Subgroups — #27271000

Order 13568.

266 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133157	—	2	1	no	—
5	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₅₃	—	2	1	no	—
7	C₄×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
8	#16810399	C₄	1	2	yes	complemented
9	C₄×#11719908	—	2	1	no	—
10	#16810377	—	2	1	no	—
11	C₂×M₃₂×C₅₃	—	2	1	no	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×C₄⋊C₁₆	—	2	1	no	—
14	C₅₃×C₄⋊C₁₆	C₄	1	2	yes	complemented
15	C₂²×#88322	—	2	1	no	—
16	C₂×C₄⋊Dic₅₃	—	2	1	no	—
17	C₂×C₄×Dic₅₃	—	2	1	no	—
18	#13133157	—	2	2	no	—
19	#13133157	—	2	2	no	—
20	C₂×#11719908	—	2	2	no	—
21	C₄⋊#88322	—	2	1	no	—
22	C₄⋊#88322	C₈	1	2	yes	—
23	C₈×Dic₅₃	—	4	1	no	—
24	C₈×Dic₅₃	—	4	1	no	—
25	#13133124	C₈	1	2	yes	—
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	M₃₂×C₅₃	—	4	1	no	—
30	M₃₂×C₅₃	—	4	1	no	—
31	C₂×C₁₆×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
39	C₂²×Dic₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
50	C₄²×C₅₃	C₄²	1	1	yes	characteristic
51	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
54	C₄⋊Dic₅₃	—	2	4	no	—
55	C₄×Dic₅₃	—	4	1	no	—
56	C₄×Dic₅₃	—	4	1	no	—
57	C₂×#88322	—	2	2	no	—
58	C₂×#88322	—	4	1	no	—
59	C₂×#88322	—	4	1	no	—
60	C₁₆×C₅₃	—	4	2	no	—
61	#11719908	—	2	4	no	—
62	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	4	1	no	—
65	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#104	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#104	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
75	C₂×Dic₅₃	—	2	4	no	—
76	C₂×Dic₅₃	—	4	2	no	—
77	C₂×Dic₅₃	—	4	2	no	—
78	C₈×C₅₃	—	2	1	no	—
79	C₈×C₅₃	—	2	1	no	—
80	C₈×C₅₃	#102	1	1	yes	characteristic
81	C₈×C₅₃	#103	1	1	yes	characteristic
82	C₈×C₅₃	#106	1	1	yes	characteristic
83	C₈×C₅₃	#107	1	1	yes	characteristic
84	#88322	—	4	2	no	—
85	#7169	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	—	2	1	no	—
87	C₂²×C₅₃	—	2	1	no	—
88	C₂²×C₅₃	D₄⋊C₈	1	1	yes	characteristic
89	C₂²×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
90	C₂²×C₅₃	#336	1	1	yes	characteristic
91	C₄×C₅₃	—	2	1	no	—
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	—	2	1	no	—
94	C₄×C₅₃	D₄⋊C₈	1	1	yes	characteristic
95	C₄×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
96	C₄×C₅₃	#339	1	1	yes	characteristic
97	C₄×C₅₃	#340	1	1	yes	characteristic
98	C₄×C₅₃	#364	1	2	yes	—
99	Dic₅₃	—	4	4	no	—
100	#2150	D₅₃	1	1	yes	characteristic
101	C₄×M₃₂	—	53	1	no	—
102	C₈×C₄⋊C₄	—	53	1	no	—
103	C₂×C₅₃	—	2	1	no	—
104	C₂×C₅₃	—	2	1	no	—
105	C₂×C₅₃	#1279	1	1	yes	characteristic
106	C₂×C₅₃	#1376	1	1	yes	characteristic
107	C₂×C₅₃	#1378	1	1	yes	characteristic
108	C₂×M₃₂	—	2	1	no	—
109	C₂×M₃₂	—	106	1	no	—
110	C₂×C₄⋊C₈	—	53	1	no	—
111	C₂×C₄×C₈	—	106	1	no	—
112	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
113	C₄×C₄⋊C₄	—	53	1	no	—
114	C₄⋊C₁₆	—	2	1	no	—
115	C₄⋊C₁₆	Dic₅₃	1	2	yes	complemented
116	#346	—	53	2	no	—
117	C₄×C₁₆	—	106	1	no	—
118	#336	—	106	1	no	—
119	C₅₃	#7169	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	—	106	1	no	—
121	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	106	1	no	—
124	C₂×C₄²	—	106	1	no	—
125	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
