Subgroups — #27277218

Order 13568.

394 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#20273463	C₂	1	1	yes	maximal, characteristic
3	#20273143	C₂	1	2	yes	maximal
4	#20272610	C₂	1	2	yes	maximal
5	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₀₆	—	2	1	no	—
8	C₈×Dic₁₀₆	—	2	1	no	—
9	C₈×Dic₁₀₆	C₄	1	2	yes	—
10	C₈×Dic₁₀₆	C₄	1	2	yes	—
11	#16810676	C₂²	1	2	yes	—
12	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
13	C₂×#13133146	—	2	1	no	—
14	C₂×C₈×Dic₅₃	—	2	1	no	—
15	#16810376	—	2	1	no	—
16	#16810374	—	2	1	no	—
17	#16810353	—	2	4	no	—
18	C₂×#108×C₅₃	—	2	1	no	—
19	C₂×#108×C₅₃	—	2	1	no	—
20	C₅₃×#423	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₀₆	Q₈	1	1	yes	characteristic
23	C₂²×#88322	—	2	1	no	—
24	C₄×Dic₁₀₆	—	2	1	no	—
25	C₄×Dic₁₀₆	—	2	1	no	—
26	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
27	C₄×Dic₁₀₆	D₄	1	2	yes	—
28	C₂×C₄⋊Dic₅₃	Q₈	1	1	yes	characteristic
29	C₂×#11719919	—	2	1	no	—
30	C₂×C₄×Dic₅₃	—	2	1	no	—
31	C₄⋊#88322	—	2	1	no	—
32	C₄⋊#88322	—	2	2	no	—
33	C₄⋊#88322	C₂×C₄	1	2	yes	—
34	#13133146	—	2	2	no	—
35	#13133146	—	2	2	no	—
36	C₂²⋊#88322	—	2	2	no	—
37	C₄×#88322	—	2	2	no	—
38	C₈×Dic₅₃	—	2	2	no	—
39	C₈×Dic₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	—	2	2	no	—
41	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	#108×C₅₃	—	4	2	no	—
45	#108×C₅₃	—	4	2	no	—
46	C₄⋊C₈×C₅₃	—	2	2	no	—
47	C₄⋊C₈×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	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
55	C₂²×Dic₅₃	—	2	1	no	—
56	C₂²×Dic₅₃	—	2	1	no	—
57	C₂×Dic₁₀₆	—	2	1	no	—
58	C₂×Dic₁₀₆	—	2	4	no	—
59	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
60	M₁₆×C₅₃	—	4	2	no	—
61	M₁₆×C₅₃	—	4	2	no	—
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	—	4	2	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	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	#41	1	1	yes	characteristic
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	C₄×Dic₅₃	—	2	2	no	—
81	C₂×#88322	—	2	2	no	—
82	C₂×#88322	—	4	1	no	—
83	C₂×#88322	—	4	1	no	—
84	C₂×#88322	—	4	2	no	—
85	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	4	1	no	—
88	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
98	C₂×Dic₅₃	—	2	4	no	—
99	C₂×Dic₅₃	—	2	2	no	—
100	C₂×Dic₅₃	—	2	2	no	—
101	C₂×Dic₅₃	—	4	2	no	—
102	Dic₁₀₆	—	2	4	no	—
103	Dic₁₀₆	—	2	2	no	—
104	Dic₁₀₆	#108	1	4	yes	complemented
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	#8659	—	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	C₂×#108	1	2	yes	—
124	C₄×C₅₃	#460	1	1	yes	characteristic
125	C₄×C₅₃	#462	1	1	yes	characteristic
126	Dic₅₃	—	2	4	no	—
127	Dic₅₃	—	4	2	no	—
128	C₂×C₈×Q₈	—	53	1	no	—
129	#1848	D₅₃	1	1	yes	characteristic
130	#1564	—	53	1	no	—
131	#1487	—	53	2	no	—
132	#1301	—	53	2	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	—	2	1	no	—
135	C₂×C₅₃	#1803	1	1	yes	characteristic
136	C₂×C₅₃	#1810	1	1	yes	characteristic
137	C₂×C₅₃	#1814	1	1	yes	characteristic
138	C₂×C₄×Q₈	—	53	1	no	—
139	C₈×Q₈	—	53	2	no	—
140	C₈×Q₈	—	53	2	no	—
141	C₈×Q₈	—	106	1	no	—
142	C₈×Q₈	—	106	1	no	—
143	#431	—	53	2	no	—
144	C₂×#108	—	2	1	no	—
145	C₂×#108	—	2	1	no	—
146	#423	C₂×D₅₃	1	2	yes	—
147	C₂×C₄⋊C₈	—	53	1	no	—
148	C₂×C₄⋊C₈	—	106	1	no	—
149	C₂×C₄×C₈	—	106	1	no	—
150	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
151	#335	—	106	1	no	—
152	#334	—	106	1	no	—
153	Q₈⋊C₈	—	106	4	no	—
154	C₅₃	#8659	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	—	53	1	no	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂²×C₈	—	106	1	no	—
158	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
160	C₄×Q₈	—	53	2	no	—
