Subgroups — #27484696

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283078	C₂	1	1	yes	maximal, characteristic
2	#20276053	C₂	1	4	yes	maximal
3	#20275681	C₂	1	2	yes	maximal
4	#20275478	C₂	1	2	yes	maximal
5	#20275473	C₂	1	2	yes	maximal
6	#20274877	C₂	1	2	yes	maximal
7	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
8	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133209	C₂²	1	2	yes	—
10	C₂×#13133150	C₂²	1	4	yes	—
11	C₂×#13133148	C₂²	1	4	yes	—
12	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
14	C₂×C₄×#88322	C₂²	1	2	yes	—
15	#16810521	—	2	2	no	—
16	#16810521	—	2	2	no	—
17	#16810493	C₂²	1	2	yes	—
18	#16810489	C₂²	1	2	yes	—
19	#16810385	C₂²	1	4	yes	—
20	#16810385	C₂²	1	4	yes	—
21	#16810378	C₂²	1	2	yes	—
22	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#389	C₂²	1	2	yes	—
24	C₅₃×#387	—	2	2	no	—
25	C₅₃×#382	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₄³×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×#88322	C₂³	1	2	yes	—
30	C₂²×#88322	C₂³	1	2	yes	—
31	#13133209	—	4	2	no	—
32	#13133209	—	4	2	no	—
33	C₂×C₄⋊Dic₅₃	—	2	2	no	—
34	C₂×C₄⋊Dic₅₃	—	2	2	no	—
35	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
36	C₂×#11719919	—	2	4	no	—
37	#13133150	—	4	4	no	—
38	#13133148	—	4	4	no	—
39	C₄⋊#88322	Q₈	1	4	yes	—
40	C₄⋊#88322	Q₈	1	4	yes	—
41	C₄×#88322	—	2	4	no	—
42	#13133124	—	2	4	no	—
43	#125×C₅₃	—	2	2	no	—
44	#125×C₅₃	Q₈	1	4	yes	—
45	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
49	C₂×C₄²×C₅₃	—	2	2	no	—
50	C₂×C₄²×C₅₃	D₄	1	2	yes	—
51	C₂×C₄²×C₅₃	D₄	1	2	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	#95×C₅₃	—	2	2	no	—
56	#95×C₅₃	—	2	2	no	—
57	#95×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₅₃	—	2	4	no	—
66	C₄⋊C₄×C₅₃	—	2	8	no	—
67	C₄⋊C₄×C₅₃	—	2	2	no	—
68	C₄⋊C₄×C₅₃	—	2	2	no	—
69	C₄⋊C₄×C₅₃	—	2	2	no	—
70	C₄⋊C₄×C₅₃	—	4	4	no	—
71	C₄⋊C₄×C₅₃	C₂×Q₈	1	4	yes	—
72	C₄²×C₅₃	—	2	4	no	—
73	C₄²×C₅₃	—	2	1	no	—
74	C₄²×C₅₃	—	2	1	no	—
75	C₄²×C₅₃	—	4	2	no	—
76	C₄²×C₅₃	—	4	4	no	—
77	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
78	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
79	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
80	C₄²×C₅₃	#41	1	2	yes	—
81	C₄⋊Dic₅₃	—	2	4	no	—
82	C₄⋊Dic₅₃	—	2	4	no	—
83	C₄⋊Dic₅₃	—	4	4	no	—
84	C₄⋊Dic₅₃	—	4	4	no	—
85	#11719919	—	4	8	no	—
86	C₂×#88322	—	2	2	no	—
87	C₂×#88322	—	2	2	no	—
88	C₂×#88322	—	2	2	no	—
89	C₂×#88322	—	2	2	no	—
90	C₂×#88322	C₂×Q₈	1	4	yes	—
91	C₂×#88322	C₂×Q₈	1	4	yes	—
92	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	4	2	no	—
104	C₂×C₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
106	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
108	C₂×C₄×C₅₃	#135	1	4	yes	—
109	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×Dic₅₃	—	4	4	no	—
116	C₂×Dic₅₃	—	4	4	no	—
117	C₂×Dic₅₃	—	4	4	no	—
118	#88322	—	2	4	no	—
119	#88322	—	2	4	no	—
120	#29210	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₂²×C₅₃	#535	1	1	yes	characteristic
124	C₂²×C₅₃	C₂×#135	1	2	yes	—
125	C₂²×C₅₃	#575	1	2	yes	—
126	C₄×C₅₃	—	2	4	no	—
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	—	4	4	no	—
129	C₄×C₅₃	—	4	2	no	—
130	C₄×C₅₃	—	8	2	no	—
131	C₄×C₅₃	#499	1	4	yes	—
132	C₄×C₅₃	#554	1	1	yes	characteristic
133	C₄×C₅₃	#554	1	1	yes	characteristic
134	C₄×C₅₃	Q₈²	1	1	yes	characteristic
135	C₄×C₅₃	Q₈²	1	1	yes	characteristic
136	Dic₅₃	—	8	4	no	—
137	#2859	—	53	1	no	—
138	C₄×#125	D₅₃	1	1	yes	characteristic
139	#2059	—	53	4	no	—
140	#1991	—	53	2	no	—
141	#1948	—	53	2	no	—
142	#1947	—	53	2	no	—
143	#1846	—	53	2	no	—
144	C₄×C₄⋊C₈	—	53	1	no	—
145	C₂×C₅₃	#2567	1	1	yes	characteristic
146	C₂×C₅₃	#3076	1	2	yes	—
147	C₂×C₅₃	C₂×#499	1	1	yes	characteristic
148	C₂×C₅₃	#3166	1	1	yes	characteristic
149	C₂×C₅₃	#3212	1	2	yes	—
150	C₂×#125	—	53	2	no	—
151	C₂×#125	C₂×D₅₃	1	1	yes	characteristic
