Subgroups — #27469921

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283326	C₂	1	1	yes	maximal, characteristic
2	#20276179	C₂	1	4	yes	maximal, complemented
3	#20275631	C₂	1	2	yes	maximal, complemented
4	#20275268	C₂	1	2	yes	maximal
5	#20275238	C₂	1	2	yes	maximal
6	#20274838	C₂	1	2	yes	maximal, complemented
7	#20274515	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#3059	C₂	1	1	yes	maximal, characteristic
9	#16811390	C₂²	1	2	yes	—
10	#16810672	—	2	2	no	—
11	#16810672	C₂²	1	2	yes	—
12	C₂×#13133153	—	2	2	no	—
13	C₂×#13133153	C₂²	1	2	yes	—
14	#16810632	C₂²	1	4	yes	—
15	#16810607	C₂²	1	1	yes	characteristic
16	C₂×#13133136	C₂²	1	4	yes	—
17	#16810597	C₂²	1	1	yes	characteristic
18	#16810578	C₂²	1	2	yes	—
19	#16810394	C₂²	1	2	yes	—
20	#16810393	C₂²	1	4	yes	—
21	#16810379	C₂²	1	4	yes	—
22	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#536	C₂²	1	2	yes	—
25	C₅₃×#470	—	2	2	no	—
26	C₅₃×#467	—	2	2	no	—
27	C₅₃×#427	C₂²	1	1	yes	characteristic
28	C₅₃×#419	C₂²	1	2	yes	—
29	#13133224	—	4	4	no	—
30	C₂×#11719917	—	2	4	no	—
31	C₂×#11719917	C₂³	1	2	yes	—
32	C₂×#11719917	C₂³	1	4	yes	—
33	#13133176	C₂³	1	2	yes	—
34	#13133153	—	2	2	no	—
35	#13133153	—	2	4	no	—
36	#13133153	—	2	2	no	—
37	#13133141	—	4	4	no	—
38	#13133136	—	2	1	no	—
39	#13133136	—	4	4	no	—
40	#13133134	—	2	1	no	—
41	#13133132	—	2	2	no	—
42	#13133127	—	2	2	no	—
43	C₈×Dic₅₃	—	2	2	no	—
44	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
46	C₂×Q₁₆×C₅₃	—	2	1	no	—
47	C₂×Q₁₆×C₅₃	—	2	1	no	—
48	C₂×Q₁₆×C₅₃	—	2	2	no	—
49	C₂×Q₁₆×C₅₃	—	2	2	no	—
50	#131×C₅₃	—	2	2	no	—
51	#131×C₅₃	—	2	2	no	—
52	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
53	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	D₄	1	2	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₅₃	—	2	2	no	—
57	#124×C₅₃	—	2	2	no	—
58	#123×C₅₃	—	2	4	no	—
59	C₂²⋊Q₈×C₅₃	—	2	4	no	—
60	#117×C₅₃	C₂³	1	2	yes	—
61	#107×C₅₃	—	2	1	no	—
62	#106×C₅₃	—	2	1	no	—
63	#103×C₅₃	—	2	4	no	—
64	#41×C₅₃	—	2	2	no	—
65	#41×C₅₃	—	2	2	no	—
66	C₂×Q₈×C₅₃	—	2	2	no	—
67	C₂×Q₈×C₅₃	—	2	2	no	—
68	C₂×Q₈×C₅₃	—	2	4	no	—
69	C₂×Q₈×C₅₃	—	2	4	no	—
70	C₂×Q₈×C₅₃	#41	1	1	yes	characteristic
71	C₂×D₄×C₅₃	#41	1	2	yes	—
72	C₂×D₄×C₅₃	#41	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	—	2	2	no	—
74	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
76	Q₁₆×C₅₃	—	4	1	no	—
77	Q₁₆×C₅₃	—	4	1	no	—
78	Q₁₆×C₅₃	—	4	2	no	—
79	M₁₆×C₅₃	—	2	2	no	—
80	M₁₆×C₅₃	—	2	4	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	#41	1	2	yes	—
87	C₂×C₈×C₅₃	#41	1	2	yes	—
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	—	4	4	no	—
90	C₂²⋊C₄×C₅₃	—	2	2	no	—
91	C₂²⋊C₄×C₅₃	—	4	4	no	—
92	C₄²×C₅₃	—	2	2	no	—
93	C₂²⋊Dic₅₃	—	2	2	no	—
94	C₄⋊Dic₅₃	—	2	2	no	—
95	C₄×Dic₅₃	—	2	2	no	—
96	#11719917	—	2	4	no	—
97	#11719917	—	2	2	no	—
98	#11719917	—	2	2	no	—
99	#11719917	—	2	4	no	—
100	#11719917	—	4	4	no	—
