Subgroups — #27300590

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133151	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810448	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic
4	#20272806	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1808	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
7	C₂²×#11719908	C₂²	1	2	yes	—
8	C₂×#13133151	—	2	4	no	—
9	C₂×#13133151	—	2	1	no	—
10	C₂×#13133151	C₂²	1	2	yes	—
11	#16810448	—	2	4	no	—
12	C₂²⋊#11719908	C₄	1	4	yes	complemented
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#380	—	2	1	no	—
17	C₅₃×#340	—	2	1	no	—
18	C₂³×Dic₅₃	—	4	1	no	—
19	C₂×C₄⋊Dic₅₃	—	2	4	no	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	4	2	no	—
22	C₂×C₄⋊Dic₅₃	—	4	1	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	—	2	2	no	—
25	C₂×#11719908	C₂×C₄	1	4	yes	complemented
26	#13133151	—	2	4	no	—
27	#13133151	—	2	2	no	—
28	#13133151	D₄	1	4	yes	—
29	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
39	#107×C₅₃	—	2	2	no	—
40	#107×C₅₃	—	2	2	no	—
41	#95×C₅₃	—	4	1	no	—
42	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	—	2	1	no	—
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	4	1	no	—
48	C₂²×C₄×C₅₃	—	4	1	no	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₅₃	—	4	4	no	—
54	C₂²×Dic₅₃	—	4	2	no	—
55	C₂²×Dic₅₃	—	8	2	no	—
56	C₂²×Dic₅₃	—	8	2	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	4	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₅₃	#41	1	2	yes	—
66	C₄⋊C₄×C₅₃	—	4	1	no	—
67	C₄⋊C₄×C₅₃	—	4	1	no	—
68	C₄⋊C₄×C₅₃	—	4	1	no	—
69	C₄⋊C₄×C₅₃	—	4	1	no	—
70	C₂²⋊C₄×C₅₃	—	8	2	no	—
71	C₄⋊Dic₅₃	—	2	4	no	—
72	C₄⋊Dic₅₃	—	4	4	no	—
73	C₄⋊Dic₅₃	—	4	2	no	—
74	#11719908	—	2	4	no	—
75	C₂³×C₅₃	—	2	1	no	—
76	C₂³×C₅₃	—	2	1	no	—
77	C₂³×C₅₃	—	2	1	no	—
78	C₂³×C₅₃	—	2	1	no	—
79	C₂³×C₅₃	#107	1	2	yes	—
80	C₂³×C₅₃	D₁₆	1	2	yes	—
81	C₂³×C₅₃	Q₃₂	1	2	yes	—
82	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	8	1	no	—
90	C₂×C₄×C₅₃	—	8	1	no	—
91	C₂×C₄×C₅₃	—	8	1	no	—
92	C₂×C₄×C₅₃	—	8	1	no	—
93	C₂×C₄×C₅₃	—	8	1	no	—
94	C₂×C₄×C₅₃	—	8	1	no	—
95	C₂×C₄×C₅₃	#107	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
102	C₂×Dic₅₃	—	4	4	no	—
103	C₂×Dic₅₃	—	8	8	no	—
104	C₂×Dic₅₃	—	8	2	no	—
105	C₂×Dic₅₃	—	8	2	no	—
106	C₈×C₅₃	—	2	2	no	—
107	C₈×C₅₃	C₄×D₄	1	2	yes	—
108	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
109	#12919	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	2	2	no	—
116	C₂²×C₅₃	#366	1	4	yes	complemented
117	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
118	C₂²×C₅₃	#447	1	1	yes	characteristic
119	C₂²×C₅₃	#480	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
122	C₂²×C₅₃	#509	1	1	yes	characteristic
123	C₂²×C₅₃	#510	1	1	yes	characteristic
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	16	1	no	—
126	C₄×C₅₃	—	16	1	no	—
127	C₄×C₅₃	#385	1	1	yes	characteristic
128	C₄×C₅₃	#428	1	1	yes	characteristic
129	C₄×C₅₃	#451	1	1	yes	characteristic
130	C₄×C₅₃	#484	1	1	yes	characteristic
131	Dic₅₃	—	8	4	no	—
132	C₂²×#107	—	53	1	no	—
133	C₂×#366	—	53	2	no	—
134	C₂×C₂²⋊C₁₆	—	53	1	no	—
135	#1808	D₅₃	1	1	yes	characteristic
136	#1380	—	53	2	no	—
137	C₂×C₅₃	—	2	4	no	—
138	C₂×C₅₃	#1852	1	1	yes	characteristic
139	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
140	C₂×C₅₃	#2159	1	1	yes	characteristic
141	C₂×C₅₃	#2184	1	1	yes	characteristic
142	C₂×C₅₃	#2188	1	1	yes	characteristic
143	C₂×C₅₃	#2232	1	1	yes	characteristic
144	C₂×C₅₃	#2236	1	1	yes	characteristic
145	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
146	C₂²×C₄⋊C₄	—	106	1	no	—
147	C₂²×C₁₆	—	53	2	no	—
148	C₂×#107	—	53	2	no	—
149	C₂×#107	—	106	4	no	—
150	C₂×#107	—	106	1	no	—
151	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
152	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
153	#380	—	2	1	no	—
154	#366	—	106	4	no	—
155	C₂²⋊C₁₆	—	53	4	no	—
