Subgroups — #20286458

Order 6784.

482 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133239	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811282	C₂	1	2	yes	maximal, complemented
4	#16811007	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810864	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810859	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810846	C₂	1	2	yes	maximal, complemented
8	#16810787	C₂	1	2	yes	maximal, complemented
9	#16810764	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	2	yes	—
13	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
14	#13133239	—	2	4	no	—
15	C₂×Dic₂₁₂	—	2	4	no	—
16	C₂×Dic₂₁₂	C₂²	1	2	yes	—
17	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
19	C₂×#11719913	C₂²	1	2	yes	—
20	C₂×#11719913	C₂²	1	2	yes	—
21	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
22	#13133205	C₂²	1	2	yes	—
23	#13133203	—	2	2	no	—
24	#13133196	C₂²	1	2	yes	—
25	#13133195	—	2	2	no	—
26	#13133191	C₂²	1	2	yes	—
27	#13133189	—	2	2	no	—
28	C₄×Dic₁₀₆	C₂²	1	2	yes	—
29	#13133177	—	2	2	no	—
30	#13133151	C₂²	1	1	yes	characteristic
31	#13133142	C₂²	1	2	yes	—
32	#13133142	C₂²	1	2	yes	—
33	#13133139	C₂²	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₅₃	—	2	2	no	—
45	C₂×C₂²⋊D₅₃	—	2	2	no	—
46	C₂²×Dic₅₃	—	2	2	no	—
47	#11719945	—	2	4	no	—
48	#11719945	—	4	2	no	—
49	C₂×D₂₁₂	C₂³	1	2	yes	—
50	C₂×C₄×D₅₃	—	2	2	no	—
51	C₂×Dic₁₀₆	—	2	4	no	—
52	C₂×Dic₁₀₆	—	2	2	no	—
53	C₂×Dic₁₀₆	—	4	2	no	—
54	C₂×Dic₁₀₆	C₂³	1	2	yes	—
55	C₂×Dic₁₀₆	C₂³	1	2	yes	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
63	C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
64	C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₂²⋊Dic₅₃	—	2	2	no	—
67	#11719921	—	2	2	no	—
68	#11719921	—	2	2	no	—
69	#11719921	—	2	2	no	—
70	C₄⋊Dic₅₃	C₂³	1	2	yes	—
71	#11719919	—	2	2	no	—
72	#11719919	—	2	2	no	—
73	C₄×Dic₅₃	—	2	2	no	—
74	Dic₂₁₂	—	2	4	no	—
75	Dic₂₁₂	—	2	2	no	—
76	Dic₂₁₂	D₄	1	4	yes	complemented
77	D₄₂₄	—	2	2	no	—
78	#11719913	—	2	4	no	—
79	#11719913	—	4	2	no	—
80	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₅₃	—	2	2	no	—
83	D₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₂²⋊D₅₃	—	4	4	no	—
94	C₂²⋊D₅₃	—	4	4	no	—
95	C₂×Dic₅₃	—	2	2	no	—
96	C₂×Dic₅₃	—	2	2	no	—
97	C₂×Dic₅₃	—	2	2	no	—
98	C₂×Dic₅₃	—	4	4	no	—
99	D₂₁₂	—	2	2	no	—
100	D₂₁₂	—	2	2	no	—
101	C₄×D₅₃	—	4	4	no	—
102	Dic₁₀₆	—	2	2	no	—
103	Dic₁₀₆	—	2	2	no	—
104	Dic₁₀₆	—	2	2	no	—
105	Dic₁₀₆	—	4	4	no	—
106	Dic₁₀₆	C₂×D₄	1	4	yes	—
107	C₈×C₅₃	—	2	1	no	—
108	C₈×C₅₃	—	2	1	no	—
109	C₈×C₅₃	—	2	1	no	—
110	C₈×C₅₃	C₂×D₄	1	2	yes	—
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	2	1	no	—
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	#135	1	2	yes	complemented
117	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₅₃	—	4	2	no	—
119	C₂×D₅₃	—	4	2	no	—
120	C₂×D₅₃	—	4	2	no	—
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
127	Dic₅₃	—	2	4	no	—
128	Dic₅₃	—	4	2	no	—
129	Dic₅₃	—	4	2	no	—
130	#3285	—	53	1	no	maximal, Sylow
131	C₂×C₅₃	—	2	2	no	—
132	C₂×C₅₃	—	4	1	no	—
133	C₂×C₅₃	D₄²	1	1	yes	characteristic
134	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
135	C₂×C₅₃	#578	1	1	yes	characteristic
136	D₅₃	—	8	2	no	—
137	C₂×#135	—	53	1	no	—
138	C₂²×Q₁₆	—	53	1	no	—
139	#548	—	53	2	no	—
140	#495	—	53	1	no	—
141	#467	—	53	1	no	—
142	#466	—	53	1	no	—
143	#464	—	53	2	no	—
144	#452	—	53	2	no	—
145	#448	—	53	2	no	—
146	C₄×Q₁₆	—	53	1	no	—
147	C₈×D₄	D₅₃	1	1	yes	characteristic, complemented
148	C₅₃	#3285	1	1	yes	characteristic, complemented, Sylow
149	C₂×#41	—	53	2	no	—
150	C₂²×Q₈	—	53	2	no	—
151	#135	—	106	4	no	—
152	C₂×Q₁₆	—	53	1	no	—
