Subgroups — #27300257

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810449	C₂	1	1	yes	maximal, characteristic, complemented
3	#20274741	C₂	1	2	yes	maximal
4	#20272806	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
7	C₂²×#11719908	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×D₅₃	—	2	2	no	—
9	C₂×C₁₆×D₅₃	—	2	2	no	—
10	C₂×C₁₆×D₅₃	C₄	1	4	yes	complemented
11	C₂×#13133151	C₂²	1	2	yes	—
12	#16810480	—	2	2	no	—
13	#16810449	—	2	2	no	—
14	#16810385	—	2	2	no	—
15	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#107×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#366	—	2	2	no	—
18	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
19	C₂²×#88322	D₄	1	1	yes	characteristic
20	C₂×C₈×D₅₃	—	2	1	no	—
21	C₂×C₈×D₅₃	—	2	1	no	—
22	C₂×C₈×D₅₃	—	2	1	no	—
23	C₂×C₈×D₅₃	—	2	1	no	—
24	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₈×D₅₃	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×#11719921	—	4	2	no	—
28	C₂×#11719908	—	2	1	no	—
29	C₂×#11719908	—	2	1	no	—
30	C₂×#11719908	C₂×C₄	1	2	yes	—
31	C₁₆×D₅₃	—	2	8	no	—
32	#13133151	—	2	2	no	—
33	#13133151	—	2	2	no	—
34	#13133124	—	4	2	no	—
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
39	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
40	#107×C₅₃	—	2	2	no	—
41	#107×C₅₃	—	2	2	no	—
42	C₂³×D₅₃	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	4	2	no	—
46	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
47	C₂×C₄×D₅₃	—	2	1	no	—
48	C₂×C₄×D₅₃	—	2	1	no	—
49	C₂×C₄×D₅₃	—	2	1	no	—
50	C₂×C₄×D₅₃	—	2	1	no	—
51	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
52	C₂×C₄×D₅₃	Q₁₆	1	2	yes	—
53	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
58	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
59	C₄⋊C₄×C₅₃	—	4	2	no	—
60	C₄⋊C₄×C₅₃	—	4	2	no	—
61	#11719921	—	8	4	no	—
62	C₄⋊Dic₅₃	—	4	2	no	—
63	C₄⋊Dic₅₃	—	4	2	no	—
64	C₂×#88322	—	2	1	no	—
65	C₂×#88322	—	2	1	no	—
66	C₂×#88322	C₄⋊C₄	1	2	yes	—
67	C₈×D₅₃	—	2	2	no	—
68	C₈×D₅₃	—	2	2	no	—
69	C₈×D₅₃	—	2	2	no	—
70	C₈×D₅₃	—	2	2	no	—
71	C₁₆×C₅₃	C₂²⋊C₄	1	4	yes	complemented
72	#11719908	—	2	2	no	—
73	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
74	C₂²×D₅₃	—	2	1	no	—
75	C₂²×D₅₃	—	2	1	no	—
76	C₂²×D₅₃	—	2	1	no	—
77	C₂²×D₅₃	—	2	1	no	—
78	C₂²×D₅₃	#107	1	2	yes	—
79	C₂²×D₅₃	D₁₆	1	2	yes	—
80	C₂²×D₅₃	Q₃₂	1	2	yes	—
81	C₂×C₄×C₅₃	—	8	2	no	—
82	C₂×C₄×C₅₃	—	8	2	no	—
83	C₂×C₄×C₅₃	—	8	2	no	—
84	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
90	C₂×Dic₅₃	—	2	1	no	—
91	C₂×Dic₅₃	—	2	1	no	—
92	C₂×Dic₅₃	—	8	2	no	—
93	C₂×Dic₅₃	—	8	2	no	—
94	C₂×Dic₅₃	—	8	2	no	—
95	C₂×Dic₅₃	#107	1	2	yes	—
96	C₄×D₅₃	—	2	2	no	—
97	C₄×D₅₃	—	2	2	no	—
98	C₄×D₅₃	—	2	2	no	—
99	C₄×D₅₃	—	2	2	no	—
100	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₈×C₅₃	#117	1	1	yes	characteristic
102	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
103	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
