Subgroups — #27273554

Order 13568.

722 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#20272550	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	—	2	1	no	—
6	#16810676	—	2	2	no	—
7	#16810676	C₄	1	4	yes	—
8	C₂×C₄⋊#88322	—	2	2	no	—
9	C₂×C₄⋊#88322	—	2	1	no	—
10	C₂×C₄⋊#88322	C₄	1	4	yes	—
11	C₂×C₄⋊#88322	C₂²	1	2	yes	—
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×C₂²⋊#88322	—	2	1	no	—
14	C₂×C₄×#88322	—	2	1	no	—
15	C₂²×M₁₆×C₅₃	C₄	1	2	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₅₃	—	2	2	no	—
18	C₄×M₁₆×C₅₃	—	2	2	no	—
19	C₂×#97×C₅₃	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
21	C₂²×#88322	—	2	4	no	—
22	C₂²×#88322	—	2	2	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₂²×#88322	—	4	4	no	—
25	C₄⋊#88322	—	2	4	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	C₂×C₄	1	4	yes	—
29	C₄⋊#88322	C₂×C₄	1	4	yes	—
30	C₂²⋊#88322	—	2	4	no	—
31	C₄×#88322	—	2	4	no	—
32	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
35	C₂×M₁₆×C₅₃	—	2	4	no	—
36	C₂×M₁₆×C₅₃	—	2	4	no	—
37	C₂×M₁₆×C₅₃	C₈	1	8	yes	complemented
38	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	2	yes	—
43	C₂×C₄²×C₅₃	D₄	1	2	yes	—
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	#97×C₅₃	—	2	4	no	—
48	C₄×C₈×C₅₃	—	2	2	no	—
49	C₄×C₈×C₅₃	—	2	2	no	—
50	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
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	2	yes	—
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
68	M₁₆×C₅₃	—	2	16	no	—
69	C₂×C₈×C₅₃	—	2	4	no	—
70	C₂×C₈×C₅₃	—	2	4	no	—
71	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	complemented
72	C₄²×C₅₃	C₄²	1	2	yes	—
73	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
77	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
78	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	—	2	4	no	—
84	C₂×#88322	—	4	8	no	—
85	C₂×#88322	—	4	2	no	—
86	C₂×#88322	—	4	4	no	—
87	C₂×#88322	—	4	2	no	—
88	C₂³×C₅₃	—	2	2	no	—
89	C₂³×C₅₃	—	2	2	no	—
90	C₂³×C₅₃	#95	1	2	yes	—
91	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
92	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	#95	1	2	yes	—
102	C₂×C₄×C₅₃	#95	1	2	yes	—
103	C₂×C₄×C₅₃	#95	1	2	yes	—
104	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
105	C₂×C₄×C₅₃	#97	1	4	yes	—
106	C₂×C₄×C₅₃	C₂²⋊C₈	1	8	yes	—
107	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
108	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
116	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₅₃	#137	1	2	yes	—
118	C₈×C₅₃	—	2	8	no	—
119	#88322	—	4	4	no	—
120	#88322	—	4	4	no	—
121	#7873	—	53	1	no	maximal, Sylow
122	C₂²×C₅₃	—	2	8	no	—
123	C₂²×C₅₃	—	2	2	no	—
124	C₂²×C₅₃	—	2	2	no	—
125	C₂²×C₅₃	#336	1	4	yes	—
126	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
127	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
128	C₂²×C₅₃	#428	1	2	yes	—
129	C₂²×C₅₃	#430	1	2	yes	—
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	#336	1	4	yes	—
135	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
136	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
137	C₄×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
138	C₄×C₅₃	#417	1	2	yes	—
139	C₄×C₅₃	#418	1	1	yes	characteristic
140	C₄×C₅₃	#419	1	1	yes	characteristic
141	C₂×#431	—	53	1	no	—
142	C₂²×C₄⋊C₈	—	53	1	no	—
143	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
144	#1277	—	53	4	no	—
145	C₂×C₅₃	—	2	4	no	—
146	C₂×C₅₃	#1565	1	4	yes	—
147	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
148	C₂×C₅₃	#1736	1	1	yes	characteristic
149	C₂×C₅₃	#1738	1	1	yes	characteristic
150	C₂²×M₁₆	Dic₅₃	1	2	yes	—
151	C₂³×C₈	—	106	1	no	—
152	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
153	#431	—	53	4	no	—
154	#431	—	106	2	no	—
155	C₂×C₄⋊C₈	—	53	4	no	—
156	C₂×C₄⋊C₈	—	53	2	no	—
157	C₂×C₄⋊C₈	—	53	2	no	—
158	C₂×C₄⋊C₈	—	106	1	no	—
159	C₂×C₄⋊C₈	—	106	2	no	—
160	C₂×C₂²⋊C₈	—	106	1	no	—
161	C₄×M₁₆	—	2	2	no	—
162	C₄×M₁₆	—	2	2	no	—
163	C₂×#97	Dic₅₃	1	2	yes	—
164	C₂×C₄×C₈	—	106	1	no	—
165	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
