Subgroups — #27282760

Order 13568.

370 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275476	C₂	1	2	yes	maximal
2	#20274582	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
4	#20272598	C₂	1	2	yes	maximal
5	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133148	—	2	2	no	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
9	C₂×#13133127	—	2	1	no	—
10	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
11	#16810385	—	2	2	no	—
12	#16810377	—	2	1	no	—
13	#16810376	C₄	1	4	yes	complemented
14	C₂×#106×C₅₃	—	2	1	no	—
15	C₂×#106×C₅₃	—	2	1	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#384	C₂²	1	2	yes	—
18	C₂²×#88322	—	2	1	no	—
19	C₂²×#88322	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	D₄	1	2	yes	—
21	C₂×C₄×Dic₅₃	—	2	1	no	—
22	#13133148	—	4	4	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	1	no	—
25	C₄⋊#88322	C₂×C₄	1	4	yes	—
26	C₄⋊#88322	Q₈	1	2	yes	—
27	#13133127	—	4	2	no	—
28	#13133124	—	2	1	no	—
29	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	#106×C₅₃	—	4	2	no	—
36	#106×C₅₃	—	4	2	no	—
37	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₃	Q₈	1	2	yes	—
39	C₂²×C₄×C₅₃	—	4	2	no	—
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂²×Dic₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
49	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
50	C₄⋊C₄×C₅₃	—	4	2	no	—
51	C₄⋊C₄×C₅₃	—	4	2	no	—
52	C₄⋊C₄×C₅₃	—	4	2	no	—
53	C₄⋊C₄×C₅₃	—	4	2	no	—
54	C₄⋊C₄×C₅₃	—	8	2	no	—
55	C₄²×C₅₃	C₂²×C₄	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	—	4	1	no	—
62	C₄×Dic₅₃	—	4	1	no	—
63	C₂×#88322	—	2	4	no	—
64	C₂×#88322	—	2	2	no	—
65	C₂×#88322	—	4	1	no	—
66	C₂×#88322	—	4	2	no	—
67	C₂×#88322	—	4	2	no	—
68	C₂×#88322	—	4	1	no	—
69	C₂³×C₅₃	#121	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	8	2	no	—
72	C₂×C₄×C₅₃	—	8	2	no	—
73	C₂×C₄×C₅₃	—	8	2	no	—
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
84	C₂×C₄×C₅₃	#137	1	2	yes	—
85	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
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	#106	1	4	yes	complemented
91	#88322	—	4	4	no	—
92	#88322	—	4	2	no	—
93	#9618	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	#389	1	1	yes	characteristic
95	C₂²×C₅₃	#460	1	1	yes	characteristic
96	C₂²×C₅₃	#461	1	1	yes	characteristic
97	C₂²×C₅₃	#468	1	2	yes	—
98	C₂²×C₅₃	#471	1	2	yes	—
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	16	2	no	—
102	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
103	C₄×C₅₃	#428	1	1	yes	characteristic
104	C₄×C₅₃	#441	1	1	yes	characteristic
105	C₄×C₅₃	#442	1	1	yes	characteristic
106	C₄×C₅₃	#475	1	1	yes	characteristic
107	C₄×C₅₃	#476	1	1	yes	characteristic
108	C₄×C₅₃	#501	1	2	yes	—
109	Dic₅₃	—	8	2	no	—
110	#1948	—	53	2	no	—
111	#1844	D₅₃	1	1	yes	characteristic
112	#1770	—	53	1	no	—
113	C₂×#335	—	53	1	no	—
114	#1296	—	53	2	no	—
115	C₂×C₅₃	#1686	1	2	yes	—
116	C₂×C₅₃	#1691	1	2	yes	—
117	C₂×C₅₃	#1943	1	1	yes	characteristic
118	C₂×C₅₃	#1948	1	1	yes	characteristic
119	C₂×C₅₃	#1951	1	1	yes	characteristic
120	C₂×#106	—	2	1	no	—
121	C₂×#106	—	2	1	no	—
122	C₂×#106	—	106	2	no	—
123	C₂×C₄⋊C₈	—	53	1	no	—
124	C₂×C₄⋊C₈	—	53	2	no	—
125	C₂×#97	—	106	1	no	—
126	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
127	#384	C₂×D₅₃	1	2	yes	—
128	C₄×C₄⋊C₄	—	53	1	no	—
129	#340	—	106	2	no	—
130	#336	—	106	1	no	—
131	#335	—	53	4	no	—
132	C₅₃	#9618	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₈	—	106	2	no	—
