Subgroups — #27299089

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆×Dic₅₃	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272793	C₂	1	1	yes	maximal, characteristic
4	#20272792	C₂	1	2	yes	maximal
5	C₅₃×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
13	C₁₆×Dic₅₃	C₄	1	8	yes	complemented
14	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
17	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
19	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
20	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	C₂×C₄	1	4	yes	—
25	C₄×#88322	—	2	2	no	—
26	C₈×Dic₅₃	—	2	1	no	—
27	C₈×Dic₅₃	—	2	1	no	—
28	C₈×Dic₅₃	—	2	2	no	—
29	C₈×Dic₅₃	C₈	1	4	yes	—
30	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	—	2	2	no	—
35	C₂×C₁₆×C₅₃	—	2	2	no	—
36	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
41	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₅₃	#41	1	2	yes	—
48	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
49	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
50	C₄²×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	—	2	1	no	—
52	C₄×Dic₅₃	—	2	1	no	—
53	C₄×Dic₅₃	—	2	1	no	—
54	C₄×Dic₅₃	—	2	4	no	—
55	C₄×Dic₅₃	C₁₆	1	8	yes	complemented
56	C₄×Dic₅₃	C₄²	1	2	yes	—
57	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
58	C₂×#88322	—	2	1	no	—
59	C₂×#88322	—	2	1	no	—
60	C₂×#88322	C₄²	1	2	yes	—
61	C₂×#88322	C₂×C₈	1	2	yes	—
62	C₂×#88322	M₁₆	1	2	yes	—
63	C₂×#88322	M₁₆	1	2	yes	—
64	C₁₆×C₅₃	—	2	4	no	—
65	#11719908	—	2	4	no	—
66	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
70	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×Dic₅₃	—	2	2	no	—
77	C₂×Dic₅₃	—	2	2	no	—
78	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
79	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
80	C₂×Dic₅₃	C₂×C₁₆	1	4	yes	—
81	C₂×Dic₅₃	C₂×C₁₆	1	4	yes	—
82	C₂×Dic₅₃	M₃₂	1	4	yes	—
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	#117	1	2	yes	—
85	C₈×C₅₃	#117	1	2	yes	—
86	#88322	—	2	2	no	—
87	#88322	#97	1	4	yes	—
88	#12590	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
90	C₂²×C₅₃	#431	1	1	yes	characteristic
91	C₂²×C₅₃	#431	1	1	yes	characteristic
92	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
93	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
94	C₄×C₅₃	—	2	4	no	—
95	C₄×C₅₃	#376	1	1	yes	characteristic
96	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
97	C₄×C₅₃	#431	1	1	yes	characteristic
98	C₄×C₅₃	#431	1	1	yes	characteristic
99	Dic₅₃	—	2	4	no	—
100	Dic₅₃	C₄×C₁₆	1	4	yes	—
101	Dic₅₃	#346	1	4	yes	—
102	C₂×C₄×C₁₆	—	53	2	no	—
103	C₄²×C₈	—	53	1	no	—
104	#1374	—	53	2	no	—
105	#1374	—	53	1	no	—
106	#1374	D₅₃	1	1	yes	characteristic
107	C₂×C₅₃	#1744	1	1	yes	characteristic
108	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
109	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
110	C₂×C₅₃	#2162	1	2	yes	—
111	C₂×C₅₃	#2162	1	2	yes	—
112	C₂²×C₁₆	—	53	2	no	—
113	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
114	C₂×C₄×C₈	—	53	1	no	—
115	C₂×C₄×C₈	—	53	1	no	—
116	C₂×C₄×C₈	—	53	1	no	—
117	C₂×C₄×C₈	—	53	1	no	—
118	C₂×C₄×C₈	—	53	1	no	—