166	C₅₃	#7873	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
168	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
169	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	2	4	no	—
172	C₂×M₁₆	#88322	1	8	yes	complemented
173	C₂²×C₈	—	106	2	no	—
174	C₂²×C₈	—	106	2	no	—
175	C₂²×C₈	—	106	4	no	—
176	C₂²×C₈	—	212	4	no	—
177	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
178	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
179	C₂×C₄²	C₄×D₅₃	1	2	yes	—
180	C₂×C₄²	C₄×D₅₃	1	2	yes	—
181	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
182	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
183	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
184	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
185	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
186	C₄⋊C₈	—	53	4	no	—
187	C₄⋊C₈	—	53	4	no	—
188	C₄⋊C₈	—	106	2	no	—
189	C₄⋊C₈	—	106	2	no	—
190	C₄⋊C₈	—	106	4	no	—
191	C₂²⋊C₈	—	106	4	no	—
192	#97	—	2	4	no	—
193	C₄×C₈	—	2	2	no	—
194	C₄×C₈	—	2	2	no	—
195	C₄×C₈	—	106	4	no	—
196	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₂×#88322	1	4	yes	—
201	C₂²×C₄	#11719917	1	4	yes	—
202	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
203	C₂²×C₄	#11719919	1	2	yes	—
204	C₂²×C₄	#11719919	1	2	yes	—
205	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
208	C₂²×C₄	#11719921	1	2	yes	—
209	C₂²×C₄	#11719921	1	2	yes	—
210	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
211	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
213	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
214	M₁₆	—	2	16	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	106	4	no	—
218	C₂×C₈	—	106	4	no	—
219	C₂×C₈	—	212	8	no	—
220	C₂×C₈	—	212	4	no	—
221	C₂×C₈	—	212	2	no	—
222	C₂×C₈	—	212	2	no	—
223	C₂×C₈	C₂×#88322	1	8	yes	complemented
224	C₄²	C₄×Dic₅₃	1	2	yes	—
225	C₄²	#11719919	1	2	yes	—
226	C₄²	#11719919	1	2	yes	—
227	C₄²	#11719921	1	2	yes	—
228	C₄²	#11719921	1	2	yes	—
229	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
230	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
231	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
232	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
233	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	#13133124	1	2	yes	—
237	C₂³	C₄⋊#88322	1	4	yes	—
238	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	#13133124	1	2	yes	—
248	C₂×C₄	#13133124	1	2	yes	—
249	C₂×C₄	#13133124	1	2	yes	—
250	C₂×C₄	C₄×#88322	1	4	yes	—
251	C₂×C₄	#13133128	1	4	yes	—
252	C₂×C₄	C₂²⋊#88322	1	8	yes	—
253	C₂×C₄	C₄⋊#88322	1	4	yes	—
254	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
255	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
256	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
259	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
260	C₂×C₄	C₂²×#88322	1	2	yes	—
261	C₂×C₄	C₂×#11719917	1	2	yes	—
262	C₂×C₄	#13133245	1	2	yes	—
263	C₂×C₄	#13133250	1	2	yes	—
264	C₈	—	2	8	no	—
265	C₈	—	212	4	no	—
266	C₈	—	212	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	8	no	—
270	C₂²	#16810378	1	4	yes	—
271	C₂²	C₂×#13133124	1	1	yes	characteristic
272	C₂²	C₂×C₄⋊#88322	1	2	yes	—
273	C₂²	#16810665	1	2	yes	—
274	C₂²	#16810672	1	2	yes	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	#16810378	1	4	yes	—
280	C₄	C₂×C₄×#88322	1	1	yes	characteristic
281	C₄	C₂×#13133128	1	1	yes	characteristic
282	C₄	C₂×C₂²⋊#88322	1	2	yes	—
283	C₄	#16810625	1	2	yes	—
284	C₄	#16810629	1	1	yes	characteristic
285	C₄	#16810633	1	1	yes	characteristic
286	C₂	—	2	4	no	—
287	C₂	#20273467	1	4	yes	—
288	C₂	C₂×#16810378	1	1	yes	characteristic
289	C₂	#20274435	1	1	yes	characteristic
290	C₂	#20274442	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	#7873
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20272550	C₂	semidirect
37. C₂×M₁₆×C₅₃	C₈	semidirect
71. C₂×C₈×C₅₃	C₂×C₈	semidirect
166. C₅₃	#7873	semidirect
172. C₂×M₁₆	#88322	semidirect
223. C₂×C₈	C₂×#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
C₂²×C₄⋊#88322	2
#20272550	2
C₂×C₄×M₁₆×C₅₃	2
#7873	53