Subgroups — C₂×#16811303 #20288216

Order 6784.

1328 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811303	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13133210	C₂	1	2	yes	maximal, complemented
3	C₂×#13133199	C₂	1	2	yes	maximal, complemented
4	C₂×#13133198	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13133193	C₂	1	2	yes	maximal, complemented
6	C₂×#13133185	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133177	C₂	1	2	yes	maximal, complemented
8	C₂×#13133174	C₂	1	2	yes	maximal, complemented
9	C₂×C₂²⋊C₄×D₅₃	C₂	1	2	yes	maximal, complemented
10	C₂×#124×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊D₅₃	C₂²	1	2	yes	—
13	C₂³×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
15	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
16	#13133210	C₂²	1	16	yes	complemented
17	#13133199	C₂²	1	16	yes	complemented
18	#13133198	C₂²	1	8	yes	complemented
19	#13133193	C₂²	1	16	yes	complemented
20	#13133185	C₂²	1	8	yes	—
21	#13133177	C₂²	1	16	yes	complemented
22	#13133174	C₂²	1	16	yes	complemented
23	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
24	C₂×#11719919	C₂²	1	2	yes	complemented
25	C₂×#11719919	C₂²	1	2	yes	complemented
26	C₂×#11719919	C₂²	1	2	yes	complemented
27	C₂×C₂²⋊Dic₅₃	C₂²	1	2	yes	—
28	C₂×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
30	C₂×#11719921	C₂²	1	2	yes	—
31	C₂×#11719921	C₂²	1	2	yes	—
32	C₂×#11719921	C₂²	1	2	yes	—
33	C₂²⋊C₄×D₅₃	C₂²	1	16	yes	complemented
34	C₂×C₄×Dic₅₃	C₂²	1	2	yes	complemented
35	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
36	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
37	#124×C₅₃	C₂²	1	8	yes	complemented
38	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
39	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
40	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
41	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
42	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
43	C₂³×D₅₃	—	2	8	no	—
44	C₂³×D₅₃	—	4	2	no	—
45	C₂³×D₅₃	C₂³	1	4	yes	—
46	C₂³×D₅₃	C₂³	1	2	yes	—
47	C₂×Q₈×C₅₃	—	4	2	no	—
48	C₂×Q₈×C₅₃	C₂³	1	4	yes	complemented
49	C₂×D₄×C₅₃	—	4	2	no	—
50	C₂×D₄×C₅₃	C₂³	1	4	yes	—
51	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
54	C₂×C₂²⋊D₅₃	—	2	8	no	—
55	C₂×C₂²⋊D₅₃	C₂³	1	8	yes	—
56	C₂²×Dic₅₃	—	4	2	no	—
57	C₂²×Dic₅₃	C₂³	1	2	yes	—
58	C₂²×Dic₅₃	C₂³	1	2	yes	—
59	C₂²×Dic₅₃	C₂³	1	2	yes	—
60	C₂²×Dic₅₃	C₂³	1	4	yes	complemented
61	C₂×C₄×D₅₃	—	2	8	no	—
62	C₂×C₄×D₅₃	—	4	4	no	—
