Subgroups — #27275222

Order 13568.

502 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133146	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133146	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
5	#20272542	C₂	1	1	yes	maximal, characteristic
6	#20272542	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133146	C₄	1	2	yes	—
9	C₂×#13133146	C₄	1	2	yes	—
10	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
14	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
15	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
16	#16810377	C₄	1	2	yes	—
17	#16810377	C₄	1	2	yes	—
18	#16810345	—	2	2	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#322	—	2	2	no	—
23	C₂²×#88322	C₂×C₄	1	2	yes	—
24	C₂²×#88322	C₂×C₄	1	2	yes	—
25	C₂²×#88322	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	#13133146	—	2	4	no	—
33	#13133146	C₈	1	8	yes	complemented
34	C₄×#88322	—	2	1	no	—
35	C₄×#88322	—	2	1	no	—
36	C₄×#88322	—	2	1	no	—
37	C₄×#88322	—	2	1	no	—
38	C₄×#88322	—	2	1	no	—
39	C₄×#88322	—	2	1	no	—
40	C₈×Dic₅₃	—	2	4	no	—
41	C₈×Dic₅₃	—	2	4	no	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₅₃	—	2	1	no	—
47	C₄×C₈×C₅₃	—	2	1	no	—
48	C₄×C₈×C₅₃	—	2	1	no	—
49	C₄×C₈×C₅₃	—	2	1	no	—
50	C₄×C₈×C₅₃	D₄	1	2	yes	—
51	C₄×C₈×C₅₃	Q₈	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
56	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
57	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
58	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
67	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
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	C₄²×C₅₃	#41	1	1	yes	characteristic
72	C₄×Dic₅₃	—	2	2	no	—
73	C₄×Dic₅₃	—	2	2	no	—
74	C₄×Dic₅₃	—	2	2	no	—
75	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
76	C₄×Dic₅₃	M₁₆	1	4	yes	—
77	C₄×Dic₅₃	M₁₆	1	4	yes	—
78	C₂×#88322	—	2	2	no	—
79	C₂×#88322	—	2	2	no	—
80	C₂×#88322	—	2	2	no	—
81	C₂×#88322	—	2	2	no	—
82	C₂×#88322	—	2	2	no	—
83	C₂×#88322	—	2	2	no	—
84	C₂×#88322	—	2	2	no	—
85	C₂×#88322	C₂×C₈	1	4	yes	—
86	C₂×#88322	M₁₆	1	4	yes	—
87	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
98	C₂×C₄×C₅₃	#131	1	2	yes	—
99	C₂×C₄×C₅₃	#131	1	2	yes	—
100	C₂×Dic₅₃	—	2	1	no	—
101	C₂×Dic₅₃	—	2	1	no	—
102	C₂×Dic₅₃	—	2	1	no	—
103	C₂×Dic₅₃	—	2	1	no	—
104	C₂×Dic₅₃	—	2	1	no	—
105	C₂×Dic₅₃	—	2	1	no	—
106	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
107	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
108	C₂×Dic₅₃	#97	1	2	yes	—
109	C₂×Dic₅₃	#97	1	2	yes	—
110	C₂×Dic₅₃	#97	1	2	yes	—
111	C₂×Dic₅₃	#97	1	2	yes	—
112	C₈×C₅₃	—	2	4	no	—
113	C₈×C₅₃	—	2	4	no	—
114	C₈×C₅₃	—	4	2	no	—
115	#88322	—	2	4	no	—
116	#88322	—	2	4	no	—
117	#88322	—	4	2	no	—
118	#8261	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
122	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
123	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
124	C₂²×C₅₃	#405	1	1	yes	characteristic
125	C₂²×C₅₃	#405	1	1	yes	characteristic
126	C₄×C₅₃	#423	1	2	yes	—
127	C₄×C₅₃	#432	1	2	yes	—
128	C₄×C₅₃	C₈×D₄	1	2	yes	—
129	C₄×C₅₃	#436	1	2	yes	—
130	C₄×C₅₃	C₈×Q₈	1	2	yes	—
131	C₄×C₅₃	#446	1	2	yes	—
132	Dic₅₃	—	2	2	no	—
133	Dic₅₃	—	2	2	no	—
134	Dic₅₃	—	2	2	no	—
135	Dic₅₃	#322	1	4	yes	complemented
136	C₄×C₄⋊C₈	—	53	1	no	—
137	C₄×C₄⋊C₈	—	53	1	no	—
138	C₄²×C₈	—	53	1	no	—
139	C₂×#322	—	53	1	no	—
140	C₂×#322	D₅₃	1	1	yes	characteristic
141	#1273	—	53	1	no	—
142	#1273	—	53	1	no	—
143	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
144	C₂×C₅₃	C₈×M₁₆	1	1	yes	characteristic
145	C₂×C₅₃	#1451	1	1	yes	characteristic
146	C₂×C₅₃	#1452	1	1	yes	characteristic
147	C₂×C₅₃	#1767	1	1	yes	characteristic
148	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₅₃	#1770	1	1	yes	characteristic
150	C₂×C₄⋊C₈	—	53	2	no	—
151	C₂×C₄⋊C₈	—	53	2	no	—
152	C₂×C₄×C₈	—	53	2	no	—
153	C₂×C₄×C₈	—	53	2	no	—
154	C₂×C₄×C₈	—	53	1	no	—
155	C₂×C₄×C₈	—	53	1	no	—
156	C₂×C₄×C₈	—	53	1	no	—
