Subgroups — #27367346

Order 13568.

582 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285868	C₂	1	2	yes	maximal
2	C₂×#16811028	C₂	1	1	yes	maximal, characteristic
3	#20283649	C₂	1	1	yes	maximal, characteristic
4	#20274244	C₂	1	4	yes	maximal, complemented
5	#20274120	C₂	1	4	yes	maximal, complemented
6	#20273395	C₂	1	2	yes	maximal
7	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
8	#16811180	C₂²	1	1	yes	characteristic
9	C₂×#13133217	C₂²	1	2	yes	—
10	#16811028	—	2	2	no	—
11	#16811028	—	2	1	no	—
12	#16811028	—	2	1	no	—
13	#16811028	C₂²	1	2	yes	—
14	#16811028	C₂²	1	2	yes	—
15	#16810974	—	2	2	no	—
16	#16810959	—	2	4	no	—
17	#16810919	C₂²	1	8	yes	—
18	#16810753	—	2	1	no	—
19	#16810753	C₂²	1	2	yes	—
20	C₈×Dic₁₀₆	—	2	1	no	—
21	C₈×Dic₁₀₆	C₂²	1	2	yes	—
22	C₂×#13133151	—	2	1	no	—
23	C₂×#13133151	—	2	1	no	—
24	#16810649	—	2	1	no	—
25	#16810647	C₂²	1	1	yes	characteristic
26	#16810578	—	2	1	no	—
27	#16810371	C₂²	1	8	yes	—
28	C₅₃×#431	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#335	C₂²	1	2	yes	—
31	C₅₃×#334	C₂²	1	2	yes	—
32	C₂²×Dic₁₀₆	—	2	2	no	—
33	C₂×#11719917	—	2	1	no	—
34	#13133217	—	2	2	no	—
35	#13133217	—	2	2	no	—
36	#13133217	—	2	2	no	—
37	#13133217	D₄	1	1	yes	characteristic
38	#13133217	D₄	1	1	yes	characteristic
39	#13133217	C₂³	1	4	yes	—
40	#13133209	—	2	1	no	—
41	#13133196	—	2	1	no	—
42	#13133196	D₄	1	2	yes	—
43	C₄×Dic₁₀₆	—	2	1	no	—
44	C₄×Dic₁₀₆	C₂³	1	2	yes	—
45	#13133176	—	2	1	no	—
46	C₂×C₄⋊Dic₅₃	—	2	2	no	—
47	C₂×C₄⋊Dic₅₃	—	2	2	no	—
48	#13133151	—	2	2	no	—
49	#13133151	—	2	2	no	—
50	#13133151	—	2	2	no	—
51	#13133151	—	2	2	no	—
52	#13133151	—	2	8	no	—
53	C₄⋊#88322	—	2	1	no	—
54	C₄⋊#88322	C₂³	1	2	yes	—
55	#13133146	—	2	2	no	—
56	#13133146	—	2	2	no	—
57	#13133142	—	2	8	no	—
58	#13133127	—	2	2	no	—
59	C₈×Dic₅₃	—	2	2	no	—
60	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
64	C₂²⋊C₈×C₅₃	—	2	2	no	—
65	C₄×C₈×C₅₃	—	2	2	no	—
66	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
67	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
68	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
73	C₂²×Dic₅₃	—	4	2	no	—
74	C₂×Dic₁₀₆	—	2	1	no	—
75	C₂×Dic₁₀₆	—	2	4	no	—
76	C₂×Dic₁₀₆	—	4	2	no	—
77	C₂×Dic₁₀₆	—	8	4	no	—
78	C₂×Dic₁₀₆	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	4	no	—
81	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
82	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₅₃	#41	1	1	yes	characteristic
89	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₂²⋊Dic₅₃	—	2	2	no	—
91	C₄⋊Dic₅₃	—	2	1	no	—
92	C₄⋊Dic₅₃	—	2	1	no	—
93	C₄⋊Dic₅₃	—	2	1	no	—
94	C₄⋊Dic₅₃	—	2	4	no	—
95	C₄⋊Dic₅₃	—	2	4	no	—
96	C₄⋊Dic₅₃	—	4	2	no	—
97	C₄⋊Dic₅₃	—	4	2	no	—
98	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
99	#11719919	—	2	2	no	—
100	#11719919	—	2	2	no	—
101	C₄×Dic₅₃	—	2	2	no	—
102	#11719917	—	4	2	no	—
103	C₂×#88322	—	2	2	no	—
104	C₂³×C₅₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄×C₅₃	#137	1	2	yes	—
