Subgroups — #27289827

Order 13568.

718 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810387	C₂	1	2	yes	maximal
3	C₂×#16810387	C₂	1	2	yes	maximal
4	#20274409	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂³×#88322	C₂²	1	1	yes	characteristic
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	—	2	4	no	—
10	C₂×C₄×#88322	C₄	1	4	yes	—
11	#16810387	—	2	4	no	—
12	#16810387	C₄	1	8	yes	complemented
13	#16810378	—	2	2	no	—
14	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#336	—	2	2	no	—
17	C₂×#11719917	—	2	2	no	—
18	C₂×#11719917	—	2	2	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	C₂×C₄	1	4	yes	—
23	C₂²×#88322	—	2	2	no	—
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	—	2	1	no	—
26	C₂²×#88322	—	2	1	no	—
27	C₂²×#88322	—	2	1	no	—
28	C₂²×#88322	—	2	1	no	—
29	C₂²×#88322	—	2	2	no	—
30	C₂²×#88322	C₂×C₄	1	2	yes	—
31	C₂²×#88322	C₂×C₄	1	4	yes	—
32	C₂²×#88322	C₂×C₄	1	2	yes	—
33	C₂²×#88322	D₄	1	4	yes	—
34	C₄×#88322	—	2	8	no	—
35	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
36	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₅₃	—	2	2	no	—
38	C₂×M₁₆×C₅₃	—	2	2	no	—
39	C₂×M₁₆×C₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	—	2	2	no	—
41	C₂×M₁₆×C₅₃	—	2	2	no	—
42	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₅₃	—	2	2	no	—
44	C₂×C₄²×C₅₃	—	2	2	no	—
45	C₂×C₄²×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₅₃	D₄	1	2	yes	—
48	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	2	4	no	—
50	C₂²×C₄×C₅₃	—	2	8	no	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	#41	1	2	yes	—
57	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	#41	1	2	yes	—
62	M₁₆×C₅₃	—	2	4	no	—
63	M₁₆×C₅₃	—	2	4	no	—
64	M₁₆×C₅₃	—	4	2	no	—
65	M₁₆×C₅₃	—	4	2	no	—
66	C₂×C₈×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	—	4	2	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	4	no	—
72	#11719917	—	2	4	no	—
73	#11719917	—	2	4	no	—
74	#11719917	—	4	2	no	—
75	#11719917	—	4	2	no	—
76	C₂×#88322	—	2	2	no	—
77	C₂×#88322	—	2	2	no	—
78	C₂×#88322	—	2	2	no	—
79	C₂×#88322	—	2	2	no	—
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	—	2	4	no	—
84	C₂×#88322	—	2	4	no	—
85	C₂×#88322	—	4	2	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂×#88322	C₄²	1	4	yes	—
88	C₂×#88322	C₂²⋊C₄	1	4	yes	—
89	C₂³×C₅₃	—	2	4	no	—
90	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
92	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
93	C₂³×C₅₃	#125	1	1	yes	characteristic
94	C₂³×C₅₃	#125	1	1	yes	characteristic
95	C₂×C₄×C₅₃	—	2	16	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	2	4	no	—
98	C₂×C₄×C₅₃	#108	1	8	yes	—
99	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#117	1	2	yes	—
103	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#117	1	2	yes	—
106	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₃	#121	1	2	yes	—
111	C₂×C₄×C₅₃	#123	1	2	yes	—
112	C₂×C₄×C₅₃	#123	1	2	yes	—
113	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#126	1	2	yes	—
116	C₂×C₄×C₅₃	#126	1	2	yes	—
117	C₈×C₅₃	—	4	4	no	—
118	#88322	—	2	4	no	—
119	#88322	—	2	4	no	—
120	#88322	—	4	4	no	—
121	#10822	—	53	1	no	maximal, Sylow
122	C₂²×C₅₃	—	2	4	no	—
123	C₂²×C₅₃	—	2	2	no	—
124	C₂²×C₅₃	—	2	2	no	—
125	C₂²×C₅₃	—	2	4	no	—
126	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₅₃	#380	1	1	yes	characteristic
128	C₂²×C₅₃	#382	1	1	yes	characteristic
129	C₂²×C₅₃	#387	1	2	yes	—
130	C₂²×C₅₃	#387	1	2	yes	—
131	C₂²×C₅₃	C₂×#108	1	4	yes	—
132	C₄×C₅₃	—	2	8	no	—
133	C₄×C₅₃	#376	1	1	yes	characteristic
134	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₅₃	#381	1	1	yes	characteristic
136	C₄×C₅₃	#383	1	1	yes	characteristic
137	C₄×C₅₃	#388	1	2	yes	—
138	C₄×C₅₃	#388	1	2	yes	—
139	C₂²×C₄×C₈	—	53	1	no	—
140	C₂×#341	—	53	2	no	—
141	C₂×#341	—	53	2	no	—
142	#1728	—	53	1	no	—
143	#1728	D₅₃	1	1	yes	characteristic
144	C₂×C₅₃	—	2	2	no	—
145	C₂×C₅₃	—	2	2	no	—
146	C₂×C₅₃	#1434	1	1	yes	characteristic
147	C₂×C₅₃	C₄×#108	1	2	yes	—
148	C₂×C₅₃	#1810	1	2	yes	—
149	C₂×C₅₃	#1837	1	2	yes	—
150	C₂²×M₁₆	—	53	2	no	—
151	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
