Subgroups — #27273565

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	#20272549	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#432	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	—	2	1	no	—
6	C₂²×#11719917	C₄	1	2	yes	—
7	#16810647	—	2	2	no	—
8	#16810647	C₄	1	4	yes	—
9	C₂×C₄⋊#88322	—	2	1	no	—
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₄×#11719917	—	2	2	no	—
12	C₄×#11719917	—	2	2	no	—
13	C₂×#13133128	C₄	1	2	yes	—
14	C₂×C₄×#88322	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#432	—	2	2	no	—
17	C₅₃×#432	C₄	1	4	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
20	C₂×#97×C₅₃	—	2	1	no	—
21	C₂×#11719917	—	2	8	no	—
22	C₂×#11719917	—	2	4	no	—
23	C₂×#11719917	—	2	4	no	—
24	C₂×#11719917	—	4	4	no	—
25	C₂²×#88322	—	2	2	no	—
26	C₂²×#88322	C₂×C₄	1	4	yes	—
27	C₄⋊#88322	—	2	4	no	—
28	C₄⋊#88322	—	2	2	no	—
29	C₄⋊#88322	C₂×C₄	1	4	yes	—
30	#13133128	—	2	4	no	—
31	C₄×#88322	—	2	2	no	—
32	C₄×#88322	—	2	2	no	—
33	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	2	yes	—
41	C₂×C₄²×C₅₃	D₄	1	2	yes	—
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₅₃	—	2	2	no	—
46	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₂²⋊C₈×C₅₃	—	2	4	no	—
48	#97×C₅₃	—	2	4	no	—
49	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	—	2	2	no	—
53	C₂²×C₄×C₅₃	—	4	2	no	—
54	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	—	4	2	no	—
69	C₄²×C₅₃	C₄²	1	2	yes	—
70	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
74	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
75	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
79	#11719917	—	2	16	no	—
80	#11719917	—	4	8	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	—	2	4	no	—
84	C₂×#88322	—	4	4	no	—
85	C₂×#88322	M₁₆	1	8	yes	complemented
86	C₂³×C₅₃	—	2	2	no	—
87	C₂³×C₅₃	—	2	2	no	—
88	C₂³×C₅₃	—	2	2	no	—
89	C₂³×C₅₃	#95	1	2	yes	—
90	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	—	2	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	—	2	4	no	—
97	C₂×C₄×C₅₃	—	2	4	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	#95	1	2	yes	—
103	C₂×C₄×C₅₃	#95	1	2	yes	—
104	C₂×C₄×C₅₃	#95	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
112	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×C₄×C₅₃	#137	1	2	yes	—
114	C₈×C₅₃	—	4	4	no	—
115	#88322	—	2	8	no	—
116	#88322	—	4	4	no	—
117	#7876	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	—	4	4	no	—
122	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
123	C₂²×C₅₃	#423	1	2	yes	—
124	C₂²×C₅₃	#428	1	2	yes	—
125	C₂²×C₅₃	#430	1	2	yes	—
126	C₄×C₅₃	—	2	2	no	—
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
132	C₄×C₅₃	#407	1	2	yes	—
133	C₄×C₅₃	#417	1	2	yes	—
134	C₄×C₅₃	#418	1	1	yes	characteristic
135	C₄×C₅₃	#419	1	1	yes	characteristic
136	C₂×#432	D₅₃	1	1	yes	characteristic
137	C₂×#423	—	53	1	no	—
138	C₂×C₄×M₁₆	—	53	1	no	—
139	#1277	—	53	4	no	—
140	C₂×C₅₃	—	4	2	no	—
141	C₂×C₅₃	#1569	1	4	yes	—
142	C₂×C₅₃	#1728	1	1	yes	characteristic
143	C₂×C₅₃	#1736	1	1	yes	characteristic
144	C₂×C₅₃	#1738	1	1	yes	characteristic
145	C₂²×M₁₆	—	53	2	no	—
146	C₂²×M₁₆	—	106	1	no	—
147	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
148	#432	—	2	2	no	—
149	#432	Dic₅₃	1	4	yes	—
150	#423	—	53	4	no	—
151	#423	—	106	2	no	—
152	C₂×C₄⋊C₈	—	53	2	no	—
153	C₂×C₄⋊C₈	—	106	1	no	—
154	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
155	C₂×C₂²⋊C₈	—	2	1	no	—
156	C₄×M₁₆	—	106	2	no	—
157	C₄×M₁₆	—	106	2	no	—
158	C₂×#97	—	2	1	no	—
159	C₂×#97	—	53	2	no	—
160	C₂×C₄×C₈	—	53	2	no	—
161	C₅₃	#7876	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
