Subgroups — #27300305

Order 13568.

454 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	C₂×#16810577	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810402	C₂	1	2	yes	maximal
3	#20272793	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272792	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11719912	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
8	C₂×#13133174	C₄	1	1	yes	characteristic
9	C₂²×#11719908	C₂²	1	2	yes	—
10	#16810577	—	2	4	no	—
11	C₂×#13133128	C₄	1	1	yes	characteristic
12	C₂×#13133127	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810402	—	2	4	no	—
15	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×#11719912	—	2	2	no	—
19	C₂×#11719912	—	2	2	no	—
20	C₂×#11719912	—	2	2	no	—
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×D₅₃	—	2	2	no	—
24	C₂×C₈×D₅₃	—	2	1	no	—
25	C₂×C₈×D₅₃	—	2	1	no	—
26	C₂×C₈×D₅₃	D₄	1	4	yes	—
27	#13133174	—	2	2	no	—
28	#13133174	—	2	2	no	—
29	C₂×#11719919	C₈	1	2	yes	—
30	C₂×#11719921	C₈	1	2	yes	—
31	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×#11719908	—	2	2	no	—
33	C₂×#11719908	—	2	2	no	—
34	C₂×#11719908	—	2	2	no	—
35	#13133128	—	2	2	no	—
36	#13133127	—	2	2	no	—
37	C₈×Dic₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₅₃	—	2	2	no	—
42	C₂×C₁₆×C₅₃	—	2	2	no	—
43	C₂×C₁₆×C₅₃	—	2	2	no	—
44	C₄×C₈×C₅₃	—	2	2	no	—
45	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
49	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₄×D₅₃	—	2	1	no	—
51	C₂×C₄×D₅₃	—	2	1	no	—
52	C₂×C₄×D₅₃	—	2	1	no	—
53	C₂×C₄×D₅₃	—	2	1	no	—
54	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
56	C₂×C₈×C₅₃	—	2	1	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
59	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₅₃	#41	1	2	yes	—
61	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
62	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
63	C₄²×C₅₃	—	2	1	no	—
64	C₄²×C₅₃	—	2	1	no	—
65	#11719921	—	2	4	no	—
66	#11719919	—	2	4	no	—
67	C₄×Dic₅₃	—	2	1	no	—
68	C₄×Dic₅₃	—	2	1	no	—
69	C₂×#88322	—	2	1	no	—
70	C₂×#88322	—	2	1	no	—
71	C₂×#88322	—	2	1	no	—
72	C₂×#88322	—	2	1	no	—
73	C₂×#88322	C₂²⋊C₄	1	2	yes	—
74	C₂×#88322	M₁₆	1	2	yes	—
75	#11719912	—	2	8	no	—
76	C₈×D₅₃	—	2	4	no	—
77	C₈×D₅₃	—	2	4	no	—
78	C₁₆×C₅₃	—	4	2	no	—
79	#11719908	—	4	2	no	—
80	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
81	C₂²×D₅₃	—	2	1	no	—
82	C₂²×D₅₃	—	2	1	no	—
83	C₂²×D₅₃	—	2	2	no	—
84	C₂²×D₅₃	—	2	2	no	—
85	C₂²×D₅₃	C₂²⋊C₈	1	2	yes	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂×Dic₅₃	—	2	1	no	—
96	C₂×Dic₅₃	—	2	1	no	—
97	C₂×Dic₅₃	—	2	2	no	—
98	C₂×Dic₅₃	—	2	2	no	—
99	C₂×Dic₅₃	—	2	2	no	—
100	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
101	C₄×D₅₃	—	2	2	no	—
102	C₄×D₅₃	—	2	2	no	—
103	C₄×D₅₃	—	2	2	no	—
104	C₄×D₅₃	—	2	2	no	—
105	C₈×C₅₃	—	2	2	no	—
106	C₈×C₅₃	#123	1	2	yes	—
107	C₈×C₅₃	#123	1	2	yes	—
108	#88322	—	2	2	no	—
109	#88322	—	2	2	no	—
110	#12845	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
112	C₂²×C₅₃	#431	1	1	yes	characteristic
113	C₂²×C₅₃	#431	1	1	yes	characteristic
114	C₂²×C₅₃	#504	1	2	yes	—
115	C₂²×C₅₃	#504	1	2	yes	—
116	C₂×D₅₃	—	2	2	no	—
117	C₂×D₅₃	—	2	4	no	—
118	C₂×D₅₃	—	4	1	no	—
119	C₂×D₅₃	—	4	2	no	—
120	C₂×D₅₃	—	4	1	no	—
121	C₄×C₅₃	—	4	2	no	—
122	C₄×C₅₃	#381	1	1	yes	characteristic
123	C₄×C₅₃	#407	1	1	yes	characteristic
124	C₄×C₅₃	#432	1	1	yes	characteristic
125	C₄×C₅₃	#432	1	1	yes	characteristic
126	Dic₅₃	—	2	2	no	—
127	Dic₅₃	—	4	2	no	—
128	C₂×#405	—	53	1	no	—
129	C₂×C₂²⋊C₁₆	—	53	2	no	—
130	#1374	—	53	2	no	—
131	#1374	—	53	1	no	—
132	#1374	D₅₃	1	1	yes	characteristic, complemented
133	C₂×C₅₃	#1817	1	1	yes	characteristic
134	C₂×C₅₃	#2113	1	2	yes	—
135	C₂×C₅₃	#2164	1	1	yes	characteristic
136	C₂×C₅₃	#2164	1	2	yes	—
137	C₂×C₅₃	#2164	1	1	yes	characteristic
138	D₅₃	—	4	2	no	—
139	C₂²×M₁₆	—	53	1	no	—
