Subgroups — #27282745

Order 13568.

382 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275478	C₂	1	2	yes	maximal
2	#20274875	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
4	#20272599	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133151	—	2	1	no	—
7	C₂×#13133151	—	2	1	no	—
8	C₂×#13133148	—	2	2	no	—
9	C₂×C₄⋊#88322	C₂²	1	2	yes	—
10	#16810493	C₂²	1	2	yes	—
11	#16810385	—	2	2	no	—
12	#16810376	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	—	2	1	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	1	no	—
18	C₂²×#88322	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	2	2	no	—
22	#13133151	—	4	2	no	—
23	#13133151	—	4	2	no	—
24	#13133148	—	4	4	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	—	2	1	no	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	1	no	—
35	C₄⋊C₈×C₅₃	Q₈	1	2	yes	—
36	C₄×C₈×C₅₃	—	4	2	no	—
37	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₃	Q₈	1	2	yes	—
39	#95×C₅₃	—	2	1	no	—
40	C₂²×C₄×C₅₃	—	2	1	no	—
41	C₂²×C₄×C₅₃	—	2	1	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	#41	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	4	2	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	4	1	no	—
50	C₂×C₈×C₅₃	—	4	1	no	—
51	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
54	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
55	C₄⋊C₄×C₅₃	—	2	1	no	—
56	C₄⋊C₄×C₅₃	—	2	1	no	—
57	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
58	C₄⋊C₄×C₅₃	Q₁₆	1	2	yes	—
59	C₄²×C₅₃	—	4	1	no	—
60	C₄²×C₅₃	—	4	1	no	—
61	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₅₃	#41	1	1	yes	characteristic
65	C₄⋊Dic₅₃	—	4	2	no	—
66	C₄⋊Dic₅₃	—	4	2	no	—
67	C₄⋊Dic₅₃	—	4	2	no	—
68	C₄⋊Dic₅₃	—	4	2	no	—
69	C₄⋊Dic₅₃	—	8	2	no	—
70	C₂×#88322	—	2	4	no	—
71	C₂×#88322	—	4	2	no	—
72	C₂×#88322	—	4	2	no	—
73	C₂³×C₅₃	#121	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	4	1	no	—
79	C₂×C₄×C₅₃	—	4	1	no	—
80	C₂×C₄×C₅₃	—	4	1	no	—
81	C₂×C₄×C₅₃	—	4	1	no	—
82	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
94	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
96	C₂×Dic₅₃	—	8	2	no	—
97	C₂×Dic₅₃	—	8	2	no	—
98	C₂×Dic₅₃	—	8	2	no	—
99	C₈×C₅₃	—	2	2	no	—
100	C₈×C₅₃	—	4	2	no	—
101	C₈×C₅₃	#106	1	4	yes	complemented
102	#88322	—	4	4	no	—
103	#9615	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	#389	1	1	yes	characteristic
105	C₂²×C₅₃	#460	1	1	yes	characteristic
106	C₂²×C₅₃	#461	1	1	yes	characteristic
107	C₂²×C₅₃	#466	1	1	yes	characteristic
108	C₂²×C₅₃	#467	1	1	yes	characteristic
109	C₂²×C₅₃	#472	1	1	yes	characteristic
110	C₂²×C₅₃	#473	1	1	yes	characteristic
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	4	2	no	—
114	C₄×C₅₃	—	4	2	no	—
115	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
116	C₄×C₅₃	#428	1	1	yes	characteristic
117	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
118	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
119	C₄×C₅₃	#475	1	1	yes	characteristic
120	C₄×C₅₃	#476	1	1	yes	characteristic
121	C₄×C₅₃	#499	1	1	yes	characteristic
122	C₄×C₅₃	#500	1	1	yes	characteristic
123	Dic₅₃	—	16	2	no	—
124	#1948	—	53	2	no	—
125	#1845	—	53	1	no	—
126	C₈×C₄⋊C₄	D₅₃	1	1	yes	characteristic
127	C₂×#335	—	53	1	no	—
128	#1296	—	53	2	no	—
129	C₂×C₅₃	#1668	1	1	yes	characteristic
130	C₂×C₅₃	#1670	1	1	yes	characteristic
131	C₂×C₅₃	#1673	1	1	yes	characteristic
132	C₂×C₅₃	#1675	1	1	yes	characteristic
133	C₂×C₅₃	#1942	1	1	yes	characteristic
134	C₂×C₅₃	#1948	1	1	yes	characteristic
135	C₂×C₅₃	#1952	1	1	yes	characteristic
136	C₂×#107	—	106	1	no	—
137	C₂×#107	—	106	1	no	—
138	C₂×#106	—	106	2	no	—
139	C₂×C₄⋊C₈	—	53	2	no	—
140	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
141	C₂×C₄×C₈	—	2	1	no	—
142	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
143	#384	—	53	2	no	—
144	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
145	#340	—	106	2	no	—
146	#336	—	2	1	no	—
147	#335	—	53	4	no	—
148	C₅₃	#9615	1	1	yes	characteristic, complemented, Sylow
