Subgroups — #27536672

Order 13568.

862 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811027	C₂	1	2	yes	maximal
2	C₂×#16811020	C₂	1	2	yes	maximal
3	#20283091	C₂	1	2	yes	maximal
4	#20274879	C₂	1	1	yes	maximal, characteristic
5	#20274878	C₂	1	2	yes	maximal
6	#20274877	C₂	1	2	yes	maximal
7	#20274876	C₂	1	2	yes	maximal
8	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic
10	C₂×#13133216	C₂²	1	4	yes	—
11	#16811027	—	2	8	no	—
12	#16811020	—	2	8	no	—
13	C₂×#13133151	C₂²	1	2	yes	—
14	C₂×#13133150	C₂²	1	4	yes	—
15	C₂×#13133149	C₂²	1	2	yes	—
16	C₂×#13133148	C₂²	1	4	yes	—
17	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
22	#16810527	—	2	8	no	—
23	#16810489	C₂²	1	2	yes	—
24	#16810488	C₂²	1	4	yes	—
25	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
26	C₂×#107×C₅₃	C₂²	1	2	yes	—
27	C₂×#106×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#382	C₂²	1	2	yes	—
30	C₂²×Dic₁₀₆	—	2	4	no	—
31	C₂²×#88322	C₂³	1	1	yes	characteristic
32	C₂²×#88322	C₂³	1	1	yes	characteristic
33	#13133216	—	2	8	no	—
34	#13133216	—	2	8	no	—
35	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
36	C₂×#11719919	—	2	4	no	—
37	C₂×C₄×Dic₅₃	D₄	1	4	yes	—
38	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
40	#13133151	—	2	4	no	—
41	#13133150	—	2	8	no	—
42	#13133149	—	2	4	no	—
43	#13133148	—	2	8	no	—
44	C₄×#88322	Q₈	1	4	yes	complemented
45	C₈×Dic₅₃	D₄	1	4	yes	—
46	C₈×Dic₅₃	D₄	1	4	yes	—
47	C₈×Dic₅₃	Q₈	1	4	yes	complemented
48	C₈×Dic₅₃	Q₈	1	4	yes	complemented
49	#13133124	—	2	4	no	—
50	#13133124	—	2	8	no	—
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	#107×C₅₃	—	2	4	no	—
56	#106×C₅₃	—	2	4	no	—
57	C₄×C₈×C₅₃	Q₈	1	4	yes	complemented
58	#95×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	—	2	4	no	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₅₃	—	2	4	no	—
63	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
64	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
65	C₂×Dic₁₀₆	—	4	8	no	—
66	C₂×Dic₁₀₆	—	4	16	no	—
67	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
73	C₄⋊C₄×C₅₃	—	2	4	no	—
74	C₄⋊C₄×C₅₃	—	2	4	no	—
75	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
76	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
77	C₄⋊Dic₅₃	—	2	4	no	—
78	C₄⋊Dic₅₃	—	2	4	no	—
79	#11719919	—	4	8	no	—
80	C₄×Dic₅₃	—	2	8	no	—
81	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
82	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
83	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
84	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
85	C₂×#88322	C₂×D₄	1	2	yes	—
86	C₂×#88322	C₂×D₄	1	2	yes	—
87	C₂×#88322	C₂×Q₈	1	2	yes	—
88	C₂×#88322	C₂×Q₈	1	2	yes	—
89	C₂×#88322	C₂×Q₈	1	2	yes	—
90	C₂×#88322	C₂×Q₈	1	2	yes	—
91	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	4	4	no	—
94	C₂×C₄×C₅₃	—	4	8	no	—
95	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
96	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
102	C₂×Dic₅₃	—	2	4	no	—
103	C₂×Dic₅₃	—	4	4	no	—
104	C₂×Dic₅₃	—	4	8	no	—
105	C₂×Dic₅₃	#127	1	4	yes	—
106	C₂×Dic₅₃	C₄⋊Q₈	1	4	yes	—
107	C₂×Dic₅₃	#135	1	8	yes	—
108	Dic₁₀₆	—	8	16	no	—
109	C₈×C₅₃	C₄⋊Q₈	1	4	yes	—
110	C₈×C₅₃	C₄⋊Q₈	1	4	yes	—
111	#88322	C₄⋊Q₈	1	4	yes	—
112	#88322	C₄⋊Q₈	1	4	yes	—
113	#33313	—	53	1	no	maximal, Sylow
114	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×#135	1	4	yes	—
118	C₄×C₅₃	—	2	4	no	—
119	C₄×C₅₃	—	8	4	no	—
120	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
124	Dic₅₃	—	2	4	no	—
125	Dic₅₃	—	8	4	no	—
126	Dic₅₃	#495	1	8	yes	—
127	C₂×#500	—	53	2	no	—
128	C₂×#498	—	53	2	no	—
129	#2862	—	53	2	no	—
130	#1846	—	53	2	no	—
131	#1846	—	53	2	no	—
132	#1846	—	53	1	no	—
133	#1846	—	53	2	no	—
134	#1846	D₅₃	1	1	yes	characteristic
135	C₄²×C₈	—	53	1	no	—
136	C₂×C₅₃	#2868	1	1	yes	characteristic
137	C₂×C₅₃	C₂×#495	1	2	yes	—
138	C₂×C₅₃	#3160	1	2	yes	—
139	C₂×C₅₃	#3160	1	2	yes	—
