Subgroups — #27539446

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811000	C₂	1	1	yes	maximal, characteristic
2	#20276958	C₂	1	4	yes	maximal, complemented
3	#20276940	C₂	1	4	yes	maximal, complemented
4	#20276430	C₂	1	4	yes	maximal, complemented
5	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄₂₄	—	2	1	no	—
8	C₂²×D₄₂₄	C₂²	1	2	yes	—
9	C₂×#13133214	—	2	1	no	—
10	C₂×#13133157	C₂²	1	2	yes	—
11	#16811000	—	2	1	no	—
12	#16811000	—	2	1	no	—
13	#16811000	C₂²	1	2	yes	—
14	#16811000	C₂²	1	2	yes	—
15	#16810423	C₂²	1	16	yes	complemented
16	#16810399	C₂²	1	2	yes	complemented
17	C₄×#11719908	C₂²	1	2	yes	complemented
18	C₅₃×#533	—	2	1	no	—
19	C₅₃×#500	C₂²	1	2	yes	complemented
20	C₅₃×#499	—	2	1	no	—
21	C₅₃×#499	C₂²	1	2	yes	complemented
22	C₅₃×#498	—	2	1	no	—
23	C₅₃×#427	—	2	1	no	—
24	C₅₃×#427	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×D₂₁₂	—	2	2	no	—
27	C₂²×D₂₁₂	—	4	2	no	—
28	C₂×D₄₂₄	—	2	2	no	—
29	C₂×D₄₂₄	—	2	2	no	—
30	C₂×D₄₂₄	—	2	2	no	—
31	C₂×D₄₂₄	—	2	8	no	—
32	C₂×D₄₂₄	—	4	4	no	—
33	C₂×D₄₂₄	C₂³	1	4	yes	—
34	#13133214	—	2	2	no	—
35	#13133214	—	2	2	no	—
36	#13133214	—	4	2	no	—
37	#13133157	—	2	4	no	—
38	C₂×#11719908	C₂³	1	4	yes	—
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₅₃	—	2	2	no	—
42	#125×C₅₃	—	2	2	no	—
43	C₂²⋊Q₈×C₅₃	—	4	2	no	—
44	C₄×Q₈×C₅₃	—	4	2	no	—
45	#117×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	#107×C₅₃	—	2	2	no	—
48	#107×C₅₃	C₂³	1	4	yes	—
49	#106×C₅₃	—	2	2	no	—
50	#106×C₅₃	—	2	2	no	—
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂³×D₅₃	—	4	2	no	—
56	C₂×Q₈×C₅₃	—	4	2	no	—
57	C₂²×C₄×C₅₃	D₈	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×D₂₁₂	—	2	4	no	—
60	C₂×D₂₁₂	—	4	4	no	—
61	C₂×D₂₁₂	—	4	4	no	—
62	C₂×D₂₁₂	—	4	4	no	—
63	C₂×D₂₁₂	—	4	4	no	—
64	C₂×D₂₁₂	—	4	2	no	—
65	C₂×D₂₁₂	—	8	8	no	—
66	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
67	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	#41	1	4	yes	—
71	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₅₃	—	2	4	no	—
73	C₄⋊C₄×C₅₃	—	4	4	no	—
74	C₄⋊C₄×C₅₃	—	4	2	no	—
75	C₂²⋊C₄×C₅₃	—	4	2	no	—
76	C₄²×C₅₃	—	4	2	no	—
77	C₄²×C₅₃	D₈	1	2	yes	—
78	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
80	D₄₂₄	—	2	8	no	—
81	D₄₂₄	—	4	4	no	—
82	D₄₂₄	—	4	4	no	—
83	D₄₂₄	—	4	4	no	—
84	#11719908	—	2	4	no	—
85	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
86	C₂²×D₅₃	—	4	4	no	—
87	C₂²×D₅₃	—	8	8	no	—
88	C₂²×D₅₃	—	8	4	no	—
89	Q₈×C₅₃	—	8	4	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	#124	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
97	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
98	D₂₁₂	—	4	4	no	—
99	D₂₁₂	—	4	4	no	—
100	D₂₁₂	—	8	8	no	—
101	D₂₁₂	—	8	4	no	—
102	D₂₁₂	—	8	8	no	—
103	C₈×C₅₃	#124	1	2	yes	—
104	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
105	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
106	C₈×C₅₃	C₂×#41	1	2	yes	—
107	#33704	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	—	2	2	no	—
109	C₂²×C₅₃	#486	1	2	yes	—
110	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
111	C₂×D₅₃	—	8	4	no	—
112	C₂×D₅₃	—	8	8	no	—
113	C₂×D₅₃	—	16	8	no	—
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	—	8	4	no	—
116	C₄×C₅₃	#486	1	2	yes	—
117	C₄×C₅₃	#509	1	4	yes	—
118	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
119	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
120	#3160	D₅₃	1	1	yes	characteristic, complemented
121	C₂×#493	—	53	1	no	—
122	#2244	—	53	4	no	—
123	#2240	—	53	4	no	—
124	#2140	—	53	4	no	—
125	C₄×M₃₂	—	53	1	no	—
126	C₂×C₅₃	—	2	2	no	—
127	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
128	C₂×C₅₃	C₂×#509	1	2	yes	—
129	D₅₃	—	16	4	no	—
130	C₂²×D₈	—	53	2	no	—
131	C₂²×D₈	—	106	1	no	—
132	#533	—	2	1	no	—
133	C₂×#127	—	106	1	no	—
134	C₂×M₃₂	—	53	2	no	—
135	#500	C₂×D₅₃	1	2	yes	complemented
136	#499	—	2	1	no	—
137	#499	C₂×D₅₃	1	2	yes	complemented
138	#498	—	2	1	no	—
139	#493	—	53	2	no	—
140	#493	—	53	2	no	—
