Subgroups — #20283409

Order 6784.

520 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	#16811134	C₂	1	1	yes	maximal, characteristic
3	#16810694	C₂	1	4	yes	maximal, complemented
4	C₈×C₂²⋊D₅₃	C₂	1	4	yes	maximal, complemented
5	#16810649	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810589	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810587	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810578	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11719945	C₂²	1	1	yes	characteristic
11	#13133223	—	2	4	no	—
12	C₂×#11719917	C₂²	1	1	yes	characteristic
13	C₂×#11719917	C₂²	1	1	yes	characteristic
14	C₂×#11719912	C₂²	1	2	yes	complemented
15	C₂×C₈×D₅₃	C₂²	1	2	yes	complemented
16	#13133200	C₄	1	2	yes	—
17	#13133196	C₄	1	2	yes	—
18	#13133191	C₄	1	2	yes	—
19	#13133186	C₄	1	2	yes	—
20	C₄×C₂²⋊D₅₃	C₂²	1	4	yes	—
21	#13133176	C₂²	1	1	yes	characteristic
22	#13133146	C₂²	1	4	yes	complemented
23	C₂²⋊#88322	C₂²	1	4	yes	—
24	#13133129	C₂²	1	4	yes	complemented
25	#13133127	C₂²	1	2	yes	complemented
26	C₈×Dic₅₃	C₂²	1	2	yes	complemented
27	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	—	2	4	no	—
29	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
31	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₅₃	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
42	#11719945	—	2	2	no	—
43	#11719945	—	2	2	no	—
44	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₅₃	C₂³	1	2	yes	—
46	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	4	no	—
49	C₂×C₈×C₅₃	—	2	4	no	—
50	C₂×C₈×C₅₃	D₄	1	4	yes	complemented
51	C₂×C₈×C₅₃	C₂³	1	2	yes	—
52	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₅₃	C₂×C₄	1	4	yes	—
56	#11719921	C₂×C₄	1	4	yes	—
57	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
58	#11719919	C₂×C₄	1	4	yes	—
59	C₄×Dic₅₃	C₂³	1	2	yes	—
60	#11719917	—	2	2	no	—
61	#11719917	—	2	2	no	—
62	C₂×#88322	C₂³	1	2	yes	—
63	C₂×#88322	C₂³	1	2	yes	—
64	#11719912	—	2	4	no	—
65	C₈×D₅₃	—	2	4	no	—
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	—	2	4	no	—
68	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
70	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	4	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₅₃	#41	1	2	yes	—
79	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₅₃	—	2	4	no	—
81	C₂²⋊D₅₃	—	2	8	no	—
82	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
84	D₂₁₂	—	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	8	no	—
94	C₂²×C₅₃	—	2	2	no	—
95	C₂²×C₅₃	—	2	2	no	—
96	C₂²×C₅₃	—	2	2	no	—
97	C₂²×C₅₃	C₄×D₄	1	2	yes	—
98	C₂²×C₅₃	#131	1	4	yes	complemented
99	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×D₅₃	—	2	2	no	—
101	C₂×D₅₃	—	2	4	no	—
102	C₄×C₅₃	—	2	2	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	4	no	—
108	#2929	—	53	1	no	maximal, Sylow
109	C₂×C₅₃	—	2	2	no	—
110	C₂×C₅₃	—	2	4	no	—
111	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂×C₅₃	C₂×#131	1	2	yes	—
113	D₅₃	—	4	2	no	—
114	C₂×#131	—	53	1	no	—
115	C₂³×C₈	D₅₃	1	1	yes	characteristic, complemented
116	#525	—	53	1	no	—
117	#435	—	53	4	no	—
118	C₈×D₄	—	53	4	no	—
119	#424	—	53	1	no	—
120	#408	—	53	1	no	—
121	#407	—	53	1	no	—
122	#405	—	53	1	no	—
