Subgroups — #27536665

Order 13568.

1006 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811028	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811026	C₂	1	2	yes	maximal
3	#20283111	C₂	1	2	yes	maximal
4	#20274873	C₂	1	8	yes	maximal
5	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133217	C₂²	1	2	yes	—
8	C₂×#13133215	—	2	4	no	—
9	#16811028	—	2	4	no	—
10	#16811026	—	2	8	no	—
11	C₂×#13133151	C₂²	1	4	yes	—
12	C₂×#13133150	C₂²	1	8	yes	—
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	4	yes	—
15	#16810492	—	2	4	no	—
16	#16810492	C₂²	1	8	yes	—
17	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#500	—	2	4	no	—
20	C₂×#107×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₀₆	—	2	2	no	—
23	C₂²×#88322	C₂³	1	2	yes	—
24	#13133217	—	2	4	no	—
25	#13133217	—	2	4	no	—
26	#13133215	—	4	16	no	—
27	C₂×C₄⋊Dic₅₃	—	2	2	no	—
28	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
29	C₂×#11719919	—	2	16	no	—
30	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
31	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
32	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
33	#13133151	—	2	8	no	—
34	#13133150	—	2	16	no	—
35	C₄×#88322	D₄	1	4	yes	—
36	C₈×Dic₅₃	Q₈	1	16	yes	complemented
37	C₂²×Q₈×C₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₅₃	—	2	4	no	—
40	C₄⋊Q₈×C₅₃	—	2	4	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	#107×C₅₃	—	2	8	no	—
45	C₄×C₈×C₅₃	D₄	1	4	yes	—
46	C₂×Q₈×C₅₃	—	4	4	no	—
47	C₂×Q₈×C₅₃	—	4	8	no	—
48	C₂²×C₄×C₅₃	—	2	4	no	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
52	C₂²×Dic₅₃	—	2	4	no	—
53	C₂²×Dic₅₃	C₂×D₄	1	4	yes	—
54	C₂×Dic₁₀₆	—	4	4	no	—
55	C₂×Dic₁₀₆	—	4	8	no	—
56	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
57	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	—
58	C₄⋊C₄×C₅₃	—	2	4	no	—
59	C₄⋊C₄×C₅₃	—	2	4	no	—
60	C₄⋊C₄×C₅₃	—	4	4	no	—
61	C₄²×C₅₃	C₂×D₄	1	2	yes	—
62	C₄²×C₅₃	C₂×D₄	1	2	yes	—
63	C₄⋊Dic₅₃	—	2	4	no	—
64	C₄⋊Dic₅₃	—	2	4	no	—
65	C₄⋊Dic₅₃	—	4	4	no	—
66	#11719919	—	4	32	no	—
67	C₄×Dic₅₃	D₈	1	8	yes	—
68	C₄×Dic₅₃	Q₁₆	1	8	yes	—
69	C₄×Dic₅₃	C₂×Q₈	1	8	yes	—
70	C₂×#88322	C₂×D₄	1	4	yes	—
71	C₂×#88322	C₂×Q₈	1	8	yes	—
72	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
73	Q₈×C₅₃	—	8	8	no	—
74	C₂×C₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	—	4	8	no	—
76	C₂×C₄×C₅₃	#127	1	2	yes	—
77	C₂×C₄×C₅₃	#127	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
79	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
80	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
83	C₂×Dic₅₃	—	4	4	no	—
84	C₂×Dic₅₃	—	4	8	no	—
85	C₂×Dic₅₃	C₄⋊Q₈	1	8	yes	—
86	C₂×Dic₅₃	C₂×D₈	1	8	yes	—
87	C₂×Dic₅₃	C₂×Q₁₆	1	8	yes	—
88	Dic₁₀₆	—	8	8	no	—
89	C₈×C₅₃	C₄⋊Q₈	1	8	yes	complemented
90	#88322	C₄⋊Q₈	1	8	yes	complemented
91	#33312	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
93	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
94	C₂²×C₅₃	C₂²×D₈	1	2	yes	—
95	C₂²×C₅₃	C₂²×Q₁₆	1	2	yes	—
96	C₄×C₅₃	—	8	4	no	—
97	C₄×C₅₃	#493	1	4	yes	—
98	C₄×C₅₃	#494	1	4	yes	—
99	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
100	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
101	Dic₅₃	—	8	4	no	—
102	Dic₅₃	#500	1	16	yes	complemented
103	C₂×#500	—	53	1	no	—
104	C₂×#500	—	53	2	no	—
105	C₂×#500	D₅₃	1	1	yes	characteristic
106	#2868	—	53	2	no	—
107	#1845	—	53	8	no	—
108	C₄²×C₈	—	53	1	no	—
109	C₂×C₅₃	#2868	1	1	yes	characteristic
110	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
