Subgroups — #27537138

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811039	C₂	1	1	yes	maximal, characteristic
2	#20283312	C₂	1	2	yes	maximal
3	#20276504	C₂	1	2	yes	maximal, complemented
4	#20276500	C₂	1	4	yes	maximal, complemented
5	#20276498	C₂	1	2	yes	maximal, complemented
6	C₂×#16810451	C₂	1	1	yes	maximal, characteristic, complemented
7	#367×D₅₃	C₂	1	2	yes	maximal, complemented
8	C₅₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133223	C₂²	1	1	yes	characteristic
10	#16811103	—	2	2	no	—
11	#16811039	C₄	1	8	yes	complemented
12	C₂×#13133157	C₂²	1	1	yes	characteristic
13	C₂×#13133156	C₂²	1	2	yes	—
14	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
15	#16810666	—	2	2	no	—
16	#16810662	—	2	4	no	—
17	#16810660	C₂²	1	4	yes	—
18	C₂×#13133151	C₂²	1	2	yes	—
19	#107×D₅₃	C₂²	1	4	yes	—
20	#16810451	C₂²	1	2	yes	—
21	#16810451	C₂²	1	2	yes	—
22	#16810450	C₂²	1	4	yes	—
23	#16810448	C₂²	1	4	yes	—
24	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#427	C₂²	1	2	yes	—
26	C₅₃×#367	C₂²	1	1	yes	characteristic
27	C₅₃×#367	C₂²	1	1	yes	characteristic
28	C₅₃×#366	C₂²	1	2	yes	—
29	C₂×#11719945	D₄	1	1	yes	characteristic
30	#13133223	C₂×C₄	1	4	yes	—
31	#13133223	Q₈	1	4	yes	—
32	C₂×#11719917	D₄	1	1	yes	characteristic
33	C₂×#11719912	D₄	1	2	yes	—
34	C₂×C₈×D₅₃	C₂³	1	2	yes	—
35	C₄×Dic₁₀₆	—	4	2	no	—
36	C₄×D₂₁₂	—	4	2	no	—
37	#13133177	—	4	4	no	—
38	#13133175	—	2	4	no	—
39	C₂×C₄⋊Dic₅₃	—	2	2	no	—
40	C₄⋊C₄×D₅₃	—	2	4	no	—
41	#13133157	C₂×C₄	1	4	yes	complemented
42	#13133156	C₂×C₄	1	8	yes	complemented
43	C₂×#11719908	C₂³	1	2	yes	—
44	C₁₆×D₅₃	C₂×C₄	1	8	yes	complemented
45	#13133151	—	2	2	no	—
46	#13133151	C₂³	1	4	yes	—
47	#13133150	—	2	4	no	—
48	#13133148	—	2	4	no	—
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	#117×C₅₃	—	2	2	no	—
51	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
52	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
54	#107×C₅₃	C₂³	1	2	yes	—
55	#107×C₅₃	C₂³	1	2	yes	—
56	#106×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₂²⋊D₅₃	D₈	1	2	yes	—
59	C₂²×Dic₅₃	—	4	2	no	—
60	#11719945	C₄⋊C₄	1	4	yes	—
61	#11719945	Q₁₆	1	4	yes	—
62	C₂×D₂₁₂	D₈	1	1	yes	characteristic
63	C₂×C₄×D₅₃	—	4	4	no	—
64	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
65	C₂×Dic₁₀₆	D₈	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₅₃	—	2	2	no	—
71	C₄⋊C₄×C₅₃	—	2	2	no	—
72	C₂²⋊C₄×C₅₃	—	4	2	no	—
73	C₄²×C₅₃	—	4	2	no	—
74	#11719921	—	4	4	no	—
75	C₄⋊Dic₅₃	—	2	4	no	—
76	C₄⋊Dic₅₃	—	4	2	no	—
77	#11719919	—	4	4	no	—
78	C₄×Dic₅₃	—	4	4	no	—
79	#11719917	C₄⋊C₄	1	4	yes	—
80	C₂×#88322	C₂×D₄	1	2	yes	—
81	#11719912	C₄⋊C₄	1	8	yes	—
82	C₈×D₅₃	C₂²×C₄	1	4	yes	—
83	C₈×D₅₃	C₂×Q₈	1	4	yes	—
84	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
85	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
86	#11719908	C₂²×C₄	1	4	yes	—
87	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
88	C₂²×D₅₃	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
94	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
95	C₂²⋊D₅₃	#107	1	8	yes	—
96	C₂×Dic₅₃	—	4	4	no	—
97	C₂×Dic₅₃	—	16	2	no	—
98	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
99	D₂₁₂	#107	1	4	yes	—
100	C₄×D₅₃	—	16	4	no	—
101	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
102	C₄×D₅₃	C₂×Q₁₆	1	4	yes	—
103	Dic₁₀₆	#107	1	4	yes	—
104	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
107	#88322	C₂×C₄⋊C₄	1	4	yes	—
108	#33411	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	—	2	2	no	—
110	C₂²×C₅₃	C₂×#107	1	2	yes	—
111	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
112	C₂×D₅₃	—	2	4	no	—
113	C₂×D₅₃	C₂×#107	1	4	yes	—
114	C₄×C₅₃	—	8	2	no	—
115	C₄×C₅₃	—	8	2	no	—
116	C₄×C₅₃	C₂×#107	1	2	yes	—
117	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
119	Dic₅₃	—	8	4	no	—
120	Dic₅₃	C₂×#107	1	4	yes	—
121	C₂×#504	—	53	1	no	—
122	#2913	—	53	2	no	—
123	#2159	—	53	2	no	—
124	#2159	—	53	4	no	—
125	#2158	—	53	2	no	—
126	#2158	D₅₃	1	1	yes	characteristic, complemented
127	C₂×#367	—	53	1	no	—
128	C₂×#367	—	53	2	no	—
129	C₂×C₅₃	—	2	2	no	—
130	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
131	C₂×C₅₃	#3416	1	2	yes	—
132	D₅₃	—	2	4	no	—
133	C₂×#131	—	53	1	no	—
134	#520	—	106	2	no	—
135	#504	—	53	8	no	—
