Subgroups — #27270656

Order 13568.

344 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274461	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133128	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133128	—	2	1	no	—
5	C₂×#13133128	C₄	1	2	yes	—
6	C₂×C₄×#88322	—	2	1	no	—
7	C₂×C₄×#88322	—	2	1	no	—
8	#16810378	—	2	1	no	—
9	#16810378	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
11	C₄³×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#336	—	2	1	no	—
13	C₅₃×#336	—	2	1	no	—
14	C₂²×#88322	—	2	2	no	—
15	C₂²×#88322	—	2	2	no	—
16	#13133128	—	2	4	no	—
17	#13133128	—	2	2	no	—
18	#13133128	C₈	1	4	yes	complemented
19	C₄×#88322	—	2	4	no	—
20	C₄×#88322	—	2	4	no	—
21	C₂²×C₈×C₅₃	—	2	2	no	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	—	4	2	no	—
30	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	—	4	2	no	—
40	C₄²×C₅₃	—	2	1	no	—
41	C₄²×C₅₃	—	2	1	no	—
42	C₄²×C₅₃	—	2	2	no	—
43	C₄²×C₅₃	—	2	2	no	—
44	C₄²×C₅₃	—	2	2	no	—
45	C₄²×C₅₃	—	2	2	no	—
46	C₄²×C₅₃	—	4	1	no	—
47	C₄²×C₅₃	C₂×C₈	1	2	yes	—
48	C₄²×C₅₃	M₁₆	1	2	yes	—
49	C₂×#88322	—	2	4	no	—
50	C₂×#88322	—	2	4	no	—
51	C₂×#88322	—	2	4	no	—
52	C₂×#88322	—	2	4	no	—
53	C₂×#88322	—	4	2	no	—
54	C₂×#88322	—	4	2	no	—
55	C₂³×C₅₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
72	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#104	1	2	yes	—
75	C₂×C₄×C₅₃	#104	1	2	yes	—
76	C₂×C₄×C₅₃	#104	1	2	yes	—
77	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
78	C₂×C₄×C₅₃	#108	1	2	yes	—
79	C₈×C₅₃	—	8	2	no	—
80	#88322	—	2	8	no	—
81	#88322	—	4	4	no	—
82	#7030	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	#336	1	1	yes	characteristic
84	C₂²×C₅₃	#339	1	1	yes	characteristic
85	C₂²×C₅₃	#339	1	1	yes	characteristic
86	C₂²×C₅₃	#339	1	1	yes	characteristic
87	C₂²×C₅₃	#341	1	1	yes	characteristic
88	C₂²×C₅₃	#343	1	1	yes	characteristic
89	C₂²×C₅₃	#344	1	1	yes	characteristic
90	C₄×C₅₃	—	2	1	no	—
91	C₄×C₅₃	—	2	1	no	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	—	4	1	no	—
97	C₄×C₅₃	#329	1	2	yes	—
98	C₄×C₅₃	#330	1	2	yes	—
99	#1831	D₅₃	1	1	yes	characteristic
100	#1744	—	53	1	no	—
101	C₄×#97	—	53	1	no	—
102	C₂×C₅₃	#1276	1	1	yes	characteristic
103	C₂×C₅₃	#1278	1	1	yes	characteristic
104	C₂×C₅₃	#1283	1	1	yes	characteristic
105	C₂×C₅₃	#1295	1	1	yes	characteristic
106	C₂×C₅₃	#1299	1	1	yes	characteristic
107	C₂×C₅₃	#1300	1	1	yes	characteristic
108	C₂×C₅₃	#1302	1	1	yes	characteristic
109	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
110	C₂×#97	—	53	2	no	—
111	C₂×#97	—	106	1	no	—
112	C₂×C₄×C₈	—	106	1	no	—
113	C₂×C₄×C₈	—	106	1	no	—
114	C₄³	C₂×D₅₃	1	1	yes	characteristic
115	#336	—	2	1	no	—
116	#336	—	2	1	no	—
117	#336	—	53	2	no	—
118	#336	—	106	1	no	—
119	C₅₃	#7030	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	—	106	2	no	—
122	C₂²×C₈	—	106	2	no	—
123	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
124	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
125	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
126	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
127	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
128	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
129	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
130	C₄⋊C₈	—	4	2	no	—
131	#97	—	53	4	no	—
132	#97	—	106	4	no	—
133	#97	—	106	2	no	—
134	C₄×C₈	—	106	4	no	—
135	C₄×C₈	—	106	4	no	—
136	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
137	C₂²×C₄	#11719919	1	1	yes	characteristic
138	C₂²×C₄	#11719919	1	1	yes	characteristic
139	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
140	C₂²×C₄	#11719921	1	1	yes	characteristic
141	C₂²×C₄	#11719921	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	106	4	no	—
147	C₂×C₈	—	106	4	no	—
148	C₂×C₈	—	106	4	no	—
149	C₂×C₈	—	106	4	no	—
150	C₂×C₈	—	212	2	no	—
151	C₂×C₈	—	212	2	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	4	1	no	—
159	C₄²	C₈×D₅₃	1	2	yes	—
160	C₄²	#11719912	1	2	yes	—
161	C₂³	#13133124	1	1	yes	characteristic
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
176	C₂×C₄	#13133127	1	1	yes	characteristic
177	C₂×C₄	#13133129	1	2	yes	—
178	C₂×C₄	#13133134	1	1	yes	characteristic
179	C₂×C₄	#13133136	1	1	yes	characteristic
180	C₂×C₄	#13133143	1	2	yes	—
181	C₂×C₄	#13133144	1	2	yes	—
182	C₂×C₄	#13133144	1	2	yes	—
183	C₂×C₄	#13133146	1	2	yes	—
184	C₂×C₄	#13133152	1	2	yes	—
185	C₈	—	8	2	no	—
186	C₈	—	106	8	no	—
187	C₈	—	212	4	no	—
188	C₂²	#16810377	1	1	yes	characteristic
189	C₂²	#16810382	1	1	yes	characteristic
190	C₂²	#16810382	1	1	yes	characteristic
191	C₂²	#16810384	1	1	yes	characteristic
192	C₂²	#16810387	1	1	yes	characteristic
193	C₂²	#16810392	1	1	yes	characteristic
194	C₂²	#16810394	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	#16810363	1	2	yes	—
203	C₄	#16810365	1	2	yes	—
204	C₂	#20272547	1	1	yes	characteristic
205	C₂	#20272551	1	1	yes	characteristic
206	C₂	#20272565	1	1	yes	characteristic
207	C₂	#20272596	1	1	yes	characteristic
208	C₂	#20272604	1	1	yes	characteristic
209	C₂	#20272607	1	1	yes	characteristic
210	C₂	#20272613	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₄³	C₂×D₅₃
Fitting	C₅₃×#1831	C₂
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄³×C₅₃	C₂²
hypercenter	#1831	D₅₃
hypocenter	C₅₃	#7030
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
18. #13133128	C₈	semidirect
119. C₅₃	#7030	semidirect

Maximal subgroups

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

subgroup	index
#20274461	2
C₄×#13133128	2
C₅₃×#1831	2
#7030	53