Subgroups — #20279292

Order 6784.

388 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	#16810565	C₂	1	2	yes	maximal
2	#16810563	C₂	1	2	yes	maximal
3	#16810530	C₂	1	1	yes	maximal, characteristic
4	#16810530	C₂	1	1	yes	maximal, characteristic
5	#16810519	C₂	1	1	yes	maximal, characteristic
6	#16810510	C₂	1	2	yes	maximal
7	#16810508	C₂	1	2	yes	maximal
8	#16810493	C₂	1	1	yes	maximal, characteristic
9	C₄⋊C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
10	C₄×#11719919	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#391	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
16	C₂×#11719919	C₂²	1	1	yes	characteristic
17	C₂×#11719919	C₂²	1	2	yes	—
18	C₂×#11719919	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
22	#13133124	C₂²	1	1	yes	characteristic
23	#13133124	C₂²	1	2	yes	—
24	#13133124	C₂²	1	2	yes	—
25	#13133124	C₂²	1	2	yes	—
26	#13133124	C₂²	1	2	yes	—
27	#13133124	C₂²	1	2	yes	—
28	#13133124	C₂²	1	2	yes	—
29	#13133124	C₂²	1	1	yes	characteristic
30	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
33	#95×C₅₃	C₂²	1	2	yes	—
34	#95×C₅₃	C₂²	1	2	yes	—
35	C₂×Q₈×C₅₃	—	2	2	no	—
36	C₂×Q₈×C₅₃	—	2	2	no	—
37	C₂×Q₈×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₅₃	C₂³	1	2	yes	—
46	C₂²×Dic₅₃	C₂³	1	2	yes	—
47	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
48	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₅₃	—	2	2	no	—
50	C₄²×C₅₃	—	2	2	no	—
51	C₄⋊Dic₅₃	—	2	2	no	—
52	C₄⋊Dic₅₃	—	2	2	no	—
53	C₄⋊Dic₅₃	—	2	4	no	—
54	C₄⋊Dic₅₃	—	4	2	no	—
55	#11719919	—	2	2	no	—
56	#11719919	—	4	2	no	—
57	#11719919	Q₈	1	4	yes	complemented
58	C₄×Dic₅₃	—	2	2	no	—
59	C₄×Dic₅₃	—	2	2	no	—
60	C₄×Dic₅₃	—	4	2	no	—
61	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
62	Q₈×C₅₃	—	4	4	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
74	C₂×C₄×C₅₃	#41	1	2	yes	—
75	C₂×C₄×C₅₃	#41	1	2	yes	—
76	C₂×C₄×C₅₃	#41	1	2	yes	—
77	C₂×C₄×C₅₃	#41	1	2	yes	—
78	C₂×Dic₅₃	—	2	1	no	—
79	C₂×Dic₅₃	—	2	1	no	—
80	C₂×Dic₅₃	—	2	1	no	—
81	C₂×Dic₅₃	—	2	1	no	—
82	C₂×Dic₅₃	—	2	1	no	—
83	C₂×Dic₅₃	—	2	1	no	—
84	C₂×Dic₅₃	—	2	1	no	—
85	C₂×Dic₅₃	—	2	2	no	—
86	C₂×Dic₅₃	—	2	2	no	—
87	C₂×Dic₅₃	—	2	2	no	—
88	C₂×Dic₅₃	—	2	2	no	—
89	C₂×Dic₅₃	—	2	2	no	—
90	C₂×Dic₅₃	—	2	1	no	—
91	C₂×Dic₅₃	—	2	2	no	—
92	C₂×Dic₅₃	—	2	1	no	—
93	C₂×Dic₅₃	—	2	1	no	—
94	C₂×Dic₅₃	C₂×Q₈	1	2	yes	—
95	C₂×Dic₅₃	C₂×Q₈	1	2	yes	—
96	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
97	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
98	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
99	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
100	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
101	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₅₃	—	2	2	no	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	4	2	no	—
106	C₄×C₅₃	—	4	2	no	—
107	C₄×C₅₃	#126	1	4	yes	—
108	Dic₅₃	—	2	2	no	—
109	Dic₅₃	—	2	2	no	—
110	Dic₅₃	—	4	1	no	—
111	Dic₅₃	—	4	1	no	—
112	Dic₅₃	—	4	2	no	—
113	Dic₅₃	—	4	2	no	—
114	#2511	—	53	1	no	maximal, Sylow
115	C₂×C₅₃	C₂×#126	1	1	yes	characteristic
116	C₂×C₅₃	#533	1	1	yes	characteristic
117	C₂×C₅₃	#542	1	1	yes	characteristic
118	C₂×C₅₃	#552	1	1	yes	characteristic
119	C₂×C₅₃	#554	1	1	yes	characteristic
120	C₂×C₅₃	#557	1	1	yes	characteristic
