Subgroups — C₂×#16810379 #20274415

Order 6784.

404 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	C₂²×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133176	C₂	1	1	yes	maximal, characteristic
3	#16810379	C₂	1	4	yes	maximal, direct product factor
4	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	—	2	1	no	—
6	C₂×#11719917	—	2	1	no	—
7	C₂×#11719917	—	2	1	no	—
8	C₂×#11719917	—	2	1	no	—
9	C₂×#11719917	C₄	1	2	yes	complemented
10	C₂×#11719917	C₂²	1	2	yes	—
11	C₂²×#88322	C₄	1	2	yes	complemented
12	#13133176	—	2	1	no	—
13	#13133176	—	2	1	no	—
14	#13133176	C₄	1	2	yes	complemented
15	#13133176	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₅₃	—	2	1	no	—
17	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
18	C₂×C₄×Dic₅₃	C₄	1	2	yes	complemented
19	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
20	#117×C₅₃	—	2	1	no	—
21	#117×C₅₃	—	2	1	no	—
22	#117×C₅₃	C₄	1	2	yes	complemented
23	#117×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
25	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
26	C₂×C₄²×C₅₃	C₄	1	2	yes	complemented
27	C₂⁴×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	—	2	2	no	—
29	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×Dic₅₃	—	2	2	no	—
44	C₄⋊C₄×C₅₃	—	2	2	no	—
45	C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₂²⋊C₄×C₅₃	—	2	2	no	—
47	C₂²⋊C₄×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	C₂×C₄	1	4	yes	complemented
50	C₂²⋊Dic₅₃	—	2	2	no	—
51	C₂²⋊Dic₅₃	—	2	2	no	—
52	C₄⋊Dic₅₃	—	2	2	no	—
53	C₄⋊Dic₅₃	—	2	2	no	—
54	C₄×Dic₅₃	—	2	2	no	—
55	C₄×Dic₅₃	C₂×C₄	1	4	yes	complemented
56	#11719917	—	2	2	no	—
57	#11719917	—	2	2	no	—
58	#11719917	—	2	2	no	—
59	#11719917	—	2	2	no	—
60	C₂×#88322	—	2	2	no	—
61	C₂×#88322	—	2	2	no	—
62	C₂×#88322	C₂×C₄	1	4	yes	complemented
63	C₂³×C₅₃	—	4	2	no	—
64	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
70	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	2	4	no	—
72	C₂×C₄×C₅₃	—	4	4	no	—
73	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
101	C₂×Dic₅₃	—	2	4	no	—
102	C₂×Dic₅₃	—	4	4	no	—
103	#88322	—	2	4	no	—
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	—	4	1	no	—
108	C₂²×C₅₃	—	4	1	no	—
109	C₂²×C₅₃	—	4	1	no	—
110	C₂²×C₅₃	—	4	1	no	—
111	C₂²×C₅₃	#95	1	1	yes	characteristic
112	C₂²×C₅₃	#95	1	1	yes	characteristic
113	C₂²×C₅₃	#95	1	1	yes	characteristic
114	C₂²×C₅₃	#95	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₅₃	—	4	4	no	—
119	C₄×C₅₃	#95	1	1	yes	characteristic
120	C₄×C₅₃	#95	1	1	yes	characteristic
121	C₄×C₅₃	#95	1	1	yes	characteristic
122	C₄×C₅₃	#95	1	1	yes	characteristic
123	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
124	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	Dic₅₃	—	4	4	no	—
128	C₂×H(3,ℤ/4)	—	53	1	no	maximal, Sylow
129	C₂×C₅₃	—	2	1	no	—
130	C₂×C₅₃	—	2	1	no	—
131	C₂×C₅₃	—	2	1	no	—
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	—	2	1	no	—
135	C₂×C₅₃	H(3,ℤ/4)	1	2	yes	complemented
136	C₂×C₅₃	C₂×#95	1	1	yes	characteristic
137	C₂²×M₁₆	—	53	1	no	—
138	C₂×#117	—	53	1	no	—
139	C₂×#117	D₅₃	1	1	yes	characteristic
140	H(3,ℤ/4)	—	53	4	no	—
141	C₅₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
143	C₂×M₁₆	—	53	2	no	—
144	C₂×M₁₆	—	53	2	no	—
145	C₂×M₁₆	—	106	1	no	—
146	C₂×M₁₆	—	106	1	no	—
147	C₂×M₁₆	—	106	1	no	—
148	C₂×M₁₆	—	106	1	no	—
149	C₂²×C₈	—	53	2	no	—
150	#117	—	2	1	no	—
151	#117	—	2	1	no	—
152	#117	—	53	2	no	—
153	#117	—	53	2	no	—
154	#117	—	106	1	no	—
155	#117	—	106	1	no	—
156	#117	Dic₅₃	1	2	yes	complemented
157	#117	C₂×D₅₃	1	2	yes	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	106	1	no	—
160	C₂×C₂²⋊C₄	—	2	1	no	—
161	C₂×C₂²⋊C₄	—	106	1	no	—
162	C₂×C₄²	—	53	2	no	—
163	C₂×C₄²	Dic₅₃	1	2	yes	complemented
164	C₂⁴	C₂²⋊D₅₃	1	1	yes	characteristic
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	106	2	no	—
167	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
168	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
170	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
