Subgroups — #27537233

Order 13568.

528 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×Dic₅₃	C₂	1	1	yes	maximal, characteristic
2	#20276528	C₂	1	4	yes	maximal, complemented
3	#20276524	C₂	1	4	yes	maximal, complemented
4	C₂×#16810443	C₂	1	2	yes	maximal, complemented
5	#20276336	C₂	1	1	yes	maximal, characteristic
6	#20276319	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2115	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133224	C₄	1	1	yes	characteristic
9	#41×Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×#13133157	C₂²	1	2	yes	—
11	#16810581	C₄	1	1	yes	characteristic
12	#16810581	C₂²	1	2	yes	—
13	#16810578	C₄	1	2	yes	—
14	#16810578	C₂²	1	1	yes	characteristic
15	M₁₆×Dic₅₃	C₄	1	1	yes	characteristic
16	M₁₆×Dic₅₃	C₂²	1	2	yes	—
17	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
18	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
19	#16810443	C₂²	1	4	yes	—
20	#16810443	C₂²	1	4	yes	—
21	#16810405	C₂²	1	4	yes	—
22	#16810400	C₂²	1	8	yes	—
23	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#349	C₂²	1	4	yes	—
26	#13133224	—	2	4	no	—
27	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
28	C₂×#11719917	C₂×C₄	1	2	yes	—
29	C₂²×#88322	C₂×C₄	1	2	yes	—
30	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
31	Q₈×Dic₅₃	—	2	1	no	—
32	D₄×Dic₅₃	—	2	1	no	—
33	D₄×Dic₅₃	—	2	2	no	—
34	#13133176	C₂×C₄	1	2	yes	—
35	#13133176	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
38	#13133157	—	2	2	no	—
39	#13133157	C₂³	1	4	yes	—
40	C₂×#11719908	—	2	2	no	—
41	#13133127	C₂×C₄	1	2	yes	—
42	#13133127	C₂×C₄	1	2	yes	—
43	#13133127	C₂³	1	2	yes	—
44	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
45	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
46	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
47	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
48	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
49	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
50	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
51	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
52	#131×C₅₃	—	2	2	no	—
53	#131×C₅₃	—	2	2	no	—
54	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
56	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
57	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
58	M₃₂×C₅₃	—	2	2	no	—
59	M₃₂×C₅₃	C₂³	1	4	yes	—
60	C₂×C₁₆×C₅₃	—	2	2	no	—
61	#41×C₅₃	—	2	2	no	—
62	#41×C₅₃	—	2	2	no	—
63	C₂×Q₈×C₅₃	M₁₆	1	1	yes	characteristic
64	C₂×D₄×C₅₃	M₁₆	1	2	yes	—
65	C₂×D₄×C₅₃	M₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₅₃	—	2	2	no	—
69	C₂²×Dic₅₃	—	2	1	no	—
70	M₁₆×C₅₃	—	2	2	no	—
71	M₁₆×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	#41	1	2	yes	—
77	C₂×C₈×C₅₃	#41	1	2	yes	—
78	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
79	C₂²⋊Dic₅₃	—	2	2	no	—
80	C₂²⋊Dic₅₃	—	2	1	no	—
81	C₄⋊Dic₅₃	—	2	2	no	—
82	C₄⋊Dic₅₃	—	2	1	no	—
83	C₄×Dic₅₃	—	2	2	no	—
84	C₄×Dic₅₃	—	2	1	no	—
85	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
87	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
88	#11719917	—	2	2	no	—
89	#11719917	—	2	4	no	—
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	M₁₆	1	4	yes	—
92	C₂×#88322	C₂²×C₄	1	2	yes	—
93	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
94	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
95	C₁₆×C₅₃	—	2	4	no	—
96	#11719908	—	2	4	no	—
97	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
98	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
99	Q₈×C₅₃	—	2	2	no	—
100	D₄×C₅₃	—	2	4	no	—
101	D₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	#117	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×Dic₅₃	—	2	1	no	—
110	C₂×Dic₅₃	—	2	2	no	—
111	C₂×Dic₅₃	—	2	1	no	—
112	C₂×Dic₅₃	—	4	2	no	—
113	C₂×Dic₅₃	—	4	4	no	—
114	C₈×C₅₃	—	2	2	no	—
115	C₈×C₅₃	#117	1	1	yes	characteristic
116	C₈×C₅₃	#117	1	1	yes	characteristic
117	C₈×C₅₃	C₂×#41	1	2	yes	—
118	#88322	—	2	2	no	—
119	#88322	C₂×M₁₆	1	2	yes	—
120	#88322	C₂×M₁₆	1	2	yes	—
121	#33428	—	53	1	no	maximal, Sylow
122	C₂²×C₅₃	—	2	2	no	—
123	C₂²×C₅₃	—	4	1	no	—
124	C₂²×C₅₃	—	4	2	no	—
125	C₂²×C₅₃	#432	1	2	yes	—
126	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	#432	1	2	yes	—
129	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
130	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
131	Dic₅₃	—	2	2	no	—
132	Dic₅₃	—	4	2	no	—
133	Dic₅₃	—	4	1	no	—
134	C₄×#131	—	53	1	no	—
135	#2166	—	53	4	no	—
136	#2165	—	53	4	no	—
137	C₂×#364	—	53	2	no	—
