Subgroups — #27277352

Order 13568.

362 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283649	C₂	1	1	yes	maximal, characteristic
2	#20274887	C₂	1	1	yes	maximal, characteristic
3	#20273403	C₂	1	2	yes	maximal
4	#20272590	C₂	1	2	yes	maximal
5	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
6	#16811180	C₂²	1	1	yes	characteristic
7	#16810753	—	2	1	no	—
8	#16810753	C₄	1	2	yes	—
9	C₈×Dic₁₀₆	—	2	1	no	—
10	C₈×Dic₁₀₆	C₄	1	2	yes	—
11	C₂×#13133153	—	2	1	no	—
12	C₂×#13133153	—	2	1	no	—
13	#16810649	—	2	1	no	—
14	#16810647	C₂²	1	2	yes	—
15	#16810647	C₂²	1	1	yes	characteristic
16	#16810578	—	2	1	no	—
17	#16810371	—	2	2	no	—
18	#16810370	—	2	2	no	—
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#335	—	2	1	no	—
22	C₅₃×#335	—	2	1	no	—
23	C₂×#11719917	—	2	1	no	—
24	C₂×#11719917	—	2	2	no	—
25	#13133217	D₄	1	1	yes	characteristic
26	#13133217	D₄	1	1	yes	characteristic
27	#13133209	—	2	1	no	—
28	#13133196	—	2	1	no	—
29	#13133196	Q₈	1	2	yes	—
30	C₄×Dic₁₀₆	—	2	1	no	—
31	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
32	#13133176	—	2	1	no	—
33	#13133153	—	4	2	no	—
34	#13133153	—	4	2	no	—
35	C₄⋊#88322	—	2	1	no	—
36	C₄⋊#88322	—	2	2	no	—
37	C₄⋊#88322	—	2	2	no	—
38	C₄⋊#88322	C₂×C₄	1	2	yes	—
39	#13133146	—	2	2	no	—
40	#13133146	—	2	2	no	—
41	#13133127	—	2	2	no	—
42	C₈×Dic₅₃	—	2	2	no	—
43	C₂²×C₈×C₅₃	—	2	2	no	—
44	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₅₃	—	2	2	no	—
48	C₄⋊C₈×C₅₃	—	2	2	no	—
49	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
56	C₂×Dic₁₀₆	—	2	1	no	—
57	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₂×C₈×C₅₃	—	8	2	no	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
67	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
68	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₃	#41	1	1	yes	characteristic
72	C₂²⋊Dic₅₃	—	2	2	no	—
73	C₄⋊Dic₅₃	—	2	1	no	—
74	C₄⋊Dic₅₃	—	2	1	no	—
75	C₄⋊Dic₅₃	—	2	1	no	—
76	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
77	#11719919	—	2	2	no	—
78	#11719919	—	2	2	no	—
79	C₄×Dic₅₃	—	2	2	no	—
80	#11719917	—	4	1	no	—
81	#11719917	—	4	1	no	—
82	#11719917	—	4	2	no	—
83	#11719917	—	4	2	no	—
84	C₂×#88322	—	2	2	no	—
85	C₂×#88322	—	4	2	no	—
86	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	4	1	no	—
90	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
100	C₂×Dic₅₃	—	2	2	no	—
101	C₂×Dic₅₃	—	2	2	no	—
102	Dic₁₀₆	—	4	1	no	—
103	Dic₁₀₆	—	4	1	no	—
104	Dic₁₀₆	—	4	2	no	—
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	—	2	1	no	—
107	C₈×C₅₃	—	8	2	no	—
108	C₈×C₅₃	#103	1	2	yes	—
109	C₈×C₅₃	#103	1	2	yes	—
110	#88322	—	4	2	no	—
111	#88322	—	8	2	no	—
112	#8685	—	53	1	no	maximal, Sylow
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	2	1	no	—
115	C₂²×C₅₃	#380	1	1	yes	characteristic
116	C₂²×C₅₃	#475	1	1	yes	characteristic
117	C₂²×C₅₃	#477	1	1	yes	characteristic
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
122	C₄×C₅₃	#417	1	1	yes	characteristic
123	C₄×C₅₃	#460	1	1	yes	characteristic
124	C₄×C₅₃	#462	1	1	yes	characteristic
125	C₄×C₅₃	#472	1	1	yes	characteristic
126	C₄×C₅₃	#473	1	1	yes	characteristic
127	Dic₅₃	—	4	2	no	—
128	Dic₅₃	—	4	2	no	—
129	#2960	—	53	1	no	—
130	#1848	—	53	1	no	—
131	C₂×#335	D₅₃	1	1	yes	characteristic
132	#1550	—	53	2	no	—
133	#1292	—	53	2	no	—
134	C₂×C₅₃	—	2	1	no	—
135	C₂×C₅₃	—	2	1	no	—
136	C₂×C₅₃	#1803	1	1	yes	characteristic
137	C₂×C₅₃	#1813	1	1	yes	characteristic
138	C₂×C₅₃	#1816	1	1	yes	characteristic
139	#533	—	53	1	no	—
140	#446	—	53	2	no	—
141	#446	—	106	1	no	—
142	C₈×Q₈	—	53	2	no	—
143	C₈×Q₈	—	106	1	no	—
144	C₂×#108	—	106	1	no	—
145	C₂×#108	—	106	1	no	—
146	#424	—	106	1	no	—
147	#423	—	53	1	no	—
148	#423	—	53	2	no	—
149	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
150	#405	—	106	1	no	—
151	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
152	#335	—	2	1	no	—
153	#335	—	2	1	no	—
154	#333	—	106	2	no	—
