Subgroups — #27282664

Order 13568.

378 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275433	C₂	1	2	yes	maximal
2	C₄×#13133149	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
4	#20272555	C₂	1	2	yes	maximal
5	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133149	C₄	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×#13133146	—	2	2	no	—
9	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	#16810386	C₄	1	1	yes	characteristic
11	#16810386	C₄	1	1	yes	characteristic
12	#16810377	—	2	2	no	—
13	#16810376	—	2	2	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#322	—	2	2	no	—
18	C₂²×#88322	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₅₃	—	2	2	no	—
22	#13133149	—	2	4	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	#13133146	—	4	4	no	—
26	#13133124	—	2	2	no	—
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₅₃	—	2	1	no	—
34	C₄×C₈×C₅₃	—	2	1	no	—
35	C₄×C₈×C₅₃	—	2	1	no	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	C₄×C₈×C₅₃	Q₈	1	2	yes	—
38	C₄×C₈×C₅₃	Q₈	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
42	C₂²×Dic₅₃	—	2	2	no	—
43	C₂²×Dic₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	2	1	no	—
50	C₂×C₈×C₅₃	—	2	1	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₅₃	#41	1	1	yes	characteristic
59	C₄⋊Dic₅₃	—	2	2	no	—
60	C₄⋊Dic₅₃	—	2	2	no	—
61	C₄⋊Dic₅₃	—	2	2	no	—
62	C₄⋊Dic₅₃	M₁₆	1	4	yes	—
63	C₄×Dic₅₃	—	4	2	no	—
64	C₄×Dic₅₃	—	4	2	no	—
65	C₂×#88322	—	4	2	no	—
66	C₂×#88322	—	4	2	no	—
67	C₂×#88322	—	4	2	no	—
68	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₅₃	#131	1	2	yes	—
79	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₅₃	#137	1	2	yes	—
81	C₂×Dic₅₃	—	2	4	no	—
82	C₂×Dic₅₃	—	4	2	no	—
83	C₂×Dic₅₃	—	4	2	no	—
84	C₂×Dic₅₃	—	4	2	no	—
85	C₂×Dic₅₃	—	4	2	no	—
86	C₂×Dic₅₃	—	4	2	no	—
87	C₈×C₅₃	—	2	4	no	—
88	C₈×C₅₃	—	2	2	no	—
89	C₈×C₅₃	—	2	2	no	—
90	C₈×C₅₃	—	4	1	no	—
91	C₈×C₅₃	—	4	1	no	—
92	#88322	—	8	2	no	—
93	#9600	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	#389	1	1	yes	characteristic
95	C₂²×C₅₃	#435	1	2	yes	—
96	C₂²×C₅₃	#440	1	2	yes	—
97	C₂²×C₅₃	#443	1	1	yes	characteristic
98	C₂²×C₅₃	#443	1	1	yes	characteristic
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	#423	1	1	yes	characteristic
102	C₄×C₅₃	#424	1	1	yes	characteristic
103	C₄×C₅₃	#446	1	2	yes	—
104	C₄×C₅₃	#469	1	1	yes	characteristic
105	C₄×C₅₃	#470	1	1	yes	characteristic
106	C₄×C₅₃	#501	1	2	yes	—
107	Dic₅₃	—	4	4	no	—
108	Dic₅₃	—	8	2	no	—
109	#1939	—	53	2	no	—
110	C₄×#106	—	53	1	no	—
111	C₂×#335	—	53	1	no	—
112	C₂×#322	D₅₃	1	1	yes	characteristic
113	#1279	—	53	2	no	—
114	C₂×C₅₃	#1585	1	2	yes	—
115	C₂×C₅₃	#1588	1	2	yes	—
116	C₂×C₅₃	#1940	1	1	yes	characteristic
117	C₂×C₅₃	#1944	1	1	yes	characteristic
118	C₂×C₅₃	#1945	1	1	yes	characteristic
119	C₂×#106	—	53	2	no	—
120	C₂×C₄⋊C₈	—	53	2	no	—
121	C₂×C₄⋊C₈	—	106	2	no	—
122	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
125	C₄×C₄⋊C₄	—	53	2	no	—
126	#340	—	53	1	no	—
127	#340	—	53	1	no	—
128	#336	—	106	2	no	—
129	#335	—	106	2	no	—
130	#322	—	2	2	no	—
131	C₅₃	#9600	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	—	2	1	no	—
