Subgroups — #27277241

Order 13568.

374 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274887	C₂	1	1	yes	maximal, characteristic
2	#20273463	C₂	1	1	yes	maximal, characteristic
3	#20273144	C₂	1	2	yes	maximal
4	#20272612	C₂	1	2	yes	maximal
5	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133153	—	2	1	no	—
8	C₂×#13133153	—	2	1	no	—
9	#16810647	C₂²	1	2	yes	—
10	#16810376	—	2	1	no	—
11	#16810374	—	2	1	no	—
12	Q₈⋊#88322	—	2	4	no	—
13	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#446	—	2	1	no	—
15	C₅₃×#446	—	2	1	no	—
16	C₅₃×#446	C₄	1	2	yes	—
17	C₅₃×#446	C₄	1	2	yes	—
18	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#97×C₅₃	—	2	1	no	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#11719917	—	2	2	no	—
23	#13133153	—	4	2	no	—
24	#13133153	—	4	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₂²⋊#88322	—	2	2	no	—
29	C₄×#88322	—	2	2	no	—
30	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	—	2	1	no	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₄×Q₈×C₅₃	—	2	1	no	—
35	C₄×Q₈×C₅₃	—	2	1	no	—
36	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₄×Q₈×C₅₃	D₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	—	2	1	no	—
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₄⋊C₈×C₅₃	—	2	1	no	—
45	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
46	#97×C₅₃	—	2	2	no	—
47	#97×C₅₃	—	2	2	no	—
48	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₂×Q₈×C₅₃	—	2	1	no	—
53	C₂×Q₈×C₅₃	—	4	2	no	—
54	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	—	2	1	no	—
56	C₂²×C₄×C₅₃	—	2	1	no	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	1	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	4	1	no	—
64	C₂×C₈×C₅₃	—	4	1	no	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
70	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
71	C₄⋊C₄×C₅₃	—	2	2	no	—
72	C₄⋊C₄×C₅₃	—	2	2	no	—
73	C₄⋊C₄×C₅₃	—	2	1	no	—
74	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
75	C₄²×C₅₃	—	2	2	no	—
76	C₄²×C₅₃	—	2	2	no	—
77	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅₃	#41	1	1	yes	characteristic
81	#11719917	—	4	2	no	—
82	#11719917	—	4	2	no	—
83	C₂×#88322	—	4	2	no	—
84	C₂×#88322	—	4	2	no	—
85	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
86	Q₈×C₅₃	—	2	4	no	—
87	Q₈×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	4	1	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	—	4	2	no	—
107	C₈×C₅₃	#103	1	2	yes	—
108	C₈×C₅₃	#103	1	2	yes	—
109	#88322	—	8	2	no	—
110	#88322	—	8	2	no	—
111	#8663	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	#380	1	1	yes	characteristic
115	C₂²×C₅₃	#475	1	1	yes	characteristic
116	C₂²×C₅₃	#477	1	1	yes	characteristic
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	4	2	no	—
121	C₄×C₅₃	—	4	2	no	—
122	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
123	C₄×C₅₃	#417	1	1	yes	characteristic
124	C₄×C₅₃	#430	1	2	yes	—
125	C₄×C₅₃	#460	1	1	yes	characteristic
126	C₄×C₅₃	#462	1	1	yes	characteristic
127	C₂×#446	D₅₃	1	1	yes	characteristic
128	#1848	—	53	1	no	—
129	#1564	—	53	1	no	—
130	#1487	—	53	2	no	—
131	#1301	—	53	2	no	—
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	#1803	1	1	yes	characteristic
135	C₂×C₅₃	#1811	1	1	yes	characteristic
136	C₂×C₅₃	#1814	1	1	yes	characteristic
137	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
138	#446	—	2	1	no	—
139	#446	—	2	1	no	—
140	#446	Dic₅₃	1	2	yes	—
141	#446	Dic₅₃	1	2	yes	—
142	#431	—	53	2	no	—
143	C₂×#108	—	106	1	no	—
144	C₂×#108	—	106	1	no	—
145	#423	—	53	2	no	—
146	C₂×C₄⋊C₈	—	2	1	no	—
147	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
148	C₂×#97	—	2	1	no	—
149	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
150	#335	—	106	1	no	—
151	#334	—	106	1	no	—
152	Q₈⋊C₈	—	106	4	no	—
153	C₅₃	#8663	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
155	C₂×M₁₆	—	106	2	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₄×Q₈	—	2	1	no	—
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	D₂₁₂	1	2	yes	—
162	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	Dic₁₀₆	1	1	yes	characteristic
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
167	#108	—	212	2	no	—
168	#108	—	212	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	106	2	no	—
173	C₄⋊C₈	—	106	2	no	—
174	C₄⋊C₈	—	106	2	no	—
175	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
176	C₂²⋊C₈	—	106	2	no	—
177	#97	—	2	2	no	—
178	#97	—	2	2	no	—
179	C₄×C₈	—	106	2	no	—
180	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
181	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
183	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
184	C₂×Q₈	—	2	1	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
190	C₂²×C₄	#11719945	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
192	M₁₆	—	212	2	no	—
193	M₁₆	—	212	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	212	2	no	—
200	C₂×C₈	—	212	2	no	—
201	C₂×C₈	#11719924	1	2	yes	—
202	C₂×C₈	#11719925	1	2	yes	—
203	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
214	C₄²	#11719945	1	1	yes	characteristic
215	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
217	C₂³	#13133196	1	1	yes	characteristic
218	Q₈	—	2	4	no	—
219	Q₈	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
228	C₂×C₄	#13133176	1	1	yes	characteristic
229	C₂×C₄	#13133191	1	1	yes	characteristic
230	C₂×C₄	#13133191	1	1	yes	characteristic
231	C₂×C₄	#13133196	1	1	yes	characteristic
232	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
233	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
234	C₂×C₄	#13133242	1	1	yes	characteristic
235	C₂×C₄	#13133247	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	4	2	no	—
239	C₈	—	424	2	no	—
240	C₈	—	424	2	no	—
241	C₈	#13133141	1	2	yes	—
242	C₈	#13133141	1	2	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	#16810483	1	1	yes	characteristic
246	C₂²	#16810908	1	1	yes	characteristic
247	C₂²	#16810918	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	4	2	no	—
252	C₄	—	4	2	no	—
253	C₄	C₂×#13133141	1	1	yes	characteristic
254	C₄	#16810624	1	1	yes	characteristic
255	C₄	#16810672	1	2	yes	—
256	C₄	#16810823	1	1	yes	characteristic
257	C₄	#16810833	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	#20274720	1	1	yes	characteristic
261	C₂	#20274757	1	1	yes	characteristic
262	C₂	#20274769	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#446	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	C₂²⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#446	D₅₃
hypocenter	C₅₃	#8663
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
153. C₅₃	#8663	semidirect

Maximal subgroups

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

subgroup	index
#20274887	2
#20273463	2
#20273144	2
#20272612	2
C₂×C₅₃×#446	2
#8663	53