171	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
173	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
174	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
175	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
176	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
178	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
181	M₁₆	—	106	2	no	—
182	M₁₆	—	106	2	no	—
183	M₁₆	—	106	2	no	—
184	M₁₆	—	106	2	no	—
185	C₂×C₈	—	53	4	no	—
186	C₂×C₈	—	106	2	no	—
187	C₂×C₈	—	106	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	106	2	no	—
191	C₄⋊C₄	—	106	2	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	106	2	no	—
195	C₂²⋊C₄	—	106	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	53	4	no	—
198	C₄²	—	106	2	no	—
199	C₄²	C₂×Dic₅₃	1	4	yes	complemented
200	C₂³	—	4	2	no	—
201	C₂³	#11719919	1	1	yes	characteristic
202	C₂³	#11719919	1	1	yes	characteristic
203	C₂³	#11719921	1	1	yes	characteristic
204	C₂³	#11719921	1	1	yes	characteristic
205	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
206	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
207	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	106	4	no	—
211	C₂×C₄	—	212	4	no	—
212	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
213	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
214	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
215	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
216	C₂×C₄	#11719919	1	1	yes	characteristic
217	C₂×C₄	#11719919	1	1	yes	characteristic
218	C₂×C₄	#11719919	1	1	yes	characteristic
219	C₂×C₄	#11719919	1	1	yes	characteristic
220	C₂×C₄	#11719919	1	1	yes	characteristic
221	C₂×C₄	#11719919	1	1	yes	characteristic
222	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
223	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
224	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
225	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
226	C₂×C₄	#11719921	1	1	yes	characteristic
227	C₂×C₄	#11719921	1	1	yes	characteristic
228	C₂×C₄	#11719921	1	1	yes	characteristic
229	C₂×C₄	#11719921	1	1	yes	characteristic
230	C₂×C₄	#11719921	1	1	yes	characteristic
231	C₂×C₄	#11719921	1	1	yes	characteristic
232	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
234	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
238	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
240	C₈	—	106	4	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	#13133124	1	1	yes	characteristic
249	C₂²	#13133124	1	1	yes	characteristic
250	C₂²	#13133124	1	1	yes	characteristic
251	C₂²	#13133124	1	1	yes	characteristic
252	C₂²	C₂×#11719921	1	1	yes	characteristic
253	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
254	C₂²	C₂×#11719919	1	1	yes	characteristic
255	C₄	—	4	4	no	—
256	C₄	—	212	4	no	—
257	C₄	#13133124	1	1	yes	characteristic
258	C₄	#13133124	1	1	yes	characteristic
259	C₄	#13133124	1	1	yes	characteristic
260	C₄	#13133124	1	1	yes	characteristic
261	C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
262	C₄	C₂×#11719921	1	1	yes	characteristic
263	C₄	C₂×#11719919	1	1	yes	characteristic
264	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	2	1	no	—
271	C₂	#16810379	1	2	yes	direct product factor
272	C₂	C₂×#13133124	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#117×C₅₃	C₂
socle	C₂²×C₅₃	#95
hypercenter	C₂×#117	D₅₃
hypocenter	C₅₃	C₂×H(3,ℤ/4)
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #16810379	C₂	direct (N × H)
9. C₂×#11719917	C₄	semidirect
11. C₂²×#88322	C₄	semidirect
14. #13133176	C₄	semidirect
18. C₂×C₄×Dic₅₃	C₄	semidirect
22. #117×C₅₃	C₄	semidirect
26. C₂×C₄²×C₅₃	C₄	semidirect
49. C₄²×C₅₃	C₂×C₄	semidirect
55. C₄×Dic₅₃	C₂×C₄	semidirect
62. C₂×#88322	C₂×C₄	semidirect
135. C₂×C₅₃	H(3,ℤ/4)	semidirect
141. C₅₃	C₂×H(3,ℤ/4)	semidirect
156. #117	Dic₅₃	semidirect
163. C₂×C₄²	Dic₅₃	semidirect
199. C₄²	C₂×Dic₅₃	semidirect
271. C₂	#16810379	direct (N × H)

Subgroups — C2×#16810379 #20274415

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810379 #20274415