Subgroups — #27283374

Order 13568.

474 conjugacy classes of subgroups.

The subgroup lattice has 244 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	#20275637	C₂	1	2	yes	maximal
3	#20274582	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810370	C₂	1	2	yes	maximal
5	C₂×C₅₃×#334	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	1	no	—
10	C₂×#13133151	—	2	1	no	—
11	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₂×#13133142	—	2	4	no	—
13	C₂×#13133127	—	2	1	no	—
14	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
15	#16810377	—	2	1	no	—
16	#16810370	—	2	4	no	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#334	—	2	2	no	—
20	C₂²×Dic₁₀₆	—	2	2	no	—
21	C₂²×#88322	—	2	1	no	—
22	#13133217	—	2	2	no	—
23	#13133217	—	2	2	no	—
24	#13133217	—	2	2	no	—
25	#13133217	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×C₄⋊Dic₅₃	—	2	2	no	—
28	C₂×C₄⋊Dic₅₃	D₄	1	2	yes	—
29	C₂×C₄×Dic₅₃	—	2	1	no	—
30	#13133151	—	2	4	no	—
31	#13133151	—	2	4	no	—
32	C₄⋊#88322	—	2	1	no	—
33	C₄⋊#88322	D₄	1	2	yes	—
34	#13133142	—	4	8	no	—
35	#13133127	—	4	2	no	—
36	#13133124	—	2	1	no	—
37	C₂²×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
48	C₂²×Dic₅₃	—	2	2	no	—
49	C₂²×Dic₅₃	—	4	2	no	—
50	C₂×Dic₁₀₆	—	2	4	no	—
51	C₂×Dic₁₀₆	—	4	4	no	—
52	C₂×Dic₁₀₆	—	4	4	no	—
53	C₂×Dic₁₀₆	—	4	2	no	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	—	2	1	no	—
56	C₂×C₈×C₅₃	—	4	2	no	—
57	C₂×C₈×C₅₃	—	4	4	no	—
58	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
61	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
62	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₅₃	#41	1	1	yes	characteristic
66	C₄⋊Dic₅₃	—	2	4	no	—
67	C₄⋊Dic₅₃	—	2	4	no	—
68	C₄⋊Dic₅₃	—	2	4	no	—
69	C₄⋊Dic₅₃	—	4	2	no	—
70	C₄⋊Dic₅₃	—	4	2	no	—
71	C₄×Dic₅₃	—	4	1	no	—
72	C₄×Dic₅₃	—	4	1	no	—
73	C₂×#88322	—	2	2	no	—
74	C₂×#88322	—	4	1	no	—
75	C₂×#88322	—	4	1	no	—
76	C₂³×C₅₃	#121	1	1	yes	characteristic
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
87	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
89	C₂×Dic₅₃	—	4	4	no	—
90	C₂×Dic₅₃	—	4	2	no	—
91	C₂×Dic₅₃	—	4	2	no	—
92	C₂×Dic₅₃	—	4	2	no	—
93	C₂×Dic₅₃	—	8	4	no	—
94	Dic₁₀₆	—	4	8	no	—
95	Dic₁₀₆	—	8	2	no	—
96	Dic₁₀₆	—	8	2	no	—
97	C₈×C₅₃	—	2	2	no	—
98	C₈×C₅₃	—	8	2	no	—
99	C₈×C₅₃	#103	1	4	yes	complemented
100	#88322	—	4	2	no	—
101	#9740	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	#390	1	1	yes	characteristic
103	C₂²×C₅₃	#461	1	1	yes	characteristic
104	C₂²×C₅₃	#462	1	1	yes	characteristic
105	C₂²×C₅₃	#469	1	2	yes	—
106	C₂²×C₅₃	#473	1	2	yes	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
110	C₄×C₅₃	#419	1	1	yes	characteristic
111	C₄×C₅₃	#442	1	2	yes	—
112	C₄×C₅₃	#461	1	1	yes	characteristic
113	C₄×C₅₃	#462	1	1	yes	characteristic
114	C₄×C₅₃	#496	1	2	yes	—
115	Dic₅₃	—	8	2	no	—
116	Dic₅₃	—	8	4	no	—
117	C₂×#500	—	53	1	no	—
118	#1981	—	53	2	no	—
119	#1770	—	53	1	no	—
120	C₂×#334	D₅₃	1	1	yes	characteristic
121	C₂×#332	—	53	2	no	—
122	C₂×C₅₃	#1693	1	2	yes	—
123	C₂×C₅₃	#1694	1	2	yes	—
124	C₂×C₅₃	#1972	1	1	yes	characteristic
125	C₂×C₅₃	#1979	1	1	yes	characteristic
126	C₂×C₅₃	#1981	1	1	yes	characteristic
127	C₂×C₄⋊Q₈	—	53	2	no	—
128	#500	—	53	4	no	—
129	#500	—	106	2	no	—
130	C₂×#107	—	106	1	no	—
131	C₂×#107	—	106	1	no	—
132	C₂×C₄⋊C₈	—	53	1	no	—
133	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
134	C₂×#103	—	106	4	no	—
135	C₂×#97	—	106	1	no	—
136	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
137	C₄×C₄⋊C₄	—	53	1	no	—
138	#336	—	106	1	no	—
139	#334	—	2	2	no	—
140	#332	—	106	4	no	—
141	C₅₃	#9740	1	1	yes	characteristic, complemented, Sylow
