Subgroups — #27277211

Order 13568.

418 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274843	C₂	1	1	yes	maximal, characteristic
2	#20274720	C₂	1	2	yes	maximal
3	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
4	#20272599	C₂	1	2	yes	maximal
5	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×#13133141	—	2	2	no	—
8	#16810493	C₂²	1	2	yes	—
9	#16810386	—	2	1	no	—
10	#16810386	—	2	1	no	—
11	#16810385	—	2	2	no	—
12	#16810374	—	2	2	no	—
13	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#446	—	2	2	no	—
15	C₅₃×#446	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	2	2	no	—
22	C₂×C₄⋊Dic₅₃	—	2	2	no	—
23	C₂×C₄⋊Dic₅₃	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	#13133141	—	4	4	no	—
27	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×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	—	2	1	no	—
32	C₄×Q₈×C₅₃	—	2	1	no	—
33	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×Q₈×C₅₃	Q₈	1	2	yes	—
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	—	2	1	no	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	—	2	4	no	—
40	C₄⋊C₈×C₅₃	—	2	1	no	—
41	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
42	#97×C₅₃	—	2	4	no	—
43	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₂×Q₈×C₅₃	—	2	1	no	—
46	C₂×Q₈×C₅₃	—	2	2	no	—
47	C₂×Q₈×C₅₃	—	2	2	no	—
48	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
54	C₂²×Dic₅₃	—	4	2	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₄⋊C₄×C₅₃	—	2	1	no	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₅₃	#41	1	1	yes	characteristic
71	C₄⋊Dic₅₃	—	4	2	no	—
72	C₄⋊Dic₅₃	—	4	2	no	—
73	C₄⋊Dic₅₃	—	4	2	no	—
74	C₄⋊Dic₅₃	—	4	2	no	—
75	C₄⋊Dic₅₃	—	8	2	no	—
76	C₂×#88322	—	4	2	no	—
77	C₂×#88322	—	4	2	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂³×C₅₃	#121	1	1	yes	characteristic
80	Q₈×C₅₃	—	2	4	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	—	2	2	no	—
85	C₂×C₄×C₅₃	—	4	1	no	—
86	C₂×C₄×C₅₃	—	4	1	no	—
87	C₂×C₄×C₅₃	—	4	2	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	C₂²⋊Q₈	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	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×C₄×C₅₃	#137	1	2	yes	—
97	C₂×C₄×C₅₃	#137	1	2	yes	—
98	C₂×Dic₅₃	—	8	2	no	—
99	C₂×Dic₅₃	—	8	2	no	—
100	C₂×Dic₅₃	—	8	2	no	—
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	—	2	2	no	—
103	C₈×C₅₃	—	4	2	no	—
104	#88322	—	8	2	no	—
105	#8657	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	#380	1	1	yes	characteristic
107	C₂²×C₅₃	#464	1	2	yes	—
108	C₂²×C₅₃	#469	1	1	yes	characteristic
109	C₂²×C₅₃	#470	1	1	yes	characteristic
110	C₂²×C₅₃	#471	1	2	yes	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	4	2	no	—
114	C₄×C₅₃	—	4	2	no	—
115	C₄×C₅₃	#416	1	1	yes	characteristic
116	C₄×C₅₃	#419	1	1	yes	characteristic
117	C₄×C₅₃	#428	1	2	yes	—
118	C₄×C₅₃	#468	1	2	yes	—
119	C₄×C₅₃	#479	1	2	yes	—
120	Dic₅₃	—	16	2	no	—
121	C₂×#446	D₅₃	1	1	yes	characteristic
122	#1834	—	53	1	no	—
123	#1803	—	53	2	no	—
124	C₂×#334	—	53	1	no	—
125	#1296	—	53	2	no	—
126	C₂×C₅₃	#1702	1	2	yes	—
127	C₂×C₅₃	#1704	1	2	yes	—
128	C₂×C₅₃	#1805	1	1	yes	characteristic
129	C₂×C₅₃	#1809	1	1	yes	characteristic
130	C₂×C₅₃	#1815	1	1	yes	characteristic
131	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
132	#446	—	2	2	no	—
133	#446	Dic₅₃	1	4	yes	complemented
134	C₂×C₄⋊C₈	—	2	1	no	—
135	C₂×C₄⋊C₈	—	53	2	no	—
136	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
137	C₂×#103	—	106	2	no	—
