Subgroups — C₄×#13133250 #20283538

Order 6784.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133250	C₂	1	1	yes	maximal, characteristic
2	C₄×#11719945	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810731	C₂	1	2	yes	maximal, complemented
5	#16810725	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₂₁₂	C₂	1	2	yes	maximal, complemented
7	C₄×#11719913	C₂	1	2	yes	maximal, complemented
8	#16810665	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810633	C₂	1	1	yes	maximal, characteristic, complemented
10	#16810625	C₂	1	1	yes	maximal, characteristic
11	C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
14	#13133250	C₄	1	8	yes	direct product factor
15	C₂×Dic₂₁₂	C₂²	1	2	yes	—
16	C₂×#11719913	C₂²	1	2	yes	—
17	C₄×Dic₁₀₆	—	2	1	no	—
18	C₄×Dic₁₀₆	—	2	1	no	—
19	C₄×Dic₁₀₆	C₂²	1	2	yes	—
20	C₄×Dic₁₀₆	C₂²	1	2	yes	—
21	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
22	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
23	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
24	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₅₃	—	2	1	no	—
26	C₄×C₂²⋊D₅₃	—	2	1	no	—
27	#13133176	C₂²	1	1	yes	characteristic
28	#13133174	—	2	1	no	—
29	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
30	C₂×#11719919	—	2	1	no	—
31	C₂×C₄×Dic₅₃	—	2	1	no	—
32	C₄²×D₅₃	—	2	1	no	—
33	#13133151	C₂²	1	2	yes	—
34	#13133149	C₂²	1	2	yes	—
35	#13133142	C₂²	1	2	yes	—
36	#13133142	C₂²	1	2	yes	—
37	#13133142	C₂²	1	2	yes	—
38	#13133139	C₂²	1	2	yes	—
39	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
42	#97×C₅₃	C₂²	1	2	yes	—
43	C₄×C₈×C₅₃	C₂²	1	2	yes	—
44	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₅₃	—	2	1	no	—
48	C₂²×Dic₅₃	—	2	1	no	—
49	C₂²×Dic₅₃	—	2	1	no	—
50	#11719945	—	2	2	no	—
51	#11719945	C₂×C₄	1	4	yes	—
52	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₅₃	—	2	1	no	—
54	C₂×C₄×D₅₃	—	2	1	no	—
55	C₂×C₄×D₅₃	—	2	1	no	—
56	C₂×Dic₁₀₆	—	2	1	no	—
57	C₂×Dic₁₀₆	—	2	2	no	—
58	C₂×Dic₁₀₆	C₂×C₄	1	4	yes	complemented
59	C₂×Dic₁₀₆	C₂³	1	2	yes	—
60	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
61	M₁₆×C₅₃	—	2	2	no	—
62	M₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
63	C₂×C₈×C₅₃	C₂³	1	2	yes	—
64	C₂×C₈×C₅₃	C₂³	1	2	yes	—
65	C₄²×C₅₃	D₄	1	1	yes	characteristic
66	C₄²×C₅₃	D₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₂²⋊Dic₅₃	—	2	1	no	—
70	#11719921	—	2	1	no	—
71	#11719921	—	2	1	no	—
72	C₄⋊Dic₅₃	—	2	1	no	—
73	C₄⋊Dic₅₃	C₂³	1	2	yes	—
74	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
76	#11719919	—	2	2	no	—
77	#11719919	—	2	2	no	—
78	#11719919	—	2	1	no	—
79	#11719919	—	2	1	no	—
80	C₄×Dic₅₃	—	2	1	no	—
81	C₄×Dic₅₃	—	2	1	no	—
82	C₄×Dic₅₃	—	2	1	no	—
83	C₄×Dic₅₃	—	2	2	no	—
84	C₄×Dic₅₃	—	2	2	no	—
85	Dic₂₁₂	C₂×C₄	1	8	yes	complemented
86	#11719913	C₂×C₄	1	8	yes	complemented
87	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×D₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	4	1	no	—
91	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#41	1	2	yes	—
