Subgroups — #27300315

Order 13568.

440 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276326	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic
3	#20272804	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₂²⋊#11719908	—	2	1	no	—
8	C₂²⋊#11719908	—	2	1	no	—
9	C₂²⋊#11719908	—	2	1	no	—
10	C₂²⋊#11719908	—	2	1	no	—
11	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
14	C₅₃×#405	—	2	1	no	—
15	C₅₃×#405	—	2	1	no	—
16	C₅₃×#405	C₄	1	2	yes	—
17	C₅₃×#405	C₂²	1	2	yes	—
18	C₂×#97×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	—	2	1	no	—
20	#13133157	—	2	4	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	—	2	4	no	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×M₁₆×C₅₃	—	2	4	no	—
26	C₂×M₁₆×C₅₃	—	2	1	no	—
27	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₅₃	—	2	4	no	—
29	C₂²×C₈×C₅₃	—	2	1	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	#117×C₅₃	—	2	1	no	—
37	#117×C₅₃	—	2	1	no	—
38	#117×C₅₃	C₈	1	2	yes	—
39	#117×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
43	C₂×C₄²×C₅₃	—	2	1	no	—
44	#97×C₅₃	—	2	2	no	—
45	#97×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	—	2	2	no	—
47	C₄×C₈×C₅₃	—	2	2	no	—
48	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	M₁₆×C₅₃	—	2	4	no	—
60	M₁₆×C₅₃	—	2	2	no	—
61	M₁₆×C₅₃	M₁₆	1	4	yes	—
62	C₂×C₈×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	#41	1	4	yes	—
74	C₄⋊C₄×C₅₃	—	2	1	no	—
75	C₄⋊C₄×C₅₃	—	2	1	no	—
76	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
77	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
78	C₂²⋊C₄×C₅₃	—	2	2	no	—
79	C₂²⋊C₄×C₅₃	—	2	2	no	—
80	C₄²×C₅₃	—	2	2	no	—
81	C₄²×C₅₃	—	2	2	no	—
82	#11719908	—	4	2	no	—
83	#11719908	—	4	2	no	—
84	C₂³×C₅₃	—	2	2	no	—
85	C₂³×C₅₃	—	2	2	no	—
86	C₂³×C₅₃	—	4	1	no	—
87	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
88	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
89	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#117	1	2	yes	—
105	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
107	C₈×C₅₃	—	2	1	no	—
108	C₈×C₅₃	—	2	1	no	—
109	C₈×C₅₃	—	2	4	no	—
110	C₈×C₅₃	#123	1	2	yes	—
111	C₈×C₅₃	#123	1	2	yes	—
112	#12848	—	53	1	no	maximal, Sylow
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	2	1	no	—
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	1	no	—
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	—	2	1	no	—
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	—	4	4	no	—
121	C₂²×C₅₃	#350	1	2	yes	—
122	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₂²×C₅₃	#432	1	2	yes	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	—	4	2	no	—
127	C₄×C₅₃	—	4	2	no	—
128	C₄×C₅₃	#381	1	1	yes	characteristic
129	C₄×C₅₃	#407	1	1	yes	characteristic
130	C₄×C₅₃	#431	1	2	yes	—
131	C₂×#405	D₅₃	1	1	yes	characteristic
132	#2112	—	53	1	no	—
133	C₂×C₂²⋊C₁₆	—	53	1	no	—
134	#1379	—	53	4	no	—
135	C₂×C₅₃	—	2	2	no	—
136	C₂×C₅₃	—	2	2	no	—
137	C₂×C₅₃	—	4	1	no	—
138	C₂×C₅₃	#1817	1	1	yes	characteristic
139	C₂×C₅₃	C₂×#350	1	1	yes	characteristic
140	C₂×C₅₃	#2117	1	1	yes	characteristic
141	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
142	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
143	C₂×#117	Dic₅₃	1	1	yes	characteristic
144	C₂×M₃₂	—	53	2	no	—
145	C₂²×C₁₆	—	53	2	no	—
146	#405	—	2	1	no	—
147	#405	—	2	1	no	—
148	#405	Dic₅₃	1	2	yes	—
149	#405	C₂×D₅₃	1	2	yes	—
150	C₂×#97	—	2	1	no	—
151	C₂×C₄×C₈	—	2	1	no	—
152	C₂²⋊C₁₆	—	106	1	no	—
153	C₂²⋊C₁₆	—	106	1	no	—
154	C₂²⋊C₁₆	—	106	1	no	—
155	C₂²⋊C₁₆	—	106	1	no	—
156	C₅₃	#12848	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
158	C₂×M₁₆	—	2	1	no	—
159	C₂×M₁₆	—	2	4	no	—
160	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	2	4	no	—
163	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
166	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
167	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
169	#117	—	2	1	no	—
170	#117	—	2	1	no	—
171	#117	#88322	1	2	yes	—
172	#117	C₂×Dic₅₃	1	2	yes	—
173	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
174	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	—	2	1	no	—
176	C₂×C₄²	—	2	1	no	—
177	M₃₂	—	106	4	no	—
178	C₂×C₁₆	—	106	4	no	—
179	C₂×C₁₆	—	106	2	no	—
180	C₂×C₁₆	—	106	2	no	—
181	#97	—	2	2	no	—
182	#97	—	2	2	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	—	2	2	no	—
185	C₂⁴	#11719917	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
191	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
192	C₂²×C₄	#11719917	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
195	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
196	M₁₆	—	2	4	no	—
197	M₁₆	—	2	2	no	—
198	M₁₆	#11719917	1	4	yes	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
208	C₂×C₈	#11719945	1	4	yes	—
209	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
210	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	C₂×#88322	1	2	yes	—
214	C₄⋊C₄	C₂×#88322	1	2	yes	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₁₆	—	212	2	no	—
220	C₁₆	—	212	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	4	1	no	—
224	C₂³	C₂²⋊#88322	1	1	yes	characteristic
225	C₂³	C₂²⋊#88322	1	1	yes	characteristic
226	C₂³	C₂×#11719917	1	1	yes	characteristic
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
239	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
241	C₂×C₄	#13133176	1	2	yes	—
242	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
243	C₂×C₄	C₂×#11719917	1	2	yes	—
244	C₈	—	2	4	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	#13133200	1	2	yes	—
248	C₈	#13133200	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	4	4	no	—
257	C₂²	#16810407	1	2	yes	—
258	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
259	C₂²	#16810680	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	2	no	—
264	C₄	#16810486	1	1	yes	characteristic
265	C₄	#16810587	1	1	yes	characteristic
266	C₄	#16810676	1	2	yes	—
267	C₂	—	2	2	no	—
268	C₂	—	2	2	no	—
269	C₂	—	4	1	no	—
270	C₂	#20274783	1	1	yes	characteristic
271	C₂	C₂×#16810407	1	1	yes	characteristic
272	C₂	#20276343	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#12848
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #20272804	C₂	semidirect
156. C₅₃	#12848	semidirect

Maximal subgroups

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

subgroup	index
#20276326	2
C₂×C₂²⋊#11719908	2
#20272804	2
C₂×C₅₃×#405	2
#12848	53