Subgroups — #27277161

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274843	C₂	1	1	yes	maximal, characteristic
2	#20274720	C₂	1	2	yes	maximal
3	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
4	#20272599	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×#13133141	—	2	2	no	—
8	#16810493	C₂²	1	2	yes	—
9	#16810386	—	2	1	no	—
10	#16810386	—	2	1	no	—
11	#16810385	—	2	2	no	—
12	#16810374	—	2	2	no	—
13	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₅₃	—	2	2	no	—
15	C₈×Q₈×C₅₃	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	2	2	no	—
22	C₂×C₄⋊Dic₅₃	—	2	2	no	—
23	C₂×C₄⋊Dic₅₃	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	#13133141	—	4	4	no	—
27	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	—	2	1	no	—
29	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄×Q₈×C₅₃	—	2	1	no	—
32	C₄×Q₈×C₅₃	—	2	1	no	—
33	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×Q₈×C₅₃	Q₈	1	2	yes	—
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	—	2	1	no	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	—	2	4	no	—
40	C₄⋊C₈×C₅₃	—	2	1	no	—
41	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₅₃	—	2	4	no	—
43	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₂×Q₈×C₅₃	—	2	1	no	—
46	C₂×Q₈×C₅₃	—	2	2	no	—
47	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₂×Q₈×C₅₃	SD₁₆	1	4	yes	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
54	C₂²×Dic₅₃	—	4	2	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₄⋊C₄×C₅₃	—	2	1	no	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₅₃	#41	1	1	yes	characteristic
71	C₄⋊Dic₅₃	—	4	2	no	—
72	C₄⋊Dic₅₃	—	4	2	no	—
73	C₄⋊Dic₅₃	—	4	2	no	—
74	C₄⋊Dic₅₃	—	4	2	no	—
75	C₄⋊Dic₅₃	—	8	2	no	—
76	C₂×#88322	—	4	2	no	—
77	C₂×#88322	—	4	2	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂³×C₅₃	#121	1	1	yes	characteristic
80	Q₈×C₅₃	—	2	2	no	—
81	Q₈×C₅₃	—	2	4	no	—
82	Q₈×C₅₃	#106	1	4	yes	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	—	4	1	no	—
90	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
97	C₂×C₄×C₅₃	#135	1	2	yes	—
98	C₂×C₄×C₅₃	#135	1	2	yes	—
99	C₂×C₄×C₅₃	#137	1	2	yes	—
100	C₂×Dic₅₃	—	8	2	no	—
101	C₂×Dic₅₃	—	8	2	no	—
102	C₂×Dic₅₃	—	8	2	no	—
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	—	2	2	no	—
105	C₈×C₅₃	—	4	2	no	—
106	#88322	—	8	2	no	—
107	#8647	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	#380	1	1	yes	characteristic
109	C₂²×C₅₃	#464	1	2	yes	—
110	C₂²×C₅₃	#465	1	2	yes	—
111	C₂²×C₅₃	#472	1	1	yes	characteristic
112	C₂²×C₅₃	#473	1	1	yes	characteristic
113	C₄×C₅₃	—	2	4	no	—
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	—	4	2	no	—
117	C₄×C₅₃	#416	1	1	yes	characteristic
118	C₄×C₅₃	#419	1	1	yes	characteristic
119	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
120	C₄×C₅₃	#427	1	1	yes	characteristic
121	C₄×C₅₃	#465	1	2	yes	—
122	C₄×C₅₃	#479	1	2	yes	—
123	Dic₅₃	—	16	2	no	—
124	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
125	#1834	—	53	1	no	—
126	#1803	—	53	2	no	—
127	C₂×#334	—	53	1	no	—
128	#1296	—	53	2	no	—
129	C₂×C₅₃	#1678	1	2	yes	—
130	C₂×C₅₃	#1680	1	2	yes	—
131	C₂×C₅₃	#1805	1	1	yes	characteristic
132	C₂×C₅₃	#1807	1	1	yes	characteristic
133	C₂×C₅₃	#1816	1	1	yes	characteristic
134	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
135	C₈×Q₈	—	2	2	no	—
136	C₈×Q₈	Dic₅₃	1	4	yes	complemented
137	C₂×C₄⋊C₈	—	2	1	no	—
138	C₂×C₄⋊C₈	—	53	2	no	—
139	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
