Subgroups — #27300020

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810575	C₂	1	1	yes	maximal, characteristic
2	#20276470	C₂	1	2	yes	maximal, complemented
3	C₂×#16810404	C₂	1	2	yes	maximal
4	#20272799	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1377	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11719912	C₄	1	1	yes	characteristic
7	C₂²×#11719912	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
9	C₂×#13133157	C₂²	1	2	yes	—
10	#16810575	—	2	4	no	—
11	C₂×#13133128	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810441	—	2	4	no	—
15	#16810404	—	2	4	no	—
16	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
17	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₅₃	C₈	1	2	yes	—
19	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×#11719912	—	2	2	no	—
21	C₂×#11719912	—	2	2	no	—
22	C₂×#11719912	—	2	2	no	—
23	C₂×#11719912	—	2	2	no	—
24	C₂×#11719912	—	2	2	no	—
25	C₂×#11719912	D₄	1	2	yes	—
26	C₂×#11719912	Q₈	1	2	yes	—
27	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
28	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
30	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×D₅₃	—	2	2	no	—
32	C₄²×D₅₃	—	2	1	no	—
33	C₄²×D₅₃	—	2	1	no	—
34	#13133157	—	2	2	no	—
35	#13133157	—	2	2	no	—
36	C₂×#11719908	—	2	2	no	—
37	#13133128	—	2	2	no	—
38	C₈×Dic₅₃	—	2	2	no	—
39	C₈×Dic₅₃	D₄	1	4	yes	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	M₃₂×C₅₃	—	2	2	no	—
44	M₃₂×C₅₃	—	2	2	no	—
45	C₂×C₁₆×C₅₃	—	2	2	no	—
46	#97×C₅₃	—	2	2	no	—
47	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
51	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
52	C₂×C₄×D₅₃	—	2	1	no	—
53	C₂×C₄×D₅₃	—	2	1	no	—
54	C₂×C₄×D₅₃	—	2	1	no	—
55	C₂×C₄×D₅₃	—	2	1	no	—
56	C₂×C₄×D₅₃	—	2	2	no	—
57	C₂×C₄×D₅₃	—	2	2	no	—
58	C₂×C₄×D₅₃	—	4	4	no	—
59	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
60	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
61	C₂×C₈×C₅₃	—	2	1	no	—
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₅₃	#41	1	2	yes	—
68	C₄²×C₅₃	—	2	1	no	—
69	C₄²×C₅₃	—	2	1	no	—
70	C₄×Dic₅₃	—	2	2	no	—
71	C₄×Dic₅₃	—	2	2	no	—
72	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
73	C₄×Dic₅₃	M₁₆	1	1	yes	characteristic
74	C₄×Dic₅₃	M₁₆	1	1	yes	characteristic
75	C₂×#88322	—	2	1	no	—
76	C₂×#88322	—	2	1	no	—
77	C₂×#88322	—	2	1	no	—
78	C₂×#88322	—	2	1	no	—
79	C₂×#88322	C₂²⋊C₄	1	2	yes	—
80	C₂×#88322	C₄⋊C₄	1	2	yes	—
81	#11719912	—	2	8	no	—
82	#11719912	—	2	4	no	—
83	#11719912	—	2	4	no	—
84	C₁₆×C₅₃	—	4	2	no	—
85	#11719908	—	4	2	no	—
86	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
87	C₂²×D₅₃	—	2	1	no	—
88	C₂²×D₅₃	—	2	1	no	—
89	C₂²×D₅₃	—	4	2	no	—
90	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×Dic₅₃	—	2	1	no	—
100	C₂×Dic₅₃	—	2	1	no	—
101	C₂×Dic₅₃	—	2	4	no	—
102	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
103	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
104	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
105	C₄×D₅₃	—	2	2	no	—
106	C₄×D₅₃	—	2	2	no	—
107	C₄×D₅₃	—	2	2	no	—
108	C₄×D₅₃	—	2	2	no	—
109	C₄×D₅₃	—	4	4	no	—
110	C₄×D₅₃	—	4	4	no	—
111	C₈×C₅₃	—	2	2	no	—
112	C₈×C₅₃	#121	1	2	yes	—
113	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
114	#88322	—	2	2	no	—
115	#88322	—	2	2	no	—
116	#12785	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	—	2	2	no	—
118	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
120	C₂²×C₅₃	#431	1	1	yes	characteristic
121	C₂×D₅₃	—	2	2	no	—
122	C₂×D₅₃	—	4	2	no	—
123	C₂×D₅₃	—	4	2	no	—
124	C₂×D₅₃	—	4	2	no	—
125	C₄×C₅₃	—	4	2	no	—
126	C₄×C₅₃	#380	1	1	yes	characteristic
127	C₄×C₅₃	#407	1	1	yes	characteristic
128	C₄×C₅₃	#423	1	1	yes	characteristic
129	C₄×C₅₃	#432	1	1	yes	characteristic
130	Dic₅₃	—	2	2	no	—
131	Dic₅₃	—	2	2	no	—
132	Dic₅₃	#349	1	4	yes	complemented
133	C₂×C₄×M₁₆	—	53	1	no	—
134	#2150	—	53	2	no	—
135	C₂×#349	—	53	2	no	—
136	#1377	—	53	1	no	—
137	#1377	D₅₃	1	1	yes	characteristic, complemented
138	C₂×C₅₃	—	2	1	no	—
139	C₂×C₅₃	—	2	1	no	—
140	C₂×C₅₃	#1797	1	1	yes	characteristic
141	C₂×C₅₃	C₂×#349	1	2	yes	—
142	D₅₃	—	4	2	no	—
143	C₂²×M₁₆	—	53	1	no	—
144	C₂²×M₁₆	—	53	1	no	—
