Subgroups — #20283801

Order 6784.

364 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	C₄×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#16810754	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810752	C₂	1	2	yes	maximal, complemented
4	#16810751	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810747	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810679	C₂	1	1	yes	maximal, characteristic
8	#16810678	C₂	1	2	yes	maximal
9	#16810675	C₂	1	2	yes	maximal
10	#16810674	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₅₃	C₄	1	1	yes	characteristic
13	Q₈×Dic₅₃	C₄	1	1	yes	characteristic
14	C₄×Dic₁₀₆	C₂²	1	2	yes	—
15	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
16	#13133182	C₄	1	2	yes	—
17	#13133182	C₄	1	1	yes	characteristic
18	C₄⋊C₄×D₅₃	C₄	1	2	yes	—
19	C₄⋊C₄×D₅₃	C₄	1	1	yes	characteristic
20	C₄²×D₅₃	C₂²	1	1	yes	characteristic
21	C₄²×D₅₃	C₂²	1	2	yes	—
22	C₄⋊#88322	C₂²	1	2	yes	—
23	C₄⋊#88322	C₂²	1	1	yes	characteristic
24	#13133146	C₂²	1	1	yes	characteristic
25	#13133146	C₂²	1	2	yes	—
26	#13133146	C₂²	1	1	yes	characteristic
27	#13133146	C₂²	1	2	yes	—
28	#13133129	C₂²	1	1	yes	characteristic
29	#13133129	C₂²	1	1	yes	characteristic
30	#13133129	C₂²	1	2	yes	—
31	#13133128	C₂²	1	2	yes	—
32	#13133127	C₂²	1	2	yes	—
33	C₄×#88322	C₂²	1	1	yes	characteristic
34	C₈×Dic₅₃	C₂²	1	2	yes	—
35	C₈×Dic₅₃	C₂²	1	2	yes	—
36	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
38	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
39	#97×C₅₃	C₂²	1	2	yes	—
40	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	Q₈×D₅₃	—	2	4	no	—
43	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
49	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂³	1	2	yes	—
53	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
54	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₄²×C₅₃	C₂³	1	2	yes	—
57	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
59	#11719919	C₂×C₄	1	2	yes	—
60	#11719919	C₂×C₄	1	2	yes	—
61	#11719919	C₂×C₄	1	2	yes	—
62	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
64	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
65	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₅₃	C₂³	1	2	yes	—
67	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
68	C₂×#88322	C₂³	1	1	yes	characteristic
69	C₂×#88322	C₂³	1	2	yes	—
70	C₂×#88322	C₂³	1	1	yes	characteristic
71	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
72	Q₈×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
78	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
80	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
83	C₄×D₅₃	—	2	2	no	—
84	C₄×D₅₃	—	2	4	no	—
85	C₄×D₅₃	—	2	2	no	—
86	C₄×D₅₃	—	2	1	no	—
87	C₄×D₅₃	—	2	1	no	—
88	C₄×D₅₃	—	2	4	no	—
89	Dic₁₀₆	—	2	4	no	—
90	Dic₁₀₆	M₁₆	1	4	yes	—
91	C₈×C₅₃	—	2	1	no	—
92	C₈×C₅₃	—	2	2	no	—
93	C₈×C₅₃	—	2	1	no	—
94	#88322	—	2	1	no	—
95	#88322	—	2	1	no	—
96	#88322	—	2	2	no	—
97	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₅₃	—	2	1	no	—
99	C₂×D₅₃	—	2	1	no	—
100	C₂×D₅₃	#131	1	2	yes	—
101	C₄×C₅₃	—	2	2	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	2	no	—
104	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
105	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
106	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
107	Dic₅₃	—	2	1	no	—
108	Dic₅₃	—	2	1	no	—
109	Dic₅₃	—	2	2	no	—
110	Dic₅₃	—	2	1	no	—
111	Dic₅₃	C₂×M₁₆	1	4	yes	—
112	#2984	—	53	1	no	maximal, Sylow
