Subgroups — #27270995

Order 13568.

286 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719908	—	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	C₄⋊#11719908	—	2	1	no	—
9	C₄⋊#11719908	C₄	1	2	yes	complemented
10	#16810377	—	2	1	no	—
11	C₂²×C₁₆×C₅₃	—	2	1	no	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₅₃	—	2	1	no	—
14	C₄×C₁₆×C₅₃	C₄	1	2	yes	complemented
15	C₂²×#88322	—	2	1	no	—
16	C₂×C₄⋊Dic₅₃	C₈	1	2	yes	—
17	C₂×C₄×Dic₅₃	—	2	1	no	—
18	C₂×#11719908	—	2	2	no	—
19	C₂×#11719908	—	4	1	no	—
20	C₂×#11719908	—	4	1	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	—	2	1	no	—
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	C₂×C₁₆×C₅₃	—	2	2	no	—
30	C₂×C₁₆×C₅₃	—	2	2	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	2	no	—
55	C₄⋊Dic₅₃	C₁₆	1	4	yes	complemented
56	C₄×Dic₅₃	—	4	1	no	—
57	C₄×Dic₅₃	—	4	1	no	—
58	C₂×#88322	—	2	2	no	—
59	C₂×#88322	—	4	1	no	—
60	C₂×#88322	—	4	1	no	—
61	C₁₆×C₅₃	—	2	4	no	—
62	#11719908	—	4	2	no	—
63	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	—	2	2	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×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
76	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
77	C₂×Dic₅₃	—	2	4	no	—
78	C₂×Dic₅₃	—	4	2	no	—
79	C₂×Dic₅₃	—	4	2	no	—
80	C₈×C₅₃	—	2	1	no	—
81	C₈×C₅₃	—	2	1	no	—
82	C₈×C₅₃	#102	1	1	yes	characteristic
83	C₈×C₅₃	#103	1	1	yes	characteristic
84	C₈×C₅₃	#106	1	1	yes	characteristic
85	C₈×C₅₃	#107	1	1	yes	characteristic
86	#88322	—	4	2	no	—
87	#7167	—	53	1	no	maximal, Sylow
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	C₂²×C₅₃	C₂²⋊C₁₆	1	2	yes	—
92	C₂²×C₅₃	#350	1	2	yes	—
93	C₄×C₅₃	—	2	1	no	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	D₄⋊C₈	1	1	yes	characteristic
96	C₄×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
97	C₄×C₅₃	#339	1	1	yes	characteristic
98	C₄×C₅₃	#340	1	1	yes	characteristic
99	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
100	C₄×C₅₃	#346	1	1	yes	characteristic
101	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
102	Dic₅₃	—	4	4	no	—
103	C₂×C₄⋊C₁₆	—	53	1	no	—
104	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
105	C₈×C₄⋊C₄	—	53	1	no	—
106	C₂×C₅₃	#1279	1	1	yes	characteristic
107	C₂×C₅₃	D₄⋊C₁₆	1	2	yes	—
108	C₂×C₅₃	Q₈⋊C₁₆	1	2	yes	—
109	C₂×C₅₃	#1374	1	1	yes	characteristic
110	C₂×C₅₃	#1379	1	1	yes	characteristic
111	C₂²×C₁₆	—	2	1	no	—
112	C₂²×C₁₆	—	106	1	no	—
113	C₂×C₄⋊C₈	—	53	1	no	—
114	C₂×C₄×C₈	—	106	1	no	—
115	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
116	C₄×C₄⋊C₄	—	53	1	no	—
117	C₄⋊C₁₆	—	53	2	no	—
118	C₄⋊C₁₆	—	106	1	no	—
119	C₄×C₁₆	—	2	1	no	—
120	C₄×C₁₆	Dic₅₃	1	2	yes	complemented
121	#336	—	106	1	no	—
122	C₅₃	#7167	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₈	—	106	1	no	—
124	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
125	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	53	2	no	—
127	C₂×C₄²	—	106	1	no	—
128	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	106	2	no	—
133	C₂×C₁₆	—	212	1	no	—
134	C₂×C₁₆	—	212	1	no	—
135	C₄⋊C₈	—	53	2	no	—
136	C₄⋊C₈	—	106	1	no	—
137	C₄×C₈	—	212	1	no	—
138	C₄×C₈	—	212	1	no	—