163	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
164	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
165	C₂×M₁₆	—	106	8	no	—
166	C₂×M₁₆	—	106	4	no	—
167	C₂×M₁₆	—	106	4	no	—
168	C₂×M₁₆	—	212	4	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	53	4	no	—
171	C₂²×C₈	—	106	2	no	—
172	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₅₃	1	2	yes	—
174	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
175	C₂×C₄²	D₂₁₂	1	2	yes	—
176	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
177	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
178	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
179	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	53	4	no	—
182	C₄⋊C₈	—	106	2	no	—
183	C₄⋊C₈	—	106	4	no	—
184	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
185	C₂²⋊C₈	—	2	4	no	—
186	#97	—	2	4	no	—
187	#97	—	106	4	no	—
188	C₄×C₈	—	106	2	no	—
189	C₄×C₈	—	106	2	no	—
190	C₂⁴	#11719919	1	1	yes	characteristic
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
196	C₂²×C₄	#11719919	1	1	yes	characteristic
197	C₂²×C₄	#11719919	1	1	yes	characteristic
198	C₂²×C₄	#11719919	1	1	yes	characteristic
199	C₂²×C₄	#11719919	1	2	yes	—
200	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
201	C₂²×C₄	#11719921	1	2	yes	—
202	C₂²×C₄	#11719921	1	2	yes	—
203	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
204	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
207	M₁₆	—	106	16	no	—
208	M₁₆	—	212	8	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	53	8	no	—
213	C₂×C₈	—	106	4	no	—
214	C₂×C₈	—	106	4	no	—
215	C₂×C₈	—	106	4	no	—
216	C₂×C₈	—	212	4	no	—
217	C₄²	C₄×Dic₅₃	1	2	yes	—
218	C₄²	#11719919	1	2	yes	—
219	C₄²	C₄⋊Dic₅₃	1	2	yes	—
220	C₄²	#11719921	1	2	yes	—
221	C₄²	#11719921	1	2	yes	—
222	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
223	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
224	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
225	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
226	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	#13133124	1	2	yes	—
231	C₂³	C₂×#11719919	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	#13133124	1	2	yes	—
244	C₂×C₄	#13133124	1	2	yes	—
245	C₂×C₄	#13133124	1	2	yes	—
246	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
247	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
248	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
250	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
252	C₂×C₄	M₁₆×D₅₃	1	4	yes	—
253	C₂×C₄	#13133242	1	2	yes	—
254	C₂×C₄	#13133247	1	2	yes	—
255	C₈	—	4	4	no	—
256	C₈	—	106	8	no	—
257	C₈	—	212	4	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₂×#13133124	1	1	yes	characteristic
263	C₂²	#16810646	1	2	yes	—
264	C₂²	#16810666	1	2	yes	—
265	C₂²	#16810671	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	M₁₆×Dic₅₃	1	2	yes	—
272	C₄	#16810586	1	2	yes	—
273	C₄	#16810624	1	2	yes	—
274	C₄	#16810627	1	1	yes	characteristic
275	C₄	#16810631	1	1	yes	characteristic
276	C₂	—	4	2	no	—
277	C₂	#20273483	1	4	yes	—
278	C₂	#20274408	1	1	yes	characteristic
279	C₂	#20274436	1	1	yes	characteristic
280	C₂	#20274441	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#432	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#432	D₅₃
hypocenter	C₅₃	#7876
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #20272549	C₂	semidirect
85. C₂×#88322	M₁₆	semidirect
161. C₅₃	#7876	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810647	2
C₂×C₄×#11719917	2
#20272549	2
C₂×C₅₃×#432	2
#7876	53