Subgroups — #27289923

Order 13568.

774 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133128	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810387	C₂	1	4	yes	maximal
4	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₂²	1	2	yes	—
6	C₂³×#88322	C₂²	1	2	yes	—
7	#16810647	—	2	2	no	—
8	#16810647	C₄	1	4	yes	—
9	C₂×C₄⋊#88322	—	2	2	no	—
10	C₂×#13133128	—	2	2	no	—
11	C₂×#13133128	—	2	2	no	—
12	C₂×#13133128	C₄	1	4	yes	—
13	#16810387	—	2	8	no	—
14	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#423	—	2	2	no	—
17	C₅₃×#423	C₄	1	4	yes	—
18	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	—	2	4	no	—
21	C₂×#11719917	—	2	4	no	—
22	C₂×#11719917	C₂×C₄	1	4	yes	—
23	C₂²×#88322	—	2	2	no	—
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	—	2	2	no	—
26	C₂²×#88322	—	2	2	no	—
27	C₂²×#88322	—	2	2	no	—
28	C₂²×#88322	C₂×C₄	1	4	yes	—
29	C₂²×#88322	D₄	1	4	yes	—
30	C₂²×#88322	Q₈	1	4	yes	—
31	C₄⋊#88322	—	2	8	no	—
32	#13133128	—	2	8	no	—
33	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
34	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₅₃	—	2	2	no	—
36	C₂×M₁₆×C₅₃	—	2	4	no	—
37	C₂×M₁₆×C₅₃	—	2	4	no	—
38	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₅₃	—	2	2	no	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅₃	D₄	1	2	yes	—
44	C₂×C₄²×C₅₃	D₄	1	2	yes	—
45	C₂×C₄²×C₅₃	D₄	1	2	yes	—
46	C₄⋊C₈×C₅₃	—	2	8	no	—
47	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	4	4	no	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	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×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₅₃	#41	1	2	yes	—
61	C₂²×C₄×C₅₃	#41	1	2	yes	—
62	M₁₆×C₅₃	—	2	8	no	—
63	M₁₆×C₅₃	—	4	4	no	—
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	4	2	no	—
66	C₂×C₈×C₅₃	—	4	2	no	—
67	C₄²×C₅₃	C₄²	1	4	yes	—
68	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
69	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
70	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
71	#11719917	—	2	8	no	—
72	#11719917	—	4	4	no	—
73	C₂×#88322	—	2	4	no	—
74	C₂×#88322	—	2	4	no	—
75	C₂×#88322	—	2	4	no	—
76	C₂×#88322	—	2	4	no	—
77	C₂×#88322	—	2	4	no	—
78	C₂×#88322	—	2	4	no	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	—	4	2	no	—
81	C₂×#88322	—	4	2	no	—
82	C₂×#88322	C₄⋊C₄	1	8	yes	—
83	C₂³×C₅₃	—	2	4	no	—
84	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂³×C₅₃	#125	1	2	yes	—
88	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
89	C₂×C₄×C₅₃	—	2	8	no	—
90	C₂×C₄×C₅₃	—	4	4	no	—
91	C₂×C₄×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	#95	1	8	yes	—
93	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₅₃	#117	1	2	yes	—
99	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
102	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
103	C₂×C₄×C₅₃	#121	1	2	yes	—
104	C₂×C₄×C₅₃	#121	1	2	yes	—
105	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
111	C₈×C₅₃	—	4	4	no	—
112	#88322	—	2	8	no	—
113	#88322	—	4	4	no	—
114	#10845	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	—	2	4	no	—
116	C₂²×C₅₃	—	2	4	no	—
117	C₂²×C₅₃	—	2	2	no	—
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
120	C₂²×C₅₃	#382	1	1	yes	characteristic
121	C₂²×C₅₃	#384	1	1	yes	characteristic
122	C₂²×C₅₃	#389	1	2	yes	—
123	C₂²×C₅₃	#389	1	2	yes	—
124	C₂²×C₅₃	#430	1	4	yes	—
125	C₄×C₅₃	—	2	8	no	—
126	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
127	C₄×C₅₃	#380	1	2	yes	—
128	C₄×C₅₃	#390	1	1	yes	characteristic
129	C₄×C₅₃	#390	1	1	yes	characteristic
130	C₄×C₅₃	#391	1	1	yes	characteristic
131	C₄×C₅₃	#391	1	1	yes	characteristic
132	C₂×#423	—	53	1	no	—
133	C₂×#423	D₅₃	1	1	yes	characteristic
134	C₂²×#97	—	53	1	no	—
135	C₂×#341	—	53	4	no	—
136	C₂×C₅₃	—	2	2	no	—
137	C₂×C₅₃	—	2	2	no	—
138	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
139	C₂×C₅₃	#1738	1	2	yes	—
140	C₂×C₅₃	#1838	1	2	yes	—
141	C₂×C₅₃	#1849	1	2	yes	—
142	C₂²×M₁₆	—	53	2	no	—
143	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
144	C₂³×C₈	—	53	2	no	—
145	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
146	#423	—	2	2	no	—
