Subgroups — #27356130

Order 13568.

644 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283782	C₂	1	2	yes	maximal, complemented
2	#20273629	C₂	1	2	yes	maximal, complemented
3	#20273545	C₂	1	4	yes	maximal, complemented
4	#20273539	C₂	1	2	yes	maximal, complemented
5	#20273463	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273132	C₂	1	2	yes	maximal, complemented
7	#20273004	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
9	#16810701	—	2	2	no	—
10	#16810701	C₂²	1	4	yes	—
11	D₄×#88322	—	2	2	no	—
12	D₄×#88322	C₂²	1	4	yes	—
13	#16810676	C₂²	1	2	yes	—
14	#16810676	C₂²	1	1	yes	characteristic
15	#16810590	—	2	4	no	—
16	C₄×#11719917	C₂²	1	1	yes	characteristic
17	#16810376	C₂²	1	2	yes	—
18	#16810374	C₂²	1	1	yes	characteristic
19	#16810374	C₂²	1	1	yes	characteristic
20	D₄⋊#88322	C₂²	1	4	yes	—
21	D₄⋊#88322	C₂²	1	4	yes	—
22	#16810345	C₂²	1	2	yes	—
23	#16810344	C₂²	1	2	yes	—
24	C₅₃×#532	C₂²	1	2	yes	—
25	C₅₃×#500	C₄	1	1	yes	characteristic
26	C₅₃×#493	C₄	1	1	yes	characteristic
27	C₅₃×#486	C₄	1	2	yes	—
28	C₅₃×#466	C₄	1	2	yes	—
29	C₅₃×#466	C₄	1	2	yes	—
30	C₄×D₈×C₅₃	C₂²	1	2	yes	—
31	C₄×D₈×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#11719917	—	2	4	no	—
34	C₂×#11719917	—	2	1	no	—
35	C₂²×#88322	—	2	4	no	—
36	C₄⋊#88322	D₄	1	2	yes	—
37	C₄⋊#88322	C₂³	1	2	yes	—
38	C₄⋊#88322	C₂³	1	2	yes	—
39	C₂²⋊#88322	—	2	4	no	—
40	C₂²⋊#88322	—	2	4	no	—
41	C₂²⋊#88322	—	2	2	no	—
42	C₂²⋊#88322	D₄	1	2	yes	—
43	#13133128	—	2	1	no	—
44	C₄×#88322	—	2	2	no	—
45	C₄×#88322	C₂³	1	2	yes	—
46	C₄×#88322	C₂³	1	2	yes	—
47	C₂×#41×C₅₃	—	2	2	no	—
48	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
53	#127×C₅₃	C₂×C₄	1	2	yes	—
54	#124×C₅₃	—	2	2	no	—
55	#121×C₅₃	—	2	2	no	—
56	#121×C₅₃	C₂×C₄	1	4	yes	—
57	C₄×D₄×C₅₃	—	2	2	no	—
58	C₄×D₄×C₅₃	C₂³	1	4	yes	—
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	#107×C₅₃	C₂×C₄	1	2	yes	—
61	#107×C₅₃	C₂×C₄	1	2	yes	—
62	#102×C₅₃	C₂×C₄	1	4	yes	—
63	#102×C₅₃	C₂×C₄	1	4	yes	—
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	#41×C₅₃	—	8	2	no	—
69	#41×C₅₃	—	8	2	no	—
70	C₂×Q₈×C₅₃	—	2	2	no	—
71	C₂×D₄×C₅₃	—	2	4	no	—
72	C₂×D₄×C₅₃	—	2	2	no	—
73	C₂×D₄×C₅₃	—	2	2	no	—
74	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₅₃	—	2	4	no	—
76	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
79	D₈×C₅₃	—	2	4	no	—
80	D₈×C₅₃	—	4	2	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	4	1	no	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
89	C₂²⋊C₄×C₅₃	—	2	4	no	—
90	C₂²⋊C₄×C₅₃	—	2	4	no	—
91	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅₃	#41	1	1	yes	characteristic
94	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
95	#11719917	—	4	2	no	—
96	#11719917	—	8	4	no	—
97	C₂×#88322	—	2	2	no	—
98	C₂×#88322	—	2	2	no	—
99	C₂×#88322	—	2	2	no	—
100	C₂×#88322	—	8	4	no	—
101	C₂×#88322	C₂×D₄	1	4	yes	—
102	C₂³×C₅₃	—	2	4	no	—
103	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
104	Q₈×C₅₃	—	8	2	no	—
105	D₄×C₅₃	—	2	4	no	—
106	D₄×C₅₃	—	2	4	no	—
107	D₄×C₅₃	—	4	4	no	—
108	D₄×C₅₃	—	8	2	no	—
109	D₄×C₅₃	—	8	4	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	4	no	—
115	C₂×C₄×C₅₃	—	4	4	no	—
116	C₂×C₄×C₅₃	—	4	4	no	—
117	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
123	C₈×C₅₃	—	2	1	no	—
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	#131	1	2	yes	—
127	#88322	—	2	4	no	—
128	#88322	—	4	2	no	—
129	#88322	—	4	2	no	—
130	#88322	—	4	2	no	—
131	#18091	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	—	2	1	no	—
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	—	4	4	no	—
136	C₂²×C₅₃	—	4	4	no	—
137	C₂²×C₅₃	—	4	4	no	—
138	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	4	4	no	—
143	C₄×C₅₃	#444	1	2	yes	—
144	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
145	C₄×C₅₃	#568	1	1	yes	characteristic
146	C₄×C₅₃	#576	1	2	yes	—
147	#3169	D₅₃	1	1	yes	characteristic
148	#2979	—	53	2	no	—
149	#1594	—	53	2	no	—
150	#1581	—	53	4	no	—
151	#1580	—	53	2	no	—
152	#1564	—	53	1	no	—
153	#1484	—	53	2	no	—
154	#1451	—	53	1	no	—
155	C₂×C₅₃	—	4	1	no	—
156	C₂×C₅₃	—	8	4	no	—
157	C₂×C₅₃	#2935	1	1	yes	characteristic
158	C₂×C₅₃	#2948	1	1	yes	characteristic
159	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
160	#532	C₂×D₅₃	1	2	yes	—
161	#500	Dic₅₃	1	1	yes	characteristic
