Subgroups — #27270667

Order 13568.

406 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274695	C₂	1	1	yes	maximal, characteristic
2	#20274409	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1797	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	C₄	1	2	yes	—
5	C₂×C₄⋊#88322	—	2	1	no	—
6	C₂×C₂²⋊#88322	C₄	1	2	yes	—
7	#16810378	—	2	1	no	—
8	#16810378	—	2	1	no	—
9	#16810378	—	2	1	no	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
12	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
13	C₅₃×#336	—	2	1	no	—
14	C₂×#11719917	—	2	2	no	—
15	C₂×#11719917	—	4	2	no	—
16	C₂×#11719917	—	4	2	no	—
17	C₂×#11719917	C₈	1	4	yes	complemented
18	C₂²×#88322	—	2	2	no	—
19	C₂²×#88322	—	2	2	no	—
20	C₄⋊#88322	—	4	2	no	—
21	C₂²⋊#88322	—	4	2	no	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	C₂×C₄²×C₅₃	—	4	2	no	—
27	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	—	4	2	no	—
32	C₂²⋊C₈×C₅₃	—	4	2	no	—
33	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	—	2	1	no	—
36	C₂²×C₄×C₅₃	—	2	1	no	—
37	C₂²×C₄×C₅₃	—	2	1	no	—
38	C₂²×C₄×C₅₃	—	2	1	no	—
39	C₂²×C₄×C₅₃	—	2	1	no	—
40	C₂²×C₄×C₅₃	—	2	1	no	—
41	C₂²×C₄×C₅₃	—	4	1	no	—
42	C₂²×C₄×C₅₃	—	4	1	no	—
43	C₂²×C₄×C₅₃	—	4	1	no	—
44	C₂²×C₄×C₅₃	—	4	1	no	—
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	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
53	C₂×C₈×C₅₃	—	4	2	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	—	4	1	no	—
59	C₄²×C₅₃	—	4	1	no	—
60	#11719917	—	4	4	no	—
61	#11719917	—	4	4	no	—
62	C₂×#88322	—	2	4	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂×#88322	—	4	2	no	—
65	C₂×#88322	—	4	2	no	—
66	C₂×#88322	—	4	2	no	—
67	C₂×#88322	—	4	2	no	—
68	C₂³×C₅₃	—	2	1	no	—
69	C₂³×C₅₃	—	2	1	no	—
70	C₂³×C₅₃	—	2	1	no	—
71	C₂³×C₅₃	—	2	1	no	—
72	C₂³×C₅₃	—	2	1	no	—
73	C₂³×C₅₃	—	2	1	no	—
74	C₂³×C₅₃	#95	1	1	yes	characteristic
75	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
76	C₂³×C₅₃	#97	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	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	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	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
98	C₂×C₄×C₅₃	#99	1	2	yes	—
99	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
106	C₈×C₅₃	—	8	2	no	—
107	#88322	—	4	4	no	—
108	#88322	—	8	2	no	—
109	#7034	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	2	1	no	—
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	4	1	no	—
117	C₂²×C₅₃	—	4	1	no	—
118	C₂²×C₅₃	—	4	1	no	—
119	C₂²×C₅₃	—	4	1	no	—
120	C₂²×C₅₃	#336	1	1	yes	characteristic
121	C₂²×C₅₃	#342	1	1	yes	characteristic
122	C₂²×C₅₃	#343	1	1	yes	characteristic
123	C₂²×C₅₃	#343	1	1	yes	characteristic
124	C₂²×C₅₃	#343	1	1	yes	characteristic
125	C₂²×C₅₃	#344	1	1	yes	characteristic
126	C₂²×C₅₃	#344	1	1	yes	characteristic
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	4	1	no	—
133	C₄×C₅₃	—	4	1	no	—
134	C₄×C₅₃	#323	1	2	yes	—
135	C₄×C₅₃	#324	1	2	yes	—
136	#1797	—	53	1	no	—
137	#1797	D₅₃	1	1	yes	characteristic
138	#1728	—	53	1	no	—
139	C₂×C₅₃	—	4	1	no	—
140	C₂×C₅₃	—	4	1	no	—
141	C₂×C₅₃	#1280	1	1	yes	characteristic
142	C₂×C₅₃	#1283	1	1	yes	characteristic
143	C₂×C₅₃	#1283	1	1	yes	characteristic
144	C₂×C₅₃	#1305	1	1	yes	characteristic
145	C₂×C₅₃	#1305	1	1	yes	characteristic
146	C₂×C₅₃	#1307	1	1	yes	characteristic
147	C₂×C₅₃	#1309	1	1	yes	characteristic
148	C₂²×M₁₆	—	53	2	no	—
149	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
