Subgroups — #27300577

Order 13568.

384 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	C₂×#16811041	C₂	1	1	yes	maximal, characteristic
2	#20276475	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810408	C₂	1	1	yes	maximal, characteristic
4	#20274552	C₂	1	1	yes	maximal, characteristic
5	#20272805	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272793	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133223	C₂²	1	1	yes	characteristic
9	#16811041	—	2	2	no	—
10	#16811041	—	2	2	no	—
11	C₂×#13133156	—	2	1	no	—
12	C₂²×#11719908	C₂²	1	1	yes	characteristic
13	C₂²×#11719908	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₅₃	—	2	1	no	—
15	C₂×#13133145	C₄	1	1	yes	characteristic
16	#16810620	C₄	1	1	yes	characteristic
17	#16810578	C₂²	1	1	yes	characteristic
18	#16810441	—	2	1	no	—
19	#16810441	—	2	1	no	—
20	#16810408	—	2	2	no	—
21	#16810388	C₄	1	1	yes	characteristic
22	#16810383	C₄	1	1	yes	characteristic
23	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#364	—	2	2	no	—
27	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
28	#13133223	—	2	1	no	—
29	#13133223	—	2	1	no	—
30	#13133223	D₄	1	2	yes	—
31	#13133223	Q₈	1	2	yes	—
32	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
33	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
34	C₂×#11719912	—	2	1	no	—
35	C₂×C₈×D₅₃	—	2	1	no	—
36	#13133176	C₂×C₄	1	1	yes	characteristic
37	#13133156	—	2	4	no	—
38	C₂×#11719908	—	2	1	no	—
39	C₂×#11719908	—	2	1	no	—
40	C₂×#11719908	—	2	1	no	—
41	C₂×#11719908	—	2	1	no	—
42	C₂×#11719908	—	2	1	no	—
43	C₂×#11719908	D₄	1	1	yes	characteristic
44	C₂×#11719908	D₄	1	1	yes	characteristic
45	C₁₆×D₅₃	—	2	4	no	—
46	#13133145	—	2	2	no	—
47	#13133142	C₈	1	1	yes	characteristic
48	#13133142	C₈	1	1	yes	characteristic
49	#13133139	C₈	1	1	yes	characteristic
50	#13133139	C₈	1	1	yes	characteristic
51	#13133127	—	2	1	no	—
52	C₈×Dic₅₃	—	2	1	no	—
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	M₃₂×C₅₃	—	2	1	no	—
57	M₃₂×C₅₃	—	2	1	no	—
58	M₃₂×C₅₃	—	2	1	no	—
59	M₃₂×C₅₃	D₄	1	2	yes	—
60	C₂×C₁₆×C₅₃	—	2	1	no	—
61	C₂×C₁₆×C₅₃	—	2	1	no	—
62	C₄×C₈×C₅₃	—	2	1	no	—
63	C₄×C₈×C₅₃	—	2	1	no	—
64	C₂²×C₄×C₅₃	—	2	1	no	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₅₃	—	2	1	no	—
67	#11719945	—	2	1	no	—
68	#11719945	—	2	1	no	—
69	#11719945	C₄⋊C₄	1	2	yes	—
70	#11719945	M₁₆	1	2	yes	—
71	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₅₃	—	2	1	no	—
73	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
74	C₂×C₈×C₅₃	—	2	1	no	—
75	C₂×C₈×C₅₃	—	2	1	no	—
76	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
84	C₄²×C₅₃	—	2	1	no	—
85	C₄²×C₅₃	—	2	1	no	—
86	C₂²⋊Dic₅₃	—	2	1	no	—
87	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
88	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
89	C₄×Dic₅₃	—	2	1	no	—
90	#11719917	—	2	1	no	—
91	#11719917	—	2	1	no	—
92	#11719917	—	2	1	no	—
93	#11719917	C₄⋊C₄	1	2	yes	—
94	C₂×#88322	—	2	1	no	—
95	C₂×#88322	—	2	1	no	—
96	#11719912	—	2	2	no	—
97	#11719912	—	2	2	no	—
98	C₈×D₅₃	—	2	2	no	—
99	C₈×D₅₃	—	2	2	no	—
100	C₁₆×C₅₃	—	2	2	no	—
101	C₁₆×C₅₃	—	4	1	no	—
102	#11719908	—	2	1	no	—
103	#11719908	—	2	1	no	—
104	#11719908	—	4	1	no	—
