Subgroups — #27272543

Order 13568.

432 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283078	C₂	1	1	yes	maximal, characteristic
2	#20274568	C₂	1	1	yes	maximal, characteristic
3	#20272537	C₂	1	4	yes	maximal
4	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133209	—	2	1	no	—
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×#13133128	—	2	1	no	—
8	#16810521	—	2	1	no	—
9	#16810521	—	2	1	no	—
10	#16810521	C₄	1	2	yes	—
11	#16810521	C₂²	1	2	yes	—
12	#16810493	C₄	1	1	yes	characteristic
13	#16810493	C₄	1	1	yes	characteristic
14	#16810489	—	2	1	no	—
15	#16810378	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	—	2	1	no	—
17	C₄³×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#336	—	2	1	no	—
19	C₅₃×#336	C₂²	1	2	yes	—
20	C₂²×#88322	—	2	2	no	—
21	#13133209	—	4	2	no	—
22	#13133209	—	4	2	no	—
23	C₂×C₄⋊Dic₅₃	—	2	1	no	—
24	C₂×C₄⋊Dic₅₃	—	2	1	no	—
25	C₂×C₄⋊Dic₅₃	—	2	1	no	—
26	C₂×C₄⋊Dic₅₃	—	2	1	no	—
27	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
29	C₂×#11719919	—	2	2	no	—
30	C₂×#11719919	—	2	2	no	—
31	C₄⋊#88322	—	2	4	no	—
32	#13133128	—	4	2	no	—
33	#13133124	—	2	2	no	—
34	#13133124	—	2	2	no	—
35	C₂²×C₈×C₅₃	—	2	2	no	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₅₃	—	2	4	no	—
44	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	2	2	no	—
52	C₂²×Dic₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	2	4	no	—
54	C₂×C₈×C₅₃	—	4	2	no	—
55	C₂×C₈×C₅₃	—	4	2	no	—
56	C₄²×C₅₃	—	2	2	no	—
57	C₄²×C₅₃	—	2	2	no	—
58	C₄²×C₅₃	—	2	2	no	—
59	C₄²×C₅₃	—	2	2	no	—
60	C₄²×C₅₃	—	2	2	no	—
61	C₄²×C₅₃	—	4	1	no	—
62	C₄²×C₅₃	D₈	1	2	yes	—
63	C₄²×C₅₃	SD₁₆	1	2	yes	—
64	C₄⋊Dic₅₃	—	2	4	no	—
65	C₄⋊Dic₅₃	—	4	2	no	—
66	C₄⋊Dic₅₃	—	4	2	no	—
67	C₄⋊Dic₅₃	—	4	2	no	—
68	C₄⋊Dic₅₃	—	4	2	no	—
69	C₄⋊Dic₅₃	—	4	2	no	—
70	#11719919	—	4	4	no	—
71	#11719919	—	4	2	no	—
72	#11719919	—	4	2	no	—
73	C₂×#88322	—	2	4	no	—
74	C₂×#88322	—	4	2	no	—
75	C₂×#88322	—	4	2	no	—
76	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	#101	1	2	yes	—
91	C₂×C₄×C₅₃	#102	1	2	yes	—
92	C₂×C₄×C₅₃	#102	1	2	yes	—
93	C₂×C₄×C₅₃	#104	1	2	yes	—
94	C₂×C₄×C₅₃	#104	1	2	yes	—
95	C₂×C₄×C₅₃	#131	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#137	1	2	yes	—
99	C₂×Dic₅₃	—	4	2	no	—
100	C₂×Dic₅₃	—	4	2	no	—
101	C₂×Dic₅₃	—	4	2	no	—
102	C₂×Dic₅₃	—	4	2	no	—
103	C₂×Dic₅₃	—	4	2	no	—
104	C₂×Dic₅₃	—	4	2	no	—
105	C₈×C₅₃	—	4	4	no	—
106	#88322	—	4	4	no	—
107	#7680	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	#408	1	1	yes	characteristic
109	C₂²×C₅₃	#410	1	1	yes	characteristic
110	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
111	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
112	C₂²×C₅₃	#419	1	1	yes	characteristic
113	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
114	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
115	C₄×C₅₃	—	2	2	no	—
116	C₄×C₅₃	—	2	2	no	—
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	4	1	no	—
122	C₄×C₅₃	#330	1	4	yes	—
123	Dic₅₃	—	8	2	no	—
124	Dic₅₃	—	8	2	no	—
125	#2859	—	53	1	no	—
126	#1767	—	53	1	no	—
127	#1744	D₅₃	1	1	yes	characteristic
128	#1271	—	53	4	no	—
129	C₂×C₅₃	#1470	1	1	yes	characteristic
130	C₂×C₅₃	#1485	1	1	yes	characteristic
131	C₂×C₅₃	#1492	1	1	yes	characteristic
132	C₂×C₅₃	#1502	1	1	yes	characteristic
133	C₂×C₅₃	#1506	1	1	yes	characteristic
134	C₂×C₅₃	C₂×#330	1	1	yes	characteristic
135	C₂×C₅₃	#1546	1	1	yes	characteristic
136	C₂×#125	—	106	1	no	—
