Subgroups — C₄×#13133247 #20283532

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×#13133247	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810728	C₂	1	2	yes	maximal, complemented
4	#16810723	C₂	1	2	yes	maximal, complemented
5	C₄×#11719925	C₂	1	2	yes	maximal, complemented
6	C₄×#11719923	C₂	1	2	yes	maximal, complemented
7	#16810666	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810631	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810624	C₂	1	1	yes	maximal, characteristic
10	C₄×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
13	#13133247	C₄	1	8	yes	direct product factor
14	C₂×#11719925	C₂²	1	2	yes	—
15	C₂×#11719923	C₂²	1	2	yes	—
16	C₂×#11719917	C₂²	1	1	yes	characteristic
17	C₂×#11719917	C₂²	1	1	yes	characteristic
18	C₄×Dic₁₀₆	—	2	1	no	—
19	C₄×Dic₁₀₆	—	2	1	no	—
20	C₄×Dic₁₀₆	C₂²	1	2	yes	—
21	C₄×Dic₁₀₆	C₂²	1	2	yes	—
22	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
23	C₂×#11719919	—	2	1	no	—
24	C₂×C₄×Dic₅₃	—	2	1	no	—
25	#13133150	C₂²	1	2	yes	—
26	#13133148	C₂²	1	2	yes	—
27	#13133141	C₂²	1	2	yes	—
28	#13133140	C₂²	1	2	yes	—
29	#13133140	C₂²	1	2	yes	—
30	#13133138	C₂²	1	2	yes	—
31	#13133128	C₂²	1	2	yes	—
32	C₄×#88322	C₂²	1	2	yes	—
33	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₅₃	—	2	1	no	—
37	C₄×D₄×C₅₃	—	2	1	no	—
38	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
40	#117×C₅₃	—	2	1	no	—
41	#117×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	—	2	1	no	—
43	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
44	#41×C₅₃	—	2	2	no	—
45	#41×C₅₃	C₂×C₄	1	4	yes	—
46	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₅₃	—	2	1	no	—
48	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
55	C₂²×Dic₅₃	—	2	1	no	—
56	C₂²×Dic₅₃	—	2	1	no	—
57	C₂×Dic₁₀₆	—	2	1	no	—
58	C₂×Dic₁₀₆	—	2	2	no	—
59	C₂×Dic₁₀₆	C₂×C₄	1	4	yes	complemented
60	C₂×Dic₁₀₆	C₂³	1	2	yes	—
61	C₄⋊C₄×C₅₃	—	2	1	no	—
62	C₄⋊C₄×C₅₃	—	2	1	no	—
63	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₂²⋊C₄×C₅₃	—	2	1	no	—
66	C₂²⋊C₄×C₅₃	—	2	1	no	—
67	C₂²⋊C₄×C₅₃	—	2	1	no	—
68	C₄²×C₅₃	—	2	1	no	—
69	C₄²×C₅₃	—	2	1	no	—
70	C₄²×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	D₄	1	1	yes	characteristic
72	C₄²×C₅₃	D₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	#11719925	C₂×C₄	1	8	yes	complemented
76	#11719923	C₂×C₄	1	8	yes	complemented
77	C₄⋊Dic₅₃	—	2	1	no	—
78	C₄⋊Dic₅₃	C₂³	1	2	yes	—
79	#11719919	—	2	2	no	—
80	#11719919	—	2	2	no	—
81	C₄×Dic₅₃	—	2	2	no	—
82	C₄×Dic₅₃	—	2	2	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	—	2	1	no	—
88	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₅₃	—	2	1	no	—
90	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
91	D₄×C₅₃	—	2	4	no	—
92	D₄×C₅₃	—	2	1	no	—
93	D₄×C₅₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	4	1	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#41	1	2	yes	—
109	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
110	C₂×Dic₅₃	—	2	4	no	—
111	C₂×Dic₅₃	—	2	2	no	—
112	C₂×Dic₅₃	—	2	2	no	—
113	C₂×Dic₅₃	—	4	2	no	—
114	Dic₁₀₆	—	2	4	no	—
115	Dic₁₀₆	—	2	2	no	—
116	Dic₁₀₆	C₂²×C₄	1	4	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	—	2	2	no	—
122	C₂²×C₅₃	—	4	1	no	—
123	C₂²×C₅₃	—	4	1	no	—
124	C₂²×C₅₃	C₄×D₄	1	2	yes	—
125	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₅₃	—	2	2	no	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	4	1	no	—
