Subgroups — #20283420

Order 6784.

492 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133223	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811135	C₂	1	1	yes	maximal, characteristic
4	#16810702	C₂	1	4	yes	maximal, complemented
5	#16810696	C₂	1	4	yes	maximal, complemented
6	#16810650	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810591	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810586	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11719945	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
12	#13133223	—	2	4	no	—
13	C₂×#11719917	C₂²	1	1	yes	characteristic
14	C₂×#11719912	—	2	4	no	—
15	C₂×#11719912	C₂²	1	2	yes	—
16	C₂×#11719912	C₂²	1	2	yes	—
17	C₂×#11719912	C₂²	1	2	yes	complemented
18	C₂²×#88322	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₅₃	C₂²	1	2	yes	complemented
20	#13133201	C₄	1	2	yes	—
21	#13133197	C₄	1	2	yes	—
22	#13133192	C₄	1	2	yes	—
23	#13133187	C₄	1	2	yes	—
24	C₄×D₂₁₂	C₂²	1	4	yes	—
25	C₄⋊#88322	C₂²	1	4	yes	complemented
26	#13133129	C₂²	1	4	yes	—
27	#13133129	C₂²	1	4	yes	complemented
28	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	#117×C₅₃	C₂²	1	1	yes	characteristic
31	#97×C₅₃	C₂²	1	2	yes	complemented
32	C₄×C₈×C₅₃	C₂²	1	2	yes	complemented
33	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
36	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	#11719945	—	2	2	no	—
38	#11719945	—	2	2	no	—
39	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
40	C₂×D₂₁₂	C₂×C₄	1	4	yes	—
41	C₂×C₄×D₅₃	—	2	2	no	—
42	C₂×C₄×D₅₃	—	2	2	no	—
43	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₅₃	C₂³	1	2	yes	—
45	C₂×C₄×D₅₃	C₂³	1	2	yes	—
46	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
47	M₁₆×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	D₄	1	4	yes	complemented
49	C₂×C₈×C₅₃	C₂³	1	2	yes	—
50	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
54	C₄²×C₅₃	C₂³	1	2	yes	—
55	#11719921	C₂×C₄	1	4	yes	—
56	#11719921	C₂×C₄	1	4	yes	—
57	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
58	#11719917	—	2	2	no	—
59	C₂×#88322	—	2	2	no	—
60	C₂×#88322	C₂³	1	2	yes	—
61	C₂×#88322	C₂³	1	2	yes	—
62	#11719912	—	2	4	no	—
63	#11719912	—	2	4	no	—
64	#11719912	—	2	4	no	—
65	C₈×D₅₃	—	2	4	no	—
66	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₅₃	—	2	2	no	—
68	C₂²×D₅₃	—	4	2	no	—
69	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
70	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
72	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₅₃	#41	1	2	yes	—
75	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
76	C₂²⋊D₅₃	—	2	4	no	—
77	C₂×Dic₅₃	—	2	2	no	—
78	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
79	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
80	D₂₁₂	—	2	2	no	—
81	D₂₁₂	—	4	4	no	—
82	C₄×D₅₃	—	2	4	no	—
83	C₄×D₅₃	—	2	2	no	—
84	C₄×D₅₃	—	2	2	no	—
85	C₄×D₅₃	—	2	2	no	—
86	C₄×D₅₃	—	2	2	no	—
87	Dic₁₀₆	—	2	2	no	—
88	C₈×C₅₃	—	2	2	no	—
89	C₈×C₅₃	C₂×D₄	1	2	yes	—
90	C₈×C₅₃	C₂×D₄	1	2	yes	—
91	#88322	—	2	2	no	—
92	#88322	—	2	2	no	—
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	C₄×D₄	1	2	yes	—
95	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₅₃	—	2	2	no	—
97	C₂×D₅₃	—	2	2	no	—
98	C₂×D₅₃	—	2	2	no	—
99	C₂×D₅₃	—	2	4	no	—
100	C₂×D₅₃	—	2	2	no	—
101	C₂×D₅₃	—	4	4	no	—
102	C₄×C₅₃	—	2	4	no	—
103	C₄×C₅₃	C₄×D₄	1	2	yes	—
104	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
106	Dic₅₃	—	2	2	no	—
107	Dic₅₃	—	2	2	no	—
108	#2930	—	53	1	no	maximal, Sylow
109	C₂×C₅₃	—	2	2	no	—
110	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂×C₅₃	#568	1	2	yes	—
112	D₅₃	—	4	2	no	—
113	D₅₃	—	4	2	no	—
114	C₂×#131	—	53	1	no	—
115	C₂²×M₁₆	—	53	1	no	—
116	#525	—	53	1	no	—
