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
120	C₄×D₅₃	—	2	1	no	—
121	C₄×D₅₃	—	2	1	no	—
122	C₄×D₅₃	—	2	2	no	—
123	C₄×D₅₃	C₂×C₁₆	1	2	yes	—
124	C₄×D₅₃	M₃₂	1	2	yes	—
125	Dic₁₀₆	—	2	1	no	—
126	Dic₁₀₆	C₂×C₁₆	1	2	yes	complemented
127	C₈×C₅₃	—	2	1	no	—
128	C₈×C₅₃	—	4	1	no	—
129	C₈×C₅₃	#135	1	1	yes	characteristic
130	C₈×C₅₃	#135	1	1	yes	characteristic
131	#88322	—	2	1	no	—
132	#88322	—	4	1	no	—
133	#88322	#104	1	2	yes	—
134	#11613	—	53	1	no	maximal, Sylow
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	—	4	1	no	—
137	C₂²×C₅₃	—	4	1	no	—
138	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₂×D₅₃	—	2	1	no	—
140	C₂×D₅₃	—	2	1	no	—
141	C₂×D₅₃	C₂²⋊C₁₆	1	2	yes	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	4	1	no	—
144	C₄×C₅₃	#416	1	1	yes	characteristic
145	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
146	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
147	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
148	Dic₅₃	—	2	2	no	—
149	Dic₅₃	—	2	1	no	—
150	Dic₅₃	—	4	1	no	—
151	Dic₅₃	C₂²⋊C₁₆	1	2	yes	—
152	Dic₅₃	#350	1	2	yes	—
153	C₈×#41	—	53	1	no	—
154	#2162	—	53	1	no	—
155	C₂×C₄×C₁₆	—	53	1	no	—
156	Q₈⋊C₁₆	—	53	1	no	—
157	Q₈⋊C₁₆	—	53	1	no	—
158	D₄⋊C₁₆	—	53	1	no	—
159	D₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
160	C₂×C₅₃	—	4	2	no	—
161	C₂×C₅₃	#1476	1	1	yes	characteristic
162	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
163	C₂×C₅₃	C₂×#350	1	1	yes	characteristic
164	D₅₃	—	2	2	no	—
165	C₄×#41	—	53	1	no	—
166	C₂²×C₁₆	—	106	1	no	—
167	C₈×Q₈	—	53	1	no	—
168	C₈×Q₈	—	53	1	no	—
169	C₈×D₄	—	53	1	no	—
170	C₈×D₄	—	106	1	no	—
171	C₈×D₄	—	106	1	no	—
172	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
173	#431	—	53	1	no	—
174	#431	—	106	1	no	—
175	C₂×C₄×C₈	—	53	1	no	—
176	C₂×C₄×C₈	—	106	1	no	—
177	C₄⋊C₁₆	—	53	1	no	—
178	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
179	C₂²⋊C₁₆	—	106	1	no	—
180	C₄×C₁₆	—	53	1	no	—
181	C₄×C₁₆	—	106	1	no	—
182	C₄×C₁₆	—	106	1	no	—
183	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
184	C₅₃	#11613	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	53	1	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	53	1	no	—
188	C₂²×C₈	—	53	1	no	—
189	C₂²×C₈	—	106	1	no	—
190	C₄×Q₈	—	53	1	no	—
191	C₄×Q₈	—	53	1	no	—
192	C₄×D₄	—	53	1	no	—
193	C₄×D₄	—	106	1	no	—
194	C₄×D₄	—	106	1	no	—
195	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
196	#117	—	53	1	no	—
197	#117	—	106	1	no	—
198	C₂×C₄²	—	53	1	no	—
199	C₂×C₄²	—	106	1	no	—
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	—	2	1	no	—
202	C₂×C₁₆	—	106	4	no	—
203	C₂×C₁₆	—	106	1	no	—
204	C₂×C₁₆	—	106	1	no	—
205	C₄⋊C₈	—	53	1	no	—
206	C₄⋊C₈	—	106	1	no	—
207	C₄⋊C₈	—	106	1	no	—
208	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	106	1	no	—
211	C₂²⋊C₈	—	106	1	no	—
212	C₄×C₈	—	53	1	no	—
213	C₄×C₈	—	53	1	no	—
214	C₄×C₈	—	53	1	no	—
215	C₄×C₈	—	106	1	no	—
216	C₄×C₈	—	106	1	no	—
217	C₄×C₈	—	106	1	no	—
218	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
219	#41	—	53	4	no	—
220	#41	—	106	2	no	—
221	C₂×Q₈	—	53	1	no	—
222	C₂×D₄	—	106	1	no	—
