Subgroups — #27300562

Order 13568.

420 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810443	C₂	1	2	yes	maximal, complemented
2	#20276326	C₂	1	1	yes	maximal, characteristic
3	C₈×C₂²⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#20272796	C₂	1	2	yes	maximal, complemented
5	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	1	yes	characteristic
7	C₂×#13133157	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊#88322	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
10	C₄×C₂²⋊Dic₅₃	C₄	1	1	yes	characteristic
11	#16810443	—	2	4	no	—
12	C₂²⋊#11719908	—	2	1	no	—
13	C₂²⋊#11719908	—	2	1	no	—
14	#16810377	C₄	1	2	yes	—
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₅₃	—	2	1	no	—
17	C₂×M₃₂×C₅₃	—	2	4	no	—
18	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
19	C₂²×C₁₆×C₅₃	—	2	1	no	—
20	C₂²×#88322	C₂×C₄	1	2	yes	—
21	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
22	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
23	#13133157	—	2	2	no	—
24	#13133157	—	2	2	no	—
25	C₂×#11719908	—	2	1	no	—
26	C₂×#11719908	—	2	1	no	—
27	C₂²⋊#88322	C₈	1	2	yes	—
28	C₂²⋊#88322	C₈	1	2	yes	—
29	C₈×Dic₅₃	—	2	2	no	—
30	C₈×Dic₅₃	—	2	2	no	—
31	#13133124	—	2	1	no	—
32	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	—	2	2	no	—
34	C₂²×C₈×C₅₃	—	2	2	no	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	M₃₂×C₅₃	—	2	4	no	—
42	M₃₂×C₅₃	—	2	2	no	—
43	M₃₂×C₅₃	D₄	1	4	yes	—
44	C₂×C₁₆×C₅₃	—	2	4	no	—
45	C₂×C₁₆×C₅₃	—	2	2	no	—
46	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	—	2	2	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	1	no	—
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₅₃	#41	1	2	yes	—
69	C₂²⋊Dic₅₃	—	2	4	no	—
70	C₄×Dic₅₃	—	2	2	no	—
71	C₄×Dic₅₃	—	2	2	no	—
72	C₂×#88322	—	2	2	no	—
73	C₂×#88322	—	2	2	no	—
74	C₂×#88322	C₂×C₈	1	4	yes	—
75	C₁₆×C₅₃	—	2	4	no	—
76	#11719908	—	4	1	no	—
77	#11719908	—	4	1	no	—
78	C₂³×C₅₃	—	2	2	no	—
79	C₂³×C₅₃	—	2	2	no	—
80	C₂³×C₅₃	—	4	1	no	—
81	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
82	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
83	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#131	1	2	yes	—
97	C₂×Dic₅₃	—	2	4	no	—
98	C₂×Dic₅₃	—	4	2	no	—
99	C₂×Dic₅₃	—	4	2	no	—
100	C₈×C₅₃	—	2	1	no	—
101	C₈×C₅₃	—	2	1	no	—
102	C₈×C₅₃	#120	1	1	yes	characteristic
103	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
104	C₈×C₅₃	#123	1	1	yes	characteristic
105	#88322	—	2	4	no	—
106	#12914	—	53	1	no	maximal, Sylow
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	4	4	no	—
115	C₂²×C₅₃	#364	1	2	yes	complemented
116	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
117	C₂²×C₅₃	#435	1	2	yes	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	#385	1	1	yes	characteristic
121	C₄×C₅₃	#424	1	1	yes	characteristic
122	C₄×C₅₃	C₈×D₄	1	2	yes	—
123	Dic₅₃	—	4	4	no	—
124	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
125	C₂×#364	—	53	2	no	—
126	#2112	—	53	1	no	—
127	C₈×C₂²⋊C₄	—	53	1	no	—
128	#1376	—	53	2	no	—
129	C₂×C₅₃	—	2	2	no	—
130	C₂×C₅₃	—	2	2	no	—
131	C₂×C₅₃	—	4	1	no	—
132	C₂×C₅₃	#1850	1	1	yes	characteristic
133	C₂×C₅₃	C₂×#364	1	1	yes	characteristic
134	C₂×C₅₃	#2153	1	1	yes	characteristic
135	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
136	C₂×M₃₂	—	2	1	no	—
137	C₂×M₃₂	—	2	4	no	—
138	C₂×M₃₂	—	53	1	no	—
139	C₂×M₃₂	—	53	1	no	—