134	C₂²×C₈	—	106	1	no	—
135	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
136	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	53	2	no	—
141	C₂×C₄²	—	106	1	no	—
142	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
143	#106	—	4	2	no	—
144	#106	—	4	2	no	—
145	#106	—	212	4	no	—
146	C₄⋊C₈	—	53	4	no	—
147	C₄⋊C₈	—	53	2	no	—
148	C₄⋊C₈	—	106	2	no	—
149	C₄⋊C₈	—	106	1	no	—
150	#97	—	212	2	no	—
151	C₄×C₈	Dic₁₀₆	1	2	yes	—
152	C₄×C₈	C₄×D₅₃	1	2	yes	—
153	#95	—	106	1	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	106	2	no	—
156	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
157	C₂²×C₄	#11719947	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	106	4	no	—
162	C₂×C₈	—	106	2	no	—
163	C₂×C₈	—	212	1	no	—
164	C₂×C₈	—	212	1	no	—
165	C₂×C₈	—	212	2	no	—
166	C₂×C₈	—	212	2	no	—
167	C₂×C₈	#11719919	1	2	yes	—
168	C₂×C₈	#11719919	1	2	yes	—
169	C₂×C₈	#11719923	1	2	yes	—
170	C₂×C₈	#11719924	1	2	yes	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	8	2	no	—
176	C₄⋊C₄	—	106	2	no	—
177	C₄⋊C₄	—	106	2	no	—
178	C₄²	—	212	1	no	—
179	C₄²	—	212	1	no	—
180	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
181	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
182	C₄²	Q₈×D₅₃	1	1	yes	characteristic
183	C₄²	#11719949	1	1	yes	characteristic
184	C₂³	#13133188	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	8	2	no	—
187	C₂×C₄	—	8	2	no	—
188	C₂×C₄	—	8	2	no	—
189	C₂×C₄	—	212	2	no	—
190	C₂×C₄	—	212	2	no	—
191	C₂×C₄	—	212	2	no	—
192	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
193	C₂×C₄	#13133179	1	1	yes	characteristic
194	C₂×C₄	#13133182	1	1	yes	characteristic
195	C₂×C₄	#13133193	1	1	yes	characteristic
196	C₂×C₄	#13133208	1	1	yes	characteristic
197	C₂×C₄	#13133216	1	1	yes	characteristic
198	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
199	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
200	C₂×C₄	#13133243	1	1	yes	characteristic
201	C₂×C₄	#13133244	1	2	yes	—
202	C₂×C₄	#13133246	1	2	yes	—
203	C₂×C₄	#13133248	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	212	2	no	—
206	C₈	—	212	4	no	—
207	C₈	#13133148	1	4	yes	complemented
208	C₂²	#16810519	1	1	yes	characteristic
209	C₂²	#16810820	1	1	yes	characteristic
210	C₂²	#16810825	1	1	yes	characteristic
211	C₂²	#16810866	1	2	yes	—
212	C₂²	#16810883	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	16	2	no	—
216	C₄	—	424	2	no	—
217	C₄	C₂×#13133148	1	1	yes	characteristic
218	C₄	#16810663	1	1	yes	characteristic
219	C₄	#16810729	1	1	yes	characteristic
220	C₄	#16810734	1	1	yes	characteristic
221	C₄	#16810905	1	1	yes	characteristic
222	C₄	#16810910	1	1	yes	characteristic
223	C₄	#16811030	1	2	yes	—
224	C₂	#20274184	1	2	yes	—
225	C₂	#20274211	1	2	yes	—
226	C₂	#20275451	1	1	yes	characteristic
227	C₂	#20275476	1	1	yes	characteristic
228	C₂	#20275493	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133188
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1844	C₂
socle, prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1844	D₅₃
hypocenter	C₅₃	#9618
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
13. #16810376	C₄	semidirect
90. C₈×C₅₃	#106	semidirect
132. C₅₃	#9618	semidirect
207. C₈	#13133148	semidirect

Maximal subgroups

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

subgroup	index
#20275476	2
#20274582	2
C₂×#16810376	2
#20272598	2
C₅₃×#1844	2
#9618	53