Subgroups — #27277126

Order 13568.

404 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20273474	C₂	1	1	yes	maximal, characteristic, complemented
2	#20273471	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273450	C₂	1	2	yes	maximal
4	#20272628	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
6	#16810647	C₂²	1	2	yes	—
7	#16810647	C₂²	1	2	yes	—
8	#16810376	—	2	1	no	—
9	#16810376	—	2	1	no	—
10	#16810374	—	2	1	no	—
11	#16810374	—	2	1	no	—
12	#16810372	—	2	2	no	—
13	#16810367	—	2	2	no	—
14	C₂×#131×C₅₃	C₄	1	2	yes	—
15	C₅₃×#532	C₂²	1	1	yes	characteristic
16	C₅₃×#436	—	2	1	no	—
17	C₅₃×#436	—	2	1	no	—
18	C₈×D₄×C₅₃	—	2	1	no	—
19	C₈×D₄×C₅₃	—	2	1	no	—
20	C₅₃×#423	C₂²	1	1	yes	characteristic
21	C₅₃×#408	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	—	2	2	no	—
29	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
30	#131×C₅₃	—	4	4	no	—
31	C₂×M₁₆×C₅₃	—	2	2	no	—
32	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₅₃	—	2	2	no	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
37	#127×C₅₃	D₄	1	1	yes	characteristic
38	#124×C₅₃	Q₈	1	2	yes	—
39	#121×C₅₃	—	2	1	no	—
40	#121×C₅₃	—	2	1	no	—
41	C₄×D₄×C₅₃	—	2	1	no	—
42	C₄×D₄×C₅₃	—	2	1	no	—
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₅₃	—	2	1	no	—
45	C₄⋊C₈×C₅₃	—	2	1	no	—
46	C₂²⋊C₈×C₅₃	—	2	2	no	—
47	C₂²⋊C₈×C₅₃	—	2	2	no	—
48	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	#41×C₅₃	—	4	2	no	—
53	#41×C₅₃	—	4	2	no	—
54	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₂×D₄×C₅₃	—	2	2	no	—
56	C₂×D₄×C₅₃	—	2	1	no	—
57	C₂×D₄×C₅₃	—	2	1	no	—
58	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
63	M₁₆×C₅₃	—	4	4	no	—
64	M₁₆×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	4	1	no	—
68	C₂×C₈×C₅₃	—	4	4	no	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₅₃	—	2	1	no	—
74	C₄⋊C₄×C₅₃	—	2	1	no	—
75	C₂²⋊C₄×C₅₃	—	2	2	no	—
76	C₂²⋊C₄×C₅₃	—	2	2	no	—
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
81	#11719917	—	8	2	no	—
82	#11719917	—	8	2	no	—
83	C₂×#88322	—	4	2	no	—
84	C₂×#88322	—	4	2	no	—
85	C₂³×C₅₃	—	2	2	no	—
86	C₂³×C₅₃	#117	1	1	yes	characteristic
87	Q₈×C₅₃	—	4	2	no	—
88	D₄×C₅₃	—	4	2	no	—
89	D₄×C₅₃	—	4	4	no	—
90	D₄×C₅₃	—	4	2	no	—
91	D₄×C₅₃	—	4	1	no	—
92	D₄×C₅₃	—	4	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₈×C₅₃	—	2	1	no	—
108	C₈×C₅₃	—	2	1	no	—
109	C₈×C₅₃	—	2	1	no	—
110	C₈×C₅₃	—	4	2	no	—
111	C₈×C₅₃	#100	1	2	yes	—
112	#88322	—	8	2	no	—
113	#88322	—	8	2	no	—
114	#8640	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	1	no	—
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	—	4	2	no	—
119	C₂²×C₅₃	—	4	2	no	—
120	C₂²×C₅₃	—	4	2	no	—
121	C₂²×C₅₃	#380	1	1	yes	characteristic
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	—	4	2	no	—
126	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
127	C₄×C₅₃	#413	1	1	yes	characteristic
128	C₄×C₅₃	#471	1	2	yes	—
129	C₄×C₅₃	#472	1	1	yes	characteristic
130	C₄×C₅₃	#473	1	1	yes	characteristic
131	#2931	D₅₃	1	1	yes	characteristic
132	#1567	—	53	1	no	—
133	#1566	—	53	1	no	—
134	#1560	—	53	2	no	—
135	#1309	—	53	2	no	—
136	C₂×C₅₃	—	4	1	no	—
137	C₂×C₅₃	—	8	2	no	—
138	C₂×C₅₃	#1800	1	1	yes	characteristic
139	C₂×C₅₃	#1815	1	1	yes	characteristic
140	C₂×C₅₃	#1816	1	1	yes	characteristic
141	C₂×#131	Dic₅₃	1	2	yes	—
142	#532	C₂×D₅₃	1	1	yes	characteristic
143	#436	—	2	1	no	—
144	#436	—	2	1	no	—
145	C₈×D₄	—	2	1	no	—
146	C₈×D₄	—	2	1	no	—
147	#423	—	53	2	no	—
148	#423	—	53	2	no	—
