Subgroups — #27537224

Order 13568.

520 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	#20283142	C₂	1	1	yes	maximal, characteristic
2	#20276530	C₂	1	4	yes	maximal, complemented
3	#20276526	C₂	1	4	yes	maximal, complemented
4	#20276481	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276335	C₂	1	2	yes	maximal
6	C₂×#16810400	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133225	C₄	1	1	yes	characteristic
9	C₂×#13133223	C₂²	1	1	yes	characteristic
10	#16811091	C₄	1	1	yes	characteristic
11	C₂×#13133157	C₂²	1	2	yes	—
12	#16810580	C₄	1	2	yes	—
13	#16810579	C₄	1	1	yes	characteristic
14	#16810577	C₂²	1	2	yes	—
15	M₁₆×Dic₅₃	C₄	1	1	yes	characteristic
16	C₄×#11719912	C₂²	1	2	yes	—
17	C₂×#13133127	C₂²	1	1	yes	characteristic
18	#97×D₅₃	C₄	1	2	yes	—
19	#16810444	C₂²	1	4	yes	—
20	#16810404	C₂²	1	8	yes	—
21	#16810400	C₂²	1	4	yes	—
22	#16810400	C₂²	1	4	yes	—
23	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
24	C₅₃×#405	C₂²	1	1	yes	characteristic
25	C₅₃×#364	C₂²	1	4	yes	—
26	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
27	#13133225	—	2	4	no	—
28	#13133223	—	2	2	no	—
29	#13133223	—	2	2	no	—
30	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
31	C₂×#11719912	C₂×C₄	1	2	yes	—
32	C₂×#11719912	C₂³	1	2	yes	—
33	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
36	#13133182	—	2	1	no	—
37	#13133179	—	2	1	no	—
38	#13133177	—	2	2	no	—
39	#13133174	C₂×C₄	1	2	yes	—
40	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄²×D₅₃	C₂×C₄	1	2	yes	—
42	#13133157	—	2	2	no	—
43	#13133157	C₂³	1	4	yes	—
44	C₂×#11719908	—	2	2	no	—
45	#13133128	C₂×C₄	1	2	yes	—
46	#13133127	C₂×C₄	1	1	yes	characteristic
47	#13133127	C₂×C₄	1	2	yes	—
48	#13133127	C₂×C₄	1	1	yes	characteristic
49	#13133127	C₂³	1	1	yes	characteristic
50	#13133127	C₂³	1	1	yes	characteristic
51	C₄×#88322	C₂×C₄	1	2	yes	—
52	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
53	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
56	M₃₂×C₅₃	—	2	2	no	—
57	M₃₂×C₅₃	C₂³	1	4	yes	—
58	C₂×C₁₆×C₅₃	—	2	2	no	—
59	#97×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₅₃	C₂³	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₃	M₁₆	1	2	yes	—
63	C₂²×Dic₅₃	—	2	1	no	—
64	#11719945	—	2	2	no	—
65	#11719945	—	2	2	no	—
66	C₂×D₂₁₂	M₁₆	1	1	yes	characteristic
67	C₂×C₄×D₅₃	—	2	2	no	—
68	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
69	C₂×Dic₁₀₆	M₁₆	1	1	yes	characteristic
70	M₁₆×C₅₃	M₁₆	1	4	yes	—
71	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₅₃	#41	1	2	yes	—
74	C₂×C₈×C₅₃	#41	1	2	yes	—
75	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
76	C₄⋊C₄×C₅₃	—	2	1	no	—
77	C₂²⋊C₄×C₅₃	—	2	1	no	—
78	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
79	#11719921	—	2	2	no	—
80	#11719919	—	2	2	no	—
81	C₄×Dic₅₃	—	2	1	no	—
82	C₄×Dic₅₃	—	2	2	no	—