123	C₅₃	#2929	1	1	yes	characteristic, complemented, Sylow
124	C₂×#41	—	53	1	no	—
125	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
126	#131	—	106	4	no	—
127	C₂×M₁₆	—	53	1	no	—
128	C₂×M₁₆	—	53	1	no	—
129	C₂×M₁₆	—	53	2	no	—
130	C₂²×C₈	—	2	4	no	—
131	C₂²×C₈	—	53	2	no	—
132	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
133	C₂²×C₈	C₂×D₅₃	1	2	yes	—
134	C₂²×C₈	C₂×D₅₃	1	2	yes	—
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	—	53	1	no	—
142	C₄⋊C₈	—	53	4	no	—
143	C₂²⋊C₈	—	53	4	no	—
144	C₂²⋊C₈	—	53	4	no	—
145	#97	—	53	2	no	—
146	C₄×C₈	—	53	2	no	—
147	C₂⁴	C₄×D₅₃	1	1	yes	characteristic
148	#41	—	106	2	no	—
149	#41	—	106	2	no	—
150	C₂×Q₈	—	53	1	no	—
151	C₂×D₄	—	53	4	no	—
152	C₂×D₄	—	53	2	no	—
153	C₂×D₄	—	53	1	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	53	2	no	—
157	C₂²×C₄	C₄×D₅₃	1	2	yes	—
158	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
160	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
161	M₁₆	—	106	2	no	—
162	M₁₆	—	106	2	no	—
163	M₁₆	—	106	4	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	53	2	no	—
168	C₂×C₈	—	53	2	no	—
169	C₂×C₈	—	106	4	no	—
170	C₂×C₈	C₂²⋊D₅₃	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	2	no	—
175	C₄⋊C₄	—	53	4	no	—
176	C₂²⋊C₄	—	53	2	no	—
177	C₂²⋊C₄	—	53	4	no	—
178	C₂²⋊C₄	—	53	4	no	—
179	C₄²	—	53	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	4	no	—
182	C₂³	—	53	2	no	—
183	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
184	C₂³	C₂×C₄×D₅₃	1	2	yes	—
185	Q₈	—	106	2	no	—
186	D₄	—	106	4	no	—
187	D₄	—	106	8	no	—
188	D₄	—	106	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	53	4	no	—
194	C₂×C₄	—	53	2	no	—
195	C₂×C₄	—	106	2	no	—
196	C₂×C₄	—	106	2	no	—
197	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₄	#11719945	1	2	yes	—
200	C₂×C₄	C₂×C₂²⋊D₅₃	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×C₂²⋊D₅₃	1	2	yes	—
206	C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	106	2	no	—
212	C₂²	—	106	4	no	—
213	C₂²	C₄×C₂²⋊D₅₃	1	2	yes	—
214	C₂²	#13133223	1	4	yes	complemented
215	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	106	2	no	—
218	C₄	—	106	4	no	—
219	C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
220	C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
221	C₄	C₂×#11719945	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	2	4	no	—
224	C₂	—	212	2	no	—
225	C₂	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
226	C₂	C₂×#13133223	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂³×C₈×C₅₃	C₂
hypercenter	C₂³×C₈	D₅₃
hypocenter	C₅₃	#2929
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#13133223	C₂	semidirect
3. #16810694	C₂	semidirect
4. C₈×C₂²⋊D₅₃	C₂	semidirect
5. #16810649	C₂	semidirect
6. #16810589	C₂	semidirect
7. #16810587	C₂	semidirect
8. #16810578	C₂	semidirect
9. C₂³×C₈×C₅₃	C₂	semidirect
14. C₂×#11719912	C₂²	semidirect
15. C₂×C₈×D₅₃	C₂²	semidirect
22. #13133146	C₂²	semidirect
24. #13133129	C₂²	semidirect
25. #13133127	C₂²	semidirect
26. C₈×Dic₅₃	C₂²	semidirect
50. C₂×C₈×C₅₃	D₄	semidirect
98. C₂²×C₅₃	#131	semidirect
115. C₂³×C₈	D₅₃	semidirect
123. C₅₃	#2929	semidirect
170. C₂×C₈	C₂²⋊D₅₃	semidirect
214. C₂²	#13133223	semidirect