140	C₂×#103	—	106	2	no	—
141	C₂×C₄×C₈	—	2	1	no	—
142	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
143	#384	—	53	2	no	—
144	#340	—	106	1	no	—
145	#340	—	106	1	no	—
146	#340	—	106	2	no	—
147	#334	—	106	2	no	—
148	C₅₃	#8647	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	—	106	2	no	—
152	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
153	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	—	2	1	no	—
156	C₄×Q₈	Dic₁₀₆	1	2	yes	—
157	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	106	2	no	—
160	C₂×C₄⋊C₄	—	106	2	no	—
161	C₂×C₄⋊C₄	—	106	2	no	—
162	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
165	C₄⋊C₈	—	2	4	no	—
166	C₄⋊C₈	—	2	1	no	—
167	C₄⋊C₈	—	106	2	no	—
168	C₄⋊C₈	—	106	2	no	—
169	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
170	#103	—	212	4	no	—
171	C₄×C₈	—	2	4	no	—
172	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
173	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
174	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	#11719913	1	4	yes	—
177	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	212	2	no	—
181	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
182	C₂²×C₄	#11719945	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	212	2	no	—
189	C₂×C₈	—	212	2	no	—
190	C₂×C₈	—	212	2	no	—
191	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
192	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	424	2	no	—
197	C₄⋊C₄	—	212	2	no	—
198	C₄⋊C₄	—	212	2	no	—
199	C₄⋊C₄	—	212	2	no	—
200	C₄⋊C₄	—	212	2	no	—
201	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
205	C₄²	#11719945	1	1	yes	characteristic
206	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
207	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
208	C₂³	#13133191	1	1	yes	characteristic
209	Q₈	—	2	4	no	—
210	Q₈	—	2	2	no	—
211	Q₈	#13133149	1	4	yes	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	424	2	no	—
220	C₂×C₄	—	424	2	no	—
221	C₂×C₄	—	424	2	no	—
222	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
224	C₂×C₄	#13133176	1	1	yes	characteristic
225	C₂×C₄	#13133191	1	1	yes	characteristic
226	C₂×C₄	#13133196	1	1	yes	characteristic
227	C₂×C₄	#13133196	1	1	yes	characteristic
228	C₂×C₄	C₂×#11719913	1	2	yes	—
229	C₂×C₄	#13133237	1	2	yes	—
230	C₂×C₄	#13133239	1	2	yes	—
231	C₂×C₄	#13133248	1	2	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	—	4	2	no	—
235	C₈	—	424	2	no	—
236	C₂²	#16810483	1	1	yes	characteristic
237	C₂²	#16810845	1	2	yes	—
238	C₂²	#16810852	1	2	yes	—
239	C₂²	#16810893	1	1	yes	characteristic
240	C₂²	#16810898	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	4	2	no	—
245	C₄	—	848	2	no	—
246	C₄	#16810619	1	1	yes	characteristic
247	C₄	#16810632	1	1	yes	characteristic
248	C₄	C₂×#13133149	1	1	yes	characteristic
249	C₄	#16810661	1	1	yes	characteristic
250	C₄	#16810852	1	2	yes	—
251	C₄	#16810930	1	2	yes	—
252	C₂	#20274131	1	2	yes	—
253	C₂	#20274145	1	2	yes	—
254	C₂	#20274730	1	1	yes	characteristic
255	C₂	#20274740	1	1	yes	characteristic
256	C₂	#20274779	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#8647
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₅₃	C₄	semidirect
136. C₈×Q₈	Dic₅₃	semidirect
148. C₅₃	#8647	semidirect

Maximal subgroups

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

subgroup	index
#20274843	2
#20274720	2
C₂×#16810374	2
#20272599	2
C₂×C₈×Q₈×C₅₃	2
#8647	53