Subgroups — #27537756

Order 13568.

444 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283647	C₂	1	1	yes	maximal, characteristic
2	#20276623	C₂	1	2	yes	maximal, complemented
3	#20276622	C₂	1	2	yes	maximal, complemented
4	#20276621	C₂	1	2	yes	maximal, complemented
5	#20276620	C₂	1	2	yes	maximal, complemented
6	#20276522	C₂	1	1	yes	maximal, characteristic
7	#20276519	C₂	1	2	yes	maximal
8	#20276515	C₂	1	2	yes	maximal
9	C₅₃×#2164	C₂	1	1	yes	maximal, characteristic
10	#16811176	C₄	1	1	yes	characteristic
11	#16810751	C₄	1	2	yes	—
12	#16810751	C₄	1	2	yes	—
13	C₈×Dic₁₀₆	C₂²	1	2	yes	—
14	C₈×Dic₁₀₆	C₂²	1	2	yes	—
15	#16810651	C₄	1	1	yes	characteristic
16	#16810650	C₄	1	1	yes	characteristic
17	#16810646	C₄	1	1	yes	characteristic
18	#16810578	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
20	C₄⋊#11719908	C₂²	1	2	yes	—
21	#16810441	C₂²	1	2	yes	—
22	#16810441	C₂²	1	2	yes	—
23	#16810441	C₂²	1	2	yes	—
24	#16810441	C₂²	1	2	yes	—
25	C₂²⋊#11719908	C₂²	1	2	yes	—
26	#16810399	C₂²	1	1	yes	characteristic
27	#16810398	C₂²	1	2	yes	—
28	#16810398	C₂²	1	2	yes	—
29	#16810398	C₂²	1	2	yes	—
30	C₄×#11719908	C₂²	1	1	yes	characteristic
31	C₁₆×Dic₅₃	C₂²	1	2	yes	—
32	C₅₃×#405	C₂²	1	1	yes	characteristic
33	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	C₅₃×#346	C₂²	1	1	yes	characteristic
36	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
38	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
39	#13133216	C₈	1	2	yes	—
40	#13133208	C₈	1	2	yes	—
41	#13133194	C₈	1	2	yes	—
42	#13133194	C₈	1	2	yes	—
43	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
44	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
45	#13133176	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×#11719908	C₂³	1	2	yes	—
48	C₂×#11719908	C₂³	1	2	yes	—
49	C₄⋊#88322	C₂×C₄	1	2	yes	—
50	C₄⋊#88322	C₂×C₄	1	2	yes	—
51	#13133146	C₂×C₄	1	2	yes	—
52	#13133146	C₂×C₄	1	2	yes	—
53	#13133146	C₂×C₄	1	2	yes	—
54	#13133146	C₂×C₄	1	2	yes	—
55	#13133127	C₂×C₄	1	2	yes	—
56	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₅₃	C₂³	1	2	yes	—
58	C₈×Dic₅₃	C₂³	1	2	yes	—
59	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
63	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
64	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
65	#97×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×Dic₅₃	C₂×C₈	1	2	yes	—
70	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
71	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
72	M₁₆×C₅₃	—	4	1	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
79	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
80	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
83	C₂²⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
84	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
85	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
86	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
87	#11719919	C₂×C₈	1	2	yes	—
88	#11719919	C₂×C₈	1	2	yes	—
89	#11719919	C₂×C₈	1	2	yes	—
90	#11719919	C₂×C₈	1	2	yes	—
91	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
92	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
93	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
94	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
95	#11719917	—	4	1	no	—
96	C₂×#88322	—	4	1	no	—
97	C₂×#88322	C₂²×C₄	1	2	yes	—
98	C₂×#88322	C₂²×C₄	1	2	yes	—
99	C₁₆×C₅₃	—	2	2	no	—
100	C₁₆×C₅₃	—	2	2	no	—
101	#11719908	—	2	2	no	—
102	#11719908	—	2	2	no	—
103	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₅₃	—	4	2	no	—
111	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
112	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
113	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
114	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
115	Dic₁₀₆	—	4	2	no	—
116	Dic₁₀₆	—	4	2	no	—
117	C₈×C₅₃	—	2	1	no	—
118	C₈×C₅₃	—	2	1	no	—
119	C₈×C₅₃	—	2	1	no	—
120	C₈×C₅₃	2₋¹⁺⁴	1	2	yes	—
121	#88322	—	2	2	no	—
122	#88322	—	2	2	no	—
123	#33501	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	—	2	2	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	#519	1	1	yes	characteristic
131	C₄×C₅₃	#568	1	1	yes	characteristic
132	Dic₅₃	—	2	2	no	—
133	Dic₅₃	—	2	2	no	—
134	Dic₅₃	—	2	2	no	—
135	Dic₅₃	—	2	2	no	—
136	#2960	—	53	1	no	—
137	#2183	—	53	2	no	—
138	#2183	—	53	2	no	—
139	#2183	—	53	2	no	—
140	#2183	—	53	2	no	—
141	#2164	—	53	1	no	—
142	#2164	—	53	2	no	—