112	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	C₂×Dic₅₃	—	2	2	no	—
116	C₂×Dic₅₃	—	2	2	no	—
117	C₂×Dic₅₃	—	4	4	no	—
118	C₂×Dic₅₃	—	8	4	no	—
119	Dic₁₀₆	—	2	4	no	—
120	Dic₁₀₆	—	4	2	no	—
121	Dic₁₀₆	—	8	4	no	—
122	Dic₁₀₆	—	8	4	no	—
123	C₈×C₅₃	—	2	1	no	—
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	—	4	4	no	—
126	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
127	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
128	#88322	—	4	2	no	—
129	#19209	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	—	2	1	no	—
131	C₂²×C₅₃	—	2	1	no	—
132	C₂²×C₅₃	#462	1	2	yes	—
133	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
134	C₄×C₅₃	—	2	1	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	#462	1	2	yes	—
137	C₄×C₅₃	#472	1	2	yes	—
138	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
139	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
140	C₄×C₅₃	#578	1	2	yes	—
141	Dic₅₃	—	4	2	no	—
142	Dic₅₃	—	4	2	no	—
143	Dic₅₃	—	8	4	no	—
144	#3230	—	53	2	no	—
145	C₂×#500	—	53	1	no	—
146	#2960	—	53	1	no	—
147	#1696	—	53	4	no	—
148	#1676	—	53	4	no	—
149	#1564	D₅₃	1	1	yes	characteristic
150	#1548	—	53	2	no	—
151	C₂×C₅₃	—	2	2	no	—
152	C₂×C₅₃	C₂×#462	1	1	yes	characteristic
153	C₂×C₅₃	#3059	1	1	yes	characteristic
154	C₂×C₅₃	C₂×#472	1	1	yes	characteristic
155	#533	—	53	1	no	—
156	C₂×C₄⋊Q₈	—	53	2	no	—
157	#500	—	53	2	no	—
158	#500	—	53	2	no	—
159	#500	—	106	2	no	—
160	#500	—	106	1	no	—
161	#500	—	106	1	no	—
162	#487	—	106	2	no	—
163	#484	—	106	4	no	—
164	#477	—	53	8	no	—
165	#446	—	53	2	no	—
166	#446	—	106	1	no	—
167	C₈×Q₈	—	53	2	no	—
168	C₈×Q₈	—	106	1	no	—
169	#431	C₂×D₅₃	1	2	yes	—
170	C₂×#107	—	106	1	no	—
171	C₂×#107	—	106	1	no	—
172	#424	—	106	1	no	—
173	#423	—	53	1	no	—
174	#405	—	106	1	no	—
175	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
176	#335	C₂×D₅₃	1	2	yes	—
177	#334	C₂×D₅₃	1	2	yes	—
178	#333	—	53	8	no	—
179	C₅₃	#19209	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	—	106	2	no	—
181	C₂×M₁₆	—	106	1	no	—
182	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
183	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
184	C₄⋊Q₈	—	53	1	no	—
185	C₄⋊Q₈	—	53	1	no	—
186	C₄⋊Q₈	—	53	4	no	—
187	C₄⋊Q₈	—	106	2	no	—
188	C₄⋊Q₈	—	106	2	no	—
189	C₄⋊Q₈	—	106	2	no	—
190	#125	—	106	1	no	—
191	C₂²⋊Q₈	—	53	2	no	—
192	C₂²⋊Q₈	—	106	1	no	—
193	C₄×Q₈	—	53	2	no	—
194	C₄×Q₈	—	106	1	no	—
195	#117	—	106	1	no	—
196	C₂×C₄⋊C₄	—	106	2	no	—
197	C₂×C₄⋊C₄	—	106	2	no	—
198	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
199	#107	—	106	8	no	—
200	#107	—	106	2	no	—
201	#107	—	106	2	no	—
202	#107	—	106	2	no	—
203	#107	—	106	2	no	—
204	C₄⋊C₈	—	53	2	no	—
205	C₄⋊C₈	—	106	2	no	—
206	C₄⋊C₈	—	106	2	no	—
207	C₄⋊C₈	—	106	1	no	—
208	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
209	#103	—	106	8	no	—
210	C₂²⋊C₈	—	2	2	no	—
211	#97	—	106	2	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	106	2	no	—
214	C₄×C₈	D₂₁₂	1	1	yes	characteristic
