Subgroups — #27282652

Order 13568.

378 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275431	C₂	1	2	yes	maximal
2	C₄×#13133151	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
4	#20272554	C₂	1	2	yes	maximal
5	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133151	C₄	1	2	yes	—
7	C₂×#13133146	—	2	2	no	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	#16810386	C₄	1	1	yes	characteristic
12	#16810386	C₄	1	1	yes	characteristic
13	#16810377	—	2	2	no	—
14	#16810345	—	2	2	no	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#335	—	2	2	no	—
18	C₂²×#88322	—	2	1	no	—
19	C₂²×#88322	D₄	1	1	yes	characteristic
20	C₂²×#88322	D₄	1	1	yes	characteristic
21	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₄×Dic₅₃	—	2	2	no	—
24	#13133151	—	2	4	no	—
25	#13133146	—	4	4	no	—
26	C₄×#88322	—	2	1	no	—
27	C₄×#88322	—	2	1	no	—
28	C₄×#88322	—	2	1	no	—
29	C₄×#88322	Q₈	1	2	yes	—
30	#13133124	—	2	2	no	—
31	C₂²×C₈×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	C₄×C₈×C₅₃	D₄	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	—	4	2	no	—
47	C₂×C₈×C₅₃	—	4	2	no	—
48	C₂×C₈×C₅₃	—	4	2	no	—
49	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₅₃	#41	1	1	yes	characteristic
55	C₄⋊Dic₅₃	—	2	2	no	—
56	C₄⋊Dic₅₃	—	2	2	no	—
57	C₄⋊Dic₅₃	—	2	2	no	—
58	C₄⋊Dic₅₃	M₁₆	1	4	yes	—
59	C₄×Dic₅₃	—	4	2	no	—
60	C₄×Dic₅₃	—	4	2	no	—
61	C₂×#88322	—	2	1	no	—
62	C₂×#88322	—	2	1	no	—
63	C₂×#88322	—	2	1	no	—
64	C₂×#88322	—	2	1	no	—
65	C₂×#88322	—	2	2	no	—
66	C₂×#88322	—	4	2	no	—
67	C₂×#88322	D₈	1	2	yes	—
68	C₂×#88322	Q₁₆	1	2	yes	—
69	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	2	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	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
83	C₂×Dic₅₃	—	2	4	no	—
84	C₂×Dic₅₃	—	4	2	no	—
85	C₂×Dic₅₃	—	4	2	no	—
86	C₂×Dic₅₃	—	4	2	no	—
87	C₂×Dic₅₃	—	4	2	no	—
88	C₂×Dic₅₃	—	4	2	no	—
89	C₈×C₅₃	—	2	2	no	—
90	C₈×C₅₃	—	2	2	no	—
91	C₈×C₅₃	—	8	2	no	—
92	#88322	—	2	2	no	—
93	#88322	—	2	2	no	—
94	#88322	—	4	2	no	—
95	#9598	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	#389	1	1	yes	characteristic
97	C₂²×C₅₃	#435	1	2	yes	—
98	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
99	C₂²×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
100	C₂²×C₅₃	#444	1	2	yes	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	#423	1	1	yes	characteristic
104	C₄×C₅₃	#424	1	1	yes	characteristic
105	C₄×C₅₃	#446	1	2	yes	—
106	C₄×C₅₃	#466	1	1	yes	characteristic
107	C₄×C₅₃	#467	1	1	yes	characteristic
108	C₄×C₅₃	#499	1	1	yes	characteristic
109	C₄×C₅₃	#500	1	1	yes	characteristic
110	Dic₅₃	—	4	4	no	—
111	Dic₅₃	—	8	2	no	—
112	#1939	—	53	2	no	—
113	C₄×#107	—	53	1	no	—
114	C₂×#335	D₅₃	1	1	yes	characteristic
115	C₂×#322	—	53	1	no	—
116	#1279	—	53	2	no	—
117	C₂×C₅₃	#1581	1	2	yes	—
118	C₂×C₅₃	#1584	1	2	yes	—
119	C₂×C₅₃	#1940	1	1	yes	characteristic
120	C₂×C₅₃	#1942	1	1	yes	characteristic
121	C₂×C₅₃	#1946	1	1	yes	characteristic
122	C₂×#107	—	53	2	no	—
123	C₂×C₄⋊C₈	—	106	2	no	—
124	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
125	C₂×C₄×C₈	—	53	1	no	—
126	C₂×C₄×C₈	—	53	1	no	—
127	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
128	C₄×C₄⋊C₄	—	53	2	no	—
129	#340	—	53	1	no	—
130	#340	—	53	1	no	—
131	#336	—	106	2	no	—
132	#335	—	2	2	no	—
133	#322	—	106	2	no	—
134	C₅₃	#9598	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	53	1	no	—