152	C₂×#107	—	53	4	no	—
153	C₂×#106	—	53	4	no	—
154	C₂×C₄⋊C₈	—	53	1	no	—
155	C₂×C₄⋊C₈	—	53	1	no	—
156	C₂×C₄×C₈	—	53	2	no	—
157	#389	—	106	2	no	—
158	#389	—	106	2	no	—
159	#389	C₂×D₅₃	1	2	yes	—
160	#387	—	2	2	no	—
161	#384	—	53	2	no	—
162	#382	—	53	2	no	—
163	#382	C₂×D₅₃	1	1	yes	characteristic
164	C₄×C₄⋊C₄	—	2	2	no	—
165	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
166	C₄³	C₂×D₅₃	1	1	yes	characteristic
167	#340	—	53	4	no	—
168	#340	—	53	4	no	—
169	#336	—	53	2	no	—
170	C₅₃	#29210	1	1	yes	characteristic, complemented, Sylow
171	C₂²×C₈	—	53	2	no	—
172	C₂²×C₈	—	53	2	no	—
173	#125	—	2	2	no	—
174	#125	—	212	2	no	—
175	#125	—	212	2	no	—
176	#125	Dic₁₀₆	1	4	yes	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	53	4	no	—
180	C₂×C₄⋊C₄	—	106	4	no	—
181	C₂×C₄⋊C₄	—	106	2	no	—
182	C₂×C₄⋊C₄	—	106	2	no	—
183	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
184	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
187	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
188	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
189	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
191	#107	—	212	4	no	—
192	#106	—	212	4	no	—
193	C₄⋊C₈	—	53	4	no	—
194	C₄⋊C₈	—	53	4	no	—
195	C₄×C₈	—	106	4	no	—
196	#95	—	2	2	no	—
197	#95	—	2	2	no	—
198	#95	—	2	2	no	—
199	#95	—	106	4	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	106	4	no	—
203	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
204	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
205	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
208	C₂×C₈	—	53	4	no	—
209	C₂×C₈	—	53	4	no	—
210	C₂×C₈	—	106	2	no	—
211	C₂×C₈	—	106	2	no	—
212	C₂×C₈	—	106	2	no	—
213	C₂×C₈	—	106	2	no	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	106	4	no	—
220	C₄⋊C₄	—	106	4	no	—
221	C₄⋊C₄	—	212	8	no	—
222	C₄⋊C₄	—	212	4	no	—
223	C₄⋊C₄	—	212	4	no	—
224	C₄⋊C₄	C₂×Dic₁₀₆	1	4	yes	—
225	C₄²	—	2	4	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	4	2	no	—
229	C₄²	—	4	4	no	—
230	C₄²	C₂×Dic₁₀₆	1	2	yes	—
231	C₄²	#11719945	1	2	yes	—
232	C₄²	Q₈×D₅₃	1	2	yes	—
233	C₄²	Q₈×D₅₃	1	2	yes	—
234	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	212	4	no	—
248	C₂×C₄	—	212	4	no	—
249	C₂×C₄	—	212	4	no	—
250	C₂×C₄	#13133196	1	4	yes	—
251	C₂×C₄	#13133215	1	2	yes	—
252	C₂×C₄	#13133215	1	2	yes	—
253	C₂×C₄	#13133237	1	4	yes	—
254	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
255	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
256	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
257	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
258	C₂×C₄	#13133263	1	1	yes	characteristic
259	C₂×C₄	#13133263	1	1	yes	characteristic
260	C₈	—	106	4	no	—
261	C₈	—	106	4	no	—
262	C₂²	C₂×#13133196	1	1	yes	characteristic
263	C₂²	C₂×#13133215	1	1	yes	characteristic
264	C₂²	#16811187	1	1	yes	characteristic
265	C₂²	C₂×#13133237	1	2	yes	—
266	C₂²	#16811421	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	2	4	no	—
269	C₄	—	4	4	no	—
270	C₄	—	4	2	no	—
271	C₄	—	8	2	no	—
272	C₄	—	424	4	no	—
273	C₄	#16811022	1	4	yes	—
274	C₄	#16811333	1	1	yes	characteristic
275	C₄	#16811333	1	1	yes	characteristic
276	C₄	Q₈×Dic₁₀₆	1	1	yes	characteristic
277	C₄	Q₈×Dic₁₀₆	1	1	yes	characteristic
278	C₂	#20279920	1	1	yes	characteristic
279	C₂	#20284503	1	2	yes	—
280	C₂	C₂×#16811022	1	1	yes	characteristic
281	C₂	#20285268	1	1	yes	characteristic
282	C₂	#20285687	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#125×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	C₄×#125	D₅₃
hypocenter	C₅₃	#29210
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
170. C₅₃	#29210	semidirect

Maximal subgroups

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

subgroup	index
#20283078	2
#20276053	2
#20275681	2
#20275478	2
#20275473	2
#20274877	2
C₄×C₄⋊#88322	2
C₄×#125×C₅₃	2
#29210	53