136	C₂×M₃₂	—	53	1	no	—
137	C₂×M₃₂	—	53	2	no	—
138	C₂²×C₁₆	—	53	2	no	—
139	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
140	#428	—	106	4	no	—
141	#428	—	106	2	no	—
142	#427	—	53	4	no	—
143	#427	C₂×D₅₃	1	2	yes	—
144	C₂×#107	—	53	2	no	—
145	C₂×#107	—	53	4	no	—
146	#367	—	53	2	no	—
147	#367	—	53	2	no	—
148	#367	—	53	4	no	—
149	#367	C₂×D₅₃	1	1	yes	characteristic
150	#367	C₂×D₅₃	1	1	yes	characteristic
151	#366	—	53	4	no	—
152	#366	C₂×D₅₃	1	2	yes	—
153	C₅₃	#33411	1	1	yes	characteristic, complemented, Sylow
154	C₂×#41	—	53	1	no	—
155	#131	—	53	4	no	—
156	#131	—	53	4	no	—
157	C₂×M₁₆	—	53	1	no	—
158	C₂×M₁₆	—	53	2	no	—
159	C₂²×C₈	—	53	2	no	—
160	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
161	C₄×Q₈	—	212	2	no	—
162	C₄×D₄	—	212	2	no	—
163	C₄×D₄	—	212	4	no	—
164	#117	—	2	2	no	—
165	#117	—	106	4	no	—
166	C₂×C₄⋊C₄	—	106	4	no	—
167	C₂×C₄⋊C₄	—	106	2	no	—
168	M₃₂	—	53	4	no	—
169	M₃₂	—	53	8	no	—
170	C₂×C₁₆	—	53	2	no	—
171	C₂×C₁₆	—	53	8	no	—
172	C₂×C₁₆	C₄×D₅₃	1	4	yes	complemented
173	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
174	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
175	#107	—	53	4	no	—
176	#107	—	106	4	no	—
177	#107	—	106	2	no	—
178	#107	C₂²×D₅₃	1	2	yes	—
179	#107	C₂²×D₅₃	1	2	yes	—
180	#106	—	2	2	no	—
181	#106	—	106	4	no	—
182	#41	—	53	4	no	—
183	#41	—	53	4	no	—
184	C₂×Q₈	—	53	1	no	—
185	C₂×D₄	—	53	2	no	—
186	C₂×D₄	—	53	1	no	—
187	C₂²×C₄	—	53	2	no	—
188	C₂²×C₄	—	212	4	no	—
189	C₂²×C₄	—	212	2	no	—
190	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
191	M₁₆	—	53	4	no	—
192	M₁₆	—	53	8	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	53	4	no	—
195	C₂×C₈	—	53	4	no	—
196	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
197	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
198	C₂×C₈	Q₈×D₅₃	1	2	yes	—
199	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	106	4	no	—
203	C₄⋊C₄	—	212	4	no	—
204	C₄⋊C₄	—	212	2	no	—
205	C₂²⋊C₄	—	4	2	no	—
206	C₂²⋊C₄	—	212	4	no	—
207	C₄²	—	4	2	no	—
208	C₄²	—	212	4	no	—
209	C₁₆	—	53	4	no	—
210	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
211	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
212	C₂³	—	53	2	no	—
213	C₂³	D₈×D₅₃	1	1	yes	characteristic
214	Q₈	—	53	4	no	—
215	D₄	—	53	8	no	—
216	D₄	—	53	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	53	2	no	—
220	C₂×C₄	—	53	4	no	—
221	C₂×C₄	—	53	4	no	—
222	C₂×C₄	—	848	4	no	—
223	C₂×C₄	—	848	2	no	—
224	C₂×C₄	—	212	4	no	—
225	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
226	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
227	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
228	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
229	C₈	—	53	4	no	—
230	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
231	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
232	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
233	C₂²	—	2	2	no	—
234	C₂²	—	53	4	no	—
235	C₂²	—	106	4	no	—
236	C₂²	#107×D₅₃	1	2	yes	—
237	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
238	C₄	—	8	2	no	—
239	C₄	—	8	2	no	—
240	C₄	—	53	4	no	—
241	C₄	—	424	4	no	—
242	C₄	#107×D₅₃	1	2	yes	—
243	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
244	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
245	C₂	—	2	2	no	—
246	C₂	—	106	4	no	—
247	C₂	C₂×#107×D₅₃	1	1	yes	characteristic
248	C₂	#20287858	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2158	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#11719945	D₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2158	D₅₃
hypocenter	C₅₃	#33411
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #20276504	C₂	semidirect
4. #20276500	C₂	semidirect
5. #20276498	C₂	semidirect
6. C₂×#16810451	C₂	semidirect
7. #367×D₅₃	C₂	semidirect
8. C₅₃×#2158	C₂	semidirect
11. #16811039	C₄	semidirect
41. #13133157	C₂×C₄	semidirect
42. #13133156	C₂×C₄	semidirect
44. C₁₆×D₅₃	C₂×C₄	semidirect
51. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
126. #2158	D₅₃	semidirect
153. C₅₃	#33411	semidirect
172. C₂×C₁₆	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811039	2
#20283312	2
#20276504	2
#20276500	2
#20276498	2
C₂×#16810451	2
#367×D₅₃	2
C₅₃×#2158	2
#33411	53