Subgroups — #20283667

Order 6784.

388 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	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	#16810645	C₂	1	2	yes	maximal
8	#16810645	C₂	1	1	yes	maximal, characteristic
9	#16810575	C₂	1	2	yes	maximal
10	C₄×#11719912	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	C₂×#11719912	C₂²	1	1	yes	characteristic
14	C₂×#11719912	C₂²	1	2	yes	—
15	C₂×#11719912	C₂²	1	1	yes	characteristic
16	Q₈×Dic₅₃	C₄	1	1	yes	characteristic
17	C₄×Dic₁₀₆	C₂²	1	2	yes	—
18	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
19	#13133182	C₄	1	2	yes	—
20	#13133182	C₄	1	1	yes	characteristic
21	C₄⋊C₄×D₅₃	C₄	1	1	yes	characteristic
22	C₄⋊C₄×D₅₃	C₄	1	2	yes	—
23	C₄²×D₅₃	C₂²	1	1	yes	characteristic
24	C₄²×D₅₃	C₂²	1	2	yes	—
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	2	yes	—
29	#13133146	C₂²	1	2	yes	—
30	#13133128	C₂²	1	2	yes	—
31	#13133127	C₂²	1	1	yes	characteristic
32	#13133127	C₂²	1	1	yes	characteristic
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	2	yes	—
52	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
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	2	yes	—
69	C₂×#88322	C₂³	1	1	yes	characteristic
70	C₂×#88322	C₂³	1	1	yes	characteristic
71	#11719912	—	2	2	no	—
72	#11719912	—	2	4	no	—
73	#11719912	Q₈	1	4	yes	complemented
74	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
83	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
86	C₄×D₅₃	—	2	2	no	—
87	C₄×D₅₃	—	2	4	no	—
88	C₄×D₅₃	—	4	1	no	—
89	C₄×D₅₃	—	4	2	no	—
90	C₄×D₅₃	C₂×Q₈	1	2	yes	—
91	C₄×D₅₃	#41	1	2	yes	—
92	Dic₁₀₆	—	2	4	no	—
93	Dic₁₀₆	M₁₆	1	4	yes	complemented
94	C₈×C₅₃	—	2	1	no	—
95	C₈×C₅₃	—	2	2	no	—
96	C₈×C₅₃	C₂×Q₈	1	2	yes	complemented
97	#88322	—	2	2	no	—
98	#88322	—	2	1	no	—
99	#88322	C₂×Q₈	1	2	yes	complemented
100	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₅₃	—	2	1	no	—
102	C₂×D₅₃	—	2	1	no	—
103	C₂×D₅₃	C₄×Q₈	1	2	yes	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
108	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
109	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
110	Dic₅₃	—	2	1	no	—
111	Dic₅₃	—	2	2	no	—
112	Dic₅₃	—	2	1	no	—
113	Dic₅₃	C₄×Q₈	1	2	yes	—
114	Dic₅₃	C₂×M₁₆	1	4	yes	—
115	Q₈×M₁₆	—	53	1	no	maximal, Sylow
116	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
117	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
118	C₂×C₅₃	#568	1	1	yes	characteristic
119	D₅₃	—	2	2	no	—
120	C₂×C₄×Q₈	—	53	1	no	—
121	#446	—	53	1	no	—
122	#446	—	53	2	no	—
123	#446	—	53	2	no	—
124	#446	D₅₃	1	1	yes	characteristic, complemented
125	C₈×Q₈	—	53	1	no	—
126	C₈×Q₈	—	53	1	no	—
127	#423	—	53	2	no	—
128	#423	—	53	1	no	—
129	C₄×M₁₆	—	53	2	no	—
130	C₄×M₁₆	—	53	1	no	—
131	C₅₃	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
132	C₂²×Q₈	—	53	1	no	—
133	C₂×M₁₆	—	53	1	no	—
134	C₂×M₁₆	—	53	2	no	—
135	C₂×M₁₆	—	53	1	no	—
136	C₄×Q₈	—	53	1	no	—
137	C₄×Q₈	—	53	1	no	—
138	C₄×Q₈	—	53	2	no	—
