Subgroups — #27299152

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133127	C₂	1	1	yes	maximal, characteristic
2	#20276326	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810400	C₂	1	2	yes	maximal, complemented
4	#20272794	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2112	C₂	1	1	yes	maximal, characteristic
6	C₂³×#88322	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
8	C₂×#13133157	C₂²	1	2	yes	—
9	C₂×#13133127	—	2	4	no	—
10	C₂×#13133127	C₄	1	1	yes	characteristic
11	C₂×#13133127	C₄	1	1	yes	characteristic
12	C₂×#13133127	C₂²	1	1	yes	characteristic
13	C₂×#13133127	C₂²	1	1	yes	characteristic
14	C₂²⋊#11719908	C₄	1	4	yes	complemented
15	#16810400	—	2	4	no	—
16	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
18	C₅₃×C₂²⋊C₁₆	C₄	1	4	yes	complemented
19	C₂³×Dic₅₃	—	2	1	no	—
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₂²×#88322	—	2	2	no	—
23	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
24	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₅₃	—	2	2	no	—
26	C₂×C₄×Dic₅₃	—	2	2	no	—
27	C₂×C₄×Dic₅₃	—	2	1	no	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	#13133157	—	2	2	no	—
31	#13133157	—	2	2	no	—
32	C₂×#11719908	C₂×C₄	1	4	yes	complemented
33	#13133127	—	2	2	no	—
34	#13133127	—	2	2	no	—
35	#13133127	—	2	2	no	—
36	#13133127	D₄	1	4	yes	—
37	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	—	2	2	no	—
39	C₂²×C₈×C₅₃	—	2	2	no	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	M₃₂×C₅₃	—	2	2	no	—
45	M₃₂×C₅₃	—	2	2	no	—
46	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
47	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
53	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₅₃	—	2	4	no	—
57	C₂²×Dic₅₃	—	2	1	no	—
58	C₂²×Dic₅₃	—	2	1	no	—
59	C₂²×Dic₅₃	—	4	4	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	1	no	—
62	C₂×C₈×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	C₄²	1	2	yes	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₅₃	#41	1	2	yes	—
69	C₄×Dic₅₃	—	2	4	no	—
70	C₄×Dic₅₃	—	2	1	no	—
71	C₄×Dic₅₃	—	2	1	no	—
72	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
73	C₄×Dic₅₃	M₁₆	1	2	yes	—
74	C₂×#88322	—	2	8	no	—
75	C₂×#88322	—	2	1	no	—
76	C₂×#88322	—	2	1	no	—
77	C₂×#88322	—	2	1	no	—
78	C₂×#88322	—	2	1	no	—
79	C₂×#88322	C₂²⋊C₄	1	2	yes	—
80	C₂×#88322	M₁₆	1	2	yes	—
81	C₁₆×C₅₃	—	2	4	no	—
82	#11719908	—	2	4	no	—
83	C₂³×C₅₃	—	2	4	no	—
84	C₂³×C₅₃	—	2	1	no	—
85	C₂³×C₅₃	—	2	1	no	—
86	C₂³×C₅₃	#97	1	2	yes	—
87	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	#97	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×Dic₅₃	—	2	2	no	—
102	C₂×Dic₅₃	—	2	2	no	—
103	C₂×Dic₅₃	—	4	8	no	—
104	C₂×Dic₅₃	—	4	2	no	—
105	C₂×Dic₅₃	—	4	2	no	—
106	C₈×C₅₃	—	2	2	no	—
107	C₈×C₅₃	C₄×D₄	1	2	yes	—
108	C₈×C₅₃	C₄×D₄	1	2	yes	—
109	#88322	—	2	2	no	—
110	#88322	—	2	2	no	—
111	#12605	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	4	no	—
113	C₂²×C₅₃	—	2	4	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	2	2	no	—
116	C₂²×C₅₃	—	2	2	no	—
117	C₂²×C₅₃	#347	1	4	yes	complemented
118	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
119	C₂²×C₅₃	#407	1	1	yes	characteristic
120	C₂²×C₅₃	#432	1	1	yes	characteristic
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
123	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
124	C₄×C₅₃	#407	1	1	yes	characteristic
125	C₄×C₅₃	#432	1	1	yes	characteristic
126	Dic₅₃	—	4	2	no	—
127	Dic₅₃	—	4	2	no	—
128	C₂²×#97	—	53	1	no	—
129	#2112	—	53	1	no	—
130	#2112	D₅₃	1	1	yes	characteristic
131	C₂×#347	—	53	2	no	—
132	#1375	—	53	2	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	—	2	4	no	—
135	C₂×C₅₃	—	2	1	no	—
136	C₂×C₅₃	#1749	1	1	yes	characteristic
137	C₂×C₅₃	C₂×#347	1	2	yes	—
138	C₂³×C₈	—	53	1	no	—
139	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
140	C₂²×C₄²	—	53	1	no	—
141	C₂×M₃₂	—	53	2	no	—
142	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
