Subgroups — #20277615

Order 6784.

776 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133179	C₂	1	2	yes	maximal
3	C₂×C₄⋊C₄×D₅₃	C₂	1	2	yes	maximal, complemented
4	#16810519	C₂	1	2	yes	maximal, complemented
5	#16810494	C₂	1	4	yes	maximal, complemented
6	#16810480	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₅₃	C₂²	1	4	yes	—
11	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
12	C₄×D₂₁₂	C₄	1	8	yes	complemented
13	#13133179	—	2	8	no	—
14	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
17	C₂×#11719919	C₂²	1	4	yes	—
18	C₄⋊C₄×D₅₃	—	2	8	no	—
19	C₂×#11719921	C₂²	1	4	yes	—
20	C₂×#11719921	C₂²	1	2	yes	—
21	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
22	#13133124	C₂²	1	4	yes	—
23	#13133124	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₂³×D₅₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
30	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₅₃	C₂³	1	2	yes	—
34	C₂²×Dic₅₃	C₂³	1	2	yes	—
35	C₂²×Dic₅₃	C₂³	1	2	yes	—
36	C₂×D₂₁₂	C₂×C₄	1	4	yes	—
37	C₂×D₂₁₂	D₄	1	4	yes	—
38	C₂×D₂₁₂	Q₈	1	4	yes	—
39	C₂×C₄×D₅₃	—	2	4	no	—
40	C₂×C₄×D₅₃	—	2	4	no	—
41	C₂×C₄×D₅₃	—	2	8	no	—
42	C₂×C₄×D₅₃	—	2	8	no	—
43	C₂×C₄×D₅₃	—	4	4	no	—
44	C₂×C₄×D₅₃	C₂×C₄	1	8	yes	complemented
45	C₄⋊C₄×C₅₃	—	2	4	no	—
46	C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₄⋊C₄×C₅₃	—	4	2	no	—
48	C₄²×C₅₃	C₂×C₄	1	4	yes	complemented
49	#11719921	—	2	8	no	—
50	#11719921	C₂×C₄	1	8	yes	complemented
51	C₄⋊Dic₅₃	—	2	2	no	—
52	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	complemented
53	C₄⋊Dic₅₃	D₄	1	4	yes	complemented
54	#11719919	—	2	8	no	—
55	C₄×Dic₅₃	—	2	4	no	—
56	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₂²×D₅₃	—	2	2	no	—
58	C₂²×D₅₃	—	2	2	no	—
59	C₂²×D₅₃	—	2	2	no	—
60	C₂²×D₅₃	—	2	2	no	—
61	C₂²×D₅₃	C₂²×C₄	1	4	yes	—
62	C₂²×D₅₃	C₂×D₄	1	4	yes	—
63	C₂²×D₅₃	C₂×Q₈	1	4	yes	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₅₃	#41	1	2	yes	—
76	C₂×Dic₅₃	—	2	2	no	—
77	C₂×Dic₅₃	—	2	2	no	—
78	C₂×Dic₅₃	—	2	2	no	—
79	C₂×Dic₅₃	—	2	2	no	—
80	C₂×Dic₅₃	—	2	2	no	—
81	C₂×Dic₅₃	—	2	2	no	—
82	C₂×Dic₅₃	—	2	2	no	—
83	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
84	C₂×Dic₅₃	C₂×D₄	1	4	yes	—
85	D₂₁₂	C₄⋊C₄	1	16	yes	complemented
86	C₄×D₅₃	—	2	16	no	—
87	C₄×D₅₃	—	4	16	no	—
88	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
89	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
90	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
91	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
92	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
93	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
94	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
95	C₂×D₅₃	—	2	4	no	—
96	C₂×D₅₃	—	2	4	no	—
97	C₂×D₅₃	—	2	4	no	—
98	C₂×D₅₃	—	2	4	no	—
99	C₂×D₅₃	—	2	4	no	—
100	C₂×D₅₃	—	2	4	no	—
101	C₂×D₅₃	C₂×C₄⋊C₄	1	8	yes	complemented
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	—	4	4	no	—
104	C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	complemented
105	C₄×C₅₃	C₄×D₄	1	4	yes	complemented
106	Dic₅₃	—	2	4	no	—
107	Dic₅₃	—	2	4	no	—
108	Dic₅₃	—	4	2	no	—
109	D₄×C₄⋊C₄	—	53	1	no	maximal, Sylow
110	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂×C₅₃	#520	1	1	yes	characteristic
113	C₂×C₅₃	D₄²	1	1	yes	characteristic
114	C₂×C₅₃	#547	1	1	yes	characteristic
115	C₂×C₅₃	D₄×Q₈	1	1	yes	characteristic
116	C₂×C₅₃	#554	1	1	yes	characteristic
117	D₅₃	—	2	8	no	—
118	C₂×C₄×D₄	—	53	2	no	—
119	C₂×C₄×D₄	—	53	1	no	—
120	C₂²×C₄⋊C₄	—	53	2	no	—
121	#389	—	53	2	no	—
122	#385	—	53	4	no	—
123	#384	D₅₃	1	1	yes	characteristic, complemented
124	#380	—	53	2	no	—
125	C₄×C₄⋊C₄	—	53	1	no	—
126	C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
127	C₂²×D₄	—	53	1	no	—
128	C₂³×C₄	—	53	2	no	—