153	C₂×Q₁₆	—	53	2	no	—
154	C₂×Q₁₆	—	53	1	no	—
155	C₂×Q₁₆	—	106	4	no	—
156	C₂×SD₁₆	—	53	2	no	—
157	C₂×SD₁₆	—	53	2	no	—
158	C₂×D₈	—	53	1	no	—
159	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
160	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
161	#124	—	53	2	no	—
162	#124	—	106	2	no	—
163	C₂²⋊Q₈	—	53	2	no	—
164	C₂²⋊Q₈	—	106	2	no	—
165	#121	—	53	2	no	—
166	#121	—	106	2	no	—
167	C₄×Q₈	—	53	2	no	—
168	C₄×D₄	—	106	2	no	—
169	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
170	#107	—	53	1	no	—
171	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
172	#103	—	53	2	no	—
173	#103	—	53	2	no	—
174	#102	—	53	2	no	—
175	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
176	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
177	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
178	#41	—	106	4	no	—
179	#41	—	212	2	no	—
180	C₂×Q₈	—	53	2	no	—
181	C₂×Q₈	—	53	2	no	—
182	C₂×Q₈	—	106	4	no	—
183	C₂×Q₈	—	106	2	no	—
184	C₂×Q₈	—	212	2	no	—
185	C₂×D₄	—	53	2	no	—
186	C₂×D₄	—	106	2	no	—
187	C₂×D₄	—	106	2	no	—
188	C₂×D₄	D₂₁₂	1	1	yes	characteristic
189	C₂²×C₄	—	106	2	no	—
190	C₂²×C₄	—	106	2	no	—
191	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
193	Q₁₆	—	53	4	no	—
194	Q₁₆	—	106	4	no	—
195	Q₁₆	—	106	2	no	—
196	SD₁₆	—	106	4	no	—
197	SD₁₆	—	212	2	no	—
198	D₈	—	106	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
204	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
205	C₄⋊C₄	—	53	2	no	—
206	C₄⋊C₄	—	106	2	no	—
207	C₄⋊C₄	—	106	2	no	—
208	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
209	C₂²⋊C₄	—	106	2	no	—
210	C₂²⋊C₄	—	106	2	no	—
211	C₂²⋊C₄	—	106	2	no	—
212	C₂²⋊C₄	—	106	2	no	—
213	C₂²⋊C₄	D₂₁₂	1	1	yes	characteristic
214	C₂²⋊C₄	D₂₁₂	1	1	yes	characteristic
215	C₄²	—	106	2	no	—
216	C₄²	C₂²×D₅₃	1	1	yes	characteristic
217	C₂³	—	106	2	no	—
218	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
219	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
220	Q₈	—	53	4	no	—
221	Q₈	—	106	2	no	—
222	Q₈	—	106	2	no	—
223	Q₈	—	106	2	no	—
224	Q₈	—	212	4	no	—
225	D₄	—	2	2	no	—
226	D₄	—	106	2	no	—
227	D₄	—	106	2	no	—
228	D₄	—	212	4	no	—
229	D₄	—	212	4	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	106	2	no	—
235	C₂×C₄	—	106	2	no	—
236	C₂×C₄	—	106	2	no	—
237	C₂×C₄	—	212	4	no	—
238	C₂×C₄	—	212	4	no	—
239	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
240	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
241	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
242	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	D₄×D₅₃	1	2	yes	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	212	2	no	—
254	C₂²	—	212	2	no	—
255	C₂²	—	212	2	no	—
256	C₂²	#13133239	1	2	yes	complemented
257	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	106	4	no	—
263	C₄	—	212	2	no	—
264	C₄	—	212	2	no	—
265	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
266	C₄	#13133263	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	4	1	no	—
269	C₂	—	424	2	no	—
270	C₂	D₄×D₂₁₂	1	1	yes	characteristic
271	C₂	C₂×#13133239	1	1	yes	characteristic
272	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	#3285
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
1. C₂×#13133239	C₂	semidirect
2. C₂²×Dic₂₁₂	C₂	semidirect
3. #16811282	C₂	semidirect
4. #16811007	C₂	semidirect
5. #16810864	C₂	semidirect
6. #16810859	C₂	semidirect
7. #16810846	C₂	semidirect
8. #16810787	C₂	semidirect
9. #16810764	C₂	semidirect
10. C₄×Dic₂₁₂	C₂	semidirect
11. C₈×D₄×C₅₃	C₂	semidirect
76. Dic₂₁₂	D₄	semidirect
116. C₂²×C₅₃	#135	semidirect
147. C₈×D₄	D₅₃	semidirect
148. C₅₃	#3285	semidirect
256. C₂²	#13133239	semidirect