Subgroups — #27271290

Order 13568.

246 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274872	C₂	1	1	yes	maximal, characteristic
2	#20273463	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic
4	#16810676	—	2	1	no	—
5	C₂×#13133149	—	2	1	no	—
6	C₂×#13133149	C₄	1	2	yes	—
7	#16810493	—	2	1	no	—
8	#16810376	C₄	1	2	yes	complemented
9	#16810374	—	2	1	no	—
10	C₂×M₃₂×C₅₃	—	2	1	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×C₄⋊C₁₆	—	2	1	no	—
13	C₅₃×C₄⋊C₁₆	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₅₃	—	2	2	no	—
15	C₂×C₄⋊Dic₅₃	—	4	1	no	—
16	C₂×C₄⋊Dic₅₃	—	4	1	no	—
17	#13133149	—	4	1	no	—
18	#13133149	—	4	1	no	—
19	#13133149	—	4	2	no	—
20	C₄⋊#88322	—	2	2	no	—
21	C₂²⋊#88322	—	4	1	no	—
22	C₄×#88322	—	4	1	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
26	M₃₂×C₅₃	—	4	1	no	—
27	M₃₂×C₅₃	—	4	1	no	—
28	C₂×C₁₆×C₅₃	—	2	2	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
36	C₂²×Dic₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
44	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
47	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
50	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
51	C₄⋊Dic₅₃	—	4	2	no	—
52	C₄⋊Dic₅₃	—	4	2	no	—
53	C₄⋊Dic₅₃	—	8	2	no	—
54	C₄⋊Dic₅₃	—	8	2	no	—
55	C₂×#88322	—	4	2	no	—
56	C₁₆×C₅₃	—	4	2	no	—
57	C₂³×C₅₃	#102	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
70	C₂×Dic₅₃	—	8	2	no	—
71	C₂×Dic₅₃	—	8	2	no	—
72	C₂×Dic₅₃	—	8	2	no	—
73	C₈×C₅₃	—	2	1	no	—
74	C₈×C₅₃	—	2	1	no	—
75	C₈×C₅₃	#102	1	1	yes	characteristic
76	C₈×C₅₃	#103	1	1	yes	characteristic
77	C₈×C₅₃	#106	1	1	yes	characteristic
78	C₈×C₅₃	#107	1	1	yes	characteristic
79	#88322	—	8	2	no	—
80	#7278	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	—	2	1	no	—
82	C₂²×C₅₃	—	2	1	no	—
83	C₂²×C₅₃	#331	1	1	yes	characteristic
84	C₂²×C₅₃	#332	1	1	yes	characteristic
85	C₂²×C₅₃	#340	1	1	yes	characteristic
86	C₄×C₅₃	—	2	1	no	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	#332	1	1	yes	characteristic
89	C₄×C₅₃	#333	1	1	yes	characteristic
90	C₄×C₅₃	#340	1	1	yes	characteristic
91	C₄×C₅₃	#343	1	1	yes	characteristic
92	C₄×C₅₃	#360	1	2	yes	—
93	C₄×C₅₃	#365	1	2	yes	—
94	Dic₅₃	—	16	2	no	—
95	#2150	D₅₃	1	1	yes	characteristic
96	#1844	—	53	1	no	—
97	#1564	—	53	1	no	—
98	C₂×C₅₃	—	2	1	no	—
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	#1296	1	1	yes	characteristic
101	C₂×C₅₃	#1383	1	1	yes	characteristic
102	C₂×C₅₃	#1386	1	1	yes	characteristic
103	C₂×M₃₂	—	2	1	no	—
104	#431	—	106	1	no	—
105	C₂×#106	—	53	2	no	—
106	C₂×#106	—	106	1	no	—
107	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
108	#384	—	106	1	no	—
109	C₄⋊C₁₆	—	2	1	no	—
110	C₄⋊C₁₆	Dic₅₃	1	2	yes	complemented
111	#335	—	53	2	no	—
112	#334	—	106	1	no	—
113	C₅₃	#7278	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	106	2	no	—
117	C₂×C₄⋊C₄	—	212	1	no	—
118	C₂×C₄⋊C₄	—	212	1	no	—
119	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
120	M₃₂	—	4	1	no	—
121	M₃₂	—	4	1	no	—
122	C₂×C₁₆	—	2	2	no	—
123	#106	—	212	1	no	—
124	#106	—	212	1	no	—
125	#106	—	212	2	no	—
126	C₄⋊C₈	—	106	2	no	—