143	#2164	D₅₃	1	1	yes	characteristic
144	#2163	—	53	2	no	—
145	C₂×C₅₃	—	4	1	no	—
146	C₂×C₅₃	#2982	1	1	yes	characteristic
147	C₂×C₅₃	#3407	1	2	yes	—
148	#533	—	53	1	no	—
149	#446	—	53	2	no	—
150	#446	—	53	2	no	—
151	C₈×Q₈	—	53	2	no	—
152	C₈×Q₈	—	53	2	no	—
153	#424	—	53	1	no	—
154	#424	—	53	1	no	—
155	#423	—	53	1	no	—
156	#405	—	53	1	no	—
157	#405	C₂×D₅₃	1	1	yes	characteristic
158	C₂×C₄×C₈	—	53	1	no	—
159	C₄⋊C₁₆	—	53	2	no	—
160	C₄⋊C₁₆	—	53	2	no	—
161	C₄⋊C₁₆	—	53	2	no	—
162	C₄⋊C₁₆	—	53	2	no	—
163	C₄⋊C₁₆	—	53	2	no	—
164	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
165	C₂²⋊C₁₆	—	53	2	no	—
166	C₂²⋊C₁₆	C₂×D₅₃	1	2	yes	—
167	#346	—	53	2	no	—
168	#346	—	53	2	no	—
169	#346	—	53	2	no	—
170	#346	—	53	1	no	—
171	#346	C₂×D₅₃	1	1	yes	characteristic
172	C₄×C₁₆	—	53	2	no	—
173	C₄×C₁₆	—	53	1	no	—
174	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
175	C₅₃	#33501	1	1	yes	characteristic, complemented, Sylow
176	C₂×M₁₆	—	53	1	no	—
177	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
178	C₂²×C₈	—	53	1	no	—
179	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
180	C₄⋊Q₈	—	53	2	no	—
181	#125	—	53	2	no	—
182	C₂²⋊Q₈	—	53	2	no	—
183	C₂²⋊Q₈	—	53	2	no	—
184	C₄×Q₈	—	53	2	no	—
185	C₄×Q₈	—	53	2	no	—
186	#117	—	53	1	no	—
187	#117	C₄×D₅₃	1	1	yes	characteristic
188	C₂×C₄²	—	53	1	no	—
189	C₂×C₁₆	—	53	2	no	—
190	C₂×C₁₆	—	53	2	no	—
191	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
192	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
193	C₄⋊C₈	—	53	2	no	—
194	C₄⋊C₈	—	53	2	no	—
195	C₄⋊C₈	—	53	2	no	—
196	C₄⋊C₈	—	53	2	no	—
197	C₄⋊C₈	—	53	2	no	—
198	C₄⋊C₈	—	53	2	no	—
199	#97	—	53	2	no	—
200	#97	C₄×D₅₃	1	1	yes	characteristic
201	#97	C₄×D₅₃	1	1	yes	characteristic
202	C₄×C₈	—	53	2	no	—
203	C₄×C₈	—	53	2	no	—
204	C₄×C₈	—	53	2	no	—
205	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
207	C₂×Q₈	—	53	2	no	—
208	C₂×Q₈	—	53	2	no	—
209	C₂²×C₄	—	53	2	no	—
210	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
211	M₁₆	—	4	1	no	—
212	M₁₆	—	212	1	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	53	2	no	—
215	C₂×C₈	—	53	2	no	—
216	C₂×C₈	—	212	1	no	—
217	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
218	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
219	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
220	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
221	C₄⋊C₄	—	53	2	no	—
222	C₄⋊C₄	—	53	2	no	—
223	C₄⋊C₄	—	53	2	no	—
224	C₄⋊C₄	—	53	2	no	—
225	C₄⋊C₄	—	53	2	no	—
226	C₄⋊C₄	—	53	2	no	—
227	C₄⋊C₄	—	53	2	no	—
228	C₄⋊C₄	C₈×D₅₃	1	2	yes	—
229	C₂²⋊C₄	—	53	1	no	—
230	C₂²⋊C₄	—	53	1	no	—
231	C₂²⋊C₄	C₈×D₅₃	1	2	yes	—
232	C₄²	—	53	1	no	—
233	C₄²	—	53	1	no	—
234	C₄²	—	53	2	no	—
235	C₄²	—	53	2	no	—
236	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
237	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
238	C₁₆	—	2	2	no	—
239	C₁₆	—	2	2	no	—
240	C₁₆	—	106	2	no	—
241	C₁₆	—	106	2	no	—
242	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
243	Q₈	—	212	2	no	—
244	Q₈	—	212	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	53	2	no	—
248	C₂×C₄	—	53	2	no	—
249	C₂×C₄	—	53	2	no	—
250	C₂×C₄	—	53	2	no	—
251	C₂×C₄	—	212	2	no	—
252	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
253	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
254	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
255	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	106	2	no	—
260	C₈	—	106	2	no	—
261	C₈	#13133264	1	2	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	106	2	no	—
269	C₄	—	106	2	no	—
270	C₄	—	106	2	no	—
271	C₄	—	106	2	no	—
272	C₄	#16811100	1	1	yes	characteristic
273	C₄	#16811389	1	1	yes	characteristic
274	C₂	—	4	1	no	—
275	C₂	#20283795	1	1	yes	characteristic
276	C₂	#20287816	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2164	C₂
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2164	D₅₃
hypocenter	C₅₃	#33501
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2164

Complements

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

N	H ≅ G/N	type
2. #20276623	C₂	semidirect
3. #20276622	C₂	semidirect
4. #20276621	C₂	semidirect
5. #20276620	C₂	semidirect
175. C₅₃	#33501	semidirect

Maximal subgroups

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

subgroup	index
#20283647	2
#20276623	2
#20276622	2
#20276621	2
#20276620	2
#20276522	2
#20276519	2
#20276515	2
C₅₃×#2164	2
#33501	53