Subgroups — #27301845

Order 13568.

332 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276475	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810408	C₂	1	1	yes	maximal, characteristic
3	#20276330	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274552	C₂	1	1	yes	maximal, characteristic
6	#20272805	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133223	C₂²	1	1	yes	characteristic
9	C₂²×#11719908	C₂²	1	1	yes	characteristic
10	C₂²×#11719908	C₂²	1	1	yes	characteristic
11	C₂×#13133145	C₄	1	1	yes	characteristic
12	#16810620	C₄	1	1	yes	characteristic
13	#16810578	C₂²	1	1	yes	characteristic
14	#16810441	—	2	2	no	—
15	#16810408	—	2	2	no	—
16	#16810402	—	2	2	no	—
17	C₄×#11719908	—	2	2	no	—
18	#16810388	C₄	1	1	yes	characteristic
19	#16810383	C₄	1	1	yes	characteristic
20	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
24	#13133223	—	2	1	no	—
25	#13133223	—	2	1	no	—
26	#13133223	—	2	1	no	—
27	#13133223	—	2	1	no	—
28	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
29	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
30	C₂×#11719912	—	2	1	no	—
31	C₂×C₈×D₅₃	—	2	1	no	—
32	#13133176	C₂×C₄	1	1	yes	characteristic
33	C₂×#11719908	—	2	1	no	—
34	C₂×#11719908	—	2	1	no	—
35	C₂×#11719908	—	2	1	no	—
36	C₂×#11719908	—	2	1	no	—
37	C₂×#11719908	D₄	1	2	yes	—
38	C₂×#11719908	D₄	1	2	yes	—
39	#13133145	—	2	2	no	—
40	#13133142	C₈	1	2	yes	—
41	#13133139	C₈	1	2	yes	—
42	#13133127	—	2	1	no	—
43	C₈×Dic₅₃	—	2	1	no	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	M₃₂×C₅₃	—	2	1	no	—
48	M₃₂×C₅₃	—	2	1	no	—
49	M₃₂×C₅₃	—	2	1	no	—
50	M₃₂×C₅₃	—	2	1	no	—
51	C₂×C₁₆×C₅₃	—	2	1	no	—
52	C₂×C₁₆×C₅₃	—	2	1	no	—
53	C₄×C₈×C₅₃	—	2	1	no	—
54	C₄×C₈×C₅₃	—	2	1	no	—
55	C₂²×C₄×C₅₃	—	2	1	no	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊D₅₃	—	2	1	no	—
58	#11719945	—	2	1	no	—
59	#11719945	—	2	1	no	—
60	#11719945	—	2	1	no	—
61	#11719945	—	2	1	no	—
62	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
63	C₂×C₄×D₅₃	—	2	1	no	—
64	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
65	C₂×C₈×C₅₃	—	2	1	no	—
66	C₂×C₈×C₅₃	—	2	1	no	—
67	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
71	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
72	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
73	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
74	C₄²×C₅₃	—	2	1	no	—
75	C₄²×C₅₃	M₁₆	1	2	yes	—
76	C₂²⋊Dic₅₃	—	2	1	no	—
77	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
78	C₄×Dic₅₃	—	2	1	no	—
79	#11719917	—	2	1	no	—
80	#11719917	—	2	1	no	—
81	#11719917	—	2	1	no	—
82	#11719917	—	2	1	no	—
83	C₂×#88322	—	2	1	no	—
84	C₂×#88322	—	2	1	no	—
85	#11719912	—	4	1	no	—
86	#11719912	—	4	1	no	—
87	C₈×D₅₃	—	4	1	no	—
88	C₈×D₅₃	—	4	1	no	—
89	C₁₆×C₅₃	—	4	1	no	—
90	C₁₆×C₅₃	—	4	1	no	—
91	#11719908	—	2	2	no	—
92	#11719908	—	2	2	no	—