129	C₂³×C₄	—	53	4	no	—
130	C₄×D₄	—	53	8	no	—
131	C₄×D₄	—	106	8	no	—
132	C₂×C₄⋊C₄	—	53	1	no	—
133	C₂×C₄⋊C₄	—	53	1	no	—
134	C₂×C₄⋊C₄	—	53	4	no	—
135	C₂×C₄⋊C₄	—	53	1	no	—
136	C₂×C₄⋊C₄	—	106	8	no	—
137	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
138	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
139	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
140	C₂×C₂²⋊C₄	—	53	2	no	—
141	C₂×C₂²⋊C₄	—	53	4	no	—
142	C₂×C₄²	—	53	2	no	—
143	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
144	#95	—	53	2	no	—
145	#95	—	53	4	no	—
146	C₂⁴	—	53	2	no	—
147	C₂×D₄	—	53	4	no	—
148	C₂×D₄	—	53	4	no	—
149	C₂×D₄	—	53	4	no	—
150	C₂²×C₄	—	53	2	no	—
151	C₂²×C₄	—	53	2	no	—
152	C₂²×C₄	—	53	8	no	—
153	C₂²×C₄	—	53	2	no	—
154	C₂²×C₄	—	106	8	no	—
155	C₂²×C₄	—	106	8	no	—
156	C₂²×C₄	—	106	4	no	—
157	C₂²×C₄	—	106	4	no	—
158	C₂²×C₄	—	212	4	no	—
159	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
160	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	53	4	no	—
167	C₄⋊C₄	—	53	4	no	—
168	C₄⋊C₄	—	106	8	no	—
169	C₄⋊C₄	—	106	2	no	—
170	C₂²⋊C₄	—	53	8	no	—
171	C₂²⋊C₄	—	106	8	no	—
172	C₄²	—	106	4	no	—
173	C₄²	C₄×D₅₃	1	4	yes	complemented
174	C₂³	—	53	4	no	—
175	C₂³	—	53	4	no	—
176	C₂³	—	53	4	no	—
177	C₂³	—	106	2	no	—
178	C₂³	—	106	2	no	—
179	C₂³	—	106	2	no	—
180	C₂³	—	106	2	no	—
181	C₂³	C₂³×D₅₃	1	1	yes	characteristic
182	D₄	—	53	16	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	53	4	no	—
189	C₂×C₄	—	53	4	no	—
190	C₂×C₄	—	106	2	no	—
191	C₂×C₄	—	106	2	no	—
192	C₂×C₄	—	106	2	no	—
193	C₂×C₄	—	106	2	no	—
194	C₂×C₄	—	106	16	no	—
195	C₂×C₄	—	106	2	no	—
196	C₂×C₄	—	106	2	no	—
197	C₂×C₄	—	106	2	no	—
198	C₂×C₄	—	212	16	no	—
199	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
201	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
202	C₂×C₄	D₄×D₅₃	1	2	yes	—
203	C₂×C₄	D₄×D₅₃	1	2	yes	—
204	C₂×C₄	Q₈×D₅₃	1	2	yes	—
205	C₂×C₄	#11719949	1	2	yes	—
206	C₂²	—	53	8	no	—
207	C₂²	—	106	4	no	—
208	C₂²	—	106	4	no	—
209	C₂²	—	106	4	no	—
210	C₂²	—	106	4	no	—
211	C₂²	—	106	4	no	—
212	C₂²	—	106	4	no	—
213	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
214	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
215	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
216	C₂²	C₂×Q₈×D₅₃	1	1	yes	characteristic
217	C₂²	C₂×#11719949	1	1	yes	characteristic
218	C₂²	#13133262	1	1	yes	characteristic
219	C₂²	#13133265	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	4	4	no	—
222	C₄	—	106	4	no	—
223	C₄	—	106	4	no	—
224	C₄	—	212	2	no	—
225	C₄	C₄⋊C₄×D₅₃	1	4	yes	complemented
226	C₄	#13133179	1	4	yes	complemented
227	C₂	—	106	8	no	—
228	C₂	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
229	C₂	C₂×#13133179	1	1	yes	characteristic
230	C₂	#16811104	1	1	yes	characteristic
231	C₂	#16811258	1	1	yes	characteristic
232	C₂	#16811276	1	1	yes	characteristic
233	C₂	#16811288	1	1	yes	characteristic
234	C₂	#16811334	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#384	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	#384	D₅₃
hypocenter	C₅₃	D₄×C₄⋊C₄
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊C₄×D₅₃	C₂	semidirect
4. #16810519	C₂	semidirect
5. #16810494	C₂	semidirect
6. #16810480	C₂	semidirect
7. C₄×C₄⋊Dic₅₃	C₂	semidirect
8. C₅₃×#384	C₂	semidirect
12. C₄×D₂₁₂	C₄	semidirect
44. C₂×C₄×D₅₃	C₂×C₄	semidirect
48. C₄²×C₅₃	C₂×C₄	semidirect
50. #11719921	C₂×C₄	semidirect
52. C₄⋊Dic₅₃	C₂×C₄	semidirect
53. C₄⋊Dic₅₃	D₄	semidirect
85. D₂₁₂	C₄⋊C₄	semidirect
101. C₂×D₅₃	C₂×C₄⋊C₄	semidirect
104. C₄×C₅₃	C₂×C₄⋊C₄	semidirect
105. C₄×C₅₃	C₄×D₄	semidirect
123. #384	D₅₃	semidirect
126. C₅₃	D₄×C₄⋊C₄	semidirect
173. C₄²	C₄×D₅₃	semidirect
225. C₄	C₄⋊C₄×D₅₃	semidirect
226. C₄	#13133179	semidirect