150	C₂×C₄⋊C₈	—	2	1	no	—
151	C₂×C₄⋊C₈	—	106	1	no	—
152	C₂×C₂²⋊C₈	—	53	2	no	—
153	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
154	#336	—	2	1	no	—
155	#336	—	106	1	no	—
156	#336	—	106	1	no	—
157	#336	—	106	1	no	—
158	C₅₃	#7034	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
160	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
161	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
162	C₂×M₁₆	—	53	4	no	—
163	C₂×M₁₆	—	106	2	no	—
164	C₂×M₁₆	—	212	2	no	—
165	C₂×M₁₆	—	212	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	106	2	no	—
168	C₂²×C₈	—	106	2	no	—
169	C₂×C₄²	—	4	2	no	—
170	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
171	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
174	C₄⋊C₈	—	4	2	no	—
175	C₄⋊C₈	—	212	2	no	—
176	C₂²⋊C₈	—	4	2	no	—
177	C₂²⋊C₈	—	212	2	no	—
178	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	4	1	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	—	4	1	no	—
190	C₂²×C₄	C₈×D₅₃	1	2	yes	—
191	C₂²×C₄	#11719912	1	2	yes	—
192	C₂²×C₄	#11719919	1	1	yes	characteristic
193	C₂²×C₄	#11719919	1	1	yes	characteristic
194	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
195	C₂²×C₄	#11719921	1	1	yes	characteristic
196	C₂²×C₄	#11719921	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
198	M₁₆	—	212	4	no	—
199	M₁₆	—	212	4	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	106	4	no	—
204	C₂×C₈	—	212	2	no	—
205	C₂×C₈	—	212	2	no	—
206	C₂×C₈	—	212	2	no	—
207	C₂×C₈	—	212	2	no	—
208	C₂×C₈	—	212	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	#13133124	1	1	yes	characteristic
220	C₂³	C₈×Dic₅₃	1	1	yes	characteristic
221	C₂³	#13133127	1	1	yes	characteristic
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	#13133129	1	2	yes	—
243	C₂×C₄	#13133131	1	2	yes	—
244	C₂×C₄	#13133133	1	1	yes	characteristic
245	C₂×C₄	#13133133	1	1	yes	characteristic
246	C₂×C₄	#13133134	1	1	yes	characteristic
247	C₂×C₄	#13133135	1	1	yes	characteristic
248	C₂×C₄	#13133135	1	1	yes	characteristic
249	C₂×C₄	#13133136	1	1	yes	characteristic
250	C₂×C₄	#13133146	1	2	yes	—
251	C₈	—	8	2	no	—
252	C₈	—	424	2	no	—
253	C₈	—	212	4	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	1	no	—
264	C₂²	#16810377	1	1	yes	characteristic
265	C₂²	#16810389	1	1	yes	characteristic
266	C₂²	#16810391	1	1	yes	characteristic
267	C₂²	#16810391	1	1	yes	characteristic
268	C₂²	#16810392	1	1	yes	characteristic
269	C₂²	#16810393	1	1	yes	characteristic
270	C₂²	#16810394	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	#16810346	1	2	yes	—
279	C₄	#16810348	1	2	yes	—
280	C₂	—	4	1	no	—
281	C₂	—	4	1	no	—
282	C₂	#20272557	1	1	yes	characteristic
283	C₂	#20272564	1	1	yes	characteristic
284	C₂	#20272565	1	1	yes	characteristic
285	C₂	#20272621	1	1	yes	characteristic
286	C₂	#20272622	1	1	yes	characteristic
287	C₂	#20272625	1	1	yes	characteristic
288	C₂	#20272628	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133124
commutator	C₂³×C₅₃	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₅₃
Fitting	C₅₃×#1797	C₂
socle	C₂³×C₅₃	#95
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₂²×C₄²×C₅₃	C₂²
hypercenter	#1797	D₅₃
hypocenter	C₅₃	#7034
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
17. C₂×#11719917	C₈	semidirect
158. C₅₃	#7034	semidirect

Maximal subgroups

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

subgroup	index
#20274695	2
#20274409	2
C₅₃×#1797	2
#7034	53