111	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
112	C₂×C₅₃	C₂×#500	1	4	yes	—
113	C₂×C₄⋊Q₈	—	53	2	no	—
114	C₂×C₄⋊Q₈	—	106	4	no	—
115	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
116	#500	—	2	4	no	—
117	#500	—	106	8	no	—
118	#500	—	106	4	no	—
119	C₂×#107	—	53	4	no	—
120	C₂×#107	—	53	8	no	—
121	C₂×#107	C₂×D₅₃	1	4	yes	—
122	C₂×C₄×C₈	—	53	4	no	—
123	C₂×C₄×C₈	—	53	1	no	—
124	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
125	#384	—	53	8	no	—
126	#384	—	106	4	no	—
127	C₄³	—	53	1	no	—
128	C₅₃	#33312	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	2	2	no	—
130	C₂²×Q₈	—	106	2	no	—
131	C₂²×C₈	—	53	2	no	—
132	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
133	C₄⋊Q₈	—	2	4	no	—
134	C₄⋊Q₈	—	2	4	no	—
135	C₄⋊Q₈	—	106	4	no	—
136	C₄⋊Q₈	—	106	4	no	—
137	C₄⋊Q₈	—	212	16	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	53	4	no	—
140	C₂×C₄⋊C₄	—	106	16	no	—
141	C₂×C₄⋊C₄	—	106	2	no	—
142	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
143	C₂×C₄²	—	53	2	no	—
144	C₂×C₄²	—	53	2	no	—
145	C₂×C₄²	—	53	2	no	—
146	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
147	#107	—	2	8	no	—
148	#107	—	106	16	no	—
149	#107	—	106	8	no	—
150	C₄×C₈	—	53	16	no	—
151	C₄×C₈	—	53	4	no	—
152	C₄×C₈	C₂²⋊D₅₃	1	4	yes	—
153	C₂×Q₈	—	4	4	no	—
154	C₂×Q₈	—	4	8	no	—
155	C₂×Q₈	—	212	4	no	—
156	C₂×Q₈	—	212	8	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	53	4	no	—
159	C₂²×C₄	—	106	4	no	—
160	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
161	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
162	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
163	C₂×C₈	—	53	4	no	—
164	C₂×C₈	—	53	8	no	—
165	C₂×C₈	Q₈×D₅₃	1	8	yes	—
166	C₂×C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	106	4	no	—
171	C₄⋊C₄	—	106	4	no	—
172	C₄⋊C₄	—	212	32	no	—
173	C₄⋊C₄	—	212	4	no	—
174	C₄²	—	53	8	no	—
175	C₄²	—	53	8	no	—
176	C₄²	—	53	8	no	—
177	C₄²	D₄×D₅₃	1	2	yes	—
178	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
179	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
180	Q₈	—	8	8	no	—
181	Q₈	—	424	8	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	53	8	no	—
185	C₂×C₄	—	53	8	no	—
186	C₂×C₄	—	53	8	no	—
187	C₂×C₄	—	212	8	no	—
188	C₂×C₄	—	212	4	no	—
189	C₂×C₄	#13133213	1	2	yes	—
190	C₂×C₄	#13133213	1	2	yes	—
191	C₂×C₄	D₈×D₅₃	1	4	yes	—
192	C₂×C₄	Q₁₆×D₅₃	1	4	yes	—
193	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
194	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
195	C₂×C₄	C₂×Q₈×D₅₃	1	4	yes	—
196	C₈	—	53	8	no	—
197	C₈	#13133215	1	8	yes	complemented
198	C₂²	C₂×#13133213	1	1	yes	characteristic
199	C₂²	C₄⋊Q₈×D₅₃	1	2	yes	—
200	C₂²	C₂×D₈×D₅₃	1	2	yes	—
201	C₂²	C₂×Q₁₆×D₅₃	1	2	yes	—
202	C₄	—	8	4	no	—
203	C₄	—	53	16	no	—
204	C₄	—	424	4	no	—
205	C₄	#16810998	1	4	yes	—
206	C₄	#16811001	1	4	yes	—
207	C₄	C₄⋊Q₈×D₅₃	1	2	yes	—
208	C₄	C₂×#13133215	1	2	yes	—
209	C₂	#20283111	1	1	yes	characteristic
210	C₂	C₂×#16810998	1	1	yes	characteristic
211	C₂	C₂×#16811001	1	1	yes	characteristic
212	C₂	#500×D₅₃	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#500	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	#33312
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
36. C₈×Dic₅₃	Q₈	semidirect
89. C₈×C₅₃	C₄⋊Q₈	semidirect
90. #88322	C₄⋊Q₈	semidirect
102. Dic₅₃	#500	semidirect
128. C₅₃	#33312	semidirect
197. C₈	#13133215	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811028	2
C₂×#16811026	2
#20283111	2
#20274873	2
C₄×C₈×Dic₅₃	2
C₂×C₅₃×#500	2
#33312	53