101	C₂×#88322	—	2	2	no	—
102	C₂×#88322	—	2	4	no	—
103	C₂×#88322	—	4	4	no	—
104	C₂³×C₅₃	#124	1	2	yes	—
105	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
106	Q₈×C₅₃	—	2	2	no	—
107	Q₈×C₅₃	—	4	2	no	—
108	Q₈×C₅₃	—	4	2	no	—
109	Q₈×C₅₃	—	4	4	no	—
110	D₄×C₅₃	—	2	4	no	—
111	D₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	—	4	4	no	—
117	C₂×C₄×C₅₃	#124	1	2	yes	—
118	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
120	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
122	C₂×Dic₅₃	—	4	2	no	—
123	C₈×C₅₃	—	2	2	no	—
124	C₈×C₅₃	#123	1	1	yes	characteristic
125	C₈×C₅₃	#123	1	1	yes	characteristic
126	C₈×C₅₃	#123	1	2	yes	—
127	#88322	—	4	4	no	—
128	#88322	—	4	2	no	—
129	#27988	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	—	4	1	no	—
132	C₂²×C₅₃	—	4	2	no	—
133	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
134	C₂²×C₅₃	#553	1	2	yes	—
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	8	2	no	—
137	C₄×C₅₃	—	8	2	no	—
138	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
139	C₄×C₅₃	#535	1	1	yes	characteristic
140	C₄×C₅₃	#553	1	2	yes	—
141	Dic₅₃	—	8	2	no	—
142	#3059	D₅₃	1	1	yes	characteristic
143	#2916	—	53	1	no	—
144	#2082	—	53	4	no	—
145	#1979	—	53	2	no	—
146	#1915	—	53	2	no	—
147	#1909	—	53	2	no	—
148	#1833	—	53	2	no	—
149	#1757	—	53	1	no	—
150	C₂×C₅₃	—	2	2	no	—
151	C₂×C₅₃	—	2	1	no	—
152	C₂×C₅₃	—	4	2	no	—
153	C₂×C₅₃	#2504	1	1	yes	characteristic
154	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
155	#568	—	53	2	no	—
156	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
157	#536	C₂×D₅₃	1	2	yes	—
158	#470	—	2	2	no	—
159	#467	—	2	2	no	—
160	#430	—	53	2	no	—
161	#430	—	106	2	no	—
162	C₂×#108	—	53	2	no	—
163	C₂×#108	—	106	2	no	—
164	#427	C₂×D₅₃	1	1	yes	characteristic
165	#419	—	53	4	no	—
166	#419	C₂×D₅₃	1	2	yes	—
167	#413	—	53	1	no	—
168	C₂×#101	—	53	4	no	—
169	#410	—	53	1	no	—
170	#405	—	53	2	no	—
171	#344	—	53	2	no	—
172	#344	—	53	4	no	—
173	H(3,ℤ/4)	—	53	4	no	—
174	C₅₃	#27988	1	1	yes	characteristic, complemented, Sylow
175	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
176	C₂²×Q₈	C₂²×D₅₃	1	2	yes	—
177	C₂×Q₁₆	—	2	2	no	—
178	C₂×Q₁₆	—	2	2	no	—
179	C₂×Q₁₆	—	2	1	no	—
180	C₂×Q₁₆	—	2	1	no	—
181	#131	—	2	2	no	—
182	#131	—	2	2	no	—
183	#131	—	212	4	no	—
184	C₂×M₁₆	—	53	4	no	—
185	C₂×M₁₆	—	53	2	no	—
186	C₂×M₁₆	—	106	4	no	—
187	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
188	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
189	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
190	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
191	C₄⋊Q₈	—	2	2	no	—
192	#124	—	2	2	no	—
193	#123	—	2	4	no	—
194	C₂²⋊Q₈	—	2	4	no	—
195	#117	—	53	2	no	—
196	#117	C₂²×D₅₃	1	2	yes	—
197	#108	—	106	2	no	—
198	#108	—	106	2	no	—
199	#108	—	106	4	no	—
200	#107	—	2	1	no	—
201	#106	—	2	1	no	—
202	#103	—	2	4	no	—
