Subgroups — #20283863

Order 6784.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811160	C₂	1	1	yes	maximal, characteristic
2	#16810754	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810702	C₂	1	2	yes	maximal, complemented
4	#16810700	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810693	C₂	1	2	yes	maximal, complemented
6	C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810682	C₂	1	2	yes	maximal, complemented
8	#16810674	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810648	C₂	1	2	yes	maximal, complemented
10	#16810645	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719912	C₂²	1	2	yes	—
13	C₂×#11719912	C₂²	1	2	yes	—
14	C₂×C₈×D₅₃	C₂²	1	2	yes	—
15	#13133212	C₄	1	2	yes	—
16	#13133208	C₄	1	1	yes	characteristic
17	#13133206	C₄	1	1	yes	characteristic
18	#13133202	C₄	1	2	yes	—
19	#13133193	C₄	1	1	yes	characteristic
20	#13133192	C₄	1	1	yes	characteristic
21	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
22	C₄×D₂₁₂	C₂²	1	2	yes	—
23	#13133174	C₂²	1	2	yes	—
24	C₄²×D₅₃	C₂²	1	1	yes	characteristic
25	C₄⋊#88322	C₂²	1	1	yes	characteristic
26	C₄⋊#88322	C₂²	1	2	yes	—
27	#13133146	C₂²	1	2	yes	—
28	#13133146	C₂²	1	1	yes	characteristic
29	#13133146	C₂²	1	1	yes	characteristic
30	#13133129	C₂²	1	1	yes	characteristic
31	#13133129	C₂²	1	1	yes	characteristic
32	#13133129	C₂²	1	2	yes	—
33	#13133129	C₂²	1	2	yes	—
34	#13133128	C₂²	1	2	yes	—
35	C₄×#88322	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
39	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₅₃	C₂²	1	2	yes	—
41	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
43	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₅₃	C₂³	1	2	yes	—
48	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂³	1	2	yes	—
51	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
53	C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₄²×C₅₃	C₂³	1	2	yes	—
55	#11719921	C₂×C₄	1	2	yes	—
56	#11719921	C₂×C₄	1	2	yes	—
57	#11719921	C₂×C₄	1	2	yes	—
58	#11719921	C₂×C₄	1	2	yes	—
59	#11719921	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
62	#11719919	C₂×C₄	1	2	yes	—
63	#11719919	C₂×C₄	1	2	yes	—
64	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
67	C₂×#88322	C₂³	1	1	yes	characteristic
68	C₂×#88322	C₂³	1	1	yes	characteristic
69	C₂×#88322	C₂³	1	2	yes	—
70	#11719912	—	4	2	no	—
71	#11719912	—	4	2	no	—
72	C₈×D₅₃	—	4	2	no	—
73	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
75	Q₈×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
85	D₂₁₂	—	2	2	no	—
86	D₂₁₂	—	4	2	no	—
87	C₄×D₅₃	—	2	2	no	—
88	C₄×D₅₃	—	2	2	no	—
89	C₄×D₅₃	—	2	1	no	—
90	C₄×D₅₃	—	2	1	no	—
91	C₄×D₅₃	—	2	2	no	—
92	C₄×D₅₃	—	4	1	no	—
93	C₈×C₅₃	—	2	1	no	—
94	C₈×C₅₃	—	2	2	no	—
95	C₈×C₅₃	—	2	1	no	—
96	#88322	—	2	2	no	—
97	#88322	—	2	1	no	—
98	#88322	—	2	1	no	—
99	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×D₅₃	—	2	2	no	—
101	C₂×D₅₃	—	2	1	no	—
102	C₂×D₅₃	—	2	1	no	—
103	C₂×D₅₃	—	2	2	no	—
104	C₂×D₅₃	—	2	1	no	—
105	C₂×D₅₃	—	2	2	no	—
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	#131	1	2	yes	—
110	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
111	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
112	Dic₅₃	—	2	2	no	—
113	Dic₅₃	—	2	1	no	—
114	Dic₅₃	—	2	1	no	—
115	Dic₅₃	—	2	1	no	—
116	#2993	—	53	1	no	maximal, Sylow
117	C₂×C₅₃	#520	1	1	yes	characteristic
118	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
119	C₂×C₅₃	#568	1	1	yes	characteristic
120	D₅₃	—	4	1	no	—
121	D₅₃	—	4	2	no	—