83	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
85	#11719917	—	2	2	no	—
86	C₂×#88322	—	2	2	no	—
87	C₂×#88322	C₂²×C₄	1	2	yes	—
88	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
89	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
90	#11719912	—	2	4	no	—
91	#11719912	—	2	2	no	—
92	#11719912	—	2	2	no	—
93	C₈×D₅₃	—	2	4	no	—
94	C₈×D₅₃	—	2	4	no	—
95	C₁₆×C₅₃	—	2	4	no	—
96	#11719908	—	2	4	no	—
97	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂²×D₅₃	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	#117	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
104	C₂²⋊D₅₃	—	2	4	no	—
105	C₂×Dic₅₃	—	2	1	no	—
106	C₂×Dic₅₃	—	2	1	no	—
107	C₂×Dic₅₃	—	8	1	no	—
108	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
109	D₂₁₂	—	2	2	no	—
110	C₄×D₅₃	—	2	2	no	—
111	C₄×D₅₃	—	2	2	no	—
112	C₄×D₅₃	—	8	2	no	—
113	Dic₁₀₆	—	2	2	no	—
114	C₈×C₅₃	#117	1	2	yes	—
115	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
116	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
117	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
118	#88322	—	2	2	no	—
119	#88322	—	2	1	no	—
120	#88322	—	2	1	no	—
121	#33426	—	53	1	no	maximal, Sylow
122	C₂²×C₅₃	—	4	1	no	—
123	C₂²×C₅₃	#432	1	2	yes	—
124	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
125	C₂×D₅₃	—	2	2	no	—
126	C₂×D₅₃	—	4	2	no	—
127	C₄×C₅₃	—	4	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	1	no	—
133	Dic₅₃	—	4	1	no	—
134	#2875	—	53	1	no	—
135	#2166	—	53	4	no	—
136	#2165	—	53	4	no	—
137	#2153	—	53	1	no	—
138	#2153	D₅₃	1	1	yes	characteristic, complemented
139	#2115	—	53	2	no	—
140	C₂×#347	—	53	2	no	—
141	C₂×C₅₃	—	2	1	no	—
142	C₂×C₅₃	—	2	2	no	—
143	C₂×C₅₃	C₂×#432	1	1	yes	characteristic
144	D₅₃	—	4	2	no	—
145	C₂×#131	—	53	1	no	—
146	C₂×#131	—	53	1	no	—
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	C₂×D₅₃	1	1	yes	characteristic
154	C₄×M₁₆	—	53	1	no	—
155	C₄×M₁₆	—	53	2	no	—
156	C₂×#97	—	53	1	no	—
157	C₂×#97	—	53	2	no	—
158	#364	—	53	4	no	—
159	#364	C₂×D₅₃	1	4	yes	—
160	#349	—	53	8	no	—
161	#347	—	53	4	no	—
162	#347	—	53	4	no	—
163	C₅₃	#33426	1	1	yes	characteristic, complemented, Sylow
164	C₂×#41	—	53	1	no	—
165	#131	—	106	4	no	—
166	#131	—	106	2	no	—
167	#131	—	106	2	no	—
168	C₂×M₁₆	—	53	1	no	—
169	C₂×M₁₆	—	53	2	no	—
170	C₂×M₁₆	—	53	2	no	—
171	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
172	C₂²×C₈	—	53	1	no	—
173	C₂²×C₈	—	53	2	no	—
174	C₂²×C₈	—	53	2	no	—
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	C₄×D₅₃	1	1	yes	characteristic
181	C₂×C₄²	—	53	1	no	—
182	C₂×C₄²	—	53	2	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	1	no	—
190	#97	—	53	2	no	—
191	#97	—	53	1	no	—
192	#97	—	53	1	no	—
193	#97	—	53	1	no	—
194	#97	—	53	2	no	—
195	#97	C₄×D₅₃	1	2	yes	—
196	C₄×C₈	—	53	2	no	—
197	C₄×C₈	—	53	2	no	—
198	C₄×C₈	C₂²×D₅₃	1	2	yes	—
199	#41	—	106	2	no	—
200	#41	—	106	2	no	—
