Subgroups — #27283849

Order 13568.

526 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	C₂×#16811028	C₂	1	1	yes	maximal, characteristic
2	#20275675	C₂	1	2	yes	maximal
3	#20275516	C₂	1	2	yes	maximal
4	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	C₂²	1	2	yes	—
7	#16811028	—	2	2	no	—
8	#16811028	C₄	1	4	yes	complemented
9	C₂×#13133151	—	2	2	no	—
10	C₂×#13133140	—	2	4	no	—
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	#16810385	—	2	4	no	—
14	C₈×#88322	—	2	2	no	—
15	C₂×#107×C₅₃	—	2	2	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#384	C₂²	1	2	yes	—
18	C₂²×Dic₁₀₆	—	2	2	no	—
19	C₂²×#88322	D₄	1	2	yes	—
20	C₂²×#88322	D₄	1	2	yes	—
21	#13133217	—	2	2	no	—
22	#13133217	—	2	2	no	—
23	#13133217	—	2	2	no	—
24	#13133217	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₅₃	—	2	2	no	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	#13133151	—	2	8	no	—
28	#13133140	—	4	8	no	—
29	C₄×#88322	—	2	1	no	—
30	C₄×#88322	—	2	1	no	—
31	C₄×#88322	—	2	1	no	—
32	C₄×#88322	—	2	1	no	—
33	C₂²×C₈×C₅₃	D₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#107×C₅₃	—	4	4	no	—
39	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₅₃	D₄	1	2	yes	—
41	C₂²×C₄×C₅₃	—	4	2	no	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	4	2	no	—
46	C₂×Dic₁₀₆	—	2	4	no	—
47	C₂×Dic₁₀₆	—	4	4	no	—
48	C₂×Dic₁₀₆	—	4	4	no	—
49	C₂×Dic₁₀₆	—	4	2	no	—
50	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
51	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
52	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
53	C₄⋊C₄×C₅₃	—	4	4	no	—
54	C₄⋊C₄×C₅₃	—	4	4	no	—
55	C₄⋊C₄×C₅₃	—	4	4	no	—
56	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₃	#41	1	1	yes	characteristic
59	C₄²×C₅₃	#41	1	1	yes	characteristic
60	C₄⋊Dic₅₃	—	2	4	no	—
61	C₄⋊Dic₅₃	—	2	4	no	—
62	C₄⋊Dic₅₃	—	4	2	no	—
63	C₄⋊Dic₅₃	—	4	2	no	—
64	C₂×#88322	—	2	2	no	—
65	C₂×#88322	—	2	2	no	—
66	C₂×#88322	—	2	2	no	—
67	C₂×#88322	—	2	2	no	—
68	C₂×#88322	D₈	1	4	yes	—
69	C₂×#88322	Q₁₆	1	4	yes	—
70	C₂³×C₅₃	#127	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	8	2	no	—
72	C₂×C₄×C₅₃	—	8	2	no	—
73	C₂×C₄×C₅₃	—	8	2	no	—
74	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄×C₅₃	#135	1	2	yes	—
85	C₂×C₄×C₅₃	#135	1	2	yes	—
86	C₂×Dic₅₃	—	4	4	no	—
87	C₂×Dic₅₃	—	8	2	no	—
88	C₂×Dic₅₃	—	8	2	no	—
89	Dic₁₀₆	—	4	8	no	—
90	Dic₁₀₆	—	8	4	no	—
91	C₈×C₅₃	#103	1	8	yes	complemented
92	#88322	—	2	4	no	—
93	#88322	—	2	4	no	—
94	#9823	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	#390	1	1	yes	characteristic
96	C₂²×C₅₃	#492	1	1	yes	characteristic
97	C₂²×C₅₃	#493	1	1	yes	characteristic
98	C₂²×C₅₃	#494	1	1	yes	characteristic
99	C₂²×C₅₃	#494	1	1	yes	characteristic
100	C₂²×C₅₃	#495	1	1	yes	characteristic
101	C₂²×C₅₃	#495	1	1	yes	characteristic
102	C₄×C₅₃	—	16	2	no	—
103	C₄×C₅₃	C₂×#103	1	2	yes	—
104	C₄×C₅₃	C₄×D₈	1	2	yes	—
105	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
106	C₄×C₅₃	#466	1	2	yes	—
107	C₄×C₅₃	#467	1	2	yes	—
108	C₄×C₅₃	#487	1	2	yes	—
109	Dic₅₃	—	8	4	no	—
110	C₂×#500	—	53	1	no	—
111	#1990	—	53	2	no	—
112	#1956	—	53	2	no	—
113	#1845	D₅₃	1	1	yes	characteristic
114	C₂×C₈²	—	53	1	no	—
115	C₂×C₅₃	#1706	1	1	yes	characteristic
116	C₂×C₅₃	#1707	1	1	yes	characteristic
117	C₂×C₅₃	#1710	1	1	yes	characteristic
118	C₂×C₅₃	#1713	1	1	yes	characteristic
119	C₂×C₅₃	#1956	1	1	yes	characteristic
120	C₂×C₅₃	#1969	1	1	yes	characteristic
121	C₂×C₅₃	#1970	1	1	yes	characteristic
122	C₂×C₄⋊Q₈	—	53	2	no	—
123	#500	—	53	4	no	—
124	#500	—	106	2	no	—
125	C₂×#107	—	2	2	no	—
126	C₂×#107	—	106	2	no	—
127	C₂×#103	—	106	4	no	—
128	C₂×C₄×C₈	—	53	1	no	—
129	C₂×C₄×C₈	—	53	1	no	—
