Subgroups — #27289920

Order 13568.

814 conjugacy classes of subgroups.

The subgroup lattice has 294 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×C₄×#88322	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×C₄×#88322	—	2	2	no	—
11	C₂×C₄×#88322	—	2	2	no	—
12	C₂×C₄×#88322	C₄	1	2	yes	—
13	C₂×C₄×#88322	C₄	1	2	yes	—
14	#16810387	—	2	8	no	—
15	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#423	—	2	2	no	—
18	C₅₃×#423	C₄	1	4	yes	—
19	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	4	no	—
22	C₂×#11719917	—	2	4	no	—
23	C₂×#11719917	C₂×C₄	1	4	yes	—
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	—	2	2	no	—
29	C₂²×#88322	C₂×C₄	1	4	yes	—
30	C₂²×#88322	D₄	1	4	yes	—
31	C₂²×#88322	Q₈	1	4	yes	—
32	C₄⋊#88322	—	2	8	no	—
33	C₄×#88322	—	2	4	no	—
34	C₄×#88322	—	2	4	no	—
35	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
36	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₅₃	—	2	2	no	—
38	C₂×M₁₆×C₅₃	—	2	4	no	—
39	C₂×M₁₆×C₅₃	—	2	4	no	—
40	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₅₃	—	2	2	no	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	D₄	1	2	yes	—
47	C₂×C₄²×C₅₃	D₄	1	2	yes	—
48	C₂×C₄²×C₅₃	D₄	1	2	yes	—
49	C₄⋊C₈×C₅₃	—	2	8	no	—
50	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	—	2	8	no	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₅₃	#41	1	2	yes	—
65	C₂²×C₄×C₅₃	#41	1	2	yes	—
66	M₁₆×C₅₃	—	2	8	no	—
67	M₁₆×C₅₃	—	4	4	no	—
68	C₂×C₈×C₅₃	—	2	4	no	—
69	C₂×C₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	4	2	no	—
71	C₄²×C₅₃	C₄²	1	2	yes	—
72	C₄²×C₅₃	C₄²	1	2	yes	—
73	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
76	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
77	#11719917	—	2	8	no	—
78	#11719917	—	4	4	no	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	—	2	4	no	—
84	C₂×#88322	—	2	4	no	—
85	C₂×#88322	—	2	4	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂×#88322	—	4	2	no	—
88	C₂×#88322	C₄⋊C₄	1	8	yes	—
89	C₂³×C₅₃	—	2	4	no	—
90	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₅₃	#125	1	2	yes	—
94	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	8	no	—
97	C₂×C₄×C₅₃	—	2	8	no	—
98	C₂×C₄×C₅₃	#95	1	4	yes	—
99	C₂×C₄×C₅₃	#95	1	4	yes	—
100	C₂×C₄×C₅₃	#108	1	8	yes	—
101	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₅₃	#117	1	2	yes	—
107	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₅₃	#121	1	2	yes	—
112	C₂×C₄×C₅₃	#121	1	2	yes	—
113	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
119	C₈×C₅₃	—	4	4	no	—
120	#88322	—	2	8	no	—
121	#88322	—	4	4	no	—
122	#10844	—	53	1	no	maximal, Sylow
123	C₂²×C₅₃	—	2	4	no	—
124	C₂²×C₅₃	—	2	4	no	—
125	C₂²×C₅₃	—	2	2	no	—
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
128	C₂²×C₅₃	#382	1	1	yes	characteristic
129	C₂²×C₅₃	#384	1	1	yes	characteristic
130	C₂²×C₅₃	#389	1	2	yes	—
131	C₂²×C₅₃	#389	1	2	yes	—
132	C₂²×C₅₃	C₂×#108	1	4	yes	—
133	C₄×C₅₃	—	2	4	no	—
134	C₄×C₅₃	#341	1	8	yes	complemented
135	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
136	C₄×C₅₃	#380	1	2	yes	—
137	C₄×C₅₃	#390	1	1	yes	characteristic
138	C₄×C₅₃	#390	1	1	yes	characteristic
139	C₄×C₅₃	#391	1	1	yes	characteristic
140	C₄×C₅₃	#391	1	1	yes	characteristic
141	C₂×#423	—	53	1	no	—
142	C₂×#423	D₅₃	1	1	yes	characteristic
143	C₂²×C₄×C₈	—	53	1	no	—
144	C₂×#341	—	53	4	no	—
145	C₂×C₅₃	—	2	2	no	—
146	C₂×C₅₃	—	2	2	no	—
147	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
148	C₂×C₅₃	C₂×#341	1	2	yes	—
149	C₂×C₅₃	#1837	1	2	yes	—
150	C₂×C₅₃	#1848	1	2	yes	—
151	C₂²×M₁₆	—	53	2	no	—
152	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
153	C₂³×C₈	—	53	2	no	—
154	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
155	#423	—	2	2	no	—
156	#423	—	53	4	no	—
157	#423	—	106	2	no	—
158	#423	Dic₅₃	1	4	yes	—
159	C₂×C₄⋊C₈	—	2	2	no	—
160	C₂×C₄⋊C₈	—	106	2	no	—
161	C₂×C₄×C₈	—	53	2	no	—
162	C₂×C₄×C₈	—	53	2	no	—
163	C₂×C₄×C₈	—	106	2	no	—
164	C₂×C₄×C₈	—	106	2	no	—
165	#341	—	106	8	no	—
166	C₅₃	#10844	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	Dic₁₀₆	1	2	yes	—
