Subgroups — #27277752

Order 13568.

422 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	#20272549	C₂	1	4	yes	maximal
4	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	1	no	—
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₄	1	2	yes	—
9	C₂×C₄×#88322	C₄	1	1	yes	characteristic
10	C₂×C₄×#88322	C₄	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
13	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	—	2	2	no	—
17	C₂²×#88322	C₂×C₄	1	2	yes	—
18	C₂²×#88322	C₂×C₄	1	2	yes	—
19	C₄⋊#88322	—	2	4	no	—
20	C₄⋊#88322	—	4	2	no	—
21	C₄⋊#88322	—	4	2	no	—
22	C₄×#88322	—	2	2	no	—
23	C₄×#88322	—	2	2	no	—
24	C₄×#88322	—	2	2	no	—
25	C₄×#88322	—	2	4	no	—
26	C₄×#88322	C₈	1	4	yes	complemented
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	—	2	4	no	—
36	C₄⋊C₈×C₅₃	—	4	2	no	—
37	C₄⋊C₈×C₅₃	—	4	2	no	—
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₅₃	—	4	2	no	—
46	C₂×C₈×C₅₃	—	4	2	no	—
47	C₂×C₈×C₅₃	—	4	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	2	2	no	—
50	C₄²×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	C₂×C₈	1	2	yes	—
52	C₄²×C₅₃	M₁₆	1	2	yes	—
53	C₄²×C₅₃	D₈	1	2	yes	—
54	C₄²×C₅₃	SD₁₆	1	2	yes	—
55	C₄²×C₅₃	SD₁₆	1	2	yes	—
56	C₄²×C₅₃	Q₁₆	1	2	yes	—
57	C₄²×C₅₃	#41	1	2	yes	—
58	C₄²×C₅₃	#41	1	2	yes	—
59	C₂×#88322	—	2	2	no	—
60	C₂×#88322	—	2	2	no	—
61	C₂×#88322	—	2	2	no	—
62	C₂×#88322	—	2	2	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂×#88322	—	4	2	no	—
65	C₂×#88322	—	4	2	no	—
66	C₂×#88322	C₂×C₈	1	4	yes	—
67	C₂×#88322	M₁₆	1	4	yes	—
68	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
76	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
77	C₂×C₄×C₅₃	#106	1	2	yes	—
78	C₂×C₄×C₅₃	#106	1	2	yes	—
79	C₂×C₄×C₅₃	#107	1	2	yes	—
80	C₂×C₄×C₅₃	#107	1	2	yes	—
81	C₂×C₄×C₅₃	#108	1	2	yes	—
82	C₂×C₄×C₅₃	#108	1	2	yes	—
83	C₂×C₄×C₅₃	#117	1	2	yes	—
84	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#125	1	2	yes	—
87	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
94	C₈×C₅₃	—	8	2	no	—
95	#88322	—	2	4	no	—
96	#88322	—	2	4	no	—
97	#88322	—	8	2	no	—
98	#8784	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	#382	1	1	yes	characteristic
100	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₂²×C₅₃	#423	1	1	yes	characteristic
102	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
103	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
104	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
105	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	#334	1	4	yes	—
110	C₄×C₅₃	#335	1	4	yes	—
111	C₄×C₅₃	#431	1	2	yes	—
112	C₄×C₅₃	#432	1	2	yes	—
113	C₄×C₅₃	#486	1	1	yes	characteristic
114	C₄×C₅₃	#486	1	1	yes	characteristic
115	C₄×C₅₃	#487	1	1	yes	characteristic
116	C₄×C₅₃	#487	1	1	yes	characteristic
117	#1842	—	53	1	no	—
118	#1842	D₅₃	1	1	yes	characteristic
119	C₄²×C₈	—	53	1	no	—
120	#1277	—	53	4	no	—
121	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
122	C₂×C₅₃	C₂×#335	1	1	yes	characteristic
123	C₂×C₅₃	#1566	1	1	yes	characteristic
124	C₂×C₅₃	#1567	1	1	yes	characteristic
125	C₂×C₅₃	#1831	1	1	yes	characteristic
126	C₂×C₅₃	#1834	1	1	yes	characteristic
127	C₂×C₅₃	#1837	1	1	yes	characteristic
128	C₂×C₄⋊C₈	—	2	1	no	—
129	C₂×C₄⋊C₈	—	2	1	no	—
130	C₂×C₄⋊C₈	—	53	2	no	—
131	C₂×C₄⋊C₈	—	106	1	no	—
132	C₂×C₄⋊C₈	—	106	1	no	—
133	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
134	C₂×C₄×C₈	—	53	1	no	—
135	C₂×C₄×C₈	—	53	2	no	—
136	C₂×C₄×C₈	—	53	1	no	—
137	C₂×C₄×C₈	—	53	2	no	—
138	C₄³	C₂×D₅₃	1	1	yes	characteristic
139	C₅₃	#8784	1	1	yes	characteristic, complemented, Sylow
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	53	2	no	—