93	C₂³×C₅₃	#131	1	1	yes	characteristic
94	C₂²×D₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
103	C₂²⋊D₅₃	—	4	1	no	—
104	C₂²⋊D₅₃	—	4	1	no	—
105	C₂×Dic₅₃	—	2	1	no	—
106	C₂×Dic₅₃	—	2	2	no	—
107	D₂₁₂	—	2	1	no	—
108	D₂₁₂	—	2	1	no	—
109	C₄×D₅₃	—	4	1	no	—
110	C₄×D₅₃	—	4	1	no	—
111	Dic₁₀₆	—	2	1	no	—
112	Dic₁₀₆	—	2	1	no	—
113	C₈×C₅₃	—	2	2	no	—
114	C₈×C₅₃	#121	1	1	yes	characteristic
115	C₈×C₅₃	#123	1	1	yes	characteristic
116	C₈×C₅₃	#124	1	1	yes	characteristic
117	C₈×C₅₃	#126	1	1	yes	characteristic
118	#88322	—	4	1	no	—
119	#88322	—	4	1	no	—
120	#13124	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	#433	1	1	yes	characteristic
123	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
124	C₂²×C₅₃	#435	1	1	yes	characteristic
125	C₂×D₅₃	—	4	2	no	—
126	C₂×D₅₃	—	4	1	no	—
127	C₄×C₅₃	—	2	4	no	—
128	C₄×C₅₃	#388	1	1	yes	characteristic
129	C₄×C₅₃	#431	1	1	yes	characteristic
130	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
131	C₄×C₅₃	#436	1	1	yes	characteristic
132	Dic₅₃	—	4	2	no	—
133	Dic₅₃	—	4	1	no	—
134	#2151	—	53	1	no	—
135	C₂×#350	—	53	1	no	—
136	#2113	—	53	1	no	—
137	C₂×C₄×C₁₆	—	53	1	no	—
138	#1764	—	53	1	no	—
139	#1379	—	53	1	no	—
140	#1376	D₅₃	1	1	yes	characteristic, complemented
141	C₂×C₅₃	—	2	1	no	—
142	C₂×C₅₃	—	2	1	no	—
143	C₂×C₅₃	#1923	1	1	yes	characteristic
144	C₂×C₅₃	#2170	1	2	yes	—
145	D₅₃	—	8	1	no	—
146	C₂×#131	—	53	1	no	—
147	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
148	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
149	C₂²×C₁₆	—	53	1	no	—
150	C₂²×C₁₆	—	53	1	no	—
151	C₂×#104	—	53	1	no	—
152	#416	—	53	1	no	—
153	#405	—	53	1	no	—
154	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
155	C₄⋊C₁₆	—	106	2	no	—
156	#350	—	106	2	no	—
157	C₂²⋊C₁₆	—	106	2	no	—
158	C₄×C₁₆	—	106	2	no	—
159	#341	—	53	1	no	—
160	#339	—	53	1	no	—
161	C₅₃	#13124	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	—	53	1	no	—
163	#131	—	106	1	no	—
164	#131	—	106	1	no	—
165	#131	—	106	1	no	—
166	#131	—	106	1	no	—
167	C₂×M₁₆	—	53	1	no	—
168	C₂×M₁₆	—	53	1	no	—
169	C₂×M₁₆	—	106	1	no	—
170	C₂²×C₈	—	106	1	no	—
171	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
173	#117	—	53	1	no	—
174	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
175	M₃₂	—	2	1	no	—
176	M₃₂	—	2	1	no	—
177	M₃₂	—	2	1	no	—
178	M₃₂	—	2	1	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	53	2	no	—
182	C₂×C₁₆	—	53	2	no	—
183	C₂×C₁₆	—	106	1	no	—
184	C₂×C₁₆	—	106	1	no	—
185	C₂×C₁₆	—	106	1	no	—
186	C₂×C₁₆	—	106	1	no	—
187	#104	—	106	2	no	—
188	#103	—	53	2	no	—
189	#102	—	53	2	no	—
190	#97	—	106	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	106	1	no	—
194	#41	—	106	1	no	—
195	#41	—	106	1	no	—
196	#41	—	106	1	no	—
197	#41	—	106	1	no	—
198	C₂×Q₈	—	53	1	no	—
199	C₂×D₄	—	53	1	no	—
200	C₂×D₄	—	106	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	106	1	no	—
