Subgroups — #27299213

Order 13568.

382 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811041	C₂	1	1	yes	maximal, characteristic
2	#20276335	C₂	1	1	yes	maximal, characteristic
3	#20276319	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274516	C₂	1	1	yes	maximal, characteristic
5	#20272805	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272804	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133223	C₂²	1	1	yes	characteristic
9	#16811041	—	2	2	no	—
10	#16811041	—	2	2	no	—
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	C₂×#13133156	C₄	1	2	yes	complemented
13	C₂²×#11719908	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₅₃	C₄	1	2	yes	complemented
15	#16810605	C₄	1	1	yes	characteristic
16	#16810596	C₄	1	1	yes	characteristic
17	#16810578	C₂²	1	1	yes	characteristic
18	#16810404	C₄	1	4	yes	complemented
19	#16810400	—	2	2	no	—
20	#16810395	C₄	1	1	yes	characteristic
21	#16810393	C₄	1	1	yes	characteristic
22	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#405	C₂²	1	1	yes	characteristic
25	C₅₃×#347	—	2	2	no	—
26	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
27	#13133223	—	2	1	no	—
28	#13133223	—	2	1	no	—
29	#13133223	—	2	1	no	—
30	#13133223	—	2	1	no	—
31	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
32	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
33	C₂×#11719912	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
35	#13133176	C₂×C₄	1	1	yes	characteristic
36	#13133157	—	2	1	no	—
37	#13133157	C₂×C₄	1	2	yes	—
38	#13133156	—	2	4	no	—
39	C₂×#11719908	—	2	1	no	—
40	C₂×#11719908	—	2	1	no	—
41	C₂×#11719908	—	2	1	no	—
42	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
43	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
44	C₁₆×D₅₃	—	2	4	no	—
45	#13133135	—	2	1	no	—
46	#13133133	—	2	1	no	—
47	#13133131	—	2	2	no	—
48	#13133127	—	2	1	no	—
49	C₈×Dic₅₃	—	2	1	no	—
50	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₅₃	—	2	1	no	—
54	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
55	M₃₂×C₅₃	D₄	1	2	yes	—
56	M₃₂×C₅₃	D₄	1	2	yes	—
57	C₂×C₁₆×C₅₃	—	2	1	no	—
58	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
59	#97×C₅₃	—	2	1	no	—
60	C₄×C₈×C₅₃	—	2	1	no	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₃	M₁₆	1	2	yes	—
63	#11719945	—	2	1	no	—
64	#11719945	—	2	1	no	—
65	#11719945	—	2	1	no	—
66	#11719945	—	2	1	no	—
67	C₂×D₂₁₂	M₁₆	1	1	yes	characteristic
68	C₂×C₄×D₅₃	C₄²	1	2	yes	—
69	C₂×Dic₁₀₆	M₁₆	1	1	yes	characteristic
70	M₁₆×C₅₃	—	2	1	no	—
71	M₁₆×C₅₃	—	2	1	no	—
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
78	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
79	C₄⋊C₄×C₅₃	—	2	1	no	—
80	C₂²⋊C₄×C₅₃	—	2	1	no	—
81	C₄²×C₅₃	—	2	1	no	—
82	C₂²⋊Dic₅₃	—	2	1	no	—
83	C₄⋊Dic₅₃	—	2	1	no	—
84	C₄×Dic₅₃	—	2	1	no	—
85	#11719917	—	2	1	no	—
86	#11719917	—	2	1	no	—
87	#11719917	—	2	1	no	—
88	#11719917	—	2	1	no	—
89	C₂×#88322	—	2	1	no	—
90	C₂×#88322	C₄²	1	2	yes	—
91	#11719912	—	2	2	no	—
92	#11719912	—	2	2	no	—
93	C₈×D₅₃	—	2	2	no	—
94	C₈×D₅₃	—	2	2	no	—
95	C₁₆×C₅₃	—	2	2	no	—
96	C₁₆×C₅₃	C₂²⋊C₄	1	4	yes	—
97	#11719908	—	2	1	no	—
