Subgroups — #20283743

Order 6784.

456 conjugacy classes of subgroups.

The subgroup lattice has 232 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	#16810696	C₂	1	4	yes	maximal, complemented
3	#16810693	C₂	1	4	yes	maximal, complemented
4	#16810680	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810585	C₂	1	2	yes	maximal, complemented
6	C₂²⋊C₈×D₅₃	C₂	1	2	yes	maximal, complemented
7	C₅₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₁₂	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
11	C₂×#11719912	C₂²	1	4	yes	complemented
12	C₂×C₈×D₅₃	C₂²	1	4	yes	complemented
13	C₄×D₂₁₂	C₄	1	2	yes	—
14	C₄×D₂₁₂	C₄	1	2	yes	—
15	C₄×D₂₁₂	C₂²	1	2	yes	—
16	C₄×D₂₁₂	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
18	C₂×#11719921	C₄	1	1	yes	characteristic
19	C₂×#11719921	C₄	1	1	yes	characteristic
20	C₄⋊#88322	C₂²	1	2	yes	complemented
21	C₂²⋊#88322	C₂²	1	2	yes	—
22	#13133129	C₂²	1	4	yes	complemented
23	#13133129	C₂²	1	4	yes	complemented
24	#13133128	C₂²	1	2	yes	complemented
25	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₅₃	C₂²	1	2	yes	complemented
27	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
28	#97×C₅₃	C₂²	1	2	yes	complemented
29	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×D₂₁₂	—	2	4	no	—
37	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
38	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₅₃	—	4	2	no	—
40	C₂×C₄×D₅₃	—	4	2	no	—
41	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₅₃	C₂³	1	2	yes	—
44	C₂×C₄×D₅₃	C₂³	1	2	yes	—
45	C₂×C₈×C₅₃	C₂³	1	4	yes	—
46	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	#11719921	C₂×C₄	1	2	yes	—
51	#11719921	C₂×C₄	1	2	yes	—
52	#11719921	C₂×C₄	1	2	yes	—
53	#11719921	C₂×C₄	1	2	yes	—
54	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
55	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
56	C₂×#88322	C₂³	1	4	yes	—
57	#11719912	—	4	4	no	—
58	C₈×D₅₃	—	4	4	no	—
59	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×D₅₃	—	2	2	no	—
61	C₂²×D₅₃	—	4	1	no	—
62	C₂²×D₅₃	—	4	1	no	—
63	C₂²×D₅₃	—	4	1	no	—
64	C₂²×D₅₃	—	4	1	no	—
65	C₂²×D₅₃	M₁₆	1	4	yes	complemented
66	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
67	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	4	1	no	—
71	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₅₃	—	4	1	no	—
78	C₂×Dic₅₃	—	4	1	no	—
79	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
80	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
81	D₂₁₂	—	2	4	no	—
82	D₂₁₂	—	2	4	no	—
83	C₄×D₅₃	—	2	4	no	—
84	C₄×D₅₃	—	2	4	no	—
85	C₄×D₅₃	—	4	2	no	—
86	C₄×D₅₃	—	4	2	no	—
87	C₈×C₅₃	—	2	4	no	—
88	#88322	—	2	4	no	—
89	C₂²×C₅₃	—	2	1	no	—
90	C₂²×C₅₃	—	2	1	no	—
91	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
92	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×D₅₃	—	2	2	no	—
94	C₂×D₅₃	—	2	2	no	—
95	C₂×D₅₃	—	2	2	no	—
96	C₂×D₅₃	—	2	2	no	—
97	C₂×D₅₃	—	2	2	no	—
98	C₂×D₅₃	—	4	4	no	—
99	C₂×D₅₃	—	4	4	no	—
100	C₂×D₅₃	C₂×M₁₆	1	4	yes	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
107	Dic₅₃	—	2	2	no	—
108	Dic₅₃	—	2	2	no	—
109	#2973	—	53	1	no	maximal, Sylow
110	C₂×C₅₃	—	2	2	no	—
111	C₂×C₅₃	#518	1	1	yes	characteristic
112	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
113	C₂×C₅₃	#568	1	1	yes	characteristic
114	D₅₃	—	2	4	no	—
115	D₅₃	—	4	2	no	—
116	C₂×C₄×D₄	—	53	1	no	—
117	#435	—	53	4	no	—
118	#435	—	53	4	no	—
119	#432	—	53	1	no	—
120	#432	D₅₃	1	1	yes	characteristic, complemented
