Subgroups — #28330830

Order 13568.

10796 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	#20291711	C₂	1	12	yes	maximal, complemented
2	#20291710	C₂	1	20	yes	maximal, complemented
3	C₂×#16811702	C₂	1	15	yes	maximal, complemented
4	D₅₃⋊#568	C₂	1	15	yes	maximal, complemented
5	#585×D₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811703	C₂²	1	30	yes	complemented
7	#16811702	C₂²	1	20	yes	complemented
8	#16811702	C₂²	1	60	yes	complemented
9	#16811702	C₂²	1	120	yes	complemented
10	#16811701	C₂²	1	90	yes	complemented
11	C₂×#13133305	C₂²	1	30	yes	complemented
12	D₅₃⋊#131	C₂²	1	120	yes	complemented
13	C₂×#13133304	C₂²	1	90	yes	complemented
14	C₂²×#11719936	C₂²	1	15	yes	complemented
15	C₂×D₅₃⋊M₁₆	C₂²	1	45	yes	complemented
16	#16811460	C₂²	1	6	yes	—
17	#16811459	C₄	1	15	yes	—
18	#16811458	C₂²	1	10	yes	—
19	2₋¹⁺⁴×D₅₃	C₄	1	6	yes	—
20	H(5,𝔽₂)×D₅₃	C₄	1	10	yes	—
21	C₂×#41×D₅₃	C₂²	1	15	yes	—
22	C₅₃×#585	C₄	1	1	yes	characteristic
23	#13133305	C₂³	1	40	yes	complemented
24	#13133305	C₂³	1	120	yes	complemented
25	#13133304	C₂³	1	360	yes	complemented
26	#13133304	C₂³	1	120	yes	complemented
27	C₂×#11719936	C₂³	1	15	yes	—
28	C₂×#11719936	C₂³	1	45	yes	—
29	C₂×#11719936	C₂³	1	90	yes	complemented
30	D₅₃⋊M₁₆	C₂³	1	60	yes	—
31	D₅₃⋊M₁₆	C₂³	1	180	yes	complemented
32	C₂×#11719940	C₂³	1	90	yes	complemented
33	C₂×D₅₃⋊C₈	C₂³	1	90	yes	complemented
34	C₂²×#88324	C₂³	1	30	yes	complemented
35	#13133265	C₂×C₄	1	30	yes	—
36	#13133264	C₂×C₄	1	60	yes	—
37	#13133263	C₂×C₄	1	60	yes	—
38	#13133262	C₂×C₄	1	90	yes	—
39	C₂×#11719949	C₂³	1	15	yes	—
40	C₂×#11719945	C₂×C₄	1	45	yes	—
41	C₂×#11719947	C₂³	1	45	yes	—
42	#41×D₅₃	C₂×C₄	1	60	yes	—
43	#41×D₅₃	C₂×C₄	1	60	yes	—
44	#41×D₅₃	C₂³	1	60	yes	—
45	#41×D₅₃	C₂³	1	20	yes	—
46	C₂×Q₈×D₅₃	C₂×C₄	1	15	yes	—
47	C₂×D₄×D₅₃	C₂×C₄	1	45	yes	—
48	C₂²×C₄×D₅₃	C₂³	1	15	yes	—
49	2₋¹⁺⁴×C₅₃	C₂×C₄	1	6	yes	—
50	H(5,𝔽₂)×C₅₃	C₂×C₄	1	10	yes	—
51	C₂×#41×C₅₃	C₂×C₄	1	15	yes	—
52	C₂³×D₅₃	C₂²×C₄	1	15	yes	—
53	#41×C₅₃	C₂²×C₄	1	20	yes	—
54	#41×C₅₃	C₂²×C₄	1	60	yes	—
55	C₂×Q₈×C₅₃	C₂²×C₄	1	15	yes	—
56	C₂×D₄×C₅₃	C₂²×C₄	1	45	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	15	yes	—
58	C₂×C₂²⋊D₅₃	C₂²×C₄	1	90	yes	—
59	C₂²×Dic₅₃	C₂⁴	1	15	yes	—
60	#11719949	C₂²×C₄	1	60	yes	—
61	#11719949	C₂⁴	1	20	yes	—
62	Q₈×D₅₃	C₂²×C₄	1	20	yes	—
63	Q₈×D₅₃	C₂²×C₄	1	60	yes	—
64	#11719947	C₂²×C₄	1	180	yes	—
65	#11719947	C₂⁴	1	60	yes	—