98	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₂²⋊D₅₃	—	2	4	no	—
100	C₂×Dic₅₃	—	2	4	no	—
101	C₂×Dic₅₃	—	2	2	no	—
102	C₂×Dic₅₃	—	2	1	no	—
103	C₂×Dic₅₃	—	2	1	no	—
104	C₂×Dic₅₃	—	2	1	no	—
105	C₂×Dic₅₃	—	2	2	no	—
106	C₂×Dic₅₃	—	4	2	no	—
107	D₂₁₂	—	2	1	no	—
108	D₂₁₂	C₂²×C₄	1	2	yes	—
109	C₄×D₅₃	—	2	4	no	—
110	C₄×D₅₃	—	4	2	no	—
111	C₄×D₅₃	—	4	2	no	—
112	Dic₁₀₆	—	2	4	no	—
113	Dic₁₀₆	—	2	2	no	—
114	Dic₁₀₆	—	2	1	no	—
115	Dic₁₀₆	C₂²×C₄	1	4	yes	—
116	Dic₁₀₆	C₂²×C₄	1	2	yes	—
117	C₈×C₅₃	—	2	2	no	—
118	C₈×C₅₃	C₂²×C₄	1	4	yes	—
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	—	2	1	no	—
121	C₂²×C₅₃	C₄×D₄	1	2	yes	—
122	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₅₃	—	2	2	no	—
124	C₂×D₅₃	—	4	1	no	—
125	C₂×D₅₃	—	4	1	no	—
126	C₄×C₅₃	—	2	1	no	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	C₄×D₄	1	2	yes	—
130	C₄×C₅₃	#137	1	2	yes	complemented
131	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
132	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
133	Dic₅₃	—	2	4	no	—
134	Dic₅₃	—	2	2	no	—
135	Dic₅₃	—	4	1	no	—
136	Dic₅₃	—	4	1	no	—
137	Dic₅₃	—	4	2	no	—
138	C₄×#137	—	53	1	no	maximal, Sylow
139	C₂×C₅₃	—	2	2	no	—
140	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂×C₅₃	C₂×#137	1	1	yes	characteristic
142	C₂×C₅₃	#575	1	1	yes	characteristic
143	D₅₃	—	4	2	no	—
144	C₂×#137	—	53	1	no	—
145	C₄×#41	—	53	1	no	—
146	C₂×C₄×Q₈	—	53	1	no	—
147	#441	—	53	2	no	—
148	#440	—	53	2	no	—
149	C₄×Q₁₆	—	53	2	no	—
150	C₄×SD₁₆	—	53	2	no	—
151	#428	—	53	1	no	—
152	#419	—	53	1	no	—
153	#417	—	53	1	no	—
154	C₄×M₁₆	D₅₃	1	1	yes	characteristic, complemented
155	C₅₃	C₄×#137	1	1	yes	characteristic, complemented, Sylow
156	C₂×#41	—	53	1	no	—
157	C₂²×Q₈	—	53	1	no	—
158	#137	—	53	8	no	—
159	C₂×Q₁₆	—	53	2	no	—
160	C₂×SD₁₆	—	53	2	no	—
161	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
162	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
163	C₄×Q₈	—	53	2	no	—
164	C₄×Q₈	—	53	2	no	—
165	C₄×Q₈	—	53	1	no	—
166	C₄×Q₈	—	53	1	no	—
167	C₄×Q₈	—	106	1	no	—
168	C₄×Q₈	—	106	1	no	—
169	C₄×D₄	—	53	1	no	—
170	C₄×D₄	—	53	1	no	—
171	C₄×D₄	—	106	1	no	—
172	C₄×D₄	—	106	1	no	—
173	#117	—	53	1	no	—
174	#117	—	106	1	no	—
175	C₂×C₄⋊C₄	—	53	1	no	—
176	C₂×C₄⋊C₄	—	106	1	no	—
177	C₂×C₄²	—	106	1	no	—
178	C₂×C₄²	—	106	1	no	—
179	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
180	#107	—	53	2	no	—
181	#106	—	53	2	no	—
182	#103	—	53	2	no	—
183	#103	—	53	2	no	—
184	#103	—	53	2	no	—
185	#102	—	53	2	no	—
186	#97	C₂×D₅₃	1	2	yes	—
187	C₄×C₈	C₂×D₅₃	1	2	yes	—
188	#41	—	53	4	no	—
189	#41	—	106	2	no	—
190	C₂×Q₈	—	53	4	no	—
191	C₂×Q₈	—	53	2	no	—
192	C₂×Q₈	—	53	1	no	—
193	C₂×Q₈	—	106	1	no	—
194	C₂×Q₈	—	106	2	no	—
195	C₂×D₄	—	53	1	no	—
196	C₂×D₄	—	106	1	no	—
197	C₂²×C₄	—	106	1	no	—
198	C₂²×C₄	—	106	1	no	—
199	C₂²×C₄	—	106	1	no	—
200	C₂²×C₄	—	106	1	no	—
201	C₂²×C₄	—	106	1	no	—
202	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
203	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
205	Q₁₆	—	53	8	no	—
