Subgroups — C₄×#13133248 #20283534

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×#13133248	C₂	1	1	yes	maximal, characteristic
2	C₄×#11719945	C₂	1	1	yes	maximal, characteristic
3	#16810728	C₂	1	2	yes	maximal, complemented
4	#16810721	C₂	1	2	yes	maximal, complemented
5	C₄×#11719925	C₂	1	2	yes	maximal, complemented
6	C₄×#11719924	C₂	1	2	yes	maximal, complemented
7	#16810663	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810632	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810622	C₂	1	1	yes	maximal, characteristic
10	C₄×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719945	C₂²	1	1	yes	characteristic
13	#13133248	C₄	1	8	yes	direct product factor
14	C₂×#11719925	C₂²	1	2	yes	—
15	C₂×#11719924	C₂²	1	2	yes	—
16	C₂×#11719917	C₂²	1	1	yes	characteristic
17	C₂×#11719917	C₂²	1	1	yes	characteristic
18	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
19	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
20	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
21	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₅₃	—	2	1	no	—
23	C₄×C₂²⋊D₅₃	—	2	1	no	—
24	#13133176	C₂²	1	1	yes	characteristic
25	#13133174	—	2	1	no	—
26	C₄²×D₅₃	—	2	1	no	—
27	#13133150	C₂²	1	2	yes	—
28	#13133148	C₂²	1	2	yes	—
29	#13133141	C₂²	1	2	yes	—
30	#13133141	C₂²	1	2	yes	—
31	#13133140	C₂²	1	2	yes	—
32	#13133137	C₂²	1	2	yes	—
33	#13133128	C₂²	1	2	yes	—
34	C₄×#88322	C₂²	1	2	yes	—
35	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₅₃	—	2	1	no	—
37	C₄×Q₈×C₅₃	—	2	1	no	—
38	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
39	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	—	2	1	no	—
43	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×Q₈×C₅₃	—	2	1	no	—
45	C₂×Q₈×C₅₃	—	2	2	no	—
46	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	complemented
47	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊D₅₃	—	2	1	no	—
54	#11719945	—	2	2	no	—
55	#11719945	C₂×C₄	1	4	yes	—
56	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₅₃	—	2	1	no	—
58	C₂×C₄×D₅₃	—	2	1	no	—
59	C₂×C₄×D₅₃	—	2	1	no	—
60	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₄⋊C₄×C₅₃	—	2	1	no	—
64	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	D₄	1	1	yes	characteristic
68	C₄²×C₅₃	D₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	C₂²⋊Dic₅₃	—	2	1	no	—
72	#11719925	C₂×C₄	1	8	yes	complemented
73	#11719924	C₂×C₄	1	8	yes	complemented
74	#11719921	—	2	1	no	—
75	#11719921	—	2	1	no	—
76	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
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	#11719917	—	2	2	no	—
84	#11719917	C₂×C₄	1	4	yes	complemented
85	C₂×#88322	C₂³	1	2	yes	—
86	C₂×#88322	C₂³	1	2	yes	—
87	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×D₅₃	—	2	1	no	—
89	Q₈×C₅₃	—	2	4	no	—
90	Q₈×C₅₃	—	2	2	no	—
91	Q₈×C₅₃	C₂²×C₄	1	4	yes	—
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	4	1	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#41	1	2	yes	—
105	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₅₃	—	2	4	no	—
107	C₂×Dic₅₃	—	2	1	no	—
108	C₂×Dic₅₃	—	2	1	no	—
109	C₂×Dic₅₃	—	2	1	no	—
110	D₂₁₂	—	2	1	no	—
111	D₂₁₂	C₂²×C₄	1	2	yes	—
112	C₄×D₅₃	—	2	4	no	—
113	C₄×D₅₃	—	4	2	no	—
114	C₄×D₅₃	—	4	2	no	—
115	Dic₁₀₆	—	2	1	no	—
116	Dic₁₀₆	C₂²×C₄	1	2	yes	—
117	#88322	—	2	2	no	—
118	#88322	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	4	no	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	4	2	no	—
131	C₄×C₅₃	C₄×D₄	1	2	yes	—
132	C₄×C₅₃	#137	1	2	yes	complemented
133	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
134	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
135	Dic₅₃	—	2	2	no	—
136	Dic₅₃	—	4	1	no	—