126	M₃₂	—	4	1	no	—
127	M₃₂	—	4	1	no	—
128	M₃₂	—	106	2	no	—
129	M₃₂	—	106	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	106	2	no	—
132	C₄⋊C₈	—	53	2	no	—
133	C₄⋊C₈	—	106	1	no	—
134	C₄×C₈	—	212	1	no	—
135	C₄×C₈	—	212	1	no	—
136	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
137	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
138	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
140	#95	—	53	2	no	—
141	C₂²×C₄	—	106	2	no	—
142	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
143	C₂²×C₄	#11719912	1	1	yes	characteristic
144	C₂²×C₄	#11719921	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	106	2	no	—
148	C₂×C₈	—	212	1	no	—
149	C₂×C₈	—	212	1	no	—
150	C₂×C₈	#11719919	1	1	yes	characteristic
151	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
152	C₂×C₈	#11719921	1	1	yes	characteristic
153	C₂×C₈	#11719922	1	1	yes	characteristic
154	C₂×C₈	#11719923	1	1	yes	characteristic
155	C₂×C₈	#11719924	1	1	yes	characteristic
156	C₂×C₈	#11719925	1	1	yes	characteristic
157	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
158	C₄⋊C₄	—	106	4	no	—
159	C₄²	—	212	1	no	—
160	C₄²	—	212	1	no	—
161	C₄²	C₈×D₅₃	1	1	yes	characteristic
162	C₄²	#11719912	1	1	yes	characteristic
163	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
164	C₄²	#11719919	1	1	yes	characteristic
165	C₁₆	—	4	2	no	—
166	C₁₆	—	106	4	no	—
167	C₂³	#13133129	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	106	4	no	—
171	C₂×C₄	—	212	2	no	—
172	C₂×C₄	—	212	2	no	—
173	C₂×C₄	#13133124	1	1	yes	characteristic
174	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
175	C₂×C₄	#13133127	1	1	yes	characteristic
176	C₂×C₄	#13133129	1	1	yes	characteristic
177	C₂×C₄	#13133137	1	1	yes	characteristic
178	C₂×C₄	#13133140	1	1	yes	characteristic
179	C₂×C₄	#13133143	1	1	yes	characteristic
180	C₂×C₄	#13133143	1	1	yes	characteristic
181	C₂×C₄	#13133146	1	1	yes	characteristic
182	C₂×C₄	#13133146	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	212	2	no	—
186	C₈	#13133138	1	1	yes	characteristic
187	C₈	#13133141	1	1	yes	characteristic
188	C₈	#13133148	1	1	yes	characteristic
189	C₈	#13133150	1	1	yes	characteristic
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	#16810350	1	1	yes	characteristic
193	C₂²	#16810353	1	1	yes	characteristic
194	C₂²	#16810377	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	212	4	no	—
199	C₄	#16810350	1	1	yes	characteristic
200	C₄	#16810353	1	1	yes	characteristic
201	C₄	#16810384	1	1	yes	characteristic
202	C₄	#16810385	1	1	yes	characteristic
203	C₄	#16810443	1	2	yes	—
204	C₂	—	2	1	no	—
205	C₂	—	2	1	no	—
206	C₂	#20272554	1	1	yes	characteristic
207	C₂	#20272796	1	1	yes	characteristic
208	C₂	#20272800	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#2150	C₂
socle	C₂²×C₅₃	#336
prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#7169
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
8. #16810399	C₄	semidirect
14. C₅₃×C₄⋊C₁₆	C₄	semidirect
115. C₄⋊C₁₆	Dic₅₃	semidirect
119. C₅₃	#7169	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133157	2
C₈×C₄⋊Dic₅₃	2
C₅₃×#2150	2
#7169	53