66	D₄×D₅₃	C₂²×C₄	1	60	yes	—
67	D₄×D₅₃	C₂²×C₄	1	180	yes	—
68	#11719945	C₂²×C₄	1	60	yes	—
69	#11719945	C₂²×C₄	1	180	yes	—
70	C₂×D₂₁₂	C₂²×C₄	1	45	yes	—
71	C₂×C₄×D₅₃	C₂²×C₄	1	45	yes	—
72	C₂×C₄×D₅₃	C₂²×C₄	1	45	yes	—
73	C₂×C₄×D₅₃	C₂⁴	1	15	yes	—
74	C₂×C₄×D₅₃	C₂⁴	1	45	yes	—
75	C₂×Dic₁₀₆	C₂²×C₄	1	45	yes	—
76	#11719940	C₂⁴	1	120	yes	—
77	C₂×#88324	C₂⁴	1	90	yes	—
78	C₂×#88324	C₂⁴	1	30	yes	—
79	#11719936	C₂⁴	1	6	yes	—
80	#11719936	C₂⁴	1	10	yes	—
81	#11719936	C₂⁴	1	30	yes	—
82	#11719936	C₂⁴	1	90	yes	—
83	D₅₃⋊C₈	C₂⁴	1	120	yes	—
84	C₂³×C₅₃	C₂³×C₄	1	15	yes	—
85	C₂²×D₅₃	—	8	15	no	—
86	C₂²×D₅₃	C₂³×C₄	1	15	yes	—
87	C₂²×D₅₃	C₂³×C₄	1	45	yes	—
88	Q₈×C₅₃	C₂³×C₄	1	20	yes	—
89	D₄×C₅₃	C₂³×C₄	1	60	yes	—
90	C₂×C₄×C₅₃	C₂³×C₄	1	45	yes	—
91	C₂×C₄×C₅₃	C₂³×C₄	1	15	yes	—
92	C₂²⋊D₅₃	C₂³×C₄	1	120	yes	—
93	C₂×Dic₅₃	C₂³×C₄	1	45	yes	—
94	C₂×Dic₅₃	C₂⁵	1	15	yes	—
95	D₂₁₂	C₂³×C₄	1	60	yes	—
96	C₄×D₅₃	C₂³×C₄	1	15	yes	—
97	C₄×D₅₃	C₂³×C₄	1	15	yes	—
98	C₄×D₅₃	C₂³×C₄	1	90	yes	—
99	C₄×D₅₃	C₂⁵	1	1	yes	characteristic
100	C₄×D₅₃	C₂⁵	1	15	yes	—
101	Dic₁₀₆	C₂³×C₄	1	60	yes	—
102	#88324	C₂⁵	1	20	yes	—
103	#88324	C₂⁵	1	12	yes	—
104	#63076	—	53	1	no	maximal, Sylow
105	C₂²×C₅₃	—	4	15	no	—
106	C₂²×C₅₃	C₂⁴×C₄	1	15	yes	—
107	C₂×D₅₃	—	4	15	no	—
108	C₂×D₅₃	—	4	45	no	—
109	C₂×D₅₃	C₂⁴×C₄	1	1	yes	characteristic
110	C₂×D₅₃	C₂⁴×C₄	1	15	yes	—
111	C₄×C₅₃	C₂⁴×C₄	1	15	yes	—
112	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
113	Dic₅₃	C₂⁴×C₄	1	15	yes	—
114	Dic₅₃	C₂⁶	1	1	yes	characteristic
115	C₂×#585	—	53	1	no	—
116	#3573	—	53	20	no	—
117	#3573	—	53	12	no	—
118	C₂×#568	—	53	15	no	—
119	C₂×#568	—	53	15	no	—
120	C₂×C₅₃	—	2	15	no	—
121	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
122	D₅₃	—	2	1	no	—
123	D₅₃	—	2	15	no	—
124	#585	—	53	6	no	—
125	#585	—	53	10	no	—
126	#585	—	53	15	no	—
127	#585	#6157	1	1	yes	characteristic
128	C₂×2₋¹⁺⁴	—	53	6	no	—
129	C₂×H(5,𝔽₂)	—	53	10	no	—
130	C₂²×#41	—	53	15	no	—
131	#568	—	53	90	no	—
132	#568	—	53	20	no	—
133	#568	—	53	60	no	—
134	#568	—	53	120	no	—
135	#568	—	53	30	no	—
136	C₂×#131	—	53	90	no	—
137	C₂×#131	—	53	120	no	—
138	C₂×#131	—	53	30	no	—
139	C₂²×M₁₆	—	53	45	no	—
140	C₂²×M₁₆	—	53	15	no	—
141	C₅₃	#63076	1	1	yes	characteristic, complemented, Sylow
142	2₋¹⁺⁴	—	53	60	no	—
143	2₋¹⁺⁴	—	53	30	no	—
144	2₋¹⁺⁴	C₂×#6157	1	6	yes	—
145	H(5,𝔽₂)	—	53	60	no	—
146	H(5,𝔽₂)	—	53	90	no	—
147	H(5,𝔽₂)	C₂×#6157	1	10	yes	—
148	C₂×#41	—	53	60	no	—
