Subgroups — #27273967

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 252 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	#20272583	C₂	1	4	yes	maximal
3	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
4	#16810491	C₄	1	4	yes	complemented
5	#16810489	—	2	2	no	—
6	#16810486	—	2	2	no	—
7	#16810483	C₂²	1	4	yes	—
8	C₄×C₄⋊Dic₅₃	—	2	2	no	—
9	C₄×C₂²⋊Dic₅₃	—	2	2	no	—
10	#16810469	C₄	1	4	yes	complemented
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#431	—	2	2	no	—
13	C₅₃×#431	—	2	2	no	—
14	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	complemented
15	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	complemented
17	C₂×C₄⋊Dic₅₃	—	2	4	no	—
18	C₂×C₂²⋊Dic₅₃	—	2	4	no	—
19	C₂×C₄×Dic₅₃	—	2	4	no	—
20	#13133124	—	2	4	no	—
21	#13133124	—	2	4	no	—
22	#13133124	—	2	4	no	—
23	#13133124	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	D₄	1	2	yes	—
33	C₂×C₄²×C₅₃	D₄	1	2	yes	—
34	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	—	2	4	no	—
37	C₂²⋊C₈×C₅₃	—	2	4	no	—
38	C₄×C₈×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	—	2	4	no	—
42	C₂²×C₄×C₅₃	—	4	2	no	—
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
56	C₂²×Dic₅₃	—	2	8	no	—
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	—	4	4	no	—
59	C₄²×C₅₃	C₄²	1	2	yes	—
60	C₄²×C₅₃	C₄²	1	1	yes	characteristic
61	C₄²×C₅₃	C₄²	1	1	yes	characteristic
62	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₂²⋊Dic₅₃	—	4	8	no	—
70	C₄⋊Dic₅₃	—	4	8	no	—
71	C₄×Dic₅₃	—	4	8	no	—
72	C₂³×C₅₃	—	4	1	no	—
73	C₂³×C₅₃	—	4	1	no	—
74	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
75	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂³×C₅₃	#135	1	4	yes	—
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	4	1	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	#95	1	2	yes	—
90	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#95	1	2	yes	—
94	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×Dic₅₃	—	4	8	no	—
99	C₂×Dic₅₃	—	4	8	no	—
100	C₂×Dic₅₃	—	4	8	no	—
101	C₈×C₅₃	—	4	4	no	—
102	#7980	—	53	1	no	maximal, Sylow
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	4	4	no	—
108	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
109	C₂²×C₅₃	#416	1	4	yes	—
110	C₂²×C₅₃	#427	1	2	yes	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	—	2	2	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	H(3,ℤ/4)	1	2	yes	—
119	C₄×C₅₃	#410	1	2	yes	—
120	Dic₅₃	—	8	8	no	—
121	C₂×#431	D₅₃	1	1	yes	characteristic
122	#2283	—	53	2	no	—
123	#1289	—	53	4	no	—
124	C₂×C₅₃	—	2	4	no	—
125	C₂×C₅₃	—	2	2	no	—
126	C₂×C₅₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
127	C₂×C₅₃	#1735	1	1	yes	characteristic
128	C₂×C₅₃	#1740	1	1	yes	characteristic
129	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
130	#431	—	2	2	no	—
131	#431	—	2	2	no	—
132	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	complemented
133	C₂×C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
134	C₂×C₄×C₈	Dic₅₃	1	2	yes	complemented
135	#383	—	53	4	no	—
136	#382	—	106	2	no	—
137	#381	—	106	2	no	—
138	#380	—	53	4	no	—
139	C₄×C₄⋊C₄	—	106	2	no	—
140	C₄×C₂²⋊C₄	—	106	2	no	—
141	#376	—	53	4	no	—
142	C₅₃	#7980	1	1	yes	characteristic, complemented, Sylow
143	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
144	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
145	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
146	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
147	C₂×C₄⋊C₄	—	106	4	no	—
148	C₂×C₂²⋊C₄	—	106	4	no	—
149	C₂×C₄²	—	106	4	no	—
150	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
151	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
152	C₂×C₄²	C₄×D₅₃	1	2	yes	—
153	C₂×C₄²	C₄×D₅₃	1	2	yes	—
154	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
155	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
156	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
157	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
158	C₄⋊C₈	—	2	4	no	—
159	C₂²⋊C₈	—	2	4	no	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	—	2	2	no	—
162	#95	—	53	8	no	—
163	#95	—	106	4	no	—
164	#95	—	106	4	no	—
165	#95	—	106	4	no	—
166	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	—	106	8	no	—
171	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
174	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
175	C₂²×C₄	#11719921	1	2	yes	—
176	C₂²×C₄	#11719921	1	2	yes	—
177	C₂²×C₄	#11719921	1	2	yes	—
178	C₂²×C₄	#11719921	1	2	yes	—
179	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	4	4	no	—
185	C₄⋊C₄	—	212	8	no	—
186	C₂²⋊C₄	—	212	8	no	—
187	C₄²	—	212	8	no	—
188	C₄²	C₄×Dic₅₃	1	2	yes	—
189	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
190	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
191	C₄²	#11719919	1	2	yes	—
192	C₄²	#11719919	1	2	yes	—
193	C₄²	#11719919	1	2	yes	—
194	C₄²	#11719919	1	2	yes	—
195	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
196	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
197	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
198	C₂³	—	4	1	no	—
199	C₂³	—	4	1	no	—
200	C₂³	C₂×#11719921	1	2	yes	—
201	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
202	C₂³	#13133239	1	4	yes	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	212	8	no	—
216	C₂×C₄	—	212	8	no	—
217	C₂×C₄	—	212	8	no	—
218	C₂×C₄	#13133124	1	1	yes	characteristic
219	C₂×C₄	#13133124	1	2	yes	—
220	C₂×C₄	#13133124	1	1	yes	characteristic
221	C₂×C₄	#13133124	1	1	yes	characteristic
222	C₂×C₄	#13133124	1	2	yes	—
223	C₂×C₄	#13133124	1	1	yes	characteristic
224	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
225	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
226	C₂×C₄	C₂×#11719919	1	2	yes	—
227	C₈	—	4	4	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	4	4	no	—
233	C₂²	C₂×#13133124	1	1	yes	characteristic
234	C₂²	#16810620	1	4	yes	—
235	C₂²	#16810661	1	2	yes	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	1	no	—
243	C₄	—	424	8	no	—
244	C₄	#16810380	1	2	yes	—
245	C₄	#16810597	1	2	yes	—
246	C₂	—	2	4	no	—
247	C₂	—	2	2	no	—
248	C₂	C₂×#16810380	1	1	yes	characteristic
249	C₂	#20274432	1	1	yes	characteristic
250	C₂	#20274448	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133124
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#7980
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
4. #16810491	C₄	semidirect
10. #16810469	C₄	semidirect
14. C₂×C₄⋊C₈×C₅₃	C₄	semidirect
16. C₂×C₄×C₈×C₅₃	C₄	semidirect
132. C₂×C₄⋊C₈	Dic₅₃	semidirect
134. C₂×C₄×C₈	Dic₅₃	semidirect
142. C₅₃	#7980	semidirect

Maximal subgroups

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

subgroup	index
#20277113	2
#20272583	2
C₂×C₅₃×#431	2
#7980	53