Subgroups — #27283322

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275639	C₂	1	2	yes	maximal
2	C₈×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810369	C₂	1	2	yes	maximal
5	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
8	C₂×#13133141	—	2	4	no	—
9	C₂×C₈×Dic₅₃	—	2	1	no	—
10	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
11	#16810377	—	2	1	no	—
12	#16810376	—	2	2	no	—
13	#16810369	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#500	—	2	2	no	—
16	C₅₃×#500	C₄	1	4	yes	complemented
17	C₂×#107×C₅₃	—	2	1	no	—
18	C₂×#107×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	1	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₂×C₄⋊Dic₅₃	D₄	1	2	yes	—
23	C₂×C₄×Dic₅₃	—	2	1	no	—
24	C₄⋊#88322	—	2	1	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	D₄	1	2	yes	—
28	#13133141	—	4	8	no	—
29	C₈×Dic₅₃	—	4	2	no	—
30	#13133124	—	2	1	no	—
31	C₂²×Q₈×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₅₃	—	2	2	no	—
35	C₄⋊Q₈×C₅₃	—	2	2	no	—
36	C₄⋊Q₈×C₅₃	—	2	2	no	—
37	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	#107×C₅₃	—	2	4	no	—
42	#107×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	4	no	—
46	C₂×Q₈×C₅₃	—	4	2	no	—
47	C₂×Q₈×C₅₃	—	4	4	no	—
48	C₂×Q₈×C₅₃	—	4	4	no	—
49	C₂²×C₄×C₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
53	C₂²×Dic₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	—	2	1	no	—
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
59	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
60	C₄⋊C₄×C₅₃	—	2	4	no	—
61	C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄⋊C₄×C₅₃	—	4	2	no	—
64	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₅₃	#41	1	1	yes	characteristic
68	C₄⋊Dic₅₃	—	2	2	no	—
69	C₄⋊Dic₅₃	D₈	1	4	yes	—
70	C₄×Dic₅₃	—	4	1	no	—
71	C₄×Dic₅₃	—	4	1	no	—
72	C₂×#88322	—	2	2	no	—
73	C₂×#88322	—	4	1	no	—
74	C₂×#88322	—	4	1	no	—
75	C₂×#88322	—	4	2	no	—
76	C₂×#88322	—	4	4	no	—
77	C₂³×C₅₃	#121	1	1	yes	characteristic
78	Q₈×C₅₃	—	4	8	no	—
79	Q₈×C₅₃	—	8	2	no	—
80	Q₈×C₅₃	—	8	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	4	4	no	—
83	C₂×C₄×C₅₃	—	8	4	no	—
84	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#135	1	2	yes	—
94	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
96	C₂×Dic₅₃	—	2	4	no	—
97	C₂×Dic₅₃	—	4	2	no	—
98	C₂×Dic₅₃	—	4	2	no	—
99	C₈×C₅₃	—	2	2	no	—
100	C₈×C₅₃	#103	1	4	yes	complemented
101	#88322	—	4	2	no	—
102	#88322	—	8	2	no	—
103	#9731	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	#390	1	1	yes	characteristic
105	C₂²×C₅₃	#461	1	1	yes	characteristic
106	C₂²×C₅₃	#462	1	1	yes	characteristic
107	C₂²×C₅₃	#466	1	2	yes	—
108	C₂²×C₅₃	#473	1	2	yes	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	8	4	no	—
112	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
113	C₄×C₅₃	#419	1	1	yes	characteristic
114	C₄×C₅₃	C₄×D₈	1	2	yes	—
115	C₄×C₅₃	#461	1	1	yes	characteristic
116	C₄×C₅₃	#462	1	1	yes	characteristic
117	C₄×C₅₃	#493	1	1	yes	characteristic
118	C₄×C₅₃	#495	1	1	yes	characteristic
119	Dic₅₃	—	4	4	no	—
120	C₂×#500	D₅₃	1	1	yes	characteristic
121	#1981	—	53	2	no	—
122	C₈×C₄⋊C₄	—	53	1	no	—
123	C₂×#335	—	53	1	no	—
124	C₂×#332	—	53	2	no	—
125	C₂×C₅₃	#1673	1	2	yes	—
126	C₂×C₅₃	#1674	1	2	yes	—
127	C₂×C₅₃	#1970	1	1	yes	characteristic
128	C₂×C₅₃	#1979	1	1	yes	characteristic
129	C₂×C₅₃	#1981	1	1	yes	characteristic
130	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
131	#500	—	2	2	no	—
132	#500	Dic₅₃	1	4	yes	complemented
133	C₂×#107	—	2	1	no	—
134	C₂×#107	—	2	1	no	—
135	C₂×C₄⋊C₈	—	53	1	no	—
136	C₂×C₄⋊C₈	—	53	2	no	—
137	C₂×#103	—	106	4	no	—
138	C₂×C₄×C₈	—	106	1	no	—
139	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
140	C₄×C₄⋊C₄	—	53	1	no	—
141	#336	—	106	1	no	—
