Subgroups — #20287084

Order 6784.

342 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811313	C₂	1	2	yes	maximal
2	#16811025	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810965	C₂	1	2	yes	maximal, complemented
4	#16810954	C₂	1	2	yes	maximal, complemented
5	#16810931	C₂	1	2	yes	maximal, complemented
6	#16810864	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810864	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810661	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133151	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
13	#13133210	—	2	2	no	—
14	#13133209	C₂²	1	2	yes	—
15	#13133207	—	2	2	no	—
16	#13133198	—	2	2	no	—
17	#13133196	C₂²	1	2	yes	—
18	#13133196	C₂²	1	2	yes	—
19	C₄×Dic₁₀₆	C₂²	1	2	yes	—
20	#13133176	C₂²	1	2	yes	—
21	#13133173	—	2	2	no	—
22	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
23	#13133151	C₂²	1	1	yes	characteristic
24	#13133151	C₂²	1	1	yes	characteristic
25	#13133151	C₂²	1	1	yes	characteristic
26	#13133151	C₂²	1	2	yes	—
27	#13133149	C₂²	1	2	yes	—
28	#13133149	C₂²	1	2	yes	—
29	#13133142	C₂²	1	2	yes	—
30	#13133142	C₂²	1	2	yes	—
31	#13133142	C₂²	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₅₃	—	2	2	no	—
43	C₂×Dic₁₀₆	C₂³	1	2	yes	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
51	C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
52	C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
53	C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²⋊Dic₅₃	—	2	2	no	—
55	C₂²⋊Dic₅₃	—	2	2	no	—
56	C₂²⋊Dic₅₃	—	2	2	no	—
57	C₄⋊Dic₅₃	—	2	2	no	—
58	C₄⋊Dic₅₃	C₂³	1	2	yes	—
59	C₄⋊Dic₅₃	C₂³	1	2	yes	—
60	C₄⋊Dic₅₃	C₂³	1	2	yes	—
61	#11719919	—	2	2	no	—
62	#11719919	—	2	2	no	—
63	#11719919	—	2	2	no	—
64	#11719919	—	2	2	no	—
65	#11719919	—	2	2	no	—
66	C₄×Dic₅₃	—	2	2	no	—
67	C₄×Dic₅₃	—	2	2	no	—
68	Dic₂₁₂	—	2	2	no	—
69	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
71	D₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
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	C₂×Dic₅₃	—	2	2	no	—
82	C₂×Dic₅₃	—	2	2	no	—
83	C₂×Dic₅₃	—	2	2	no	—
84	C₂×Dic₅₃	—	2	2	no	—
85	C₂×Dic₅₃	—	4	4	no	—
86	Dic₁₀₆	—	2	2	no	—
87	Dic₁₀₆	—	2	2	no	—
88	C₈×C₅₃	—	2	1	no	—
89	C₈×C₅₃	—	2	1	no	—
90	C₈×C₅₃	—	2	1	no	—
91	C₈×C₅₃	#41	1	2	yes	—
92	C₂²×C₅₃	—	2	1	no	—
93	C₂²×C₅₃	—	2	1	no	—
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	#135	1	2	yes	—
98	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
104	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
105	Dic₅₃	—	4	2	no	—
106	Dic₅₃	—	4	2	no	—
107	Dic₅₃	—	4	2	no	—
108	Dic₅₃	—	4	2	no	—
109	#3336	—	53	1	no	maximal, Sylow
110	C₂×C₅₃	—	2	2	no	—
111	C₂×C₅₃	—	4	1	no	—
112	C₂×C₅₃	#547	1	1	yes	characteristic
113	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
114	C₂×C₅₃	#578	1	1	yes	characteristic
115	#552	—	53	2	no	—
116	#499	—	53	1	no	—
117	#485	—	53	2	no	—
118	#483	—	53	2	no	—
119	#479	—	53	2	no	—
120	#467	—	53	1	no	—
121	#467	—	53	1	no	—
122	C₄×Q₁₆	—	53	1	no	—
123	C₈×D₄	D₅₃	1	1	yes	characteristic
124	#427	—	53	1	no	—
125	C₂×#107	—	53	1	no	—
126	C₅₃	#3336	1	1	yes	characteristic, complemented, Sylow
127	C₂×Q₁₆	—	53	1	no	—
128	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