206	SD₁₆	—	53	8	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	C₄×D₅₃	1	4	yes	complemented
209	C₂×C₈	C₂²×D₅₃	1	2	yes	—
210	C₂×C₈	C₂²×D₅₃	1	2	yes	—
211	C₄⋊C₄	—	53	2	no	—
212	C₄⋊C₄	—	53	1	no	—
213	C₄⋊C₄	—	53	1	no	—
214	C₄⋊C₄	—	106	1	no	—
215	C₄⋊C₄	—	106	2	no	—
216	C₄⋊C₄	—	106	2	no	—
217	C₄⋊C₄	—	106	1	no	—
218	C₄⋊C₄	—	106	1	no	—
219	C₂²⋊C₄	—	106	1	no	—
220	C₂²⋊C₄	—	106	1	no	—
221	C₂²⋊C₄	—	106	1	no	—
222	C₄²	—	106	1	no	—
223	C₄²	—	106	1	no	—
224	C₄²	—	106	2	no	—
225	C₄²	—	106	1	no	—
226	C₄²	—	106	2	no	—
227	C₄²	D₂₁₂	1	1	yes	characteristic
228	C₄²	D₂₁₂	1	1	yes	characteristic
229	C₄²	C₂²×D₅₃	1	1	yes	characteristic
230	C₄²	C₂²×D₅₃	1	1	yes	characteristic
231	C₂³	—	106	1	no	—
232	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
233	Q₈	—	53	4	no	—
234	Q₈	—	53	2	no	—
235	Q₈	—	106	4	no	—
236	Q₈	—	106	2	no	—
237	Q₈	—	106	1	no	—
238	D₄	—	53	2	no	—
239	D₄	—	106	4	no	—
240	D₄	—	106	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	106	4	no	—
244	C₂×C₄	—	106	1	no	—
245	C₂×C₄	—	106	2	no	—
246	C₂×C₄	—	106	4	no	—
247	C₂×C₄	—	106	1	no	—
248	C₂×C₄	—	106	1	no	—
249	C₂×C₄	—	106	2	no	—
250	C₂×C₄	—	212	2	no	—
251	C₂×C₄	—	212	2	no	—
252	C₂×C₄	—	212	2	no	—
253	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
254	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
255	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
256	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
257	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
258	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
259	C₂×C₄	#11719945	1	2	yes	—
260	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	C₂×C₄×D₅₃	1	4	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	106	2	no	—
266	C₂²	—	212	1	no	—
267	C₂²	—	212	1	no	—
268	C₂²	C₄×D₂₁₂	1	2	yes	—
269	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	106	2	no	—
274	C₄	—	106	4	no	—
275	C₄	—	212	1	no	—
276	C₄	—	212	1	no	—
277	C₄	—	212	2	no	—
278	C₄	C₄×D₂₁₂	1	2	yes	—
279	C₄	#13133250	1	2	yes	direct product factor
280	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
281	C₄	C₂×#11719945	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	212	2	no	—
284	C₂	C₂×C₄×D₂₁₂	1	1	yes	characteristic
285	C₂	C₂×#13133250	1	1	yes	characteristic
286	C₂	#16811423	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₄×M₁₆	D₅₃
hypocenter	C₅₃	C₄×#137
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Dic₁₀₆	C₂	semidirect
4. #16810731	C₂	semidirect
5. #16810725	C₂	semidirect
6. C₄×Dic₂₁₂	C₂	semidirect
7. C₄×#11719913	C₂	semidirect
8. #16810665	C₂	semidirect
9. #16810633	C₂	semidirect
11. C₄×M₁₆×C₅₃	C₂	semidirect
14. #13133250	C₄	direct (N × H)
58. C₂×Dic₁₀₆	C₂×C₄	semidirect
62. M₁₆×C₅₃	C₂×C₄	semidirect
85. Dic₂₁₂	C₂×C₄	semidirect
86. #11719913	C₂×C₄	semidirect
130. C₄×C₅₃	#137	semidirect
154. C₄×M₁₆	D₅₃	semidirect
155. C₅₃	C₄×#137	semidirect
208. M₁₆	C₄×D₅₃	semidirect
279. C₄	#13133250	direct (N × H)

Subgroups — C4×#13133250 #20283538

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#13133250 #20283538