Subgroups — #27277171

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 268 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×#16810376	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	#16810376	—	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	1	no	—
47	C₂×Q₈×C₅₃	—	2	1	no	—
48	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂×Q₈×C₅₃	D₈	1	2	yes	—
50	C₂×Q₈×C₅₃	Q₁₆	1	2	yes	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
56	C₂²×Dic₅₃	—	4	2	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	—	2	1	no	—
65	C₄⋊C₄×C₅₃	—	2	2	no	—
66	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	—	2	2	no	—
69	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₅₃	#41	1	1	yes	characteristic
73	C₄⋊Dic₅₃	—	4	2	no	—
74	C₄⋊Dic₅₃	—	4	2	no	—
75	C₄⋊Dic₅₃	—	4	2	no	—
76	C₄⋊Dic₅₃	—	4	2	no	—
77	C₄⋊Dic₅₃	—	8	2	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂×#88322	—	4	2	no	—
80	C₂×#88322	—	4	2	no	—
81	C₂³×C₅₃	#121	1	1	yes	characteristic
82	Q₈×C₅₃	—	2	2	no	—
83	Q₈×C₅₃	—	2	2	no	—
84	Q₈×C₅₃	—	2	2	no	—
85	Q₈×C₅₃	#107	1	4	yes	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#135	1	2	yes	—
101	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#137	1	2	yes	—
104	C₂×Dic₅₃	—	8	2	no	—
105	C₂×Dic₅₃	—	8	2	no	—
106	C₂×Dic₅₃	—	8	2	no	—
107	C₈×C₅₃	—	2	2	no	—
108	C₈×C₅₃	—	2	2	no	—
109	C₈×C₅₃	—	4	2	no	—
110	#88322	—	8	2	no	—
111	#8649	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	#380	1	1	yes	characteristic
113	C₂²×C₅₃	#464	1	2	yes	—
114	C₂²×C₅₃	#466	1	1	yes	characteristic
115	C₂²×C₅₃	#467	1	1	yes	characteristic
116	C₂²×C₅₃	#471	1	2	yes	—
117	C₄×C₅₃	—	2	4	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	4	2	no	—
121	C₄×C₅₃	#416	1	1	yes	characteristic
122	C₄×C₅₃	#419	1	1	yes	characteristic
123	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
124	C₄×C₅₃	#427	1	1	yes	characteristic
125	C₄×C₅₃	#466	1	1	yes	characteristic
126	C₄×C₅₃	#467	1	1	yes	characteristic
127	C₄×C₅₃	#479	1	2	yes	—
128	Dic₅₃	—	16	2	no	—
129	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
130	#1834	—	53	1	no	—
131	#1803	—	53	2	no	—
132	C₂×#335	—	53	1	no	—
133	#1296	—	53	2	no	—
134	C₂×C₅₃	#1682	1	2	yes	—
135	C₂×C₅₃	#1684	1	2	yes	—
136	C₂×C₅₃	#1805	1	1	yes	characteristic
137	C₂×C₅₃	#1808	1	1	yes	characteristic
138	C₂×C₅₃	#1815	1	1	yes	characteristic
139	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
140	C₈×Q₈	—	2	2	no	—
141	C₈×Q₈	Dic₅₃	1	4	yes	complemented
142	C₂×C₄⋊C₈	—	2	1	no	—
143	C₂×C₄⋊C₈	—	53	2	no	—
144	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
145	C₂×#103	—	106	2	no	—
146	C₂×C₄×C₈	—	2	1	no	—
147	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
148	#384	—	53	2	no	—
149	#340	—	106	1	no	—
150	#340	—	106	1	no	—
151	#340	—	106	2	no	—
152	#335	—	106	2	no	—
153	C₅₃	#8649	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	106	2	no	—
157	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₄×Q₈	—	2	1	no	—
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	Dic₁₀₆	1	2	yes	—
162	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	106	2	no	—
165	C₂×C₄⋊C₄	—	106	2	no	—
166	C₂×C₄⋊C₄	—	106	2	no	—
167	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
168	C₂×C₄²	—	2	1	no	—
169	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	106	2	no	—
173	C₄⋊C₈	—	106	2	no	—
174	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
175	#103	—	212	4	no	—
176	C₄×C₈	—	2	4	no	—
177	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
178	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
179	C₂×Q₈	—	2	1	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	D₄₂₄	1	2	yes	—
183	C₂×Q₈	Dic₂₁₂	1	2	yes	—
184	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	212	2	no	—
188	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
189	C₂²×C₄	#11719945	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	212	2	no	—
196	C₂×C₈	—	212	2	no	—
197	C₂×C₈	—	212	2	no	—
198	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
199	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	424	2	no	—
204	C₄⋊C₄	—	212	2	no	—
205	C₄⋊C₄	—	212	2	no	—
206	C₄⋊C₄	—	212	2	no	—
207	C₄⋊C₄	—	212	2	no	—
208	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
212	C₄²	#11719945	1	1	yes	characteristic
213	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
215	C₂³	#13133191	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	Q₈	—	2	2	no	—
218	Q₈	—	2	2	no	—
219	Q₈	#13133151	1	4	yes	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	424	2	no	—
227	C₂×C₄	—	424	2	no	—
228	C₂×C₄	—	424	2	no	—
229	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
231	C₂×C₄	#13133176	1	1	yes	characteristic
232	C₂×C₄	#13133191	1	1	yes	characteristic
233	C₂×C₄	#13133196	1	1	yes	characteristic
234	C₂×C₄	#13133196	1	1	yes	characteristic
235	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
236	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
237	C₂×C₄	#13133237	1	2	yes	—
238	C₂×C₄	#13133239	1	1	yes	characteristic
239	C₂×C₄	#13133239	1	1	yes	characteristic
240	C₂×C₄	#13133248	1	2	yes	—
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	424	2	no	—
245	C₂²	#16810483	1	1	yes	characteristic
246	C₂²	#16810845	1	2	yes	—
247	C₂²	#16810859	1	1	yes	characteristic
248	C₂²	#16810864	1	1	yes	characteristic
249	C₂²	#16810886	1	2	yes	—
250	C₄	—	2	4	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	848	2	no	—
255	C₄	#16810619	1	1	yes	characteristic
256	C₄	#16810632	1	1	yes	characteristic
257	C₄	C₂×#13133151	1	1	yes	characteristic
258	C₄	#16810661	1	1	yes	characteristic
259	C₄	#16810859	1	1	yes	characteristic
260	C₄	#16810864	1	1	yes	characteristic
261	C₄	#16810930	1	2	yes	—
262	C₂	#20274159	1	2	yes	—
263	C₂	#20274173	1	2	yes	—
264	C₂	#20274730	1	1	yes	characteristic
265	C₂	#20274744	1	1	yes	characteristic
266	C₂	#20274774	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	#8649
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
141. C₈×Q₈	Dic₅₃	semidirect
153. C₅₃	#8649	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×#16810376	2
#20272599	2
C₂×C₈×Q₈×C₅₃	2
#8649	53