98	#11719908	—	2	1	no	—
99	#11719908	—	2	2	no	—
100	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×D₅₃	#97	1	2	yes	—
102	C₂×C₄×C₅₃	—	4	1	no	—
103	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
109	C₂²⋊D₅₃	—	2	2	no	—
110	C₂²⋊D₅₃	—	2	2	no	—
111	C₂×Dic₅₃	—	4	1	no	—
112	C₂×Dic₅₃	#97	1	2	yes	—
113	D₂₁₂	—	2	1	no	—
114	D₂₁₂	—	2	1	no	—
115	C₄×D₅₃	—	2	2	no	—
116	C₄×D₅₃	—	2	2	no	—
117	Dic₁₀₆	—	2	1	no	—
118	Dic₁₀₆	—	2	1	no	—
119	C₈×C₅₃	—	4	1	no	—
120	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₈×C₅₃	#117	1	1	yes	characteristic
122	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
124	#88322	—	2	2	no	—
125	#88322	—	4	1	no	—
126	#12617	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	—	4	1	no	—
128	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
129	C₂²×C₅₃	#435	1	1	yes	characteristic
130	C₂²×C₅₃	#435	1	1	yes	characteristic
131	C₂×D₅₃	—	2	2	no	—
132	C₂×D₅₃	—	4	2	no	—
133	C₄×C₅₃	—	8	1	no	—
134	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₅₃	#405	1	1	yes	characteristic
136	C₄×C₅₃	#435	1	1	yes	characteristic
137	C₄×C₅₃	#435	1	1	yes	characteristic
138	Dic₅₃	—	2	2	no	—
139	Dic₅₃	—	8	1	no	—
140	C₂×#504	—	53	1	no	—
141	#2115	—	53	1	no	—
142	#2110	—	53	1	no	—
143	#2110	D₅₃	1	1	yes	characteristic, complemented
144	#1757	—	53	1	no	—
145	#1379	—	53	1	no	—
146	#1379	—	53	1	no	—
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	#1752	1	1	yes	characteristic
151	D₅₃	—	4	2	no	—
152	C₂×#131	—	53	1	no	—
153	#504	—	106	2	no	—
154	#504	—	106	2	no	—
155	C₂×M₃₂	—	53	2	no	—
156	C₂×M₃₂	—	53	1	no	—
157	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
158	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
159	C₂²×C₁₆	—	53	1	no	—
160	C₂²×C₁₆	—	53	2	no	—
161	#413	—	53	1	no	—
162	#410	—	53	1	no	—
163	#405	—	53	1	no	—
164	#405	C₂×D₅₃	1	1	yes	characteristic
165	#349	—	53	4	no	—
166	#347	—	2	2	no	—
167	#347	—	106	2	no	—
168	#344	—	53	1	no	—
169	#344	—	53	1	no	—
170	C₅₃	#12617	1	1	yes	characteristic, complemented, Sylow
171	C₂×#41	—	53	1	no	—
172	#131	—	106	1	no	—
173	#131	—	106	1	no	—
174	#131	—	106	1	no	—
175	#131	—	106	1	no	—
176	C₂×M₁₆	—	53	1	no	—
177	C₂×M₁₆	—	53	2	no	—
178	C₂×M₁₆	—	53	1	no	—
179	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
180	C₂²×C₈	—	53	2	no	—
181	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
182	#117	—	53	1	no	—
183	#117	C₄×D₅₃	1	1	yes	characteristic
184	M₃₂	—	2	1	no	—
185	M₃₂	—	53	2	no	—
186	M₃₂	—	106	1	no	—
187	M₃₂	—	106	4	no	—
188	M₃₂	C₄×D₅₃	1	2	yes	—
189	M₃₂	D₂₁₂	1	2	yes	—
190	M₃₂	C₂²⋊D₅₃	1	2	yes	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	53	1	no	—
193	C₂×C₁₆	—	53	1	no	—
194	C₂×C₁₆	—	106	1	no	—
195	C₂×C₁₆	—	106	1	no	—
196	C₂×C₁₆	—	106	1	no	—
197	C₂×C₁₆	—	106	4	no	—
198	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
199	#101	—	106	1	no	—
200	#100	—	106	1	no	—
201	#99	—	106	2	no	—
202	#97	—	2	1	no	—
203	#97	—	106	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	106	1	no	—
206	#41	—	106	1	no	—
207	#41	—	106	1	no	—
208	#41	—	106	1	no	—
209	#41	—	106	1	no	—
210	C₂×Q₈	—	53	1	no	—
