Subgroups — #27397971

Order 13568.

846 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283427	C₂	1	2	yes	maximal, complemented
2	#20275544	C₂	1	2	yes	maximal, complemented
3	#20275044	C₂	1	8	yes	maximal, complemented
4	#20274853	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274460	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133224	C₂²	1	4	yes	—
8	#16810701	—	2	2	no	—
9	#16810701	—	2	2	no	—
10	D₄×#88322	—	2	2	no	—
11	D₄×#88322	—	2	2	no	—
12	#16810647	C₂²	1	2	yes	—
13	C₂×#13133144	C₂²	1	8	yes	—
14	#16810590	C₂²	1	4	yes	—
15	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
16	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
17	#16810388	C₂²	1	4	yes	—
18	#16810377	C₂²	1	4	yes	—
19	C₂×#135×C₅₃	C₂²	1	4	yes	—
20	C₅₃×#532	C₂²	1	2	yes	—
21	C₅₃×#495	C₄	1	4	yes	complemented
22	C₅₃×#494	C₄	1	1	yes	characteristic, complemented
23	C₅₃×#493	C₄	1	1	yes	characteristic, complemented
24	C₅₃×#492	C₄	1	2	yes	complemented
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	#13133224	—	4	8	no	—
27	C₂×#11719917	—	2	4	no	—
28	C₂×#11719917	C₂³	1	4	yes	—
29	C₂²×#88322	—	2	4	no	—
30	C₂²×#88322	C₂³	1	2	yes	—
31	C₂×C₄×Dic₅₃	—	2	1	no	—
32	C₂×C₄×Dic₅₃	—	2	1	no	—
33	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
34	C₄⋊#88322	—	2	2	no	—
35	C₄⋊#88322	—	2	2	no	—
36	#13133144	—	4	8	no	—
37	C₂²⋊#88322	—	2	4	no	—
38	C₂²⋊#88322	—	2	4	no	—
39	C₄×#88322	D₄	1	1	yes	characteristic
40	C₄×#88322	D₄	1	2	yes	—
41	C₄×#88322	D₄	1	1	yes	characteristic
42	C₂×#41×C₅₃	C₂³	1	4	yes	—
43	#135×C₅₃	—	4	8	no	—
44	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
45	C₂×SD₁₆×C₅₃	C₂×C₄	1	8	yes	—
46	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
49	#127×C₅₃	C₂×C₄	1	2	yes	—
50	#124×C₅₃	C₂×C₄	1	4	yes	—
51	#121×C₅₃	—	2	2	no	—
52	#121×C₅₃	—	2	2	no	—
53	C₄×D₄×C₅₃	—	2	2	no	—
54	C₄×D₄×C₅₃	—	2	2	no	—
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
59	#41×C₅₃	—	2	8	no	—
60	#41×C₅₃	—	2	8	no	—
61	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
62	C₂×D₄×C₅₃	—	2	4	no	—
63	C₂×D₄×C₅₃	—	2	2	no	—
64	C₂×D₄×C₅₃	—	2	2	no	—
65	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₅₃	—	2	4	no	—
67	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₅₃	—	2	2	no	—
71	Q₁₆×C₅₃	—	4	4	no	—
72	SD₁₆×C₅₃	—	4	8	no	—
73	D₈×C₅₃	—	4	4	no	—
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
77	C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₄⋊C₄×C₅₃	—	2	2	no	—
79	C₂²⋊C₄×C₅₃	—	2	4	no	—
80	C₂²⋊C₄×C₅₃	—	2	4	no	—
81	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₄×Dic₅₃	—	2	4	no	—
86	C₄×Dic₅₃	—	2	4	no	—
87	C₄×Dic₅₃	—	4	2	no	—
88	#11719917	—	2	8	no	—
89	#11719917	—	4	8	no	—
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	4	8	no	—
93	C₂×#88322	C₂×D₄	1	2	yes	—
94	C₂×#88322	C₂×D₄	1	2	yes	—
95	C₂³×C₅₃	—	2	4	no	—
96	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
97	Q₈×C₅₃	—	2	8	no	—
98	D₄×C₅₃	—	2	8	no	—
99	D₄×C₅₃	—	4	8	no	—
100	D₄×C₅₃	—	4	4	no	—
101	D₄×C₅₃	—	4	4	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	4	4	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
114	C₂×Dic₅₃	—	2	4	no	—
115	C₂×Dic₅₃	—	4	2	no	—
116	C₂×Dic₅₃	—	4	2	no	—
117	C₈×C₅₃	—	2	2	no	—
118	C₈×C₅₃	—	2	2	no	—
119	#88322	—	2	2	no	—
120	#88322	—	2	2	no	—
121	#88322	—	4	4	no	—
122	#22280	—	53	1	no	maximal, Sylow
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	4	4	no	—
126	C₂²×C₅₃	—	4	4	no	—
127	C₂²×C₅₃	—	4	4	no	—
128	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂²×C₅₃	#540	1	2	yes	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	4	4	no	—
133	C₄×C₅₃	#443	1	2	yes	—
134	C₄×C₅₃	#444	1	2	yes	—
135	C₄×C₅₃	#518	1	2	yes	—
136	C₄×C₅₃	#535	1	2	yes	—
137	Dic₅₃	—	4	4	no	—
138	#3147	D₅₃	1	1	yes	characteristic
139	#2931	—	53	2	no	—
140	#1962	—	53	2	no	—
141	#1876	—	53	8	no	—
142	#1837	—	53	1	no	—
143	#1744	—	53	1	no	—
144	C₂×C₅₃	—	2	2	no	—
145	C₂×C₅₃	—	8	4	no	—
146	C₂×C₅₃	#2328	1	1	yes	characteristic
147	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
148	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
149	C₂×#135	C₂×D₅₃	1	4	yes	—
150	C₂×#131	—	53	4	no	—
151	#532	C₂×D₅₃	1	2	yes	—
152	#495	Dic₅₃	1	4	yes	complemented
153	#494	Dic₅₃	1	1	yes	characteristic, complemented
154	#493	Dic₅₃	1	1	yes	characteristic, complemented
155	#492	Dic₅₃	1	2	yes	complemented