140	C₂×C₄⋊Q₈	—	53	4	no	—
141	#500	—	106	8	no	—
142	#498	—	106	8	no	—
143	C₂×#107	—	53	2	no	—
144	C₂×#107	—	53	4	no	—
145	C₂×#107	C₂×D₅₃	1	2	yes	—
146	C₂×#106	—	53	2	no	—
147	C₂×#106	—	53	4	no	—
148	C₂×#106	C₂×D₅₃	1	2	yes	—
149	C₂×C₄×C₈	—	53	1	no	—
150	C₂×C₄×C₈	—	53	1	no	—
151	C₂×C₄×C₈	—	53	1	no	—
152	C₂×C₄×C₈	—	53	1	no	—
153	C₂×C₄×C₈	—	53	1	no	—
154	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
155	#391	—	106	8	no	—
156	#382	—	53	4	no	—
157	#382	—	53	2	no	—
158	#382	C₂×D₅₃	1	2	yes	—
159	C₄³	—	53	1	no	—
160	C₅₃	#33313	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₈	—	106	4	no	—
162	C₂²×C₈	—	53	1	no	—
163	C₂²×C₈	—	53	1	no	—
164	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
166	C₄⋊Q₈	—	106	8	no	—
167	C₄⋊Q₈	—	106	8	no	—
168	C₂×C₄⋊C₄	—	53	4	no	—
169	C₂×C₄⋊C₄	—	106	4	no	—
170	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
171	C₂×C₄²	—	53	1	no	—
172	C₂×C₄²	—	53	1	no	—
173	C₂×C₄²	—	53	4	no	—
174	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
175	#107	—	2	4	no	—
176	#107	—	106	8	no	—
177	#107	—	106	4	no	—
178	#106	—	2	4	no	—
179	#106	—	106	8	no	—
180	#106	—	106	4	no	—
181	C₄×C₈	—	53	4	no	—
182	C₄×C₈	—	53	4	no	—
183	C₄×C₈	—	53	4	no	—
184	C₄×C₈	—	53	4	no	—
185	C₄×C₈	—	53	4	no	—
186	C₄×C₈	Dic₁₀₆	1	4	yes	complemented
187	#95	—	2	4	no	—
188	#95	—	106	8	no	—
189	#95	—	106	4	no	—
190	C₂×Q₈	—	212	8	no	—
191	C₂×Q₈	—	212	16	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	53	2	no	—
194	C₂²×C₄	—	53	2	no	—
195	C₂²×C₄	—	106	4	no	—
196	C₂²×C₄	D₄×D₅₃	1	2	yes	—
197	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
198	C₂×C₈	—	53	2	no	—
199	C₂×C₈	—	53	2	no	—
200	C₂×C₈	—	53	2	no	—
201	C₂×C₈	—	53	2	no	—
202	C₂×C₈	—	53	2	no	—
203	C₂×C₈	—	53	2	no	—
204	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
205	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
206	C₂×C₈	D₄×D₅₃	1	2	yes	—
207	C₂×C₈	D₄×D₅₃	1	2	yes	—
208	C₂×C₈	Q₈×D₅₃	1	2	yes	—
209	C₂×C₈	Q₈×D₅₃	1	2	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	106	4	no	—
213	C₄⋊C₄	—	106	4	no	—
214	C₄⋊C₄	—	212	8	no	—
215	C₄²	—	53	2	no	—
216	C₄²	—	53	2	no	—
217	C₄²	—	53	2	no	—
218	C₄²	—	53	2	no	—
219	C₄²	—	106	8	no	—
220	C₄²	C₂×Dic₁₀₆	1	2	yes	—
221	C₄²	Q₈×D₅₃	1	2	yes	—
222	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
223	Q₈	—	424	16	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	53	8	no	—
228	C₂×C₄	—	53	4	no	—
229	C₂×C₄	—	53	4	no	—
230	C₂×C₄	—	106	4	no	—
231	C₂×C₄	—	212	8	no	—
232	C₂×C₄	—	212	4	no	—
233	C₂×C₄	#13133216	1	4	yes	—
234	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
239	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
240	C₈	—	53	4	no	—
241	C₈	—	53	4	no	—
242	C₈	#13133216	1	4	yes	—
243	C₈	#13133216	1	4	yes	—
244	C₂²	#127×D₅₃	1	1	yes	characteristic
245	C₂²	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
246	C₂²	C₂×#13133216	1	1	yes	characteristic
247	C₂²	#135×D₅₃	1	4	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	8	4	no	—
250	C₄	—	53	8	no	—
251	C₄	—	106	4	no	—
252	C₄	—	424	4	no	—
253	C₄	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
254	C₄	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
255	C₄	C₂×#13133216	1	1	yes	characteristic
256	C₄	C₂×#13133216	1	1	yes	characteristic
257	C₂	#20283112	1	1	yes	characteristic
258	C₂	#495×D₅₃	1	2	yes	—
259	C₂	#20285227	1	2	yes	—
260	C₂	#20285228	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1846	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#1846	D₅₃
hypocenter	C₅₃	#33313
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
44. C₄×#88322	Q₈	semidirect
47. C₈×Dic₅₃	Q₈	semidirect
48. C₈×Dic₅₃	Q₈	semidirect
57. C₄×C₈×C₅₃	Q₈	semidirect
160. C₅₃	#33313	semidirect
186. C₄×C₈	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811027	2
C₂×#16811020	2
#20283091	2
#20274879	2
#20274878	2
#20274877	2
#20274876	2
C₄×C₈×Dic₅₃	2
C₅₃×#1846	2
#33313	53