161	C₄×Q₈	—	53	2	no	—
162	C₄×Q₈	—	106	1	no	—
163	C₄×Q₈	—	106	1	no	—
164	C₂×C₄⋊C₄	—	53	1	no	—
165	C₂×C₄⋊C₄	—	106	1	no	—
166	C₂×C₄²	—	106	1	no	—
167	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
168	#108	—	4	2	no	—
169	#108	—	4	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	53	2	no	—
173	C₄⋊C₈	—	106	2	no	—
174	C₄⋊C₈	—	106	2	no	—
175	C₄⋊C₈	—	106	1	no	—
176	C₄⋊C₈	—	106	2	no	—
177	C₂²⋊C₈	—	106	2	no	—
178	C₄×C₈	—	106	2	no	—
179	C₄×C₈	—	106	2	no	—
180	C₄×C₈	—	106	2	no	—
181	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
182	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
183	C₄×C₈	D₂₁₂	1	1	yes	characteristic
184	C₄×C₈	D₂₁₂	1	1	yes	characteristic
185	C₂×Q₈	—	53	2	no	—
186	C₂×Q₈	—	106	4	no	—
187	C₂×Q₈	—	106	1	no	—
188	C₂²×C₄	—	106	1	no	—
189	C₂²×C₄	—	106	1	no	—
190	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
191	C₂²×C₄	#11719949	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
193	M₁₆	—	4	2	no	—
194	M₁₆	—	4	2	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	106	2	no	—
199	C₂×C₈	—	212	2	no	—
200	C₂×C₈	—	212	1	no	—
201	C₂×C₈	—	212	1	no	—
202	C₂×C₈	#11719921	1	1	yes	characteristic
203	C₂×C₈	#11719921	1	1	yes	characteristic
204	C₂×C₈	#11719921	1	1	yes	characteristic
205	C₂×C₈	#11719921	1	1	yes	characteristic
206	C₂×C₈	#11719923	1	2	yes	—
207	C₂×C₈	#11719925	1	2	yes	—
208	C₄⋊C₄	—	53	2	no	—
209	C₄⋊C₄	—	106	2	no	—
210	C₄⋊C₄	—	106	2	no	—
211	C₄⋊C₄	—	106	1	no	—
212	C₄²	—	106	2	no	—
213	C₄²	—	106	2	no	—
214	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
215	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
216	C₄²	D₄×D₅₃	1	1	yes	characteristic
217	C₄²	#11719947	1	1	yes	characteristic
218	C₂³	#13133193	1	1	yes	characteristic
219	Q₈	—	53	4	no	—
220	Q₈	—	106	4	no	—
221	Q₈	—	106	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	106	4	no	—
225	C₂×C₄	—	106	2	no	—
226	C₂×C₄	—	106	2	no	—
227	C₂×C₄	—	212	2	no	—
228	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
229	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
230	C₂×C₄	#13133175	1	1	yes	characteristic
231	C₂×C₄	#13133188	1	1	yes	characteristic
232	C₂×C₄	#13133192	1	1	yes	characteristic
233	C₂×C₄	#13133197	1	1	yes	characteristic
234	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
235	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
236	C₂×C₄	#13133243	1	1	yes	characteristic
237	C₂×C₄	#13133248	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	8	2	no	—
241	C₈	—	424	2	no	—
242	C₈	—	212	2	no	—
243	C₈	#13133140	1	2	yes	—
244	C₈	#13133140	1	2	yes	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	#16810480	1	1	yes	characteristic
248	C₂²	#16810905	1	1	yes	characteristic
249	C₂²	#16810915	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	106	4	no	—
254	C₄	—	212	2	no	—
255	C₄	C₂×#13133140	1	1	yes	characteristic
256	C₄	#16810622	1	1	yes	characteristic
257	C₄	#108×D₅₃	1	2	yes	—
258	C₄	#16810820	1	1	yes	characteristic
259	C₄	#16810830	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	#20274717	1	1	yes	characteristic
263	C₂	#20274750	1	1	yes	characteristic
264	C₂	#20274770	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133188
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1848	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	C₂²⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1848	D₅₃
hypocenter	C₅₃	#8659
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
104. Dic₁₀₆	#108	semidirect
154. C₅₃	#8659	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₀₆	2
#20273463	2
#20273143	2
#20272610	2
C₅₃×#1848	2
#8659	53