Subgroups — #20285354

Order 6784.

362 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811244	C₂	1	2	yes	maximal, complemented
2	#16811028	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811007	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810983	C₂	1	2	yes	maximal, complemented
5	#16810869	C₂	1	2	yes	maximal, complemented
6	#16810864	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810859	C₂	1	1	yes	maximal, characteristic
8	#16810725	C₂	1	2	yes	maximal, complemented
9	C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
13	C₂×#11719913	C₂²	1	2	yes	—
14	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
15	#13133217	C₂²	1	2	yes	—
16	#13133212	—	2	2	no	—
17	#13133205	C₂²	1	2	yes	—
18	#13133203	—	2	2	no	—
19	#13133199	—	2	2	no	—
20	#13133196	C₂²	1	2	yes	—
21	#13133195	—	2	2	no	—
22	#13133191	C₂²	1	2	yes	—
23	C₄×Dic₁₀₆	C₂²	1	2	yes	—
24	C₄×D₂₁₂	C₂²	1	2	yes	—
25	#13133151	C₂²	1	1	yes	characteristic
26	#13133151	C₂²	1	1	yes	characteristic
27	#13133151	C₂²	1	2	yes	—
28	#13133142	C₂²	1	2	yes	—
29	#13133142	C₂²	1	2	yes	—
30	#13133139	C₂²	1	2	yes	—
31	#13133139	C₂²	1	2	yes	—
32	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	#117×C₅₃	C₂²	1	1	yes	characteristic
35	#97×C₅₃	C₂²	1	1	yes	characteristic
36	#97×C₅₃	C₂²	1	1	yes	characteristic
37	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₅₃	—	2	2	no	—
41	C₂×D₂₁₂	C₂³	1	2	yes	—
42	C₂×C₄×D₅₃	—	2	2	no	—
43	C₂×Dic₁₀₆	—	2	2	no	—
44	C₂×Dic₁₀₆	C₂³	1	2	yes	—
45	M₁₆×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
52	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
53	C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
54	C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
55	C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₅₃	—	2	2	no	—
58	#11719921	—	2	2	no	—
59	#11719921	—	2	2	no	—
60	#11719921	—	2	2	no	—
61	#11719921	—	2	2	no	—
62	C₄⋊Dic₅₃	—	2	2	no	—
63	C₄⋊Dic₅₃	C₂³	1	2	yes	—
64	C₄⋊Dic₅₃	C₂³	1	2	yes	—
65	#11719919	—	2	2	no	—
66	#11719919	—	2	2	no	—
67	C₄×Dic₅₃	—	2	2	no	—
68	Dic₂₁₂	—	2	2	no	—
69	D₄₂₄	—	2	2	no	—
70	#11719913	—	2	4	no	—
71	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×D₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₂²⋊D₅₃	—	8	2	no	—
82	C₂×Dic₅₃	—	2	2	no	—
83	C₂×Dic₅₃	—	2	2	no	—
84	C₂×Dic₅₃	—	2	2	no	—
85	D₂₁₂	—	2	2	no	—
86	D₂₁₂	—	2	2	no	—
87	C₄×D₅₃	—	8	2	no	—
88	Dic₁₀₆	—	2	2	no	—
89	Dic₁₀₆	—	2	2	no	—
90	Dic₁₀₆	—	8	2	no	—
91	C₈×C₅₃	—	2	1	no	—
92	C₈×C₅₃	—	2	1	no	—
93	C₈×C₅₃	#41	1	2	yes	—
94	C₈×C₅₃	#41	1	2	yes	—
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×D₅₃	—	4	2	no	—
100	C₂×D₅₃	—	4	2	no	—
101	C₂×D₅₃	—	4	2	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
108	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
109	Dic₅₃	—	4	2	no	—
110	Dic₅₃	—	4	2	no	—
111	Dic₅₃	—	4	2	no	—
112	#3175	—	53	1	no	maximal, Sylow
113	C₂×C₅₃	—	4	1	no	—
114	C₂×C₅₃	#532	1	1	yes	characteristic
115	C₂×C₅₃	#576	1	1	yes	characteristic
116	C₂×C₅₃	#578	1	1	yes	characteristic
117	D₅₃	—	8	2	no	—
118	#542	—	53	2	no	—
119	#500	—	53	1	no	—
120	#495	—	53	1	no	—
121	#489	—	53	2	no	—
122	#468	—	53	2	no	—
123	#467	—	53	1	no	—
124	#466	—	53	1	no	—
125	#440	—	53	2	no	—
126	C₄×Q₁₆	—	53	1	no	—
127	C₄×D₈	—	53	1	no	—
128	#405	D₅₃	1	1	yes	characteristic, complemented
129	C₅₃	#3175	1	1	yes	characteristic, complemented, Sylow