138	#2115	—	53	1	no	—
139	#2115	D₅₃	1	1	yes	characteristic
140	#2110	—	53	2	no	—
141	C₂×C₅₃	—	2	1	no	—
142	C₂×C₅₃	—	2	2	no	—
143	C₂×C₅₃	—	4	2	no	—
144	C₂×C₅₃	C₂×#432	1	1	yes	characteristic
145	C₂×#131	—	53	1	no	—
146	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
147	C₄×#41	—	53	1	no	—
148	C₂×M₃₂	—	53	2	no	—
149	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
150	#405	—	53	1	no	—
151	#405	—	53	2	no	—
152	#405	—	53	2	no	—
153	#405	—	53	1	no	—
154	C₄×M₁₆	—	53	1	no	—
155	C₄×M₁₆	—	53	2	no	—
156	C₂×C₄×C₈	—	53	2	no	—
157	C₂×C₄×C₈	—	53	1	no	—
158	#364	—	53	4	no	—
159	#364	—	53	4	no	—
160	#349	—	53	4	no	—
161	#349	C₂×D₅₃	1	4	yes	—
162	#347	—	53	8	no	—
163	C₅₃	#33428	1	1	yes	characteristic, complemented, Sylow
164	C₂×#41	C₄×D₅₃	1	1	yes	characteristic
165	#131	—	2	2	no	—
166	#131	—	2	2	no	—
167	#131	—	106	4	no	—
168	C₂×M₁₆	—	53	2	no	—
169	C₂×M₁₆	—	53	1	no	—
170	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
171	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
172	C₂²×C₈	—	53	1	no	—
173	C₂²×C₈	—	53	2	no	—
174	C₂²×C₈	C₄×D₅₃	1	2	yes	—
175	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
176	C₄×Q₈	—	106	1	no	—
177	C₄×D₄	—	106	2	no	—
178	C₄×D₄	—	106	1	no	—
179	#117	—	53	2	no	—
180	#117	—	53	1	no	—
181	C₂×C₄²	—	53	2	no	—
182	C₂×C₄²	—	53	1	no	—
183	M₃₂	—	2	2	no	—
184	M₃₂	—	53	4	no	—
185	M₃₂	—	106	2	no	—
186	M₃₂	C₂²×D₅₃	1	4	yes	—
187	C₂×C₁₆	—	2	2	no	—
188	C₂×C₁₆	—	106	2	no	—
189	#97	—	53	2	no	—
190	#97	—	53	2	no	—
191	#97	—	53	2	no	—
192	C₄×C₈	—	53	1	no	—
193	C₄×C₈	—	53	2	no	—
194	C₄×C₈	—	53	2	no	—
195	C₄×C₈	—	53	1	no	—
196	C₄×C₈	—	53	2	no	—
197	C₄×C₈	—	53	1	no	—
198	C₄×C₈	—	53	1	no	—
199	#41	—	2	2	no	—
200	#41	—	2	2	no	—
201	C₂×Q₈	#11719912	1	1	yes	characteristic
202	C₂×D₄	#11719912	1	2	yes	—
203	C₂×D₄	#11719912	1	1	yes	characteristic
204	C₂²×C₄	—	106	2	no	—
205	C₂²×C₄	—	106	1	no	—
206	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
207	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
208	M₁₆	—	2	2	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	106	4	no	—
211	M₁₆	—	106	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	53	4	no	—
215	C₂×C₈	—	53	1	no	—
216	C₂×C₈	—	53	1	no	—
217	C₂×C₈	—	53	2	no	—
218	C₂×C₈	—	106	4	no	—
219	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
220	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
221	C₂×C₈	#11719947	1	2	yes	—
222	C₂×C₈	#11719947	1	2	yes	—
223	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
224	C₄⋊C₄	—	106	2	no	—
225	C₄⋊C₄	—	106	1	no	—
226	C₂²⋊C₄	—	106	2	no	—
227	C₂²⋊C₄	—	106	1	no	—
228	C₄²	—	53	1	no	—
229	C₄²	—	53	2	no	—
230	C₄²	—	53	1	no	—
231	C₄²	—	106	2	no	—
232	C₄²	—	106	1	no	—
233	C₁₆	—	2	4	no	—
234	C₁₆	—	106	4	no	—
235	C₂³	C₂×#11719912	1	2	yes	—
236	C₂³	C₂×#11719912	1	1	yes	characteristic
237	Q₈	—	2	2	no	—
238	D₄	—	2	4	no	—
239	D₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	106	1	no	—
243	C₂×C₄	—	106	2	no	—
244	C₂×C₄	—	106	1	no	—
245	C₂×C₄	—	212	2	no	—
246	C₂×C₄	—	212	4	no	—
247	C₂×C₄	#13133173	1	2	yes	—
248	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
249	C₂×C₄	C₂×#11719912	1	2	yes	—
250	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	53	2	no	—
254	C₈	—	53	2	no	—
255	C₈	—	106	2	no	—
256	C₈	#13133173	1	1	yes	characteristic
257	C₈	#13133173	1	1	yes	characteristic
258	C₈	C₂×#11719947	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	2	no	—
262	C₂²	#16810677	1	2	yes	—
263	C₂²	C₂²×#11719912	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	106	2	no	—
266	C₄	—	212	1	no	—
267	C₄	—	212	2	no	—
268	C₄	#16810677	1	2	yes	—
269	C₄	C₂×#13133173	1	1	yes	characteristic
270	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
271	C₂	—	2	1	no	—
272	C₂	—	2	2	no	—
273	C₂	—	4	2	no	—
274	C₂	C₂×#16810677	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133173
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2115	C₂
socle	C₂×C₅₃	C₂×#432
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2115	D₅₃
hypocenter	C₅₃	#33428
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2115

Complements

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

N	H ≅ G/N	type
2. #20276528	C₂	semidirect
3. #20276524	C₂	semidirect
4. C₂×#16810443	C₂	semidirect
6. #20276319	C₂	semidirect
163. C₅₃	#33428	semidirect

Maximal subgroups

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

subgroup	index
#131×Dic₅₃	2
#20276528	2
#20276524	2
C₂×#16810443	2
#20276336	2
#20276319	2
C₅₃×#2115	2
#33428	53