139	C₄×Q₈	—	53	1	no	—
140	C₄×Q₈	—	53	2	no	—
141	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	53	2	no	—
143	C₂×C₄⋊C₄	—	53	1	no	—
144	C₂×C₄²	—	53	2	no	—
145	C₂×C₄²	—	53	1	no	—
146	C₄⋊C₈	—	53	2	no	—
147	C₄⋊C₈	—	53	2	no	—
148	C₄⋊C₈	—	53	2	no	—
149	C₄⋊C₈	—	53	1	no	—
150	C₄⋊C₈	—	53	2	no	—
151	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
152	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
153	#97	—	53	1	no	—
154	#97	—	53	1	no	—
155	#97	—	53	2	no	—
156	#97	C₂×D₅₃	1	2	yes	—
157	C₄×C₈	—	53	2	no	—
158	C₄×C₈	—	53	2	no	—
159	C₄×C₈	—	53	1	no	—
160	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
161	C₂×Q₈	—	53	1	no	—
162	C₂×Q₈	—	53	2	no	—
163	C₂×Q₈	—	106	4	no	—
164	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
165	C₂²×C₄	—	53	1	no	—
166	C₂²×C₄	—	53	2	no	—
167	C₂²×C₄	—	53	1	no	—
168	C₂²×C₄	—	53	2	no	—
169	C₂²×C₄	—	53	1	no	—
170	M₁₆	—	53	4	no	—
171	M₁₆	—	106	4	no	—
172	M₁₆	—	106	2	no	—
173	C₂×C₈	—	53	1	no	—
174	C₂×C₈	—	53	2	no	—
175	C₂×C₈	—	53	1	no	—
176	C₂×C₈	C₂²×D₅₃	1	2	yes	—
177	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
179	C₄⋊C₄	—	53	2	no	—
180	C₄⋊C₄	—	53	2	no	—
181	C₄⋊C₄	—	53	2	no	—
182	C₄⋊C₄	—	53	2	no	—
183	C₄⋊C₄	—	53	1	no	—
184	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
185	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
186	C₄²	—	53	2	no	—
187	C₄²	—	53	2	no	—
188	C₄²	—	53	1	no	—
189	C₄²	—	53	1	no	—
190	C₄²	—	53	2	no	—
191	C₄²	—	53	1	no	—
192	C₄²	C₂²×D₅₃	1	1	yes	characteristic
193	C₄²	C₂²×D₅₃	1	2	yes	—
194	C₂³	—	53	1	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	53	4	no	—
197	Q₈	—	106	4	no	—
198	C₂×C₄	—	53	1	no	—
199	C₂×C₄	—	53	2	no	—
200	C₂×C₄	—	53	2	no	—
201	C₂×C₄	—	53	2	no	—
202	C₂×C₄	—	53	2	no	—
203	C₂×C₄	—	53	1	no	—
204	C₂×C₄	—	53	1	no	—
205	C₂×C₄	—	106	4	no	—
206	C₂×C₄	—	106	2	no	—
207	C₂×C₄	—	212	1	no	—
208	C₂×C₄	—	212	2	no	—
209	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
211	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
213	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	2	1	no	—
216	C₈	—	53	2	no	—
217	C₈	—	106	2	no	—
218	C₈	—	106	1	no	—
219	C₈	Q₈×D₅₃	1	2	yes	complemented
220	C₂²	—	53	2	no	—
221	C₂²	—	106	1	no	—
222	C₂²	—	106	1	no	—
223	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	2	no	—
227	C₄	—	53	2	no	—
228	C₄	—	53	4	no	—
229	C₄	—	106	1	no	—
230	C₄	—	106	2	no	—
231	C₄	—	106	1	no	—
232	C₄	M₁₆×D₅₃	1	2	yes	—
233	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
234	C₄	#41×D₅₃	1	1	yes	characteristic
235	C₂	—	106	2	no	—
236	C₂	C₄×Q₈×D₅₃	1	1	yes	characteristic
237	C₂	C₂×M₁₆×D₅₃	1	1	yes	characteristic
238	C₂	#16811388	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	Q₈×M₁₆
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
73. #11719912	Q₈	semidirect
93. Dic₁₀₆	M₁₆	semidirect
96. C₈×C₅₃	C₂×Q₈	semidirect
99. #88322	C₂×Q₈	semidirect
124. #446	D₅₃	semidirect
131. C₅₃	Q₈×M₁₆	semidirect
219. C₈	Q₈×D₅₃	semidirect