Subgroups — #27536667

Order 13568.

1006 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811027	C₂	1	4	yes	maximal
2	#20283112	C₂	1	2	yes	maximal
3	#20274875	C₂	1	1	yes	maximal, characteristic
4	#20274874	C₂	1	2	yes	maximal
5	#20274873	C₂	1	4	yes	maximal
6	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133216	—	2	4	no	—
9	C₂×#13133216	C₂²	1	4	yes	—
10	#16811027	—	2	16	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	2	yes	—
15	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
16	#16810493	C₂²	1	2	yes	—
17	#16810492	—	2	4	no	—
18	#16810492	C₂²	1	4	yes	—
19	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
20	C₂×#107×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#384	C₂²	1	2	yes	—
23	C₂²×Dic₁₀₆	—	2	4	no	—
24	C₂²×#88322	C₂³	1	2	yes	—
25	#13133216	—	2	16	no	—
26	#13133216	—	4	16	no	—
27	C₂×C₄⋊Dic₅₃	—	2	4	no	—
28	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
29	C₂×#11719919	—	2	8	no	—
30	C₂×#11719919	—	2	4	no	—
31	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
32	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
33	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
35	#13133151	—	2	8	no	—
36	#13133150	—	2	16	no	—
37	C₄×#88322	Q₈	1	4	yes	complemented
38	C₈×Dic₅₃	D₄	1	8	yes	—
39	C₈×Dic₅₃	Q₈	1	8	yes	complemented
40	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₅₃	—	2	4	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	Q₈	1	4	yes	complemented
46	C₂²×C₄×C₅₃	—	2	4	no	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	2	4	no	—
50	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
53	C₂×Dic₁₀₆	—	4	8	no	—
54	C₂×Dic₁₀₆	—	4	16	no	—
55	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
56	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
57	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
58	C₄⋊C₄×C₅₃	—	2	8	no	—
59	C₄⋊C₄×C₅₃	—	4	8	no	—
60	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
61	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
62	C₄⋊Dic₅₃	—	2	8	no	—
63	C₄⋊Dic₅₃	—	4	8	no	—
64	#11719919	—	4	8	no	—
65	#11719919	—	4	16	no	—
66	C₄×Dic₅₃	D₈	1	8	yes	—
67	C₄×Dic₅₃	Q₁₆	1	8	yes	—
68	C₄×Dic₅₃	C₂×D₄	1	4	yes	—
69	C₄×Dic₅₃	C₂×Q₈	1	4	yes	—
70	C₂×#88322	C₂×D₄	1	4	yes	—
71	C₂×#88322	C₂×Q₈	1	4	yes	—
72	C₂×#88322	C₂×Q₈	1	4	yes	—
73	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	—	4	8	no	—
76	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
77	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
78	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
79	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
80	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
82	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
83	C₂×Dic₅₃	—	4	4	no	—
84	C₂×Dic₅₃	—	4	8	no	—
85	C₂×Dic₅₃	#127	1	2	yes	—
86	C₂×Dic₅₃	#127	1	2	yes	—
87	C₂×Dic₅₃	C₄⋊Q₈	1	4	yes	—
88	C₂×Dic₅₃	C₂×D₈	1	4	yes	—
89	C₂×Dic₅₃	C₂×D₈	1	4	yes	—
90	C₂×Dic₅₃	C₂×Q₁₆	1	4	yes	—
91	C₂×Dic₅₃	C₂×Q₁₆	1	4	yes	—
92	Dic₁₀₆	—	8	16	no	—
93	C₈×C₅₃	C₄⋊Q₈	1	8	yes	complemented
94	#88322	C₄⋊Q₈	1	8	yes	complemented
95	#33312	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
97	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
98	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
99	C₂²×C₅₃	C₂²×D₈	1	2	yes	—
100	C₂²×C₅₃	C₂²×Q₁₆	1	2	yes	—
101	C₄×C₅₃	—	8	4	no	—
102	C₄×C₅₃	#500	1	8	yes	complemented
103	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
104	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
105	Dic₅₃	—	8	4	no	—
106	Dic₅₃	#493	1	4	yes	—
107	Dic₅₃	#494	1	4	yes	—
108	Dic₅₃	#500	1	8	yes	complemented
109	C₂×#500	—	53	4	no	—
110	#2868	—	53	2	no	—
111	#1845	—	53	4	no	—
112	#1845	—	53	1	no	—
113	#1845	—	53	2	no	—
114	#1845	D₅₃	1	1	yes	characteristic
115	C₄²×C₈	—	53	1	no	—
116	C₂×C₅₃	#2868	1	1	yes	characteristic
117	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
118	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
