Subgroups — #27302646

Order 13568.

332 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285229	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810444	C₂	1	1	yes	maximal, characteristic
3	#20272829	C₂	1	2	yes	maximal
4	#20272816	C₂	1	2	yes	maximal
5	C₅₃×#1772	C₂	1	1	yes	maximal, characteristic
6	#16811180	—	2	1	no	—
7	C₂×#13133157	C₂²	1	2	yes	—
8	#16811028	—	2	1	no	—
9	#16811025	—	2	1	no	—
10	#16811025	C₄	1	2	yes	complemented
11	#16811021	C₄	1	2	yes	complemented
12	#16810661	—	2	1	no	—
13	#16810661	C₂²	1	2	yes	—
14	#16810444	—	2	1	no	—
15	#16810444	—	2	1	no	—
16	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#427	C₂²	1	1	yes	characteristic
18	C₅₃×#405	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#339	—	2	1	no	—
21	#13133217	—	2	2	no	—
22	#13133209	—	2	2	no	—
23	#13133196	—	4	2	no	—
24	C₄×Dic₁₀₆	—	4	2	no	—
25	#13133176	—	2	2	no	—
26	#13133157	—	2	2	no	—
27	#13133157	—	2	2	no	—
28	C₂×#11719908	—	2	2	no	—
29	#13133151	—	2	2	no	—
30	#13133151	—	2	2	no	—
31	#13133149	—	2	2	no	—
32	#13133149	C₂×C₄	1	4	yes	—
33	C₂×M₁₆×C₅₃	—	2	1	no	—
34	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	#117×C₅₃	—	2	1	no	—
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	#108×C₅₃	—	2	1	no	—
39	#108×C₅₃	Q₈	1	2	yes	complemented
40	#107×C₅₃	—	2	1	no	—
41	#106×C₅₃	Q₈	1	2	yes	complemented
42	#97×C₅₃	—	4	1	no	—
43	C₄×C₈×C₅₃	—	4	1	no	—
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₁₀₆	—	4	2	no	—
51	M₁₆×C₅₃	—	2	2	no	—
52	M₁₆×C₅₃	—	4	1	no	—
53	C₂×C₈×C₅₃	—	4	1	no	—
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
57	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
58	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
59	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
65	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
66	C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₂²⋊C₄×C₅₃	—	4	1	no	—
68	C₄²×C₅₃	—	2	1	no	—
69	C₄²×C₅₃	—	4	1	no	—
70	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²⋊Dic₅₃	—	4	2	no	—
73	C₄⋊Dic₅₃	—	2	4	no	—
74	C₄⋊Dic₅₃	—	4	2	no	—
75	#11719919	—	4	4	no	—
76	C₄×Dic₅₃	—	4	2	no	—
77	#11719908	—	4	2	no	—
78	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	4	2	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×Dic₅₃	—	4	4	no	—
90	Dic₁₀₆	—	8	4	no	—
91	C₈×C₅₃	—	4	2	no	—
92	C₈×C₅₃	#103	1	1	yes	characteristic
93	C₈×C₅₃	#103	1	1	yes	characteristic
94	C₈×C₅₃	#103	1	1	yes	characteristic
95	C₈×C₅₃	#103	1	1	yes	characteristic
96	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
97	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
98	C₈×C₅₃	#124	1	1	yes	characteristic
99	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
100	#13271	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	—	4	1	no	—
102	C₂²×C₅₃	#418	1	1	yes	characteristic
103	C₂²×C₅₃	#476	1	1	yes	characteristic
104	C₂²×C₅₃	#477	1	1	yes	characteristic
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	4	1	no	—
107	C₄×C₅₃	—	8	2	no	—
108	C₄×C₅₃	#391	1	1	yes	characteristic
109	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
110	C₄×C₅₃	#476	1	1	yes	characteristic
111	C₄×C₅₃	#477	1	1	yes	characteristic
112	Dic₅₃	—	8	4	no	—
113	#3160	—	53	1	no	—
114	C₂×#364	—	53	1	no	—
115	#1772	D₅₃	1	1	yes	characteristic
116	#1389	—	53	2	no	—
117	#1384	—	53	2	no	—
118	C₂×C₅₃	—	2	1	no	—
119	C₂×C₅₃	—	2	1	no	—
120	C₂×C₅₃	—	2	1	no	—
121	C₂×C₅₃	#1987	1	1	yes	characteristic
122	#533	—	106	1	no	—
123	C₂×M₃₂	—	53	2	no	—
124	#500	—	106	1	no	—
125	#499	—	53	2	no	—
126	#499	—	106	1	no	—
127	#498	—	53	2	no	—
128	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
129	#427	—	53	2	no	—
130	#427	—	106	1	no	—
131	#427	C₂×D₅₃	1	1	yes	characteristic
132	#405	—	2	1	no	—
133	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
