Subgroups — #27273478

Order 13568.

734 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	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#20272542	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₄	1	2	yes	—
6	#16810676	—	2	2	no	—
7	#16810676	—	2	2	no	—
8	C₂×C₄⋊#88322	C₄	1	2	yes	—
9	C₂×C₂²⋊#88322	C₄	1	2	yes	—
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	C₂²	1	2	yes	—
14	C₂×C₄×#88322	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#431	—	2	2	no	—
17	C₅₃×#431	—	2	2	no	—
18	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
19	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
21	C₂²×#88322	—	2	4	no	—
22	C₂²×#88322	—	2	4	no	—
23	C₂²×#88322	C₈	1	8	yes	complemented
24	C₂²×#88322	C₂×C₄	1	4	yes	—
25	C₂²×#88322	C₂×C₄	1	4	yes	—
26	C₄⋊#88322	—	2	4	no	—
27	C₂²⋊#88322	—	2	4	no	—
28	C₄×#88322	—	2	4	no	—
29	C₄×#88322	—	2	2	no	—
30	C₄×#88322	—	2	2	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	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	2	yes	—
41	C₂×C₄²×C₅₃	D₄	1	2	yes	—
42	C₂×C₄²×C₅₃	D₄	1	2	yes	—
43	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₅₃	—	2	4	no	—
46	C₂²⋊C₈×C₅₃	—	2	4	no	—
47	C₄×C₈×C₅₃	—	2	2	no	—
48	C₄×C₈×C₅₃	—	2	2	no	—
49	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	—	2	4	no	—
52	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
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	2	yes	—
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	—	2	8	no	—
68	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
72	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
73	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
78	C₂×#88322	—	2	16	no	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	—	2	8	no	—
83	C₂×#88322	C₂×C₈	1	8	yes	—
84	C₂³×C₅₃	—	2	2	no	—
85	C₂³×C₅₃	—	2	2	no	—
86	C₂³×C₅₃	#95	1	2	yes	—
87	C₂³×C₅₃	C₄×C₈	1	2	yes	—
88	C₂³×C₅₃	#97	1	2	yes	—
89	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	2	4	no	—
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	#95	1	2	yes	—
97	C₂×C₄×C₅₃	#95	1	2	yes	—
98	C₂×C₄×C₅₃	#95	1	2	yes	—
99	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
100	C₂×C₄×C₅₃	#97	1	2	yes	—
101	C₂×C₄×C₅₃	C₂²⋊C₈	1	8	yes	—
102	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	—
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×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₅₃	#131	1	4	yes	—
112	C₂×C₄×C₅₃	#131	1	4	yes	—
113	C₈×C₅₃	—	4	4	no	—
114	#88322	—	2	8	no	—
115	#88322	—	4	4	no	—
116	#7849	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	—	2	4	no	—
118	C₂²×C₅₃	—	2	4	no	—
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	#336	1	4	yes	—
122	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
123	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
124	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
125	C₂²×C₅₃	#405	1	2	yes	—
126	C₂²×C₅₃	#405	1	2	yes	—
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	#336	1	4	yes	—
130	C₄×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
131	C₄×C₅₃	#408	1	2	yes	—
132	C₄×C₅₃	#408	1	2	yes	—
133	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
134	C₄×C₅₃	#424	1	2	yes	—
135	C₄×C₅₃	#424	1	2	yes	—
136	C₂×#431	—	53	1	no	—
137	C₂×#431	D₅₃	1	1	yes	characteristic
138	C₂²×C₄×C₈	—	53	1	no	—
139	#1273	—	53	4	no	—
140	C₂×C₅₃	—	2	4	no	—
141	C₂×C₅₃	#1450	1	2	yes	—
142	C₂×C₅₃	#1452	1	2	yes	—
143	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
144	C₂×C₅₃	#1729	1	1	yes	characteristic
145	C₂×C₅₃	#1729	1	1	yes	characteristic
146	C₂³×C₈	—	53	2	no	—
147	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
148	#431	—	2	2	no	—
149	#431	—	2	2	no	—
150	#431	—	106	2	no	—
151	#431	—	106	2	no	—
152	C₂×C₄⋊C₈	—	53	2	no	—
153	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
154	C₂×C₂²⋊C₈	—	53	2	no	—
155	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
156	C₂×C₄×C₈	—	53	2	no	—
157	C₂×C₄×C₈	—	53	2	no	—
158	C₂×C₄×C₈	—	53	2	no	—
159	C₂×C₄×C₈	—	106	2	no	—
160	C₂×C₄×C₈	—	106	2	no	—
161	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
