Subgroups — #20283578

Order 6784.

456 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#16811099	C₂	1	2	yes	maximal
3	#16810728	C₂	1	2	yes	maximal, complemented
4	#16810727	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810717	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810660	C₂	1	1	yes	maximal, characteristic
9	#16810630	C₂	1	2	yes	maximal
10	#16810577	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₅₃	C₂²	1	2	yes	—
13	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
14	C₂×#11719925	C₂²	1	2	yes	—
15	Q₁₆×D₅₃	C₄	1	8	yes	complemented
16	C₂×#11719912	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
18	Q₈×Dic₅₃	C₂²	1	2	yes	—
19	C₄×Dic₁₀₆	—	2	2	no	—
20	C₄×Dic₁₀₆	C₂²	1	2	yes	—
21	#13133182	—	2	2	no	—
22	#13133182	C₂²	1	2	yes	—
23	#13133175	—	2	2	no	—
24	#13133175	C₂²	1	2	yes	—
25	#13133174	—	2	2	no	—
26	#13133174	C₂²	1	1	yes	characteristic
27	#13133151	C₂²	1	1	yes	characteristic
28	#13133148	C₂²	1	2	yes	—
29	#13133142	C₂²	1	2	yes	—
30	#13133141	C₂²	1	2	yes	—
31	#13133140	C₂²	1	2	yes	—
32	#13133128	C₂²	1	1	yes	characteristic
33	#13133127	C₂²	1	1	yes	characteristic
34	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
37	#107×C₅₃	C₂²	1	1	yes	characteristic
38	#103×C₅₃	C₂²	1	2	yes	—
39	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
41	Q₈×D₅₃	—	2	4	no	—
42	Q₈×D₅₃	C₂×C₄	1	8	yes	—
43	C₂×C₄×D₅₃	—	2	2	no	—
44	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₀₆	—	2	2	no	—
46	C₂×Dic₁₀₆	C₂³	1	2	yes	—
47	Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
48	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
52	C₄²×C₅₃	—	2	2	no	—
53	C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	#11719925	C₂×C₄	1	8	yes	—
55	#11719921	—	2	2	no	—
56	#11719921	D₄	1	1	yes	characteristic
57	#11719921	D₄	1	1	yes	characteristic
58	C₄⋊Dic₅₃	—	2	2	no	—
59	C₄⋊Dic₅₃	C₂³	1	2	yes	—
60	#11719919	—	2	2	no	—
61	#11719919	—	2	2	no	—
62	#11719919	D₄	1	1	yes	characteristic
63	#11719919	D₄	1	1	yes	characteristic
64	C₄×Dic₅₃	—	2	2	no	—
65	C₄×Dic₅₃	—	2	2	no	—
66	C₄×Dic₅₃	—	2	2	no	—
67	C₄×Dic₅₃	—	2	2	no	—
68	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
69	C₂×#88322	C₂³	1	1	yes	characteristic
70	C₂×#88322	C₂³	1	1	yes	characteristic
71	Dic₂₁₂	C₂×C₄	1	4	yes	—
72	#11719912	—	2	2	no	—
73	C₈×D₅₃	C₂×C₄	1	4	yes	—
74	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
75	Q₈×C₅₃	—	2	2	no	—
76	Q₈×C₅₃	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₅₃	—	2	2	no	—
83	C₂×Dic₅₃	—	2	2	no	—
84	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
87	C₄×D₅₃	—	2	8	no	—
88	C₄×D₅₃	C₂²×C₄	1	2	yes	—
89	C₄×D₅₃	#41	1	2	yes	—
90	Dic₁₀₆	—	2	2	no	—
91	Dic₁₀₆	—	2	8	no	—
92	Dic₁₀₆	C₂²×C₄	1	4	yes	—
93	C₈×C₅₃	—	2	1	no	—
94	C₈×C₅₃	C₂²×C₄	1	2	yes	—
95	#88322	—	2	1	no	—
96	#88322	C₂²×C₄	1	2	yes	—
97	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×D₅₃	—	2	1	no	—
99	C₂×D₅₃	—	2	1	no	—
100	C₂×D₅₃	C₄×D₄	1	2	yes	—
101	C₄×C₅₃	—	2	4	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	4	2	no	—
105	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
107	Dic₅₃	—	2	4	no	—
108	Dic₅₃	—	2	1	no	—
109	Dic₅₃	—	2	1	no	—
110	Dic₅₃	—	4	2	no	—
111	Dic₅₃	C₄×D₄	1	2	yes	—
112	#2950	—	53	1	no	maximal, Sylow
113	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
114	C₂×C₅₃	#578	1	1	yes	characteristic
115	C₂×C₅₃	#578	1	1	yes	characteristic
116	D₅₃	—	2	2	no	—
117	C₂²×Q₁₆	—	53	1	no	—
118	#519	—	53	2	no	—
119	#441	—	53	2	no	—