211	C₂×D₄	—	53	2	no	—
212	C₂×D₄	—	53	1	no	—
213	C₂²×C₄	—	53	2	no	—
214	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
215	M₁₆	—	2	1	no	—
216	M₁₆	—	2	1	no	—
217	M₁₆	—	106	1	no	—
218	M₁₆	—	106	1	no	—
219	M₁₆	—	106	2	no	—
220	M₁₆	—	106	2	no	—
221	M₁₆	—	106	1	no	—
222	M₁₆	—	106	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	53	2	no	—
225	C₂×C₈	—	106	2	no	—
226	C₂×C₈	—	106	2	no	—
227	C₂×C₈	—	106	1	no	—
228	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
230	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
231	C₂×C₈	#11719945	1	1	yes	characteristic
232	C₂×C₈	#11719945	1	1	yes	characteristic
233	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	106	1	no	—
236	C₂²⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	106	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	106	1	no	—
240	C₁₆	—	2	2	no	—
241	C₁₆	—	106	1	no	—
242	C₁₆	—	106	1	no	—
243	C₁₆	—	106	2	no	—
244	C₁₆	#11719921	1	4	yes	—
245	C₂³	—	53	2	no	—
246	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
247	Q₈	—	106	1	no	—
248	Q₈	—	106	1	no	—
249	D₄	—	106	2	no	—
250	D₄	—	106	2	no	—
251	D₄	—	106	1	no	—
252	D₄	—	106	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	53	2	no	—
255	C₂×C₄	—	106	2	no	—
256	C₂×C₄	—	106	2	no	—
257	C₂×C₄	—	212	1	no	—
258	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
259	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
260	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
261	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
262	C₂×C₄	#13133225	1	1	yes	characteristic
263	C₂×C₄	#13133225	1	1	yes	characteristic
264	C₈	—	4	1	no	—
265	C₈	—	106	2	no	—
266	C₈	—	212	1	no	—
267	C₈	C₂×#11719921	1	1	yes	characteristic
268	C₈	#13133174	1	1	yes	characteristic
269	C₈	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
270	C₈	C₄×D₂₁₂	1	1	yes	characteristic
271	C₂²	—	4	1	no	—
272	C₂²	—	106	2	no	—
273	C₂²	—	212	2	no	—
274	C₂²	#97×D₅₃	1	1	yes	characteristic
275	C₂²	#16810692	1	1	yes	characteristic
276	C₂²	#16810696	1	1	yes	characteristic
277	C₄	—	8	1	no	—
278	C₄	—	106	2	no	—
279	C₄	—	424	1	no	—
280	C₄	C₄×#11719921	1	1	yes	characteristic
281	C₄	#16810580	1	1	yes	characteristic
282	C₄	#16810692	1	1	yes	characteristic
283	C₄	#16810696	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	212	2	no	—
288	C₂	#20274492	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133174
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2110	C₂
socle	C₂×C₅₃	#1752
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2110	D₅₃
hypocenter	C₅₃	#12617
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ #2110

Complements

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

N	H ≅ G/N	type
3. #20276319	C₂	semidirect
5. #20272805	C₂	semidirect
6. #20272804	C₂	semidirect
7. C₅₃×#2110	C₂	semidirect
12. C₂×#13133156	C₄	semidirect
14. C₂×C₁₆×D₅₃	C₄	semidirect
18. #16810404	C₄	semidirect
143. #2110	D₅₃	semidirect
170. C₅₃	#12617	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811041	2
#20276335	2
#20276319	2
#20274516	2
#20272805	2
#20272804	2
C₅₃×#2110	2
#12617	53