Subgroups — #27300014

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810577	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810402	C₂	1	2	yes	maximal
4	#20272792	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11719912	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
8	C₂×#13133174	C₄	1	1	yes	characteristic
9	C₂²×#11719908	C₂²	1	2	yes	—
10	#16810577	—	2	2	no	—
11	#16810577	—	2	2	no	—
12	C₂×#13133128	C₄	1	1	yes	characteristic
13	C₂×#13133127	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊#11719908	—	2	2	no	—
16	#16810402	—	2	4	no	—
17	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×C₄⋊C₁₆	—	2	2	no	—
20	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×#11719912	—	2	2	no	—
22	C₂×#11719912	—	2	2	no	—
23	C₂×#11719912	—	2	2	no	—
24	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₅₃	—	2	1	no	—
27	C₂×C₈×D₅₃	—	2	1	no	—
28	C₂×C₈×D₅₃	—	2	1	no	—
29	C₂×C₈×D₅₃	—	2	1	no	—
30	C₂×C₈×D₅₃	D₄	1	2	yes	—
31	C₂×C₈×D₅₃	Q₈	1	2	yes	—
32	#13133174	—	2	2	no	—
33	#13133174	—	2	2	no	—
34	C₂×#11719919	C₈	1	1	yes	characteristic
35	C₂×#11719919	C₈	1	1	yes	characteristic
36	C₂×#11719921	C₈	1	1	yes	characteristic
37	C₂×#11719921	C₈	1	1	yes	characteristic
38	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×#11719908	—	2	2	no	—
40	C₂×#11719908	—	2	2	no	—
41	C₂×#11719908	—	2	2	no	—
42	#13133128	—	2	2	no	—
43	#13133127	—	2	2	no	—
44	C₈×Dic₅₃	—	2	1	no	—
45	C₈×Dic₅₃	—	2	1	no	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₅₃	—	2	2	no	—
50	C₂×C₁₆×C₅₃	—	2	2	no	—
51	C₂×C₁₆×C₅₃	—	2	2	no	—
52	C₄×C₈×C₅₃	D₄	1	2	yes	—
53	C₄×C₈×C₅₃	D₄	1	2	yes	—
54	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂×C₄×D₅₃	—	2	1	no	—
62	C₂×C₄×D₅₃	—	2	1	no	—
63	C₂×C₄×D₅₃	—	2	1	no	—
64	C₂×C₄×D₅₃	—	2	1	no	—
65	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	—	2	1	no	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
75	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
76	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₅₃	M₁₆	1	2	yes	—
78	#11719921	—	2	2	no	—
79	#11719921	—	2	2	no	—
80	#11719919	—	2	2	no	—
81	#11719919	—	2	2	no	—
82	C₄×Dic₅₃	—	2	1	no	—
83	C₄×Dic₅₃	—	2	1	no	—
84	C₂×#88322	—	2	1	no	—
85	C₂×#88322	—	2	1	no	—
86	C₂×#88322	—	2	1	no	—
87	C₂×#88322	—	2	1	no	—
88	C₂×#88322	C₄⋊C₄	1	2	yes	—
89	C₂×#88322	M₁₆	1	2	yes	—
90	#11719912	—	2	8	no	—
91	C₈×D₅₃	—	2	2	no	—
92	C₈×D₅₃	—	2	2	no	—
93	C₈×D₅₃	—	2	2	no	—
94	C₈×D₅₃	—	2	2	no	—
95	C₁₆×C₅₃	—	4	2	no	—
96	#11719908	—	4	2	no	—
97	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
98	C₂²×D₅₃	—	2	1	no	—
99	C₂²×D₅₃	—	2	1	no	—
100	C₂²×D₅₃	—	2	2	no	—
101	C₂²×D₅₃	—	2	2	no	—
102	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
113	C₂×Dic₅₃	—	2	1	no	—
114	C₂×Dic₅₃	—	2	1	no	—
115	C₂×Dic₅₃	—	2	1	no	—
116	C₂×Dic₅₃	—	2	1	no	—
117	C₂×Dic₅₃	—	2	2	no	—
118	C₂×Dic₅₃	—	2	2	no	—
119	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
120	C₄×D₅₃	—	2	2	no	—
121	C₄×D₅₃	—	2	2	no	—
122	C₄×D₅₃	—	2	2	no	—
123	C₄×D₅₃	—	2	2	no	—