130	C₂×Q₁₆	—	53	1	no	—
131	C₂×SD₁₆	—	53	2	no	—
132	C₂×D₈	—	53	1	no	—
133	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
134	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
135	C₄⋊Q₈	—	53	2	no	—
136	#126	—	106	2	no	—
137	#124	—	53	2	no	—
138	#124	—	106	2	no	—
139	#123	—	106	2	no	—
140	C₂²⋊Q₈	—	53	2	no	—
141	C₂²⋊Q₈	—	106	2	no	—
142	#121	—	53	2	no	—
143	C₄×Q₈	—	53	2	no	—
144	C₄×D₄	—	53	2	no	—
145	#117	C₂×D₅₃	1	1	yes	characteristic
146	#107	—	53	1	no	—
147	#107	—	53	1	no	—
148	#107	—	53	2	no	—
149	#103	—	53	2	no	—
150	#103	—	53	2	no	—
151	#102	—	53	2	no	—
152	#102	—	53	2	no	—
153	#97	C₂×D₅₃	1	1	yes	characteristic
154	#97	C₂×D₅₃	1	1	yes	characteristic
155	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
156	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
157	C₂×Q₈	—	53	2	no	—
158	C₂×Q₈	—	106	2	no	—
159	C₂×D₄	—	53	2	no	—
160	C₂×D₄	—	106	2	no	—
161	C₂²×C₄	—	106	2	no	—
162	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
163	Q₁₆	—	106	2	no	—
164	SD₁₆	—	106	4	no	—
165	D₈	—	106	2	no	—
166	M₁₆	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
169	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
170	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
172	C₄⋊C₄	—	53	2	no	—
173	C₄⋊C₄	—	53	2	no	—
174	C₄⋊C₄	—	106	2	no	—
175	C₄⋊C₄	—	106	2	no	—
176	C₄⋊C₄	—	106	2	no	—
177	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
178	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
179	C₂²⋊C₄	—	106	2	no	—
180	C₂²⋊C₄	—	106	2	no	—
181	C₂²⋊C₄	—	106	2	no	—
182	C₂²⋊C₄	—	106	2	no	—
183	C₂²⋊C₄	—	106	2	no	—
184	C₂²⋊C₄	D₂₁₂	1	1	yes	characteristic
185	C₂²⋊C₄	D₂₁₂	1	1	yes	characteristic
186	C₄²	—	106	2	no	—
187	C₄²	C₂²×D₅₃	1	1	yes	characteristic
188	C₄²	C₂²×D₅₃	1	1	yes	characteristic
189	C₂³	—	106	2	no	—
190	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
191	Q₈	—	106	2	no	—
192	Q₈	—	106	2	no	—
193	Q₈	—	424	2	no	—
194	D₄	—	106	2	no	—
195	D₄	—	106	2	no	—
196	D₄	—	424	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	106	2	no	—
200	C₂×C₄	—	106	2	no	—
201	C₂×C₄	—	106	2	no	—
202	C₂×C₄	—	424	2	no	—
203	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
204	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
205	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
206	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
207	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	#11719945	1	2	yes	—
212	C₈	#11719945	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	212	2	no	—
217	C₂²	—	212	2	no	—
218	C₂²	—	212	2	no	—
219	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	212	2	no	—
226	C₄	—	212	2	no	—
227	C₄	—	212	2	no	—
228	C₄	C₂×#11719945	1	1	yes	characteristic
229	C₄	C₂×#11719945	1	1	yes	characteristic
230	C₂	—	4	1	no	—
231	C₂	—	424	2	no	—
232	C₂	#16811173	1	1	yes	characteristic
233	C₂	#16811428	1	1	yes	characteristic
234	C₂	#16811440	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#405	D₅₃
hypocenter	C₅₃	#3175
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
1. #16811244	C₂	semidirect
2. #16811028	C₂	semidirect
3. #16811007	C₂	semidirect
4. #16810983	C₂	semidirect
5. #16810869	C₂	semidirect
6. #16810864	C₂	semidirect
8. #16810725	C₂	semidirect
9. C₄×Dic₂₁₂	C₂	semidirect
10. C₄×D₄₂₄	C₂	semidirect
11. C₅₃×#405	C₂	semidirect
128. #405	D₅₃	semidirect
129. C₅₃	#3175	semidirect