142	C₂²×C₈	—	53	2	no	—
143	C₂²×C₈	—	106	2	no	—
144	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
145	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
146	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
147	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
148	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
149	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
150	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	—	4	2	no	—
153	C₄⋊C₈	—	4	2	no	—
154	C₄⋊C₈	—	106	4	no	—
155	C₄⋊C₈	—	212	2	no	—
156	C₄⋊C₈	—	212	2	no	—
157	C₄×C₈	—	53	4	no	—
158	C₄×C₈	—	106	4	no	—
159	C₄×C₈	—	106	2	no	—
160	C₄×C₈	—	106	2	no	—
161	C₄×C₈	—	106	2	no	—
162	C₂²×C₄	#11719919	1	1	yes	characteristic
163	C₂²×C₄	#11719919	1	1	yes	characteristic
164	C₂²×C₄	#11719919	1	1	yes	characteristic
165	C₂²×C₄	#11719919	1	1	yes	characteristic
166	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	53	4	no	—
173	C₂×C₈	—	53	4	no	—
174	C₂×C₈	—	106	2	no	—
175	C₂×C₈	—	106	2	no	—
176	C₂×C₈	—	106	2	no	—
177	C₂×C₈	—	106	2	no	—
178	C₂×C₈	—	212	2	no	—
179	C₂×C₈	—	212	2	no	—
180	C₂×C₈	—	212	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₄²	C₈×D₅₃	1	2	yes	—
185	C₄²	#11719912	1	2	yes	—
186	C₄²	#11719922	1	2	yes	—
187	C₄²	#11719923	1	2	yes	—
188	C₄²	#11719924	1	2	yes	—
189	C₄²	#11719925	1	2	yes	—
190	C₄²	#11719947	1	2	yes	—
191	C₄²	#11719949	1	2	yes	—
192	C₂³	C₂×#11719919	1	1	yes	characteristic
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	#13133146	1	2	yes	—
200	C₂×C₄	#13133146	1	2	yes	—
201	C₂×C₄	#13133148	1	2	yes	—
202	C₂×C₄	#13133148	1	2	yes	—
203	C₂×C₄	#13133150	1	2	yes	—
204	C₂×C₄	#13133150	1	2	yes	—
205	C₂×C₄	#13133152	1	2	yes	—
206	C₂×C₄	#13133152	1	2	yes	—
207	C₂×C₄	#13133175	1	2	yes	—
208	C₂×C₄	#13133204	1	1	yes	characteristic
209	C₂×C₄	#13133206	1	1	yes	characteristic
210	C₂×C₄	#13133208	1	2	yes	—
211	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
212	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
213	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
214	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
215	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
216	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
217	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
218	C₈	—	8	2	no	—
219	C₈	—	106	4	no	—
220	C₈	—	106	4	no	—
221	C₈	—	424	2	no	—
222	C₂²	#16810488	1	1	yes	characteristic
223	C₂²	C₂×#13133146	1	1	yes	characteristic
224	C₂²	#16810646	1	1	yes	characteristic
225	C₂²	C₂×#13133148	1	1	yes	characteristic
226	C₂²	C₂×#13133150	1	1	yes	characteristic
227	C₂²	C₂×#13133152	1	1	yes	characteristic
228	C₂²	C₂×#13133152	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	#16810373	1	4	yes	—
233	C₄	#16810375	1	4	yes	—
234	C₄	#16810675	1	2	yes	—
235	C₄	#16810679	1	2	yes	—
236	C₄	#16810969	1	1	yes	characteristic
237	C₄	#16810971	1	1	yes	characteristic
238	C₄	#16810973	1	1	yes	characteristic
239	C₄	#16810975	1	1	yes	characteristic
240	C₂	C₂×#16810373	1	1	yes	characteristic
241	C₂	C₂×#16810375	1	1	yes	characteristic
242	C₂	#20273470	1	1	yes	characteristic
243	C₂	#20273473	1	1	yes	characteristic
244	C₂	#20274831	1	1	yes	characteristic
245	C₂	#20274842	1	1	yes	characteristic
246	C₂	#20274852	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1842	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1842	D₅₃
hypocenter	C₅₃	#8784
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
26. C₄×#88322	C₈	semidirect
139. C₅₃	#8784	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
C₄²×#88322	2
#20272549	2
C₅₃×#1842	2
#8784	53