122	#529	—	53	1	no	—
123	#446	—	53	1	no	—
124	C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
125	#436	—	53	1	no	—
126	#436	—	53	2	no	—
127	#435	—	53	2	no	—
128	C₈×D₄	—	53	1	no	—
129	#433	—	53	2	no	—
130	#431	—	53	1	no	—
131	#424	—	53	2	no	—
132	#423	—	53	1	no	—
133	C₅₃	#2993	1	1	yes	characteristic, complemented, Sylow
134	C₂×M₁₆	—	53	2	no	—
135	C₂×M₁₆	—	53	2	no	—
136	C₂²×C₈	—	53	2	no	—
137	#126	—	53	2	no	—
138	#125	—	53	1	no	—
139	#124	—	53	1	no	—
140	#123	—	53	2	no	—
141	C₂²⋊Q₈	—	53	1	no	—
142	#121	—	53	1	no	—
143	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
144	C₄×D₄	—	53	1	no	—
145	C₄×D₄	—	53	2	no	—
146	#117	—	53	2	no	—
147	C₂×C₄²	—	53	1	no	—
148	C₄⋊C₈	—	53	2	no	—
149	C₄⋊C₈	—	53	1	no	—
150	C₄⋊C₈	—	53	1	no	—
151	C₄⋊C₈	—	53	1	no	—
152	C₄⋊C₈	—	53	2	no	—
153	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
155	C₂²⋊C₈	—	53	1	no	—
156	C₂²⋊C₈	—	53	1	no	—
157	C₂²⋊C₈	—	53	2	no	—
158	C₂²⋊C₈	—	53	2	no	—
159	#97	—	53	2	no	—
160	C₄×C₈	—	53	1	no	—
161	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
162	C₄×C₈	C₂×D₅₃	1	2	yes	—
163	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
164	C₂×D₄	—	53	1	no	—
165	C₂×D₄	—	53	2	no	—
166	C₂²×C₄	—	53	1	no	—
167	C₂²×C₄	—	53	1	no	—
168	C₂²×C₄	—	53	2	no	—
169	C₂²×C₄	—	53	1	no	—
170	M₁₆	—	212	2	no	—
171	M₁₆	—	212	2	no	—
172	C₂×C₈	—	53	1	no	—
173	C₂×C₈	—	53	1	no	—
174	C₂×C₈	—	53	2	no	—
175	C₂×C₈	—	212	2	no	—
176	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₅₃	1	2	yes	—
179	C₄⋊C₄	—	53	2	no	—
180	C₄⋊C₄	—	53	2	no	—
181	C₄⋊C₄	—	53	1	no	—
182	C₄⋊C₄	—	53	2	no	—
183	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
184	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
185	C₂²⋊C₄	—	53	2	no	—
186	C₂²⋊C₄	—	53	2	no	—
187	C₂²⋊C₄	—	53	2	no	—
188	C₂²⋊C₄	—	53	2	no	—
189	C₂²⋊C₄	—	53	2	no	—
190	C₄²	—	53	1	no	—
191	C₄²	—	53	1	no	—
192	C₄²	—	53	1	no	—
193	C₄²	C₂²×D₅₃	1	1	yes	characteristic
194	C₄²	C₂²×D₅₃	1	2	yes	—
195	C₂³	—	53	2	no	—
196	C₂³	—	53	1	no	—
197	Q₈	—	2	2	no	—
198	D₄	—	106	2	no	—
199	D₄	—	212	2	no	—
200	C₂×C₄	—	53	1	no	—
201	C₂×C₄	—	53	1	no	—
202	C₂×C₄	—	53	1	no	—
203	C₂×C₄	—	53	2	no	—
204	C₂×C₄	—	106	2	no	—
205	C₂×C₄	—	106	2	no	—
206	C₂×C₄	—	106	1	no	—
207	C₂×C₄	—	106	1	no	—
208	C₂×C₄	—	106	2	no	—
209	C₂×C₄	—	212	1	no	—
210	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
213	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	2	no	—
217	C₈	—	2	1	no	—
218	C₈	—	106	2	no	—
219	C₈	—	106	1	no	—
220	C₈	—	106	1	no	—
221	C₂²	—	106	2	no	—
222	C₂²	—	106	1	no	—
223	C₂²	—	106	1	no	—
224	C₂²	—	106	2	no	—
225	C₂²	—	106	1	no	—
226	C₂²	—	106	2	no	—
227	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	106	1	no	—
232	C₄	—	106	2	no	—
233	C₄	—	106	1	no	—
234	C₄	—	106	1	no	—
235	C₄	#13133223	1	2	yes	—
236	C₄	#13133263	1	1	yes	characteristic
237	C₄	#13133265	1	1	yes	characteristic
238	C₂	—	212	1	no	—
239	C₂	—	212	2	no	—
240	C₂	#16811105	1	1	yes	characteristic
241	C₂	C₂×#13133223	1	1	yes	characteristic
242	C₂	#16811388	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₈×Q₈×C₅₃	C₂
hypercenter	C₈×Q₈	D₅₃
hypocenter	C₅₃	#2993
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #16810754	C₂	semidirect
3. #16810702	C₂	semidirect
4. #16810700	C₂	semidirect
5. #16810693	C₂	semidirect
6. C₈×D₂₁₂	C₂	semidirect
7. #16810682	C₂	semidirect
8. #16810674	C₂	semidirect
9. #16810648	C₂	semidirect
10. #16810645	C₂	semidirect
11. C₈×Q₈×C₅₃	C₂	semidirect
124. C₈×Q₈	D₅₃	semidirect
133. C₅₃	#2993	semidirect