137	C₂²×C₈	—	53	1	no	—
138	C₂²×C₈	—	106	1	no	—
139	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
140	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	53	2	no	—
142	C₂×C₄⋊C₄	—	53	2	no	—
143	C₂×C₄²	—	106	2	no	—
144	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
145	#107	—	106	4	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	212	4	no	—
149	C₄×C₈	—	53	2	no	—
150	C₄×C₈	—	106	1	no	—
151	C₄×C₈	—	106	1	no	—
152	C₄×C₈	—	106	1	no	—
153	C₄×C₈	Dic₁₀₆	1	2	yes	—
154	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
155	#95	—	106	2	no	—
156	C₂²×C₄	—	106	2	no	—
157	C₂²×C₄	—	106	2	no	—
158	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
159	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
160	C₂²×C₄	#11719949	1	1	yes	characteristic
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	53	2	no	—
167	C₂×C₈	—	53	2	no	—
168	C₂×C₈	—	106	1	no	—
169	C₂×C₈	—	106	1	no	—
170	C₂×C₈	—	106	1	no	—
171	C₂×C₈	—	106	1	no	—
172	C₂×C₈	—	106	2	no	—
173	C₂×C₈	—	212	2	no	—
174	C₂×C₈	#11719919	1	2	yes	—
175	C₂×C₈	#11719919	1	2	yes	—
176	C₄⋊C₄	—	53	4	no	—
177	C₄⋊C₄	—	106	2	no	—
178	C₄⋊C₄	—	106	2	no	—
179	C₄⋊C₄	—	106	2	no	—
180	C₄²	—	212	2	no	—
181	C₄²	—	212	2	no	—
182	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
183	C₄²	#11719947	1	1	yes	characteristic
184	C₄²	Q₈×D₅₃	1	1	yes	characteristic
185	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
186	C₂³	#13133179	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	106	4	no	—
189	C₂×C₄	—	212	2	no	—
190	C₂×C₄	—	212	2	no	—
191	C₂×C₄	—	212	2	no	—
192	C₂×C₄	—	212	2	no	—
193	C₂×C₄	—	212	2	no	—
194	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
195	C₂×C₄	#13133182	1	1	yes	characteristic
196	C₂×C₄	#13133188	1	1	yes	characteristic
197	C₂×C₄	#13133193	1	1	yes	characteristic
198	C₂×C₄	#13133208	1	1	yes	characteristic
199	C₂×C₄	#13133216	1	1	yes	characteristic
200	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
201	C₂×C₄	#13133223	1	2	yes	—
202	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
203	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
204	C₂×C₄	#13133236	1	1	yes	characteristic
205	C₂×C₄	#13133240	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	8	2	no	—
209	C₈	—	106	2	no	—
210	C₈	—	106	2	no	—
211	C₈	—	212	2	no	—
212	C₂²	#16810519	1	1	yes	characteristic
213	C₂²	#16810693	1	2	yes	—
214	C₂²	#16810705	1	1	yes	characteristic
215	C₂²	#16810717	1	1	yes	characteristic
216	C₂²	#16810744	1	2	yes	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	424	2	no	—
220	C₄	—	212	4	no	—
221	C₄	#16810646	1	1	yes	characteristic
222	C₄	#16810649	1	1	yes	characteristic
223	C₄	#16810752	1	2	yes	—
224	C₄	#16810856	1	1	yes	characteristic
225	C₄	#16810861	1	1	yes	characteristic
226	C₄	#16811024	1	1	yes	characteristic
227	C₄	#16811027	1	1	yes	characteristic
228	C₂	#20273543	1	2	yes	—
229	C₂	#20273564	1	2	yes	—
230	C₂	#20275436	1	1	yes	characteristic
231	C₂	#20275446	1	1	yes	characteristic
232	C₂	#20275466	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	#9598
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
134. C₅₃	#9598	semidirect

Maximal subgroups

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

subgroup	index
#20275431	2
C₄×#13133151	2
C₂×#16810345	2
#20272554	2
C₂×C₅₃×#335	2
#9598	53