149	C₂×#41	—	53	20	no	—
150	C₂×#41	—	53	60	no	—
151	C₂×#41	—	53	60	no	—
152	C₂×#41	—	53	15	no	—
153	C₂×#41	—	53	45	no	—
154	C₂×#41	—	53	45	no	—
155	C₂×#41	C₂×#6157	1	15	yes	—
156	C₂²×Q₈	—	53	15	no	—
157	C₂²×D₄	—	53	45	no	—
158	C₂³×C₄	—	53	15	no	—
159	#131	—	53	360	no	—
160	#131	—	53	120	no	—
161	#131	—	53	40	no	—
162	#131	—	53	120	no	—
163	C₂×M₁₆	—	53	60	no	—
164	C₂×M₁₆	—	53	15	no	—
165	C₂×M₁₆	—	53	45	no	—
166	C₂×M₁₆	—	53	90	no	—
167	C₂×M₁₆	—	53	180	no	—
168	C₂×M₁₆	—	53	90	no	—
169	C₂²×C₈	—	53	30	no	—
170	C₂²×C₈	—	53	90	no	—
171	C₂⁴	—	53	15	no	—
172	#41	—	53	60	no	—
173	#41	—	53	20	no	—
174	#41	—	53	180	no	—
175	#41	—	53	60	no	—
176	#41	—	53	60	no	—
177	#41	—	53	180	no	—
178	#41	C₂²×#6157	1	20	yes	—
179	#41	C₂²×#6157	1	60	yes	—
180	C₂×Q₈	—	53	20	no	—
181	C₂×Q₈	—	53	60	no	—
182	C₂×Q₈	—	53	45	no	—
183	C₂×Q₈	C₂²×#6157	1	15	yes	—
184	C₂×D₄	—	53	90	no	—
185	C₂×D₄	—	53	60	no	—
186	C₂×D₄	—	53	180	no	—
187	C₂×D₄	—	53	45	no	—
188	C₂×D₄	C₂²×#6157	1	45	yes	—
189	C₂²×C₄	—	53	15	no	—
190	C₂²×C₄	—	53	45	no	—
191	C₂²×C₄	—	53	45	no	—
192	C₂²×C₄	—	53	45	no	—
193	C₂²×C₄	—	53	15	no	—
194	C₂²×C₄	C₂²×#6157	1	15	yes	—
195	M₁₆	—	53	6	no	—
196	M₁₆	—	53	10	no	—
197	M₁₆	—	53	30	no	—
198	M₁₆	—	53	90	no	—
199	M₁₆	—	53	120	no	—
200	C₂×C₈	—	53	90	no	—
201	C₂×C₈	—	53	30	no	—
202	C₂×C₈	—	53	120	no	—
203	C₂³	—	53	15	no	—
204	C₂³	—	53	45	no	—
205	C₂³	—	424	15	no	—
206	C₂³	C₂³×#6157	1	15	yes	—
207	Q₈	—	53	60	no	—
208	Q₈	C₂³×#6157	1	20	yes	—
209	D₄	—	53	120	no	—
210	D₄	—	53	60	no	—
211	D₄	C₂³×#6157	1	60	yes	—
212	C₂×C₄	—	53	45	no	—
213	C₂×C₄	—	53	15	no	—
214	C₂×C₄	—	53	15	no	—
215	C₂×C₄	—	53	15	no	—
216	C₂×C₄	—	53	90	no	—
217	C₂×C₄	—	53	15	no	—
218	C₂×C₄	—	53	1	no	—
219	C₂×C₄	C₂³×#6157	1	15	yes	—
220	C₂×C₄	C₂³×#6157	1	45	yes	—
221	C₈	—	53	20	no	—
222	C₈	—	53	12	no	—
223	C₂²	—	4	15	no	—
224	C₂²	—	53	1	no	—
225	C₂²	—	53	15	no	—
226	C₂²	—	212	15	no	—
227	C₂²	—	212	45	no	—
228	C₂²	C₂⁴×#6157	1	15	yes	—
229	C₄	—	53	1	no	—
230	C₄	—	53	15	no	—
231	C₄	C₂⁴×#6157	1	1	yes	characteristic
232	C₄	C₂⁴×#6157	1	15	yes	—
233	C₂	—	2	15	no	—
234	C₂	—	106	1	no	—
235	C₂	—	106	15	no	—
236	C₂	C₂⁵×#6157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×#6157
commutator, socle	C₂×C₅₃	C₂⁵×C₄
Fitting	C₅₃×#585	C₄
prime radical	#585×D₅₃	C₂
cosocle	Dic₅₃	C₂⁶
hypercenter	#585	#6157
hypocenter	C₅₃	#63076
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ #585