142	#335	—	106	2	no	—
143	#332	—	106	4	no	—
144	C₅₃	#9731	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	—	2	2	no	—
146	C₂²×C₈	—	106	1	no	—
147	C₂²×C₈	—	106	2	no	—
148	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
149	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	C₂×Dic₅₃	1	4	yes	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	53	2	no	—
157	C₂×C₄²	—	106	1	no	—
158	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
159	#107	—	2	4	no	—
160	#107	—	2	4	no	—
161	C₄⋊C₈	—	53	2	no	—
162	C₄⋊C₈	—	106	2	no	—
163	C₄⋊C₈	—	106	1	no	—
164	C₄⋊C₈	—	106	2	no	—
165	#103	—	212	8	no	—
166	C₄×C₈	—	212	2	no	—
167	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
168	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
169	#95	—	106	1	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	4	no	—
173	C₂×Q₈	—	4	4	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	106	2	no	—
176	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
177	C₂²×C₄	#11719947	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	106	2	no	—
182	C₂×C₈	—	212	2	no	—
183	C₂×C₈	—	212	4	no	—
184	C₂×C₈	—	212	1	no	—
185	C₂×C₈	—	212	1	no	—
186	C₂×C₈	#11719924	1	2	yes	—
187	C₂×C₈	#11719925	1	2	yes	—
188	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
189	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	53	4	no	—
195	C₄⋊C₄	—	106	2	no	—
196	C₄²	—	212	1	no	—
197	C₄²	—	212	1	no	—
198	C₄²	D₄×D₅₃	1	1	yes	characteristic
199	C₄²	#11719947	1	1	yes	characteristic
200	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
202	C₂³	#13133188	1	1	yes	characteristic
203	Q₈	—	4	8	no	—
204	Q₈	—	8	2	no	—
205	Q₈	—	8	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	4	no	—
209	C₂×C₄	—	106	4	no	—
210	C₂×C₄	—	212	2	no	—
211	C₂×C₄	—	212	2	no	—
212	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
213	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
214	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
215	C₂×C₄	#13133188	1	1	yes	characteristic
216	C₂×C₄	#13133204	1	1	yes	characteristic
217	C₂×C₄	#13133213	1	1	yes	characteristic
218	C₂×C₄	D₈×D₅₃	1	2	yes	—
219	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
220	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
221	C₂×C₄	#13133236	1	2	yes	—
222	C₂×C₄	#13133248	1	1	yes	characteristic
223	C₂×C₄	#13133248	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	424	2	no	—
226	C₈	—	212	2	no	—
227	C₈	#13133141	1	4	yes	complemented
228	C₂²	#16810526	1	1	yes	characteristic
229	C₂²	#16810825	1	1	yes	characteristic
230	C₂²	#16810830	1	1	yes	characteristic
231	C₂²	#16810856	1	2	yes	—
232	C₂²	#16810895	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	8	4	no	—
236	C₄	—	212	4	no	—
237	C₄	C₂×#13133141	1	1	yes	characteristic
238	C₄	#16810632	1	1	yes	characteristic
239	C₄	D₈×Dic₅₃	1	2	yes	—
240	C₄	#16810825	1	1	yes	characteristic
241	C₄	#16810830	1	1	yes	characteristic
242	C₄	#16810998	1	1	yes	characteristic
243	C₄	#16811006	1	1	yes	characteristic
244	C₂	#20274101	1	2	yes	—
245	C₂	#20274106	1	2	yes	—
246	C₂	#20275582	1	1	yes	characteristic
247	C₂	#20275631	1	1	yes	characteristic
248	C₂	#20275639	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133188
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#500	C₂
socle, prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	#9731
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
16. C₅₃×#500	C₄	semidirect
100. C₈×C₅₃	#103	semidirect
132. #500	Dic₅₃	semidirect
144. C₅₃	#9731	semidirect
227. C₈	#13133141	semidirect

Maximal subgroups

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

subgroup	index
#20275639	2
C₈×C₄⋊Dic₅₃	2
C₂×#16810376	2
C₂×#16810369	2
C₂×C₅₃×#500	2
#9731	53