121	#407	—	53	2	no	—
122	C₂×C₂²⋊C₈	—	53	2	no	—
123	C₅₃	#2973	1	1	yes	characteristic, complemented, Sylow
124	C₂²×D₄	—	53	1	no	—
125	C₂³×C₄	—	53	1	no	—
126	C₂³×C₄	—	53	1	no	—
127	C₂×M₁₆	—	53	4	no	—
128	C₂²×C₈	—	53	4	no	—
129	C₄×D₄	—	53	2	no	—
130	C₄×D₄	—	53	2	no	—
131	C₄×D₄	—	53	2	no	—
132	C₄×D₄	—	53	2	no	—
133	C₂×C₄⋊C₄	—	53	1	no	—
134	C₂×C₂²⋊C₄	—	53	1	no	—
135	C₂×C₂²⋊C₄	—	53	1	no	—
136	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
137	C₄⋊C₈	—	53	2	no	—
138	C₄⋊C₈	C₂×D₅₃	1	2	yes	complemented
139	C₂²⋊C₈	—	53	2	no	—
140	C₂²⋊C₈	—	53	4	no	—
141	C₂²⋊C₈	—	53	4	no	—
142	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
143	#97	—	53	2	no	—
144	#97	C₂×D₅₃	1	2	yes	complemented
145	C₂⁴	—	53	1	no	—
146	C₂⁴	—	53	1	no	—
147	C₂×D₄	—	53	2	no	—
148	C₂×D₄	—	53	2	no	—
149	C₂×D₄	—	106	4	no	—
150	C₂²×C₄	—	53	1	no	—
151	C₂²×C₄	—	53	1	no	—
152	C₂²×C₄	—	53	2	no	—
153	C₂²×C₄	—	53	2	no	—
154	C₂²×C₄	—	53	2	no	—
155	C₂²×C₄	—	53	2	no	—
156	C₂²×C₄	—	212	2	no	—
157	C₂²×C₄	—	212	2	no	—
158	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
159	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
160	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
161	M₁₆	—	212	4	no	—
162	C₂×C₈	—	53	4	no	—
163	C₂×C₈	—	212	4	no	—
164	C₂×C₈	C₂²×D₅₃	1	4	yes	—
165	C₄⋊C₄	—	53	2	no	—
166	C₄⋊C₄	—	53	2	no	—
167	C₂²⋊C₄	—	53	2	no	—
168	C₂²⋊C₄	—	53	2	no	—
169	C₂²⋊C₄	—	53	2	no	—
170	C₂²⋊C₄	—	53	2	no	—
171	C₄²	C₄×D₅₃	1	1	yes	characteristic
172	C₄²	C₄×D₅₃	1	1	yes	characteristic
173	C₄²	C₂²×D₅₃	1	1	yes	characteristic
174	C₄²	C₂²×D₅₃	1	1	yes	characteristic
175	C₂³	—	53	2	no	—
176	C₂³	—	53	4	no	—
177	C₂³	—	53	2	no	—
178	C₂³	—	106	2	no	—
179	C₂³	—	212	1	no	—
180	C₂³	—	212	1	no	—
181	C₂³	—	212	1	no	—
182	C₂³	—	212	1	no	—
183	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
184	D₄	—	106	4	no	—
185	D₄	—	106	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	53	2	no	—
190	C₂×C₄	—	53	2	no	—
191	C₂×C₄	—	106	4	no	—
192	C₂×C₄	—	106	4	no	—
193	C₂×C₄	—	212	2	no	—
194	C₂×C₄	—	212	2	no	—
195	C₂×C₄	—	212	1	no	—
196	C₂×C₄	—	212	1	no	—
197	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
203	C₈	—	2	4	no	—
204	C₈	—	106	4	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	53	4	no	—
208	C₂²	—	106	2	no	—
209	C₂²	—	106	2	no	—
210	C₂²	—	106	2	no	—
211	C₂²	—	106	2	no	—
212	C₂²	—	106	2	no	—
213	C₂²	—	212	4	no	—
214	C₂²	—	212	4	no	—
215	C₂²	M₁₆×D₅₃	1	2	yes	—
216	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	106	2	no	—
223	C₄	—	106	2	no	—
224	C₄	#13133263	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	—	106	4	no	—
227	C₂	—	212	2	no	—
228	C₂	#16811097	1	1	yes	characteristic
229	C₂	C₂×M₁₆×D₅₃	1	1	yes	characteristic
230	C₂	#16811389	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#432	C₂
hypercenter	#432	D₅₃
hypocenter	C₅₃	#2973
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #16810696	C₂	semidirect
3. #16810693	C₂	semidirect
4. #16810680	C₂	semidirect
5. #16810585	C₂	semidirect
6. C₂²⋊C₈×D₅₃	C₂	semidirect
7. C₅₃×#432	C₂	semidirect
11. C₂×#11719912	C₂²	semidirect
12. C₂×C₈×D₅₃	C₂²	semidirect
20. C₄⋊#88322	C₂²	semidirect
22. #13133129	C₂²	semidirect
23. #13133129	C₂²	semidirect
24. #13133128	C₂²	semidirect
26. C₄⋊C₈×C₅₃	C₂²	semidirect
28. #97×C₅₃	C₂²	semidirect
65. C₂²×D₅₃	M₁₆	semidirect
120. #432	D₅₃	semidirect
123. C₅₃	#2973	semidirect
138. C₄⋊C₈	C₂×D₅₃	semidirect
144. #97	C₂×D₅₃	semidirect