Subgroups — #27270646

Order 13568.

442 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic
2	#20274409	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	C₄	1	2	yes	—
5	C₂×C₂²⋊#88322	—	2	1	no	—
6	C₂×C₄×#88322	—	2	1	no	—
7	#16810378	—	2	1	no	—
8	#16810378	—	2	1	no	—
9	#16810378	C₄	1	2	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
12	C₂×C₄×C₈×C₅₃	—	2	1	no	—
13	C₅₃×#336	C₄	1	2	yes	—
14	C₂×#11719917	—	2	2	no	—
15	C₂×#11719917	—	4	2	no	—
16	C₂×#11719917	—	4	2	no	—
17	C₂×#11719917	C₈	1	4	yes	complemented
18	C₂²×#88322	—	2	2	no	—
19	C₂²×#88322	—	2	2	no	—
20	C₂²⋊#88322	—	2	4	no	—
21	C₄×#88322	—	4	2	no	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	C₂×C₄²×C₅₃	—	4	2	no	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₂²⋊C₈×C₅₃	—	2	4	no	—
32	C₄×C₈×C₅₃	—	4	2	no	—
33	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	—	2	1	no	—
36	C₂²×C₄×C₅₃	—	2	1	no	—
37	C₂²×C₄×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	—	4	1	no	—
42	C₂²×C₄×C₅₃	—	4	1	no	—
43	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
51	C₂×C₈×C₅₃	—	2	4	no	—
52	C₂×C₈×C₅₃	—	4	2	no	—
53	C₂×C₈×C₅₃	—	4	2	no	—
54	C₄²×C₅₃	—	2	2	no	—
55	C₄²×C₅₃	—	2	2	no	—
56	C₄²×C₅₃	—	4	1	no	—
57	C₄²×C₅₃	—	4	1	no	—
58	#11719917	—	4	4	no	—
59	#11719917	—	4	4	no	—
60	C₂×#88322	—	2	4	no	—
61	C₂×#88322	—	2	4	no	—
62	C₂×#88322	—	4	2	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂×#88322	—	4	2	no	—
65	C₂×#88322	—	4	2	no	—
66	C₂³×C₅₃	—	2	1	no	—
67	C₂³×C₅₃	—	2	1	no	—
68	C₂³×C₅₃	—	2	1	no	—
69	C₂³×C₅₃	—	2	1	no	—
70	C₂³×C₅₃	—	2	1	no	—
71	C₂³×C₅₃	—	2	1	no	—
72	C₂³×C₅₃	#95	1	1	yes	characteristic
73	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	2	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	—	4	4	no	—
89	C₂×C₄×C₅₃	—	4	4	no	—
90	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
93	C₂×C₄×C₅₃	#99	1	2	yes	—
94	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#104	1	2	yes	—
97	C₂×C₄×C₅₃	#104	1	2	yes	—
98	C₂×C₄×C₅₃	#108	1	2	yes	—
99	C₂×C₄×C₅₃	#108	1	2	yes	—
100	C₈×C₅₃	—	4	4	no	—
101	#88322	—	4	4	no	—
102	#88322	—	4	4	no	—
103	#7026	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	—	4	1	no	—
108	C₂²×C₅₃	—	4	1	no	—
109	C₂²×C₅₃	—	4	1	no	—
110	C₂²×C₅₃	—	4	1	no	—
111	C₂²×C₅₃	#336	1	1	yes	characteristic
112	C₂²×C₅₃	#339	1	1	yes	characteristic
113	C₂²×C₅₃	#339	1	1	yes	characteristic
114	C₂²×C₅₃	#341	1	1	yes	characteristic
115	C₂²×C₅₃	#341	1	1	yes	characteristic
116	C₂²×C₅₃	#342	1	1	yes	characteristic
117	C₂²×C₅₃	#343	1	1	yes	characteristic
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	—	2	2	no	—
124	C₄×C₅₃	#323	1	2	yes	—
125	C₄×C₅₃	#324	1	2	yes	—
126	C₄×C₂²⋊C₈	—	53	1	no	—
127	C₄×C₂²⋊C₈	D₅₃	1	1	yes	characteristic
128	#1728	—	53	1	no	—
129	C₂×C₅₃	—	4	2	no	—
130	C₂×C₅₃	#1278	1	1	yes	characteristic
131	C₂×C₅₃	#1278	1	1	yes	characteristic
132	C₂×C₅₃	#1280	1	1	yes	characteristic
133	C₂×C₅₃	#1297	1	1	yes	characteristic
134	C₂×C₅₃	#1298	1	1	yes	characteristic
135	C₂×C₅₃	#1299	1	1	yes	characteristic
136	C₂×C₅₃	#1299	1	1	yes	characteristic
137	C₂²×M₁₆	—	53	2	no	—
138	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
139	C₂×C₂²⋊C₈	—	2	1	no	—
140	C₂×C₂²⋊C₈	—	106	1	no	—
141	C₂×C₄×C₈	—	2	1	no	—
142	C₂×C₄×C₈	—	106	1	no	—
143	#336	—	53	2	no	—
144	#336	—	106	1	no	—