134	#364	—	106	1	no	—
135	#364	—	106	1	no	—
136	#339	—	2	1	no	—
137	C₅₃	#13271	1	1	yes	characteristic, complemented, Sylow
138	C₂×M₁₆	—	2	1	no	—
139	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
141	C₄⋊Q₈	—	106	2	no	—
142	#125	—	106	2	no	—
143	C₂²⋊Q₈	—	212	2	no	—
144	C₄×Q₈	—	212	2	no	—
145	#117	—	2	1	no	—
146	#117	—	106	2	no	—
147	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
148	M₃₂	—	106	2	no	—
149	M₃₂	—	106	2	no	—
150	C₂×C₁₆	—	106	2	no	—
151	#108	—	2	1	no	—
152	#108	Dic₁₀₆	1	2	yes	complemented
153	#107	—	2	1	no	—
154	#107	—	106	2	no	—
155	#107	—	106	2	no	—
156	#106	—	53	4	no	—
157	#106	—	106	2	no	—
158	#106	Dic₁₀₆	1	2	yes	complemented
159	#97	—	4	1	no	—
160	C₄×C₈	—	4	1	no	—
161	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
162	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
163	C₄×C₈	D₂₁₂	1	1	yes	characteristic
164	C₄×C₈	D₂₁₂	1	1	yes	characteristic
165	C₂×Q₈	—	212	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
168	M₁₆	—	2	2	no	—
169	M₁₆	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	#11719921	1	1	yes	characteristic
172	C₂×C₈	#11719921	1	1	yes	characteristic
173	C₂×C₈	#11719923	1	1	yes	characteristic
174	C₂×C₈	#11719923	1	1	yes	characteristic
175	C₂×C₈	#11719925	1	1	yes	characteristic
176	C₂×C₈	#11719925	1	1	yes	characteristic
177	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
178	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
179	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
180	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
181	C₂×C₈	#11719945	1	1	yes	characteristic
182	C₂×C₈	#11719945	1	1	yes	characteristic
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	106	4	no	—
185	C₄⋊C₄	—	212	4	no	—
186	C₄⋊C₄	—	212	2	no	—
187	C₂²⋊C₄	—	4	1	no	—
188	C₂²⋊C₄	—	212	2	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	4	1	no	—
191	C₄²	—	212	2	no	—
192	C₄²	#11719921	1	1	yes	characteristic
193	C₄²	#11719921	1	1	yes	characteristic
194	C₁₆	—	212	2	no	—
195	C₂³	#13133242	1	1	yes	characteristic
196	Q₈	—	424	4	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	212	4	no	—
202	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
203	C₂×C₄	#13133196	1	1	yes	characteristic
204	C₂×C₄	#13133196	1	1	yes	characteristic
205	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
206	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
207	C₂×C₄	#13133242	1	1	yes	characteristic
208	C₈	—	4	2	no	—
209	C₈	#13133140	1	1	yes	characteristic
210	C₈	#13133140	1	1	yes	characteristic
211	C₈	#13133140	1	1	yes	characteristic
212	C₈	#13133140	1	1	yes	characteristic
213	C₈	C₄×Dic₁₀₆	1	1	yes	characteristic
214	C₈	C₄×Dic₁₀₆	1	1	yes	characteristic
215	C₈	#13133205	1	1	yes	characteristic
216	C₈	#13133217	1	1	yes	characteristic
217	C₂²	—	4	1	no	—
218	C₂²	#16810627	1	1	yes	characteristic
219	C₂²	#16810913	1	1	yes	characteristic
220	C₂²	#16810918	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	8	2	no	—
224	C₄	—	424	4	no	—
225	C₄	#16810529	1	1	yes	characteristic
226	C₄	C₂×#13133140	1	1	yes	characteristic
227	C₄	#16810913	1	1	yes	characteristic
228	C₄	#16810918	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	#20275663	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810913
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1772	C₂
socle	C₂×C₅₃	#1987
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1772	D₅₃
hypocenter	C₅₃	#13271
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
10. #16811025	C₄	semidirect
11. #16811021	C₄	semidirect
39. #108×C₅₃	Q₈	semidirect
41. #106×C₅₃	Q₈	semidirect
137. C₅₃	#13271	semidirect
152. #108	Dic₁₀₆	semidirect
158. #106	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20285229	2
C₂×#16810444	2
#20272829	2
#20272816	2
C₅₃×#1772	2
#13271	53