203	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
204	M₁₆	—	106	1	no	—
205	M₁₆	—	106	1	no	—
206	M₁₆	—	106	1	no	—
207	M₁₆	—	106	1	no	—
208	M₁₆	—	212	1	no	—
209	M₁₆	—	212	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	106	1	no	—
213	C₂×C₈	—	106	1	no	—
214	C₂×C₈	—	212	1	no	—
215	C₂×C₈	—	212	1	no	—
216	C₂×C₈	C₈×D₅₃	1	2	yes	—
217	C₂×C₈	#11719945	1	1	yes	characteristic
218	C₂×C₈	#11719945	1	1	yes	characteristic
219	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
220	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
221	C₂×C₈	#11719947	1	1	yes	characteristic
222	C₂×C₈	#11719949	1	1	yes	characteristic
223	C₄⋊C₄	—	53	2	no	—
224	C₂²⋊C₄	—	106	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	106	1	no	—
227	C₄²	#11719912	1	2	yes	—
228	C₁₆	—	4	1	no	—
229	C₁₆	—	4	1	no	—
230	C₁₆	—	106	2	no	—
231	C₁₆	—	106	2	no	—
232	C₂³	—	106	1	no	—
233	C₂³	#13133225	1	1	yes	characteristic
234	Q₈	—	106	1	no	—
235	Q₈	—	106	1	no	—
236	D₄	—	106	1	no	—
237	D₄	—	106	1	no	—
238	D₄	—	212	1	no	—
239	D₄	—	212	1	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	106	2	no	—
243	C₂×C₄	—	106	1	no	—
244	C₂×C₄	—	212	1	no	—
245	C₂×C₄	—	212	1	no	—
246	C₂×C₄	#13133174	1	1	yes	characteristic
247	C₂×C₄	#13133177	1	1	yes	characteristic
248	C₂×C₄	#13133179	1	1	yes	characteristic
249	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
250	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
251	C₂×C₄	#13133225	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	212	1	no	—
254	C₈	—	212	1	no	—
255	C₈	#13133190	1	1	yes	characteristic
256	C₈	#13133202	1	1	yes	characteristic
257	C₈	#13133203	1	1	yes	characteristic
258	C₈	#13133212	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	212	2	no	—
261	C₂²	—	212	1	no	—
262	C₂²	#16810685	1	1	yes	characteristic
263	C₂²	#16810688	1	1	yes	characteristic
264	C₂²	#16810691	1	1	yes	characteristic
265	C₄	—	2	4	no	—
266	C₄	—	212	2	no	—
267	C₄	—	212	1	no	—
268	C₄	#16810512	1	1	yes	characteristic
269	C₄	#16810674	1	1	yes	characteristic
270	C₄	#16810686	1	1	yes	characteristic
271	C₄	#16810700	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	424	1	no	—
275	C₂	#20275319	1	1	yes	characteristic
276	C₂	#20276551	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1376	C₂
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1376	D₅₃
hypocenter	C₅₃	#13124
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
1. #20276475	C₂	semidirect
4. C₂×C₄×#11719908	C₂	semidirect
6. #20272805	C₂	semidirect
7. C₅₃×#1376	C₂	semidirect
140. #1376	D₅₃	semidirect
161. C₅₃	#13124	semidirect

Maximal subgroups

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

subgroup	index
#20276475	2
C₂×#16810408	2
#20276330	2
C₂×C₄×#11719908	2
#20274552	2
#20272805	2
C₅₃×#1376	2
#13124	53