155	#332	—	106	2	no	—
156	C₅₃	#8685	1	1	yes	characteristic, complemented, Sylow
157	C₂×M₁₆	—	106	1	no	—
158	C₂×M₁₆	—	106	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
161	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
162	C₄⋊Q₈	—	53	1	no	—
163	C₄⋊Q₈	—	53	1	no	—
164	#125	—	106	1	no	—
165	C₂²⋊Q₈	—	53	2	no	—
166	C₂²⋊Q₈	—	106	1	no	—
167	C₄×Q₈	—	53	2	no	—
168	C₄×Q₈	—	106	1	no	—
169	#117	—	106	1	no	—
170	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
171	#108	—	212	2	no	—
172	#108	—	212	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	53	2	no	—
176	C₄⋊C₈	—	106	2	no	—
177	C₄⋊C₈	—	106	2	no	—
178	C₄⋊C₈	—	106	2	no	—
179	C₄⋊C₈	—	106	2	no	—
180	C₄⋊C₈	—	106	1	no	—
181	#97	—	106	2	no	—
182	C₄×C₈	—	106	2	no	—
183	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
184	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
185	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
186	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
187	C₂×Q₈	—	53	2	no	—
188	C₂×Q₈	—	106	1	no	—
189	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
190	C₂²×C₄	#11719947	1	1	yes	characteristic
191	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
192	M₁₆	—	212	2	no	—
193	M₁₆	—	212	2	no	—
194	M₁₆	—	212	1	no	—
195	M₁₆	—	212	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	8	2	no	—
200	C₂×C₈	—	106	2	no	—
201	C₂×C₈	—	212	2	no	—
202	C₂×C₈	#11719913	1	2	yes	—
203	C₂×C₈	Dic₂₁₂	1	2	yes	—
204	C₂×C₈	#11719921	1	1	yes	characteristic
205	C₂×C₈	#11719921	1	1	yes	characteristic
206	C₂×C₈	#11719921	1	1	yes	characteristic
207	C₂×C₈	#11719921	1	1	yes	characteristic
208	C₄⋊C₄	—	53	2	no	—
209	C₄⋊C₄	—	106	2	no	—
210	C₄⋊C₄	—	106	1	no	—
211	C₄⋊C₄	—	106	1	no	—
212	C₄⋊C₄	—	106	1	no	—
213	C₄⋊C₄	—	106	2	no	—
214	C₂²⋊C₄	—	106	2	no	—
215	C₄²	—	106	2	no	—
216	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
217	C₄²	D₄×D₅₃	1	1	yes	characteristic
218	C₄²	#11719949	1	1	yes	characteristic
219	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
220	C₂³	#13133197	1	1	yes	characteristic
221	Q₈	—	212	2	no	—
222	Q₈	—	212	1	no	—
223	Q₈	—	212	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	106	2	no	—
228	C₂×C₄	—	106	2	no	—
229	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
230	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
231	C₂×C₄	#13133175	1	1	yes	characteristic
232	C₂×C₄	#13133188	1	1	yes	characteristic
233	C₂×C₄	#13133192	1	1	yes	characteristic
234	C₂×C₄	#13133193	1	1	yes	characteristic
235	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
236	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
237	C₂×C₄	#13133245	1	1	yes	characteristic
238	C₂×C₄	#13133250	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	8	2	no	—
242	C₈	—	424	2	no	—
243	C₈	—	212	2	no	—
244	C₈	#13133142	1	2	yes	—
245	C₈	#13133142	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	#16810480	1	1	yes	characteristic
249	C₂²	#16810909	1	1	yes	characteristic
250	C₂²	#16810919	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	212	2	no	—
255	C₄	—	212	2	no	—
256	C₄	C₂×#13133142	1	1	yes	characteristic
257	C₄	#16810625	1	1	yes	characteristic
258	C₄	#16810824	1	1	yes	characteristic
259	C₄	#16810834	1	1	yes	characteristic
260	C₄	#16810890	1	1	yes	characteristic
261	C₄	#16810895	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	#20274721	1	1	yes	characteristic
265	C₂	#20274762	1	1	yes	characteristic
266	C₂	#20274780	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810480
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	C₂²⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	#8685
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
156. C₅₃	#8685	semidirect

Maximal subgroups

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

subgroup	index
#20283649	2
#20274887	2
#20273403	2
#20272590	2
C₂×C₅₃×#335	2
#8685	53