201	C₂×Q₈	—	53	1	no	—
202	C₂×D₄	—	53	2	no	—
203	C₂×D₄	—	53	1	no	—
204	C₂²×C₄	—	53	2	no	—
205	C₂²×C₄	—	106	2	no	—
206	C₂²×C₄	—	106	1	no	—
207	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
208	M₁₆	—	106	2	no	—
209	M₁₆	—	106	4	no	—
210	M₁₆	—	106	2	no	—
211	M₁₆	—	106	2	no	—
212	M₁₆	#11719912	1	4	yes	—
213	C₂×C₈	—	53	2	no	—
214	C₂×C₈	—	53	1	no	—
215	C₂×C₈	—	53	1	no	—
216	C₂×C₈	—	106	2	no	—
217	C₂×C₈	—	106	4	no	—
218	C₂×C₈	—	106	4	no	—
219	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
220	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
221	C₂×C₈	#11719947	1	2	yes	—
222	C₂×C₈	#11719949	1	2	yes	—
223	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	106	2	no	—
226	C₂²⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	106	2	no	—
228	C₄²	—	53	1	no	—
229	C₄²	—	53	1	no	—
230	C₄²	—	106	2	no	—
231	C₄²	—	106	1	no	—
232	C₄²	C₂×C₄×D₅₃	1	2	yes	—
233	C₁₆	—	2	4	no	—
234	C₁₆	—	106	4	no	—
235	C₂³	—	53	2	no	—
236	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
237	Q₈	—	106	2	no	—
238	D₄	—	106	4	no	—
239	D₄	—	106	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	53	2	no	—
242	C₂×C₄	—	106	2	no	—
243	C₂×C₄	—	106	2	no	—
244	C₂×C₄	—	106	1	no	—
245	C₂×C₄	—	106	1	no	—
246	C₂×C₄	—	424	2	no	—
247	C₂×C₄	—	424	1	no	—
248	C₂×C₄	#13133175	1	2	yes	—
249	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
250	C₂×C₄	C₂×#11719912	1	2	yes	—
251	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
252	C₈	—	106	2	no	—
253	C₈	—	106	1	no	—
254	C₈	—	106	1	no	—
255	C₈	#13133175	1	2	yes	—
256	C₈	C₂×#11719912	1	4	yes	—
257	C₈	C₂×#11719947	1	1	yes	characteristic
258	C₈	C₂×#11719949	1	1	yes	characteristic
259	C₂²	—	4	1	no	—
260	C₂²	—	106	2	no	—
261	C₂²	—	212	2	no	—
262	C₂²	#16810679	1	2	yes	—
263	C₂²	C₂×M₁₆×D₅₃	1	1	yes	characteristic
264	C₄	—	4	2	no	—
265	C₄	—	106	2	no	—
266	C₄	—	212	1	no	—
267	C₄	—	212	1	no	—
268	C₄	#16810679	1	2	yes	—
269	C₄	C₂×#13133175	1	1	yes	characteristic
270	C₄	C₂²×#11719912	1	1	yes	characteristic
271	C₂	—	2	1	no	—
272	C₂	—	2	2	no	—
273	C₂	—	212	2	no	—
274	C₂	C₂×#16810679	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13133175
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2153	C₂
socle	C₂×C₅₃	C₂×#432
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2153	D₅₃
hypocenter	C₅₃	#33426
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #2153

Complements

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

N	H ≅ G/N	type
2. #20276530	C₂	semidirect
3. #20276526	C₂	semidirect
4. #20276481	C₂	semidirect
6. C₂×#16810400	C₂	semidirect
7. C₅₃×#2153	C₂	semidirect
138. #2153	D₅₃	semidirect
163. C₅₃	#33426	semidirect

Maximal subgroups

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

subgroup	index
#20283142	2
#20276530	2
#20276526	2
#20276481	2
#20276335	2
C₂×#16810400	2
C₅₃×#2153	2
#33426	53