Subgroups — #27302647

Order 13568.

332 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810444	C₂	1	1	yes	maximal, characteristic
2	#20274594	C₂	1	1	yes	maximal, characteristic
3	#20272831	C₂	1	2	yes	maximal
4	#20272816	C₂	1	2	yes	maximal
5	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133153	C₂²	1	1	yes	characteristic
8	#16810661	C₂²	1	1	yes	characteristic
9	#16810578	—	2	1	no	—
10	#16810444	—	2	1	no	—
11	#16810444	—	2	1	no	—
12	#16810383	—	2	1	no	—
13	C₅₃×#533	—	2	1	no	—
14	C₅₃×#500	—	2	1	no	—
15	C₅₃×#499	—	2	1	no	—
16	C₅₃×#499	C₄	1	2	yes	complemented
17	C₅₃×#498	C₄	1	2	yes	complemented
18	C₅₃×#427	—	2	1	no	—
19	C₅₃×#427	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#11719917	—	2	1	no	—
22	#13133176	—	2	1	no	—
23	#13133157	—	2	2	no	—
24	#13133157	—	2	2	no	—
25	C₂×#11719908	—	2	2	no	—
26	#13133153	—	2	1	no	—
27	#13133153	Q₈	1	2	yes	complemented
28	#13133151	—	2	1	no	—
29	#13133149	Q₈	1	2	yes	complemented
30	#13133127	—	4	1	no	—
31	C₈×Dic₅₃	—	4	1	no	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₅₃	—	2	2	no	—
35	#125×C₅₃	—	2	2	no	—
36	C₂²⋊Q₈×C₅₃	—	4	2	no	—
37	C₄×Q₈×C₅₃	—	4	2	no	—
38	#117×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	#107×C₅₃	—	2	2	no	—
41	#107×C₅₃	—	2	2	no	—
42	#106×C₅₃	—	2	2	no	—
43	#106×C₅₃	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₂×Q₈×C₅₃	—	4	2	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
56	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
62	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
63	C₄⋊C₄×C₅₃	—	2	4	no	—
64	C₄⋊C₄×C₅₃	—	4	4	no	—
65	C₄⋊C₄×C₅₃	—	4	2	no	—
66	C₂²⋊C₄×C₅₃	—	4	2	no	—
67	C₄²×C₅₃	—	2	1	no	—
68	C₄²×C₅₃	—	4	2	no	—
69	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²⋊Dic₅₃	—	4	1	no	—
72	C₄⋊Dic₅₃	—	2	2	no	—
73	C₄×Dic₅₃	—	4	1	no	—
74	#11719917	—	2	2	no	—
75	#11719917	—	4	1	no	—
76	C₂×#88322	—	4	1	no	—
77	#11719908	—	4	2	no	—
78	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	Q₈×C₅₃	—	8	4	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	—	4	4	no	—
84	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₂×Dic₅₃	—	4	2	no	—
91	C₈×C₅₃	#103	1	1	yes	characteristic
92	C₈×C₅₃	#103	1	1	yes	characteristic
93	C₈×C₅₃	#103	1	1	yes	characteristic
94	C₈×C₅₃	#103	1	1	yes	characteristic
95	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
96	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
97	C₈×C₅₃	#124	1	1	yes	characteristic
98	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
99	#88322	—	4	2	no	—
100	#13271	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	—	4	1	no	—
102	C₂²×C₅₃	#418	1	1	yes	characteristic
103	C₂²×C₅₃	#476	1	1	yes	characteristic
104	C₂²×C₅₃	#477	1	1	yes	characteristic
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	4	1	no	—
107	C₄×C₅₃	—	8	4	no	—
108	C₄×C₅₃	#391	1	1	yes	characteristic
109	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
110	C₄×C₅₃	#476	1	1	yes	characteristic
111	C₄×C₅₃	#477	1	1	yes	characteristic
112	Dic₅₃	—	8	2	no	—
113	#3160	D₅₃	1	1	yes	characteristic
114	C₂×#364	—	53	1	no	—
115	#1772	—	53	1	no	—
116	#1389	—	53	2	no	—
117	#1384	—	53	2	no	—
118	C₂×C₅₃	—	2	1	no	—
119	C₂×C₅₃	—	2	1	no	—
120	C₂×C₅₃	—	2	1	no	—
121	C₂×C₅₃	#1987	1	1	yes	characteristic
122	#533	—	2	1	no	—
123	C₂×M₃₂	—	53	2	no	—
124	#500	—	2	1	no	—
125	#499	—	2	1	no	—
126	#499	Dic₅₃	1	2	yes	complemented
127	#498	Dic₅₃	1	2	yes	complemented
128	C₂×#108	—	53	1	no	—
129	#427	—	2	1	no	—
130	#427	—	53	1	no	—
131	#427	C₂×D₅₃	1	2	yes	—
132	#405	—	106	1	no	—
133	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
134	#364	—	106	1	no	—
135	#364	—	106	1	no	—