129	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
130	#126	—	106	2	no	—
131	#125	—	53	2	no	—
132	#125	—	106	2	no	—
133	#123	—	106	2	no	—
134	C₂²⋊Q₈	—	53	2	no	—
135	C₂²⋊Q₈	—	53	2	no	—
136	C₄×Q₈	—	53	2	no	—
137	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
138	#117	—	53	2	no	—
139	#117	—	106	2	no	—
140	C₂×C₄⋊C₄	—	53	2	no	—
141	#107	—	53	1	no	—
142	#107	—	53	1	no	—
143	#107	—	53	2	no	—
144	#107	—	53	1	no	—
145	#106	—	53	2	no	—
146	#106	—	53	2	no	—
147	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
148	#103	—	53	2	no	—
149	#103	—	53	2	no	—
150	#103	—	53	2	no	—
151	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
152	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
153	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
154	C₂×Q₈	—	53	2	no	—
155	C₂×D₄	D₂₁₂	1	1	yes	characteristic
156	C₂²×C₄	—	106	2	no	—
157	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
159	Q₁₆	—	106	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
163	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
164	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
165	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
166	C₄⋊C₄	—	53	2	no	—
167	C₄⋊C₄	—	53	2	no	—
168	C₄⋊C₄	—	53	2	no	—
169	C₄⋊C₄	—	106	2	no	—
170	C₄⋊C₄	—	106	2	no	—
171	C₄⋊C₄	—	106	2	no	—
172	C₄⋊C₄	—	106	2	no	—
173	C₄⋊C₄	—	106	2	no	—
174	C₄⋊C₄	—	106	2	no	—
175	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
176	C₂²⋊C₄	—	106	2	no	—
177	C₂²⋊C₄	—	106	2	no	—
178	C₂²⋊C₄	—	106	2	no	—
179	C₂²⋊C₄	D₂₁₂	1	1	yes	characteristic
180	C₂²⋊C₄	D₂₁₂	1	1	yes	characteristic
181	C₄²	—	106	2	no	—
182	C₄²	—	106	2	no	—
183	C₄²	C₂²×D₅₃	1	1	yes	characteristic
184	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
185	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
186	Q₈	—	106	2	no	—
187	Q₈	—	106	2	no	—
188	D₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	106	2	no	—
194	C₂×C₄	—	106	2	no	—
195	C₂×C₄	—	106	2	no	—
196	C₂×C₄	—	106	2	no	—
197	C₂×C₄	—	212	4	no	—
198	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
199	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
200	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
201	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	#11719947	1	2	yes	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	#13133239	1	2	yes	—
213	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	212	2	no	—
219	C₄	—	212	2	no	—
220	C₄	—	212	2	no	—
221	C₄	—	212	2	no	—
222	C₄	C₂×#11719947	1	1	yes	characteristic
223	C₄	#13133264	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	4	1	no	—
226	C₂	#16811277	1	1	yes	characteristic
227	C₂	C₂×#13133239	1	1	yes	characteristic
228	C₂	#16811440	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₈×D₄×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₈×D₄	D₅₃
hypocenter	C₅₃	#3336
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #16811025	C₂	semidirect
3. #16810965	C₂	semidirect
4. #16810954	C₂	semidirect
5. #16810931	C₂	semidirect
6. #16810864	C₂	semidirect
7. #16810864	C₂	semidirect
8. C₄×Dic₂₁₂	C₂	semidirect
9. #16810661	C₂	semidirect
10. C₂×#13133151	C₂	semidirect
126. C₅₃	#3336	semidirect