223	C₂×D₄	C₈×D₅₃	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	53	1	no	—
226	C₂²×C₄	—	53	1	no	—
227	C₂²×C₄	—	53	1	no	—
228	C₂²×C₄	—	106	1	no	—
229	C₂²×C₄	—	106	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	53	1	no	—
234	C₂×C₈	—	53	1	no	—
235	C₂×C₈	—	106	2	no	—
236	C₂×C₈	—	106	1	no	—
237	C₂×C₈	—	106	1	no	—
238	C₂×C₈	—	106	1	no	—
239	C₂×C₈	—	212	2	no	—
240	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
241	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
242	C₄⋊C₄	—	53	1	no	—
243	C₄⋊C₄	—	106	1	no	—
244	C₄⋊C₄	—	106	1	no	—
245	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	106	1	no	—
248	C₂²⋊C₄	—	106	1	no	—
249	C₄²	—	53	1	no	—
250	C₄²	—	53	1	no	—
251	C₄²	—	53	1	no	—
252	C₄²	—	106	1	no	—
253	C₄²	—	106	1	no	—
254	C₄²	—	106	1	no	—
255	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
256	C₁₆	—	2	2	no	—
257	C₁₆	—	4	1	no	—
258	C₁₆	—	106	2	no	—
259	C₁₆	—	212	1	no	—
260	C₂³	—	2	1	no	—
261	C₂³	—	53	1	no	—
262	Q₈	—	53	2	no	—
263	Q₈	—	106	1	no	—
264	D₄	—	2	1	no	—
265	D₄	—	106	4	no	—
266	D₄	C₁₆×D₅₃	1	2	yes	complemented
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	53	1	no	—
271	C₂×C₄	—	53	1	no	—
272	C₂×C₄	—	53	2	no	—
273	C₂×C₄	—	53	2	no	—
274	C₂×C₄	—	53	1	no	—
275	C₂×C₄	—	106	2	no	—
276	C₂×C₄	—	106	4	no	—
277	C₂×C₄	—	106	1	no	—
278	C₂×C₄	—	106	1	no	—
279	C₂×C₄	—	106	1	no	—
280	C₂×C₄	—	106	1	no	—
281	C₂×C₄	—	212	2	no	—
282	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
283	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
284	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	4	1	no	—
287	C₈	—	53	2	no	—
288	C₈	—	106	1	no	—
289	C₈	—	212	1	no	—
290	C₈	#13133236	1	1	yes	characteristic
291	C₈	#13133238	1	1	yes	characteristic
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	53	2	no	—
296	C₂²	—	106	1	no	—
297	C₂²	—	106	1	no	—
298	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	53	2	no	—
302	C₄	—	53	2	no	—
303	C₄	—	106	2	no	—
304	C₄	—	106	1	no	—
305	C₄	—	212	1	no	—
306	C₄	#16810616	1	1	yes	characteristic
307	C₄	#104×D₅₃	1	1	yes	characteristic
308	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
309	C₄	M₃₂×D₅₃	1	1	yes	characteristic
310	C₂	—	4	2	no	—
311	C₂	—	106	2	no	—
312	C₂	#20273094	1	1	yes	characteristic
313	C₂	C₂²⋊C₁₆×D₅₃	1	1	yes	characteristic
314	C₂	#350×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×D₄⋊C₁₆	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂×#11719947	C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	D₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#11613
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ D₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #20276513	C₂	semidirect
3. C₄×C₁₆×D₅₃	C₂	semidirect
4. #20272695	C₂	semidirect
5. #20272693	C₂	semidirect
6. D₄⋊#11719908	C₂	semidirect
7. C₅₃×D₄⋊C₁₆	C₂	semidirect
67. #11719947	C₁₆	semidirect
105. D₄×C₅₃	C₂×C₁₆	semidirect
126. Dic₁₀₆	C₂×C₁₆	semidirect
159. D₄⋊C₁₆	D₅₃	semidirect
184. C₅₃	#11613	semidirect
266. D₄	C₁₆×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11719947	2
#20276513	2
C₄×C₁₆×D₅₃	2
#20272695	2
#20272693	2
D₄⋊#11719908	2
C₅₃×D₄⋊C₁₆	2
#11613	53