Subgroups — #27384639

Order 13568.

894 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20277113	C₂	1	2	yes	maximal
2	#20274879	C₂	1	2	yes	maximal, complemented
3	#20274846	C₂	1	2	yes	maximal, complemented
4	#20274744	C₂	1	2	yes	maximal
5	#20274740	C₂	1	2	yes	maximal
6	C₄×#13133151	C₂	1	2	yes	maximal, complemented
7	C₄×#13133149	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133151	C₂²	1	4	yes	—
10	C₂×#13133149	C₂²	1	4	yes	—
11	#16810491	—	2	2	no	—
12	#16810489	C₂²	1	4	yes	—
13	#16810486	—	2	2	no	—
14	#16810483	C₂²	1	4	yes	—
15	C₄×C₄⋊Dic₅₃	C₂²	1	4	yes	—
16	C₄×C₂²⋊Dic₅₃	—	2	2	no	—
17	#16810469	—	2	2	no	—
18	#16810386	C₂²	1	4	yes	—
19	#16810386	C₂²	1	4	yes	—
20	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₄	1	8	yes	complemented
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₅₃	C₂³	1	8	yes	—
29	C₂×C₂²⋊Dic₅₃	—	2	4	no	—
30	C₂×C₄×Dic₅₃	—	2	4	no	—
31	#13133151	—	2	8	no	—
32	#13133149	—	2	8	no	—
33	#13133124	—	2	4	no	—
34	#13133124	—	2	4	no	—
35	#13133124	—	2	4	no	—
36	#13133124	—	2	4	no	—
37	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	—	2	4	no	—
41	C₂²×C₈×C₅₃	—	2	4	no	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
55	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
56	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
57	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
58	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	—	2	4	no	—
60	C₂²×C₄×C₅₃	—	2	4	no	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₅₃	—	2	8	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	8	no	—
77	C₂×C₈×C₅₃	—	2	8	no	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₃	#41	1	4	yes	—
83	C₂×C₈×C₅₃	#41	1	4	yes	—
84	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
85	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
86	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
87	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
89	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
90	C₂²⋊Dic₅₃	—	4	8	no	—
91	C₄⋊Dic₅₃	—	2	8	no	—
92	C₄⋊Dic₅₃	—	2	8	no	—
93	C₄×Dic₅₃	—	4	8	no	—
94	C₂³×C₅₃	—	2	2	no	—
95	C₂³×C₅₃	—	2	2	no	—
96	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₅₃	#135	1	4	yes	—
98	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	8	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	2	8	no	—
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₅₃	#135	1	4	yes	—
114	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
117	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
118	C₂×Dic₅₃	—	4	8	no	—
119	C₂×Dic₅₃	—	4	8	no	—
120	C₂×Dic₅₃	—	4	8	no	—
121	C₈×C₅₃	—	2	2	no	—
122	C₈×C₅₃	—	2	2	no	—
123	C₈×C₅₃	#117	1	4	yes	—
124	C₈×C₅₃	#117	1	4	yes	—
125	#20818	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	—	2	8	no	—
128	C₂²×C₅₃	—	2	2	no	—
129	C₂²×C₅₃	#427	1	4	yes	—
130	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₅₃	#532	1	2	yes	—
132	C₂²×C₅₃	C₂×#135	1	2	yes	—
133	C₂²×C₅₃	C₂×#135	1	2	yes	—
134	C₄×C₅₃	—	2	2	no	—
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	#427	1	4	yes	—
139	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
140	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
141	C₄×C₅₃	#533	1	2	yes	—
142	Dic₅₃	—	8	8	no	—
143	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
144	#2283	—	53	2	no	—
145	#1846	—	53	2	no	—
146	#1835	—	53	2	no	—
147	#1808	—	53	2	no	—
148	#1807	—	53	2	no	—
149	C₄×#107	—	53	2	no	—
150	C₄×#106	—	53	2	no	—
151	C₂×C₅₃	—	2	4	no	—
152	C₂×C₅₃	#2285	1	1	yes	characteristic
153	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
154	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
155	C₂×C₅₃	#3168	1	4	yes	—
156	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
157	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
158	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
159	C₂×#107	—	53	4	no	—
160	C₂×#106	—	53	4	no	—
161	C₂×C₄×C₈	Dic₅₃	1	8	yes	complemented
162	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
166	#383	—	106	2	no	—
167	#382	—	53	4	no	—
168	#381	—	106	2	no	—
169	#380	—	53	4	no	—
170	C₄×C₄⋊C₄	—	53	4	no	—
171	C₄×C₂²⋊C₄	—	106	2	no	—
172	#376	—	106	2	no	—
