Subgroups — #27304162

Order 13568.

378 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	C₂×#16810451	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810449	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810425	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810425	C₂	1	1	yes	maximal, characteristic
5	#20275581	C₂	1	1	yes	maximal, characteristic
6	#20274879	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
9	C₂×#13133151	C₂²	1	1	yes	characteristic
10	C₂×#13133151	C₂²	1	1	yes	characteristic
11	C₂×#13133151	C₂²	1	1	yes	characteristic
12	C₂×#13133149	—	2	1	no	—
13	C₂×#13133139	—	2	1	no	—
14	#16810525	—	2	1	no	—
15	#16810489	—	2	1	no	—
16	#16810451	—	2	2	no	—
17	#16810449	—	2	2	no	—
18	#16810425	—	2	2	no	—
19	#16810425	—	2	2	no	—
20	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×D₂₁₂	—	2	1	no	—
24	C₂×D₄₂₄	—	2	1	no	—
25	C₂×D₄₂₄	—	2	1	no	—
26	C₂×D₄₂₄	—	2	1	no	—
27	C₂×D₄₂₄	—	2	1	no	—
28	C₂×D₄₂₄	—	2	1	no	—
29	C₂×D₄₂₄	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₅₃	—	2	1	no	—
31	C₂×C₄⋊Dic₅₃	—	2	1	no	—
32	C₂×C₄⋊Dic₅₃	—	2	1	no	—
33	C₂×#11719921	—	4	2	no	—
34	#13133151	—	2	1	no	—
35	#13133151	—	2	1	no	—
36	#13133151	—	2	1	no	—
37	#13133151	—	2	1	no	—
38	#13133151	—	2	1	no	—
39	#13133151	—	2	1	no	—
40	#13133149	—	4	2	no	—
41	#13133139	—	4	2	no	—
42	#13133124	—	4	2	no	—
43	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₁₆×C₅₃	—	2	1	no	—
47	C₂×C₁₆×C₅₃	—	2	1	no	—
48	C₂×C₁₆×C₅₃	—	2	1	no	—
49	C₂×C₁₆×C₅₃	—	2	1	no	—
50	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
51	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
52	C₄×C₈×C₅₃	—	2	1	no	—
53	C₄×C₈×C₅₃	—	2	1	no	—
54	C₂³×D₅₃	—	4	1	no	—
55	C₂²×C₄×C₅₃	D₈	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₅₃	—	4	1	no	—
58	C₂²×Dic₅₃	—	4	1	no	—
59	C₂²×Dic₅₃	—	4	1	no	—
60	C₂×D₂₁₂	—	2	2	no	—
61	C₂×D₂₁₂	—	4	1	no	—
62	C₂×D₂₁₂	—	4	1	no	—
63	C₂×D₂₁₂	—	4	1	no	—
64	C₂×D₂₁₂	—	4	1	no	—
65	C₂×D₂₁₂	—	4	1	no	—
66	C₂×C₈×C₅₃	—	2	1	no	—
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	D₈	1	2	yes	—
69	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
73	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
74	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
75	C₄²×C₅₃	—	2	1	no	—
76	C₄²×C₅₃	—	2	1	no	—
77	#11719921	—	8	4	no	—
78	C₄⋊Dic₅₃	—	4	1	no	—
79	C₄⋊Dic₅₃	—	4	1	no	—
80	C₄⋊Dic₅₃	—	4	1	no	—
81	C₄⋊Dic₅₃	—	4	1	no	—
82	C₄⋊Dic₅₃	—	4	1	no	—
83	C₄⋊Dic₅₃	—	4	1	no	—
84	D₄₂₄	—	2	2	no	—
85	D₄₂₄	—	2	2	no	—
86	D₄₂₄	—	4	1	no	—
87	D₄₂₄	—	4	1	no	—
88	C₁₆×C₅₃	—	2	2	no	—
89	C₁₆×C₅₃	—	2	2	no	—
90	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
91	C₂²×D₅₃	—	4	2	no	—
92	C₂²×D₅₃	—	8	1	no	—
93	C₂²×D₅₃	—	8	1	no	—
94	C₂²×D₅₃	—	8	2	no	—
95	C₂²×D₅₃	—	8	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
105	C₂×Dic₅₃	—	8	1	no	—
106	C₂×Dic₅₃	—	8	1	no	—