137	Dic₅₃	—	4	1	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	D₅₃	1	1	yes	characteristic, complemented
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	—	53	1	no	—
155	C₅₃	C₄×#137	1	1	yes	characteristic, complemented, Sylow
156	C₂×#41	—	53	1	no	—
157	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
158	#137	—	53	8	no	—
159	C₂×Q₁₆	—	53	2	no	—
160	C₂×SD₁₆	—	53	2	no	—
161	C₂×M₁₆	—	53	1	no	—
162	C₂×M₁₆	—	53	1	no	—
163	C₄×Q₈	—	2	1	no	—
164	C₄×Q₈	—	2	1	no	—
165	C₄×Q₈	—	53	1	no	—
166	C₄×Q₈	—	53	1	no	—
167	C₄×Q₈	C₂×D₅₃	1	2	yes	—
168	C₄×Q₈	C₂×D₅₃	1	2	yes	—
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	—	2	1	no	—
176	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
177	C₂×C₄²	—	2	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	—	53	2	no	—
187	C₄×C₈	—	53	2	no	—
188	#41	—	53	4	no	—
189	#41	—	106	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×Q₈	—	53	1	no	—
193	C₂×Q₈	C₄×D₅₃	1	4	yes	complemented
194	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
195	C₂×D₄	—	53	1	no	—
196	C₂×D₄	—	106	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	106	1	no	—
200	C₂²×C₄	—	106	1	no	—
201	C₂²×C₄	—	106	1	no	—
202	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊D₅₃	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	—	53	4	no	—
208	M₁₆	—	106	2	no	—
209	C₂×C₈	—	53	2	no	—
210	C₂×C₈	—	53	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	53	1	no	—
215	C₄⋊C₄	—	53	1	no	—
216	C₄⋊C₄	—	106	1	no	—
217	C₄⋊C₄	—	106	1	no	—
218	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
219	C₂²⋊C₄	—	106	1	no	—
220	C₂²⋊C₄	—	106	1	no	—
221	C₂²⋊C₄	—	106	1	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	106	1	no	—
225	C₄²	—	106	1	no	—
226	C₄²	—	106	1	no	—
227	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
228	C₄²	C₂²⋊D₅₃	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×C₂²⋊D₅₃	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	53	2	no	—
236	Q₈	—	106	1	no	—
237	Q₈	C₂×C₄×D₅₃	1	4	yes	—
238	D₄	—	53	2	no	—
239	D₄	—	106	4	no	—
240	D₄	—	106	1	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	106	1	no	—
248	C₂×C₄	—	106	4	no	—
249	C₂×C₄	—	106	1	no	—
250	C₂×C₄	—	106	1	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	#11719945	1	2	yes	—
255	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
261	C₈	—	53	4	no	—
262	C₈	—	106	2	no	—
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×C₂²⋊D₅₃	1	2	yes	—
269	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
270	C₄	—	2	4	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	2	no	—
275	C₄	—	106	2	no	—
276	C₄	—	212	1	no	—
277	C₄	—	212	1	no	—
278	C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
279	C₄	#13133248	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×C₂²⋊D₅₃	1	1	yes	characteristic
285	C₂	C₂×#13133248	1	1	yes	characteristic
286	C₂	#16811421	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×C₄×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	C₄×#137
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
3. #16810728	C₂	semidirect
4. #16810721	C₂	semidirect
5. C₄×#11719925	C₂	semidirect
6. C₄×#11719924	C₂	semidirect
7. #16810663	C₂	semidirect
8. #16810632	C₂	semidirect
10. C₄×#11719917	C₂	semidirect
11. C₂×C₄×Q₈×C₅₃	C₂	semidirect
13. #13133248	C₄	direct (N × H)
46. C₂×Q₈×C₅₃	C₂×C₄	semidirect
72. #11719925	C₂×C₄	semidirect
73. #11719924	C₂×C₄	semidirect
84. #11719917	C₂×C₄	semidirect
132. C₄×C₅₃	#137	semidirect
146. C₂×C₄×Q₈	D₅₃	semidirect
155. C₅₃	C₄×#137	semidirect
193. C₂×Q₈	C₄×D₅₃	semidirect
279. C₄	#13133248	direct (N × H)

Subgroups — C4×#13133248 #20283534

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#13133248 #20283534