117	#436	—	53	4	no	—
118	#435	—	53	4	no	—
119	#424	—	53	1	no	—
120	#408	—	53	1	no	—
121	#407	—	53	1	no	—
122	#405	D₅₃	1	1	yes	characteristic, complemented
123	C₅₃	#2930	1	1	yes	characteristic, complemented, Sylow
124	C₂×#41	—	53	1	no	—
125	C₂³×C₄	—	53	1	no	—
126	#131	—	106	4	no	—
127	C₂×M₁₆	—	53	1	no	—
128	C₂×M₁₆	—	53	2	no	—
129	C₂×M₁₆	—	53	2	no	—
130	C₂×M₁₆	—	53	2	no	—
131	C₂×M₁₆	—	106	4	no	—
132	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
133	C₂²×C₈	—	53	1	no	—
134	C₂²×C₈	—	53	2	no	—
135	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
136	#123	—	53	2	no	—
137	C₂²⋊Q₈	—	53	2	no	—
138	#121	—	53	2	no	—
139	#120	—	53	2	no	—
140	C₄×D₄	—	53	4	no	—
141	#117	C₂×D₅₃	1	1	yes	characteristic
142	C₄⋊C₈	—	53	4	no	—
143	C₂²⋊C₈	—	53	4	no	—
144	C₂²⋊C₈	—	53	4	no	—
145	#97	C₂×D₅₃	1	2	yes	complemented
146	C₄×C₈	C₂×D₅₃	1	2	yes	complemented
147	C₂⁴	—	53	1	no	—
148	#41	—	106	2	no	—
149	#41	—	106	2	no	—
150	C₂×Q₈	—	53	1	no	—
151	C₂×D₄	—	53	2	no	—
152	C₂×D₄	—	53	1	no	—
153	C₂×D₄	—	53	4	no	—
154	C₂²×C₄	—	53	1	no	—
155	C₂²×C₄	—	53	2	no	—
156	C₂²×C₄	—	53	2	no	—
157	C₂²×C₄	—	53	2	no	—
158	C₂²×C₄	—	106	2	no	—
159	C₂²×C₄	—	106	2	no	—
160	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	106	2	no	—
163	M₁₆	—	106	4	no	—
164	M₁₆	—	106	4	no	—
165	M₁₆	—	106	4	no	—
166	C₂×C₈	—	53	2	no	—
167	C₂×C₈	—	53	2	no	—
168	C₂×C₈	—	106	2	no	—
169	C₂×C₈	—	106	4	no	—
170	C₂×C₈	D₂₁₂	1	4	yes	complemented
171	C₂×C₈	C₂²×D₅₃	1	2	yes	—
172	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
174	C₄⋊C₄	—	53	4	no	—
175	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
176	C₂²⋊C₄	—	53	4	no	—
177	C₂²⋊C₄	—	53	4	no	—
178	C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
179	C₄²	C₂²×D₅₃	1	2	yes	—
180	C₂³	—	53	2	no	—
181	C₂³	—	53	2	no	—
182	C₂³	—	106	2	no	—
183	C₂³	—	212	2	no	—
184	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
185	Q₈	—	106	2	no	—
186	D₄	—	106	4	no	—
187	D₄	—	106	2	no	—
188	D₄	—	212	4	no	—
189	C₂×C₄	—	53	2	no	—
190	C₂×C₄	—	53	2	no	—
191	C₂×C₄	—	106	2	no	—
192	C₂×C₄	—	106	4	no	—
193	C₂×C₄	—	106	2	no	—
194	C₂×C₄	—	106	2	no	—
195	C₂×C₄	—	106	2	no	—
196	C₂×C₄	—	106	2	no	—
197	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
198	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
200	C₂×C₄	#11719945	1	2	yes	—
201	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	106	2	no	—
204	C₈	—	106	2	no	—
205	C₈	C₂×D₂₁₂	1	2	yes	—
206	C₈	C₂×D₂₁₂	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	106	2	no	—
209	C₂²	—	106	2	no	—
210	C₂²	—	106	2	no	—
211	C₂²	—	106	4	no	—
212	C₂²	—	106	2	no	—
213	C₂²	—	212	4	no	—
214	C₂²	C₄×D₂₁₂	1	2	yes	—
215	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	106	2	no	—
218	C₄	—	106	2	no	—
219	C₄	C₄×D₂₁₂	1	2	yes	—
220	C₄	C₂²×D₂₁₂	1	1	yes	characteristic
221	C₄	C₂×#11719945	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	212	2	no	—
224	C₂	—	212	2	no	—
225	C₂	C₂×C₄×D₂₁₂	1	1	yes	characteristic
226	C₂	#16811388	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#405	C₂
hypercenter	#405	D₅₃
hypocenter	C₅₃	#2930
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
1. C₂×#13133223	C₂	semidirect
2. C₂²×#11719912	C₂	semidirect
4. #16810702	C₂	semidirect
5. #16810696	C₂	semidirect
6. #16810650	C₂	semidirect
7. #16810591	C₂	semidirect
8. #16810586	C₂	semidirect
9. C₅₃×#405	C₂	semidirect
17. C₂×#11719912	C₂²	semidirect
19. C₂×C₈×D₅₃	C₂²	semidirect
25. C₄⋊#88322	C₂²	semidirect
27. #13133129	C₂²	semidirect
31. #97×C₅₃	C₂²	semidirect
32. C₄×C₈×C₅₃	C₂²	semidirect
48. C₂×C₈×C₅₃	D₄	semidirect
122. #405	D₅₃	semidirect
123. C₅₃	#2930	semidirect
145. #97	C₂×D₅₃	semidirect
146. C₄×C₈	C₂×D₅₃	semidirect
170. C₂×C₈	D₂₁₂	semidirect