Subgroups — #27355907

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285301	C₂	1	1	yes	maximal, characteristic
2	#20283658	C₂	1	2	yes	maximal
3	#20273623	C₂	1	2	yes	maximal, complemented
4	#20273585	C₂	1	4	yes	maximal, complemented
5	#20273533	C₂	1	2	yes	maximal, complemented
6	#20273117	C₂	1	2	yes	maximal
7	#20273004	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
9	#16811180	C₂²	1	2	yes	—
10	#16811028	C₄	1	1	yes	characteristic
11	#16811003	C₄	1	1	yes	characteristic
12	#16810974	C₄	1	2	yes	—
13	#16810864	C₄	1	2	yes	—
14	#16810864	C₄	1	2	yes	—
15	#16810752	—	2	4	no	—
16	#16810750	C₂²	1	4	yes	—
17	#16810747	C₂²	1	4	yes	—
18	C₄×Dic₂₁₂	C₂²	1	2	yes	—
19	C₄×Dic₂₁₂	C₂²	1	2	yes	—
20	#16810676	C₂²	1	1	yes	characteristic
21	#16810649	—	2	2	no	—
22	#16810578	—	2	2	no	—
23	C₄×#11719917	C₂²	1	1	yes	characteristic
24	#16810353	C₂²	1	4	yes	—
25	#16810353	C₂²	1	4	yes	—
26	#16810345	C₂²	1	2	yes	—
27	#16810344	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#334	C₂²	1	1	yes	characteristic
31	C₅₃×#334	C₂²	1	2	yes	—
32	C₅₃×#334	C₂²	1	1	yes	characteristic
33	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
35	C₂×#11719917	D₄	1	2	yes	—
36	#13133217	C₂×C₄	1	2	yes	—
37	#13133217	C₂×C₄	1	2	yes	—
38	#13133209	—	2	2	no	—
39	#13133196	—	2	2	no	—
40	#13133196	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₀₆	—	2	2	no	—
42	C₄×Dic₁₀₆	C₂³	1	4	yes	—
43	#13133176	—	2	2	no	—
44	#13133151	C₂×C₄	1	2	yes	—
45	#13133151	C₂×C₄	1	2	yes	—
46	C₄⋊#88322	—	2	1	no	—
47	#13133146	—	2	4	no	—
48	#13133146	—	2	4	no	—
49	#13133142	C₂×C₄	1	4	yes	—
50	#13133142	C₂×C₄	1	4	yes	—
51	C₂²⋊#88322	—	2	1	no	—
52	#13133128	D₄	1	2	yes	—
53	#13133127	—	2	4	no	—
54	C₄×#88322	C₂³	1	2	yes	—
55	C₄×#88322	C₂³	1	2	yes	—
56	C₈×Dic₅₃	—	2	4	no	—
57	C₂²×C₈×C₅₃	—	2	2	no	—
58	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₅₃	—	2	2	no	—
62	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
63	C₄⋊C₈×C₅₃	C₂³	1	2	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	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
71	C₂×Dic₁₀₆	—	2	2	no	—
72	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	—	4	4	no	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×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	#41	1	1	yes	characteristic
85	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₂²⋊Dic₅₃	—	2	4	no	—
87	C₄⋊Dic₅₃	—	2	2	no	—
88	C₄⋊Dic₅₃	—	2	2	no	—
89	C₄⋊Dic₅₃	—	2	2	no	—
90	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
91	#11719919	—	2	4	no	—
92	#11719919	—	2	4	no	—
93	C₄×Dic₅₃	—	2	4	no	—
94	#11719917	—	2	4	no	—
95	C₂×#88322	—	2	2	no	—
96	C₂×#88322	C₂×D₄	1	4	yes	—
97	Dic₂₁₂	—	2	4	no	—
98	Dic₂₁₂	—	4	2	no	—
99	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	4	1	no	—
103	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#137	1	4	yes	—
107	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
110	C₂×Dic₅₃	—	2	4	no	—
111	C₂×Dic₅₃	—	2	4	no	—
112	Dic₁₀₆	—	2	4	no	—
113	Dic₁₀₆	—	2	4	no	—
114	Dic₁₀₆	—	8	2	no	—
115	C₈×C₅₃	—	2	1	no	—
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	—	2	1	no	—
118	C₈×C₅₃	—	4	2	no	—
119	C₈×C₅₃	—	4	2	no	—
120	C₈×C₅₃	#131	1	2	yes	—
121	#88322	—	2	4	no	—
122	#88322	—	4	2	no	—
123	#18072	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	#441	1	2	yes	—
127	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	#441	1	2	yes	—
132	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
133	C₄×C₅₃	#568	1	1	yes	characteristic
134	C₄×C₅₃	C₂×#137	1	2	yes	—
135	Dic₅₃	—	4	4	no	—
136	Dic₅₃	—	4	4	no	—
137	#3170	—	53	1	no	—
138	#2962	—	53	2	no	—
139	#1593	—	53	2	no	—
140	#1587	—	53	4	no	—
141	#1579	—	53	2	no	—
142	C₂×#334	D₅₃	1	1	yes	characteristic
143	#1481	—	53	2	no	—
144	#1451	—	53	1	no	—
145	C₂×C₅₃	—	2	2	no	—
146	C₂×C₅₃	#2935	1	1	yes	characteristic
147	C₂×C₅₃	C₂×#441	1	1	yes	characteristic
148	C₂×C₅₃	#2957	1	1	yes	characteristic
149	#533	—	53	2	no	—
150	#500	—	53	1	no	—
151	#494	—	53	1	no	—