137	C₂×C₄⋊C₈	—	106	1	no	—
138	C₂×#97	—	106	1	no	—
139	C₂×C₄×C₈	—	2	1	no	—
140	#389	—	53	2	no	—
141	#389	—	53	2	no	—
142	#389	—	106	1	no	—
143	#389	—	106	1	no	—
144	#384	—	53	1	no	—
145	#384	—	53	1	no	—
146	#382	—	106	1	no	—
147	C₄³	C₂×D₅₃	1	1	yes	characteristic
148	#336	—	2	1	no	—
149	#336	—	53	2	no	—
150	#336	C₂×D₅₃	1	2	yes	—
151	C₅₃	#7680	1	1	yes	characteristic, complemented, Sylow
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	106	2	no	—
154	#125	—	212	2	no	—
155	#125	—	212	2	no	—
156	C₂×C₄⋊C₄	—	53	2	no	—
157	C₂×C₄⋊C₄	—	53	2	no	—
158	C₂×C₄⋊C₄	—	106	2	no	—
159	C₂×C₄⋊C₄	—	106	2	no	—
160	C₂×C₄⋊C₄	—	106	1	no	—
161	C₂×C₄⋊C₄	—	106	1	no	—
162	C₂×C₄⋊C₄	—	106	1	no	—
163	C₂×C₄⋊C₄	—	106	1	no	—
164	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
165	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
166	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
167	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
171	C₄⋊C₈	—	106	4	no	—
172	#97	—	212	2	no	—
173	C₄×C₈	—	2	4	no	—
174	#95	—	106	2	no	—
175	#95	—	106	2	no	—
176	C₂²×C₄	—	106	2	no	—
177	C₂²×C₄	—	106	2	no	—
178	C₂²×C₄	#11719921	1	1	yes	characteristic
179	C₂²×C₄	#11719921	1	1	yes	characteristic
180	C₂²×C₄	#11719921	1	1	yes	characteristic
181	C₂²×C₄	#11719921	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	106	4	no	—
189	C₂×C₈	—	212	2	no	—
190	C₂×C₈	—	212	2	no	—
191	C₄⋊C₄	—	106	4	no	—
192	C₄⋊C₄	—	212	2	no	—
193	C₄⋊C₄	—	212	4	no	—
194	C₄⋊C₄	—	212	2	no	—
195	C₄⋊C₄	—	212	2	no	—
196	C₄⋊C₄	—	212	2	no	—
197	C₄⋊C₄	—	212	2	no	—
198	C₄⋊C₄	—	212	2	no	—
199	C₄⋊C₄	—	212	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	4	1	no	—
206	C₄²	#11719913	1	2	yes	—
207	C₄²	D₄₂₄	1	2	yes	—
208	C₂³	C₂×#11719921	1	1	yes	characteristic
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	212	2	no	—
223	C₂×C₄	—	212	2	no	—
224	C₂×C₄	—	212	2	no	—
225	C₂×C₄	—	212	2	no	—
226	C₂×C₄	—	212	2	no	—
227	C₂×C₄	—	212	2	no	—
228	C₂×C₄	#13133135	1	2	yes	—
229	C₂×C₄	#13133139	1	2	yes	—
230	C₂×C₄	#13133139	1	2	yes	—
231	C₂×C₄	#13133143	1	2	yes	—
232	C₂×C₄	#13133145	1	2	yes	—
233	C₂×C₄	#13133223	1	2	yes	—
234	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
235	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
236	C₂×C₄	#13133247	1	2	yes	—
237	C₈	—	4	4	no	—
238	C₈	—	212	4	no	—
239	C₂²	#16810589	1	1	yes	characteristic
240	C₂²	#16810596	1	1	yes	characteristic
241	C₂²	C₂×#13133135	1	1	yes	characteristic
242	C₂²	C₂×#13133139	1	1	yes	characteristic
243	C₂²	#16810631	1	1	yes	characteristic
244	C₂²	C₂×#13133143	1	1	yes	characteristic
245	C₂²	C₂×#13133145	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	1	no	—
253	C₄	—	424	2	no	—
254	C₄	—	424	2	no	—
255	C₄	#16810364	1	4	yes	—
256	C₂	#20273070	1	1	yes	characteristic
257	C₂	#20273135	1	1	yes	characteristic
258	C₂	#20273167	1	1	yes	characteristic
259	C₂	#20273211	1	1	yes	characteristic
260	C₂	#20273226	1	1	yes	characteristic
261	C₂	C₂×#16810364	1	1	yes	characteristic
262	C₂	#20273386	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719921
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1744	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#7680
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
151. C₅₃	#7680	semidirect

Maximal subgroups

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

subgroup	index
#20283078	2
#20274568	2
#20272537	2
C₅₃×#1744	2
#7680	53