105	C₂³×C₅₃	#131	1	1	yes	characteristic
106	C₂²×D₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	4	1	no	—
109	C₂×C₄×C₅₃	—	4	1	no	—
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
116	C₂²⋊D₅₃	—	2	2	no	—
117	C₂²⋊D₅₃	—	2	2	no	—
118	C₂×Dic₅₃	—	2	1	no	—
119	C₂×Dic₅₃	—	2	1	no	—
120	C₂×Dic₅₃	—	2	1	no	—
121	D₂₁₂	—	2	1	no	—
122	D₂₁₂	C₄⋊C₈	1	2	yes	—
123	C₄×D₅₃	—	2	2	no	—
124	C₄×D₅₃	—	2	2	no	—
125	Dic₁₀₆	—	2	1	no	—
126	Dic₁₀₆	C₄⋊C₈	1	2	yes	—
127	C₈×C₅₃	—	2	1	no	—
128	C₈×C₅₃	—	2	1	no	—
129	C₈×C₅₃	#120	1	1	yes	characteristic
130	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
131	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
132	C₈×C₅₃	#123	1	1	yes	characteristic
133	#88322	—	2	2	no	—
134	#88322	—	4	1	no	—
135	#12917	—	53	1	no	maximal, Sylow
136	C₂²×C₅₃	—	2	1	no	—
137	C₂²×C₅₃	—	2	1	no	—
138	C₂²×C₅₃	#424	1	1	yes	characteristic
139	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
140	C₂²×C₅₃	#435	1	1	yes	characteristic
141	C₂×D₅₃	—	2	2	no	—
142	C₂×D₅₃	—	4	1	no	—
143	C₂×D₅₃	—	4	1	no	—
144	C₄×C₅₃	—	4	2	no	—
145	C₄×C₅₃	#385	1	1	yes	characteristic
146	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
147	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
148	C₄×C₅₃	#435	1	1	yes	characteristic
149	Dic₅₃	—	2	2	no	—
150	Dic₅₃	—	4	1	no	—
151	Dic₅₃	—	4	1	no	—
152	C₂×#504	—	53	1	no	—
153	C₂×#364	D₅₃	1	1	yes	characteristic, complemented
154	#2151	—	53	1	no	—
155	C₂×#350	—	53	1	no	—
156	#1764	—	53	1	no	—
157	#1379	—	53	1	no	—
158	#1374	—	53	1	no	—
159	C₂×C₅₃	—	2	1	no	—
160	C₂×C₅₃	—	2	1	no	—
161	C₂×C₅₃	#1850	1	1	yes	characteristic
162	C₂×C₅₃	#2170	1	1	yes	characteristic
163	C₂×C₅₃	#2171	1	1	yes	characteristic
164	D₅₃	—	4	2	no	—
165	C₂×#131	—	53	1	no	—
166	#504	—	106	2	no	—
167	#504	—	106	2	no	—
168	C₂×M₃₂	—	106	1	no	—
169	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
170	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
171	C₂²×C₁₆	—	53	1	no	—
172	C₂²×C₁₆	—	53	1	no	—
173	C₂²×C₁₆	—	106	1	no	—
174	C₂×#104	—	53	1	no	—
175	#416	—	53	1	no	—
176	#405	—	53	1	no	—
177	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
178	#364	—	2	2	no	—
179	C₄⋊C₁₆	—	106	1	no	—
180	C₄⋊C₁₆	—	106	1	no	—
181	#350	—	106	2	no	—
182	#341	—	53	1	no	—
183	#339	—	53	1	no	—
184	C₅₃	#12917	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	53	1	no	—
186	#131	—	53	2	no	—
187	#131	—	53	2	no	—
188	#131	—	106	1	no	—
189	#131	—	106	1	no	—
190	C₂×M₁₆	—	53	1	no	—
191	C₂×M₁₆	—	53	1	no	—
192	C₂×M₁₆	—	106	1	no	—
193	C₂²×C₈	—	106	1	no	—
194	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
196	#117	—	53	1	no	—
197	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
198	M₃₂	—	2	1	no	—
199	M₃₂	—	2	1	no	—
200	M₃₂	—	2	1	no	—
201	M₃₂	—	106	4	no	—
202	M₃₂	C₂²⋊D₅₃	1	2	yes	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	53	1	no	—
206	C₂×C₁₆	—	53	1	no	—
207	C₂×C₁₆	—	106	4	no	—
208	C₂×C₁₆	—	106	1	no	—
209	C₂×C₁₆	—	106	1	no	—
210	C₂×C₁₆	—	106	1	no	—
211	C₂×C₁₆	—	106	1	no	—
212	C₂×C₁₆	—	106	1	no	—
213	#104	—	106	2	no	—
214	#103	—	53	1	no	—
215	#103	—	53	1	no	—
216	#102	—	53	1	no	—
217	#102	—	53	1	no	—
218	#97	—	106	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	106	1	no	—