130	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
131	#384	C₂×D₅₃	1	2	yes	—
132	#340	—	106	4	no	—
133	C₈²	—	106	2	no	—
134	C₅₃	#9823	1	1	yes	characteristic, complemented, Sylow
135	C₂²×Q₈	—	106	2	no	—
136	C₂²×C₈	—	53	2	no	—
137	C₂²×C₈	—	53	2	no	—
138	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
139	C₄⋊Q₈	—	53	4	no	—
140	C₄⋊Q₈	—	106	2	no	—
141	C₄⋊Q₈	—	106	2	no	—
142	C₄⋊Q₈	—	106	2	no	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	106	2	no	—
147	C₂×C₄⋊C₄	—	106	2	no	—
148	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
149	#107	—	4	4	no	—
150	#107	—	106	8	no	—
151	#103	—	212	8	no	—
152	C₄×C₈	—	106	1	no	—
153	C₄×C₈	—	106	1	no	—
154	C₄×C₈	—	106	1	no	—
155	C₄×C₈	—	106	1	no	—
156	C₄×C₈	C₄×D₅₃	1	2	yes	—
157	C₄×C₈	D₂₁₂	1	2	yes	—
158	C₂×Q₈	—	106	4	no	—
159	C₂×Q₈	—	212	4	no	—
160	C₂×Q₈	—	212	4	no	—
161	C₂×Q₈	—	212	2	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	—	212	2	no	—
164	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
165	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
167	C₂×C₈	—	53	4	no	—
168	C₂×C₈	—	53	4	no	—
169	C₂×C₈	—	106	2	no	—
170	C₂×C₈	—	106	2	no	—
171	C₂×C₈	—	106	2	no	—
172	C₂×C₈	—	106	2	no	—
173	C₂×C₈	#11719921	1	4	yes	—
174	C₂×C₈	#11719923	1	4	yes	—
175	C₂×C₈	#11719925	1	4	yes	—
176	C₄⋊C₄	—	4	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	4	4	no	—
179	C₄⋊C₄	—	106	4	no	—
180	C₄⋊C₄	—	106	4	no	—
181	C₄⋊C₄	—	212	2	no	—
182	C₄⋊C₄	—	212	2	no	—
183	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
184	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
185	C₄²	#11719949	1	1	yes	characteristic
186	C₄²	#11719949	1	1	yes	characteristic
187	C₂³	#13133213	1	1	yes	characteristic
188	Q₈	—	424	4	no	—
189	Q₈	—	212	8	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	424	2	no	—
194	C₂×C₄	—	424	2	no	—
195	C₂×C₄	—	212	4	no	—
196	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
197	C₂×C₄	#13133179	1	1	yes	characteristic
198	C₂×C₄	#13133179	1	1	yes	characteristic
199	C₂×C₄	#13133192	1	1	yes	characteristic
200	C₂×C₄	#13133192	1	1	yes	characteristic
201	C₂×C₄	#13133206	1	1	yes	characteristic
202	C₂×C₄	D₈×D₅₃	1	2	yes	—
203	C₂×C₄	C₂×#11719923	1	2	yes	—
204	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
205	C₂×C₄	C₂×#11719925	1	2	yes	—
206	C₂×C₄	#13133236	1	2	yes	—
207	C₂×C₄	#13133240	1	2	yes	—
208	C₈	—	106	4	no	—
209	C₈	—	106	4	no	—
210	C₈	#13133140	1	8	yes	complemented
211	C₂²	#16810524	1	1	yes	characteristic
212	C₂²	#16810995	1	1	yes	characteristic
213	C₂²	#16810998	1	1	yes	characteristic
214	C₂²	#16811001	1	1	yes	characteristic
215	C₂²	#16811002	1	1	yes	characteristic
216	C₂²	#16811006	1	1	yes	characteristic
217	C₂²	#16811008	1	1	yes	characteristic
218	C₄	—	16	2	no	—
219	C₄	—	424	4	no	—
220	C₄	C₂×#13133140	1	2	yes	—
221	C₄	#16810705	1	2	yes	—
222	C₄	#16810717	1	2	yes	—
223	C₄	#16810860	1	2	yes	—
224	C₄	#16810865	1	2	yes	—
225	C₄	#16810975	1	2	yes	—
226	C₂	#20274304	1	1	yes	characteristic
227	C₂	#20274309	1	1	yes	characteristic
228	C₂	#20274324	1	1	yes	characteristic
229	C₂	#20274332	1	1	yes	characteristic
230	C₂	#20275516	1	1	yes	characteristic
231	C₂	#20275574	1	1	yes	characteristic
232	C₂	#20275579	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1845	C₂
socle, prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1845	D₅₃
hypocenter	C₅₃	#9823
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
8. #16811028	C₄	semidirect
91. C₈×C₅₃	#103	semidirect
134. C₅₃	#9823	semidirect
210. C₈	#13133140	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
#20275675	2
#20275516	2
C₂×C₈×#88322	2
C₅₃×#1845	2
#9823	53