107	C₂×Dic₅₃	—	8	1	no	—
108	C₂×Dic₅₃	—	8	2	no	—
109	C₂×Dic₅₃	—	8	2	no	—
110	C₂×Dic₅₃	—	8	2	no	—
111	D₂₁₂	—	4	2	no	—
112	D₂₁₂	—	4	2	no	—
113	D₂₁₂	—	8	1	no	—
114	D₂₁₂	—	8	1	no	—
115	C₈×C₅₃	—	2	2	no	—
116	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
118	C₈×C₅₃	#123	1	1	yes	characteristic
119	C₈×C₅₃	#123	1	1	yes	characteristic
120	#13528	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	#474	1	1	yes	characteristic
122	C₂²×C₅₃	#480	1	1	yes	characteristic
123	C₂²×C₅₃	#493	1	1	yes	characteristic
124	C₂²×C₅₃	#508	1	2	yes	—
125	C₂²×C₅₃	#509	1	2	yes	—
126	C₂×D₅₃	—	8	2	no	—
127	C₂×D₅₃	—	8	4	no	—
128	C₂×D₅₃	—	16	2	no	—
129	C₂×D₅₃	—	16	1	no	—
130	C₂×D₅₃	—	16	1	no	—
131	C₄×C₅₃	—	4	2	no	—
132	C₄×C₅₃	#399	1	1	yes	characteristic
133	C₄×C₅₃	#474	1	1	yes	characteristic
134	C₄×C₅₃	#480	1	1	yes	characteristic
135	C₄×C₅₃	#497	1	1	yes	characteristic
136	Dic₅₃	—	16	1	no	—
137	Dic₅₃	—	16	1	no	—
138	Dic₅₃	—	16	1	no	—
139	C₂×#367	—	53	1	no	—
140	C₂×#366	—	53	1	no	—
141	C₂×#357	—	53	1	no	—
142	C₂×#357	—	53	1	no	—
143	#1970	—	53	1	no	—
144	#1846	—	53	1	no	—
145	#1374	D₅₃	1	1	yes	characteristic, complemented
146	C₂×C₅₃	#2093	1	1	yes	characteristic
147	C₂×C₅₃	#2228	1	2	yes	—
148	C₂×C₅₃	#2234	1	2	yes	—
149	C₂×C₅₃	#2249	1	1	yes	characteristic
150	C₂×C₅₃	#2250	1	1	yes	characteristic
151	D₅₃	—	16	2	no	—
152	C₂²×D₈	—	53	1	no	—
153	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
154	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
155	C₂×#107	—	53	1	no	—
156	C₂×#107	—	53	1	no	—
157	C₂×#107	—	53	1	no	—
158	C₂×#106	—	106	1	no	—
159	C₂×#102	—	106	1	no	—
160	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
161	#390	—	106	1	no	—
162	#382	—	106	1	no	—
163	#367	—	106	2	no	—
164	#366	—	106	2	no	—
165	#357	—	106	2	no	—
166	#357	—	106	2	no	—
167	C₅₃	#13528	1	1	yes	characteristic, complemented, Sylow
168	C₂²×D₄	—	106	1	no	—
169	C₂×D₈	—	53	2	no	—
170	C₂×D₈	—	106	1	no	—
171	C₂×D₈	—	106	1	no	—
172	C₂×D₈	—	106	1	no	—
173	C₂×D₈	—	106	1	no	—
174	C₂×D₈	—	106	1	no	—
175	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
177	C₂×C₄⋊C₄	—	106	1	no	—
178	C₂×C₄⋊C₄	—	106	1	no	—
179	C₂×C₄⋊C₄	—	106	1	no	—
180	C₂×C₂²⋊C₄	—	212	2	no	—
181	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	—	2	1	no	—
186	C₂×C₁₆	D₂₁₂	1	2	yes	—
187	C₂×C₁₆	D₂₁₂	1	2	yes	—
188	#107	—	106	1	no	—
189	#107	—	106	1	no	—
190	#107	—	106	1	no	—
191	#107	—	106	1	no	—
192	#107	—	106	1	no	—
193	#107	—	106	1	no	—
194	#106	—	212	2	no	—
195	#102	—	212	2	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	#95	—	212	2	no	—
199	C₂⁴	—	212	1	no	—
200	C₂×D₄	—	106	2	no	—
201	C₂×D₄	—	212	1	no	—
202	C₂×D₄	—	212	1	no	—
203	C₂×D₄	—	212	1	no	—
204	C₂×D₄	—	212	1	no	—
205	C₂×D₄	—	212	1	no	—
206	C₂²×C₄	—	212	1	no	—
207	C₂²×C₄	—	212	1	no	—
208	C₂²×C₄	—	212	1	no	—
209	C₂²×C₄	D₄₂₄	1	2	yes	—