142	C₂²×Q₈	—	106	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	106	1	no	—
145	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
146	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
147	C₄⋊Q₈	—	53	4	no	—
148	C₄⋊Q₈	—	106	2	no	—
149	C₄⋊Q₈	—	106	2	no	—
150	C₄⋊Q₈	—	106	2	no	—
151	C₂×C₄⋊C₄	—	53	2	no	—
152	C₂×C₄⋊C₄	—	106	2	no	—
153	C₂×C₄⋊C₄	—	106	2	no	—
154	C₂×C₄²	—	106	1	no	—
155	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
156	#107	—	106	4	no	—
157	#107	—	106	4	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	53	2	no	—
161	C₄⋊C₈	—	106	1	no	—
162	#103	—	212	8	no	—
163	#97	—	212	2	no	—
164	C₄×C₈	C₄×D₅₃	1	2	yes	—
165	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
166	#95	—	106	1	no	—
167	C₂×Q₈	—	106	4	no	—
168	C₂×Q₈	—	212	4	no	—
169	C₂×Q₈	—	212	4	no	—
170	C₂×Q₈	—	212	2	no	—
171	C₂²×C₄	—	106	2	no	—
172	C₂²×C₄	—	212	2	no	—
173	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
174	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
175	C₂²×C₄	#11719949	1	1	yes	characteristic
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	4	no	—
180	C₂×C₈	—	106	2	no	—
181	C₂×C₈	—	212	1	no	—
182	C₂×C₈	—	212	1	no	—
183	C₂×C₈	#11719913	1	2	yes	—
184	C₂×C₈	Dic₂₁₂	1	2	yes	—
185	C₂×C₈	#11719921	1	2	yes	—
186	C₂×C₈	#11719921	1	2	yes	—
187	C₄⋊C₄	—	106	4	no	—
188	C₄⋊C₄	—	106	4	no	—
189	C₄⋊C₄	—	106	4	no	—
190	C₄⋊C₄	—	212	2	no	—
191	C₄⋊C₄	—	212	2	no	—
192	C₄²	—	212	1	no	—
193	C₄²	—	212	1	no	—
194	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
195	C₄²	D₄×D₅₃	1	1	yes	characteristic
196	C₄²	#11719949	1	1	yes	characteristic
197	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
198	C₂³	#13133192	1	1	yes	characteristic
199	Q₈	—	424	2	no	—
200	Q₈	—	424	2	no	—
201	Q₈	—	212	8	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	424	4	no	—
204	C₂×C₄	—	212	4	no	—
205	C₂×C₄	—	212	2	no	—
206	C₂×C₄	—	212	2	no	—
207	C₂×C₄	—	212	2	no	—
208	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
209	C₂×C₄	#13133179	1	1	yes	characteristic
210	C₂×C₄	#13133179	1	1	yes	characteristic
211	C₂×C₄	#13133192	1	1	yes	characteristic
212	C₂×C₄	#13133206	1	1	yes	characteristic
213	C₂×C₄	#13133213	1	1	yes	characteristic
214	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
215	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
216	C₂×C₄	#13133244	1	4	yes	—
217	C₂×C₄	#13133250	1	1	yes	characteristic
218	C₂×C₄	#13133250	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	—	8	2	no	—
221	C₈	—	212	2	no	—
222	C₈	#13133142	1	4	yes	complemented
223	C₂²	#16810524	1	1	yes	characteristic
224	C₂²	#16810829	1	1	yes	characteristic
225	C₂²	#16810834	1	1	yes	characteristic
226	C₂²	#16810877	1	2	yes	—
227	C₂²	#16810899	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	424	2	no	—
231	C₄	—	424	4	no	—
232	C₄	C₂×#13133142	1	1	yes	characteristic
233	C₄	#16810633	1	1	yes	characteristic
234	C₄	#16810734	1	2	yes	—
235	C₄	#16810829	1	1	yes	characteristic
236	C₄	#16810834	1	1	yes	characteristic
237	C₄	#16811009	1	2	yes	—
238	C₂	#20274229	1	2	yes	—
239	C₂	#20274234	1	2	yes	—
240	C₂	#20275589	1	1	yes	characteristic
241	C₂	#20275628	1	1	yes	characteristic
242	C₂	#20275637	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133192
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#334	C₂
socle, prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#334	D₅₃
hypocenter	C₅₃	#9740
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. #16811028	C₄	semidirect
99. C₈×C₅₃	#103	semidirect
141. C₅₃	#9740	semidirect
222. C₈	#13133142	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
#20275637	2
#20274582	2
C₂×#16810370	2
C₂×C₅₃×#334	2
#9740	53