Subgroups — #20274692

Order 6784.

400 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133129	C₂	1	2	yes	maximal
4	#16810377	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₄×D₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
9	C₂²×#88322	C₂²	1	2	yes	—
10	C₂×C₄×Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₄²×D₅₃	—	2	2	no	—
14	C₄²×D₅₃	—	2	2	no	—
15	C₄⋊#88322	—	2	2	no	—
16	#13133129	—	2	4	no	—
17	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₅₃	—	2	2	no	—
20	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₅₃	—	2	2	no	—
28	C₂×C₄×D₅₃	—	2	2	no	—
29	C₂×C₄×D₅₃	—	2	2	no	—
30	C₂×C₄×D₅₃	—	2	2	no	—
31	C₂×C₄×D₅₃	—	2	1	no	—
32	C₂×C₄×D₅₃	—	2	1	no	—
33	C₂×C₄×D₅₃	—	2	1	no	—
34	C₂×C₄×D₅₃	—	2	1	no	—
35	C₂×C₄×D₅₃	—	2	2	no	—
36	C₂×C₄×D₅₃	C₈	1	4	yes	complemented
37	C₂×C₄×D₅₃	D₄	1	2	yes	—
38	C₂×C₄×D₅₃	Q₈	1	2	yes	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₄²×C₅₃	D₄	1	2	yes	—
43	C₄²×C₅₃	D₄	1	2	yes	—
44	C₄×Dic₅₃	—	2	2	no	—
45	C₄×Dic₅₃	—	2	2	no	—
46	C₄×Dic₅₃	—	2	1	no	—
47	C₄×Dic₅₃	—	2	1	no	—
48	C₂×#88322	—	2	2	no	—
49	C₂×#88322	—	2	2	no	—
50	C₂×#88322	—	2	2	no	—
51	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×D₅₃	—	2	1	no	—
53	C₂²×D₅₃	—	2	1	no	—
54	C₂²×D₅₃	—	2	1	no	—
55	C₂²×D₅₃	—	2	1	no	—
56	C₂²×D₅₃	C₄⋊C₄	1	2	yes	—
57	C₂²×D₅₃	C₂×C₈	1	2	yes	—
58	C₂²×D₅₃	M₁₆	1	2	yes	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
64	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
71	C₂×Dic₅₃	—	2	1	no	—
72	C₂×Dic₅₃	—	2	1	no	—
73	C₂×Dic₅₃	—	2	1	no	—
74	C₂×Dic₅₃	—	2	1	no	—
75	C₂×Dic₅₃	—	2	1	no	—
76	C₂×Dic₅₃	—	2	1	no	—
77	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
78	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
79	C₂×Dic₅₃	M₁₆	1	2	yes	—
80	C₄×D₅₃	—	2	8	no	—
81	C₄×D₅₃	—	2	8	no	—
82	C₄×D₅₃	—	2	2	no	—
83	C₄×D₅₃	—	2	2	no	—
84	C₄×D₅₃	—	2	2	no	—
85	C₄×D₅₃	—	2	2	no	—
86	C₈×C₅₃	—	4	2	no	—
87	#88322	—	4	2	no	—
88	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₅₃	#117	1	1	yes	characteristic
91	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
92	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
93	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂×D₅₃	—	2	2	no	—
96	C₂×D₅₃	—	2	2	no	—
97	C₂×D₅₃	—	2	2	no	—
98	C₂×D₅₃	—	2	2	no	—
99	C₂×D₅₃	—	2	2	no	—
100	C₂×D₅₃	—	2	2	no	—
101	C₂×D₅₃	C₄⋊C₈	1	4	yes	complemented
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	C₂²⋊C₈	1	4	yes	complemented
104	C₄×C₅₃	#121	1	1	yes	characteristic
105	C₄×C₅₃	#121	1	1	yes	characteristic
106	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
108	Dic₅₃	—	2	2	no	—
109	Dic₅₃	—	2	2	no	—
110	Dic₅₃	—	2	2	no	—
111	#1797	—	53	1	no	maximal, Sylow
112	C₂×C₅₃	#380	1	1	yes	characteristic
113	C₂×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₂×C₅₃	#407	1	1	yes	characteristic
115	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₂×C₅₃	#423	1	1	yes	characteristic
117	C₂×C₅₃	#431	1	1	yes	characteristic
118	C₂×C₅₃	#432	1	1	yes	characteristic
119	D₅₃	—	2	4	no	—
120	C₂²×C₄²	—	53	1	no	—
121	C₂×C₄⋊C₈	—	53	1	no	—
122	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic, complemented
123	C₂×C₂²⋊C₈	—	53	2	no	—
124	#336	—	53	2	no	—
125	C₅₃	#1797	1	1	yes	characteristic, complemented, Sylow
126	C₂³×C₄	—	53	1	no	—
127	C₂³×C₄	—	53	1	no	—
128	C₂³×C₄	—	53	1	no	—
129	C₂²×C₈	—	53	2	no	—
130	C₂²×C₈	C₂×D₅₃	1	2	yes	—
131	C₂×C₄²	—	53	1	no	—