141	#493	—	106	1	no	—
142	#493	—	106	1	no	—
143	#427	—	2	1	no	—
144	#427	C₂×D₅₃	1	2	yes	—
145	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
146	#357	—	53	16	no	—
147	#346	—	53	2	no	—
148	C₄×C₁₆	—	53	2	no	—
149	C₅₃	#33704	1	1	yes	characteristic, complemented, Sylow
150	C₂²×D₄	—	106	2	no	—
151	C₂²×D₄	—	212	2	no	—
152	C₂×D₈	—	53	4	no	—
153	C₂×D₈	—	106	8	no	—
154	C₂×D₈	—	106	2	no	—
155	C₂×D₈	—	106	2	no	—
156	C₂×D₈	—	106	2	no	—
157	C₂×D₈	—	212	4	no	—
158	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
160	C₄⋊Q₈	—	2	2	no	—
161	#127	—	106	2	no	—
162	#127	—	106	2	no	—
163	#127	—	212	2	no	—
164	#125	—	2	2	no	—
165	C₂²⋊Q₈	—	4	2	no	—
166	C₄×Q₈	—	4	2	no	—
167	#117	—	2	2	no	—
168	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
169	M₃₂	—	106	4	no	—
170	C₂×C₁₆	—	53	4	no	—
171	#107	—	2	2	no	—
172	#107	C₂²×D₅₃	1	4	yes	—
173	#106	—	2	2	no	—
174	#106	—	2	2	no	—
175	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
176	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
177	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
178	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
179	C₂⁴	—	212	2	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×D₄	—	106	4	no	—
182	C₂×D₄	—	424	8	no	—
183	C₂×D₄	—	212	4	no	—
184	C₂×D₄	—	212	4	no	—
185	C₂×D₄	—	212	4	no	—
186	C₂×D₄	—	212	2	no	—
187	C₂×D₄	—	212	4	no	—
188	C₂²×C₄	#11719922	1	2	yes	—
189	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
190	D₈	—	106	8	no	—
191	D₈	—	212	4	no	—
192	D₈	—	212	4	no	—
193	D₈	—	212	4	no	—
194	C₂×C₈	#11719924	1	4	yes	—
195	C₂×C₈	#11719949	1	4	yes	—
196	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
197	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
198	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
199	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	4	4	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₄²	—	4	2	no	—
205	C₄²	#11719922	1	2	yes	—
206	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
207	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
208	C₁₆	—	106	4	no	—
209	C₂³	—	424	8	no	—
210	C₂³	—	424	4	no	—
211	C₂³	—	212	4	no	—
212	C₂³	C₂×#11719922	1	1	yes	characteristic
213	Q₈	—	8	4	no	—
214	D₄	—	424	8	no	—
215	D₄	—	424	8	no	—
216	D₄	—	424	4	no	—
217	D₄	—	212	4	no	—
218	D₄	—	212	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	#13133206	1	2	yes	—
222	C₂×C₄	C₂×#11719922	1	2	yes	—
223	C₂×C₄	C₂×#11719922	1	2	yes	—
224	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
225	C₂×C₄	C₂×#11719924	1	2	yes	—
226	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
227	C₈	#13133206	1	2	yes	—
228	C₈	C₂×#11719924	1	2	yes	—
229	C₈	C₂×#11719924	1	2	yes	—
230	C₈	C₂×#11719949	1	2	yes	—
231	C₂²	—	2	2	no	—
232	C₂²	—	848	8	no	—
233	C₂²	—	424	8	no	—
234	C₂²	—	424	4	no	—
235	C₂²	#16810971	1	2	yes	—
236	C₂²	C₂²×#11719922	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	8	4	no	—
239	C₄	#16810971	1	2	yes	—
240	C₄	#16811058	1	4	yes	—
241	C₄	C₂×#13133206	1	1	yes	characteristic
242	C₄	C₂²×#11719924	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	848	4	no	—
245	C₂	C₂×#16810971	1	1	yes	characteristic
246	C₂	C₂×#16811058	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3160	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#16811000	C₂
hypercenter	#3160	D₅₃
hypocenter	C₅₃	#33704
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #20276958	C₂	semidirect
3. #20276940	C₂	semidirect
4. #20276430	C₂	semidirect
5. C₄×#13133157	C₂	semidirect
6. C₅₃×#3160	C₂	semidirect
15. #16810423	C₂²	semidirect
16. #16810399	C₂²	semidirect
17. C₄×#11719908	C₂²	semidirect
19. C₅₃×#500	C₂²	semidirect
21. C₅₃×#499	C₂²	semidirect
120. #3160	D₅₃	semidirect
135. #500	C₂×D₅₃	semidirect
137. #499	C₂×D₅₃	semidirect
149. C₅₃	#33704	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811000	2
#20276958	2
#20276940	2
#20276430	2
C₄×#13133157	2
C₅₃×#3160	2
#33704	53