119	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
120	C₄³	—	53	1	no	—
121	C₄×C₁₆	—	53	8	no	—
122	C₅₃	#12590	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₈	—	53	1	no	—
124	C₂²×C₈	—	53	1	no	—
125	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
127	C₂×C₄²	—	53	1	no	—
128	C₂×C₄²	—	53	2	no	—
129	C₂×C₄²	—	53	1	no	—
130	C₂×C₄²	—	53	2	no	—
131	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
132	C₂×C₁₆	—	2	2	no	—
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	—	53	4	no	—
135	C₂×C₁₆	—	106	2	no	—
136	C₂×C₁₆	—	106	2	no	—
137	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	53	4	no	—
140	C₄×C₈	—	53	4	no	—
141	C₄×C₈	—	106	2	no	—
142	C₄×C₈	—	106	2	no	—
143	C₄×C₈	—	106	1	no	—
144	C₄×C₈	—	106	1	no	—
145	C₂²×C₄	—	53	1	no	—
146	C₂²×C₄	—	53	1	no	—
147	C₂²×C₄	—	53	2	no	—
148	C₂²×C₄	C₈×D₅₃	1	2	yes	—
149	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	53	2	no	—
153	C₂×C₈	—	53	2	no	—
154	C₂×C₈	—	53	2	no	—
155	C₂×C₈	—	53	2	no	—
156	C₂×C₈	—	106	1	no	—
157	C₂×C₈	—	106	1	no	—
158	C₂×C₈	C₈×D₅₃	1	2	yes	—
159	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
160	C₂×C₈	#11719947	1	1	yes	characteristic
161	C₂×C₈	#11719947	1	2	yes	—
162	C₂×C₈	#11719949	1	1	yes	characteristic
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₄²	—	53	8	no	—
166	C₄²	—	53	2	no	—
167	C₄²	—	53	2	no	—
168	C₄²	—	106	4	no	—
169	C₄²	—	106	1	no	—
170	C₄²	—	106	1	no	—
171	C₁₆	—	2	4	no	—
172	C₁₆	—	106	4	no	—
173	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	53	4	no	—
177	C₂×C₄	—	53	4	no	—
178	C₂×C₄	—	53	2	no	—
179	C₂×C₄	—	53	2	no	—
180	C₂×C₄	—	53	4	no	—
181	C₂×C₄	—	106	2	no	—
182	C₂×C₄	—	106	2	no	—
183	C₂×C₄	C₁₆×D₅₃	1	4	yes	—
184	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
185	C₂×C₄	#13133173	1	1	yes	characteristic
186	C₂×C₄	#13133175	1	1	yes	characteristic
187	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
188	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
189	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
190	C₈	—	2	2	no	—
191	C₈	—	53	4	no	—
192	C₈	—	106	2	no	—
193	C₈	#13133173	1	2	yes	—
194	C₈	#13133175	1	2	yes	—
195	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
196	C₂²	#16810673	1	1	yes	characteristic
197	C₂²	#16810675	1	1	yes	characteristic
198	C₂²	C₂×C₁₆×D₅₃	1	2	yes	—
199	C₂²	M₃₂×D₅₃	1	2	yes	—
200	C₄	—	2	4	no	—
201	C₄	—	53	4	no	—
202	C₄	—	53	4	no	—
203	C₄	—	106	4	no	—
204	C₄	#16810468	1	1	yes	characteristic
205	C₄	#97×D₅₃	1	1	yes	characteristic
206	C₄	#16810673	1	1	yes	characteristic
207	C₄	#16810675	1	1	yes	characteristic
208	C₂	#20274459	1	1	yes	characteristic
209	C₂	C₄×C₁₆×D₅₃	1	1	yes	characteristic
210	C₂	#346×D₅₃	1	1	yes	characteristic
211	C₂	#20276511	1	2	yes	—
212	C₂	#20276513	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Fitting	C₅₃×#1374	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1374	D₅₃
hypocenter	C₅₃	#12590
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
13. C₁₆×Dic₅₃	C₄	semidirect
55. C₄×Dic₅₃	C₁₆	semidirect
122. C₅₃	#12590	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₆×Dic₅₃	2
C₄×C₈×Dic₅₃	2
#20272793	2
#20272792	2
C₅₃×#1374	2
#12590	53