152	C₂³×C₈	—	53	1	no	—
153	C₂³×C₈	—	53	1	no	—
154	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
155	C₂×C₄×C₈	—	53	4	no	—
156	C₂×C₄×C₈	—	106	4	no	—
157	#341	—	53	8	no	—
158	#341	—	106	4	no	—
159	#336	—	2	2	no	—
160	#336	—	106	2	no	—
161	C₅₃	#10822	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₄	Dic₁₀₆	1	2	yes	—
163	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	53	4	no	—
170	C₂×M₁₆	—	106	2	no	—
171	C₂×M₁₆	—	106	2	no	—
172	C₂×M₁₆	—	106	2	no	—
173	C₂×M₁₆	—	106	2	no	—
174	C₂×M₁₆	—	106	2	no	—
175	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	53	4	no	—
178	C₂²×C₈	—	53	2	no	—
179	C₂²×C₈	—	53	2	no	—
180	C₂²×C₈	—	53	4	no	—
181	C₂²×C₈	—	106	2	no	—
182	C₂²×C₈	—	106	2	no	—
183	C₂²×C₈	—	106	1	no	—
184	C₂²×C₈	—	106	1	no	—
185	C₂²×C₈	—	106	1	no	—
186	C₂²×C₈	—	106	1	no	—
187	C₂²×C₈	—	106	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	C₄×D₅₃	1	2	yes	—
191	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
192	C₄×C₈	—	106	8	no	—
193	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	#11719919	1	4	yes	—
197	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
199	C₂²×C₄	#11719945	1	2	yes	—
200	C₂²×C₄	D₄×D₅₃	1	2	yes	—
201	C₂²×C₄	#11719947	1	2	yes	—
202	C₂²×C₄	#11719947	1	1	yes	characteristic
203	C₂²×C₄	#11719947	1	1	yes	characteristic
204	C₂²×C₄	#11719949	1	1	yes	characteristic
205	C₂²×C₄	#11719949	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	4	2	no	—
211	M₁₆	—	106	4	no	—
212	M₁₆	—	106	4	no	—
213	M₁₆	—	212	2	no	—
214	M₁₆	—	212	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	53	4	no	—
219	C₂×C₈	—	53	4	no	—
220	C₂×C₈	—	106	2	no	—
221	C₂×C₈	—	106	2	no	—
222	C₂×C₈	—	106	2	no	—
223	C₂×C₈	—	106	2	no	—
224	C₂×C₈	—	106	4	no	—
225	C₂×C₈	—	106	4	no	—
226	C₂×C₈	—	106	4	no	—
227	C₂×C₈	—	106	4	no	—
228	C₂×C₈	—	106	4	no	—
229	C₂×C₈	—	212	2	no	—
230	C₂×C₈	—	212	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	4	no	—
234	C₂³	—	2	4	no	—
235	C₂³	C₂×#11719919	1	1	yes	characteristic
236	C₂³	C₄×Dic₁₀₆	1	2	yes	—
237	C₂³	#13133194	1	2	yes	—
238	C₂³	#13133208	1	1	yes	characteristic
239	C₂³	#13133208	1	1	yes	characteristic
240	C₂×C₄	—	2	16	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	#13133152	1	8	yes	—
244	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
245	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
246	C₂×C₄	#13133173	1	2	yes	—
247	C₂×C₄	#13133173	1	1	yes	characteristic
248	C₂×C₄	#13133174	1	1	yes	characteristic
249	C₂×C₄	#13133175	1	2	yes	—
250	C₂×C₄	#13133175	1	1	yes	characteristic
251	C₂×C₄	#13133175	1	1	yes	characteristic
252	C₂×C₄	#13133177	1	2	yes	—
253	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
254	C₂×C₄	#13133179	1	2	yes	—
255	C₂×C₄	#13133189	1	2	yes	—
256	C₂×C₄	#13133199	1	2	yes	—
257	C₂×C₄	#13133202	1	2	yes	—
258	C₂×C₄	#13133204	1	1	yes	characteristic
259	C₂×C₄	#13133206	1	1	yes	characteristic
260	C₂×C₄	#13133210	1	2	yes	—
261	C₂×C₄	#13133212	1	2	yes	—
262	C₈	—	4	4	no	—
263	C₈	—	106	4	no	—
264	C₈	—	106	4	no	—
265	C₈	—	212	4	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	C₄×#11719919	1	1	yes	characteristic
271	C₂²	#16810481	1	1	yes	characteristic
272	C₂²	#16810488	1	1	yes	characteristic
273	C₂²	#16810506	1	2	yes	—
274	C₂²	#16810510	1	2	yes	—
275	C₂²	C₂×#13133152	1	4	yes	—
276	C₄	—	2	8	no	—
277	C₄	#16810468	1	1	yes	characteristic
278	C₄	#16810470	1	1	yes	characteristic
279	C₄	#16810485	1	1	yes	characteristic
280	C₄	#16810490	1	1	yes	characteristic
281	C₄	#16810513	1	2	yes	—
282	C₄	#16810517	1	2	yes	—
283	C₂	—	2	2	no	—
284	C₂	—	2	2	no	—
285	C₂	#20272941	1	1	yes	characteristic
286	C₂	C₄×#13133152	1	2	yes	—
287	C₂	#20274751	1	2	yes	—
288	C₂	#20274852	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1728	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1728	D₅₃
hypocenter	C₅₃	#10822
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
12. #16810387	C₄	semidirect
161. C₅₃	#10822	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×#88322	2
C₂×#16810387	2
#20274409	2
C₅₃×#1728	2
#10822	53