149	#423	C₂×D₅₃	1	1	yes	characteristic
150	#408	Dic₅₃	1	2	yes	—
151	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
152	#335	—	106	1	no	—
153	#335	—	106	1	no	—
154	#334	—	106	1	no	—
155	#334	—	106	1	no	—
156	#333	—	106	2	no	—
157	#331	—	106	2	no	—
158	C₅₃	#8640	1	1	yes	characteristic, complemented, Sylow
159	C₂×#41	C₂×Dic₅₃	1	2	yes	—
160	#131	—	4	4	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	106	2	no	—
163	C₂×M₁₆	—	106	2	no	—
164	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
167	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
168	C₄⋊Q₈	D₂₁₂	1	1	yes	characteristic
169	#127	D₂₁₂	1	1	yes	characteristic
170	#124	Dic₁₀₆	1	2	yes	—
171	#121	—	2	1	no	—
172	#121	—	2	1	no	—
173	C₄×D₄	—	2	1	no	—
174	C₄×D₄	—	2	1	no	—
175	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	—	2	1	no	—
178	C₄⋊C₈	—	106	2	no	—
179	C₄⋊C₈	—	106	2	no	—
180	C₄⋊C₈	—	106	2	no	—
181	C₄⋊C₈	—	106	2	no	—
182	C₂²⋊C₈	—	2	2	no	—
183	C₂²⋊C₈	—	2	2	no	—
184	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
185	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
186	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
187	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
188	#41	—	4	2	no	—
189	#41	—	4	2	no	—
190	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	C₄⋊Dic₅₃	1	2	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	#11719945	1	1	yes	characteristic
197	C₂²×C₄	#11719945	1	1	yes	characteristic
198	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
199	M₁₆	—	4	4	no	—
200	M₁₆	—	4	2	no	—
201	M₁₆	—	424	2	no	—
202	M₁₆	—	424	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	212	2	no	—
208	C₂×C₈	—	212	2	no	—
209	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
218	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
219	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
220	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	C₂³	#13133176	1	1	yes	characteristic
223	Q₈	—	4	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	4	4	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	1	no	—
228	D₄	—	4	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
239	C₂×C₄	#13133191	1	1	yes	characteristic
240	C₂×C₄	#13133191	1	1	yes	characteristic
241	C₂×C₄	#13133196	1	1	yes	characteristic
242	C₂×C₄	#13133196	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₈	—	424	2	no	—
248	C₈	—	424	2	no	—
249	C₈	#13133134	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	#16810483	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	2	no	—
261	C₄	C₂×#13133134	1	1	yes	characteristic
262	C₄	#16810607	1	1	yes	characteristic
263	C₄	#16810886	1	2	yes	—
264	C₄	#16810893	1	1	yes	characteristic
265	C₄	#16810898	1	1	yes	characteristic
266	C₂	—	4	1	no	—
267	C₂	—	8	2	no	—
268	C₂	#20274707	1	1	yes	characteristic
269	C₂	#20274774	1	1	yes	characteristic
270	C₂	#20274779	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810483
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2931	C₂
socle	C₂²×C₅₃	#380
prime radical	C₅₃×#532	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2931	D₅₃
hypocenter	C₅₃	#8640
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #20273474	C₂	semidirect
2. #20273471	C₂	semidirect
4. #20272628	C₂	semidirect
158. C₅₃	#8640	semidirect

Maximal subgroups

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

subgroup	index
#20273474	2
#20273471	2
#20273450	2
#20272628	2
C₅₃×#2931	2
#8640	53