139	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
140	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
141	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
143	#95	—	106	1	no	—
144	C₂²×C₄	—	106	2	no	—
145	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
146	C₂²×C₄	#11719912	1	1	yes	characteristic
147	C₂²×C₄	#11719921	1	1	yes	characteristic
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	106	2	no	—
151	C₂×C₈	—	212	1	no	—
152	C₂×C₈	—	212	1	no	—
153	C₂×C₈	#11719913	1	1	yes	characteristic
154	C₂×C₈	#11719913	1	1	yes	characteristic
155	C₂×C₈	D₄₂₄	1	1	yes	characteristic
156	C₂×C₈	Dic₂₁₂	1	1	yes	characteristic
157	C₂×C₈	#11719919	1	1	yes	characteristic
158	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
159	C₂×C₈	#11719921	1	1	yes	characteristic
160	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
161	C₄⋊C₄	—	53	4	no	—
162	C₄⋊C₄	—	106	2	no	—
163	C₄²	—	212	1	no	—
164	C₄²	—	212	1	no	—
165	C₄²	C₈×D₅₃	1	1	yes	characteristic
166	C₄²	#11719912	1	1	yes	characteristic
167	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
168	C₄²	#11719919	1	1	yes	characteristic
169	C₁₆	—	2	4	no	—
170	C₁₆	—	212	2	no	—
171	C₂³	#13133129	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	106	4	no	—
174	C₂×C₄	—	212	2	no	—
175	C₂×C₄	—	212	2	no	—
176	C₂×C₄	#13133124	1	1	yes	characteristic
177	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
178	C₂×C₄	#13133127	1	1	yes	characteristic
179	C₂×C₄	#13133129	1	1	yes	characteristic
180	C₂×C₄	#13133139	1	1	yes	characteristic
181	C₂×C₄	#13133142	1	1	yes	characteristic
182	C₂×C₄	#13133145	1	1	yes	characteristic
183	C₂×C₄	#13133145	1	1	yes	characteristic
184	C₂×C₄	#13133146	1	1	yes	characteristic
185	C₂×C₄	#13133146	1	1	yes	characteristic
186	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
187	C₂×C₄	#13133156	1	2	yes	—
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	212	2	no	—
191	C₈	#13133139	1	1	yes	characteristic
192	C₈	#13133142	1	1	yes	characteristic
193	C₈	#13133149	1	1	yes	characteristic
194	C₈	#13133151	1	1	yes	characteristic
195	C₂²	#16810352	1	1	yes	characteristic
196	C₂²	#16810355	1	1	yes	characteristic
197	C₂²	#16810377	1	1	yes	characteristic
198	C₂²	#16810402	1	2	yes	—
199	C₂²	#16810406	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	212	4	no	—
203	C₄	#16810352	1	1	yes	characteristic
204	C₄	#16810355	1	1	yes	characteristic
205	C₄	#16810383	1	1	yes	characteristic
206	C₄	#16810386	1	1	yes	characteristic
207	C₄	C₁₆×Dic₅₃	1	1	yes	characteristic
208	C₄	#16810398	1	1	yes	characteristic
209	C₄	#16810441	1	2	yes	—
210	C₂	#20272555	1	1	yes	characteristic
211	C₂	#20272671	1	2	yes	—
212	C₂	#20272695	1	2	yes	—
213	C₂	#20272792	1	1	yes	characteristic
214	C₂	#20272803	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	D₂₁₂
commutator	C₄×C₅₃	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#7167
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
9. C₄⋊#11719908	C₄	semidirect
14. C₄×C₁₆×C₅₃	C₄	semidirect
55. C₄⋊Dic₅₃	C₁₆	semidirect
120. C₄×C₁₆	Dic₅₃	semidirect
122. C₅₃	#7167	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719908	2
C₈×C₄⋊Dic₅₃	2
C₂×C₄×C₁₆×C₅₃	2
#7167	53