104	#88322	—	2	2	no	—
105	#12834	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
107	C₂²×C₅₃	#466	1	1	yes	characteristic
108	C₂²×C₅₃	#466	1	1	yes	characteristic
109	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
110	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
111	C₂²×C₅₃	#508	1	1	yes	characteristic
112	C₂²×C₅₃	#508	1	1	yes	characteristic
113	C₂×D₅₃	—	2	2	no	—
114	C₂×D₅₃	—	2	2	no	—
115	C₂×D₅₃	—	2	2	no	—
116	C₂×D₅₃	—	2	2	no	—
117	C₂×D₅₃	—	2	2	no	—
118	C₂×D₅₃	—	2	2	no	—
119	C₂×D₅₃	#366	1	4	yes	complemented
120	C₄×C₅₃	—	16	2	no	—
121	C₄×C₅₃	#380	1	1	yes	characteristic
122	C₄×C₅₃	#427	1	1	yes	characteristic
123	C₄×C₅₃	#467	1	1	yes	characteristic
124	C₄×C₅₃	#467	1	1	yes	characteristic
125	Dic₅₃	—	2	2	no	—
126	Dic₅₃	—	16	2	no	—
127	C₂³×C₁₆	—	53	1	no	—
128	C₂×#366	—	53	1	no	—
129	C₂×#366	D₅₃	1	1	yes	characteristic, complemented
130	#1808	—	53	2	no	—
131	#1380	—	53	2	no	—
132	C₂×C₅₃	#1808	1	1	yes	characteristic
133	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
134	C₂×C₅₃	#2158	1	1	yes	characteristic
135	C₂×C₅₃	#2215	1	1	yes	characteristic
136	C₂×C₅₃	#2215	1	1	yes	characteristic
137	C₂×C₅₃	#2216	1	1	yes	characteristic
138	C₂×C₅₃	#2216	1	1	yes	characteristic
139	D₅₃	—	2	4	no	—
140	C₂³×C₈	—	53	1	no	—
141	C₂²×C₁₆	—	53	1	no	—
142	C₂²×C₁₆	—	53	4	no	—
143	C₂²×C₁₆	—	106	2	no	—
144	C₂²×C₁₆	—	106	2	no	—
145	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
146	C₂×#107	—	53	2	no	—
147	C₂×#107	C₂×D₅₃	1	2	yes	—
148	#380	—	106	2	no	—
149	#366	—	2	2	no	—
150	#366	—	106	2	no	—
151	#340	—	106	2	no	—
152	C₅₃	#12834	1	1	yes	characteristic, complemented, Sylow
153	C₂³×C₄	—	53	1	no	—
154	C₂²×C₈	—	53	1	no	—
155	C₂²×C₈	—	53	2	no	—
156	C₂²×C₈	—	53	2	no	—
157	C₂²×C₈	—	106	1	no	—
158	C₂²×C₈	—	106	1	no	—
159	C₂²×C₈	—	106	1	no	—
160	C₂²×C₈	—	106	1	no	—
161	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	106	2	no	—
164	C₂×C₂²⋊C₄	—	212	2	no	—
165	C₂×C₁₆	—	53	2	no	—
166	C₂×C₁₆	—	106	1	no	—
167	C₂×C₁₆	—	106	1	no	—
168	C₂×C₁₆	—	106	8	no	—
169	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
170	C₂×C₁₆	D₂₁₂	1	2	yes	—
171	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
172	#107	—	2	2	no	—
173	#107	—	2	2	no	—
174	#107	—	106	2	no	—
175	#107	—	106	2	no	—
176	#95	—	212	2	no	—
177	C₂⁴	—	53	1	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	53	1	no	—
180	C₂²×C₄	—	53	2	no	—
181	C₂²×C₄	—	53	2	no	—
182	C₂²×C₄	—	106	1	no	—
183	C₂²×C₄	—	106	1	no	—
184	C₂²×C₄	—	106	1	no	—
185	C₂²×C₄	—	106	1	no	—
186	C₂²×C₄	—	212	2	no	—
187	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
188	C₂×C₈	—	53	2	no	—
189	C₂×C₈	—	106	1	no	—
190	C₂×C₈	—	106	1	no	—
191	C₂×C₈	—	106	2	no	—
192	C₂×C₈	—	106	2	no	—
193	C₂×C₈	—	106	2	no	—
194	C₂×C₈	—	106	2	no	—
195	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
196	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