140	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
141	C₂²×C₁₆	—	2	1	no	—
142	C₂×C₂²⋊C₈	—	53	1	no	—
143	C₂×C₄×C₈	—	53	2	no	—
144	C₄×C₂²⋊C₄	—	53	1	no	—
145	#364	—	106	4	no	—
146	C₂²⋊C₁₆	—	106	1	no	—
147	C₂²⋊C₁₆	—	106	1	no	—
148	#336	—	53	2	no	—
149	C₅₃	#12914	1	1	yes	characteristic, complemented, Sylow
150	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	53	2	no	—
154	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
155	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
156	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
157	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	—	106	1	no	—
161	C₂×C₄²	—	53	2	no	—
162	M₃₂	—	2	4	no	—
163	M₃₂	—	2	2	no	—
164	M₃₂	—	106	2	no	—
165	M₃₂	—	106	2	no	—
166	M₃₂	C₂²⋊D₅₃	1	4	yes	—
167	C₂×C₁₆	—	2	4	no	—
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	106	1	no	—
170	C₂×C₁₆	—	106	1	no	—
171	C₂²⋊C₈	—	53	2	no	—
172	C₂²⋊C₈	—	53	2	no	—
173	C₄×C₈	—	106	2	no	—
174	C₄×C₈	—	106	2	no	—
175	#95	—	106	1	no	—
176	C₂⁴	#11719912	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	106	2	no	—
180	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
182	C₂²×C₄	#11719912	1	1	yes	characteristic
183	C₂²×C₄	#11719919	1	1	yes	characteristic
184	C₂²×C₄	#11719919	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	53	4	no	—
194	C₂×C₈	—	106	2	no	—
195	C₂×C₈	—	106	2	no	—
196	C₂×C₈	#11719919	1	1	yes	characteristic
197	C₂×C₈	#11719919	1	1	yes	characteristic
198	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
199	C₂×C₈	#11719945	1	2	yes	—
200	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
201	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
202	C₂²⋊C₄	—	106	4	no	—
203	C₄²	—	106	2	no	—
204	C₄²	—	106	2	no	—
205	C₁₆	—	2	4	no	—
206	C₁₆	—	212	1	no	—
207	C₁₆	—	212	1	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	1	no	—
211	C₂³	#13133146	1	1	yes	characteristic
212	C₂³	#13133146	1	1	yes	characteristic
213	C₂³	C₂×#11719912	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	106	4	no	—
222	C₂×C₄	—	212	2	no	—
223	C₂×C₄	—	212	2	no	—
224	C₂×C₄	#13133146	1	1	yes	characteristic
225	C₂×C₄	#13133146	1	1	yes	characteristic
226	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
227	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
228	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
229	C₂×C₄	#13133223	1	2	yes	—
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	106	4	no	—
233	C₈	#13133186	1	1	yes	characteristic
234	C₈	#13133196	1	2	yes	—
235	C₈	#13133200	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	#16810443	1	2	yes	complemented
245	C₂²	C₂×#13133146	1	1	yes	characteristic
246	C₂²	#16810694	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	212	4	no	—
250	C₄	#16810496	1	1	yes	characteristic
251	C₄	#16810649	1	1	yes	characteristic
252	C₄	C₈×C₂²⋊D₅₃	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	4	1	no	—
256	C₂	#20274893	1	1	yes	characteristic
257	C₂	C₂×#16810443	1	1	yes	characteristic
258	C₂	#20276480	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	#12914
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#16810443	C₂	semidirect
4. #20272796	C₂	semidirect
115. C₂²×C₅₃	#364	semidirect
149. C₅₃	#12914	semidirect
244. C₂²	#16810443	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810443	2
#20276326	2
C₈×C₂²⋊Dic₅₃	2
#20272796	2
C₂²×M₃₂×C₅₃	2
#12914	53