Subgroups — #27277361

Order 13568.

346 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274853	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
3	#20273405	C₂	1	2	yes	maximal
4	#20272589	C₂	1	2	yes	maximal
5	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic
6	#16810647	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	#16810388	—	2	1	no	—
9	#16810388	—	2	1	no	—
10	#16810374	—	2	1	no	—
11	#16810374	—	2	1	no	—
12	#16810372	—	2	2	no	—
13	#16810369	—	2	2	no	—
14	C₅₃×#533	C₂²	1	1	yes	characteristic
15	C₅₃×#446	—	2	1	no	—
16	C₅₃×#446	C₄	1	2	yes	—
17	C₈×Q₈×C₅₃	—	2	1	no	—
18	C₈×Q₈×C₅₃	C₄	1	2	yes	—
19	C₅₃×#424	—	2	1	no	—
20	C₅₃×#423	C₂²	1	1	yes	characteristic
21	C₅₃×#405	—	2	1	no	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#11719917	—	2	2	no	—
24	C₂²×#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	—	2	2	no	—
29	C₂×M₁₆×C₅₃	—	2	1	no	—
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
34	#125×C₅₃	—	2	1	no	—
35	C₂²⋊Q₈×C₅₃	—	2	1	no	—
36	C₂²⋊Q₈×C₅₃	Q₈	1	2	yes	—
37	C₄×Q₈×C₅₃	—	2	1	no	—
38	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
39	#117×C₅₃	—	2	1	no	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	—	2	1	no	—
44	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
45	#97×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	—	2	2	no	—
47	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₂×Q₈×C₅₃	—	2	1	no	—
52	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
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	M₁₆×C₅₃	—	4	2	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₄⋊C₄×C₅₃	—	2	1	no	—
66	C₄⋊C₄×C₅₃	—	2	1	no	—
67	C₄⋊C₄×C₅₃	—	2	1	no	—
68	C₄⋊C₄×C₅₃	—	2	2	no	—
69	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₂²⋊C₄×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	#41	1	1	yes	characteristic
76	#11719917	—	4	2	no	—
77	#11719917	—	4	2	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂×#88322	—	4	2	no	—
80	C₂×#88322	—	8	2	no	—
81	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
82	Q₈×C₅₃	—	4	2	no	—
83	Q₈×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
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	#135	1	2	yes	—
95	C₂×C₄×C₅₃	#137	1	2	yes	—
96	C₈×C₅₃	—	2	1	no	—
97	C₈×C₅₃	—	2	1	no	—
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	—	2	1	no	—
100	C₈×C₅₃	—	4	2	no	—
101	#88322	—	8	2	no	—
102	#88322	—	8	2	no	—
103	#8687	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	#380	1	1	yes	characteristic
107	C₂²×C₅₃	#479	1	2	yes	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	4	2	no	—
112	C₄×C₅₃	—	4	2	no	—
113	C₄×C₅₃	#416	1	1	yes	characteristic
114	C₄×C₅₃	#419	1	1	yes	characteristic
115	C₄×C₅₃	#464	1	2	yes	—
116	C₄×C₅₃	#471	1	2	yes	—
117	#2960	D₅₃	1	1	yes	characteristic
118	#1837	—	53	1	no	—
119	C₂×#334	—	53	1	no	—
120	#1550	—	53	2	no	—
121	#1292	—	53	2	no	—
122	C₂×C₅₃	—	2	2	no	—
123	C₂×C₅₃	#1805	1	1	yes	characteristic
124	C₂×C₅₃	#1813	1	1	yes	characteristic
125	C₂×C₅₃	#1815	1	1	yes	characteristic
126	#533	C₂×D₅₃	1	1	yes	characteristic
127	#446	—	2	1	no	—
128	#446	Dic₅₃	1	2	yes	—
129	C₈×Q₈	—	2	1	no	—
130	C₈×Q₈	Dic₅₃	1	2	yes	—
131	#424	—	2	1	no	—
132	#423	—	53	2	no	—
133	#423	C₂×D₅₃	1	1	yes	characteristic
134	C₂×C₄⋊C₈	—	53	2	no	—
135	#405	—	2	1	no	—
136	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
137	#341	—	106	1	no	—
138	#341	—	106	1	no	—
139	#334	—	106	1	no	—
140	#334	—	106	1	no	—
141	#333	—	106	2	no	—
142	#332	—	106	2	no	—
143	C₅₃	#8687	1	1	yes	characteristic, complemented, Sylow
144	C₂×M₁₆	—	2	1	no	—
145	C₂×M₁₆	—	106	2	no	—
146	C₂²×C₈	—	106	2	no	—
147	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
148	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
149	C₄⋊Q₈	D₂₁₂	1	1	yes	characteristic
150	C₄⋊Q₈	D₂₁₂	1	1	yes	characteristic
151	#125	—	2	1	no	—
152	C₂²⋊Q₈	—	2	1	no	—
153	C₂²⋊Q₈	Dic₁₀₆	1	2	yes	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
156	#117	—	2	1	no	—
157	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	—	106	2	no	—
162	C₄⋊C₈	—	106	2	no	—
163	C₄⋊C₈	—	106	2	no	—
164	C₄⋊C₈	—	106	2	no	—
165	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
166	#97	—	2	2	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
169	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
170	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
171	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
174	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
175	C₂²×C₄	#11719945	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
177	M₁₆	—	4	2	no	—
178	M₁₆	—	212	2	no	—
179	M₁₆	—	212	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	424	2	no	—
184	C₂×C₈	—	212	2	no	—
185	C₂×C₈	—	212	2	no	—
186	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
187	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
188	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
199	C₄²	#11719945	1	1	yes	characteristic
200	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
202	C₂³	#13133196	1	1	yes	characteristic
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
211	C₂×C₄	#13133176	1	1	yes	characteristic
212	C₂×C₄	#13133191	1	1	yes	characteristic
213	C₂×C₄	#13133191	1	1	yes	characteristic
214	C₂×C₄	#13133196	1	1	yes	characteristic
215	C₂×C₄	#13133237	1	2	yes	—
216	C₂×C₄	#13133248	1	2	yes	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	4	2	no	—
222	C₈	—	424	2	no	—
223	C₈	—	424	2	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	#16810483	1	1	yes	characteristic
227	C₂²	#16810930	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	#16810619	1	1	yes	characteristic
234	C₄	#16810632	1	1	yes	characteristic
235	C₄	#16810845	1	2	yes	—
236	C₄	#16810886	1	2	yes	—
237	C₂	—	2	2	no	—
238	C₂	#20274730	1	1	yes	characteristic
239	C₂	#20274765	1	1	yes	characteristic
240	C₂	#20274774	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810483
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2960	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	C₂²⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#8687
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
143. C₅₃	#8687	semidirect

Maximal subgroups

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

subgroup	index
#20274853	2
C₂×#16810374	2
#20273405	2
#20272589	2
C₅₃×#2960	2
#8687	53