Subgroups — #20283276

Order 6784.

484 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11719912	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#117×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810651	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810650	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810649	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810648	C₂	1	4	yes	maximal, complemented
8	#16810645	C₂	1	4	yes	maximal, complemented
9	C₅₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
11	C₂×#11719917	C₂²	1	1	yes	characteristic
12	C₂×#11719912	—	2	4	no	—
13	C₂×#11719912	C₂²	1	2	yes	—
14	C₂×#11719912	C₂²	1	2	yes	—
15	C₂×#11719912	C₂²	1	2	yes	—
16	M₁₆×D₅₃	—	2	4	no	—
17	C₂²×#88322	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₅₃	C₂²	1	2	yes	—
19	#13133176	C₂²	1	1	yes	characteristic
20	#13133175	C₄	1	2	yes	—
21	#13133174	C₂²	1	2	yes	—
22	#13133173	C₄	1	2	yes	—
23	C₄⋊C₄×D₅₃	C₄	1	2	yes	—
24	C₂²⋊C₄×D₅₃	C₄	1	2	yes	—
25	C₄²×D₅₃	C₂²	1	2	yes	—
26	C₄⋊#88322	C₂²	1	4	yes	complemented
27	#13133146	C₂²	1	4	yes	complemented
28	#13133146	C₂²	1	4	yes	complemented
29	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	#117×C₅₃	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₅₃	C₂²	1	4	yes	complemented
33	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₅₃	C₂×C₄	1	4	yes	—
37	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₅₃	D₄	1	4	yes	—
40	C₂×C₄×D₅₃	Q₈	1	4	yes	—
41	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
43	M₁₆×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	C₂³	1	2	yes	—
46	C₂×C₈×C₅₃	C₂³	1	2	yes	—
47	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₄²×C₅₃	C₂³	1	2	yes	—
50	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
51	#11719921	C₂×C₄	1	4	yes	—
52	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
53	#11719919	C₂×C₄	1	4	yes	—
54	C₄×Dic₅₃	C₂×C₄	1	4	yes	—
55	C₄×Dic₅₃	C₂³	1	2	yes	—
56	#11719917	—	2	2	no	—
57	C₂×#88322	—	2	2	no	—
58	C₂×#88322	C₂³	1	2	yes	—
59	C₂×#88322	C₂³	1	2	yes	—
60	#11719912	—	2	4	no	—
61	#11719912	—	2	4	no	—
62	#11719912	—	2	4	no	—
63	C₈×D₅₃	—	2	4	no	—
64	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×D₅₃	—	4	2	no	—
66	C₂²×D₅₃	C₄⋊C₄	1	4	yes	—
67	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
70	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₅₃	C₄⋊C₄	1	4	yes	—
75	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
76	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₄×D₅₃	—	4	4	no	—
79	C₄×D₅₃	C₄⋊C₄	1	4	yes	—
80	C₄×D₅₃	C₄⋊C₄	1	4	yes	—
81	C₄×D₅₃	C₂×D₄	1	2	yes	—
82	C₄×D₅₃	C₂×D₄	1	2	yes	—
83	C₄×D₅₃	C₂×Q₈	1	2	yes	—
84	C₄×D₅₃	C₂×Q₈	1	2	yes	—
85	C₈×C₅₃	—	2	2	no	—
86	C₈×C₅₃	—	2	2	no	—
87	#88322	—	2	2	no	—
88	#88322	—	2	2	no	—
89	C₂²×C₅₃	—	2	2	no	—
90	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
92	C₂×D₅₃	—	2	2	no	—
93	C₂×D₅₃	—	2	2	no	—
94	C₂×D₅₃	—	4	4	no	—
95	C₂×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₄×C₅₃	—	2	4	no	—
98	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
100	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
101	Dic₅₃	—	2	4	no	—
102	Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
103	Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
104	#2905	—	53	1	no	maximal, Sylow
105	C₂×C₅₃	—	2	2	no	—
106	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₅₃	#568	1	2	yes	—
108	D₅₃	—	2	2	no	—
109	D₅₃	—	2	2	no	—
110	C₂²×M₁₆	—	53	1	no	—
111	C₂²×M₁₆	—	53	1	no	—