168	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	2	4	no	—
172	C₂×M₁₆	—	53	4	no	—
173	C₂×M₁₆	—	106	2	no	—
174	C₂×M₁₆	—	106	4	no	—
175	C₂×M₁₆	—	106	4	no	—
176	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	53	4	no	—
179	C₂²×C₈	—	53	4	no	—
180	C₂²×C₈	—	53	4	no	—
181	C₂²×C₈	—	106	2	no	—
182	C₂²×C₈	—	106	2	no	—
183	C₂²×C₈	—	106	2	no	—
184	C₂²×C₈	—	106	2	no	—
185	C₂²×C₈	—	106	2	no	—
186	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
187	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
188	C₂×C₄²	C₄×D₅₃	1	2	yes	—
189	C₂×C₄²	D₂₁₂	1	2	yes	—
190	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
191	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
192	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
193	C₄⋊C₈	—	2	8	no	—
194	C₄⋊C₈	—	106	8	no	—
195	C₄×C₈	—	106	4	no	—
196	C₄×C₈	—	106	4	no	—
197	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	#11719919	1	2	yes	—
200	C₂²×C₄	#11719919	1	4	yes	—
201	C₂²×C₄	#11719919	1	2	yes	—
202	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
203	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₅₃	1	2	yes	—
207	C₂²×C₄	#11719947	1	2	yes	—
208	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
209	C₂²×C₄	#11719949	1	2	yes	—
210	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
213	M₁₆	—	2	8	no	—
214	M₁₆	—	4	4	no	—
215	M₁₆	—	106	8	no	—
216	M₁₆	—	212	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	53	8	no	—
221	C₂×C₈	—	106	4	no	—
222	C₂×C₈	—	106	4	no	—
223	C₂×C₈	—	106	4	no	—
224	C₂×C₈	—	106	4	no	—
225	C₂×C₈	—	106	4	no	—
226	C₂×C₈	—	106	4	no	—
227	C₂×C₈	—	106	4	no	—
228	C₂×C₈	—	212	2	no	—
229	C₂×C₈	—	212	2	no	—
230	C₄²	C₄×Dic₅₃	1	2	yes	—
231	C₄²	C₄×Dic₅₃	1	2	yes	—
232	C₄²	#11719921	1	2	yes	—
233	C₄²	#11719921	1	2	yes	—
234	C₄²	#11719921	1	4	yes	—
235	C₄²	C₂²⋊Dic₅₃	1	4	yes	—
236	C₂³	—	2	4	no	—
237	C₂³	C₂×#11719919	1	1	yes	characteristic
238	C₂³	C₂×#11719919	1	1	yes	characteristic
239	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
240	C₂³	#13133208	1	2	yes	—
241	C₂³	#13133216	1	2	yes	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	#13133124	1	4	yes	—
246	C₂×C₄	#13133124	1	4	yes	—
247	C₂×C₄	#13133152	1	8	yes	—
248	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
249	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
250	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
252	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
253	C₂×C₄	#13133175	1	2	yes	—
254	C₂×C₄	#13133179	1	2	yes	—
255	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
256	C₂×C₄	#13133182	1	2	yes	—
257	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
258	C₂×C₄	#13133188	1	2	yes	—
259	C₂×C₄	#13133192	1	2	yes	—
260	C₂×C₄	#13133193	1	2	yes	—
261	C₂×C₄	#13133197	1	2	yes	—
262	C₂×C₄	#13133204	1	1	yes	characteristic
263	C₂×C₄	#13133206	1	1	yes	characteristic
264	C₂×C₄	#13133213	1	1	yes	characteristic
265	C₂×C₄	#13133215	1	1	yes	characteristic
266	C₈	—	4	4	no	—
267	C₈	—	106	8	no	—
268	C₈	—	212	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	C₂×#13133124	1	1	yes	characteristic
274	C₂²	#16810488	1	1	yes	characteristic
275	C₂²	#16810492	1	1	yes	characteristic
276	C₂²	#16810519	1	2	yes	—
277	C₂²	#16810522	1	2	yes	—
278	C₂²	C₂×#13133152	1	4	yes	—
279	C₄	—	2	4	no	—
280	C₄	#16810387	1	8	yes	complemented
281	C₄	C₄⋊C₄×Dic₅₃	1	2	yes	—
282	C₄	#16810480	1	2	yes	—
283	C₄	#16810524	1	1	yes	characteristic
284	C₄	#16810526	1	1	yes	characteristic
285	C₄	#16810528	1	1	yes	characteristic
286	C₄	#16810530	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	2	no	—
289	C₂	#20272944	1	1	yes	characteristic
290	C₂	C₂×#16810387	1	2	yes	—
291	C₂	#20274852	1	2	yes	—
292	C₂	#20274886	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	#10844
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
134. C₄×C₅₃	#341	semidirect
166. C₅₃	#10844	semidirect
280. C₄	#16810387	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×C₄×#88322	2
C₂×#16810387	2
C₂×C₅₃×#423	2
#10844	53