145	#336	—	106	1	no	—
146	#336	Dic₅₃	1	2	yes	—
147	C₅₃	#7026	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
149	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
150	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
151	C₂×M₁₆	—	53	4	no	—
152	C₂×M₁₆	—	106	2	no	—
153	C₂×M₁₆	—	212	2	no	—
154	C₂×M₁₆	—	212	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	106	2	no	—
157	C₂²×C₈	—	106	2	no	—
158	C₂×C₄²	—	4	2	no	—
159	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
160	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
161	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
162	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
163	C₂²⋊C₈	—	2	4	no	—
164	C₂²⋊C₈	—	106	4	no	—
165	C₄×C₈	—	4	2	no	—
166	C₄×C₈	—	212	2	no	—
167	C₂⁴	#11719919	1	1	yes	characteristic
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	4	1	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	C₈×D₅₃	1	2	yes	—
178	C₂²×C₄	#11719912	1	2	yes	—
179	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
180	C₂²×C₄	#11719919	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
182	C₂²×C₄	#11719921	1	1	yes	characteristic
183	C₂²×C₄	#11719921	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
185	M₁₆	—	212	4	no	—
186	M₁₆	—	212	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	106	4	no	—
191	C₂×C₈	—	106	4	no	—
192	C₂×C₈	—	212	2	no	—
193	C₂×C₈	—	212	2	no	—
194	C₂×C₈	—	212	2	no	—
195	C₂×C₈	—	212	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	1	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	1	no	—
206	C₂³	#13133124	1	1	yes	characteristic
207	C₂³	#13133146	1	2	yes	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
225	C₂×C₄	#13133127	1	1	yes	characteristic
226	C₂×C₄	#13133129	1	2	yes	—
227	C₂×C₄	#13133131	1	2	yes	—
228	C₂×C₄	#13133133	1	1	yes	characteristic
229	C₂×C₄	#13133135	1	1	yes	characteristic
230	C₂×C₄	#13133144	1	2	yes	—
231	C₂×C₄	#13133145	1	2	yes	—
232	C₂×C₄	#13133152	1	2	yes	—
233	C₂×C₄	#13133153	1	2	yes	—
234	C₈	—	4	4	no	—
235	C₈	—	212	4	no	—
236	C₈	—	212	4	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	#16810377	1	1	yes	characteristic
245	C₂²	#16810382	1	1	yes	characteristic
246	C₂²	#16810383	1	1	yes	characteristic
247	C₂²	#16810387	1	1	yes	characteristic
248	C₂²	#16810388	1	1	yes	characteristic
249	C₂²	#16810389	1	1	yes	characteristic
250	C₂²	#16810391	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	#16810346	1	2	yes	—
258	C₄	#16810348	1	2	yes	—
259	C₂	—	4	2	no	—
260	C₂	#20272551	1	1	yes	characteristic
261	C₂	#20272552	1	1	yes	characteristic
262	C₂	#20272557	1	1	yes	characteristic
263	C₂	#20272601	1	1	yes	characteristic
264	C₂	#20272603	1	1	yes	characteristic
265	C₂	#20272605	1	1	yes	characteristic
266	C₂	#20272606	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11719921
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₅₃
Fitting	C₄×C₂²⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	#95
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂²×C₄²×C₅₃	C₂²
hypercenter	C₄×C₂²⋊C₈	D₅₃
hypocenter	C₅₃	#7026
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
17. C₂×#11719917	C₈	semidirect
147. C₅₃	#7026	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊#88322	2
#20274409	2
C₄×C₂²⋊C₈×C₅₃	2
#7026	53