133	C₂²×C₈	—	106	2	no	—
134	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
135	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
136	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
137	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	53	2	no	—
139	C₂×C₄⋊C₄	—	53	2	no	—
140	C₂×C₄²	—	106	2	no	—
141	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
142	#106	—	106	4	no	—
143	C₄⋊C₈	—	106	2	no	—
144	C₄⋊C₈	—	106	2	no	—
145	C₄⋊C₈	—	212	4	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	—	2	1	no	—
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	Dic₁₀₆	1	2	yes	—
150	C₄×C₈	Dic₁₀₆	1	2	yes	—
151	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
152	#95	—	106	2	no	—
153	C₂²×C₄	—	106	2	no	—
154	C₂²×C₄	—	106	2	no	—
155	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
157	C₂²×C₄	#11719945	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	212	2	no	—
168	C₂×C₈	—	212	2	no	—
169	C₂×C₈	—	212	2	no	—
170	C₂×C₈	#11719919	1	2	yes	—
171	C₂×C₈	#11719919	1	2	yes	—
172	C₄⋊C₄	—	53	4	no	—
173	C₄⋊C₄	—	106	2	no	—
174	C₄⋊C₄	—	106	2	no	—
175	C₄⋊C₄	—	106	2	no	—
176	C₄²	—	212	2	no	—
177	C₄²	—	212	2	no	—
178	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
179	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
180	C₄²	#11719945	1	1	yes	characteristic
181	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
182	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	106	4	no	—
186	C₂×C₄	—	212	2	no	—
187	C₂×C₄	—	212	2	no	—
188	C₂×C₄	—	212	2	no	—
189	C₂×C₄	—	212	2	no	—
190	C₂×C₄	—	212	2	no	—
191	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
192	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
193	C₂×C₄	#13133191	1	1	yes	characteristic
194	C₂×C₄	#13133196	1	1	yes	characteristic
195	C₂×C₄	#13133209	1	1	yes	characteristic
196	C₂×C₄	#13133217	1	1	yes	characteristic
197	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
198	C₂×C₄	#13133225	1	2	yes	—
199	C₂×C₄	#13133245	1	2	yes	—
200	C₂×C₄	#13133250	1	2	yes	—
201	C₈	—	2	4	no	—
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₈	—	4	1	no	—
205	C₈	—	4	1	no	—
206	C₈	—	424	2	no	—
207	C₂²	#16810521	1	1	yes	characteristic
208	C₂²	#16810696	1	2	yes	—
209	C₂²	#16810725	1	2	yes	—
210	C₂²	#16810741	1	1	yes	characteristic
211	C₂²	#16810741	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	424	2	no	—
215	C₄	—	212	4	no	—
216	C₄	#16810646	1	1	yes	characteristic
217	C₄	#16810651	1	1	yes	characteristic
218	C₄	#16810751	1	2	yes	—
219	C₄	#16810876	1	1	yes	characteristic
220	C₄	#16810881	1	1	yes	characteristic
221	C₄	#16811032	1	2	yes	—
222	C₂	#20273574	1	2	yes	—
223	C₂	#20273595	1	2	yes	—
224	C₂	#20275438	1	1	yes	characteristic
225	C₂	#20275458	1	1	yes	characteristic
226	C₂	#20275463	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#9600
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
131. C₅₃	#9600	semidirect

Maximal subgroups

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

subgroup	index
#20275433	2
C₄×#13133149	2
C₂×#16810376	2
#20272555	2
C₂×C₅₃×#322	2
#9600	53