173	#340	—	53	4	no	—
174	#340	—	53	4	no	—
175	C₅₃	#20818	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
177	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
178	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
179	C₂²×C₈	—	2	4	no	—
180	C₂²×C₈	—	2	4	no	—
181	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
182	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
183	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
187	C₂×C₄⋊C₄	—	53	8	no	—
188	C₂×C₂²⋊C₄	—	106	4	no	—
189	C₂×C₄²	—	106	4	no	—
190	C₂×C₄²	Dic₁₀₆	1	4	yes	—
191	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
192	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
193	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
194	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
195	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
196	#107	—	106	8	no	—
197	#106	—	106	8	no	—
198	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
199	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
200	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
201	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
202	#95	—	106	4	no	—
203	#95	—	106	4	no	—
204	#95	—	106	4	no	—
205	#95	—	106	4	no	—
206	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	106	8	no	—
212	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
213	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
214	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
215	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
220	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
221	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	#11719945	1	4	yes	—
227	C₂×C₈	#11719945	1	4	yes	—
228	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
229	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
230	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
231	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
232	C₄⋊C₄	—	106	8	no	—
233	C₄⋊C₄	—	106	8	no	—
234	C₂²⋊C₄	—	212	8	no	—
235	C₄²	—	212	8	no	—
236	C₄²	C₄⋊Dic₅₃	1	4	yes	—
237	C₄²	C₄⋊Dic₅₃	1	4	yes	—
238	C₄²	C₂×Dic₁₀₆	1	2	yes	—
239	C₄²	C₂×Dic₁₀₆	1	2	yes	—
240	C₄²	C₂²×Dic₅₃	1	2	yes	—
241	C₄²	C₂²×Dic₅₃	1	2	yes	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
245	C₂³	#13133239	1	4	yes	—
246	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	212	8	no	—
257	C₂×C₄	—	212	8	no	—
258	C₂×C₄	—	212	8	no	—
259	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
260	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
261	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
262	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
263	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
264	C₂×C₄	#13133239	1	4	yes	—
265	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
266	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
267	C₂×C₄	C₂×#11719945	1	2	yes	—
268	C₂×C₄	C₂×#11719945	1	2	yes	—
269	C₈	—	2	2	no	—
270	C₈	—	2	2	no	—
271	C₈	#13133176	1	4	yes	—
272	C₈	#13133176	1	4	yes	—
273	C₂²	—	2	8	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	#16810661	1	4	yes	—
277	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
278	C₂²	#16811173	1	2	yes	—
279	C₂²	C₂×#13133239	1	2	yes	—
280	C₂²	C₂×#13133239	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	424	8	no	—
286	C₄	#16810661	1	4	yes	—
287	C₄	C₂×#13133176	1	1	yes	characteristic
288	C₄	C₂×#13133176	1	1	yes	characteristic
289	C₄	#16811180	1	2	yes	—
290	C₂	—	2	4	no	—
291	C₂	#20277122	1	1	yes	characteristic
292	C₂	C₂×#16810661	1	1	yes	characteristic
293	C₂	C₂×#16810661	1	1	yes	characteristic
294	C₂	#20285285	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	#20818
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20274879	C₂	semidirect
3. #20274846	C₂	semidirect
6. C₄×#13133151	C₂	semidirect
7. C₄×#13133149	C₂	semidirect
23. C₂×C₄×C₈×C₅₃	C₄	semidirect
161. C₂×C₄×C₈	Dic₅₃	semidirect
175. C₅₃	#20818	semidirect

Maximal subgroups

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

subgroup	index
#20277113	2
#20274879	2
#20274846	2
#20274744	2
#20274740	2
C₄×#13133151	2
C₄×#13133149	2
C₂²×C₄×C₈×C₅₃	2
#20818	53