Subgroups — #27273567

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
3	#20272549	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	—	2	1	no	—
6	#16810676	—	2	2	no	—
7	#16810676	—	2	2	no	—
8	#16810647	—	2	2	no	—
9	#16810647	C₄	1	4	yes	—
10	C₂×C₄⋊#88322	—	2	1	no	—
11	C₂×C₄⋊#88322	C₄	1	2	yes	—
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×C₂²⋊#88322	C₄	1	2	yes	—
14	C₂×C₄×#88322	C₂²	1	2	yes	—
15	C₂³×C₈×C₅₃	—	2	1	no	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
18	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
19	C₂×C₄⋊C₈×C₅₃	C₄	1	4	yes	—
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×#11719917	—	2	4	no	—
22	C₂×#11719917	—	4	2	no	—
23	C₂×#11719917	—	4	2	no	—
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	C₂×C₄	1	4	yes	—
26	C₄⋊#88322	—	2	4	no	—
27	C₄⋊#88322	—	2	4	no	—
28	C₄⋊#88322	—	2	2	no	—
29	C₄⋊#88322	C₂×C₄	1	4	yes	—
30	C₂²⋊#88322	—	2	4	no	—
31	C₄×#88322	—	2	2	no	—
32	C₄×#88322	—	2	2	no	—
33	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	—	2	4	no	—
37	C₂²×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	—	4	2	no	—
39	C₂²×C₈×C₅₃	—	4	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	1	yes	characteristic
43	C₂×C₄²×C₅₃	D₄	1	2	yes	—
44	C₂×C₄²×C₅₃	D₄	1	2	yes	—
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₅₃	—	2	4	no	—
49	C₄⋊C₈×C₅₃	—	2	2	no	—
50	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
51	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	—	4	2	no	—
54	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	D₈	1	2	yes	—
64	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
66	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₈×C₅₃	—	2	4	no	—
71	C₂×C₈×C₅₃	—	4	8	no	—
72	C₂×C₈×C₅₃	—	4	2	no	—
73	C₂×C₈×C₅₃	—	4	2	no	—
74	C₄²×C₅₃	C₄²	1	2	yes	—
75	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
79	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
80	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	#11719917	—	4	8	no	—
85	C₂×#88322	—	2	4	no	—
86	C₂×#88322	—	2	4	no	—
87	C₂×#88322	—	2	8	no	—
88	C₂×#88322	—	4	2	no	—
89	C₂×#88322	—	4	2	no	—
90	C₂³×C₅₃	—	2	2	no	—
91	C₂³×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	#95	1	2	yes	—
93	C₂³×C₅₃	#106	1	2	yes	—
94	C₂³×C₅₃	#107	1	2	yes	—
95	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	#95	1	2	yes	—
104	C₂×C₄×C₅₃	#95	1	2	yes	—
105	C₂×C₄×C₅₃	#95	1	2	yes	—
106	C₂×C₄×C₅₃	#102	1	4	yes	—
107	C₂×C₄×C₅₃	#102	1	4	yes	—
108	C₂×C₄×C₅₃	#103	1	4	yes	—
109	C₂×C₄×C₅₃	#103	1	4	yes	—
110	C₂×C₄×C₅₃	#106	1	2	yes	—
111	C₂×C₄×C₅₃	#107	1	2	yes	—
112	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#131	1	4	yes	—
119	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
123	C₈×C₅₃	—	4	4	no	—
124	#88322	—	4	4	no	—
125	#88322	—	4	4	no	—
126	#7877	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	—	2	4	no	—
128	C₂²×C₅₃	—	2	4	no	—
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	#340	1	4	yes	—
132	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
133	C₂²×C₅₃	#424	1	2	yes	—
134	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
135	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
136	C₂²×C₅₃	#430	1	2	yes	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	#340	1	4	yes	—
142	C₄×C₅₃	#405	1	2	yes	—
143	C₄×C₅₃	#408	1	2	yes	—
144	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
145	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
146	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
147	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
148	C₂×#431	—	53	1	no	—
149	C₂×#423	—	53	1	no	—
150	C₂²×C₄⋊C₈	D₅₃	1	1	yes	characteristic
151	#1277	—	53	4	no	—
152	C₂×C₅₃	—	2	4	no	—
153	C₂×C₅₃	#1570	1	2	yes	—
154	C₂×C₅₃	#1571	1	2	yes	—
155	C₂×C₅₃	#1729	1	1	yes	characteristic
156	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
157	C₂×C₅₃	#1738	1	1	yes	characteristic
158	C₂²×M₁₆	—	106	1	no	—
159	C₂³×C₈	—	2	1	no	—
160	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
161	#431	—	106	2	no	—
162	#431	—	106	2	no	—
163	#423	—	53	4	no	—
164	#423	—	106	2	no	—
165	C₂×C₄⋊C₈	—	2	2	no	—
166	C₂×C₄⋊C₈	—	2	1	no	—
167	C₂×C₄⋊C₈	—	53	2	no	—
168	C₂×C₄⋊C₈	—	53	2	no	—
169	C₂×C₄⋊C₈	—	106	1	no	—
170	C₂×C₄⋊C₈	Dic₅₃	1	4	yes	—
171	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
172	C₂×C₂²⋊C₈	—	53	2	no	—
173	C₂×C₄×C₈	—	53	2	no	—
174	C₅₃	#7877	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
