Subgroups — #27298936

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810578	C₂	1	1	yes	maximal, characteristic
3	#20272792	C₂	1	4	yes	maximal, complemented
4	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₄	1	1	yes	characteristic
6	C₂×#13133176	C₄	1	1	yes	characteristic
7	C₂²×#11719908	—	2	2	no	—
8	C₂²×#11719908	—	2	2	no	—
9	C₂²×#11719908	C₄	1	4	yes	complemented
10	C₂²×#11719908	C₂²	1	2	yes	—
11	#16810578	—	2	2	no	—
12	#16810578	—	2	2	no	—
13	C₂×#13133127	C₄	1	2	yes	—
14	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₅₃	—	2	2	no	—
17	C₂×M₃₂×C₅₃	—	2	2	no	—
18	C₂×M₃₂×C₅₃	C₄	1	4	yes	complemented
19	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	C₈	1	4	yes	—
23	C₂²×#88322	C₂×C₄	1	2	yes	—
24	#13133176	—	2	2	no	—
25	#13133176	—	2	2	no	—
26	C₂×C₄⋊Dic₅₃	C₈	1	2	yes	—
27	C₂×C₂²⋊Dic₅₃	C₈	1	2	yes	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
29	C₂×#11719908	—	2	8	no	—
30	C₂×#11719908	—	2	2	no	—
31	C₂×#11719908	—	2	2	no	—
32	C₂×#11719908	C₂×C₄	1	4	yes	—
33	#13133127	—	2	4	no	—
34	C₈×Dic₅₃	—	2	2	no	—
35	C₈×Dic₅₃	—	2	2	no	—
36	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	D₄	1	2	yes	—
41	C₂²×C₈×C₅₃	D₄	1	2	yes	—
42	C₂²×C₈×C₅₃	D₄	1	2	yes	—
43	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	M₃₂×C₅₃	—	2	8	no	—
46	C₂×C₁₆×C₅₃	—	2	2	no	—
47	C₂×C₁₆×C₅₃	—	2	2	no	—
48	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
49	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
56	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
59	C₂×C₈×C₅₃	C₄²	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
63	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
65	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
72	C₂²⋊Dic₅₃	—	2	4	no	—
73	C₄⋊Dic₅₃	—	2	4	no	—
74	C₄×Dic₅₃	—	2	2	no	—
75	C₄×Dic₅₃	—	2	2	no	—
76	#11719917	—	2	8	no	—
77	C₂×#88322	—	2	2	no	—
78	C₂×#88322	—	2	2	no	—
79	C₂×#88322	C₂×C₈	1	4	yes	—
80	C₁₆×C₅₃	—	2	4	no	—
81	#11719908	—	2	4	no	—
82	C₂³×C₅₃	—	2	4	no	—
83	C₂³×C₅₃	C₄×C₈	1	2	yes	—
84	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
85	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
86	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#95	1	4	yes	—
88	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
89	C₂×C₄×C₅₃	#97	1	4	yes	—
90	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
91	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
92	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
94	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×Dic₅₃	—	2	4	no	—
101	C₂×Dic₅₃	—	2	8	no	—
102	C₈×C₅₃	#95	1	4	yes	—
103	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	#88322	—	2	4	no	—
108	#12545	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	—	2	4	no	—
110	C₂²×C₅₃	—	2	4	no	—
111	C₂²×C₅₃	—	2	4	no	—
112	C₂²×C₅₃	#336	1	4	yes	—
113	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
114	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₂²×C₅₃	#504	1	4	yes	—
117	C₄×C₅₃	#336	1	4	yes	—
118	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
119	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
120	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
122	Dic₅₃	—	4	4	no	—
123	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
124	C₂³×C₁₆	—	53	1	no	—
125	C₂×#405	—	53	1	no	—
126	#1374	—	53	4	no	—
127	C₂×C₅₃	—	2	4	no	—
128	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
129	C₂×C₅₃	#2108	1	2	yes	—
130	C₂×C₅₃	#2113	1	2	yes	—
131	C₂×C₅₃	#2151	1	2	yes	—
132	C₂²×M₁₆	—	53	1	no	—
133	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
134	C₂×#117	—	53	1	no	—
135	C₂×M₃₂	—	2	2	no	—
136	C₂×M₃₂	—	2	2	no	—
137	C₂×M₃₂	Dic₅₃	1	4	yes	complemented
138	C₂²×C₁₆	—	53	4	no	—
139	C₂²×C₁₆	—	53	2	no	—
140	C₂²×C₁₆	—	106	2	no	—
141	C₂²×C₁₆	—	106	2	no	—
142	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
143	#405	—	106	2	no	—
144	#405	—	106	2	no	—
145	C₂×#97	—	53	2	no	—
146	C₂×C₄×C₈	—	53	2	no	—
147	C₅₃	#12545	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