112	C₂×#117	—	53	1	no	—
113	#424	—	53	1	no	—
114	#424	—	53	1	no	—
115	#424	—	53	1	no	—
116	#424	—	53	4	no	—
117	#424	D₅₃	1	1	yes	characteristic, complemented
118	#423	—	53	4	no	—
119	C₅₃	#2905	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₄	—	53	1	no	—
121	C₂×M₁₆	—	53	1	no	—
122	C₂×M₁₆	—	53	2	no	—
123	C₂×M₁₆	—	53	2	no	—
124	C₂×M₁₆	—	53	2	no	—
125	C₂×M₁₆	—	106	4	no	—
126	C₂×M₁₆	—	106	4	no	—
127	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
128	C₂²×C₈	—	53	1	no	—
129	C₂²×C₈	—	53	2	no	—
130	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
131	#117	—	53	1	no	—
132	#117	—	53	2	no	—
133	#117	—	53	2	no	—
134	#117	—	53	2	no	—
135	#117	C₂×D₅₃	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	53	2	no	—
137	C₂×C₂²⋊C₄	—	53	2	no	—
138	C₂×C₄²	—	53	2	no	—
139	C₄⋊C₈	—	53	4	no	—
140	C₄⋊C₈	—	53	4	no	—
141	C₄⋊C₈	—	53	4	no	—
142	C₄⋊C₈	C₂×D₅₃	1	4	yes	complemented
143	C₂⁴	—	53	1	no	—
144	C₂²×C₄	—	53	1	no	—
145	C₂²×C₄	—	53	1	no	—
146	C₂²×C₄	—	53	1	no	—
147	C₂²×C₄	—	53	4	no	—
148	C₂²×C₄	—	53	4	no	—
149	C₂²×C₄	—	53	1	no	—
150	C₂²×C₄	—	53	1	no	—
151	C₂²×C₄	—	53	4	no	—
152	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
153	M₁₆	—	2	2	no	—
154	M₁₆	—	106	4	no	—
155	M₁₆	—	106	4	no	—
156	M₁₆	—	106	4	no	—
157	M₁₆	—	106	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	53	2	no	—
160	C₂×C₈	—	53	2	no	—
161	C₂×C₈	—	106	4	no	—
162	C₂×C₈	—	106	2	no	—
163	C₂×C₈	C₂²×D₅₃	1	2	yes	—
164	C₂×C₈	C₂²×D₅₃	1	2	yes	—
165	C₄⋊C₄	—	53	4	no	—
166	C₄⋊C₄	—	53	2	no	—
167	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
168	C₂²⋊C₄	—	53	2	no	—
169	C₂²⋊C₄	—	53	4	no	—
170	C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
171	C₄²	—	53	4	no	—
172	C₄²	—	53	2	no	—
173	C₄²	C₂²×D₅₃	1	2	yes	—
174	C₂³	—	53	1	no	—
175	C₂³	—	53	1	no	—
176	C₂³	—	53	4	no	—
177	C₂³	—	212	2	no	—
178	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
179	C₂×C₄	—	53	4	no	—
180	C₂×C₄	—	53	1	no	—
181	C₂×C₄	—	53	1	no	—
182	C₂×C₄	—	53	4	no	—
183	C₂×C₄	—	53	4	no	—
184	C₂×C₄	—	53	2	no	—
185	C₂×C₄	—	53	2	no	—
186	C₂×C₄	—	53	2	no	—
187	C₂×C₄	—	53	2	no	—
188	C₂×C₄	—	53	4	no	—
189	C₂×C₄	—	212	4	no	—
190	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
191	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
192	C₂×C₄	D₄×D₅₃	1	2	yes	—
193	C₂×C₄	Q₈×D₅₃	1	2	yes	—
194	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	—	2	2	no	—
197	C₈	—	106	2	no	—
198	C₈	—	106	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	53	2	no	—
201	C₂²	—	53	2	no	—
202	C₂²	—	106	2	no	—
203	C₂²	—	106	2	no	—
204	C₂²	—	212	4	no	—
205	C₂²	C₄⋊C₄×D₅₃	1	2	yes	—
206	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	53	2	no	—
209	C₄	—	53	2	no	—
210	C₄	—	106	4	no	—
211	C₄	C₄⋊C₄×D₅₃	1	2	yes	—
212	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
213	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	—	106	2	no	—
216	C₂	—	106	2	no	—
217	C₂	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
218	C₂	#16811388	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#424	C₂
hypercenter	#424	D₅₃
hypocenter	C₅₃	#2905
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
4. #16810651	C₂	semidirect
5. #16810650	C₂	semidirect
6. #16810649	C₂	semidirect
7. #16810648	C₂	semidirect
8. #16810645	C₂	semidirect
9. C₅₃×#424	C₂	semidirect
26. C₄⋊#88322	C₂²	semidirect
27. #13133146	C₂²	semidirect
28. #13133146	C₂²	semidirect
32. C₄⋊C₈×C₅₃	C₂²	semidirect
117. #424	D₅₃	semidirect
119. C₅₃	#2905	semidirect
142. C₄⋊C₈	C₂×D₅₃	semidirect