157	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
160	C₄³	—	53	1	no	—
161	#336	—	53	2	no	—
162	#336	—	53	2	no	—
163	#322	—	2	2	no	—
164	#322	—	106	2	no	—
165	C₅₃	#8261	1	1	yes	characteristic, complemented, Sylow
166	C₂²×C₈	—	53	2	no	—
167	C₂²×C₈	—	53	2	no	—
168	C₂²×C₈	—	53	2	no	—
169	C₂²×C₈	C₄×D₅₃	1	2	yes	—
170	C₂²×C₈	C₄×D₅₃	1	2	yes	—
171	C₂²×C₈	C₄×D₅₃	1	2	yes	—
172	C₂×C₄²	—	53	1	no	—
173	C₂×C₄²	—	53	1	no	—
174	C₂×C₄²	—	53	1	no	—
175	C₂×C₄²	—	53	1	no	—
176	C₂×C₄²	—	53	1	no	—
177	C₂×C₄²	—	53	1	no	—
178	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
179	C₄⋊C₈	—	53	8	no	—
180	C₄⋊C₈	—	106	4	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	106	4	no	—
186	C₄×C₈	—	106	4	no	—
187	C₄×C₈	—	106	1	no	—
188	C₄×C₈	—	106	1	no	—
189	C₄×C₈	—	106	1	no	—
190	C₄×C₈	—	106	1	no	—
191	C₄×C₈	—	106	1	no	—
192	C₄×C₈	—	106	1	no	—
193	C₄×C₈	Dic₁₀₆	1	2	yes	—
194	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
195	C₂²×C₄	—	53	1	no	—
196	C₂²×C₄	—	53	1	no	—
197	C₂²×C₄	—	53	1	no	—
198	C₂²×C₄	—	53	1	no	—
199	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	53	4	no	—
210	C₂×C₈	—	53	4	no	—
211	C₂×C₈	—	106	2	no	—
212	C₂×C₈	—	106	2	no	—
213	C₂×C₈	—	106	2	no	—
214	C₂×C₈	—	106	2	no	—
215	C₂×C₈	—	106	2	no	—
216	C₂×C₈	—	106	2	no	—
217	C₂×C₈	—	106	2	no	—
218	C₂×C₈	C₈×D₅₃	1	4	yes	—
219	C₂×C₈	#11719919	1	4	yes	—
220	C₄²	—	53	4	no	—
221	C₄²	—	53	4	no	—
222	C₄²	—	53	4	no	—
223	C₄²	—	106	2	no	—
224	C₄²	—	106	2	no	—
225	C₄²	—	106	2	no	—
226	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
227	C₄²	#11719945	1	1	yes	characteristic
228	C₄²	#11719945	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
230	C₂³	C₄²×D₅₃	1	1	yes	characteristic
231	C₂×C₄	—	53	2	no	—
232	C₂×C₄	—	53	2	no	—
233	C₂×C₄	—	53	2	no	—
234	C₂×C₄	—	53	2	no	—
235	C₂×C₄	—	53	2	no	—
236	C₂×C₄	—	53	2	no	—
237	C₂×C₄	—	106	1	no	—
238	C₂×C₄	—	106	1	no	—
239	C₂×C₄	—	106	1	no	—
240	C₂×C₄	—	106	1	no	—
241	C₂×C₄	—	106	1	no	—
242	C₂×C₄	—	106	1	no	—
243	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
244	C₂×C₄	#13133174	1	1	yes	characteristic
245	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
246	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
247	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
248	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
249	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
250	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
251	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
252	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
253	C₂×C₄	#13133223	1	2	yes	—
254	C₂×C₄	#13133225	1	2	yes	—
255	C₈	—	2	4	no	—
256	C₈	—	2	4	no	—
257	C₈	—	4	2	no	—
258	C₈	—	106	4	no	—
259	C₈	—	106	4	no	—
260	C₈	—	212	2	no	—
261	C₂²	C₄×#11719919	1	1	yes	characteristic
262	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
263	C₂²	#97×D₅₃	1	1	yes	characteristic
264	C₂²	#97×D₅₃	1	1	yes	characteristic
265	C₂²	#16810575	1	1	yes	characteristic
266	C₂²	#16810577	1	1	yes	characteristic
267	C₂²	#16810580	1	1	yes	characteristic
268	C₄	—	53	4	no	—
269	C₄	—	106	2	no	—
270	C₄	—	106	2	no	—
271	C₄	—	106	2	no	—
272	C₄	#16810646	1	2	yes	—
273	C₄	#16810678	1	2	yes	—
274	C₄	C₈×C₂²⋊D₅₃	1	2	yes	—
275	C₄	#16810699	1	2	yes	—
276	C₄	C₈×Dic₁₀₆	1	2	yes	—
277	C₄	#16810751	1	2	yes	—
278	C₂	#322×D₅₃	1	1	yes	characteristic
279	C₂	#20272994	1	1	yes	characteristic
280	C₂	#20273003	1	1	yes	characteristic
281	C₂	#20273006	1	1	yes	characteristic
282	C₂	#20274569	1	1	yes	characteristic
283	C₂	C₈×#11719919	1	1	yes	characteristic
284	C₂	#20274583	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₅₃×#322	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#8261
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
33. #13133146	C₈	semidirect
135. Dic₅₃	#322	semidirect
165. C₅₃	#8261	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133146	2
C₄×C₈×Dic₅₃	2
C₂×#16810345	2
#20272542	2
C₂×C₅₃×#322	2
#8261	53