156	#436	—	106	2	no	—
157	#436	—	106	2	no	—
158	C₈×D₄	—	106	2	no	—
159	C₈×D₄	—	106	2	no	—
160	#423	—	53	2	no	—
161	C₂×#104	—	53	8	no	—
162	#408	—	53	4	no	—
163	C₂×C₄×C₈	—	53	1	no	—
164	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
165	C₄³	—	53	1	no	—
166	#341	—	53	4	no	—
167	#336	—	53	4	no	—
168	C₅₃	#22280	1	1	yes	characteristic, complemented, Sylow
169	C₂×#41	C₂²×D₅₃	1	4	yes	—
170	#135	—	4	8	no	—
171	C₂×Q₁₆	C₂×Dic₅₃	1	4	yes	—
172	C₂×SD₁₆	C₂×Dic₅₃	1	8	yes	—
173	C₂×D₈	C₂×Dic₅₃	1	4	yes	—
174	#131	—	212	8	no	—
175	C₂×M₁₆	—	53	4	no	—
176	C₂×M₁₆	—	106	4	no	—
177	C₂²×C₈	—	53	2	no	—
178	C₂²×C₈	—	106	4	no	—
179	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
180	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
181	#127	C₂×Dic₅₃	1	2	yes	—
182	#124	C₂×Dic₅₃	1	4	yes	—
183	#121	—	2	2	no	—
184	#121	—	2	2	no	—
185	C₄×D₄	—	2	2	no	—
186	C₄×D₄	—	2	2	no	—
187	C₂×C₄²	—	53	2	no	—
188	C₂×C₄²	—	106	1	no	—
189	C₂×C₄²	—	106	1	no	—
190	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
191	C₄⋊C₈	—	106	2	no	—
192	C₄⋊C₈	—	106	2	no	—
193	#104	—	212	8	no	—
194	C₂²⋊C₈	—	106	4	no	—
195	C₂²⋊C₈	—	106	4	no	—
196	C₄×C₈	—	53	1	no	—
197	C₄×C₈	—	53	2	no	—
198	C₄×C₈	—	53	1	no	—
199	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
200	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
201	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
202	#41	—	2	8	no	—
203	#41	—	2	8	no	—
204	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	C₂²×Dic₅₃	1	4	yes	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	106	2	no	—
211	C₂²×C₄	#11719947	1	1	yes	characteristic
212	C₂²×C₄	#11719947	1	1	yes	characteristic
213	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
214	Q₁₆	—	4	4	no	—
215	SD₁₆	—	4	8	no	—
216	D₈	—	4	4	no	—
217	M₁₆	—	106	8	no	—
218	M₁₆	—	212	8	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	53	2	no	—
221	C₂×C₈	—	53	2	no	—
222	C₂×C₈	—	106	4	no	—
223	C₂×C₈	—	106	4	no	—
224	C₂×C₈	—	212	8	no	—
225	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
226	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	4	no	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₄²	—	106	4	no	—
232	C₄²	—	106	4	no	—
233	C₄²	—	212	2	no	—
234	C₄²	D₄×D₅₃	1	1	yes	characteristic
235	C₄²	D₄×D₅₃	1	1	yes	characteristic
236	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
237	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
238	C₂³	—	2	4	no	—
239	C₂³	C₂×#11719947	1	1	yes	characteristic
240	Q₈	—	2	8	no	—
241	D₄	—	2	8	no	—
242	D₄	—	4	8	no	—
243	D₄	—	4	4	no	—
244	D₄	—	4	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	106	4	no	—
251	C₂×C₄	—	212	2	no	—
252	C₂×C₄	—	212	2	no	—
253	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
254	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
255	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
256	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
259	C₂×C₄	#13133262	1	4	yes	—
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₈	—	106	2	no	—
263	C₈	—	106	2	no	—
264	C₈	—	212	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	4	4	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
271	C₂²	#16811227	1	2	yes	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	4	no	—
275	C₄	—	212	4	no	—
276	C₄	#16810740	1	2	yes	—
277	C₄	#16810745	1	2	yes	—
278	C₄	#16811098	1	2	yes	—
279	C₄	#16811186	1	2	yes	—
280	C₂	—	2	2	no	—
281	C₂	—	8	4	no	—
282	C₂	#20277441	1	1	yes	characteristic
283	C₂	C₂×#16810740	1	1	yes	characteristic
284	C₂	C₂×#16810745	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3147	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#3147	D₅₃
hypocenter	C₅₃	#22280
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #20283427	C₂	semidirect
2. #20275544	C₂	semidirect
3. #20275044	C₂	semidirect
4. #20274853	C₂	semidirect
5. #20274460	C₂	semidirect
21. C₅₃×#495	C₄	semidirect
22. C₅₃×#494	C₄	semidirect
23. C₅₃×#493	C₄	semidirect
24. C₅₃×#492	C₄	semidirect
152. #495	Dic₅₃	semidirect
153. #494	Dic₅₃	semidirect
154. #493	Dic₅₃	semidirect
155. #492	Dic₅₃	semidirect
168. C₅₃	#22280	semidirect

Maximal subgroups

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

subgroup	index
#20283427	2
#20275544	2
#20275044	2
#20274853	2
#20274460	2
C₅₃×#3147	2
#22280	53