Subgroups — #27300035

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810577	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810444	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276325	C₂	1	2	yes	maximal
4	#20272796	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11719912	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
8	C₂×#13133174	C₄	1	1	yes	characteristic
9	C₂×#13133157	C₂²	1	2	yes	—
10	#16810577	—	2	2	no	—
11	#16810577	—	2	2	no	—
12	C₂×#13133128	C₄	1	1	yes	characteristic
13	C₂×#13133127	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	#16810444	—	2	2	no	—
16	#16810402	—	2	4	no	—
17	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#364	—	2	2	no	—
20	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×#11719912	—	2	2	no	—
22	C₂×#11719912	—	2	2	no	—
23	C₂×#11719912	C₈	1	4	yes	—
24	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₅₃	—	2	1	no	—
27	C₂×C₈×D₅₃	—	2	1	no	—
28	C₂×C₈×D₅₃	—	2	1	no	—
29	C₂×C₈×D₅₃	—	2	1	no	—
30	C₂×C₈×D₅₃	D₄	1	2	yes	—
31	C₂×C₈×D₅₃	Q₈	1	2	yes	—
32	#13133174	—	2	2	no	—
33	#13133174	—	2	2	no	—
34	C₂×#11719919	—	2	1	no	—
35	C₂×#11719921	—	2	1	no	—
36	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	#13133157	—	2	2	no	—
38	#13133157	—	2	2	no	—
39	C₂×#11719908	—	2	2	no	—
40	#13133128	—	2	2	no	—
41	#13133127	—	2	2	no	—
42	C₈×Dic₅₃	—	2	1	no	—
43	C₈×Dic₅₃	—	2	1	no	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	M₃₂×C₅₃	—	2	2	no	—
48	M₃₂×C₅₃	—	2	2	no	—
49	C₂×C₁₆×C₅₃	—	2	2	no	—
50	C₄×C₈×C₅₃	D₄	1	2	yes	—
51	C₄×C₈×C₅₃	D₄	1	2	yes	—
52	C₂³×D₅₃	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₅₃	—	2	1	no	—
56	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
57	C₂×C₄×D₅₃	—	2	1	no	—
58	C₂×C₄×D₅₃	—	2	1	no	—
59	C₂×C₄×D₅₃	—	2	1	no	—
60	C₂×C₄×D₅₃	—	2	1	no	—
61	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
66	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
71	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
72	C₄²×C₅₃	—	2	1	no	—
73	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
74	#11719921	—	2	4	no	—
75	#11719919	—	2	4	no	—
76	C₄×Dic₅₃	—	2	1	no	—
77	C₄×Dic₅₃	M₁₆	1	2	yes	—
78	C₂×#88322	—	2	1	no	—
79	C₂×#88322	—	2	1	no	—
80	C₂×#88322	—	2	1	no	—
81	C₂×#88322	—	2	1	no	—
82	C₂×#88322	C₄⋊C₄	1	2	yes	—
83	C₂×#88322	C₂×C₈	1	2	yes	—
84	#11719912	—	2	8	no	—
85	C₈×D₅₃	—	2	2	no	—
86	C₈×D₅₃	—	2	2	no	—
87	C₈×D₅₃	—	2	2	no	—
88	C₈×D₅₃	—	2	2	no	—
89	C₁₆×C₅₃	—	4	2	no	—
90	#11719908	—	4	2	no	—
91	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
92	C₂²×D₅₃	—	2	1	no	—
93	C₂²×D₅₃	—	2	1	no	—
94	C₂²×D₅₃	—	4	2	no	—
95	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×Dic₅₃	—	2	1	no	—
105	C₂×Dic₅₃	—	2	1	no	—
106	C₂×Dic₅₃	—	2	2	no	—
107	C₂×Dic₅₃	—	4	2	no	—
108	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
109	C₄×D₅₃	—	2	2	no	—
110	C₄×D₅₃	—	2	2	no	—
111	C₄×D₅₃	—	2	2	no	—
112	C₄×D₅₃	—	2	2	no	—
113	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
114	C₈×C₅₃	#121	1	1	yes	characteristic
115	C₈×C₅₃	#121	1	1	yes	characteristic
116	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
118	#88322	—	2	2	no	—
119	#88322	—	2	2	no	—
120	#12789	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	#407	1	1	yes	characteristic
123	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
124	C₂²×C₅₃	#432	1	1	yes	characteristic
125	C₂×D₅₃	—	2	2	no	—
126	C₂×D₅₃	—	2	4	no	—
127	C₂×D₅₃	—	4	2	no	—
128	C₂×D₅₃	—	4	2	no	—
129	C₄×C₅₃	—	4	2	no	—
130	C₄×C₅₃	#380	1	1	yes	characteristic
131	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₄×C₅₃	#423	1	1	yes	characteristic
133	C₄×C₅₃	#431	1	1	yes	characteristic
134	Dic₅₃	—	2	2	no	—
135	Dic₅₃	—	4	2	no	—
136	C₂×#405	—	53	1	no	—
137	C₂×#364	—	53	1	no	—
138	C₂×#364	D₅₃	1	1	yes	characteristic, complemented
139	#2112	—	53	2	no	—
140	#1376	—	53	2	no	—
141	C₂×C₅₃	—	2	1	no	—
142	C₂×C₅₃	—	2	1	no	—
143	C₂×C₅₃	#1797	1	1	yes	characteristic
144	C₂×C₅₃	#2153	1	2	yes	—
145	D₅₃	—	4	2	no	—
146	C₂²×M₁₆	—	53	1	no	—
147	C₂³×C₈	—	53	1	no	—
148	C₂×#117	—	53	1	no	—
149	C₂×M₃₂	—	53	2	no	—
150	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
151	#405	—	106	2	no	—