145	C₂²×C₄²	—	53	1	no	—
146	C₂×M₃₂	—	53	2	no	—
147	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
148	C₄×M₁₆	—	106	4	no	—
149	C₂×#97	—	53	1	no	—
150	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
151	C₂×C₄×C₈	—	53	1	no	—
152	C₂×C₄×C₈	—	53	1	no	—
153	C₄⋊C₁₆	—	106	4	no	—
154	#349	—	106	4	no	—
155	C₅₃	#12785	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₄	—	53	1	no	—
157	C₂³×C₄	—	53	2	no	—
158	C₂×M₁₆	—	53	2	no	—
159	C₂×M₁₆	—	53	2	no	—
160	C₂×M₁₆	—	106	2	no	—
161	C₂×M₁₆	—	106	2	no	—
162	C₂×M₁₆	—	106	2	no	—
163	C₂×M₁₆	—	106	2	no	—
164	C₂×M₁₆	—	106	2	no	—
165	C₂²×C₈	—	53	1	no	—
166	C₂²×C₈	—	53	1	no	—
167	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
169	C₂×C₄²	—	53	1	no	—
170	C₂×C₄²	—	53	2	no	—
171	C₂×C₄²	—	106	2	no	—
172	C₂×C₄²	—	106	1	no	—
173	C₂×C₄²	—	106	1	no	—
174	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
175	M₃₂	—	2	2	no	—
176	M₃₂	—	2	2	no	—
177	M₃₂	—	106	2	no	—
178	M₃₂	—	106	2	no	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	106	2	no	—
181	#97	—	2	2	no	—
182	#97	—	106	2	no	—
183	C₄×C₈	—	53	4	no	—
184	C₄×C₈	—	106	2	no	—
185	C₂⁴	—	53	1	no	—
186	C₂²×C₄	—	53	2	no	—
187	C₂²×C₄	—	53	1	no	—
188	C₂²×C₄	—	53	2	no	—
189	C₂²×C₄	—	53	2	no	—
190	C₂²×C₄	—	106	2	no	—
191	C₂²×C₄	—	106	2	no	—
192	C₂²×C₄	—	106	1	no	—
193	C₂²×C₄	—	106	1	no	—
194	C₂²×C₄	—	106	1	no	—
195	C₂²×C₄	—	106	1	no	—
196	C₂²×C₄	—	212	4	no	—
197	C₂²×C₄	C₈×D₅₃	1	2	yes	—
198	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
199	M₁₆	—	106	8	no	—
200	M₁₆	—	106	4	no	—
201	M₁₆	—	106	4	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	53	2	no	—
205	C₂×C₈	—	53	2	no	—
206	C₂×C₈	—	106	1	no	—
207	C₂×C₈	—	106	1	no	—
208	C₂×C₈	—	106	1	no	—
209	C₂×C₈	—	106	1	no	—
210	C₂×C₈	#11719912	1	2	yes	—
211	C₂×C₈	#11719945	1	2	yes	—
212	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
213	C₂×C₈	D₄×D₅₃	1	2	yes	—
214	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
215	C₄²	—	2	1	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	53	2	no	—
218	C₄²	—	53	1	no	—
219	C₄²	—	53	1	no	—
220	C₄²	—	106	2	no	—
221	C₄²	—	106	2	no	—
222	C₁₆	—	4	2	no	—
223	C₁₆	—	212	2	no	—
224	C₂³	—	53	2	no	—
225	C₂³	—	106	1	no	—
226	C₂³	—	106	1	no	—
227	C₂³	—	212	2	no	—
228	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	53	2	no	—
232	C₂×C₄	—	53	2	no	—
233	C₂×C₄	—	53	2	no	—
234	C₂×C₄	—	106	4	no	—
235	C₂×C₄	—	106	1	no	—
236	C₂×C₄	—	106	1	no	—
237	C₂×C₄	—	106	2	no	—
238	C₂×C₄	—	106	2	no	—
239	C₂×C₄	—	106	2	no	—
240	C₂×C₄	—	106	2	no	—
241	C₂×C₄	—	212	4	no	—
242	C₂×C₄	—	212	4	no	—
243	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
244	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
245	C₂×C₄	#13133174	1	1	yes	characteristic
246	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
247	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	106	2	no	—
251	C₈	—	106	2	no	—
252	C₈	#13133190	1	2	yes	—
253	C₈	#13133195	1	2	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	106	2	no	—
256	C₂²	—	212	2	no	—
257	C₂²	—	212	2	no	—
258	C₂²	—	212	2	no	—
259	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
260	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
261	C₂²	#16810674	1	1	yes	characteristic
262	C₄	—	4	2	no	—
263	C₄	—	53	4	no	—
264	C₄	—	106	2	no	—
265	C₄	—	106	2	no	—
266	C₄	#16810482	1	1	yes	characteristic
267	C₄	#16810585	1	1	yes	characteristic
268	C₄	#16810645	1	1	yes	characteristic
269	C₄	#16810678	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	212	2	no	—
273	C₂	#20274694	1	1	yes	characteristic
274	C₂	#349×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1377	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1377	D₅₃
hypocenter	C₅₃	#12785
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1377

Complements

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

N	H ≅ G/N	type
2. #20276470	C₂	semidirect
4. #20272799	C₂	semidirect
5. C₅₃×#1377	C₂	semidirect
132. Dic₅₃	#349	semidirect
137. #1377	D₅₃	semidirect
155. C₅₃	#12785	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810575	2
#20276470	2
C₂×#16810404	2
#20272799	2
C₅₃×#1377	2
#12785	53