63	C₂×C₄×D₅₃	C₂³	1	8	yes	—
64	C₂×C₄×D₅₃	C₂³	1	8	yes	—
65	C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
66	C₂²⋊C₄×C₅₃	C₂³	1	8	yes	complemented
67	C₄²×C₅₃	C₂³	1	4	yes	complemented
68	C₂²⋊Dic₅₃	C₂³	1	8	yes	—
69	C₂²⋊Dic₅₃	C₂³	1	4	yes	—
70	C₂²⋊Dic₅₃	C₂³	1	4	yes	complemented
71	#11719921	C₂³	1	8	yes	—
72	#11719921	C₂³	1	8	yes	—
73	#11719921	C₂³	1	8	yes	—
74	C₄⋊Dic₅₃	C₂³	1	8	yes	complemented
75	#11719919	C₂³	1	8	yes	complemented
76	#11719919	C₂³	1	8	yes	complemented
77	#11719919	C₂³	1	8	yes	complemented
78	C₄×Dic₅₃	C₂³	1	8	yes	complemented
79	C₂³×C₅₃	—	2	4	no	—
80	C₂³×C₅₃	—	2	4	no	—
81	C₂³×C₅₃	—	2	2	no	—
82	C₂³×C₅₃	—	2	2	no	—
83	C₂³×C₅₃	C₂⁴	1	2	yes	—
84	C₂³×C₅₃	C₂⁴	1	2	yes	—
85	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₂²×D₅₃	—	2	4	no	—
87	C₂²×D₅₃	—	2	4	no	—
88	C₂²×D₅₃	—	2	16	no	—
89	C₂²×D₅₃	—	4	4	no	—
90	C₂²×D₅₃	—	4	16	no	—
91	C₂²×D₅₃	#41	1	8	yes	—
92	C₂²×D₅₃	C₂⁴	1	4	yes	—
93	Q₈×C₅₃	—	4	4	no	—
94	D₄×C₅₃	—	4	4	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
99	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
100	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
101	C₂²⋊D₅₃	—	2	16	no	—
102	C₂×Dic₅₃	—	2	4	no	—
103	C₂×Dic₅₃	—	2	4	no	—
104	C₂×Dic₅₃	—	2	4	no	—
105	C₂×Dic₅₃	—	4	4	no	—
106	C₂×Dic₅₃	C₂⁴	1	4	yes	—
107	C₂×Dic₅₃	C₂⁴	1	4	yes	—
108	C₂×Dic₅₃	C₂⁴	1	4	yes	—
109	C₄×D₅₃	—	2	16	no	—
110	C₄×D₅₃	—	4	8	no	—
111	C₂²×C₅₃	—	2	4	no	—
112	C₂²×C₅₃	—	2	4	no	—
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	4	4	no	—
116	C₂²×C₅₃	—	4	4	no	—
117	C₂²×C₅₃	C₂×#41	1	4	yes	—
118	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
119	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
120	C₂×D₅₃	—	2	16	no	—
121	C₂×D₅₃	—	2	4	no	—
122	C₂×D₅₃	—	2	4	no	—
123	C₂×D₅₃	—	4	16	no	—
124	C₂×D₅₃	C₂×#41	1	8	yes	—
125	C₄×C₅₃	—	2	4	no	—
126	C₄×C₅₃	—	2	4	no	—
127	C₄×C₅₃	—	2	2	no	—
128	Dic₅₃	—	2	4	no	—
129	Dic₅₃	—	2	4	no	—
130	Dic₅₃	—	2	4	no	—
131	C₂×#551	—	53	1	no	maximal, Sylow
132	C₂×C₅₃	—	4	2	no	—
133	C₂×C₅₃	—	4	2	no	—
134	C₂×C₅₃	#551	1	4	yes	complemented
135	C₂×C₅₃	C₂²×#41	1	2	yes	—
136	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
137	D₅₃	—	2	8	no	—
138	#551	—	53	16	no	—
139	C₂×#126	—	53	2	no	—
140	C₂×#124	D₅₃	1	1	yes	characteristic, complemented
141	C₂×#123	—	53	1	no	—
142	C₂×#123	—	53	2	no	—
143	C₂×C₂²⋊Q₈	—	53	2	no	—
144	C₂×#120	—	53	1	no	—
145	C₂×C₄×D₄	—	53	2	no	—
146	C₂×#117	—	53	2	no	—
147	C₂²×C₂²⋊C₄	—	53	2	no	—
