Subgroups — #27292941

Order 13568.

574 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	C₂×#16810672	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810393	C₂	1	2	yes	maximal
3	C₂×#16810388	C₂	1	2	yes	maximal
4	C₂×#16810381	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂²×#11719917	C₂²	1	2	yes	—
8	#16810672	—	2	2	no	—
9	#16810672	C₄	1	4	yes	complemented
10	C₂×#13133153	—	2	2	no	—
11	#16810393	—	2	4	no	—
12	#16810388	—	2	4	no	—
13	#16810381	—	2	2	no	—
14	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	2	no	—
18	C₂×#11719917	—	2	2	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	2	no	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	C₂×C₄	1	4	yes	—
29	C₂×#11719917	D₄	1	4	yes	—
30	C₂²×#88322	—	2	2	no	—
31	C₂²×#88322	—	2	2	no	—
32	#13133153	—	2	8	no	—
33	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₅₃	—	2	2	no	—
35	C₂×M₁₆×C₅₃	—	2	2	no	—
36	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
38	C₂²×C₈×C₅₃	—	2	1	no	—
39	C₂²×C₈×C₅₃	—	2	2	no	—
40	C₂²×C₈×C₅₃	—	2	2	no	—
41	C₂²×C₈×C₅₃	—	2	1	no	—
42	C₂²×C₈×C₅₃	D₄	1	2	yes	—
43	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
44	#117×C₅₃	—	2	2	no	—
45	#117×C₅₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
47	C₂×C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
48	C₂×C₄²×C₅₃	—	2	1	no	—
49	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	#41	1	2	yes	—
57	C₂²×C₄×C₅₃	#41	1	2	yes	—
58	C₂²×C₄×C₅₃	#41	1	2	yes	—
59	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
61	M₁₆×C₅₃	—	2	4	no	—
62	M₁₆×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	—	2	4	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	4	1	no	—
68	C₂×C₈×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	—	4	1	no	—
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
71	C₄⋊C₄×C₅₃	—	4	2	no	—
72	C₂²⋊C₄×C₅₃	—	2	4	no	—
73	C₄²×C₅₃	—	4	1	no	—
74	C₄²×C₅₃	—	4	1	no	—
75	#11719917	—	2	4	no	—
76	#11719917	—	2	4	no	—
77	#11719917	—	2	4	no	—
78	#11719917	—	2	4	no	—
79	#11719917	—	4	2	no	—
80	#11719917	—	4	2	no	—
81	#11719917	—	4	2	no	—
82	#11719917	—	4	2	no	—
83	C₂×#88322	—	2	4	no	—
84	C₂×#88322	—	2	4	no	—
85	C₂×#88322	—	4	2	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂×#88322	—	4	2	no	—
88	C₂×#88322	—	4	2	no	—
89	C₂³×C₅₃	—	4	2	no	—
90	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
91	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
92	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂³×C₅₃	#125	1	2	yes	—
94	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	—	4	4	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#117	1	2	yes	—
99	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#121	1	2	yes	—
102	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#123	1	2	yes	—
107	C₂×C₄×C₅₃	#123	1	2	yes	—
108	C₂×C₄×C₅₃	#123	1	2	yes	—
109	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#125	1	2	yes	—
112	C₂×C₄×C₅₃	#126	1	2	yes	—
113	C₂×C₄×C₅₃	#126	1	2	yes	—
114	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
115	C₈×C₅₃	—	2	4	no	—
116	C₈×C₅₃	—	4	2	no	—
117	#88322	—	4	4	no	—
118	#88322	—	4	4	no	—
119	#11368	—	53	1	no	maximal, Sylow
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	—	2	4	no	—
122	C₂²×C₅₃	—	4	1	no	—
123	C₂²×C₅₃	—	4	2	no	—
124	C₂²×C₅₃	—	4	1	no	—
125	C₂²×C₅₃	#387	1	2	yes	—
126	C₂²×C₅₃	#391	1	1	yes	characteristic
127	C₂²×C₅₃	#398	1	2	yes	—
128	C₂²×C₅₃	#399	1	1	yes	characteristic
129	C₂²×C₅₃	#400	1	1	yes	characteristic
130	C₄×C₅₃	—	8	2	no	—
131	C₄×C₅₃	#381	1	1	yes	characteristic
132	C₄×C₅₃	#387	1	2	yes	—
133	C₄×C₅₃	#390	1	1	yes	characteristic
134	C₄×C₅₃	#397	1	2	yes	—
135	C₄×C₅₃	#399	1	1	yes	characteristic
136	C₄×C₅₃	#400	1	1	yes	characteristic
137	C₂×#430	—	53	1	no	—
138	C₂×#344	—	53	2	no	—
139	C₂×#341	—	53	2	no	—
140	C₂×#338	—	53	1	no	—
141	#1729	D₅₃	1	1	yes	characteristic
142	C₂×C₅₃	—	2	1	no	—
143	C₂×C₅₃	—	2	2	no	—
144	C₂×C₅₃	—	2	1	no	—
145	C₂×C₅₃	—	2	2	no	—
146	C₂×C₅₃	#1445	1	1	yes	characteristic
147	C₂×C₅₃	#2103	1	2	yes	—
