Subgroups — #27536663

Order 13568.

830 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811024	C₂	1	4	yes	maximal
2	#20283076	C₂	1	2	yes	maximal
3	#20274878	C₂	1	4	yes	maximal
4	#20274875	C₂	1	1	yes	maximal, characteristic
5	#20274871	C₂	1	2	yes	maximal
6	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133208	C₂²	1	4	yes	—
9	#16811024	—	2	16	no	—
10	C₂×#13133151	C₂²	1	4	yes	—
11	C₂×#13133148	C₂²	1	8	yes	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
14	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
15	#16810522	—	2	4	no	—
16	#16810519	—	2	4	no	—
17	#16810493	C₂²	1	2	yes	—
18	#16810488	C₂²	1	4	yes	—
19	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
20	C₂×#107×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#384	C₂²	1	2	yes	—
23	C₂²×#88322	C₂³	1	2	yes	—
24	#13133208	—	2	16	no	—
25	C₂×C₄⋊Dic₅₃	—	2	4	no	—
26	C₂×C₄⋊Dic₅₃	—	2	4	no	—
27	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
28	C₂×#11719919	—	2	4	no	—
29	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
30	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
31	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
33	#13133151	—	2	8	no	—
34	#13133148	—	2	16	no	—
35	C₄×#88322	Q₈	1	4	yes	—
36	C₈×Dic₅₃	D₄	1	8	yes	—
37	C₈×Dic₅₃	Q₈	1	8	yes	—
38	#13133124	—	2	8	no	—
39	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
41	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	#107×C₅₃	—	2	8	no	—
44	C₄×C₈×C₅₃	Q₈	1	4	yes	—
45	C₂²×C₄×C₅₃	—	2	4	no	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₂²×Dic₅₃	—	2	4	no	—
49	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
53	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
54	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
55	C₄⋊C₄×C₅₃	—	2	8	no	—
56	C₄⋊C₄×C₅₃	—	4	8	no	—
57	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
58	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
59	C₄⋊Dic₅₃	—	2	8	no	—
60	C₄⋊Dic₅₃	—	4	8	no	—
61	C₄⋊Dic₅₃	—	4	8	no	—
62	#11719919	—	4	8	no	—
63	C₄×Dic₅₃	—	2	4	no	—
64	C₄×Dic₅₃	—	2	4	no	—
65	C₄×Dic₅₃	C₂×D₄	1	4	yes	—
66	C₄×Dic₅₃	C₂×Q₈	1	4	yes	—
67	C₂×#88322	C₂×D₄	1	4	yes	—
68	C₂×#88322	C₂×Q₈	1	4	yes	—
69	C₂×#88322	C₂×Q₈	1	4	yes	—
70	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	4	4	no	—
72	C₂×C₄×C₅₃	—	4	8	no	—
73	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
74	C₂×C₄×C₅₃	#135	1	4	yes	—
75	C₂×C₄×C₅₃	#135	1	4	yes	—
76	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
79	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
80	C₂×Dic₅₃	—	2	2	no	—
81	C₂×Dic₅₃	—	2	2	no	—
82	C₂×Dic₅₃	—	2	2	no	—
83	C₂×Dic₅₃	—	2	2	no	—
84	C₂×Dic₅₃	—	4	4	no	—
85	C₂×Dic₅₃	—	4	8	no	—
86	C₂×Dic₅₃	#127	1	2	yes	—
87	C₂×Dic₅₃	#127	1	2	yes	—
88	C₂×Dic₅₃	C₄⋊Q₈	1	4	yes	—
89	C₈×C₅₃	C₄⋊Q₈	1	8	yes	—
90	#88322	C₄⋊Q₈	1	8	yes	—
91	#33311	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
93	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
94	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
95	C₂²×C₅₃	C₂×#135	1	2	yes	—
96	C₂²×C₅₃	C₂×#135	1	2	yes	—
97	C₄×C₅₃	—	8	4	no	—
98	C₄×C₅₃	#499	1	8	yes	complemented
99	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
100	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
101	Dic₅₃	—	2	2	no	—
102	Dic₅₃	—	2	2	no	—
103	Dic₅₃	—	2	4	no	—
104	Dic₅₃	—	8	4	no	—
105	C₂×#499	—	53	4	no	—
106	#2859	—	53	2	no	—
107	#1846	—	53	4	no	—
108	#1845	—	53	1	no	—
109	#1845	D₅₃	1	1	yes	characteristic
110	#1844	—	53	2	no	—
111	C₄²×C₈	—	53	1	no	—
112	C₂×C₅₃	#2868	1	1	yes	characteristic
113	C₂×C₅₃	#3147	1	1	yes	characteristic
