Subgroups — #27392390

Order 13568.

742 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	C₄×#13133209	C₂	1	1	yes	maximal, characteristic
2	#20275431	C₂	1	2	yes	maximal
3	#20275270	C₂	1	4	yes	maximal
4	#20274580	C₂	1	2	yes	maximal
5	#20274576	C₂	1	4	yes	maximal
6	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133209	C₄	1	1	yes	characteristic
9	C₂×#13133146	C₂²	1	4	yes	—
10	C₂×C₄×#88322	C₂²	1	2	yes	—
11	C₂×C₄×#88322	C₂²	1	2	yes	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	4	yes	—
15	#16810521	C₄	1	2	yes	—
16	#16810507	C₄	1	4	yes	—
17	#16810489	C₄	1	1	yes	characteristic
18	C₄×#11719919	C₂²	1	4	yes	—
19	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
20	#16810377	C₂²	1	4	yes	—
21	#16810377	C₂²	1	4	yes	—
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
23	C₄³×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#336	C₂²	1	4	yes	—
25	C₂²×#88322	C₂³	1	1	yes	characteristic
26	C₂²×#88322	C₂³	1	2	yes	—
27	C₂²×#88322	C₂³	1	1	yes	characteristic
28	#13133209	C₈	1	8	yes	complemented
29	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
31	C₂×#11719919	C₂×C₄	1	4	yes	—
32	C₂×#11719919	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₅₃	C₂³	1	4	yes	—
34	#13133146	—	2	8	no	—
35	C₄×#88322	—	2	2	no	—
36	C₄×#88322	—	2	4	no	—
37	C₄×#88322	—	2	4	no	—
38	C₄×#88322	Q₈	1	4	yes	—
39	C₈×Dic₅₃	—	2	8	no	—
40	#13133124	C₂×C₄	1	4	yes	—
41	#13133124	C₂×C₄	1	4	yes	—
42	#13133124	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₅₃	—	2	4	no	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
55	C₂²×Dic₅₃	C₂²×C₄	1	4	yes	—
56	C₂²×Dic₅₃	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	—	2	4	no	—
59	C₂×C₈×C₅₃	—	2	4	no	—
60	C₄²×C₅₃	—	2	4	no	—
61	C₄²×C₅₃	—	2	2	no	—
62	C₄²×C₅₃	—	2	2	no	—
63	C₄²×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	C₂×C₈	1	4	yes	—
65	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
66	C₄²×C₅₃	#41	1	2	yes	—
67	C₄⋊Dic₅₃	—	2	4	no	—
68	C₄⋊Dic₅₃	C₂×C₈	1	8	yes	—
69	#11719919	—	2	8	no	—
70	#11719919	C₂×C₈	1	16	yes	—
71	C₄×Dic₅₃	—	2	4	no	—
72	C₄×Dic₅₃	—	2	4	no	—
73	C₂×#88322	—	2	2	no	—
74	C₂×#88322	—	2	2	no	—
75	C₂×#88322	—	2	2	no	—
76	C₂×#88322	C₂×Q₈	1	2	yes	—
77	C₂×#88322	C₂×Q₈	1	2	yes	—
78	C₂×#88322	#41	1	4	yes	—
79	C₂×#88322	#41	1	4	yes	—
80	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
87	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
88	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
89	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₅₃	#131	1	4	yes	—
91	C₂×C₄×C₅₃	#131	1	2	yes	—
92	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
96	C₂×Dic₅₃	—	2	4	no	—
97	C₂×Dic₅₃	—	2	4	no	—
98	C₂×Dic₅₃	—	2	4	no	—
99	C₂×Dic₅₃	—	2	4	no	—
100	C₂×Dic₅₃	—	2	4	no	—
101	C₂×Dic₅₃	C₂²×C₈	1	8	yes	—
102	C₈×C₅₃	—	4	4	no	—
103	#88322	—	2	4	no	—
104	#88322	—	2	2	no	—
105	#88322	#125	1	4	yes	complemented
106	C₈×#125	—	53	1	no	maximal, Sylow
107	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
108	C₂²×C₅₃	C₄×#41	1	2	yes	—
109	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
110	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
111	C₂²×C₅₃	C₂×#131	1	2	yes	—
112	C₄×C₅₃	—	2	2	no	—
113	C₄×C₅₃	—	2	4	no	—
114	C₄×C₅₃	—	2	4	no	—
115	C₄×C₅₃	C₈×Q₈	1	4	yes	—
116	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
117	C₄×C₅₃	#529	1	2	yes	—
118	C₄×C₅₃	#533	1	1	yes	characteristic
119	Dic₅₃	—	2	8	no	—
120	Dic₅₃	—	4	4	no	—
121	C₄×#125	—	53	1	no	—
122	#1939	—	53	2	no	—
123	#1916	—	53	4	no	—
124	#1831	D₅₃	1	1	yes	characteristic
125	C₈×C₄⋊C₄	—	53	2	no	—
126	C₈×C₄⋊C₄	—	53	4	no	—
127	C₄²×C₈	—	53	1	no	—
128	C₂×C₅₃	C₄×#125	1	1	yes	characteristic
129	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
130	C₂×C₅₃	#2960	1	1	yes	characteristic
