Subgroups — #27301460

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810427	C₂	1	2	yes	maximal, complemented
3	#20274744	C₂	1	1	yes	maximal, characteristic
4	#20272806	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#467	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	—	2	4	no	—
7	C₂²×#11719908	—	2	2	no	—
8	C₂²×#11719908	C₂²	1	2	yes	—
9	C₂×#13133151	C₂²	1	2	yes	—
10	#16810483	—	2	1	no	—
11	#16810427	—	2	4	no	—
12	#16810386	—	2	1	no	—
13	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
16	C₅₃×#467	—	2	2	no	—
17	C₅₃×#467	C₄	1	4	yes	complemented
18	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#103×C₅₃	—	2	1	no	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×C₂²⋊Dic₅₃	—	4	1	no	—
22	C₂×#11719908	—	2	8	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	—	2	2	no	—
25	C₂×#11719908	D₄	1	4	yes	—
26	#13133151	—	2	2	no	—
27	#13133151	—	2	2	no	—
28	#13133124	—	4	1	no	—
29	C₂²×Q₈×C₅₃	—	2	1	no	—
30	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂×Q₁₆×C₅₃	—	2	1	no	—
32	C₂×Q₁₆×C₅₃	—	2	2	no	—
33	C₂×Q₁₆×C₅₃	—	2	2	no	—
34	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₃	—	2	2	no	—
36	C₂²×C₈×C₅₃	—	2	2	no	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₅₃	D₄	1	2	yes	—
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²⋊Q₈×C₅₃	—	2	4	no	—
42	C₂²⋊Q₈×C₅₃	—	4	2	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
45	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
46	#107×C₅₃	—	2	1	no	—
47	#107×C₅₃	C₂×C₄	1	2	yes	—
48	#103×C₅₃	—	2	4	no	—
49	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
50	C₂×Q₈×C₅₃	—	2	2	no	—
51	C₂×Q₈×C₅₃	—	4	1	no	—
52	C₂×Q₈×C₅₃	—	4	2	no	—
53	C₂×Q₈×C₅₃	—	4	2	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	4	1	no	—
57	C₂²×C₄×C₅₃	—	4	1	no	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₅₃	—	4	2	no	—
63	Q₁₆×C₅₃	—	2	4	no	—
64	Q₁₆×C₅₃	—	4	1	no	—
65	Q₁₆×C₅₃	—	4	1	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	4	no	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	#41	1	2	yes	—
75	C₄⋊C₄×C₅₃	—	2	2	no	—
76	C₄⋊C₄×C₅₃	—	4	4	no	—
77	C₄⋊C₄×C₅₃	—	4	1	no	—
78	C₂²⋊C₄×C₅₃	—	4	4	no	—
79	C₂²⋊Dic₅₃	—	8	2	no	—
80	C₄⋊Dic₅₃	—	4	2	no	—
81	C₄⋊Dic₅₃	—	4	2	no	—
82	#11719908	—	2	4	no	—
83	C₂³×C₅₃	—	2	2	no	—
84	C₂³×C₅₃	—	2	2	no	—
85	C₂³×C₅₃	#102	1	2	yes	—
86	C₂³×C₅₃	SD₃₂	1	2	yes	—
87	C₂³×C₅₃	Q₃₂	1	2	yes	—
88	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
89	Q₈×C₅₃	—	4	2	no	—
90	Q₈×C₅₃	—	4	2	no	—
91	Q₈×C₅₃	—	8	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	8	2	no	—
99	C₂×C₄×C₅₃	—	8	2	no	—
100	C₂×C₄×C₅₃	#102	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#121	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#135	1	2	yes	—
105	C₂×Dic₅₃	—	8	2	no	—
106	C₂×Dic₅₃	—	8	2	no	—
107	C₂×Dic₅₃	—	8	2	no	—
108	C₈×C₅₃	—	2	2	no	—
109	C₈×C₅₃	C₄×D₄	1	2	yes	—
110	C₈×C₅₃	#120	1	1	yes	characteristic
111	C₈×C₅₃	#123	1	1	yes	characteristic
112	#13060	—	53	1	no	maximal, Sylow
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	2	4	no	—
116	C₂²×C₅₃	—	2	4	no	—
117	C₂²×C₅₃	#358	1	4	yes	complemented
118	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
119	C₂²×C₅₃	#466	1	2	yes	—
120	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
122	C₂²×C₅₃	#508	1	2	yes	—
123	C₄×C₅₃	—	2	2	no	—
124	C₄×C₅₃	—	8	2	no	—
125	C₄×C₅₃	—	8	2	no	—
126	C₄×C₅₃	#386	1	1	yes	characteristic
127	C₄×C₅₃	#416	1	1	yes	characteristic
128	C₄×C₅₃	#465	1	2	yes	—
129	Dic₅₃	—	16	2	no	—
130	C₂³×C₁₆	—	53	1	no	—
131	C₂×#467	D₅₃	1	1	yes	characteristic
132	C₂×#358	—	53	2	no	—
133	#1808	—	53	1	no	—
134	#1380	—	53	2	no	—
135	C₂×C₅₃	—	2	4	no	—
136	C₂×C₅₃	#1893	1	1	yes	characteristic
137	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
138	C₂×C₅₃	#2138	1	1	yes	characteristic
139	C₂×C₅₃	#2216	1	2	yes	—
140	C₂×C₅₃	#2217	1	2	yes	—