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	—	106	2	no	—
151	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
152	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	106	2	no	—
154	C₂×C₄⋊C₄	—	106	2	no	—
155	C₂×C₄⋊C₄	—	106	2	no	—
156	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
157	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
158	C₂×C₄²	—	2	1	no	—
159	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
160	#107	—	212	2	no	—
161	#107	—	212	2	no	—
162	#106	—	212	4	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	—	53	4	no	—
165	C₄⋊C₈	—	106	2	no	—
166	C₄⋊C₈	Dic₁₀₆	1	2	yes	—
167	C₄×C₈	—	4	2	no	—
168	C₄×C₈	Dic₁₀₆	1	2	yes	—
169	C₄×C₈	C₄×D₅₃	1	2	yes	—
170	#95	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	212	2	no	—
174	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
175	C₂²×C₄	#11719945	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	106	4	no	—
183	C₂×C₈	—	212	2	no	—
184	C₂×C₈	—	212	2	no	—
185	C₂×C₈	#11719919	1	2	yes	—
186	C₂×C₈	#11719919	1	2	yes	—
187	C₂×C₈	#11719923	1	2	yes	—
188	C₂×C₈	#11719924	1	2	yes	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	424	2	no	—
192	C₄⋊C₄	—	212	2	no	—
193	C₄⋊C₄	—	212	2	no	—
194	C₄⋊C₄	—	212	2	no	—
195	C₄⋊C₄	—	212	2	no	—
196	C₄⋊C₄	D₄₂₄	1	2	yes	—
197	C₄⋊C₄	Dic₂₁₂	1	2	yes	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
201	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
202	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
203	C₄²	#11719945	1	1	yes	characteristic
204	C₂³	#13133191	1	1	yes	characteristic
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	424	2	no	—
214	C₂×C₄	—	424	2	no	—
215	C₂×C₄	—	424	2	no	—
216	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
217	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
218	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
219	C₂×C₄	#13133196	1	1	yes	characteristic
220	C₂×C₄	#13133209	1	1	yes	characteristic
221	C₂×C₄	#13133217	1	1	yes	characteristic
222	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
223	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
224	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
225	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
226	C₂×C₄	#13133239	1	1	yes	characteristic
227	C₂×C₄	#13133239	1	1	yes	characteristic
228	C₂×C₄	#13133242	1	1	yes	characteristic
229	C₂×C₄	#13133247	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	4	2	no	—
232	C₈	—	212	4	no	—
233	C₈	#13133148	1	4	yes	complemented
234	C₂²	#16810521	1	1	yes	characteristic
235	C₂²	#16810823	1	1	yes	characteristic
236	C₂²	#16810828	1	1	yes	characteristic
237	C₂²	#16810859	1	1	yes	characteristic
238	C₂²	#16810864	1	1	yes	characteristic
239	C₂²	#16810893	1	1	yes	characteristic
240	C₂²	#16810898	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	848	2	no	—
246	C₄	C₂×#13133148	1	1	yes	characteristic
247	C₄	#16810666	1	1	yes	characteristic
248	C₄	C₄×D₄₂₄	1	1	yes	characteristic
249	C₄	C₄×Dic₂₁₂	1	1	yes	characteristic
250	C₄	#16810908	1	1	yes	characteristic
251	C₄	#16810913	1	1	yes	characteristic
252	C₄	#16811025	1	1	yes	characteristic
253	C₄	#16811028	1	1	yes	characteristic
254	C₂	#20274075	1	1	yes	characteristic
255	C₂	#20274085	1	1	yes	characteristic
256	C₂	#20274104	1	1	yes	characteristic
257	C₂	#20274114	1	1	yes	characteristic
258	C₂	#20275448	1	1	yes	characteristic
259	C₂	#20275478	1	1	yes	characteristic
260	C₂	#20275500	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₈×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₈×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#9615
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
12. #16810376	C₄	semidirect
101. C₈×C₅₃	#106	semidirect
148. C₅₃	#9615	semidirect
233. C₈	#13133148	semidirect

Maximal subgroups

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

subgroup	index
#20275478	2
#20274875	2
C₂×#16810376	2
#20272599	2
C₈×C₄⋊C₄×C₅₃	2
#9615	53