136	#339	—	106	1	no	—
137	C₅₃	#13271	1	1	yes	characteristic, complemented, Sylow
138	C₂×M₁₆	—	106	1	no	—
139	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	2	no	—
142	#125	—	2	2	no	—
143	C₂²⋊Q₈	—	4	2	no	—
144	C₄×Q₈	—	4	2	no	—
145	#117	—	2	2	no	—
146	#117	—	106	1	no	—
147	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
148	M₃₂	—	106	2	no	—
149	M₃₂	—	106	2	no	—
150	C₂×C₁₆	—	106	2	no	—
151	#108	—	53	2	no	—
152	#108	—	106	1	no	—
153	#107	—	2	2	no	—
154	#107	—	2	2	no	—
155	#107	—	106	1	no	—
156	#106	—	2	2	no	—
157	#106	—	53	2	no	—
158	#106	C₂×Dic₅₃	1	4	yes	—
159	#97	—	212	1	no	—
160	C₄×C₈	—	212	1	no	—
161	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
162	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
165	C₂×Q₈	—	4	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
168	M₁₆	—	106	2	no	—
169	M₁₆	—	212	1	no	—
170	C₂×C₈	—	212	1	no	—
171	C₂×C₈	#11719924	1	1	yes	characteristic
172	C₂×C₈	#11719924	1	1	yes	characteristic
173	C₂×C₈	#11719925	1	1	yes	characteristic
174	C₂×C₈	#11719925	1	1	yes	characteristic
175	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
176	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
177	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
178	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
179	C₂×C₈	#11719949	1	1	yes	characteristic
180	C₂×C₈	#11719949	1	1	yes	characteristic
181	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
182	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	106	2	no	—
187	C₂²⋊C₄	—	4	2	no	—
188	C₂²⋊C₄	—	212	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	4	2	no	—
191	C₄²	—	212	1	no	—
192	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
193	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
194	C₁₆	—	212	2	no	—
195	C₂³	#13133243	1	1	yes	characteristic
196	Q₈	—	8	4	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	212	2	no	—
202	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
203	C₂×C₄	#13133193	1	1	yes	characteristic
204	C₂×C₄	#13133193	1	1	yes	characteristic
205	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
206	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
207	C₂×C₄	#13133243	1	1	yes	characteristic
208	C₈	—	212	2	no	—
209	C₈	#13133141	1	1	yes	characteristic
210	C₈	#13133141	1	1	yes	characteristic
211	C₈	#13133141	1	1	yes	characteristic
212	C₈	#13133141	1	1	yes	characteristic
213	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
214	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
215	C₈	#13133206	1	1	yes	characteristic
216	C₈	#13133215	1	1	yes	characteristic
217	C₂²	—	4	1	no	—
218	C₂²	#16810628	1	1	yes	characteristic
219	C₂²	#16810910	1	1	yes	characteristic
220	C₂²	#16810915	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	8	4	no	—
224	C₄	—	424	2	no	—
225	C₄	#16810530	1	1	yes	characteristic
226	C₄	C₂×#13133141	1	1	yes	characteristic
227	C₄	#16810910	1	1	yes	characteristic
228	C₄	#16810915	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	#20275664	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810910
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#3160	C₂
socle	C₂×C₅₃	#1987
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#3160	D₅₃
hypocenter	C₅₃	#13271
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
16. C₅₃×#499	C₄	semidirect
17. C₅₃×#498	C₄	semidirect
27. #13133153	Q₈	semidirect
29. #13133149	Q₈	semidirect
126. #499	Dic₅₃	semidirect
127. #498	Dic₅₃	semidirect
137. C₅₃	#13271	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810444	2
#20274594	2
#20272831	2
#20272816	2
C₅₃×#3160	2
#13271	53