162	#493	Dic₅₃	1	1	yes	characteristic
163	#486	Dic₅₃	1	2	yes	—
164	#466	Dic₅₃	1	2	yes	—
165	#466	Dic₅₃	1	2	yes	—
166	C₄×D₈	C₂×D₅₃	1	2	yes	—
167	C₄×D₈	C₂×D₅₃	1	2	yes	—
168	#436	—	53	4	no	—
169	#436	—	106	2	no	—
170	C₈×D₄	—	53	4	no	—
171	C₈×D₄	—	106	2	no	—
172	#431	—	53	1	no	—
173	#431	—	53	2	no	—
174	#408	—	106	4	no	—
175	C₄×M₁₆	—	53	1	no	—
176	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
177	#335	—	53	2	no	—
178	#334	—	53	1	no	—
179	#334	—	53	1	no	—
180	D₄⋊C₈	—	53	4	no	—
181	D₄⋊C₈	—	53	4	no	—
182	#322	—	53	2	no	—
183	#322	—	53	2	no	—
184	C₅₃	#18091	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	2	2	no	—
186	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
187	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
188	C₂×M₁₆	—	106	4	no	—
189	C₂×M₁₆	—	106	1	no	—
190	C₂²×C₈	—	106	4	no	—
191	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
192	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
193	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
194	#127	C₂×Dic₅₃	1	2	yes	—
195	#124	—	2	2	no	—
196	#121	—	2	2	no	—
197	#121	C₂×Dic₅₃	1	4	yes	—
198	C₄×D₄	—	2	2	no	—
199	C₄×D₄	C₂²×D₅₃	1	4	yes	—
200	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
201	#107	C₂×Dic₅₃	1	2	yes	—
202	#107	C₂×Dic₅₃	1	2	yes	—
203	C₄⋊C₈	—	53	2	no	—
204	C₄⋊C₈	—	53	2	no	—
205	C₄⋊C₈	—	53	2	no	—
206	#102	C₂×Dic₅₃	1	4	yes	—
207	#102	C₂×Dic₅₃	1	4	yes	—
208	C₂²⋊C₈	—	53	2	no	—
209	C₂²⋊C₈	—	106	4	no	—
210	C₂²⋊C₈	—	106	4	no	—
211	C₂²⋊C₈	—	106	2	no	—
212	#97	—	106	1	no	—
213	C₄×C₈	—	53	2	no	—
214	C₄×C₈	—	53	2	no	—
215	C₄×C₈	—	106	2	no	—
216	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
217	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
220	#41	—	8	2	no	—
221	#41	—	8	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	C₂²×Dic₅₃	1	4	yes	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
229	C₂²×C₄	#11719947	1	1	yes	characteristic
230	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
231	D₈	—	2	4	no	—
232	D₈	—	4	2	no	—
233	M₁₆	—	424	4	no	—
234	M₁₆	—	212	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	53	4	no	—
238	C₂×C₈	—	106	2	no	—
239	C₂×C₈	—	106	2	no	—
240	C₂×C₈	—	106	2	no	—
241	C₂×C₈	—	424	4	no	—
242	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
243	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
244	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
245	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₄²	D₄×D₅₃	1	1	yes	characteristic
251	C₄²	#11719947	1	1	yes	characteristic
252	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
253	C₄²	C₂³×D₅₃	1	1	yes	characteristic
254	C₂³	—	2	4	no	—
255	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
256	Q₈	—	8	2	no	—
257	D₄	—	2	4	no	—
258	D₄	—	2	4	no	—
259	D₄	—	4	4	no	—
260	D₄	—	8	2	no	—
261	D₄	—	8	4	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
270	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
271	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
272	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
274	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	106	4	no	—
279	C₈	—	212	2	no	—
280	C₈	—	212	2	no	—
281	C₈	—	212	2	no	—
282	C₈	#13133224	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	4	no	—
289	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	4	no	—
294	C₄	#16810745	1	2	yes	—
295	C₄	C₂×#13133224	1	1	yes	characteristic
296	C₄	#16811390	1	1	yes	characteristic
297	C₄	#16811424	1	2	yes	—
298	C₂	—	4	1	no	—
299	C₂	—	8	4	no	—
300	C₂	#20283459	1	1	yes	characteristic
301	C₂	#20283567	1	1	yes	characteristic
302	C₂	C₂×#16810745	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3169	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#3169	D₅₃
hypocenter	C₅₃	#18091
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #20283782	C₂	semidirect
2. #20273629	C₂	semidirect
3. #20273545	C₂	semidirect
4. #20273539	C₂	semidirect
5. #20273463	C₂	semidirect
6. #20273132	C₂	semidirect
7. #20273004	C₂	semidirect
184. C₅₃	#18091	semidirect

Maximal subgroups

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

subgroup	index
#20283782	2
#20273629	2
#20273545	2
#20273539	2
#20273463	2
#20273132	2
#20273004	2
C₅₃×#3169	2
#18091	53