Complements

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

N	H ≅ G/N	type
1. #20291711	C₂	semidirect
2. #20291710	C₂	semidirect
3. C₂×#16811702	C₂	semidirect
4. D₅₃⋊#568	C₂	semidirect
6. #16811703	C₂²	semidirect
7. #16811702	C₂²	semidirect
8. #16811702	C₂²	semidirect
9. #16811702	C₂²	semidirect
10. #16811701	C₂²	semidirect
11. C₂×#13133305	C₂²	semidirect
12. D₅₃⋊#131	C₂²	semidirect
13. C₂×#13133304	C₂²	semidirect
14. C₂²×#11719936	C₂²	semidirect
15. C₂×D₅₃⋊M₁₆	C₂²	semidirect
23. #13133305	C₂³	semidirect
24. #13133305	C₂³	semidirect
25. #13133304	C₂³	semidirect
26. #13133304	C₂³	semidirect
29. C₂×#11719936	C₂³	semidirect
31. D₅₃⋊M₁₆	C₂³	semidirect
32. C₂×#11719940	C₂³	semidirect
33. C₂×D₅₃⋊C₈	C₂³	semidirect
34. C₂²×#88324	C₂³	semidirect
141. C₅₃	#63076	semidirect

Maximal subgroups

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

subgroup	index
#20291711	2
#20291710	2
C₂×#16811702	2
D₅₃⋊#568	2
#585×D₅₃	2
#63076	53