Subgroups — #27274039

Order 13568.

542 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	2	yes	maximal
2	#20272590	C₂	1	4	yes	maximal, complemented
3	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
4	#16810680	—	2	2	no	—
5	#16810680	C₄	1	4	yes	—
6	#16810676	—	2	2	no	—
7	#16810676	C₄	1	4	yes	—
8	#16810647	C₂²	1	4	yes	—
9	#16810587	—	2	2	no	—
10	C₄×#11719917	—	2	2	no	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#431	—	2	2	no	—
13	C₅₃×#431	—	2	2	no	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
17	C₂×#11719917	—	2	4	no	—
18	C₄⋊#88322	—	2	4	no	—
19	C₄⋊#88322	C₂×C₄	1	8	yes	—
20	C₂²⋊#88322	—	2	4	no	—
21	C₂²⋊#88322	—	2	4	no	—
22	#13133128	—	2	4	no	—
23	C₄×#88322	—	2	4	no	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	D₄	1	2	yes	—
33	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	—	2	4	no	—
37	C₂²⋊C₈×C₅₃	—	2	4	no	—
38	C₄×C₈×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	—	2	4	no	—
42	C₂²×C₄×C₅₃	—	4	2	no	—
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₅₃	—	2	4	no	—
57	C₂×C₈×C₅₃	—	4	4	no	—
58	C₄²×C₅₃	C₄²	1	1	yes	characteristic
59	C₄²×C₅₃	C₄²	1	1	yes	characteristic
60	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
67	C₄²×C₅₃	C₂×D₄	1	2	yes	—
68	#11719917	—	4	8	no	—
69	C₂×#88322	—	2	8	no	—
70	C₂³×C₅₃	—	2	2	no	—
71	C₂³×C₅₃	—	4	1	no	—
72	C₂³×C₅₃	—	4	1	no	—
73	C₂³×C₅₃	#95	1	2	yes	—
74	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	4	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	4	1	no	—
85	C₂×C₄×C₅₃	—	4	1	no	—
86	C₂×C₄×C₅₃	#95	1	2	yes	—
87	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₅₃	#135	1	4	yes	—
96	C₈×C₅₃	—	4	4	no	—
97	#88322	—	4	8	no	—
98	#8000	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	—	2	1	no	—
101	C₂²×C₅₃	—	2	2	no	—
102	C₂²×C₅₃	—	2	2	no	—
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	4	4	no	—
105	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
106	C₂²×C₅₃	#427	1	2	yes	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	#338	1	2	yes	—
114	C₄×C₅₃	#413	1	2	yes	—
115	C₄×C₅₃	#416	1	4	yes	—
116	#2920	—	53	2	no	—
117	C₂×#431	D₅₃	1	1	yes	characteristic
118	#1292	—	53	4	no	—
119	C₂×C₅₃	—	2	2	no	—
120	C₂×C₅₃	—	4	2	no	—
121	C₂×C₅₃	C₂×#338	1	1	yes	characteristic
122	C₂×C₅₃	#1735	1	1	yes	characteristic
123	C₂×C₅₃	#1742	1	1	yes	characteristic
124	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
125	#432	—	53	4	no	—
126	#432	—	106	2	no	—
127	#431	—	2	2	no	—
128	#431	—	2	2	no	—
129	#431	—	53	4	no	—
130	#431	—	106	2	no	—
131	#423	—	53	4	no	—
132	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
133	#407	—	106	2	no	—
134	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
135	C₄×M₁₆	—	106	2	no	—
136	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
137	C₅₃	#8000	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
139	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
140	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
141	C₂×M₁₆	—	106	4	no	—
142	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
143	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
144	C₂×C₄²	C₄×D₅₃	1	2	yes	—
145	C₂×C₄²	C₄×D₅₃	1	2	yes	—
146	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
147	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
148	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
149	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
150	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	—	53	8	no	—
153	C₄⋊C₈	—	106	4	no	—
154	C₂²⋊C₈	—	2	4	no	—
155	C₂²⋊C₈	—	106	4	no	—
156	C₂²⋊C₈	—	106	4	no	—
157	#97	—	106	4	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	106	4	no	—
161	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
166	C₂²×C₄	#11719919	1	2	yes	—
167	C₂²×C₄	#11719919	1	2	yes	—
168	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
169	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
170	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
171	C₂²×C₄	#11719921	1	2	yes	—
172	C₂²×C₄	#11719921	1	2	yes	—
173	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
174	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
175	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
176	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
177	M₁₆	—	212	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	4	4	no	—
180	C₂×C₈	—	106	8	no	—
181	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
182	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
183	C₄²	#11719919	1	2	yes	—
184	C₄²	#11719919	1	2	yes	—
185	C₄²	#11719921	1	2	yes	—
186	C₄²	#11719921	1	2	yes	—
187	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
188	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
189	C₄²	C₂×C₄×D₅₃	1	2	yes	—
190	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
191	C₂³	—	2	2	no	—
192	C₂³	—	4	1	no	—
193	C₂³	—	4	1	no	—
194	C₂³	#13133124	1	2	yes	—
195	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	#13133124	1	1	yes	characteristic
208	C₂×C₄	#13133124	1	1	yes	characteristic
209	C₂×C₄	#13133124	1	1	yes	characteristic
210	C₂×C₄	#13133124	1	2	yes	—
211	C₂×C₄	#13133124	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
213	C₂×C₄	C₂×#11719921	1	2	yes	—
214	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
215	C₂×C₄	C₂×#11719919	1	2	yes	—
216	C₂×C₄	#13133239	1	4	yes	—
217	C₈	—	4	4	no	—
218	C₈	—	212	8	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	4	no	—
225	C₂²	C₂×#13133124	1	1	yes	characteristic
226	C₂²	#16810661	1	2	yes	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	2	no	—
233	C₄	#16810381	1	2	yes	—
234	C₄	#16810607	1	2	yes	—
235	C₄	#16810620	1	4	yes	—
236	C₂	—	2	2	no	—
237	C₂	—	4	2	no	—
238	C₂	C₂×#16810381	1	1	yes	characteristic
239	C₂	#20274432	1	1	yes	characteristic
240	C₂	#20274454	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133124
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#8000
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. #20272590	C₂	semidirect
137. C₅₃	#8000	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
#20272590	2
C₂×C₅₃×#431	2
#8000	53