143	C₂×#97	—	53	1	no	—
144	C₂×#97	—	53	1	no	—
145	C₂×#97	—	53	1	no	—
146	C₂×#97	—	53	1	no	—
147	C₂×#97	—	106	4	no	—
148	C₂²⋊C₁₆	—	53	4	no	—
149	C₂²⋊C₁₆	Dic₅₃	1	4	yes	complemented
150	#347	—	106	4	no	—
151	C₅₃	#12605	1	1	yes	characteristic, complemented, Sylow
152	C₂³×C₄	—	106	1	no	—
153	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	53	1	no	—
157	C₂²×C₈	—	53	1	no	—
158	C₂²×C₈	—	106	2	no	—
159	C₂²×C₈	—	106	2	no	—
160	C₂²×C₈	—	106	2	no	—
161	C₂²×C₈	C₄×D₅₃	1	2	yes	—
162	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
163	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
164	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
165	C₂×C₄²	—	53	1	no	—
166	C₂×C₄²	—	53	1	no	—
167	C₂×C₄²	—	106	2	no	—
168	C₂×C₄²	—	106	2	no	—
169	C₂×C₄²	—	106	1	no	—
170	M₃₂	—	2	2	no	—
171	M₃₂	—	2	2	no	—
172	M₃₂	—	106	2	no	—
173	M₃₂	—	106	2	no	—
174	C₂×C₁₆	—	53	4	no	—
175	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	complemented
176	#97	—	53	4	no	—
177	#97	—	106	2	no	—
178	#97	—	106	2	no	—
179	#97	—	106	2	no	—
180	C₂⁴	#11719912	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	106	4	no	—
186	C₂²×C₄	—	106	1	no	—
187	C₂²×C₄	—	106	1	no	—
188	C₂²×C₄	—	212	4	no	—
189	C₂²×C₄	#11719912	1	1	yes	characteristic
190	C₂²×C₄	#11719912	1	2	yes	—
191	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
192	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	53	2	no	—
198	C₂×C₈	—	53	2	no	—
199	C₂×C₈	—	106	8	no	—
200	C₂×C₈	—	106	1	no	—
201	C₂×C₈	—	106	1	no	—
202	C₂×C₈	—	106	1	no	—
203	C₂×C₈	—	106	1	no	—
204	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
205	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
206	C₂×C₈	D₄×D₅₃	1	2	yes	—
207	C₂×C₈	#11719947	1	2	yes	—
208	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
209	C₄²	—	53	2	no	—
210	C₄²	—	53	2	no	—
211	C₄²	—	106	4	no	—
212	C₄²	—	106	1	no	—
213	C₄²	—	106	1	no	—
214	C₁₆	—	2	4	no	—
215	C₁₆	—	106	4	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	1	no	—
219	C₂³	#13133127	1	2	yes	—
220	C₂³	C₂×#11719912	1	1	yes	characteristic
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	106	2	no	—
228	C₂×C₄	—	106	2	no	—
229	C₂×C₄	—	212	8	no	—
230	C₂×C₄	—	212	2	no	—
231	C₂×C₄	—	212	2	no	—
232	C₂×C₄	#13133127	1	2	yes	—
233	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
234	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
235	C₂×C₄	#13133173	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	#13133177	1	2	yes	—
243	C₈	D₄×Dic₅₃	1	2	yes	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	#16810400	1	4	yes	complemented
250	C₂²	C₂×#13133127	1	1	yes	characteristic
251	C₂²	#16810585	1	1	yes	characteristic
252	C₂²	#16810677	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	212	2	no	—
255	C₄	—	212	2	no	—
256	C₄	C₂²⋊C₄×Dic₅₃	1	1	yes	characteristic
257	C₄	M₁₆×Dic₅₃	1	1	yes	characteristic
258	C₄	#16810585	1	1	yes	characteristic
259	C₄	#16810677	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	4	no	—
262	C₂	—	2	1	no	—
263	C₂	#20274479	1	1	yes	characteristic
264	C₂	C₂×#16810400	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2112	C₂
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2112	D₅₃
hypocenter	C₅₃	#12605
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #2112

Complements

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

N	H ≅ G/N	type
3. C₂×#16810400	C₂	semidirect
4. #20272794	C₂	semidirect
14. C₂²⋊#11719908	C₄	semidirect
18. C₅₃×C₂²⋊C₁₆	C₄	semidirect
32. C₂×#11719908	C₂×C₄	semidirect
46. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
117. C₂²×C₅₃	#347	semidirect
149. C₂²⋊C₁₆	Dic₅₃	semidirect
151. C₅₃	#12605	semidirect
175. C₂×C₁₆	C₂×Dic₅₃	semidirect
249. C₂²	#16810400	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133127	2
#20276326	2
C₂×#16810400	2
#20272794	2
C₅₃×#2112	2
#12605	53