147	#423	—	53	4	no	—
148	#423	—	106	2	no	—
149	#423	Dic₅₃	1	4	yes	—
150	C₂×C₄⋊C₈	—	2	2	no	—
151	C₂×C₄⋊C₈	—	106	2	no	—
152	C₂×#97	—	53	4	no	—
153	C₂×#97	—	106	2	no	—
154	C₂×#97	—	106	2	no	—
155	#341	—	106	8	no	—
156	C₅₃	#10845	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₄	Dic₁₀₆	1	2	yes	—
158	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	4	no	—
161	C₂×M₁₆	—	2	4	no	—
162	C₂×M₁₆	—	53	4	no	—
163	C₂×M₁₆	—	106	2	no	—
164	C₂×M₁₆	—	106	4	no	—
165	C₂×M₁₆	—	106	4	no	—
166	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	53	4	no	—
169	C₂²×C₈	—	53	4	no	—
170	C₂²×C₈	—	53	4	no	—
171	C₂²×C₈	—	106	2	no	—
172	C₂²×C₈	—	106	2	no	—
173	C₂²×C₈	—	106	2	no	—
174	C₂²×C₈	—	106	2	no	—
175	C₂²×C₈	—	106	2	no	—
176	C₂×C₄²	C₄×D₅₃	1	2	yes	—
177	C₂×C₄²	C₄×D₅₃	1	2	yes	—
178	C₂×C₄²	D₂₁₂	1	2	yes	—
179	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
180	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
181	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
182	C₄⋊C₈	—	2	8	no	—
183	C₄⋊C₈	—	106	8	no	—
184	#97	—	106	8	no	—
185	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	#11719919	1	4	yes	—
188	C₂²×C₄	#11719919	1	4	yes	—
189	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
190	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
193	C₂²×C₄	D₄×D₅₃	1	2	yes	—
194	C₂²×C₄	#11719947	1	2	yes	—
195	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
196	C₂²×C₄	#11719949	1	2	yes	—
197	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
200	M₁₆	—	2	8	no	—
201	M₁₆	—	4	4	no	—
202	M₁₆	—	106	8	no	—
203	M₁₆	—	212	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	53	8	no	—
208	C₂×C₈	—	106	4	no	—
209	C₂×C₈	—	106	4	no	—
210	C₂×C₈	—	106	4	no	—
211	C₂×C₈	—	106	4	no	—
212	C₂×C₈	—	106	4	no	—
213	C₂×C₈	—	106	4	no	—
214	C₂×C₈	—	106	4	no	—
215	C₂×C₈	—	212	2	no	—
216	C₂×C₈	—	212	2	no	—
217	C₄²	C₄×Dic₅₃	1	4	yes	—
218	C₄²	#11719921	1	4	yes	—
219	C₄²	#11719921	1	4	yes	—
220	C₄²	C₂²⋊Dic₅₃	1	4	yes	—
221	C₂³	—	2	4	no	—
222	C₂³	C₂×#11719919	1	1	yes	characteristic
223	C₂³	C₂×#11719919	1	1	yes	characteristic
224	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
225	C₂³	#13133208	1	2	yes	—
226	C₂³	#13133216	1	2	yes	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	#13133124	1	8	yes	—
231	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
232	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
233	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
235	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
236	C₂×C₄	#13133175	1	2	yes	—
237	C₂×C₄	#13133179	1	2	yes	—
238	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
239	C₂×C₄	#13133182	1	2	yes	—
240	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
241	C₂×C₄	#13133188	1	2	yes	—
242	C₂×C₄	#13133192	1	2	yes	—
243	C₂×C₄	#13133193	1	2	yes	—
244	C₂×C₄	#13133197	1	2	yes	—
245	C₂×C₄	#13133204	1	1	yes	characteristic
246	C₂×C₄	#13133206	1	1	yes	characteristic
247	C₂×C₄	#13133213	1	1	yes	characteristic
248	C₂×C₄	#13133215	1	1	yes	characteristic
249	C₈	—	4	4	no	—
250	C₈	—	106	8	no	—
251	C₈	—	212	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	4	no	—
256	C₂²	C₂×#13133124	1	1	yes	characteristic
257	C₂²	#16810488	1	1	yes	characteristic
258	C₂²	#16810492	1	1	yes	characteristic
259	C₂²	#16810519	1	2	yes	—
260	C₂²	#16810522	1	2	yes	—
261	C₂²	#16810671	1	4	yes	—
262	C₄	—	2	8	no	—
263	C₄	C₄⋊C₄×Dic₅₃	1	2	yes	—
264	C₄	#16810480	1	2	yes	—
265	C₄	#16810524	1	1	yes	characteristic
266	C₄	#16810526	1	1	yes	characteristic
267	C₄	#16810528	1	1	yes	characteristic
268	C₄	#16810530	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	2	2	no	—
271	C₂	#20272944	1	1	yes	characteristic
272	C₂	#20274441	1	2	yes	—
273	C₂	#20274855	1	2	yes	—
274	C₂	#20274889	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#423	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#423	D₅₃
hypocenter	C₅₃	#10845
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
156. C₅₃	#10845	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810647	2
C₂²×#13133128	2
C₂×#16810387	2
C₂×C₅₃×#423	2
#10845	53