120	#441	—	53	2	no	—
121	C₄×Q₁₆	—	53	1	no	—
122	C₄×Q₁₆	—	53	1	no	—
123	C₄×Q₁₆	—	53	1	no	—
124	C₄×Q₁₆	D₅₃	1	1	yes	characteristic, complemented
125	#427	—	53	1	no	—
126	#419	—	53	2	no	—
127	#405	—	53	1	no	—
128	C₅₃	#2950	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	53	2	no	—
130	C₂×Q₁₆	—	53	2	no	—
131	C₂×Q₁₆	—	53	8	no	—
132	C₂×Q₁₆	—	53	1	no	—
133	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
134	C₂×M₁₆	—	53	1	no	—
135	C₂²×C₈	—	53	1	no	—
136	C₄×Q₈	—	53	2	no	—
137	C₄×Q₈	—	53	2	no	—
138	C₄×Q₈	—	53	2	no	—
139	C₄×Q₈	—	106	2	no	—
140	C₄×Q₈	—	106	2	no	—
141	C₄×Q₈	C₂×D₅₃	1	2	yes	—
142	#117	—	53	1	no	—
143	#117	—	53	2	no	—
144	#117	—	106	2	no	—
145	#117	—	106	2	no	—
146	#107	—	53	1	no	—
147	#107	C₂×D₅₃	1	1	yes	characteristic
148	#106	—	53	2	no	—
149	#103	—	53	2	no	—
150	#103	—	53	2	no	—
151	#103	—	53	2	no	—
152	#103	C₂×D₅₃	1	2	yes	—
153	#97	—	53	1	no	—
154	#97	—	53	1	no	—
155	C₄×C₈	—	53	1	no	—
156	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
157	C₂×Q₈	—	53	8	no	—
158	C₂×Q₈	—	53	2	no	—
159	C₂×Q₈	—	106	2	no	—
160	C₂×Q₈	—	106	4	no	—
161	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
162	C₂²×C₄	—	53	1	no	—
163	C₂²×C₄	—	106	2	no	—
164	Q₁₆	—	53	8	no	—
165	Q₁₆	—	53	4	no	—
166	Q₁₆	C₄×D₅₃	1	4	yes	—
167	M₁₆	—	106	2	no	—
168	C₂×C₈	—	53	1	no	—
169	C₂×C₈	—	53	4	no	—
170	C₂×C₈	—	53	1	no	—
171	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	53	1	no	—
175	C₄⋊C₄	—	53	1	no	—
176	C₄⋊C₄	—	53	2	no	—
177	C₄⋊C₄	—	106	2	no	—
178	C₄⋊C₄	—	106	2	no	—
179	C₄⋊C₄	—	106	2	no	—
180	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
181	C₂²⋊C₄	—	53	1	no	—
182	C₂²⋊C₄	—	53	1	no	—
183	C₂²⋊C₄	—	106	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	53	1	no	—
186	C₄²	—	106	2	no	—
187	C₄²	—	106	2	no	—
188	C₄²	—	106	2	no	—
189	C₄²	—	106	2	no	—
190	C₄²	C₂²×D₅₃	1	1	yes	characteristic
191	C₂³	—	53	1	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	53	4	no	—
194	Q₈	—	106	8	no	—
195	Q₈	—	106	2	no	—
196	Q₈	C₂×C₄×D₅₃	1	4	yes	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	53	1	no	—
200	C₂×C₄	—	53	2	no	—
201	C₂×C₄	—	53	2	no	—
202	C₂×C₄	—	53	1	no	—
203	C₂×C₄	—	53	1	no	—
204	C₂×C₄	—	106	8	no	—
205	C₂×C₄	—	106	2	no	—
206	C₂×C₄	—	106	2	no	—
207	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
208	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
209	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	53	2	no	—
212	C₈	—	106	1	no	—
213	C₈	C₂×C₄×D₅₃	1	2	yes	—
214	C₂²	—	53	2	no	—
215	C₂²	—	106	1	no	—
216	C₂²	—	106	1	no	—
217	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
218	C₄	—	2	4	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	2	no	—
222	C₄	—	53	2	no	—
223	C₄	—	106	4	no	—
224	C₄	—	106	1	no	—
225	C₄	—	106	1	no	—
226	C₄	—	212	2	no	—
227	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
228	C₄	#41×D₅₃	1	1	yes	characteristic
229	C₂	—	106	2	no	—
230	C₂	C₄×D₄×D₅₃	1	1	yes	characteristic
231	C₂	#16811436	1	1	yes	characteristic
232	C₂	#16811437	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₄×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₄×Q₁₆	D₅₃
hypocenter	C₅₃	#2950
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #16810728	C₂	semidirect
4. #16810727	C₂	semidirect
5. C₄×Dic₂₁₂	C₂	semidirect
6. Q₁₆×Dic₅₃	C₂	semidirect
7. #16810717	C₂	semidirect
11. C₄×Q₁₆×C₅₃	C₂	semidirect
15. Q₁₆×D₅₃	C₄	semidirect
124. C₄×Q₁₆	D₅₃	semidirect
128. C₅₃	#2950	semidirect