132	C₂×C₄²	—	53	1	no	—
133	C₂×C₄²	—	53	1	no	—
134	C₂×C₄²	—	106	2	no	—
135	C₂×C₄²	—	106	2	no	—
136	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	106	2	no	—
139	C₂²⋊C₈	—	106	4	no	—
140	C₂⁴	—	53	1	no	—
141	C₂²×C₄	—	53	1	no	—
142	C₂²×C₄	—	53	1	no	—
143	C₂²×C₄	—	53	1	no	—
144	C₂²×C₄	—	53	2	no	—
145	C₂²×C₄	—	53	2	no	—
146	C₂²×C₄	—	53	4	no	—
147	C₂²×C₄	—	106	2	no	—
148	C₂²×C₄	—	106	2	no	—
149	C₂²×C₄	—	106	1	no	—
150	C₂²×C₄	—	106	1	no	—
151	C₂²×C₄	—	106	1	no	—
152	C₂²×C₄	—	106	1	no	—
153	C₂²×C₄	—	106	2	no	—
154	C₂²×C₄	—	106	2	no	—
155	C₂²×C₄	—	106	2	no	—
156	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
157	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	106	2	no	—
163	C₂×C₈	—	106	2	no	—
164	C₂×C₈	—	106	2	no	—
165	C₄²	—	106	1	no	—
166	C₄²	—	106	1	no	—
167	C₄²	—	106	2	no	—
168	C₄²	—	106	2	no	—
169	C₄²	D₂₁₂	1	2	yes	—
170	C₄²	C₂²⋊D₅₃	1	2	yes	—
171	C₂³	—	53	2	no	—
172	C₂³	—	53	2	no	—
173	C₂³	—	53	2	no	—
174	C₂³	—	106	1	no	—
175	C₂³	—	106	1	no	—
176	C₂³	—	106	1	no	—
177	C₂³	—	106	1	no	—
178	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	53	2	no	—
182	C₂×C₄	—	53	2	no	—
183	C₂×C₄	—	53	2	no	—
184	C₂×C₄	—	106	8	no	—
185	C₂×C₄	—	106	1	no	—
186	C₂×C₄	—	106	1	no	—
187	C₂×C₄	—	106	1	no	—
188	C₂×C₄	—	106	1	no	—
189	C₂×C₄	—	106	1	no	—
190	C₂×C₄	—	106	1	no	—
191	C₂×C₄	—	106	8	no	—
192	C₂×C₄	—	106	2	no	—
193	C₂×C₄	—	106	2	no	—
194	C₂×C₄	—	106	2	no	—
195	C₂×C₄	—	106	2	no	—
196	C₂×C₄	C₈×D₅₃	1	2	yes	—
197	C₂×C₄	#11719912	1	2	yes	—
198	C₂×C₄	#11719921	1	2	yes	—
199	C₂×C₄	#11719921	1	2	yes	—
200	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
201	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
202	C₂×C₄	#11719947	1	1	yes	characteristic
203	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
204	C₂×C₄	#11719949	1	1	yes	characteristic
205	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
206	C₈	—	4	2	no	—
207	C₈	—	212	2	no	—
208	C₂²	—	53	4	no	—
209	C₂²	—	106	2	no	—
210	C₂²	—	106	2	no	—
211	C₂²	—	106	2	no	—
212	C₂²	—	106	2	no	—
213	C₂²	—	106	2	no	—
214	C₂²	—	106	2	no	—
215	C₂²	C₂×#11719921	1	1	yes	characteristic
216	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
217	C₂²	#13133175	1	1	yes	characteristic
218	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
219	C₂²	M₁₆×D₅₃	1	1	yes	characteristic
220	C₂²	M₁₆×D₅₃	1	1	yes	characteristic
221	C₂²	C₂×#11719912	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	106	2	no	—
224	C₄	—	106	2	no	—
225	C₄	—	106	2	no	—
226	C₄	#13133129	1	4	yes	complemented
227	C₄	#13133188	1	1	yes	characteristic
228	C₄	#13133192	1	1	yes	characteristic
229	C₄	#13133193	1	1	yes	characteristic
230	C₄	#13133197	1	1	yes	characteristic
231	C₂	—	106	4	no	—
232	C₂	#16810480	1	1	yes	characteristic
233	C₂	C₂×#13133129	1	1	yes	characteristic
234	C₂	#16810586	1	1	yes	characteristic
235	C₂	C₄⋊C₈×D₅₃	1	1	yes	characteristic
236	C₂	#16810645	1	1	yes	characteristic
237	C₂	#16810675	1	1	yes	characteristic
238	C₂	#16810679	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₈
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#1797
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊#88322	C₂	semidirect
4. #16810377	C₂	semidirect
5. C₂×C₄⋊C₈×C₅₃	C₂	semidirect
36. C₂×C₄×D₅₃	C₈	semidirect
101. C₂×D₅₃	C₄⋊C₈	semidirect
103. C₄×C₅₃	C₂²⋊C₈	semidirect
122. C₂×C₄⋊C₈	D₅₃	semidirect
125. C₅₃	#1797	semidirect
226. C₄	#13133129	semidirect