124	C₈×C₅₃	—	2	2	no	—
125	C₈×C₅₃	#121	1	1	yes	characteristic
126	C₈×C₅₃	#121	1	1	yes	characteristic
127	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
128	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
129	#88322	—	2	2	no	—
130	#88322	—	2	2	no	—
131	#12784	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
134	C₂²×C₅₃	#431	1	1	yes	characteristic
135	C₂²×C₅₃	#504	1	2	yes	—
136	C₂²×C₅₃	#504	1	2	yes	—
137	C₂×D₅₃	—	2	2	no	—
138	C₂×D₅₃	—	2	4	no	—
139	C₂×D₅₃	—	4	1	no	—
140	C₂×D₅₃	—	4	1	no	—
141	C₂×D₅₃	—	4	2	no	—
142	C₄×C₅₃	—	2	2	no	—
143	C₄×C₅₃	—	2	2	no	—
144	C₄×C₅₃	#380	1	1	yes	characteristic
145	C₄×C₅₃	#407	1	1	yes	characteristic
146	C₄×C₅₃	#423	1	1	yes	characteristic
147	C₄×C₅₃	#432	1	1	yes	characteristic
148	Dic₅₃	—	2	2	no	—
149	Dic₅₃	—	4	1	no	—
150	Dic₅₃	—	4	1	no	—
151	C₂×#405	—	53	1	no	—
152	C₂×C₄⋊C₁₆	—	53	1	no	—
153	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
154	C₂×C₂²⋊C₁₆	—	53	2	no	—
155	#1374	—	53	2	no	—
156	C₂×C₅₃	#1797	1	1	yes	characteristic
157	C₂×C₅₃	#2113	1	1	yes	characteristic
158	C₂×C₅₃	#2113	1	1	yes	characteristic
159	C₂×C₅₃	#2151	1	2	yes	—
160	C₂×C₅₃	#2164	1	2	yes	—
161	D₅₃	—	4	2	no	—
162	C₂²×M₁₆	—	53	1	no	—
163	C₂³×C₈	—	53	1	no	—
164	C₂×#117	—	53	1	no	—
165	C₂²×C₁₆	—	53	2	no	—
166	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
167	#405	—	106	2	no	—
168	#405	—	106	2	no	—
169	C₂×#97	—	53	1	no	—
170	C₂×#97	—	53	1	no	—
171	C₂×C₄×C₈	—	53	1	no	—
172	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
173	C₄⋊C₁₆	—	2	2	no	—
174	C₄⋊C₁₆	—	106	2	no	—
175	C₂²⋊C₁₆	—	106	4	no	—
176	C₅₃	#12784	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₄	—	53	1	no	—
178	C₂×M₁₆	—	106	2	no	—
179	C₂×M₁₆	—	106	2	no	—
180	C₂×M₁₆	—	106	2	no	—
181	C₂²×C₈	—	53	1	no	—
182	C₂²×C₈	—	53	1	no	—
183	C₂²×C₈	—	53	2	no	—
184	C₂²×C₈	—	53	2	no	—
185	C₂²×C₈	—	106	1	no	—
186	C₂²×C₈	—	106	1	no	—
187	C₂²×C₈	—	106	1	no	—
188	C₂²×C₈	—	106	1	no	—
189	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
191	#117	—	106	2	no	—
192	#117	—	106	2	no	—
193	C₂×C₄⋊C₄	—	53	1	no	—
194	C₂×C₄⋊C₄	—	53	1	no	—
195	C₂×C₂²⋊C₄	—	53	1	no	—
196	C₂×C₂²⋊C₄	—	53	1	no	—
197	C₂×C₄²	—	53	1	no	—
198	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₁₆	—	2	2	no	—
200	C₂×C₁₆	—	2	2	no	—
201	C₂×C₁₆	—	2	2	no	—
202	C₂×C₁₆	—	106	2	no	—
203	C₂×C₁₆	—	106	2	no	—
204	C₂×C₁₆	—	106	2	no	—
205	#97	—	106	2	no	—
206	#97	—	106	2	no	—
207	C₄×C₈	—	106	1	no	—
208	C₄×C₈	—	106	1	no	—
209	C₄×C₈	D₂₁₂	1	2	yes	—
210	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
211	C₂⁴	—	53	1	no	—
212	C₂²×C₄	—	53	1	no	—
213	C₂²×C₄	—	53	2	no	—
214	C₂²×C₄	—	53	2	no	—
215	C₂²×C₄	—	53	1	no	—
216	C₂²×C₄	—	53	1	no	—
217	C₂²×C₄	—	106	1	no	—
218	C₂²×C₄	—	106	1	no	—
219	C₂²×C₄	—	106	1	no	—
220	C₂²×C₄	—	106	1	no	—
221	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
222	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
224	M₁₆	—	106	8	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	53	2	no	—
228	C₂×C₈	—	53	2	no	—
229	C₂×C₈	—	106	1	no	—