149	C₂×M₁₆	—	53	4	no	—
150	C₂×M₁₆	—	106	2	no	—
151	C₂×M₁₆	—	106	2	no	—
152	C₂²×C₈	—	53	2	no	—
153	C₂²×C₈	Dic₁₀₆	1	2	yes	—
154	C₂²×C₈	C₄×D₅₃	1	2	yes	—
155	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
156	C₂²×C₈	D₂₁₂	1	2	yes	—
157	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
158	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
159	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
160	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
161	#117	—	106	2	no	—
162	#117	—	106	2	no	—
163	C₂×C₄⋊C₄	—	53	2	no	—
164	C₂×C₂²⋊C₄	—	53	2	no	—
165	C₂×C₄²	—	53	2	no	—
166	M₃₂	—	2	8	no	—
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	53	4	no	—
170	C₂×C₁₆	—	106	2	no	—
171	C₂×C₁₆	—	106	2	no	—
172	C₂×C₁₆	—	106	8	no	—
173	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
174	#97	—	106	4	no	—
175	C₄×C₈	—	106	2	no	—
176	C₄×C₈	—	106	2	no	—
177	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
178	C₂²×C₄	—	53	4	no	—
179	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₈×D₅₃	1	2	yes	—
181	C₂²×C₄	#11719912	1	2	yes	—
182	C₂²×C₄	#11719912	1	2	yes	—
183	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
184	C₂²×C₄	#11719919	1	2	yes	—
185	C₂²×C₄	#11719921	1	2	yes	—
186	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
187	M₁₆	—	106	8	no	—
188	C₂×C₈	—	53	4	no	—
189	C₂×C₈	—	106	2	no	—
190	C₂×C₈	—	106	2	no	—
191	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
192	C₂×C₈	#11719919	1	2	yes	—
193	C₂×C₈	#11719919	1	4	yes	—
194	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
195	C₂×C₈	#11719921	1	2	yes	—
196	C₂×C₈	#11719921	1	4	yes	—
197	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
198	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
200	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
201	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
204	C₄⋊C₄	—	106	4	no	—
205	C₂²⋊C₄	—	106	4	no	—
206	C₄²	—	106	2	no	—
207	C₄²	—	106	2	no	—
208	C₁₆	—	2	4	no	—
209	C₁₆	—	106	4	no	—
210	C₂³	—	2	4	no	—
211	C₂³	C₈×Dic₅₃	1	2	yes	—
212	C₂³	#13133129	1	2	yes	—
213	C₂³	#13133146	1	2	yes	—
214	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
215	C₂×C₄	—	106	8	no	—
216	C₂×C₄	—	106	4	no	—
217	C₂×C₄	#13133124	1	4	yes	—
218	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
219	C₂×C₄	#13133127	1	4	yes	—
220	C₂×C₄	#13133129	1	4	yes	—
221	C₂×C₄	#13133129	1	2	yes	—
222	C₂×C₄	#13133146	1	2	yes	—
223	C₂×C₄	#13133146	1	4	yes	—
224	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
225	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
226	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
227	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
228	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
229	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
230	C₈	—	106	4	no	—
231	C₈	#13133124	1	4	yes	—
232	C₈	C₂×#11719921	1	1	yes	characteristic
233	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
234	C₈	C₂×#11719919	1	1	yes	characteristic
235	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	#16810377	1	4	yes	—
240	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
241	C₂²	C₂×#13133129	1	1	yes	characteristic
242	C₂²	C₂×#13133146	1	1	yes	characteristic
243	C₂²	#16811041	1	4	yes	—
244	C₄	—	212	4	no	—
245	C₄	#16810377	1	4	yes	—
246	C₄	C₂×#13133124	1	1	yes	characteristic
247	C₄	C₂×#13133127	1	1	yes	characteristic
248	C₄	C₂×#13133129	1	1	yes	characteristic
249	C₄	C₂×#13133146	1	1	yes	characteristic
250	C₂	—	2	4	no	—
251	C₂	C₂×#16810377	1	1	yes	characteristic
252	C₂	#20276314	1	2	yes	—
253	C₂	#20276330	1	2	yes	—
254	C₂	#20276475	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	#12545
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20272792	C₂	semidirect
9. C₂²×#11719908	C₄	semidirect
18. C₂×M₃₂×C₅₃	C₄	semidirect
137. C₂×M₃₂	Dic₅₃	semidirect
147. C₅₃	#12545	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#11719908	2
C₂×#16810578	2
#20272792	2
C₂²×M₃₂×C₅₃	2
#12545	53