Subgroups — #27277321

Order 13568.

350 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274846	C₂	1	1	yes	maximal, characteristic
2	#20273463	C₂	1	1	yes	maximal, characteristic
3	#20273279	C₂	1	2	yes	maximal
4	#20272575	C₂	1	2	yes	maximal
5	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	#16810489	C₂²	1	2	yes	—
8	#16810386	—	2	1	no	—
9	#16810386	—	2	1	no	—
10	#16810376	—	2	1	no	—
11	#16810374	—	2	1	no	—
12	#16810360	—	2	4	no	—
13	C₅₃×#533	C₂²	1	1	yes	characteristic
14	C₅₃×#446	—	2	1	no	—
15	C₅₃×#446	C₄	1	2	yes	complemented
16	C₈×Q₈×C₅₃	—	2	1	no	—
17	C₈×Q₈×C₅₃	C₄	1	2	yes	complemented
18	C₅₃×#424	—	2	1	no	—
19	C₅₃×#423	C₂²	1	1	yes	characteristic
20	C₅₃×#405	—	2	1	no	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₅₃	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₂²⋊#88322	—	2	2	no	—
25	C₄×#88322	—	2	2	no	—
26	#13133124	—	2	2	no	—
27	#13133124	—	2	2	no	—
28	C₂×M₁₆×C₅₃	—	2	1	no	—
29	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₅₃	Q₈	1	1	yes	characteristic
32	C₄⋊Q₈×C₅₃	Q₈	1	1	yes	characteristic
33	#125×C₅₃	—	2	1	no	—
34	C₂²⋊Q₈×C₅₃	—	2	1	no	—
35	C₂²⋊Q₈×C₅₃	D₄	1	2	yes	—
36	C₄×Q₈×C₅₃	—	2	1	no	—
37	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
38	#117×C₅₃	—	2	1	no	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	—	2	1	no	—
43	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
44	#97×C₅₃	—	2	2	no	—
45	C₄×C₈×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂×Q₈×C₅₃	—	2	1	no	—
51	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
55	C₂²×Dic₅₃	—	4	2	no	—
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	2	no	—
66	C₄⋊C₄×C₅₃	—	2	1	no	—
67	C₄⋊C₄×C₅₃	—	2	1	no	—
68	C₄⋊C₄×C₅₃	—	2	1	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×Q₈	1	1	yes	characteristic
75	C₄²×C₅₃	#41	1	1	yes	characteristic
76	C₄⋊Dic₅₃	—	4	2	no	—
77	C₄⋊Dic₅₃	—	4	2	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂³×C₅₃	#121	1	1	yes	characteristic
80	Q₈×C₅₃	—	4	2	no	—
81	Q₈×C₅₃	—	4	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	4	1	no	—
85	C₂×C₄×C₅₃	—	4	1	no	—
86	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#135	1	2	yes	—
93	C₂×C₄×C₅₃	#137	1	2	yes	—
94	C₂×Dic₅₃	—	8	2	no	—
95	C₂×Dic₅₃	—	8	2	no	—
96	C₂×Dic₅₃	—	8	2	no	—
97	C₈×C₅₃	—	2	1	no	—
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	—	2	1	no	—
100	C₈×C₅₃	—	2	1	no	—
101	C₈×C₅₃	—	4	2	no	—
102	#88322	—	8	2	no	—
103	#8679	—	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	#464	1	2	yes	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	4	2	no	—
113	C₄×C₅₃	—	4	2	no	—
114	C₄×C₅₃	#416	1	1	yes	characteristic
115	C₄×C₅₃	#419	1	1	yes	characteristic
116	C₄×C₅₃	#479	1	2	yes	—
117	Dic₅₃	—	16	2	no	—
118	#2960	D₅₃	1	1	yes	characteristic
119	#1835	—	53	1	no	—
120	#1564	—	53	1	no	—
121	#1518	—	53	2	no	—
122	#1286	—	53	2	no	—
123	C₂×C₅₃	—	2	2	no	—
124	C₂×C₅₃	#1805	1	1	yes	characteristic
125	C₂×C₅₃	#1812	1	1	yes	characteristic
126	C₂×C₅₃	#1814	1	1	yes	characteristic
127	#533	C₂×D₅₃	1	1	yes	characteristic
128	#446	—	2	1	no	—
129	#446	Dic₅₃	1	2	yes	complemented
130	C₈×Q₈	—	2	1	no	—
131	C₈×Q₈	Dic₅₃	1	2	yes	complemented
132	#431	—	53	2	no	—
133	#424	—	2	1	no	—
