Subgroups — #27283369

Order 13568.

442 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811028	C₂	1	1	yes	maximal, characteristic
2	#20275662	C₂	1	2	yes	maximal
3	#20274582	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810371	C₂	1	2	yes	maximal
5	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	C₂²	1	2	yes	—
7	#16811028	—	2	2	no	—
8	#16811028	C₄	1	4	yes	complemented
9	C₂×#13133151	—	2	1	no	—
10	C₂×#13133151	—	2	1	no	—
11	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₂×#13133127	—	2	1	no	—
13	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810386	—	2	4	no	—
15	#16810377	—	2	1	no	—
16	#16810371	—	2	4	no	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#334	—	2	2	no	—
20	C₂²×Dic₁₀₆	—	2	2	no	—
21	C₂²×#88322	—	2	1	no	—
22	#13133217	—	2	2	no	—
23	#13133217	—	2	2	no	—
24	#13133217	—	2	2	no	—
25	#13133217	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×C₄⋊Dic₅₃	—	2	2	no	—
28	C₂×C₄⋊Dic₅₃	D₄	1	2	yes	—
29	C₂×C₄×Dic₅₃	—	2	1	no	—
30	#13133151	—	2	4	no	—
31	#13133151	—	2	4	no	—
32	C₄⋊#88322	—	2	1	no	—
33	C₄⋊#88322	D₄	1	2	yes	—
34	#13133127	—	4	2	no	—
35	#13133124	—	2	1	no	—
36	C₂²×C₈×C₅₃	—	2	2	no	—
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₅₃	D₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
47	C₂²×Dic₅₃	—	2	2	no	—
48	C₂²×Dic₅₃	—	4	2	no	—
49	C₂×Dic₁₀₆	—	2	4	no	—
50	C₂×Dic₁₀₆	—	4	2	no	—
51	C₂×Dic₁₀₆	—	8	4	no	—
52	C₂×C₈×C₅₃	—	2	1	no	—
53	C₂×C₈×C₅₃	—	2	1	no	—
54	C₂×C₈×C₅₃	—	4	2	no	—
55	C₂×C₈×C₅₃	—	4	4	no	—
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
59	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
60	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₅₃	#41	1	1	yes	characteristic
64	C₄⋊Dic₅₃	—	2	4	no	—
65	C₄⋊Dic₅₃	—	2	4	no	—
66	C₄⋊Dic₅₃	—	2	4	no	—
67	C₄⋊Dic₅₃	—	4	2	no	—
68	C₄⋊Dic₅₃	—	4	2	no	—
69	C₄×Dic₅₃	—	4	1	no	—
70	C₄×Dic₅₃	—	4	1	no	—
71	C₂×#88322	—	2	2	no	—
72	C₂×#88322	—	4	1	no	—
73	C₂×#88322	—	4	1	no	—
74	C₂³×C₅₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#137	1	4	yes	—
85	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
87	C₂×Dic₅₃	—	4	4	no	—
88	C₂×Dic₅₃	—	4	2	no	—
89	C₂×Dic₅₃	—	4	2	no	—
90	C₂×Dic₅₃	—	4	2	no	—
91	C₂×Dic₅₃	—	8	4	no	—
92	Dic₁₀₆	—	8	4	no	—
93	Dic₁₀₆	—	8	4	no	—
94	C₈×C₅₃	—	2	2	no	—
95	C₈×C₅₃	—	8	2	no	—
96	C₈×C₅₃	#103	1	4	yes	—
97	#88322	—	4	2	no	—
98	#9739	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	#390	1	1	yes	characteristic
100	C₂²×C₅₃	#461	1	1	yes	characteristic
101	C₂²×C₅₃	#462	1	1	yes	characteristic
102	C₂²×C₅₃	#470	1	2	yes	—
103	C₂²×C₅₃	#472	1	2	yes	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
107	C₄×C₅₃	#419	1	1	yes	characteristic
108	C₄×C₅₃	#441	1	2	yes	—
109	C₄×C₅₃	#461	1	1	yes	characteristic
110	C₄×C₅₃	#462	1	1	yes	characteristic
111	C₄×C₅₃	#497	1	2	yes	—
112	Dic₅₃	—	8	2	no	—
113	Dic₅₃	—	8	4	no	—
114	C₂×#500	—	53	1	no	—
115	#1987	—	53	2	no	—
116	#1770	—	53	1	no	—
117	C₂×#334	D₅₃	1	1	yes	characteristic
118	C₂×#333	—	53	2	no	—
119	C₂×C₅₃	#1695	1	2	yes	—
120	C₂×C₅₃	#1696	1	2	yes	—
121	C₂×C₅₃	#1971	1	1	yes	characteristic
122	C₂×C₅₃	#1978	1	1	yes	characteristic
123	C₂×C₅₃	#1981	1	1	yes	characteristic
124	C₂×C₄⋊Q₈	—	53	2	no	—
125	#500	—	53	4	no	—
126	#500	—	106	2	no	—
127	C₂×#107	—	106	1	no	—
128	C₂×#107	—	106	1	no	—
129	C₂×C₄⋊C₈	—	53	1	no	—
130	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
131	C₂×#97	—	106	1	no	—
132	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
133	C₄×C₄⋊C₄	—	53	1	no	—
134	#340	—	106	4	no	—
135	#336	—	106	1	no	—
136	#334	—	2	2	no	—
137	#333	—	106	4	no	—