131	C₂×C₅₃	C₈×#41	1	2	yes	—
132	C₂×C₅₃	#2966	1	2	yes	—
133	C₂×#125	—	53	1	no	—
134	C₂×C₄⋊C₈	—	53	4	no	—
135	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
136	C₂×C₄×C₈	—	53	1	no	—
137	C₂×C₄×C₈	—	53	4	no	—
138	C₂×C₄×C₈	—	53	2	no	—
139	C₂×C₄×C₈	—	53	1	no	—
140	C₂×C₄×C₈	—	53	2	no	—
141	#389	—	53	2	no	—
142	#387	—	53	4	no	—
143	#382	—	53	1	no	—
144	C₄×C₄⋊C₄	—	53	4	no	—
145	C₄×C₄⋊C₄	—	53	2	no	—
146	C₄³	C₂×D₅₃	1	1	yes	characteristic
147	#336	—	53	4	no	—
148	#336	—	53	4	no	—
149	#336	C₂×D₅₃	1	4	yes	—
150	C₅₃	C₈×#125	1	1	yes	characteristic, complemented, Sylow
151	C₂²×C₈	—	53	1	no	—
152	C₂²×C₈	—	53	1	no	—
153	C₂²×C₈	—	53	2	no	—
154	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
155	#125	—	53	8	no	—
156	C₂×C₄⋊C₄	—	53	4	no	—
157	C₂×C₄⋊C₄	—	53	2	no	—
158	C₂×C₄⋊C₄	—	53	4	no	—
159	C₂×C₄⋊C₄	—	53	2	no	—
160	C₂×C₄²	—	53	4	no	—
161	C₂×C₄²	C₄×D₅₃	1	2	yes	—
162	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
163	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
164	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
165	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	—	106	8	no	—
168	C₄×C₈	—	53	4	no	—
169	C₄×C₈	—	106	4	no	—
170	C₄×C₈	—	106	4	no	—
171	C₄×C₈	—	106	8	no	—
172	C₄×C₈	—	106	2	no	—
173	#95	—	53	4	no	—
174	#95	—	53	4	no	—
175	#95	—	53	4	no	—
176	C₂²×C₄	—	53	4	no	—
177	C₂²×C₄	—	53	4	no	—
178	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
180	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
181	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	53	2	no	—
187	C₂×C₈	—	53	4	no	—
188	C₂×C₈	—	53	2	no	—
189	C₂×C₈	—	53	4	no	—
190	C₂×C₈	—	106	2	no	—
191	C₂×C₈	—	106	2	no	—
192	C₂×C₈	—	106	2	no	—
193	C₄⋊C₄	—	53	16	no	—
194	C₄⋊C₄	—	53	8	no	—
195	C₄⋊C₄	—	106	8	no	—
196	C₄⋊C₄	—	106	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	106	4	no	—
202	C₄²	—	106	4	no	—
203	C₄²	C₈×D₅₃	1	4	yes	—
204	C₄²	#11719947	1	2	yes	—
205	C₄²	Q₈×D₅₃	1	2	yes	—
206	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	53	8	no	—
213	C₂×C₄	—	106	4	no	—
214	C₂×C₄	—	106	4	no	—
215	C₂×C₄	—	106	4	no	—
216	C₂×C₄	—	106	4	no	—
217	C₂×C₄	—	106	4	no	—
218	C₂×C₄	#13133182	1	2	yes	—
219	C₂×C₄	#13133182	1	2	yes	—
220	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
221	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
222	C₂×C₄	#13133225	1	4	yes	—
223	C₂×C₄	#13133225	1	2	yes	—
224	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
225	C₂×C₄	#41×D₅₃	1	2	yes	—
226	C₂×C₄	#41×D₅₃	1	2	yes	—
227	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
228	C₈	—	4	4	no	—
229	C₈	—	53	4	no	—
230	C₈	—	106	4	no	—
231	C₈	—	106	2	no	—
232	C₂²	C₂×#13133182	1	1	yes	characteristic
233	C₂²	#16811091	1	2	yes	—
234	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
235	C₂²	C₂×#13133225	1	2	yes	—
236	C₂²	C₂×#13133225	1	1	yes	characteristic
237	C₄	—	2	4	no	—
238	C₄	—	2	4	no	—
239	C₄	—	2	2	no	—
240	C₄	—	106	8	no	—
241	C₄	—	212	4	no	—
242	C₄	#16810747	1	4	yes	—
243	C₄	#125×D₅₃	1	1	yes	characteristic
244	C₄	#16811152	1	2	yes	—
245	C₄	#16811179	1	1	yes	characteristic
246	C₂	#20277270	1	1	yes	characteristic
247	C₂	C₂×#16810747	1	1	yes	characteristic
248	C₂	#20283645	1	1	yes	characteristic
249	C₂	#20283671	1	2	yes	—
250	C₂	#20283684	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#1831	C₂
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1831	D₅₃
hypocenter	C₅₃	C₈×#125
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1831

Complements

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

N	H ≅ G/N	type
28. #13133209	C₈	semidirect
105. #88322	#125	semidirect
150. C₅₃	C₈×#125	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133209	2
#20275431	2
#20275270	2
#20274580	2
#20274576	2
C₄²×#88322	2
C₅₃×#1831	2
C₈×#125	53