134	#423	C₂×D₅₃	1	1	yes	characteristic
135	#405	—	2	1	no	—
136	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
137	#382	—	53	2	no	—
138	#340	—	106	1	no	—
139	#340	—	106	1	no	—
140	#335	—	106	1	no	—
141	#334	—	106	1	no	—
142	#328	—	106	4	no	—
143	C₅₃	#8679	1	1	yes	characteristic, complemented, Sylow
144	C₂×M₁₆	—	2	1	no	—
145	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
147	C₄⋊Q₈	Dic₁₀₆	1	1	yes	characteristic
148	C₄⋊Q₈	Dic₁₀₆	1	1	yes	characteristic
149	#125	—	2	1	no	—
150	C₂²⋊Q₈	—	2	1	no	—
151	C₂²⋊Q₈	D₂₁₂	1	2	yes	—
152	C₄×Q₈	—	2	1	no	—
153	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
154	#117	—	2	1	no	—
155	C₂×C₄⋊C₄	—	106	2	no	—
156	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	—	106	2	no	—
161	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
162	C₂²⋊C₈	—	106	2	no	—
163	#97	—	2	2	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	106	2	no	—
166	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
167	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
170	#95	—	106	2	no	—
171	#95	—	106	2	no	—
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
174	C₂²×C₄	—	212	2	no	—
175	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
176	C₂²×C₄	#11719945	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
178	M₁₆	—	4	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	212	2	no	—
183	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
184	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	212	2	no	—
193	C₄⋊C₄	—	212	2	no	—
194	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
198	C₄²	#11719945	1	1	yes	characteristic
199	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
200	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
201	C₂³	#13133191	1	1	yes	characteristic
202	Q₈	—	4	2	no	—
203	Q₈	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	424	2	no	—
209	C₂×C₄	—	424	2	no	—
210	C₂×C₄	—	424	2	no	—
211	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
213	C₂×C₄	#13133176	1	1	yes	characteristic
214	C₂×C₄	#13133191	1	1	yes	characteristic
215	C₂×C₄	#13133196	1	1	yes	characteristic
216	C₂×C₄	#13133196	1	1	yes	characteristic
217	C₂×C₄	#13133237	1	2	yes	—
218	C₂×C₄	#13133248	1	2	yes	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	4	2	no	—
224	C₈	—	424	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	#16810483	1	1	yes	characteristic
228	C₂²	#16810845	1	2	yes	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	848	2	no	—
236	C₄	#16810619	1	1	yes	characteristic
237	C₄	#16810632	1	1	yes	characteristic
238	C₄	#16810930	1	2	yes	—
239	C₂	—	2	2	no	—
240	C₂	#20274730	1	1	yes	characteristic
241	C₂	#20274761	1	1	yes	characteristic
242	C₂	#20274769	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	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#8679
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
15. C₅₃×#446	C₄	semidirect
17. C₈×Q₈×C₅₃	C₄	semidirect
129. #446	Dic₅₃	semidirect
131. C₈×Q₈	Dic₅₃	semidirect
143. C₅₃	#8679	semidirect

Maximal subgroups

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

subgroup	index
#20274846	2
#20273463	2
#20273279	2
#20272575	2
C₅₃×#2960	2
#8679	53