140	C₂³×C₈	—	53	1	no	—
141	C₂×#117	—	53	1	no	—
142	C₂²×C₁₆	—	53	2	no	—
143	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
144	#405	—	106	4	no	—
145	C₂×#97	—	53	1	no	—
146	C₂×#97	—	53	1	no	—
147	C₂×C₄×C₈	—	53	1	no	—
148	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
149	C₂²⋊C₁₆	—	106	4	no	—
150	C₅₃	#12845	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₄	—	53	1	no	—
152	C₂×M₁₆	—	106	2	no	—
153	C₂×M₁₆	—	106	2	no	—
154	C₂×M₁₆	—	106	2	no	—
155	C₂²×C₈	—	53	1	no	—
156	C₂²×C₈	—	53	1	no	—
157	C₂²×C₈	—	53	4	no	—
158	C₂²×C₈	—	106	2	no	—
159	C₂²×C₈	—	106	1	no	—
160	C₂²×C₈	—	106	1	no	—
161	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
163	#117	—	106	2	no	—
164	#117	—	106	2	no	—
165	C₂×C₄⋊C₄	—	53	2	no	—
166	C₂×C₂²⋊C₄	—	53	2	no	—
167	C₂×C₄²	—	53	1	no	—
168	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	106	2	no	—
173	C₂×C₁₆	—	106	2	no	—
174	C₂×C₁₆	—	106	2	no	—
175	#97	—	106	2	no	—
176	#97	—	106	2	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	106	2	no	—
179	C₂⁴	—	53	1	no	—
180	C₂²×C₄	—	53	1	no	—
181	C₂²×C₄	—	53	2	no	—
182	C₂²×C₄	—	53	2	no	—
183	C₂²×C₄	—	53	2	no	—
184	C₂²×C₄	—	106	1	no	—
185	C₂²×C₄	—	106	1	no	—
186	C₂²×C₄	—	106	1	no	—
187	C₂²×C₄	—	106	1	no	—
188	C₂²×C₄	C₈×D₅₃	1	2	yes	—
189	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
190	M₁₆	—	106	8	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	53	2	no	—
195	C₂×C₈	—	106	1	no	—
196	C₂×C₈	—	106	1	no	—
197	C₂×C₈	—	106	1	no	—
198	C₂×C₈	—	106	1	no	—
199	C₂×C₈	—	106	4	no	—
200	C₂×C₈	—	106	4	no	—
201	C₂×C₈	#11719912	1	2	yes	—
202	C₂×C₈	#11719945	1	2	yes	—
203	C₂×C₈	D₄×D₅₃	1	2	yes	—
204	C₂×C₈	#11719947	1	1	yes	characteristic
205	C₂×C₈	#11719949	1	1	yes	characteristic
206	C₄⋊C₄	—	106	4	no	—
207	C₂²⋊C₄	—	106	4	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	106	1	no	—
211	C₄²	—	106	1	no	—
212	C₁₆	—	4	2	no	—
213	C₁₆	—	212	2	no	—
214	C₂³	—	53	2	no	—
215	C₂³	—	106	2	no	—
216	C₂³	—	106	2	no	—
217	C₂³	—	106	1	no	—
218	C₂³	—	106	1	no	—
219	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	53	2	no	—
224	C₂×C₄	—	106	1	no	—
225	C₂×C₄	—	106	1	no	—
226	C₂×C₄	—	106	2	no	—
227	C₂×C₄	—	106	2	no	—
228	C₂×C₄	—	106	2	no	—
229	C₂×C₄	—	106	2	no	—
230	C₂×C₄	—	106	2	no	—
231	C₂×C₄	—	106	2	no	—
232	C₂×C₄	—	106	2	no	—
233	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
234	C₂×C₄	#13133174	1	1	yes	characteristic
235	C₂×C₄	#13133175	1	1	yes	characteristic
236	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
237	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	106	2	no	—
241	C₈	—	106	2	no	—
242	C₈	#13133199	1	2	yes	—
243	C₈	#13133202	1	2	yes	—
244	C₂²	—	106	2	no	—
245	C₂²	—	106	4	no	—
246	C₂²	—	212	1	no	—
247	C₂²	—	212	2	no	—
248	C₂²	—	212	1	no	—
249	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
250	C₂²	#16810674	1	1	yes	characteristic
251	C₂²	#16810675	1	1	yes	characteristic
252	C₂²	#16811039	1	2	yes	—
253	C₂²	#16811041	1	2	yes	—
254	C₄	—	4	2	no	—
255	C₄	—	106	2	no	—
256	C₄	—	212	2	no	—
257	C₄	#16810485	1	1	yes	characteristic
258	C₄	#16810585	1	1	yes	characteristic
259	C₄	#16810678	1	1	yes	characteristic
260	C₄	#16810679	1	1	yes	characteristic
261	C₂	—	212	2	no	—
262	C₂	#20274782	1	1	yes	characteristic
263	C₂	#20276327	1	2	yes	—
264	C₂	#20276520	1	1	yes	characteristic
265	C₂	#20276521	1	2	yes	—
266	C₂	#20276523	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂²×C₈
Fitting	C₅₃×#1374	C₂
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1374	D₅₃
hypocenter	C₅₃	#12845
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
3. #20272793	C₂	semidirect
4. #20272792	C₂	semidirect
5. C₅₃×#1374	C₂	semidirect
132. #1374	D₅₃	semidirect
150. C₅₃	#12845	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810577	2
C₂×#16810402	2
#20272793	2
#20272792	2
C₅₃×#1374	2
#12845	53