127	C₂²⋊C₈	—	212	1	no	—
128	C₄×C₈	—	212	1	no	—
129	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
130	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
131	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
132	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
133	C₂²×C₄	—	212	2	no	—
134	C₂²×C₄	#11719913	1	1	yes	characteristic
135	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
136	C₂²×C₄	#11719921	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	212	2	no	—
140	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
141	C₂×C₈	#11719919	1	1	yes	characteristic
142	C₂×C₈	#11719919	1	1	yes	characteristic
143	C₂×C₈	#11719922	1	1	yes	characteristic
144	C₂×C₈	#11719923	1	1	yes	characteristic
145	C₂×C₈	#11719924	1	1	yes	characteristic
146	C₂×C₈	#11719925	1	1	yes	characteristic
147	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
148	C₄⋊C₄	—	424	2	no	—
149	C₄⋊C₄	—	424	2	no	—
150	C₄⋊C₄	—	212	2	no	—
151	C₄⋊C₄	—	212	2	no	—
152	C₄²	#11719913	1	1	yes	characteristic
153	C₄²	Dic₂₁₂	1	1	yes	characteristic
154	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
155	C₄²	#11719921	1	1	yes	characteristic
156	C₁₆	—	4	2	no	—
157	C₂³	#13133139	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	424	2	no	—
161	C₂×C₄	—	424	2	no	—
162	C₂×C₄	—	424	2	no	—
163	C₂×C₄	#13133124	1	1	yes	characteristic
164	C₂×C₄	#13133133	1	1	yes	characteristic
165	C₂×C₄	#13133135	1	1	yes	characteristic
166	C₂×C₄	#13133137	1	1	yes	characteristic
167	C₂×C₄	#13133139	1	1	yes	characteristic
168	C₂×C₄	#13133140	1	1	yes	characteristic
169	C₂×C₄	#13133142	1	1	yes	characteristic
170	C₂×C₄	#13133142	1	1	yes	characteristic
171	C₂×C₄	#13133149	1	1	yes	characteristic
172	C₂×C₄	#13133151	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	424	2	no	—
176	C₈	#13133138	1	1	yes	characteristic
177	C₈	#13133141	1	1	yes	characteristic
178	C₈	#13133148	1	1	yes	characteristic
179	C₈	#13133150	1	1	yes	characteristic
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	#16810366	1	1	yes	characteristic
183	C₂²	#16810368	1	1	yes	characteristic
184	C₂²	#16810386	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	848	2	no	—
188	C₄	#16810370	1	1	yes	characteristic
189	C₄	#16810371	1	1	yes	characteristic
190	C₄	#16810385	1	1	yes	characteristic
191	C₄	#16810391	1	1	yes	characteristic
192	C₄	#16810434	1	2	yes	—
193	C₄	#16810446	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#20272599	1	1	yes	characteristic
197	C₂	#20272814	1	1	yes	characteristic
198	C₂	#20272821	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810386
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#2150	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#102
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#7278
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
8. #16810376	C₄	semidirect
13. C₅₃×C₄⋊C₁₆	C₄	semidirect
110. C₄⋊C₁₆	Dic₅₃	semidirect
113. C₅₃	#7278	semidirect

Maximal subgroups

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

subgroup	index
#20274872	2
#20273463	2
C₅₃×#2150	2
#7278	53