Subgroups — #27298930

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133156	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272793	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272792	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
9	C₂×#13133156	—	2	4	no	—
10	C₂×#13133156	—	2	4	no	—
11	C₂×#13133156	C₄	1	8	yes	complemented
12	C₂²×#11719908	C₂²	1	2	yes	—
13	C₂×C₄×#88322	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₅₃	—	2	4	no	—
17	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄×D₅₃	C₈	1	2	yes	—
20	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×D₅₃	—	2	2	no	—
24	C₂×C₈×D₅₃	—	2	2	no	—
25	C₂×C₈×D₅₃	C₈	1	4	yes	—
26	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
27	C₂×C₈×D₅₃	D₄	1	4	yes	—
28	C₂×C₈×D₅₃	D₄	1	2	yes	—
29	C₂×C₈×D₅₃	Q₈	1	2	yes	—
30	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
31	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄²×D₅₃	—	2	2	no	—
33	C₄²×D₅₃	—	2	2	no	—
34	#13133156	—	2	16	no	—
35	C₂×#11719908	—	2	2	no	—
36	C₂×#11719908	—	2	2	no	—
37	C₂×#11719908	C₂×C₄	1	4	yes	complemented
38	C₄×#88322	—	2	2	no	—
39	C₈×Dic₅₃	—	2	2	no	—
40	C₈×Dic₅₃	—	2	2	no	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₆×C₅₃	—	2	2	no	—
45	C₂×C₁₆×C₅₃	—	2	2	no	—
46	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
47	C₄×C₈×C₅₃	—	2	2	no	—
48	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
52	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×D₅₃	—	2	4	no	—
54	C₂×C₄×D₅₃	—	2	4	no	—
55	C₂×C₄×D₅₃	—	2	4	no	—
56	C₂×C₄×D₅₃	C₄²	1	2	yes	—
57	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
60	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
61	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₅₃	—	2	1	no	—
70	C₄²×C₅₃	—	2	1	no	—
71	C₄×Dic₅₃	—	2	1	no	—
72	C₄×Dic₅₃	—	2	1	no	—
73	C₄×Dic₅₃	—	2	4	no	—
74	C₂×#88322	—	2	1	no	—
75	C₂×#88322	—	2	1	no	—
76	C₂×#88322	C₄²	1	2	yes	—
77	C₂×#88322	C₂²⋊C₄	1	2	yes	—
78	C₂×#88322	C₄⋊C₄	1	2	yes	—
79	C₂×#88322	C₂×C₈	1	2	yes	—
80	C₈×D₅₃	—	2	8	no	—
81	C₈×D₅₃	C₂²⋊C₄	1	8	yes	—
82	C₈×D₅₃	C₄⋊C₄	1	8	yes	—
83	C₁₆×C₅₃	—	2	4	no	—
84	#11719908	—	2	4	no	—
85	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
86	C₂²×D₅₃	—	2	2	no	—
87	C₂²×D₅₃	—	2	2	no	—
88	C₂²×D₅₃	C₄×C₈	1	2	yes	—
89	C₂²×D₅₃	C₂²⋊C₈	1	2	yes	—
90	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	M₃₂	1	4	yes	complemented
94	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×Dic₅₃	—	2	2	no	—
101	C₂×Dic₅₃	—	2	2	no	—
102	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
103	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
104	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
105	C₂×Dic₅₃	M₃₂	1	4	yes	complemented
106	C₄×D₅₃	—	2	16	no	—
107	C₄×D₅₃	#95	1	4	yes	—
108	C₄×D₅₃	#97	1	4	yes	—
109	C₄×D₅₃	C₂²⋊C₈	1	4	yes	—
110	C₄×D₅₃	C₄⋊C₈	1	4	yes	—
111	C₈×C₅₃	—	2	2	no	—
112	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
113	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
114	#88322	—	2	2	no	—
115	#88322	#95	1	4	yes	—
116	#12544	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
118	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
121	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
122	C₂×D₅₃	—	2	2	no	—
123	C₂×D₅₃	—	2	2	no	—
124	C₂×D₅₃	—	2	4	no	—
125	C₂×D₅₃	—	2	4	no	—
126	C₂×D₅₃	#336	1	4	yes	—
127	C₄×C₅₃	—	2	4	no	—
128	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
129	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
130	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
132	Dic₅₃	—	2	4	no	—
133	Dic₅₃	#336	1	4	yes	—
134	C₂²×M₃₂	—	53	2	no	—
135	C₂²×C₄×C₈	—	53	1	no	—
136	#1374	—	53	2	no	—
137	#1374	—	53	1	no	—
138	#1374	D₅₃	1	1	yes	characteristic, complemented
139	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
140	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
141	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
142	C₂×C₅₃	#2112	1	2	yes	—
143	C₂×C₅₃	#2150	1	2	yes	—
144	D₅₃	—	2	4	no	—
145	C₂³×C₈	—	53	1	no	—
146	C₂³×C₈	—	53	1	no	—
147	C₂²×C₄²	—	53	1	no	—
148	C₂×M₃₂	—	53	8	no	—
149	C₂×M₃₂	—	106	4	no	—
150	C₂×M₃₂	—	106	4	no	—
151	C₂²×C₁₆	—	53	2	no	—
152	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
153	C₂×C₄×C₈	—	53	1	no	—