148	C₂²×M₁₆	—	53	2	no	—
149	C₂²×M₁₆	—	53	2	no	—
150	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
151	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
152	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
153	#430	—	53	4	no	—
154	#430	—	106	2	no	—
155	C₂×#108	—	106	2	no	—
156	#344	—	106	4	no	—
157	#341	—	106	4	no	—
158	#338	—	106	2	no	—
159	#336	—	2	2	no	—
160	C₅₃	#11368	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	53	4	no	—
165	C₂×M₁₆	—	53	4	no	—
166	C₂×M₁₆	—	106	2	no	—
167	C₂×M₁₆	—	106	2	no	—
168	C₂×M₁₆	—	106	2	no	—
169	C₂×M₁₆	—	106	2	no	—
170	C₂×M₁₆	—	106	2	no	—
171	C₂×M₁₆	—	106	2	no	—
172	C₂×M₁₆	—	106	2	no	—
173	C₂×M₁₆	—	106	2	no	—
174	C₂×M₁₆	—	106	2	no	—
175	C₂×M₁₆	—	106	2	no	—
176	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
177	C₂×M₁₆	D₂₁₂	1	2	yes	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	106	2	no	—
183	C₂²×C₈	—	106	2	no	—
184	C₂²×C₈	Dic₁₀₆	1	2	yes	—
185	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₂²⋊C₄	Dic₁₀₆	1	2	yes	—
190	C₂×C₄²	—	2	1	no	—
191	#108	—	106	8	no	—
192	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
197	C₂²×C₄	#11719945	1	1	yes	characteristic
198	C₂²×C₄	#11719945	1	1	yes	characteristic
199	C₂²×C₄	D₄×D₅₃	1	2	yes	—
200	C₂²×C₄	#11719947	1	2	yes	—
201	C₂²×C₄	#11719947	1	2	yes	—
202	C₂²×C₄	#11719949	1	2	yes	—
203	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
204	M₁₆	—	2	4	no	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	106	4	no	—
207	M₁₆	—	106	4	no	—
208	M₁₆	—	106	4	no	—
209	M₁₆	—	106	4	no	—
210	M₁₆	—	212	2	no	—
211	M₁₆	—	212	2	no	—
212	M₁₆	—	212	2	no	—
213	M₁₆	—	212	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	106	4	no	—
222	C₂×C₈	—	106	4	no	—
223	C₂×C₈	—	212	2	no	—
224	C₂×C₈	—	212	2	no	—
225	C₂×C₈	—	212	2	no	—
226	C₂×C₈	—	212	2	no	—
227	C₂×C₈	#11719921	1	4	yes	—
228	C₄⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	2	4	no	—
230	C₄²	—	4	1	no	—
231	C₄²	—	4	1	no	—
232	C₂³	—	4	2	no	—
233	C₂³	C₄×Dic₁₀₆	1	1	yes	characteristic
234	C₂³	#13133194	1	2	yes	—
235	C₂³	#13133196	1	1	yes	characteristic
236	C₂³	#13133208	1	2	yes	—
237	C₂³	#13133217	1	1	yes	characteristic
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
241	C₂×C₄	#13133173	1	2	yes	—
242	C₂×C₄	#13133179	1	2	yes	—
243	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
244	C₂×C₄	#13133192	1	2	yes	—
245	C₂×C₄	#13133194	1	2	yes	—
246	C₂×C₄	#13133196	1	1	yes	characteristic
247	C₂×C₄	#13133198	1	1	yes	characteristic
248	C₂×C₄	#13133198	1	1	yes	characteristic
249	C₂×C₄	#13133199	1	2	yes	—
250	C₂×C₄	#13133199	1	2	yes	—
251	C₂×C₄	#13133201	1	2	yes	—
252	C₂×C₄	#13133205	1	1	yes	characteristic
253	C₂×C₄	#13133206	1	1	yes	characteristic
254	C₂×C₄	#13133208	1	2	yes	—
255	C₂×C₄	#13133212	1	2	yes	—
256	C₂×C₄	#13133212	1	2	yes	—
257	C₂×C₄	#13133213	1	1	yes	characteristic
258	C₈	—	2	4	no	—
259	C₈	—	4	2	no	—
260	C₈	—	212	4	no	—
261	C₈	—	212	4	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	#16810506	1	2	yes	—
268	C₂²	#16810529	1	1	yes	characteristic
269	C₂²	#16810555	1	2	yes	—
270	C₂²	#16810559	1	1	yes	characteristic
271	C₂²	#16810562	1	1	yes	characteristic
272	C₄	—	8	2	no	—
273	C₄	#16810484	1	1	yes	characteristic
274	C₄	#16810506	1	2	yes	—
275	C₄	#16810524	1	1	yes	characteristic
276	C₄	#16810550	1	2	yes	—
277	C₄	#16810559	1	1	yes	characteristic
278	C₄	#16810562	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	2	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	2	no	—
283	C₂	#20272984	1	1	yes	characteristic
284	C₂	#20276295	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133206
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1729	C₂
socle	C₂²×C₅₃	#400
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1729	D₅₃
hypocenter	C₅₃	#11368
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
9. #16810672	C₄	semidirect
160. C₅₃	#11368	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810672	2
C₂×#16810393	2
C₂×#16810388	2
C₂×#16810381	2
C₅₃×#1729	2
#11368	53