Subgroups — #27302718

Order 13568.

354 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	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
2	#20274879	C₂	1	1	yes	maximal, characteristic
3	#20272820	C₂	1	4	yes	maximal
4	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂×#13133151	—	2	1	no	—
7	C₂×#13133149	C₂²	1	2	yes	—
8	#16810489	—	2	1	no	—
9	#16810399	—	2	1	no	—
10	C₄×#11719908	—	2	1	no	—
11	C₅₃×#533	—	2	1	no	—
12	C₅₃×#500	C₄	1	2	yes	complemented
13	C₅₃×#499	—	2	1	no	—
14	C₅₃×#499	C₄	1	2	yes	complemented
15	C₅₃×#498	—	2	1	no	—
16	C₅₃×#427	—	2	1	no	—
17	C₅₃×#427	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	#13133157	—	2	2	no	—
21	#13133157	Q₈	1	4	yes	complemented
22	C₂×#11719908	—	2	2	no	—
23	#13133151	—	4	2	no	—
24	#13133149	—	2	4	no	—
25	#13133124	—	4	2	no	—
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₄⋊Q₈×C₅₃	—	2	2	no	—
29	#125×C₅₃	—	2	2	no	—
30	C₂²⋊Q₈×C₅₃	—	4	2	no	—
31	C₄×Q₈×C₅₃	—	4	2	no	—
32	#117×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
34	#107×C₅₃	—	2	2	no	—
35	#107×C₅₃	C₂×C₄	1	4	yes	—
36	#106×C₅₃	—	2	2	no	—
37	#106×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₅₃	—	4	2	no	—
43	C₂²×C₄×C₅₃	D₈	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	4	2	no	—
46	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
49	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
50	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₅₃	#41	1	2	yes	—
54	C₄⋊C₄×C₅₃	—	2	4	no	—
55	C₄⋊C₄×C₅₃	—	4	4	no	—
56	C₄⋊C₄×C₅₃	—	4	2	no	—
57	C₂²⋊C₄×C₅₃	—	4	2	no	—
58	C₄²×C₅₃	—	4	2	no	—
59	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₅₃	SD₁₆	1	2	yes	—
62	C₄⋊Dic₅₃	—	4	2	no	—
63	C₄⋊Dic₅₃	—	4	2	no	—
64	#11719908	—	2	4	no	—
65	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
66	Q₈×C₅₃	—	8	4	no	—
67	C₂×C₄×C₅₃	—	4	2	no	—
68	C₂×C₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	#102	1	2	yes	—
70	C₂×C₄×C₅₃	#102	1	2	yes	—
71	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×Dic₅₃	—	8	4	no	—
78	C₂×Dic₅₃	—	8	2	no	—
79	C₈×C₅₃	#103	1	2	yes	—
80	C₈×C₅₃	#103	1	2	yes	—
81	C₈×C₅₃	C₄×Q₈	1	2	yes	—
82	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
83	#13288	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	—	2	2	no	—
85	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
86	C₂²×C₅₃	#486	1	1	yes	characteristic
87	C₂²×C₅₃	#500	1	1	yes	characteristic
88	C₄×C₅₃	—	2	2	no	—
89	C₄×C₅₃	—	2	2	no	—
90	C₄×C₅₃	—	8	4	no	—
91	C₄×C₅₃	#391	1	1	yes	characteristic
92	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
93	C₄×C₅₃	#487	1	1	yes	characteristic
94	C₄×C₅₃	#498	1	1	yes	characteristic
95	Dic₅₃	—	16	2	no	—
96	#3160	D₅₃	1	1	yes	characteristic
97	C₄×M₃₂	—	53	1	no	—
98	#1846	—	53	1	no	—
99	#1386	—	53	4	no	—
100	C₂×C₅₃	—	2	1	no	—
101	C₂×C₅₃	—	2	1	no	—
102	C₂×C₅₃	#1990	1	1	yes	characteristic
103	C₂×C₅₃	#2145	1	2	yes	—
104	#533	—	2	1	no	—
105	C₂×M₃₂	—	53	2	no	—
106	#500	Dic₅₃	1	2	yes	complemented
107	#499	—	2	1	no	—
108	#499	Dic₅₃	1	2	yes	complemented
109	#498	—	2	1	no	—
110	#427	—	2	1	no	—
111	#427	C₂×D₅₃	1	2	yes	—
112	C₂×#107	—	106	1	no	—
113	C₂×#106	—	53	2	no	—
114	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
115	#382	—	106	1	no	—
116	#346	—	106	1	no	—
117	C₄×C₁₆	—	106	1	no	—
118	C₅₃	#13288	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
121	C₄⋊Q₈	—	2	2	no	—
122	#125	—	2	2	no	—
123	C₂²⋊Q₈	—	4	2	no	—
124	C₄×Q₈	—	4	2	no	—
125	#117	—	2	2	no	—
126	C₂×C₄⋊C₄	—	106	2	no	—
127	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
128	M₃₂	—	53	4	no	—
129	M₃₂	—	106	2	no	—
130	C₂×C₁₆	—	106	2	no	—
131	#107	—	2	2	no	—
132	#107	—	212	2	no	—
133	#107	C₂×Dic₅₃	1	4	yes	—
134	#106	—	2	2	no	—
135	#106	—	2	2	no	—
136	#106	—	106	4	no	—
137	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
138	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
140	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
141	#95	—	212	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂²×C₄	—	212	2	no	—
144	C₂²×C₄	#11719922	1	2	yes	—
145	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
146	C₂×C₈	#11719924	1	2	yes	—
147	C₂×C₈	#11719925	1	2	yes	—
148	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
149	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
150	C₂×C₈	Q₈×D₅₃	1	2	yes	—
151	C₂×C₈	#11719949	1	2	yes	—
152	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
153	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	212	2	no	—
158	C₄⋊C₄	—	212	2	no	—
159	C₂²⋊C₄	—	4	2	no	—
160	C₄²	—	4	2	no	—
161	C₄²	#11719923	1	2	yes	—
162	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
163	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
164	C₁₆	—	106	4	no	—
165	C₂³	C₂×#11719922	1	1	yes	characteristic
166	Q₈	—	8	4	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	424	4	no	—
170	C₂×C₄	—	424	2	no	—
171	C₂×C₄	#13133138	1	2	yes	—
172	C₂×C₄	#13133138	1	2	yes	—
173	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
174	C₂×C₄	#13133206	1	1	yes	characteristic
175	C₂×C₄	#13133215	1	1	yes	characteristic
176	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
177	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
178	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
179	C₈	#13133141	1	2	yes	—
180	C₈	#13133141	1	2	yes	—
181	C₈	Q₈×Dic₅₃	1	2	yes	—
182	C₈	#13133193	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	C₂×#13133138	1	1	yes	characteristic
185	C₂²	#16810971	1	1	yes	characteristic
186	C₂²	#16811026	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	8	4	no	—
190	C₄	—	848	2	no	—
191	C₄	#16810530	1	1	yes	characteristic
192	C₄	C₂×#13133141	1	1	yes	characteristic
193	C₄	#16810975	1	1	yes	characteristic
194	C₄	#16811019	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	#20275677	1	1	yes	characteristic
198	C₂	#20276451	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#3160	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#3160	D₅₃
hypocenter	C₅₃	#13288
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
12. C₅₃×#500	C₄	semidirect
14. C₅₃×#499	C₄	semidirect
21. #13133157	Q₈	semidirect
106. #500	Dic₅₃	semidirect
108. #499	Dic₅₃	semidirect
118. C₅₃	#13288	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
#20274879	2
#20272820	2
C₅₃×#3160	2
#13288	53