222	#41	—	53	2	no	—
223	#41	—	53	2	no	—
224	#41	—	106	1	no	—
225	#41	—	106	1	no	—
226	C₂×Q₈	—	53	1	no	—
227	C₂×D₄	—	53	1	no	—
228	C₂×D₄	—	106	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	106	1	no	—
231	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
232	M₁₆	—	53	2	no	—
233	M₁₆	—	106	1	no	—
234	M₁₆	—	106	2	no	—
235	M₁₆	—	106	2	no	—
236	M₁₆	—	106	1	no	—
237	M₁₆	—	106	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	106	1	no	—
241	C₂×C₈	—	106	2	no	—
242	C₂×C₈	—	106	2	no	—
243	C₂×C₈	—	106	1	no	—
244	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
245	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
246	C₂×C₈	#11719945	1	1	yes	characteristic
247	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
248	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
249	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
250	C₂×C₈	#11719949	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
252	C₄⋊C₄	—	53	1	no	—
253	C₄⋊C₄	—	53	1	no	—
254	C₂²⋊C₄	—	106	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	106	1	no	—
258	C₁₆	—	2	2	no	—
259	C₁₆	—	4	1	no	—
260	C₁₆	—	106	1	no	—
261	C₁₆	—	106	1	no	—
262	C₁₆	—	212	1	no	—
263	C₂³	—	106	1	no	—
264	C₂³	#13133225	1	1	yes	characteristic
265	Q₈	—	53	2	no	—
266	Q₈	—	106	1	no	—
267	D₄	—	53	2	no	—
268	D₄	—	106	2	no	—
269	D₄	—	106	2	no	—
270	D₄	—	106	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	106	1	no	—
275	C₂×C₄	—	106	2	no	—
276	C₂×C₄	—	106	2	no	—
277	C₂×C₄	—	106	1	no	—
278	C₂×C₄	—	106	1	no	—
279	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
280	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
281	C₂×C₄	#13133179	1	1	yes	characteristic
282	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
283	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
284	C₂×C₄	#13133223	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	106	2	no	—
288	C₈	—	212	1	no	—
289	C₈	#13133185	1	1	yes	characteristic
290	C₈	#13133193	1	1	yes	characteristic
291	C₈	#13133195	1	1	yes	characteristic
292	C₈	#13133202	1	1	yes	characteristic
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	106	2	no	—
296	C₂²	—	212	1	no	—
297	C₂²	—	212	1	no	—
298	C₂²	#16810648	1	1	yes	characteristic
299	C₂²	#16810688	1	1	yes	characteristic
300	C₂²	#16810692	1	1	yes	characteristic
301	C₄	—	4	2	no	—
302	C₄	—	106	2	no	—
303	C₄	—	212	1	no	—
304	C₄	—	212	1	no	—
305	C₄	#16810494	1	1	yes	characteristic
306	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
307	C₄	C₈×C₂²⋊D₅₃	1	1	yes	characteristic
308	C₄	#16810693	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	212	2	no	—
312	C₂	#20274891	1	1	yes	characteristic
313	C₂	#20276551	1	1	yes	characteristic
314	C₂	#20276556	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#12917
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. #20276475	C₂	semidirect
5. #20272805	C₂	semidirect
6. #20272793	C₂	semidirect
7. C₂×C₅₃×#364	C₂	semidirect
153. C₂×#364	D₅₃	semidirect
184. C₅₃	#12917	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811041	2
#20276475	2
C₂×#16810408	2
#20274552	2
#20272805	2
#20272793	2
C₂×C₅₃×#364	2
#12917	53