203	#103	—	212	4	no	—
204	#101	—	106	1	no	—
205	#101	—	212	4	no	—
206	#100	—	106	1	no	—
207	#99	—	106	2	no	—
208	#97	—	106	2	no	—
209	C₄×C₈	—	106	2	no	—
210	#41	—	2	2	no	—
211	#41	—	2	2	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	#11719947	1	1	yes	characteristic
217	C₂×D₄	#11719947	1	2	yes	—
218	C₂×D₄	#11719947	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
221	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
222	Q₁₆	—	4	1	no	—
223	Q₁₆	—	4	1	no	—
224	Q₁₆	—	4	2	no	—
225	M₁₆	—	2	2	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	106	4	no	—
228	M₁₆	—	106	2	no	—
229	M₁₆	—	106	2	no	—
230	M₁₆	—	106	4	no	—
231	M₁₆	—	212	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	106	2	no	—
235	C₂×C₈	—	106	4	no	—
236	C₂×C₈	—	212	4	no	—
237	C₂×C₈	#11719947	1	2	yes	—
238	C₂×C₈	#11719947	1	2	yes	—
239	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
240	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
241	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	106	2	no	—
245	C₂²⋊C₄	—	2	2	no	—
246	C₂²⋊C₄	—	4	4	no	—
247	C₂²⋊C₄	—	106	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	106	2	no	—
250	C₂³	#13133204	1	2	yes	—
251	C₂³	C₂×#11719947	1	1	yes	characteristic
252	Q₈	—	2	2	no	—
253	Q₈	—	4	4	no	—
254	Q₈	—	4	2	no	—
255	Q₈	—	4	2	no	—
256	D₄	—	2	4	no	—
257	D₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	212	2	no	—
264	C₂×C₄	#13133204	1	2	yes	—
265	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
266	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
267	C₂×C₄	C₂×#11719947	1	2	yes	—
268	C₂×C₄	#13133263	1	2	yes	—
269	C₈	—	2	2	no	—
270	C₈	—	212	2	no	—
271	C₈	—	212	4	no	—
272	C₈	#13133198	1	1	yes	characteristic
273	C₈	#13133198	1	1	yes	characteristic
274	C₈	#13133198	1	2	yes	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	2	no	—
278	C₂²	C₂×#13133204	1	1	yes	characteristic
279	C₂²	#16811325	1	2	yes	—
280	C₄	—	2	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	424	2	no	—
284	C₄	C₂×#13133198	1	1	yes	characteristic
285	C₄	#16811187	1	1	yes	characteristic
286	C₄	#16811325	1	2	yes	—
287	C₂	—	2	1	no	—
288	C₂	—	2	2	no	—
289	C₂	—	4	2	no	—
290	C₂	#20279208	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20279208
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3059	C₂
socle	C₂×C₅₃	#2504
prime radical	C₂×#41×C₅₃	C₂³
hypercenter	#3059	D₅₃
hypocenter	C₅₃	#27988
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3059

Complements

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

N	H ≅ G/N	type
2. #20276179	C₂	semidirect
3. #20275631	C₂	semidirect
6. #20274838	C₂	semidirect
174. C₅₃	#27988	semidirect

Maximal subgroups

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

subgroup	index
#20283326	2
#20276179	2
#20275631	2
#20275268	2
#20275238	2
#20274838	2
#20274515	2
C₅₃×#3059	2
#27988	53