152	#487	—	53	2	no	—
153	#467	—	53	2	no	—
154	#467	—	53	2	no	—
155	#446	—	53	4	no	—
156	#446	—	106	4	no	—
157	C₈×Q₈	—	53	4	no	—
158	C₄×Q₁₆	—	53	2	no	—
159	C₄×Q₁₆	—	53	2	no	—
160	#431	—	53	1	no	—
161	#424	—	106	2	no	—
162	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
163	#405	—	106	2	no	—
164	C₄×M₁₆	—	53	1	no	—
165	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
166	#334	C₂×D₅₃	1	1	yes	characteristic
167	#334	C₂×D₅₃	1	2	yes	—
168	#334	C₂×D₅₃	1	1	yes	characteristic
169	Q₈⋊C₈	—	53	4	no	—
170	Q₈⋊C₈	—	53	4	no	—
171	#322	—	53	2	no	—
172	#322	—	53	2	no	—
173	C₅₃	#18072	1	1	yes	characteristic, complemented, Sylow
174	C₂×Q₁₆	—	53	2	no	—
175	C₂×Q₁₆	—	53	2	no	—
176	C₂×M₁₆	—	53	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
179	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
180	C₄⋊Q₈	—	53	2	no	—
181	C₄⋊Q₈	—	53	2	no	—
182	#125	—	106	2	no	—
183	C₂²⋊Q₈	—	53	4	no	—
184	C₂²⋊Q₈	—	106	2	no	—
185	C₄×Q₈	—	53	4	no	—
186	C₄×Q₈	—	106	2	no	—
187	#117	—	106	2	no	—
188	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
189	#107	—	53	2	no	—
190	#107	—	53	2	no	—
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	106	4	no	—
193	C₄⋊C₈	—	106	4	no	—
194	C₄⋊C₈	—	106	1	no	—
195	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
196	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
197	#103	—	53	4	no	—
198	#103	—	53	4	no	—
199	C₂²⋊C₈	—	106	1	no	—
200	#97	—	53	2	no	—
201	#97	—	106	4	no	—
202	C₄×C₈	—	53	2	no	—
203	C₄×C₈	—	53	2	no	—
204	C₄×C₈	—	106	4	no	—
205	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
206	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
209	C₂×Q₈	—	53	4	no	—
210	C₂×Q₈	—	106	2	no	—
211	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
212	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
213	C₂²×C₄	#11719949	1	1	yes	characteristic
214	Q₁₆	—	106	4	no	—
215	Q₁₆	—	212	2	no	—
216	M₁₆	—	106	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	—	53	4	no	—
223	C₂×C₈	—	106	2	no	—
224	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
228	C₄⋊C₄	—	53	4	no	—
229	C₄⋊C₄	—	106	2	no	—
230	C₄⋊C₄	—	106	2	no	—
231	C₄⋊C₄	—	106	4	no	—
232	C₄⋊C₄	—	106	2	no	—
233	C₄⋊C₄	—	106	4	no	—
234	C₂²⋊C₄	—	106	4	no	—
235	C₄²	—	106	4	no	—
236	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
237	C₄²	D₄×D₅₃	1	1	yes	characteristic
238	C₄²	#11719949	1	1	yes	characteristic
239	C₄²	C₂³×D₅₃	1	1	yes	characteristic
240	C₂³	#13133179	1	1	yes	characteristic
241	Q₈	—	106	4	no	—
242	Q₈	—	106	4	no	—
243	Q₈	—	424	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	106	4	no	—
248	C₂×C₄	—	106	4	no	—
249	C₂×C₄	#13133179	1	1	yes	characteristic
250	C₂×C₄	#13133179	1	1	yes	characteristic
251	C₂×C₄	#13133179	1	1	yes	characteristic
252	C₂×C₄	#13133246	1	4	yes	—
253	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
255	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	4	2	no	—
260	C₈	—	4	2	no	—
261	C₈	—	106	4	no	—
262	C₈	—	212	2	no	—
263	C₈	#13133223	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#16810729	1	2	yes	—
267	C₂²	C₂×#13133179	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	212	4	no	—
272	C₄	—	212	4	no	—
273	C₄	#16810729	1	2	yes	—
274	C₄	C₂×#13133223	1	1	yes	characteristic
275	C₄	#16811389	1	1	yes	characteristic
276	C₄	C₂×#13133246	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	#20283458	1	1	yes	characteristic
279	C₂	C₂×#16810729	1	1	yes	characteristic
280	C₂	#20283631	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#334	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#334	D₅₃
hypocenter	C₅₃	#18072
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. #20273623	C₂	semidirect
4. #20273585	C₂	semidirect
5. #20273533	C₂	semidirect
173. C₅₃	#18072	semidirect

Maximal subgroups

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

subgroup	index
#20285301	2
#20283658	2
#20273623	2
#20273585	2
#20273533	2
#20273117	2
#20273004	2
C₂×C₅₃×#334	2
#18072	53