Subgroups — #27277741

Order 13568.

310 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274401	C₂	1	1	yes	maximal, characteristic
2	#20273463	C₂	1	1	yes	maximal, characteristic
3	#20272559	C₂	1	2	yes	maximal
4	#20272543	C₂	1	2	yes	maximal
5	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133128	C₂²	1	1	yes	characteristic
8	C₂×#13133127	C₄	1	1	yes	characteristic
9	C₂×#13133127	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
12	#16810469	C₂²	1	1	yes	characteristic
13	#16810376	—	2	1	no	—
14	#16810374	—	2	1	no	—
15	C₅₃×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#335	—	2	1	no	—
18	C₅₃×#334	—	2	1	no	—
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄⋊#88322	—	2	2	no	—
24	C₂²⋊#88322	—	2	2	no	—
25	#13133128	—	2	1	no	—
26	#13133128	—	2	1	no	—
27	#13133127	—	2	2	no	—
28	#13133127	C₈	1	4	yes	complemented
29	C₄×#88322	—	2	2	no	—
30	C₈×Dic₅₃	—	2	2	no	—
31	C₈×Dic₅₃	—	2	2	no	—
32	#13133124	—	2	1	no	—
33	#13133124	—	2	1	no	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	—	2	2	no	—
38	C₂²⋊C₈×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
43	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
47	C₂²×Dic₅₃	—	2	1	no	—
48	C₂²×Dic₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	—	2	1	no	—
50	C₂×C₈×C₅₃	—	2	1	no	—
51	C₂×C₈×C₅₃	—	4	2	no	—
52	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
58	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₅₃	#41	1	1	yes	characteristic
61	C₄²×C₅₃	#41	1	1	yes	characteristic
62	C₄×Dic₅₃	—	2	1	no	—
63	C₄×Dic₅₃	—	2	1	no	—
64	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
65	C₄×Dic₅₃	M₁₆	1	2	yes	—
66	C₂×#88322	—	2	1	no	—
67	C₂×#88322	—	2	1	no	—
68	C₂×#88322	—	2	1	no	—
69	C₂×#88322	—	2	1	no	—
70	C₂×#88322	—	4	2	no	—
71	C₂×#88322	C₂×C₈	1	2	yes	—
72	C₂×#88322	M₁₆	1	2	yes	—
73	C₂³×C₅₃	#117	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	4	1	no	—
75	C₂×C₄×C₅₃	—	4	1	no	—
76	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×Dic₅₃	—	2	2	no	—
87	C₂×Dic₅₃	—	2	2	no	—
88	C₂×Dic₅₃	—	4	1	no	—
89	C₂×Dic₅₃	—	4	1	no	—
90	C₂×Dic₅₃	—	4	1	no	—
91	C₂×Dic₅₃	—	4	1	no	—
92	C₈×C₅₃	—	2	1	no	—
93	C₈×C₅₃	—	2	1	no	—
94	C₈×C₅₃	—	8	2	no	—
95	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
96	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
97	#88322	—	2	2	no	—
98	#88322	—	2	2	no	—
99	#88322	—	8	2	no	—
100	#8781	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	#382	1	1	yes	characteristic
104	C₂²×C₅₃	#431	1	1	yes	characteristic
105	C₂²×C₅₃	#432	1	1	yes	characteristic
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
111	C₄×C₅₃	#423	1	1	yes	characteristic
112	C₄×C₅₃	#431	1	1	yes	characteristic
113	C₄×C₅₃	#432	1	1	yes	characteristic
114	Dic₅₃	—	4	2	no	—
115	Dic₅₃	—	4	2	no	—
116	#1726	—	53	1	no	—
117	#1564	—	53	1	no	—
118	#1564	D₅₃	1	1	yes	characteristic
119	#1281	—	53	2	no	—
120	#1274	—	53	2	no	—
121	C₂×C₅₃	—	2	1	no	—
122	C₂×C₅₃	—	2	1	no	—
123	C₂×C₅₃	#1831	1	1	yes	characteristic
124	C₂×C₅₃	#1833	1	1	yes	characteristic
125	C₂×C₅₃	#1833	1	1	yes	characteristic
126	#431	—	53	2	no	—
127	#431	C₂×D₅₃	1	2	yes	—
128	C₂×#97	—	53	1	no	—
129	C₂×#97	—	53	1	no	—
130	C₂×#97	—	53	1	no	—
131	C₂×C₄×C₈	—	53	1	no	—
132	C₂×C₄×C₈	—	53	1	no	—
133	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
134	#376	—	53	1	no	—
135	#335	—	2	1	no	—
136	#335	—	106	1	no	—
137	#334	—	2	1	no	—
138	#334	—	106	1	no	—
139	C₅₃	#8781	1	1	yes	characteristic, complemented, Sylow
