Subgroups — #27293415

Order 13568.

390 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276518	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276514	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272639	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11719908	—	2	1	no	—
6	C₄⋊#11719908	—	2	1	no	—
7	C₂²⋊#11719908	C₂²	1	2	yes	—
8	C₂²⋊#11719908	C₂²	1	2	yes	—
9	#16810399	—	2	1	no	—
10	C₄×#11719908	—	2	1	no	—
11	C₂²×M₁₆×C₅₃	C₄	1	2	yes	—
12	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
13	C₅₃×#436	—	2	1	no	—
14	C₅₃×#436	—	2	1	no	—
15	C₅₃×#436	—	2	1	no	—
16	C₅₃×#436	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₂²×D₄×C₅₃	C₈	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂×M₁₆×C₅₃	—	4	4	no	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₄×D₄×C₅₃	—	2	1	no	—
31	C₄×D₄×C₅₃	—	2	1	no	—
32	C₄×D₄×C₅₃	—	2	1	no	—
33	C₄×D₄×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	—	2	1	no	—
38	C₄⋊C₈×C₅₃	—	2	1	no	—
39	C₂²⋊C₈×C₅₃	—	2	2	no	—
40	C₂²⋊C₈×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂⁴×C₅₃	C₂×C₈	1	2	yes	—
46	C₂×D₄×C₅₃	—	2	1	no	—
47	C₂×D₄×C₅₃	—	2	1	no	—
48	C₂×D₄×C₅₃	—	4	2	no	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	—	4	2	no	—
52	C₂²×C₄×C₅₃	—	4	2	no	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	M₁₆×C₅₃	—	4	4	no	—
58	M₁₆×C₅₃	—	4	4	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	4	1	no	—
62	C₂×C₈×C₅₃	—	4	2	no	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₅₃	—	2	1	no	—
68	C₄⋊C₄×C₅₃	—	2	1	no	—
69	C₂²⋊C₄×C₅₃	—	2	2	no	—
70	C₂²⋊C₄×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
74	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
75	#11719908	—	4	2	no	—
76	#11719908	—	4	2	no	—
77	C₂³×C₅₃	—	2	2	no	—
78	C₂³×C₅₃	—	2	4	no	—
79	C₂³×C₅₃	—	4	2	no	—
80	C₂³×C₅₃	—	4	2	no	—
81	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	D₄×C₅₃	—	4	2	no	—
83	D₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	4	4	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
99	C₈×C₅₃	—	2	1	no	—
100	C₈×C₅₃	—	2	1	no	—
101	C₈×C₅₃	—	2	1	no	—
102	C₈×C₅₃	—	4	2	no	—
103	C₈×C₅₃	#100	1	2	yes	—
104	#11489	—	53	1	no	maximal, Sylow
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	4	2	no	—
108	C₂²×C₅₃	—	4	2	no	—
109	C₂²×C₅₃	—	4	2	no	—
110	C₂²×C₅₃	—	8	4	no	—
111	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
112	C₂²×C₅₃	#504	1	2	yes	—
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	4	2	no	—
118	C₄×C₅₃	#349	1	2	yes	—
119	C₄×C₅₃	#410	1	1	yes	characteristic
120	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
121	C₂×#436	D₅₃	1	1	yes	characteristic
122	#2163	—	53	1	no	—
123	#2162	—	53	1	no	—
124	#1314	—	53	4	no	—
125	C₂×C₅₃	—	2	2	no	—
126	C₂×C₅₃	—	8	2	no	—
127	C₂×C₅₃	#1460	1	1	yes	characteristic
128	C₂×C₅₃	#2113	1	1	yes	characteristic
129	C₂×C₅₃	C₂×#349	1	1	yes	characteristic
130	C₂²×M₁₆	Dic₅₃	1	2	yes	—
131	C₂×C₄×D₄	Dic₅₃	1	1	yes	characteristic
132	#436	—	2	1	no	—
133	#436	—	2	1	no	—
134	#436	—	2	1	no	—
135	#436	—	2	1	no	—
136	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
137	C₂×C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
138	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
139	C₄⋊C₁₆	—	106	1	no	—
140	C₄⋊C₁₆	—	106	1	no	—
141	C₂²⋊C₁₆	—	53	2	no	—
142	C₂²⋊C₁₆	—	53	2	no	—
143	#346	—	106	1	no	—
144	C₄×C₁₆	—	106	1	no	—
145	C₅₃	#11489	1	1	yes	characteristic, complemented, Sylow
146	C₂²×D₄	#88322	1	1	yes	characteristic
147	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	4	4	no	—
151	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
152	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
153	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
154	C₄×D₄	—	2	1	no	—
155	C₄×D₄	—	2	1	no	—
156	C₄×D₄	—	2	1	no	—
157	C₄×D₄	—	2	1	no	—
158	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
159	C₂×C₂²⋊C₄	#88322	1	2	yes	—
160	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
161	C₂×C₁₆	—	106	2	no	—
162	C₂×C₁₆	—	106	2	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	—	2	1	no	—
165	C₂²⋊C₈	—	2	2	no	—
166	C₂²⋊C₈	—	2	2	no	—
167	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
170	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
171	C₂⁴	C₂×#88322	1	2	yes	—
172	C₂×D₄	—	2	1	no	—
173	C₂×D₄	—	2	1	no	—
174	C₂×D₄	—	4	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
180	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
181	C₂²×C₄	C₂×#88322	1	2	yes	—
182	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
183	M₁₆	—	4	4	no	—
184	M₁₆	—	4	4	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
191	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
192	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₄²	#11719917	1	1	yes	characteristic
198	C₄²	#11719917	1	1	yes	characteristic
199	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
200	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
201	C₁₆	—	212	2	no	—
202	C₁₆	—	212	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	4	no	—
205	C₂³	—	4	2	no	—
206	C₂³	—	4	2	no	—
207	C₂³	C₂²×#88322	1	1	yes	characteristic
208	D₄	—	4	2	no	—
209	D₄	—	4	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
220	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
221	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
222	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
223	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
224	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	2	1	no	—
228	C₈	—	4	2	no	—
229	C₈	#13133134	1	2	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	8	4	no	—
236	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
237	C₂²	#16811040	1	2	yes	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	4	2	no	—
243	C₄	#16810405	1	2	yes	—
244	C₄	#16810597	1	1	yes	characteristic
245	C₄	C₂×#13133134	1	1	yes	characteristic
246	C₂	—	2	2	no	—
247	C₂	—	8	2	no	—
248	C₂	#20273036	1	1	yes	characteristic
249	C₂	#20276329	1	1	yes	characteristic
250	C₂	C₂×#16810405	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	#11489
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. #20276518	C₂	semidirect
2. #20276514	C₂	semidirect
3. #20272639	C₂	semidirect
145. C₅₃	#11489	semidirect

Maximal subgroups

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

subgroup	index
#20276518	2
#20276514	2
#20272639	2
C₂×C₅₃×#436	2
#11489	53