148	C₅₃	C₂×#551	1	1	yes	characteristic, complemented, Sylow
149	C₂⁵	—	53	1	no	—
150	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
151	C₂²×D₄	—	53	2	no	—
152	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
153	C₂³×C₄	—	53	1	no	—
154	C₂³×C₄	—	53	2	no	—
155	C₂³×C₄	—	53	2	no	—
156	#126	—	53	16	no	—
157	#124	C₂×D₅₃	1	8	yes	complemented
158	#123	—	53	8	no	—
159	#123	—	53	16	no	—
160	C₂²⋊Q₈	—	53	16	no	—
161	#120	—	53	8	no	—
162	C₄×D₄	—	53	16	no	—
163	#117	—	53	16	no	—
164	C₂×C₄⋊C₄	—	53	2	no	—
165	C₂×C₄⋊C₄	—	53	2	no	—
166	C₂×C₄⋊C₄	—	53	2	no	—
167	C₂×C₄⋊C₄	—	53	2	no	—
168	C₂×C₂²⋊C₄	—	53	2	no	—
169	C₂×C₂²⋊C₄	—	53	1	no	—
170	C₂×C₂²⋊C₄	—	53	1	no	—
171	C₂×C₂²⋊C₄	—	53	2	no	—
172	C₂×C₂²⋊C₄	—	53	2	no	—
173	C₂×C₂²⋊C₄	—	53	2	no	—
174	C₂×C₂²⋊C₄	—	53	16	no	—
175	C₂×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂×D₅₃	1	2	yes	complemented
177	C₂×C₄²	—	53	2	no	—
178	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
179	C₂⁴	—	53	2	no	—
180	C₂⁴	—	53	4	no	—
181	C₂⁴	—	106	8	no	—
182	C₂⁴	—	212	2	no	—
183	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
184	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	C₂²×D₅₃	1	4	yes	complemented
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	53	8	no	—
189	C₂×D₄	—	106	8	no	—
190	C₂×D₄	C₂²×D₅₃	1	4	yes	—
191	C₂²×C₄	—	53	2	no	—
192	C₂²×C₄	—	53	2	no	—
193	C₂²×C₄	—	53	2	no	—
194	C₂²×C₄	—	53	4	no	—
195	C₂²×C₄	—	53	8	no	—
196	C₂²×C₄	—	53	8	no	—
197	C₂²×C₄	—	106	8	no	—
198	C₂²×C₄	—	212	4	no	—
199	C₂²×C₄	—	212	2	no	—
200	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
201	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
203	C₄⋊C₄	—	53	8	no	—
204	C₄⋊C₄	—	53	8	no	—
205	C₄⋊C₄	—	53	8	no	—
206	C₄⋊C₄	—	53	8	no	—
207	C₂²⋊C₄	—	53	8	no	—
208	C₂²⋊C₄	—	53	4	no	—
209	C₂²⋊C₄	—	53	4	no	—
210	C₂²⋊C₄	—	53	8	no	—
211	C₂²⋊C₄	—	53	8	no	—
212	C₂²⋊C₄	—	53	8	no	—
213	C₂²⋊C₄	C₂²×D₅₃	1	8	yes	—
214	C₂²⋊C₄	C₂²×D₅₃	1	8	yes	complemented
215	C₄²	—	53	8	no	—
216	C₄²	C₂²×D₅₃	1	4	yes	complemented
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	53	8	no	—
222	C₂³	—	53	4	no	—
223	C₂³	—	106	16	no	—
224	C₂³	—	106	4	no	—
225	C₂³	—	106	4	no	—
226	C₂³	—	212	4	no	—
227	C₂³	—	212	16	no	—
228	C₂³	C₂³×D₅₃	1	2	yes	—
229	C₂³	C₂³×D₅₃	1	2	yes	—
230	C₂³	C₂³×D₅₃	1	1	yes	characteristic
231	Q₈	—	4	4	no	—
232	D₄	—	4	4	no	—
233	D₄	—	106	16	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	53	4	no	—
238	C₂×C₄	—	53	4	no	—
239	C₂×C₄	—	53	4	no	—
240	C₂×C₄	—	106	16	no	—
241	C₂×C₄	—	106	4	no	—