215	C₄×C₈	D₂₁₂	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
218	C₂×Q₈	—	53	2	no	—
219	C₂×Q₈	—	106	1	no	—
220	C₂×Q₈	—	106	4	no	—
221	C₂×Q₈	—	424	4	no	—
222	C₂×Q₈	—	212	2	no	—
223	C₂²×C₄	—	212	2	no	—
224	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
225	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
226	C₂²×C₄	#11719949	1	1	yes	characteristic
227	M₁₆	—	212	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	106	2	no	—
231	C₂×C₈	Dic₂₁₂	1	4	yes	—
232	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
233	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
234	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
235	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
236	C₄⋊C₄	—	53	2	no	—
237	C₄⋊C₄	—	106	1	no	—
238	C₄⋊C₄	—	106	1	no	—
239	C₄⋊C₄	—	106	2	no	—
240	C₄⋊C₄	—	106	1	no	—
241	C₄⋊C₄	—	106	2	no	—
242	C₄⋊C₄	—	106	4	no	—
243	C₄⋊C₄	—	106	4	no	—
244	C₄⋊C₄	—	212	2	no	—
245	C₄⋊C₄	—	212	2	no	—
246	C₂²⋊C₄	—	106	2	no	—
247	C₄²	—	106	2	no	—
248	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
249	C₄²	D₄×D₅₃	1	1	yes	characteristic
250	C₄²	#11719949	1	1	yes	characteristic
251	C₄²	C₂³×D₅₃	1	1	yes	characteristic
252	C₂³	#13133192	1	1	yes	characteristic
253	Q₈	—	106	4	no	—
254	Q₈	—	424	4	no	—
255	Q₈	—	424	4	no	—
256	Q₈	—	212	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	106	2	no	—
260	C₂×C₄	—	106	2	no	—
261	C₂×C₄	—	424	4	no	—
262	C₂×C₄	—	212	4	no	—
263	C₂×C₄	#13133192	1	1	yes	characteristic
264	C₂×C₄	#13133192	1	1	yes	characteristic
265	C₂×C₄	#13133192	1	1	yes	characteristic
266	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
267	C₂×C₄	#13133250	1	2	yes	—
268	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
269	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
270	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	4	4	no	—
274	C₈	—	212	2	no	—
275	C₈	C₂×Dic₂₁₂	1	2	yes	—
276	C₈	C₂×Dic₂₁₂	1	2	yes	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	#16810834	1	2	yes	—
280	C₂²	C₂×#13133192	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	424	4	no	—
284	C₄	—	212	2	no	—
285	C₄	—	212	2	no	—
286	C₄	#16810834	1	2	yes	—
287	C₄	#16810894	1	2	yes	—
288	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
289	C₄	C₂×#13133250	1	1	yes	characteristic
290	C₄	#16811437	1	2	yes	—
291	C₂	—	2	2	no	—
292	C₂	C₂×#16810834	1	1	yes	characteristic
293	C₂	#20284371	1	1	yes	characteristic
294	C₂	C₂×#16810894	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133192
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂³×C₅₃	#121
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#19209
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
4. #20274244	C₂	semidirect
5. #20274120	C₂	semidirect
179. C₅₃	#19209	semidirect

Maximal subgroups

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

subgroup	index
#20285868	2
C₂×#16811028	2
#20283649	2
#20274244	2
#20274120	2
#20273395	2
C₅₃×#1564	2
#19209	53