152	#405	—	106	2	no	—
153	C₂×#97	—	53	1	no	—
154	C₂×#97	—	53	1	no	—
155	C₂×C₄×C₈	—	53	1	no	—
156	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
157	#364	—	2	2	no	—
158	#364	—	106	2	no	—
159	C₂²⋊C₁₆	—	106	4	no	—
160	C₅₃	#12789	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	—	53	1	no	—
162	C₂×M₁₆	—	53	4	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	—	53	2	no	—
168	C₂²×C₈	—	53	2	no	—
169	C₂²×C₈	—	106	1	no	—
170	C₂²×C₈	—	106	1	no	—
171	C₂²×C₈	—	106	1	no	—
172	C₂²×C₈	—	106	1	no	—
173	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
175	#117	—	106	2	no	—
176	#117	—	106	2	no	—
177	C₂×C₄⋊C₄	—	106	1	no	—
178	C₂×C₂²⋊C₄	—	106	1	no	—
179	C₂×C₄²	—	53	1	no	—
180	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
181	M₃₂	—	2	2	no	—
182	M₃₂	—	2	2	no	—
183	M₃₂	—	106	2	no	—
184	M₃₂	—	106	2	no	—
185	C₂×C₁₆	—	2	2	no	—
186	C₂×C₁₆	—	106	2	no	—
187	#97	—	106	2	no	—
188	#97	—	106	2	no	—
189	C₄×C₈	—	106	1	no	—
190	C₄×C₈	—	106	1	no	—
191	C₄×C₈	D₂₁₂	1	2	yes	—
192	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
193	C₂⁴	—	53	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	53	1	no	—
196	C₂²×C₄	—	53	2	no	—
197	C₂²×C₄	—	53	2	no	—
198	C₂²×C₄	—	106	1	no	—
199	C₂²×C₄	—	106	1	no	—
200	C₂²×C₄	—	106	1	no	—
201	C₂²×C₄	—	106	1	no	—
202	C₂²×C₄	—	106	1	no	—
203	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
204	M₁₆	—	106	8	no	—
205	C₂×C₈	—	53	2	no	—
206	C₂×C₈	—	53	2	no	—
207	C₂×C₈	—	106	1	no	—
208	C₂×C₈	—	106	1	no	—
209	C₂×C₈	—	106	1	no	—
210	C₂×C₈	—	106	1	no	—
211	C₂×C₈	—	106	2	no	—
212	C₂×C₈	—	106	2	no	—
213	C₂×C₈	—	106	2	no	—
214	C₂×C₈	—	106	2	no	—
215	C₂×C₈	C₈×D₅₃	1	2	yes	—
216	C₂×C₈	#11719912	1	2	yes	—
217	C₂×C₈	#11719921	1	2	yes	—
218	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
219	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
220	C₂×C₈	#11719947	1	1	yes	characteristic
221	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
222	C₂×C₈	#11719949	1	1	yes	characteristic
223	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
224	C₄⋊C₄	—	106	4	no	—
225	C₂²⋊C₄	—	106	4	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	53	2	no	—
228	C₄²	—	106	1	no	—
229	C₄²	#11719921	1	2	yes	—
230	C₁₆	—	4	2	no	—
231	C₁₆	—	212	2	no	—
232	C₂³	—	53	2	no	—
233	C₂³	—	106	1	no	—
234	C₂³	—	106	1	no	—
235	C₂³	—	212	2	no	—
236	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	53	2	no	—
240	C₂×C₄	—	106	1	no	—
241	C₂×C₄	—	106	1	no	—
242	C₂×C₄	—	106	2	no	—
243	C₂×C₄	—	106	2	no	—
244	C₂×C₄	—	106	2	no	—
245	C₂×C₄	—	106	2	no	—
246	C₂×C₄	—	106	2	no	—
247	C₂×C₄	—	212	2	no	—
248	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
249	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
250	C₂×C₄	#13133175	1	1	yes	characteristic
251	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
252	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
253	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
254	C₈	—	106	2	no	—
255	C₈	—	106	2	no	—
256	C₈	#13133129	1	4	yes	—
257	C₈	#13133188	1	1	yes	characteristic
258	C₈	#13133192	1	1	yes	characteristic
259	C₈	#13133193	1	1	yes	characteristic
260	C₈	#13133197	1	1	yes	characteristic
261	C₂²	—	2	2	no	—
262	C₂²	—	106	2	no	—
263	C₂²	—	106	4	no	—
264	C₂²	—	212	2	no	—
265	C₂²	—	212	2	no	—
266	C₂²	#16810586	1	1	yes	characteristic
267	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
268	C₂²	#16810679	1	1	yes	characteristic
269	C₄	—	4	2	no	—
270	C₄	—	106	2	no	—
271	C₄	—	212	2	no	—
272	C₄	#16810480	1	1	yes	characteristic
273	C₄	C₂×#13133129	1	1	yes	characteristic
274	C₄	#16810645	1	1	yes	characteristic
275	C₄	#16810675	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	212	2	no	—
279	C₂	#20274692	1	1	yes	characteristic
280	C₂	#20276479	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#12789
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. C₂×#16810444	C₂	semidirect
4. #20272796	C₂	semidirect
5. C₂×C₅₃×#364	C₂	semidirect
138. C₂×#364	D₅₃	semidirect
160. C₅₃	#12789	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810577	2
C₂×#16810444	2
#20276325	2
#20272796	2
C₂×C₅₃×#364	2
#12789	53