154	C₂×C₄×C₈	—	53	1	no	—
155	C₂×C₄×C₈	—	53	1	no	—
156	C₂×C₄×C₈	—	106	4	no	—
157	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
158	C₅₃	#12544	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	—	53	1	no	—
160	C₂³×C₄	—	53	2	no	—
161	C₂²×C₈	—	53	1	no	—
162	C₂²×C₈	—	53	4	no	—
163	C₂²×C₈	—	53	4	no	—
164	C₂²×C₈	—	53	2	no	—
165	C₂²×C₈	—	53	2	no	—
166	C₂²×C₈	—	53	1	no	—
167	C₂²×C₈	—	53	4	no	—
168	C₂²×C₈	—	106	2	no	—
169	C₂²×C₈	—	106	2	no	—
170	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
172	C₂×C₄²	—	53	1	no	—
173	C₂×C₄²	—	53	2	no	—
174	C₂×C₄²	—	106	2	no	—
175	C₂×C₄²	—	106	2	no	—
176	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
177	M₃₂	—	106	16	no	—
178	C₂×C₁₆	—	2	2	no	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	53	4	no	—
181	C₂×C₁₆	—	106	2	no	—
182	C₂×C₁₆	—	106	2	no	—
183	C₂×C₁₆	C₄×D₅₃	1	4	yes	complemented
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	106	2	no	—
186	C₄×C₈	—	106	2	no	—
187	C₄×C₈	—	106	2	no	—
188	C₂⁴	—	53	1	no	—
189	C₂²×C₄	—	53	1	no	—
190	C₂²×C₄	—	53	2	no	—
191	C₂²×C₄	—	53	2	no	—
192	C₂²×C₄	—	53	2	no	—
193	C₂²×C₄	—	53	2	no	—
194	C₂²×C₄	—	53	2	no	—
195	C₂²×C₄	—	53	2	no	—
196	C₂²×C₄	—	53	2	no	—
197	C₂²×C₄	—	106	4	no	—
198	C₂²×C₄	—	106	4	no	—
199	C₂²×C₄	—	106	4	no	—
200	C₂²×C₄	C₈×D₅₃	1	2	yes	—
201	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	53	2	no	—
205	C₂×C₈	—	53	2	no	—
206	C₂×C₈	—	53	2	no	—
207	C₂×C₈	—	53	8	no	—
208	C₂×C₈	—	53	8	no	—
209	C₂×C₈	—	53	2	no	—
210	C₂×C₈	—	106	8	no	—
211	C₂×C₈	—	106	1	no	—
212	C₂×C₈	—	106	1	no	—
213	C₂×C₈	C₈×D₅₃	1	2	yes	—
214	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
215	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
216	C₂×C₈	D₄×D₅₃	1	2	yes	—
217	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	106	4	no	—
221	C₄²	—	106	1	no	—
222	C₄²	—	106	1	no	—
223	C₁₆	—	2	4	no	—
224	C₁₆	—	106	4	no	—
225	C₂³	—	53	2	no	—
226	C₂³	—	53	2	no	—
227	C₂³	—	53	2	no	—
228	C₂³	—	106	2	no	—
229	C₂³	—	106	2	no	—
230	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	53	2	no	—
234	C₂×C₄	—	53	2	no	—
235	C₂×C₄	—	53	2	no	—
236	C₂×C₄	—	53	4	no	—
237	C₂×C₄	—	53	4	no	—
238	C₂×C₄	—	53	4	no	—
239	C₂×C₄	—	53	4	no	—
240	C₂×C₄	—	53	4	no	—
241	C₂×C₄	—	106	16	no	—
242	C₂×C₄	—	106	2	no	—
243	C₂×C₄	—	106	2	no	—
244	C₂×C₄	#13133156	1	4	yes	complemented
245	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
246	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
247	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
249	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
250	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	53	4	no	—
253	C₈	—	106	2	no	—
254	C₈	C₂²⋊C₄×D₅₃	1	2	yes	—
255	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
256	C₂²	—	53	4	no	—
257	C₂²	—	106	2	no	—
258	C₂²	—	106	2	no	—
259	C₂²	—	106	4	no	—
260	C₂²	—	106	4	no	—
261	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
262	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
263	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
264	C₂²	M₃₂×D₅₃	1	2	yes	—
265	C₂²	C₂×#13133156	1	2	yes	—
266	C₄	—	2	4	no	—
267	C₄	—	53	4	no	—
268	C₄	—	106	4	no	—
269	C₄	#95×D₅₃	1	1	yes	characteristic
270	C₄	#97×D₅₃	1	1	yes	characteristic
271	C₄	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
272	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
273	C₂	—	106	4	no	—
274	C₂	#336×D₅₃	1	1	yes	characteristic
275	C₂	C₄×#13133156	1	1	yes	characteristic
276	C₂	#20276308	1	1	yes	characteristic
277	C₂	#20276324	1	2	yes	—
278	C₂	#20276469	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#1374	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1374	D₅₃
hypocenter	C₅₃	#12544
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
3. #20272793	C₂	semidirect
4. #20272792	C₂	semidirect
5. C₅₃×#1374	C₂	semidirect
11. C₂×#13133156	C₄	semidirect
37. C₂×#11719908	C₂×C₄	semidirect
46. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
93. C₂×C₄×C₅₃	M₃₂	semidirect
105. C₂×Dic₅₃	M₃₂	semidirect
138. #1374	D₅₃	semidirect
158. C₅₃	#12544	semidirect
183. C₂×C₁₆	C₄×D₅₃	semidirect
244. C₂×C₄	#13133156	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133156	2
C₂×C₄×C₈×D₅₃	2
#20272793	2
#20272792	2
C₅₃×#1374	2
#12544	53