242	C₂×C₄	—	106	4	no	—
243	C₂×C₄	—	106	4	no	—
244	C₂×C₄	—	212	8	no	—
245	C₂×C₄	—	212	4	no	—
246	C₂×C₄	C₂³×D₅₃	1	4	yes	—
247	C₂×C₄	C₂³×D₅₃	1	2	yes	—
248	C₂×C₄	C₂³×D₅₃	1	4	yes	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	53	8	no	—
256	C₂²	—	106	16	no	—
257	C₂²	—	106	4	no	—
258	C₂²	—	106	4	no	—
259	C₂²	—	212	16	no	—
260	C₂²	#41×D₅₃	1	4	yes	—
261	C₂²	#13133262	1	2	yes	—
262	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	106	4	no	—
267	C₄	—	106	4	no	—
268	C₄	—	106	4	no	—
269	C₂	—	4	2	no	—
270	C₂	—	4	2	no	—
271	C₂	—	106	8	no	—
272	C₂	#16811303	1	4	yes	direct product factor
273	C₂	C₂×#41×D₅₃	1	2	yes	—
274	C₂	C₂×#13133262	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂×#124×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×#124	D₅₃
hypocenter	C₅₃	C₂×#551
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. #16811303	C₂	direct (N × H)
2. C₂×#13133210	C₂	semidirect
3. C₂×#13133199	C₂	semidirect
4. C₂×#13133198	C₂	semidirect
5. C₂×#13133193	C₂	semidirect
7. C₂×#13133177	C₂	semidirect
8. C₂×#13133174	C₂	semidirect
9. C₂×C₂²⋊C₄×D₅₃	C₂	semidirect
10. C₂×#124×C₅₃	C₂	semidirect
13. C₂³×Dic₅₃	C₂²	semidirect
16. #13133210	C₂²	semidirect
17. #13133199	C₂²	semidirect
18. #13133198	C₂²	semidirect
19. #13133193	C₂²	semidirect
21. #13133177	C₂²	semidirect
22. #13133174	C₂²	semidirect
23. C₂×C₄⋊Dic₅₃	C₂²	semidirect
24. C₂×#11719919	C₂²	semidirect
25. C₂×#11719919	C₂²	semidirect
26. C₂×#11719919	C₂²	semidirect
29. C₂×C₂²⋊Dic₅₃	C₂²	semidirect
33. C₂²⋊C₄×D₅₃	C₂²	semidirect
34. C₂×C₄×Dic₅₃	C₂²	semidirect
35. C₂²×Q₈×C₅₃	C₂²	semidirect
37. #124×C₅₃	C₂²	semidirect
39. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
40. C₂×C₄²×C₅₃	C₂²	semidirect
48. C₂×Q₈×C₅₃	C₂³	semidirect
60. C₂²×Dic₅₃	C₂³	semidirect
66. C₂²⋊C₄×C₅₃	C₂³	semidirect
67. C₄²×C₅₃	C₂³	semidirect
70. C₂²⋊Dic₅₃	C₂³	semidirect
74. C₄⋊Dic₅₃	C₂³	semidirect
75. #11719919	C₂³	semidirect
76. #11719919	C₂³	semidirect
77. #11719919	C₂³	semidirect
78. C₄×Dic₅₃	C₂³	semidirect
134. C₂×C₅₃	#551	semidirect
140. C₂×#124	D₅₃	semidirect
148. C₅₃	C₂×#551	semidirect
150. C₂²×Q₈	C₂×D₅₃	semidirect
157. #124	C₂×D₅₃	semidirect
176. C₂×C₂²⋊C₄	C₂×D₅₃	semidirect
178. C₂×C₄²	C₂×D₅₃	semidirect
186. C₂×Q₈	C₂²×D₅₃	semidirect
214. C₂²⋊C₄	C₂²×D₅₃	semidirect
216. C₄²	C₂²×D₅₃	semidirect
272. C₂	#16811303	direct (N × H)

Subgroups — C2×#16811303 #20288216

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811303 #20288216