114	C₂×C₅₃	#3147	1	1	yes	characteristic
115	C₂×C₅₃	C₂×#499	1	2	yes	—
116	C₂×C₅₃	#3160	1	2	yes	—
117	C₂×#125	—	53	4	no	—
118	#499	—	106	16	no	—
119	C₂×#107	—	53	4	no	—
120	C₂×#107	C₂×D₅₃	1	4	yes	—
121	C₂×#106	—	53	8	no	—
122	C₂×C₄×C₈	—	53	2	no	—
123	C₂×C₄×C₈	—	53	2	no	—
124	C₂×C₄×C₈	—	53	1	no	—
125	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
126	#389	—	106	4	no	—
127	#389	—	106	4	no	—
128	#384	—	53	2	no	—
129	#384	C₂×D₅₃	1	2	yes	—
130	#382	—	53	4	no	—
131	C₄³	—	53	1	no	—
132	C₅₃	#33311	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₈	—	53	2	no	—
134	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
135	#125	—	106	16	no	—
136	C₂×C₄⋊C₄	—	2	4	no	—
137	C₂×C₄⋊C₄	—	53	4	no	—
138	C₂×C₄⋊C₄	—	106	4	no	—
139	C₂×C₄⋊C₄	—	106	4	no	—
140	C₂×C₄⋊C₄	—	106	4	no	—
141	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
142	C₂×C₄²	—	53	1	no	—
143	C₂×C₄²	—	53	1	no	—
144	C₂×C₄²	—	53	2	no	—
145	C₂×C₄²	—	53	2	no	—
146	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
147	#107	—	2	8	no	—
148	#107	—	106	8	no	—
149	#106	—	106	16	no	—
150	C₄×C₈	—	53	8	no	—
151	C₄×C₈	—	53	8	no	—
152	C₄×C₈	—	53	4	no	—
153	C₄×C₈	Dic₁₀₆	1	4	yes	—
154	#95	—	106	8	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	53	2	no	—
157	C₂²×C₄	—	53	1	no	—
158	C₂²×C₄	—	53	1	no	—
159	C₂²×C₄	—	106	4	no	—
160	C₂²×C₄	D₄×D₅₃	1	2	yes	—
161	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
162	C₂×C₈	—	53	4	no	—
163	C₂×C₈	—	53	4	no	—
164	C₂×C₈	—	53	4	no	—
165	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
166	C₂×C₈	D₄×D₅₃	1	4	yes	—
167	C₂×C₈	Q₈×D₅₃	1	4	yes	—
168	C₄⋊C₄	—	2	8	no	—
169	C₄⋊C₄	—	4	8	no	—
170	C₄⋊C₄	—	106	8	no	—
171	C₄⋊C₄	—	212	8	no	—
172	C₄⋊C₄	—	212	8	no	—
173	C₄⋊C₄	—	212	8	no	—
174	C₄²	—	53	4	no	—
175	C₄²	—	53	4	no	—
176	C₄²	—	106	4	no	—
177	C₄²	—	106	4	no	—
178	C₄²	C₂×Dic₁₀₆	1	2	yes	—
179	C₄²	Q₈×D₅₃	1	2	yes	—
180	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
181	C₂×C₄	—	4	8	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	53	2	no	—
184	C₂×C₄	—	53	2	no	—
185	C₂×C₄	—	53	4	no	—
186	C₂×C₄	—	106	2	no	—
187	C₂×C₄	—	106	2	no	—
188	C₂×C₄	—	106	2	no	—
189	C₂×C₄	—	106	2	no	—
190	C₂×C₄	—	212	8	no	—
191	C₂×C₄	—	212	4	no	—
192	C₂×C₄	#13133216	1	4	yes	—
193	C₂×C₄	#13133236	1	4	yes	—
194	C₂×C₄	#13133240	1	4	yes	—
195	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
196	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
197	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
198	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
199	C₈	—	53	8	no	—
200	C₈	#13133216	1	8	yes	—
201	C₂²	#127×D₅₃	1	1	yes	characteristic
202	C₂²	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
203	C₂²	C₂×#13133216	1	1	yes	characteristic
204	C₂²	C₂×#13133236	1	2	yes	—
205	C₂²	C₂×#13133240	1	2	yes	—
206	C₄	—	8	4	no	—
207	C₄	—	106	4	no	—
208	C₄	—	106	2	no	—
209	C₄	—	106	2	no	—
210	C₄	—	424	4	no	—
211	C₄	#16811024	1	8	yes	complemented
212	C₄	C₄⋊Q₈×D₅₃	1	2	yes	—
213	C₄	C₂×#13133216	1	2	yes	—
214	C₂	#20283112	1	1	yes	characteristic
215	C₂	#20285142	1	1	yes	characteristic
216	C₂	#20285144	1	1	yes	characteristic
217	C₂	C₂×#16811024	1	2	yes	—
218	C₂	#20285226	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1845	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#1845	D₅₃
hypocenter	C₅₃	#33311
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
98. C₄×C₅₃	#499	semidirect
132. C₅₃	#33311	semidirect
211. C₄	#16811024	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811024	2
#20283076	2
#20274878	2
#20274875	2
#20274871	2
C₄×C₈×Dic₅₃	2
C₅₃×#1845	2
#33311	53