162	C₅₃	#7849	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
164	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
165	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
166	C₂²×C₈	—	53	4	no	—
167	C₂²×C₈	—	53	8	no	—
168	C₂²×C₈	—	53	4	no	—
169	C₂²×C₈	—	106	4	no	—
170	C₂²×C₈	—	106	4	no	—
171	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
172	C₂×C₄²	Dic₁₀₆	1	2	yes	—
173	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
174	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
175	C₂×C₄²	D₂₁₂	1	2	yes	—
176	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
177	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
178	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
179	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
180	C₄⋊C₈	—	2	4	no	—
181	C₄⋊C₈	—	106	4	no	—
182	C₂²⋊C₈	—	2	4	no	—
183	C₂²⋊C₈	—	106	4	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	—	106	2	no	—
187	C₄×C₈	—	106	2	no	—
188	C₄×C₈	—	106	2	no	—
189	C₄×C₈	—	106	2	no	—
190	C₄×C₈	—	106	4	no	—
191	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	C₈×D₅₃	1	4	yes	—
195	C₂²×C₄	#11719912	1	4	yes	—
196	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
197	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
198	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
199	C₂²×C₄	#11719919	1	2	yes	—
200	C₂²×C₄	#11719919	1	2	yes	—
201	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
202	C₂²×C₄	#11719921	1	2	yes	—
203	C₂²×C₄	#11719921	1	2	yes	—
204	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
205	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	8	no	—
210	C₂×C₈	—	53	8	no	—
211	C₂×C₈	—	106	16	no	—
212	C₂×C₈	—	106	4	no	—
213	C₂×C₈	—	106	4	no	—
214	C₂×C₈	—	106	4	no	—
215	C₂×C₈	—	106	8	no	—
216	C₄²	#11719919	1	2	yes	—
217	C₄²	#11719919	1	2	yes	—
218	C₄²	C₄⋊Dic₅₃	1	2	yes	—
219	C₄²	#11719921	1	2	yes	—
220	C₄²	#11719921	1	2	yes	—
221	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
222	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
223	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
224	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	#13133124	1	2	yes	—
229	C₂³	C₈×Dic₅₃	1	2	yes	—
230	C₂³	#13133127	1	2	yes	—
231	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	#13133124	1	2	yes	—
239	C₂×C₄	#13133124	1	2	yes	—
240	C₂×C₄	#13133124	1	2	yes	—
241	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
242	C₂×C₄	#13133127	1	2	yes	—
243	C₂×C₄	#13133129	1	8	yes	—
244	C₂×C₄	#13133146	1	8	yes	—
245	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
246	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
247	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
249	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
250	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
252	C₂×C₄	C₂×#11719912	1	2	yes	—
253	C₂×C₄	#13133224	1	4	yes	—
254	C₂×C₄	#13133225	1	4	yes	—
255	C₈	—	4	4	no	—
256	C₈	—	106	8	no	—
257	C₈	—	212	4	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	#16810377	1	4	yes	—
263	C₂²	C₂×#13133124	1	1	yes	characteristic
264	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
265	C₂²	C₂×#13133127	1	1	yes	characteristic
266	C₂²	#16810579	1	2	yes	—
267	C₂²	#16810581	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	#16810377	1	4	yes	—
271	C₄	C₂×#13133129	1	2	yes	—
272	C₄	#16810590	1	2	yes	—
273	C₄	#16810591	1	2	yes	—
274	C₄	C₂×#13133146	1	2	yes	—
275	C₄	#16810650	1	2	yes	—
276	C₄	#16810651	1	2	yes	—
277	C₂	—	2	4	no	—
278	C₂	#20272999	1	2	yes	—
279	C₂	#20273008	1	2	yes	—
280	C₂	C₂×#16810377	1	1	yes	characteristic
281	C₂	#20274412	1	1	yes	characteristic
282	C₂	#20274413	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₅₃×#431	C₂
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#7849
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20272542	C₂	semidirect
23. C₂²×#88322	C₈	semidirect
162. C₅₃	#7849	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
C₂²×C₄×#88322	2
#20272542	2
C₂×C₅₃×#431	2
#7849	53