138	C₅₃	#9739	1	1	yes	characteristic, complemented, Sylow
139	C₂²×Q₈	—	106	2	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	106	1	no	—
142	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
143	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
144	C₄⋊Q₈	—	53	4	no	—
145	C₄⋊Q₈	—	106	2	no	—
146	C₄⋊Q₈	—	106	2	no	—
147	C₄⋊Q₈	—	106	2	no	—
148	C₂×C₄⋊C₄	—	53	2	no	—
149	C₂×C₄⋊C₄	—	106	2	no	—
150	C₂×C₄⋊C₄	—	106	2	no	—
151	C₂×C₄²	—	106	1	no	—
152	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
153	#107	—	106	4	no	—
154	#107	—	106	4	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	53	2	no	—
158	C₄⋊C₈	—	106	1	no	—
159	#97	—	212	2	no	—
160	C₄×C₈	C₄×D₅₃	1	2	yes	—
161	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
162	#95	—	106	1	no	—
163	C₂×Q₈	—	106	4	no	—
164	C₂×Q₈	—	424	4	no	—
165	C₂×Q₈	—	212	2	no	—
166	C₂²×C₄	—	106	2	no	—
167	C₂²×C₄	—	212	2	no	—
168	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
169	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
170	C₂²×C₄	#11719949	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	4	no	—
175	C₂×C₈	—	106	2	no	—
176	C₂×C₈	—	212	1	no	—
177	C₂×C₈	—	212	1	no	—
178	C₂×C₈	#11719913	1	2	yes	—
179	C₂×C₈	Dic₂₁₂	1	2	yes	—
180	C₂×C₈	#11719921	1	2	yes	—
181	C₂×C₈	#11719921	1	2	yes	—
182	C₄⋊C₄	—	106	4	no	—
183	C₄⋊C₄	—	106	4	no	—
184	C₄⋊C₄	—	106	4	no	—
185	C₄⋊C₄	—	212	2	no	—
186	C₄⋊C₄	—	212	2	no	—
187	C₄²	—	212	1	no	—
188	C₄²	—	212	1	no	—
189	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
190	C₄²	D₄×D₅₃	1	1	yes	characteristic
191	C₄²	#11719949	1	1	yes	characteristic
192	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
193	C₂³	#13133192	1	1	yes	characteristic
194	Q₈	—	424	4	no	—
195	Q₈	—	424	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	424	4	no	—
198	C₂×C₄	—	212	4	no	—
199	C₂×C₄	—	212	2	no	—
200	C₂×C₄	—	212	2	no	—
201	C₂×C₄	—	212	2	no	—
202	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
203	C₂×C₄	#13133179	1	1	yes	characteristic
204	C₂×C₄	#13133179	1	1	yes	characteristic
205	C₂×C₄	#13133192	1	1	yes	characteristic
206	C₂×C₄	#13133206	1	1	yes	characteristic
207	C₂×C₄	#13133213	1	1	yes	characteristic
208	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
209	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
210	C₂×C₄	#13133246	1	4	yes	—
211	C₂×C₄	#13133250	1	1	yes	characteristic
212	C₂×C₄	#13133250	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	8	2	no	—
215	C₈	—	212	2	no	—
216	C₈	#13133142	1	4	yes	—
217	C₂²	#16810524	1	1	yes	characteristic
218	C₂²	#16810829	1	1	yes	characteristic
219	C₂²	#16810834	1	1	yes	characteristic
220	C₂²	#16810882	1	2	yes	—
221	C₂²	#16810894	1	2	yes	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	424	2	no	—
225	C₄	—	424	4	no	—
226	C₄	C₂×#13133142	1	1	yes	characteristic
227	C₄	#16810633	1	1	yes	characteristic
228	C₄	#16810729	1	2	yes	—
229	C₄	#16810829	1	1	yes	characteristic
230	C₄	#16810834	1	1	yes	characteristic
231	C₄	#16811014	1	2	yes	—
232	C₂	#20274239	1	2	yes	—
233	C₂	#20274244	1	2	yes	—
234	C₂	#20275584	1	1	yes	characteristic
235	C₂	#20275623	1	1	yes	characteristic
236	C₂	#20275637	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133192
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#334	C₂
socle, prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#334	D₅₃
hypocenter	C₅₃	#9739
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. #16811028	C₄	semidirect
138. C₅₃	#9739	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811028	2
#20275662	2
#20274582	2
C₂×#16810371	2
C₂×C₅₃×#334	2
#9739	53