Subgroups — #27537231

Order 13568.

524 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276531	C₂	1	4	yes	maximal, complemented
2	#20276527	C₂	1	4	yes	maximal, complemented
3	#20276481	C₂	1	2	yes	maximal, complemented
4	C₂×#16810405	C₂	1	2	yes	maximal
5	#20276320	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2873	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133157	C₂²	1	4	yes	—
8	#16810444	C₂²	1	8	yes	complemented
9	#16810405	C₂²	1	4	yes	—
10	#16810405	C₂²	1	4	yes	—
11	#16810401	C₂²	1	8	yes	complemented
12	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
13	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
15	C₅₃×#405	C₄	1	1	yes	characteristic
16	C₅₃×#405	C₄	1	2	yes	—
17	C₅₃×#405	C₄	1	1	yes	characteristic
18	C₅₃×#405	C₂²	1	1	yes	characteristic
19	C₅₃×#405	C₂²	1	2	yes	—
20	C₅₃×#405	C₂²	1	1	yes	characteristic
21	C₄×M₁₆×C₅₃	C₄	1	2	yes	—
22	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
23	#13133157	—	2	4	no	—
24	#13133157	C₂³	1	8	yes	—
25	C₂×#11719908	—	2	4	no	—
26	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₅₃	—	2	4	no	—
28	C₂×M₁₆×C₅₃	—	2	4	no	—
29	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	#117×C₅₃	—	2	1	no	—
40	#117×C₅₃	—	2	1	no	—
41	#117×C₅₃	C₂×C₄	1	2	yes	—
42	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
43	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
47	#97×C₅₃	C₂×C₄	1	2	yes	—
48	#97×C₅₃	C₂×C₄	1	2	yes	—
49	#97×C₅₃	C₂×C₄	1	2	yes	—
50	#97×C₅₃	C₂³	1	2	yes	—
51	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₅₃	C₂³	1	2	yes	—
55	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	M₁₆×C₅₃	—	2	2	no	—
66	M₁₆×C₅₃	—	2	2	no	—
67	M₁₆×C₅₃	—	2	2	no	—
68	M₁₆×C₅₃	—	2	4	no	—
69	M₁₆×C₅₃	—	2	2	no	—
70	M₁₆×C₅₃	—	2	2	no	—
71	M₁₆×C₅₃	M₁₆	1	4	yes	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₅₃	#41	1	4	yes	—
80	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₄⋊C₄×C₅₃	—	2	1	no	—
83	C₄⋊C₄×C₅₃	—	2	1	no	—
84	C₂²⋊C₄×C₅₃	—	2	2	no	—
85	C₂²⋊C₄×C₅₃	—	2	1	no	—
86	C₂²⋊C₄×C₅₃	—	2	1	no	—
87	C₄²×C₅₃	—	2	2	no	—
88	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
90	#11719908	—	2	8	no	—
91	C₂³×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	—	8	1	no	—
93	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	8	2	no	—
106	C₂×C₄×C₅₃	#117	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
112	C₈×C₅₃	—	2	1	no	—
113	C₈×C₅₃	—	2	1	no	—
114	C₈×C₅₃	—	2	2	no	—
115	C₈×C₅₃	#117	1	2	yes	—
116	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
117	C₈×C₅₃	C₂×#41	1	2	yes	—
118	#33427	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	—	2	1	no	—
121	C₂²×C₅₃	—	4	1	no	—
122	C₂²×C₅₃	—	4	1	no	—
123	C₂²×C₅₃	—	4	1	no	—
124	C₂²×C₅₃	—	8	2	no	—
125	C₂²×C₅₃	#432	1	2	yes	—
126	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	4	2	no	—
130	C₄×C₅₃	—	4	2	no	—
131	C₄×C₅₃	#432	1	2	yes	—
132	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
133	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
134	#2873	D₅₃	1	1	yes	characteristic
135	#2166	—	53	4	no	—
136	#2165	—	53	4	no	—
137	#2153	—	53	2	no	—
138	C₂×#349	—	53	2	no	—
139	#2110	—	53	2	no	—
140	C₂×C₅₃	—	2	1	no	—
141	C₂×C₅₃	—	2	2	no	—
142	C₂×C₅₃	—	4	1	no	—
143	C₂×C₅₃	—	4	1	no	—
144	C₂×C₅₃	C₂×#432	1	1	yes	characteristic
145	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
146	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
147	C₂×#117	Dic₅₃	1	1	yes	characteristic
148	C₂×M₃₂	—	53	4	no	—
149	#405	Dic₅₃	1	1	yes	characteristic
150	#405	Dic₅₃	1	2	yes	—
151	#405	Dic₅₃	1	1	yes	characteristic
152	#405	C₂×D₅₃	1	2	yes	—
153	#405	C₂×D₅₃	1	1	yes	characteristic
154	#405	C₂×D₅₃	1	1	yes	characteristic
155	C₄×M₁₆	Dic₅₃	1	2	yes	—
156	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
157	#364	—	53	8	no	—
158	#349	—	53	4	no	—
159	#349	—	53	4	no	—