113	C₂×C₅₃	#519	1	1	yes	characteristic
114	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
115	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
116	D₅₃	—	2	2	no	—
117	C₂×C₄×Q₈	—	53	1	no	—
118	#446	—	53	1	no	—
119	#446	—	53	2	no	—
120	#446	—	53	2	no	—
121	#446	D₅₃	1	1	yes	characteristic, complemented
122	C₈×Q₈	—	53	1	no	—
123	C₈×Q₈	—	53	1	no	—
124	#432	—	53	1	no	—
125	#432	—	53	2	no	—
126	#431	—	53	2	no	—
127	#431	—	53	1	no	—
128	C₅₃	#2984	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	53	1	no	—
130	C₄×Q₈	—	53	1	no	—
131	C₄×Q₈	—	53	1	no	—
132	C₄×Q₈	—	53	2	no	—
133	C₄×Q₈	—	53	1	no	—
134	C₄×Q₈	—	53	2	no	—
135	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	53	2	no	—
137	C₂×C₄⋊C₄	—	53	1	no	—
138	C₂×C₄²	—	53	2	no	—
139	C₂×C₄²	—	53	1	no	—
140	C₄⋊C₈	—	53	1	no	—
141	C₄⋊C₈	—	53	1	no	—
142	C₄⋊C₈	—	53	2	no	—
143	C₄⋊C₈	—	53	2	no	—
144	C₄⋊C₈	—	53	2	no	—
145	C₄⋊C₈	—	53	1	no	—
146	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
147	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
148	C₂²⋊C₈	—	53	1	no	—
149	C₂²⋊C₈	—	53	2	no	—
150	C₂²⋊C₈	—	53	1	no	—
151	#97	—	53	2	no	—
152	#97	—	53	2	no	—
153	#97	C₂×D₅₃	1	2	yes	—
154	C₄×C₈	—	53	2	no	—
155	C₄×C₈	—	53	2	no	—
156	C₄×C₈	—	53	1	no	—
157	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
158	C₂×Q₈	—	53	1	no	—
159	C₂×Q₈	—	53	2	no	—
160	C₂×Q₈	—	106	4	no	—
161	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
162	C₂²×C₄	—	53	2	no	—
163	C₂²×C₄	—	53	1	no	—
164	C₂²×C₄	—	53	2	no	—
165	C₂²×C₄	—	53	1	no	—
166	C₂²×C₄	—	53	1	no	—
167	C₂×C₈	—	53	1	no	—
168	C₂×C₈	—	53	1	no	—
169	C₂×C₈	—	53	2	no	—
170	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₅₃	1	2	yes	—
172	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
173	C₄⋊C₄	—	53	2	no	—
174	C₄⋊C₄	—	53	2	no	—
175	C₄⋊C₄	—	53	2	no	—
176	C₄⋊C₄	—	53	2	no	—
177	C₄⋊C₄	—	53	1	no	—
178	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
179	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
180	C₄²	—	53	2	no	—
181	C₄²	—	53	2	no	—
182	C₄²	—	53	1	no	—
183	C₄²	—	53	1	no	—
184	C₄²	—	53	2	no	—
185	C₄²	—	53	1	no	—
186	C₄²	C₂²×D₅₃	1	1	yes	characteristic
187	C₄²	C₂²×D₅₃	1	2	yes	—
188	C₂³	—	53	1	no	—
189	Q₈	—	2	2	no	—
190	Q₈	—	53	4	no	—
191	Q₈	—	106	4	no	—
192	C₂×C₄	—	53	2	no	—
193	C₂×C₄	—	53	1	no	—
194	C₂×C₄	—	53	2	no	—
195	C₂×C₄	—	53	1	no	—
196	C₂×C₄	—	53	1	no	—
197	C₂×C₄	—	106	4	no	—
198	C₂×C₄	—	106	2	no	—
199	C₂×C₄	—	106	4	no	—
200	C₂×C₄	—	106	2	no	—
201	C₂×C₄	—	106	1	no	—
202	C₂×C₄	—	106	1	no	—
203	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	2	no	—
210	C₈	—	2	1	no	—
211	C₈	—	106	1	no	—
212	C₈	—	106	1	no	—
213	C₈	—	106	2	no	—
214	C₂²	—	53	2	no	—
215	C₂²	—	106	1	no	—
216	C₂²	—	106	1	no	—
217	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	53	4	no	—
222	C₄	—	106	1	no	—
223	C₄	—	106	1	no	—
224	C₄	—	106	2	no	—
225	C₄	—	106	1	no	—
226	C₄	M₁₆×D₅₃	1	2	yes	—
227	C₄	#13133264	1	1	yes	characteristic
228	C₄	#13133265	1	1	yes	characteristic
229	C₂	—	106	2	no	—
230	C₂	#16811099	1	1	yes	characteristic
231	C₂	C₂×M₁₆×D₅₃	1	1	yes	characteristic
232	C₂	#131×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#446	C₂
hypercenter	#446	D₅₃
hypocenter	C₅₃	#2984
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #16810754	C₂	semidirect
3. #16810752	C₂	semidirect
4. #16810751	C₂	semidirect
5. C₈×Dic₁₀₆	C₂	semidirect
6. #16810747	C₂	semidirect
11. C₅₃×#446	C₂	semidirect
121. #446	D₅₃	semidirect
128. C₅₃	#2984	semidirect