Subgroups — #27302268

Order 13568.

466 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276471	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810405	C₂	1	4	yes	maximal, complemented
3	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₄⋊#11719908	—	2	1	no	—
8	C₄⋊#11719908	—	2	1	no	—
9	#16810405	—	2	8	no	—
10	#16810399	—	2	1	no	—
11	C₄×#11719908	—	2	1	no	—
12	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
14	C₅₃×#436	—	2	1	no	—
15	C₅₃×#436	—	2	1	no	—
16	C₅₃×#436	—	2	1	no	—
17	C₅₃×#436	—	2	1	no	—
18	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
19	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	#13133157	—	2	2	no	—
22	#13133157	—	2	4	no	—
23	#13133157	D₄	1	4	yes	complemented
24	C₂×#11719908	—	2	2	no	—
25	C₂×#11719908	—	2	2	no	—
26	C₂²×D₄×C₅₃	C₈	1	1	yes	characteristic
27	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×M₁₆×C₅₃	—	2	4	no	—
29	C₂×M₁₆×C₅₃	—	2	2	no	—
30	C₂×M₁₆×C₅₃	—	2	2	no	—
31	C₂×M₁₆×C₅₃	—	2	4	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₄×D₄×C₅₃	—	2	1	no	—
36	C₄×D₄×C₅₃	—	2	1	no	—
37	C₄×D₄×C₅₃	—	2	1	no	—
38	C₄×D₄×C₅₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	—	2	1	no	—
43	C₄⋊C₈×C₅₃	—	2	1	no	—
44	C₂²⋊C₈×C₅₃	—	2	2	no	—
45	C₂²⋊C₈×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂⁴×C₅₃	C₂×C₈	1	2	yes	—
51	C₂×D₄×C₅₃	—	2	1	no	—
52	C₂×D₄×C₅₃	—	2	1	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	M₁₆×C₅₃	—	4	4	no	—
63	M₁₆×C₅₃	—	4	2	no	—
64	M₁₆×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	4	no	—
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	#41	1	2	yes	—
74	C₄⋊C₄×C₅₃	—	2	1	no	—
75	C₄⋊C₄×C₅₃	—	2	1	no	—
76	C₂²⋊C₄×C₅₃	—	2	2	no	—
77	C₂²⋊C₄×C₅₃	—	2	2	no	—
78	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
81	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
82	#11719908	—	2	4	no	—
83	#11719908	—	4	2	no	—
84	C₂³×C₅₃	—	2	2	no	—
85	C₂³×C₅₃	—	2	4	no	—
86	C₂³×C₅₃	—	4	2	no	—
87	C₂³×C₅₃	—	4	2	no	—
88	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
89	D₄×C₅₃	—	4	2	no	—
90	D₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	4	no	—
94	C₂×C₄×C₅₃	—	4	1	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	4	no	—
98	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#131	1	2	yes	—
106	C₈×C₅₃	—	2	1	no	—
107	C₈×C₅₃	—	2	1	no	—
108	C₈×C₅₃	—	4	2	no	—
109	C₈×C₅₃	#121	1	2	yes	—
110	C₈×C₅₃	#124	1	1	yes	characteristic
111	C₈×C₅₃	#127	1	1	yes	characteristic
112	#13194	—	53	1	no	maximal, Sylow
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	2	1	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	—	4	2	no	—
117	C₂²×C₅₃	—	4	2	no	—
118	C₂²×C₅₃	—	8	4	no	—
119	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₂²×C₅₃	C₈×D₄	1	2	yes	—
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	4	2	no	—
125	C₄×C₅₃	#349	1	2	yes	complemented
126	C₄×C₅₃	#390	1	1	yes	characteristic
127	C₄×C₅₃	#408	1	1	yes	characteristic
128	C₄×C₅₃	#436	1	2	yes	—
129	C₂×#436	D₅₃	1	1	yes	characteristic
130	#2150	—	53	1	no	—
131	C₂×#349	—	53	4	no	—
132	C₄×M₃₂	—	53	1	no	—
133	C₂×C₅₃	—	2	2	no	—
134	C₂×C₅₃	—	8	2	no	—
135	C₂×C₅₃	#1954	1	1	yes	characteristic
136	C₂×C₅₃	C₂×#349	1	1	yes	characteristic
137	C₂×C₅₃	#2115	1	1	yes	characteristic
138	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
139	C₂×C₄×D₄	Dic₅₃	1	1	yes	characteristic
140	C₂×M₃₂	—	53	2	no	—
141	C₂×M₃₂	—	53	2	no	—
142	#436	—	2	1	no	—
143	#436	—	2	1	no	—
144	#436	—	2	1	no	—
145	#436	—	2	1	no	—
146	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
147	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
148	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
149	C₄⋊C₁₆	—	106	1	no	—
150	C₄⋊C₁₆	—	106	1	no	—
151	#349	—	106	8	no	—
152	#346	—	106	1	no	—
153	C₄×C₁₆	—	106	1	no	—
154	C₅₃	#13194	1	1	yes	characteristic, complemented, Sylow
155	C₂²×D₄	#88322	1	1	yes	characteristic
156	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
162	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	—	2	1	no	—
168	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	#88322	1	2	yes	—
170	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
171	M₃₂	—	53	4	no	—
172	M₃₂	—	106	2	no	—
173	M₃₂	—	106	4	no	—
174	C₂×C₁₆	—	106	2	no	—
175	C₂×C₁₆	—	106	2	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	—	2	1	no	—
178	C₂²⋊C₈	—	2	2	no	—
179	C₂²⋊C₈	—	2	2	no	—
180	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
183	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
184	C₂⁴	C₂×#88322	1	2	yes	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	C₂×#88322	1	2	yes	—
193	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
194	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
195	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
196	M₁₆	—	4	4	no	—
197	M₁₆	—	4	2	no	—
198	M₁₆	—	4	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
204	C₂×C₈	D₄×D₅₃	1	2	yes	—
205	C₂×C₈	#11719947	1	2	yes	—
206	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
207	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₄²	#11719917	1	1	yes	characteristic
213	C₄²	#11719917	1	1	yes	characteristic
214	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
215	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
216	C₁₆	—	106	4	no	—
217	C₁₆	—	212	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	2	no	—
222	C₂³	C₂²×#88322	1	1	yes	characteristic
223	D₄	—	4	2	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
233	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
234	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
235	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
237	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
238	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
239	C₂×C₄	#13133224	1	2	yes	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	4	2	no	—
243	C₈	#13133188	1	2	yes	—
244	C₈	#13133204	1	1	yes	characteristic
245	C₈	#13133213	1	1	yes	characteristic
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	8	4	no	—
252	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
253	C₂²	D₄×#88322	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	2	no	—
258	C₄	#16810405	1	2	yes	complemented
259	C₄	#16810526	1	1	yes	characteristic
260	C₄	#16810590	1	1	yes	characteristic
261	C₄	#16810701	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	8	2	no	—
264	C₂	#20275510	1	1	yes	characteristic
265	C₂	C₂×#16810405	1	1	yes	characteristic
266	C₂	#20276336	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#436	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₅₃	C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#436	D₅₃
hypocenter	C₅₃	#13194
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. #20276471	C₂	semidirect
2. C₂×#16810405	C₂	semidirect
3. C₄×#13133157	C₂	semidirect
23. #13133157	D₄	semidirect
125. C₄×C₅₃	#349	semidirect
154. C₅₃	#13194	semidirect
258. C₄	#16810405	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
C₂×#16810405	2
C₄×#13133157	2
C₂×C₅₃×#436	2
#13194	53