Subgroups — #27283927

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275677	C₂	1	2	yes	maximal
2	#20275518	C₂	1	2	yes	maximal
3	#20274872	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810344	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133149	—	2	2	no	—
7	C₂×#13133141	—	2	4	no	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	#16810493	C₂²	1	2	yes	—
11	#16810385	—	2	4	no	—
12	#16810344	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#500	—	2	2	no	—
15	C₅₃×#500	C₄	1	4	yes	complemented
16	C₂×#107×C₅₃	—	2	2	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×#88322	D₄	1	2	yes	—
19	C₂²×#88322	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	2	2	no	—
22	C₂×C₄⋊Dic₅₃	—	2	2	no	—
23	#13133149	—	4	4	no	—
24	#13133141	—	4	8	no	—
25	C₄×#88322	—	2	1	no	—
26	C₄×#88322	—	2	1	no	—
27	C₄×#88322	—	2	1	no	—
28	C₄×#88322	—	2	1	no	—
29	C₂²×Q₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	D₄	1	2	yes	—
31	C₄⋊Q₈×C₅₃	—	2	2	no	—
32	C₄⋊Q₈×C₅₃	—	2	2	no	—
33	C₄⋊Q₈×C₅₃	—	2	2	no	—
34	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#107×C₅₃	—	2	8	no	—
39	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₅₃	D₄	1	2	yes	—
41	C₂×Q₈×C₅₃	—	2	4	no	—
42	C₂×Q₈×C₅₃	—	4	2	no	—
43	C₂×Q₈×C₅₃	—	4	4	no	—
44	C₂×Q₈×C₅₃	—	4	4	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	4	2	no	—
50	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
51	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
52	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
53	C₄⋊C₄×C₅₃	—	2	4	no	—
54	C₄⋊C₄×C₅₃	—	2	4	no	—
55	C₄⋊C₄×C₅₃	—	4	2	no	—
56	C₄⋊C₄×C₅₃	—	4	2	no	—
57	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅₃	#41	1	1	yes	characteristic
60	C₄²×C₅₃	#41	1	1	yes	characteristic
61	C₄⋊Dic₅₃	—	4	4	no	—
62	C₄⋊Dic₅₃	—	4	4	no	—
63	C₄⋊Dic₅₃	—	4	4	no	—
64	C₂×#88322	—	2	2	no	—
65	C₂×#88322	—	2	2	no	—
66	C₂×#88322	—	2	4	no	—
67	C₂×#88322	—	2	4	no	—
68	C₂³×C₅₃	#127	1	1	yes	characteristic
69	Q₈×C₅₃	—	4	8	no	—
70	Q₈×C₅₃	—	8	4	no	—
71	C₂×C₄×C₅₃	—	4	4	no	—
72	C₂×C₄×C₅₃	—	8	2	no	—
73	C₂×C₄×C₅₃	—	8	2	no	—
74	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
83	C₂×C₄×C₅₃	#137	1	4	yes	—
84	C₂×Dic₅₃	—	8	2	no	—
85	C₂×Dic₅₃	—	8	2	no	—
86	C₂×Dic₅₃	—	8	2	no	—
87	C₈×C₅₃	#103	1	8	yes	complemented
88	#88322	—	4	2	no	—
89	#88322	—	4	2	no	—
90	#9836	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	#390	1	1	yes	characteristic
92	C₂²×C₅₃	#492	1	1	yes	characteristic
93	C₂²×C₅₃	#494	1	1	yes	characteristic
94	C₂²×C₅₃	#496	1	2	yes	—
95	C₂²×C₅₃	#497	1	2	yes	—
96	C₄×C₅₃	—	8	4	no	—
97	C₄×C₅₃	C₂×#103	1	2	yes	—
98	C₄×C₅₃	#441	1	2	yes	—
99	C₄×C₅₃	#442	1	2	yes	—
100	C₄×C₅₃	#469	1	2	yes	—
101	C₄×C₅₃	#470	1	2	yes	—
102	C₄×C₅₃	#487	1	2	yes	—
103	Dic₅₃	—	16	2	no	—
104	C₂×#500	D₅₃	1	1	yes	characteristic
105	#1990	—	53	2	no	—
106	#1956	—	53	2	no	—
107	#1844	—	53	1	no	—
108	C₂×#322	—	53	1	no	—
109	C₂×C₅₃	#1714	1	2	yes	—
110	C₂×C₅₃	#1723	1	2	yes	—
111	C₂×C₅₃	#1956	1	1	yes	characteristic
112	C₂×C₅₃	#1971	1	1	yes	characteristic
113	C₂×C₅₃	#1972	1	1	yes	characteristic
114	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
115	#500	—	2	2	no	—
116	#500	Dic₅₃	1	4	yes	complemented
117	C₂×#107	—	2	2	no	—
118	C₂×#106	—	106	2	no	—