119	C₂×C₅₃	C₂×#500	1	2	yes	—
120	C₂×C₅₃	C₂×#500	1	2	yes	—
121	C₂×C₄⋊Q₈	—	53	4	no	—
122	C₂×C₄⋊Q₈	—	106	4	no	—
123	#500	—	106	16	no	—
124	C₂×#107	—	53	4	no	—
125	C₂×#107	—	53	8	no	—
126	C₂×#107	C₂×D₅₃	1	4	yes	—
127	C₂×C₄×C₈	—	53	2	no	—
128	C₂×C₄×C₈	—	53	2	no	—
129	C₂×C₄×C₈	—	53	1	no	—
130	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
131	#384	—	53	4	no	—
132	#384	—	53	2	no	—
133	#384	—	106	4	no	—
134	#384	C₂×D₅₃	1	2	yes	—
135	C₄³	—	53	1	no	—
136	C₅₃	#33312	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₈	—	106	4	no	—
138	C₂²×C₈	—	53	2	no	—
139	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
140	C₄⋊Q₈	—	106	16	no	—
141	C₄⋊Q₈	—	212	16	no	—
142	C₂×C₄⋊C₄	—	2	4	no	—
143	C₂×C₄⋊C₄	—	53	4	no	—
144	C₂×C₄⋊C₄	—	106	4	no	—
145	C₂×C₄⋊C₄	—	106	8	no	—
146	C₂×C₄⋊C₄	—	106	4	no	—
147	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
148	C₂×C₄²	—	53	2	no	—
149	C₂×C₄²	—	53	2	no	—
150	C₂×C₄²	—	53	1	no	—
151	C₂×C₄²	—	53	1	no	—
152	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
153	#107	—	2	8	no	—
154	#107	—	106	16	no	—
155	#107	—	106	8	no	—
156	C₄×C₈	—	53	8	no	—
157	C₄×C₈	—	53	8	no	—
158	C₄×C₈	—	53	4	no	—
159	C₄×C₈	Dic₁₀₆	1	4	yes	complemented
160	C₂×Q₈	—	212	8	no	—
161	C₂×Q₈	—	212	16	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	53	2	no	—
164	C₂²×C₄	—	53	1	no	—
165	C₂²×C₄	—	53	1	no	—
166	C₂²×C₄	—	106	4	no	—
167	C₂²×C₄	D₄×D₅₃	1	2	yes	—
168	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
169	C₂×C₈	—	53	4	no	—
170	C₂×C₈	—	53	4	no	—
171	C₂×C₈	—	53	4	no	—
172	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
173	C₂×C₈	D₄×D₅₃	1	4	yes	—
174	C₂×C₈	Q₈×D₅₃	1	4	yes	—
175	C₄⋊C₄	—	2	8	no	—
176	C₄⋊C₄	—	4	8	no	—
177	C₄⋊C₄	—	106	8	no	—
178	C₄⋊C₄	—	212	8	no	—
179	C₄⋊C₄	—	212	16	no	—
180	C₄⋊C₄	—	212	8	no	—
181	C₄²	—	53	8	no	—
182	C₄²	—	53	8	no	—
183	C₄²	—	53	4	no	—
184	C₄²	—	53	4	no	—
185	C₄²	C₂×Dic₁₀₆	1	2	yes	—
186	C₄²	Q₈×D₅₃	1	2	yes	—
187	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
188	Q₈	—	424	16	no	—
189	C₂×C₄	—	4	8	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	53	4	no	—
192	C₂×C₄	—	53	4	no	—
193	C₂×C₄	—	53	4	no	—
194	C₂×C₄	—	53	4	no	—
195	C₂×C₄	—	53	2	no	—
196	C₂×C₄	—	53	2	no	—
197	C₂×C₄	—	53	4	no	—
198	C₂×C₄	—	212	8	no	—
199	C₂×C₄	—	212	4	no	—
200	C₂×C₄	#13133216	1	4	yes	—
201	C₂×C₄	D₈×D₅₃	1	4	yes	—
202	C₂×C₄	Q₁₆×D₅₃	1	4	yes	—
203	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
204	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
205	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
206	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
207	C₈	—	53	8	no	—
208	C₈	#13133216	1	8	yes	complemented
209	C₂²	#127×D₅₃	1	1	yes	characteristic
210	C₂²	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
211	C₂²	C₂×#13133216	1	1	yes	characteristic
212	C₂²	C₂×D₈×D₅₃	1	2	yes	—
213	C₂²	C₂×Q₁₆×D₅₃	1	2	yes	—
214	C₄	—	8	4	no	—
215	C₄	—	53	8	no	—
216	C₄	—	53	4	no	—
217	C₄	—	53	4	no	—
218	C₄	—	424	4	no	—
219	C₄	#16811027	1	8	yes	complemented
220	C₄	C₄⋊Q₈×D₅₃	1	2	yes	—
221	C₄	C₂×#13133216	1	2	yes	—
222	C₂	#20283112	1	1	yes	characteristic
223	C₂	#493×D₅₃	1	1	yes	characteristic
224	C₂	#494×D₅₃	1	1	yes	characteristic
225	C₂	#500×D₅₃	1	2	yes	—
226	C₂	C₂×#16811027	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1845	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#1845	D₅₃
hypocenter	C₅₃	#33312
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
37. C₄×#88322	Q₈	semidirect
39. C₈×Dic₅₃	Q₈	semidirect
45. C₄×C₈×C₅₃	Q₈	semidirect
93. C₈×C₅₃	C₄⋊Q₈	semidirect
94. #88322	C₄⋊Q₈	semidirect
102. C₄×C₅₃	#500	semidirect
108. Dic₅₃	#500	semidirect
136. C₅₃	#33312	semidirect
159. C₄×C₈	Dic₁₀₆	semidirect
208. C₈	#13133216	semidirect
219. C₄	#16811027	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811027	2
#20283112	2
#20274875	2
#20274874	2
#20274873	2
C₄×C₈×Dic₅₃	2
C₅₃×#1845	2
#33312	53