141	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
142	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
143	C₂×C₂²⋊Q₈	—	2	1	no	—
144	C₂²×C₁₆	—	53	2	no	—
145	C₂²×C₁₆	—	106	4	no	—
146	C₂²×C₁₆	—	106	2	no	—
147	#467	—	2	2	no	—
148	#467	Dic₅₃	1	4	yes	complemented
149	C₂×#107	—	53	2	no	—
150	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
151	C₂×#103	—	2	1	no	—
152	#380	—	106	1	no	—
153	#358	—	106	4	no	—
154	#340	—	106	1	no	—
155	C₅₃	#13060	1	1	yes	characteristic, complemented, Sylow
156	C₂²×Q₈	—	2	1	no	—
157	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
158	C₂×Q₁₆	—	2	2	no	—
159	C₂×Q₁₆	—	2	2	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
165	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
166	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
168	C₂²⋊Q₈	—	2	4	no	—
169	C₂²⋊Q₈	—	4	2	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	4	1	no	—
172	C₂×C₄⋊C₄	—	106	2	no	—
173	C₂×C₂²⋊C₄	—	4	1	no	—
174	C₂×C₂²⋊C₄	—	212	1	no	—
175	C₂×C₁₆	—	53	4	no	—
176	C₂×C₁₆	—	106	8	no	—
177	C₂×C₁₆	—	106	2	no	—
178	C₂×C₁₆	—	106	2	no	—
179	#107	—	2	1	no	—
180	#107	—	106	2	no	—
181	#107	—	106	2	no	—
182	#107	C₂×Dic₅₃	1	2	yes	—
183	#103	—	2	4	no	—
184	#95	—	212	1	no	—
185	C₂⁴	#11719922	1	1	yes	characteristic
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	4	1	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	212	2	no	—
195	C₂²×C₄	#11719922	1	1	yes	characteristic
196	C₂²×C₄	#11719923	1	2	yes	—
197	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
198	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
199	Q₁₆	—	2	4	no	—
200	Q₁₆	—	4	1	no	—
201	Q₁₆	—	4	1	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
207	C₂×C₈	D₄×D₅₃	1	2	yes	—
208	C₂×C₈	#11719947	1	2	yes	—
209	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
210	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	212	2	no	—
215	C₄⋊C₄	—	212	2	no	—
216	C₂²⋊C₄	—	4	4	no	—
217	C₂²⋊C₄	—	424	2	no	—
218	C₁₆	—	106	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	#13133138	1	2	yes	—
222	C₂³	#13133160	1	2	yes	—
223	C₂³	#13133163	1	2	yes	—
224	C₂³	C₂×#11719922	1	1	yes	characteristic
225	Q₈	—	4	2	no	—
226	Q₈	—	4	2	no	—
227	Q₈	—	8	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	424	2	no	—
237	C₂×C₄	—	424	2	no	—
238	C₂×C₄	—	424	2	no	—
239	C₂×C₄	#13133138	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
241	C₂×C₄	#13133189	1	2	yes	—
242	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
243	C₂×C₄	#13133237	1	2	yes	—
244	C₈	—	2	2	no	—
245	C₈	D₄×Dic₅₃	1	2	yes	—
246	C₈	#13133185	1	1	yes	characteristic
247	C₈	#13133198	1	1	yes	characteristic
248	C₂²	—	2	4	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	#16810427	1	4	yes	complemented
253	C₂²	C₂×#13133138	1	1	yes	characteristic
254	C₂²	#16810857	1	2	yes	—
255	C₂²	C₂×#13133160	1	1	yes	characteristic
256	C₂²	C₂×#13133163	1	1	yes	characteristic
257	C₂²	#16811053	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	8	2	no	—
260	C₄	—	8	2	no	—
261	C₄	—	848	2	no	—
262	C₄	#16810503	1	1	yes	characteristic
263	C₄	#16810619	1	1	yes	characteristic
264	C₄	#16810850	1	2	yes	—
265	C₂	—	2	4	no	—
266	C₂	#20275144	1	1	yes	characteristic
267	C₂	C₂×#16810427	1	1	yes	characteristic
268	C₂	#20276422	1	1	yes	characteristic
269	C₂	#20276804	1	2	yes	—
270	C₂	#20276813	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#467	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#467	D₅₃
hypocenter	C₅₃	#13060
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
2. C₂×#16810427	C₂	semidirect
4. #20272806	C₂	semidirect
17. C₅₃×#467	C₄	semidirect
117. C₂²×C₅₃	#358	semidirect
148. #467	Dic₅₃	semidirect
155. C₅₃	#13060	semidirect
252. C₂²	#16810427	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#11719908	2
C₂×#16810427	2
#20274744	2
#20272806	2
C₂×C₅₃×#467	2
#13060	53