119	C₂×#103	—	106	4	no	—
120	C₂×C₄×C₈	—	53	1	no	—
121	C₂×C₄×C₈	—	53	1	no	—
122	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
123	#384	—	53	2	no	—
124	#340	—	106	4	no	—
125	#322	—	106	2	no	—
126	C₅₃	#9836	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₈	—	2	2	no	—
128	C₂²×C₈	—	53	2	no	—
129	C₂²×C₈	—	53	2	no	—
130	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	C₂×Dic₅₃	1	4	yes	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	106	2	no	—
138	C₂×C₄⋊C₄	—	106	2	no	—
139	C₂×C₄⋊C₄	—	106	2	no	—
140	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
141	#107	—	2	8	no	—
142	#106	—	212	4	no	—
143	#103	—	212	8	no	—
144	C₄×C₈	—	106	1	no	—
145	C₄×C₈	—	106	1	no	—
146	C₄×C₈	—	106	1	no	—
147	C₄×C₈	—	106	1	no	—
148	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
149	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	4	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	212	2	no	—
156	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
157	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂×C₈	—	106	2	no	—
160	C₂×C₈	—	106	2	no	—
161	C₂×C₈	—	106	4	no	—
162	C₂×C₈	—	106	4	no	—
163	C₂×C₈	#11719924	1	4	yes	—
164	C₂×C₈	#11719925	1	4	yes	—
165	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	212	4	no	—
171	C₄⋊C₄	—	212	4	no	—
172	C₄⋊C₄	—	212	4	no	—
173	C₄²	#11719947	1	1	yes	characteristic
174	C₄²	#11719947	1	1	yes	characteristic
175	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
176	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
177	C₂³	#13133213	1	1	yes	characteristic
178	Q₈	—	4	8	no	—
179	Q₈	—	8	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	424	2	no	—
184	C₂×C₄	—	424	2	no	—
185	C₂×C₄	—	424	2	no	—
186	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
188	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
189	C₂×C₄	#13133188	1	1	yes	characteristic
190	C₂×C₄	#13133188	1	1	yes	characteristic
191	C₂×C₄	#13133204	1	1	yes	characteristic
192	C₂×C₄	C₂×#11719924	1	2	yes	—
193	C₂×C₄	C₂×#11719925	1	2	yes	—
194	C₂×C₄	#13133241	1	4	yes	—
195	C₂×C₄	#13133249	1	4	yes	—
196	C₈	—	212	2	no	—
197	C₈	—	212	2	no	—
198	C₈	#13133141	1	8	yes	complemented
199	C₂²	#16810526	1	1	yes	characteristic
200	C₂²	#16810997	1	1	yes	characteristic
201	C₂²	#16811002	1	1	yes	characteristic
202	C₂²	#16811011	1	2	yes	—
203	C₂²	#16811016	1	2	yes	—
204	C₄	—	8	4	no	—
205	C₄	—	848	2	no	—
206	C₄	C₂×#13133141	1	2	yes	—
207	C₄	#16810730	1	2	yes	—
208	C₄	#16810735	1	2	yes	—
209	C₄	#16810873	1	2	yes	—
210	C₄	#16810878	1	2	yes	—
211	C₄	#16810973	1	2	yes	—
212	C₂	#20274338	1	2	yes	—
213	C₂	#20274393	1	2	yes	—
214	C₂	#20275518	1	1	yes	characteristic
215	C₂	#20275587	1	1	yes	characteristic
216	C₂	#20275592	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#500	C₂
socle, prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	#9836
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
15. C₅₃×#500	C₄	semidirect
87. C₈×C₅₃	#103	semidirect
116. #500	Dic₅₃	semidirect
126. C₅₃	#9836	semidirect
198. C₈	#13133141	semidirect

Maximal subgroups

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

subgroup	index
#20275677	2
#20275518	2
#20274872	2
C₂×#16810344	2
C₂×C₅₃×#500	2
#9836	53