140	C₂²×C₈	—	53	1	no	—
141	C₂²×C₈	—	53	1	no	—
142	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
143	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
144	C₂×C₄²	—	53	1	no	—
145	C₂×C₄²	—	53	1	no	—
146	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	106	2	no	—
149	C₂²⋊C₈	—	2	2	no	—
150	C₂²⋊C₈	—	106	2	no	—
151	#97	—	53	4	no	—
152	#97	—	106	2	no	—
153	#97	—	106	1	no	—
154	#97	—	106	1	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	106	2	no	—
157	C₄×C₈	—	106	2	no	—
158	C₄×C₈	—	106	2	no	—
159	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
160	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
163	#95	—	106	1	no	—
164	#95	—	106	1	no	—
165	C₂²×C₄	—	106	1	no	—
166	C₂²×C₄	—	106	1	no	—
167	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
168	C₂²×C₄	#11719947	1	1	yes	characteristic
169	C₂²×C₄	#11719949	1	1	yes	characteristic
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	53	2	no	—
174	C₂×C₈	—	53	2	no	—
175	C₂×C₈	—	106	1	no	—
176	C₂×C₈	—	106	1	no	—
177	C₂×C₈	—	106	1	no	—
178	C₂×C₈	—	106	1	no	—
179	C₂×C₈	—	212	2	no	—
180	C₂×C₈	C₈×D₅₃	1	2	yes	—
181	C₂×C₈	#11719912	1	2	yes	—
182	C₂×C₈	#11719919	1	1	yes	characteristic
183	C₂×C₈	#11719919	1	1	yes	characteristic
184	C₂×C₈	#11719919	1	1	yes	characteristic
185	C₂×C₈	#11719919	1	1	yes	characteristic
186	C₄²	—	53	2	no	—
187	C₄²	—	53	2	no	—
188	C₄²	—	106	1	no	—
189	C₄²	—	106	1	no	—
190	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
191	C₄²	#11719947	1	1	yes	characteristic
192	C₄²	#11719949	1	1	yes	characteristic
193	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
194	C₂³	#13133175	1	1	yes	characteristic
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	106	2	no	—
198	C₂×C₄	—	106	2	no	—
199	C₂×C₄	—	212	1	no	—
200	C₂×C₄	—	212	1	no	—
201	C₂×C₄	—	212	1	no	—
202	C₂×C₄	—	212	1	no	—
203	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
204	C₂×C₄	#13133175	1	1	yes	characteristic
205	C₂×C₄	#13133204	1	1	yes	characteristic
206	C₂×C₄	#13133206	1	1	yes	characteristic
207	C₂×C₄	#13133208	1	1	yes	characteristic
208	C₂×C₄	#13133208	1	1	yes	characteristic
209	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
210	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
211	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
212	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	—	8	2	no	—
216	C₈	—	106	2	no	—
217	C₈	—	106	2	no	—
218	C₈	—	424	2	no	—
219	C₈	#13133146	1	2	yes	—
220	C₈	#13133146	1	2	yes	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	#16810488	1	1	yes	characteristic
224	C₂²	#16810675	1	1	yes	characteristic
225	C₂²	#16810679	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	212	2	no	—
231	C₄	—	212	2	no	—
232	C₄	C₂×#13133146	1	1	yes	characteristic
233	C₄	#16810646	1	1	yes	characteristic
234	C₄	#16810675	1	1	yes	characteristic
235	C₄	#16810679	1	1	yes	characteristic
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	#20274831	1	1	yes	characteristic
239	C₂	#20274838	1	1	yes	characteristic
240	C₂	#20274840	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133175
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#8781
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
28. #13133127	C₈	semidirect
139. C₅₃	#8781	semidirect

Maximal subgroups

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

subgroup	index
#20274401	2
#20273463	2
#20272559	2
#20272543	2
C₅₃×#1564	2
#8781	53