160	#347	—	53	8	no	—
161	C₅₃	#33427	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
166	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
167	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
168	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
169	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
170	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
171	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
172	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
173	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
175	#117	—	2	1	no	—
176	#117	—	2	1	no	—
177	#117	C₂×Dic₅₃	1	2	yes	—
178	#117	C₂×Dic₅₃	1	1	yes	characteristic
179	#117	C₂×Dic₅₃	1	1	yes	characteristic
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₂²⋊C₄	—	2	1	no	—
182	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
183	M₃₂	—	53	8	no	—
184	M₃₂	—	106	4	no	—
185	C₂×C₁₆	—	106	4	no	—
186	#97	C₂×Dic₅₃	1	2	yes	—
187	#97	C₂×Dic₅₃	1	2	yes	—
188	#97	C₂×Dic₅₃	1	2	yes	—
189	#97	C₂²×D₅₃	1	2	yes	—
190	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
191	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
192	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
193	C₄×C₈	C₂²×D₅₃	1	2	yes	—
194	C₂⁴	#11719917	1	1	yes	characteristic
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	#11719917	1	1	yes	characteristic
199	C₂²×C₄	#11719917	1	1	yes	characteristic
200	C₂²×C₄	#11719917	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	#11719917	1	4	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	#11719945	1	4	yes	—
214	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
215	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
216	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
217	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
218	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
219	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₂²⋊C₄	—	2	1	no	—
226	C₄²	—	2	2	no	—
227	C₄²	C₂²×Dic₅₃	1	2	yes	—
228	C₄²	C₂²×Dic₅₃	1	2	yes	—
229	C₁₆	—	106	8	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	8	1	no	—
232	C₂³	C₂×#11719917	1	1	yes	characteristic
233	C₂³	C₂×#11719917	1	1	yes	characteristic
234	C₂³	C₂×#11719917	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	#13133176	1	2	yes	—
246	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
247	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
248	C₂×C₄	C₂×#11719917	1	2	yes	—
249	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
250	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	—	2	2	no	—
254	C₈	#13133176	1	2	yes	—
255	C₈	C₂×#11719917	1	4	yes	—
256	C₈	C₂×#11719945	1	2	yes	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	2	no	—
263	C₂²	#16810680	1	2	yes	—
264	C₂²	C₂²×#11719917	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	#16810680	1	2	yes	—
270	C₄	C₂×#13133176	1	1	yes	characteristic
271	C₄	C₂²×#11719917	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	2	no	—
274	C₂	—	4	1	no	—
275	C₂	—	4	1	no	—
276	C₂	C₂×#16810680	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13133176
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2873	C₂
socle	C₂×C₅₃	C₂×#432
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2873	D₅₃
hypocenter	C₅₃	#33427
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #405 ◁ #2873

Complements

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

N	H ≅ G/N	type
1. #20276531	C₂	semidirect
2. #20276527	C₂	semidirect
3. #20276481	C₂	semidirect
5. #20276320	C₂	semidirect
8. #16810444	C₂²	semidirect
11. #16810401	C₂²	semidirect
161. C₅₃	#33427	semidirect

Maximal subgroups

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

subgroup	index
#20276531	2
#20276527	2
#20276481	2
C₂×#16810405	2
#20276320	2
C₅₃×#2873	2
#33427	53