176	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
177	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
178	C₂×M₁₆	—	106	4	no	—
179	C₂×M₁₆	—	212	2	no	—
180	C₂×M₁₆	—	212	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	4	no	—
183	C₂²×C₈	—	4	2	no	—
184	C₂²×C₈	—	4	2	no	—
185	C₂²×C₈	—	53	4	no	—
186	C₂²×C₈	—	106	2	no	—
187	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
188	C₂×C₄²	C₄×D₅₃	1	2	yes	—
189	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
190	C₂×C₄²	D₂₁₂	1	2	yes	—
191	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
192	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
193	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
194	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	4	no	—
197	C₄⋊C₈	—	53	4	no	—
198	C₄⋊C₈	—	106	2	no	—
199	C₄⋊C₈	—	106	4	no	—
200	C₄⋊C₈	—	106	4	no	—
201	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
202	C₂²⋊C₈	—	106	4	no	—
203	C₄×C₈	—	106	2	no	—
204	C₄×C₈	—	106	2	no	—
205	C₂⁴	#11719919	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
209	C₂²×C₄	#11719919	1	1	yes	characteristic
210	C₂²×C₄	#11719919	1	1	yes	characteristic
211	C₂²×C₄	#11719919	1	2	yes	—
212	C₂²×C₄	#11719919	1	1	yes	characteristic
213	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
214	C₂²×C₄	#11719921	1	2	yes	—
215	C₂²×C₄	#11719921	1	2	yes	—
216	C₂²×C₄	#11719922	1	2	yes	—
217	C₂²×C₄	#11719923	1	2	yes	—
218	C₂²×C₄	#11719924	1	2	yes	—
219	C₂²×C₄	#11719925	1	2	yes	—
220	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
221	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
224	M₁₆	—	212	8	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	4	8	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	106	4	no	—
230	C₂×C₈	—	106	4	no	—
231	C₂×C₈	—	106	8	no	—
232	C₂×C₈	—	212	2	no	—
233	C₂×C₈	—	212	2	no	—
234	C₄²	C₄×Dic₅₃	1	2	yes	—
235	C₄²	#11719919	1	2	yes	—
236	C₄²	C₄⋊Dic₅₃	1	2	yes	—
237	C₄²	#11719921	1	2	yes	—
238	C₄²	#11719921	1	2	yes	—
239	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
240	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
241	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
242	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
243	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	#13133124	1	2	yes	—
247	C₂³	#13133148	1	2	yes	—
248	C₂³	#13133150	1	2	yes	—
249	C₂³	C₂×#11719919	1	1	yes	characteristic
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	#13133124	1	2	yes	—
258	C₂×C₄	#13133124	1	2	yes	—
259	C₂×C₄	#13133124	1	2	yes	—
260	C₂×C₄	#13133137	1	4	yes	—
261	C₂×C₄	#13133138	1	4	yes	—
262	C₂×C₄	#13133140	1	4	yes	—
263	C₂×C₄	#13133141	1	4	yes	—
264	C₂×C₄	#13133148	1	2	yes	—
265	C₂×C₄	#13133150	1	2	yes	—
266	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
268	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
270	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
272	C₂×C₄	#13133223	1	4	yes	—
273	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
274	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
275	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
276	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
277	C₈	—	4	4	no	—
278	C₈	—	212	4	no	—
279	C₈	—	212	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	2	no	—
284	C₂²	#16810385	1	4	yes	—
285	C₂²	C₂×#13133124	1	1	yes	characteristic
286	C₂²	#16810649	1	2	yes	—
287	C₂²	C₂×#13133148	1	1	yes	characteristic
288	C₂²	C₂×#13133150	1	1	yes	characteristic
289	C₂²	#16810671	1	2	yes	—
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	2	no	—
294	C₄	#16810385	1	4	yes	—
295	C₄	#16810578	1	2	yes	—
296	C₄	#16810589	1	2	yes	—
297	C₄	C₂×#13133137	1	1	yes	characteristic
298	C₄	C₂×#13133138	1	1	yes	characteristic
299	C₄	C₂×#13133140	1	1	yes	characteristic
300	C₄	C₂×#13133141	1	1	yes	characteristic
301	C₂	—	2	4	no	—
302	C₂	#20273485	1	2	yes	—
303	C₂	#20273489	1	2	yes	—
304	C₂	#20274411	1	1	yes	characteristic
305	C₂	C₂×#16810385	1	1	yes	characteristic
306	C₂	#20274441	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂²×C₄⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂²×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#7877
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #20272549	C₂	semidirect
174. C₅₃	#7877	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
C₂×#16810647	2
#20272549	2
C₂²×C₄⋊C₈×C₅₃	2
#7877	53