230	C₂×C₈	—	106	1	no	—
231	C₂×C₈	—	106	1	no	—
232	C₂×C₈	—	106	1	no	—
233	C₂×C₈	—	106	2	no	—
234	C₂×C₈	—	106	2	no	—
235	C₂×C₈	—	106	2	no	—
236	C₂×C₈	—	106	2	no	—
237	C₂×C₈	#11719921	1	2	yes	—
238	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
239	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
240	C₂×C₈	#11719947	1	1	yes	characteristic
241	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
242	C₂×C₈	#11719949	1	1	yes	characteristic
243	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
244	C₄⋊C₄	—	106	2	no	—
245	C₄⋊C₄	—	106	2	no	—
246	C₂²⋊C₄	—	106	2	no	—
247	C₂²⋊C₄	—	106	2	no	—
248	C₄²	—	106	1	no	—
249	C₄²	—	106	1	no	—
250	C₄²	#11719912	1	2	yes	—
251	C₄²	#11719921	1	2	yes	—
252	C₁₆	—	4	2	no	—
253	C₁₆	—	212	2	no	—
254	C₂³	—	53	2	no	—
255	C₂³	—	106	2	no	—
256	C₂³	—	106	2	no	—
257	C₂³	—	106	1	no	—
258	C₂³	—	106	1	no	—
259	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	53	2	no	—
263	C₂×C₄	—	106	1	no	—
264	C₂×C₄	—	106	1	no	—
265	C₂×C₄	—	106	2	no	—
266	C₂×C₄	—	106	2	no	—
267	C₂×C₄	—	106	2	no	—
268	C₂×C₄	—	106	2	no	—
269	C₂×C₄	—	106	2	no	—
270	C₂×C₄	—	106	2	no	—
271	C₂×C₄	—	106	1	no	—
272	C₂×C₄	—	106	1	no	—
273	C₂×C₄	#13133129	1	2	yes	—
274	C₂×C₄	#13133129	1	2	yes	—
275	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
276	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
277	C₂×C₄	#13133175	1	1	yes	characteristic
278	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
279	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
280	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	—	106	2	no	—
283	C₈	—	106	2	no	—
284	C₈	#13133188	1	1	yes	characteristic
285	C₈	#13133192	1	1	yes	characteristic
286	C₈	#13133193	1	1	yes	characteristic
287	C₈	#13133197	1	1	yes	characteristic
288	C₂²	—	106	2	no	—
289	C₂²	—	106	4	no	—
290	C₂²	—	212	2	no	—
291	C₂²	—	212	1	no	—
292	C₂²	—	212	1	no	—
293	C₂²	C₂×#13133129	1	1	yes	characteristic
294	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
295	C₂²	#16810675	1	1	yes	characteristic
296	C₂²	#16811039	1	2	yes	—
297	C₂²	#16811041	1	2	yes	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	106	2	no	—
301	C₄	—	212	1	no	—
302	C₄	—	212	1	no	—
303	C₄	#16810480	1	1	yes	characteristic
304	C₄	#16810586	1	1	yes	characteristic
305	C₄	#16810645	1	1	yes	characteristic
306	C₄	#16810679	1	1	yes	characteristic
307	C₂	—	212	2	no	—
308	C₂	#20274692	1	1	yes	characteristic
309	C₂	#20276328	1	1	yes	characteristic
310	C₂	#20276330	1	1	yes	characteristic
311	C₂	#20276472	1	2	yes	—
312	C₂	#20276521	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂²×C₈
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#12784
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊#11719908	C₂	semidirect
4. #20272792	C₂	semidirect
5. C₂×C₅₃×C₄⋊C₁₆	C₂	semidirect
153. C₂×C₄⋊C₁₆	D₅₃	semidirect
176. C₅₃	#12784	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810577	2
C₂×C₄⋊#11719908	2
C₂×#16810402	2
#20272792	2
C₂×C₅₃×C₄⋊C₁₆	2
#12784	53