156	#340	—	2	1	no	—
157	C₅₃	#12919	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₄	—	212	1	no	—
159	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	Dic₁₀₆	1	2	yes	—
163	C₂²×C₈	C₄×D₅₃	1	2	yes	—
164	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	—	106	4	no	—
169	C₂×C₄⋊C₄	—	106	2	no	—
170	C₂×C₄⋊C₄	—	212	2	no	—
171	C₂×C₄⋊C₄	—	212	1	no	—
172	C₂×C₂²⋊C₄	—	4	1	no	—
173	C₂×C₁₆	—	53	4	no	—
174	C₂×C₁₆	—	106	2	no	—
175	C₂×C₁₆	—	106	2	no	—
176	#107	—	2	2	no	—
177	#107	—	2	2	no	—
178	#107	—	53	4	no	—
179	#107	—	106	4	no	—
180	#107	—	106	2	no	—
181	#95	—	4	1	no	—
182	C₂⁴	#11719922	1	1	yes	characteristic
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	4	1	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	—	424	2	no	—
190	C₂²×C₄	—	424	2	no	—
191	C₂²×C₄	—	212	4	no	—
192	C₂²×C₄	—	212	2	no	—
193	C₂²×C₄	#11719919	1	2	yes	—
194	C₂²×C₄	#11719922	1	1	yes	characteristic
195	C₂²×C₄	#11719925	1	2	yes	—
196	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	#11719919	1	2	yes	—
202	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
203	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
204	C₂×C₈	D₄×D₅₃	1	2	yes	—
205	C₂×C₈	#11719947	1	2	yes	—
206	C₄⋊C₄	—	4	1	no	—
207	C₄⋊C₄	—	4	1	no	—
208	C₄⋊C₄	—	4	1	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₄⋊C₄	—	106	4	no	—
211	C₄⋊C₄	—	212	4	no	—
212	C₄⋊C₄	—	212	2	no	—
213	C₂²⋊C₄	—	8	2	no	—
214	C₁₆	—	106	4	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	#13133150	1	2	yes	—
220	C₂³	#13133158	1	2	yes	—
221	C₂³	#13133163	1	2	yes	—
222	C₂³	C₂×#11719922	1	1	yes	characteristic
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	8	1	no	—
230	C₂×C₄	—	8	1	no	—
231	C₂×C₄	—	8	1	no	—
232	C₂×C₄	—	8	1	no	—
233	C₂×C₄	—	8	1	no	—
234	C₂×C₄	—	8	1	no	—
235	C₂×C₄	—	424	8	no	—
236	C₂×C₄	—	424	2	no	—
237	C₂×C₄	—	424	2	no	—
238	C₂×C₄	—	212	4	no	—
239	C₂×C₄	#13133150	1	2	yes	—
240	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
241	C₂×C₄	#13133185	1	1	yes	characteristic
242	C₂×C₄	#13133198	1	1	yes	characteristic
243	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
244	C₂×C₄	#13133242	1	1	yes	characteristic
245	C₂×C₄	#13133247	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	#13133177	1	2	yes	—
248	C₈	#13133194	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	#16810448	1	4	yes	complemented
256	C₂²	C₂×#13133150	1	1	yes	characteristic
257	C₂²	#16810756	1	1	yes	characteristic
258	C₂²	#16810934	1	1	yes	characteristic
259	C₂²	C₂×#13133158	1	1	yes	characteristic
260	C₂²	C₂×#13133163	1	1	yes	characteristic
261	C₂²	#16811058	1	1	yes	characteristic
262	C₂²	#16811063	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	16	1	no	—
265	C₄	—	16	1	no	—
266	C₄	—	424	4	no	—
267	C₄	#16810495	1	1	yes	characteristic
268	C₄	#16810666	1	1	yes	characteristic
269	C₄	#16810778	1	1	yes	characteristic
270	C₄	#16810956	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	#20274902	1	1	yes	characteristic
273	C₂	C₂×#16810448	1	1	yes	characteristic
274	C₂	#20276504	1	1	yes	characteristic
275	C₂	#20276626	1	1	yes	characteristic
276	C₂	#20276648	1	1	yes	characteristic
277	C₂	#20276893	1	1	yes	characteristic
278	C₂	#20276915	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1808	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1808	D₅₃
hypocenter	C₅₃	#12919
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1808

Complements

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

N	H ≅ G/N	type
2. C₂×#16810448	C₂	semidirect
4. #20272806	C₂	semidirect
12. C₂²⋊#11719908	C₄	semidirect
25. C₂×#11719908	C₂×C₄	semidirect
116. C₂²×C₅₃	#366	semidirect
157. C₅₃	#12919	semidirect
255. C₂²	#16810448	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133151	2
C₂×#16810448	2
C₂×C₂²⋊#11719908	2
#20272806	2
C₅₃×#1808	2
#12919	53