131	C₄×C₅₃	—	4	1	no	—
132	C₄×C₅₃	C₄×D₄	1	2	yes	—
133	C₄×C₅₃	#137	1	2	yes	complemented
134	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
135	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
136	Dic₅₃	—	2	4	no	—
137	Dic₅₃	—	4	2	no	—
138	C₄×#137	—	53	1	no	maximal, Sylow
139	C₂×C₅₃	—	2	2	no	—
140	C₂×C₅₃	—	4	2	no	—
141	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂×C₅₃	C₂×#137	1	1	yes	characteristic
143	C₂×C₅₃	#575	1	1	yes	characteristic
144	C₂×#137	—	53	1	no	—
145	C₄×#41	D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
155	C₅₃	C₄×#137	1	1	yes	characteristic, complemented, Sylow
156	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
162	C₂×M₁₆	—	53	1	no	—
163	C₄×Q₈	—	53	2	no	—
164	C₄×Q₈	—	53	2	no	—
165	C₄×Q₈	—	106	1	no	—
166	C₄×Q₈	—	106	1	no	—
167	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
168	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
172	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
173	#117	—	2	1	no	—
174	#117	C₂×D₅₃	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	53	1	no	—
176	C₂×C₄⋊C₄	—	106	1	no	—
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	—	2	2	no	—
189	#41	C₄×D₅₃	1	4	yes	—
190	C₂×Q₈	—	53	4	no	—
191	C₂×Q₈	—	53	2	no	—
192	C₂×Q₈	—	106	2	no	—
193	C₂×Q₈	—	106	1	no	—
194	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	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	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	53	2	no	—
214	C₄⋊C₄	—	106	1	no	—
215	C₄⋊C₄	—	106	2	no	—
216	C₄⋊C₄	—	106	2	no	—
217	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
218	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	106	2	no	—
226	C₄²	—	106	2	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	—	2	1	no	—
232	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
233	Q₈	—	2	1	no	—
234	Q₈	—	53	4	no	—
235	Q₈	—	106	4	no	—
236	Q₈	—	106	2	no	—
237	Q₈	C₂×C₄×D₅₃	1	2	yes	—
238	D₄	—	2	4	no	—
239	D₄	—	2	1	no	—
240	D₄	C₂×C₄×D₅₃	1	2	yes	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	106	4	no	—
250	C₂×C₄	—	106	2	no	—
251	C₂×C₄	—	106	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	—	2	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	C₄×C₂²⋊D₅₃	1	2	yes	—
269	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	106	4	no	—
277	C₄	—	212	2	no	—
278	C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
279	C₄	#13133247	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	—	4	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
285	C₂	C₂×#13133247	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	C₄×#41×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₄×#41	D₅₃
hypocenter	C₅₃	C₄×#137
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₁₀₆	C₂	semidirect
3. #16810728	C₂	semidirect
4. #16810723	C₂	semidirect
5. C₄×#11719925	C₂	semidirect
6. C₄×#11719923	C₂	semidirect
7. #16810666	C₂	semidirect
8. #16810631	C₂	semidirect
10. C₄×#11719917	C₂	semidirect
13. #13133247	C₄	direct (N × H)
59. C₂×Dic₁₀₆	C₂×C₄	semidirect
75. #11719925	C₂×C₄	semidirect
76. #11719923	C₂×C₄	semidirect
84. #11719917	C₂×C₄	semidirect
133. C₄×C₅₃	#137	semidirect
155. C₅₃	C₄×#137	semidirect
279. C₄	#13133247	direct (N × H)

Subgroups — C4×#13133247 #20283532

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#13133247 #20283532