197	C₂×C₈	#11719947	1	1	yes	characteristic
198	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
199	C₂×C₈	#11719949	1	1	yes	characteristic
200	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	212	2	no	—
204	C₄⋊C₄	—	212	2	no	—
205	C₂²⋊C₄	—	424	4	no	—
206	C₁₆	—	106	2	no	—
207	C₁₆	#11719921	1	4	yes	complemented
208	C₂³	—	53	2	no	—
209	C₂³	—	53	2	no	—
210	C₂³	—	53	2	no	—
211	C₂³	—	106	1	no	—
212	C₂³	—	106	1	no	—
213	C₂³	—	106	1	no	—
214	C₂³	—	106	1	no	—
215	C₂³	D₈×D₅₃	1	1	yes	characteristic
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	53	2	no	—
220	C₂×C₄	—	106	1	no	—
221	C₂×C₄	—	106	1	no	—
222	C₂×C₄	—	106	2	no	—
223	C₂×C₄	—	106	2	no	—
224	C₂×C₄	—	106	2	no	—
225	C₂×C₄	—	106	2	no	—
226	C₂×C₄	—	424	2	no	—
227	C₂×C₄	—	424	2	no	—
228	C₂×C₄	—	424	2	no	—
229	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
230	C₂×C₄	#13133188	1	1	yes	characteristic
231	C₂×C₄	#13133192	1	1	yes	characteristic
232	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
233	C₂×C₄	#13133236	1	1	yes	characteristic
234	C₂×C₄	#13133240	1	1	yes	characteristic
235	C₈	—	106	2	no	—
236	C₈	C₂×#11719921	1	1	yes	characteristic
237	C₈	#13133175	1	1	yes	characteristic
238	C₈	#13133193	1	1	yes	characteristic
239	C₈	#13133197	1	1	yes	characteristic
240	C₂²	—	53	4	no	—
241	C₂²	—	106	2	no	—
242	C₂²	—	106	2	no	—
243	C₂²	—	106	2	no	—
244	C₂²	—	106	2	no	—
245	C₂²	—	106	2	no	—
246	C₂²	—	106	2	no	—
247	C₂²	#107×D₅₃	1	1	yes	characteristic
248	C₂²	#16810856	1	1	yes	characteristic
249	C₂²	#16810860	1	1	yes	characteristic
250	C₂²	D₁₆×D₅₃	1	1	yes	characteristic
251	C₂²	Q₃₂×D₅₃	1	1	yes	characteristic
252	C₂²	#16811052	1	1	yes	characteristic
253	C₂²	#16811056	1	1	yes	characteristic
254	C₄	—	16	2	no	—
255	C₄	—	848	2	no	—
256	C₄	—	106	2	no	—
257	C₄	#16810480	1	1	yes	characteristic
258	C₄	#16810660	1	1	yes	characteristic
259	C₄	#16810861	1	1	yes	characteristic
260	C₄	#16810865	1	1	yes	characteristic
261	C₂	—	106	4	no	—
262	C₂	#20274741	1	1	yes	characteristic
263	C₂	#366×D₅₃	1	1	yes	characteristic
264	C₂	#20276496	1	1	yes	characteristic
265	C₂	#20276798	1	1	yes	characteristic
266	C₂	#20276802	1	1	yes	characteristic
267	C₂	#20276803	1	1	yes	characteristic
268	C₂	#20276807	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#366	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂³×D₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#366	D₅₃
hypocenter	C₅₃	#12834
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
2. C₂×#16810449	C₂	semidirect
4. #20272806	C₂	semidirect
5. C₂×C₅₃×#366	C₂	semidirect
10. C₂×C₁₆×D₅₃	C₄	semidirect
71. C₁₆×C₅₃	C₂²⋊C₄	semidirect
119. C₂×D₅₃	#366	semidirect
129. C₂×#366	D₅₃	semidirect
152. C₅₃	#12834	semidirect
207. C₁₆	#11719921	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₅₃	2
C₂×#16810449	2
#20274741	2
#20272806	2
C₂×C₅₃×#366	2
#12834	53