121	C₂×C₅₃	#559	1	1	yes	characteristic
122	#400	—	53	2	no	—
123	#400	—	53	2	no	—
124	#391	—	53	1	no	—
125	#391	—	53	1	no	—
126	#391	D₅₃	1	1	yes	characteristic
127	#389	—	53	1	no	—
128	#387	—	53	2	no	—
129	#387	—	53	2	no	—
130	#384	—	53	1	no	—
131	C₄×C₄⋊C₄	—	53	1	no	—
132	C₄×C₄⋊C₄	—	53	1	no	—
133	C₅₃	#2511	1	1	yes	characteristic, complemented, Sylow
134	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	53	1	no	—
136	C₂×C₄⋊C₄	—	53	1	no	—
137	C₂×C₄⋊C₄	—	53	1	no	—
138	C₂×C₄⋊C₄	—	53	2	no	—
139	C₂×C₄⋊C₄	—	53	1	no	—
140	C₂×C₄⋊C₄	—	53	2	no	—
141	C₂×C₄⋊C₄	—	53	2	no	—
142	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
143	C₂×C₄²	—	53	1	no	—
144	C₂×C₄²	—	53	1	no	—
145	C₂×C₄²	—	53	2	no	—
146	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
147	#95	—	53	1	no	—
148	#95	—	53	1	no	—
149	#95	—	53	2	no	—
150	#95	—	53	2	no	—
151	#95	—	53	2	no	—
152	#95	—	53	2	no	—
153	#95	—	53	2	no	—
154	#95	—	53	2	no	—
155	#95	C₂×D₅₃	1	2	yes	—
156	#95	C₂×D₅₃	1	2	yes	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂²×C₄	—	53	1	no	—
161	C₂²×C₄	—	53	1	no	—
162	C₂²×C₄	—	53	1	no	—
163	C₂²×C₄	—	53	1	no	—
164	C₂²×C₄	—	53	2	no	—
165	C₂²×C₄	—	53	2	no	—
166	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
167	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
168	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	53	4	no	—
173	C₄⋊C₄	—	106	2	no	—
174	C₄⋊C₄	—	106	2	no	—
175	C₄⋊C₄	—	106	2	no	—
176	C₄⋊C₄	—	106	4	no	—
177	C₄⋊C₄	—	212	2	no	—
178	C₄⋊C₄	—	212	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	106	2	no	—
181	C₄²	—	106	2	no	—
182	C₄²	—	212	2	no	—
183	C₂³	C₂³×D₅₃	1	1	yes	characteristic
184	Q₈	—	4	4	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	53	2	no	—
196	C₂×C₄	—	53	2	no	—
197	C₂×C₄	—	106	1	no	—
198	C₂×C₄	—	106	1	no	—
199	C₂×C₄	—	106	1	no	—
200	C₂×C₄	—	106	1	no	—
201	C₂×C₄	—	106	1	no	—
202	C₂×C₄	—	106	1	no	—
203	C₂×C₄	—	106	2	no	—
204	C₂×C₄	—	106	2	no	—
205	C₂×C₄	—	106	1	no	—
206	C₂×C₄	—	106	2	no	—
207	C₂×C₄	—	106	2	no	—
208	C₂×C₄	—	106	2	no	—
209	C₂×C₄	—	106	2	no	—
210	C₂×C₄	—	106	1	no	—
211	C₂×C₄	—	106	1	no	—
212	C₂×C₄	—	106	1	no	—
213	C₂×C₄	#11719945	1	2	yes	—
214	C₂×C₄	#11719947	1	2	yes	—
215	C₂×C₄	#11719947	1	2	yes	—
216	C₂×C₄	Q₈×D₅₃	1	2	yes	—
217	C₂×C₄	#11719949	1	2	yes	—
218	C₂²	C₂×Q₈×D₅₃	1	1	yes	characteristic
219	C₂²	C₂×#11719947	1	1	yes	characteristic
220	C₂²	C₂×#11719947	1	1	yes	characteristic
221	C₂²	C₂×#11719945	1	1	yes	characteristic
222	C₂²	C₂×#11719949	1	1	yes	characteristic
223	C₂²	#13133263	1	1	yes	characteristic
224	C₂²	#13133264	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	106	2	no	—
230	C₄	—	106	2	no	—
231	C₄	—	212	1	no	—
232	C₄	—	212	2	no	—
233	C₄	—	212	2	no	—
234	C₄	—	212	1	no	—
235	C₄	#13133210	1	4	yes	—
236	C₂	C₂×#13133210	1	1	yes	characteristic
237	C₂	#16811178	1	1	yes	characteristic
238	C₂	#16811242	1	1	yes	characteristic
239	C₂	#16811315	1	1	yes	characteristic
240	C₂	#16811331	1	1	yes	characteristic
241	C₂	#16811351	1	1	yes	characteristic
242	C₂	#16811358	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₅₃×#391	C₂
hypercenter	#391	D₅₃
hypocenter	C₅₃	#2511
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
57. #11719919	Q₈	semidirect
133. C₅₃	#2511	semidirect