138	C₂×#97	—	2	1	no	—
139	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
140	#384	—	53	2	no	—
141	#340	—	106	1	no	—
142	#340	—	106	1	no	—
143	#340	—	106	2	no	—
144	#334	—	106	2	no	—
145	C₅₃	#8657	1	1	yes	characteristic, complemented, Sylow
146	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	106	2	no	—
149	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
150	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
151	C₄×Q₈	—	2	1	no	—
152	C₄×Q₈	—	2	1	no	—
153	C₄×Q₈	Dic₁₀₆	1	2	yes	—
154	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	106	2	no	—
157	C₂×C₄⋊C₄	—	106	2	no	—
158	C₂×C₄⋊C₄	—	106	2	no	—
159	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
160	C₂×C₄²	—	2	1	no	—
161	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
162	C₄⋊C₈	—	2	4	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	—	106	2	no	—
165	C₄⋊C₈	—	106	2	no	—
166	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
167	#103	—	212	4	no	—
168	#97	—	2	4	no	—
169	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
170	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	212	2	no	—
178	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
179	C₂²×C₄	#11719945	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	212	2	no	—
186	C₂×C₈	—	212	2	no	—
187	C₂×C₈	—	212	2	no	—
188	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
189	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	424	2	no	—
194	C₄⋊C₄	—	212	2	no	—
195	C₄⋊C₄	—	212	2	no	—
196	C₄⋊C₄	—	212	2	no	—
197	C₄⋊C₄	—	212	2	no	—
198	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
202	C₄²	#11719945	1	1	yes	characteristic
203	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
204	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
205	C₂³	#13133191	1	1	yes	characteristic
206	Q₈	—	2	4	no	—
207	Q₈	—	4	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	424	2	no	—
215	C₂×C₄	—	424	2	no	—
216	C₂×C₄	—	424	2	no	—
217	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
219	C₂×C₄	#13133176	1	1	yes	characteristic
220	C₂×C₄	#13133191	1	1	yes	characteristic
221	C₂×C₄	#13133196	1	1	yes	characteristic
222	C₂×C₄	#13133196	1	1	yes	characteristic
223	C₂×C₄	#13133237	1	2	yes	—
224	C₂×C₄	#13133245	1	2	yes	—
225	C₂×C₄	#13133248	1	2	yes	—
226	C₂×C₄	#13133250	1	2	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	4	2	no	—
230	C₈	—	424	2	no	—
231	C₂²	#16810483	1	1	yes	characteristic
232	C₂²	#16810845	1	2	yes	—
233	C₂²	#16810876	1	1	yes	characteristic
234	C₂²	#16810881	1	1	yes	characteristic
235	C₂²	#16810886	1	2	yes	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	848	2	no	—
241	C₄	#16810619	1	1	yes	characteristic
242	C₄	#16810632	1	1	yes	characteristic
243	C₄	#16810665	1	2	yes	—
244	C₄	#16810869	1	2	yes	—
245	C₄	#16810930	1	2	yes	—
246	C₂	#20274283	1	2	yes	—
247	C₂	#20274297	1	2	yes	—
248	C₂	#20274730	1	1	yes	characteristic
249	C₂	#20274748	1	1	yes	characteristic
250	C₂	#20274774	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#446	C₂
socle, prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#446	D₅₃
hypocenter	C₅₃	#8657
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
15. C₅₃×#446	C₄	semidirect
133. #446	Dic₅₃	semidirect
145. C₅₃	#8657	semidirect

Maximal subgroups

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

subgroup	index
#20274843	2
#20274720	2
C₂×#16810374	2
#20272599	2
C₂×C₅₃×#446	2
#8657	53