210	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
211	D₈	—	106	2	no	—
212	D₈	—	106	2	no	—
213	D₈	—	212	1	no	—
214	D₈	—	212	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	D₄₂₄	1	2	yes	—
218	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
219	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
220	C₂×C₈	#11719947	1	1	yes	characteristic
221	C₂×C₈	#11719947	1	1	yes	characteristic
222	C₂×C₈	#11719947	1	1	yes	characteristic
223	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
224	C₄⋊C₄	—	212	1	no	—
225	C₄⋊C₄	—	212	1	no	—
226	C₄⋊C₄	—	212	1	no	—
227	C₄⋊C₄	—	212	1	no	—
228	C₄⋊C₄	—	212	1	no	—
229	C₄⋊C₄	—	212	1	no	—
230	C₂²⋊C₄	—	424	4	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₁₆	—	2	2	no	—
234	C₁₆	—	2	2	no	—
235	C₂³	—	424	2	no	—
236	C₂³	—	424	2	no	—
237	C₂³	—	424	1	no	—
238	C₂³	—	424	1	no	—
239	C₂³	—	212	2	no	—
240	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
241	D₄	—	424	1	no	—
242	D₄	—	424	1	no	—
243	D₄	—	212	2	no	—
244	D₄	—	212	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	424	2	no	—
249	C₂×C₄	—	424	2	no	—
250	C₂×C₄	—	424	2	no	—
251	C₂×C₄	—	424	1	no	—
252	C₂×C₄	—	424	1	no	—
253	C₂×C₄	—	424	1	no	—
254	C₂×C₄	#13133197	1	1	yes	characteristic
255	C₂×C₄	#13133201	1	1	yes	characteristic
256	C₂×C₄	#13133214	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
258	C₂×C₄	#13133250	1	1	yes	characteristic
259	C₂×C₄	#13133250	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	#13133197	1	1	yes	characteristic
262	C₈	#13133197	1	1	yes	characteristic
263	C₈	#13133201	1	1	yes	characteristic
264	C₈	#13133201	1	1	yes	characteristic
265	C₂²	—	848	2	no	—
266	C₂²	—	848	1	no	—
267	C₂²	—	848	1	no	—
268	C₂²	—	424	2	no	—
269	C₂²	—	424	4	no	—
270	C₂²	#16810904	1	1	yes	characteristic
271	C₂²	#16810937	1	1	yes	characteristic
272	C₂²	#16811000	1	1	yes	characteristic
273	C₂²	#16811055	1	2	yes	—
274	C₂²	#16811061	1	2	yes	—
275	C₄	—	4	2	no	—
276	C₄	—	848	1	no	—
277	C₄	—	848	1	no	—
278	C₄	—	848	1	no	—
279	C₄	#16810560	1	1	yes	characteristic
280	C₄	#16810904	1	1	yes	characteristic
281	C₄	#16810937	1	1	yes	characteristic
282	C₄	#16811017	1	1	yes	characteristic
283	C₂	—	848	2	no	—
284	C₂	#20276242	1	1	yes	characteristic
285	C₂	#20276873	1	2	yes	—
286	C₂	#20276907	1	2	yes	—
287	C₂	#20276976	1	1	yes	characteristic
288	C₂	#20276981	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1374	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂²×D₄₂₄	C₂²
hypercenter	#1374	D₅₃
hypocenter	C₅₃	#13528
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
1. C₂×#16810451	C₂	semidirect
2. C₂×#16810449	C₂	semidirect
6. #20274879	C₂	semidirect
7. C₅₃×#1374	C₂	semidirect
145. #1374	D₅₃	semidirect
167. C₅₃	#13528	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810451	2
C₂×#16810449	2
C₂×#16810425	2
#20275581	2
#20274879	2
C₅₃×#1374	2
#13528	53