Subgroups — C₂×#16810395 #20274458

Order 6784.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#16810395	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×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	—
15	#13133176	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₅₃	—	2	1	no	—
17	C₂×C₂²⋊Dic₅₃	C₄	1	2	yes	—
18	C₂×C₄×Dic₅₃	—	2	1	no	—
19	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₅₃	—	2	1	no	—
21	C₂×M₁₆×C₅₃	—	2	1	no	—
22	C₂×M₁₆×C₅₃	—	2	1	no	—
23	C₂×M₁₆×C₅₃	C₄	1	2	yes	complemented
24	C₂×M₁₆×C₅₃	C₄	1	2	yes	complemented
25	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
26	C₂²×C₈×C₅₃	—	2	1	no	—
27	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	D₄	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	Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₅₃	—	2	2	no	—
43	M₁₆×C₅₃	—	2	2	no	—
44	M₁₆×C₅₃	—	2	2	no	—
45	M₁₆×C₅₃	—	2	2	no	—
46	M₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂²⋊Dic₅₃	—	2	2	no	—
51	C₂²⋊Dic₅₃	C₂×C₄	1	4	yes	—
52	C₄⋊Dic₅₃	—	2	2	no	—
53	C₄⋊Dic₅₃	—	2	2	no	—
54	C₄×Dic₅₃	—	2	2	no	—
55	C₄×Dic₅₃	—	2	2	no	—
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	C₄²	1	1	yes	characteristic
65	C₂³×C₅₃	C₄²	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	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²⋊C₄	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	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
99	C₂×Dic₅₃	—	2	4	no	—
100	C₂×Dic₅₃	—	4	4	no	—
101	C₈×C₅₃	—	2	4	no	—
102	#88322	—	2	4	no	—
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	4	1	no	—
107	C₂²×C₅₃	—	4	1	no	—
108	C₂²×C₅₃	—	4	1	no	—
109	C₂²×C₅₃	—	4	1	no	—
110	C₂²×C₅₃	#95	1	1	yes	characteristic
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	C₂×C₄²	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₅₃	#95	1	1	yes	characteristic
118	C₄×C₅₃	#95	1	1	yes	characteristic
119	C₄×C₅₃	#95	1	1	yes	characteristic
120	C₄×C₅₃	#95	1	1	yes	characteristic
121	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	Dic₅₃	—	4	4	no	—
126	C₂×#344	—	53	1	no	maximal, Sylow
127	C₂×C₅₃	—	2	1	no	—
128	C₂×C₅₃	—	2	1	no	—
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	#344	1	2	yes	complemented
134	C₂×C₅₃	C₂×#95	1	1	yes	characteristic
135	C₂²×M₁₆	—	53	1	no	—
136	C₂²×M₁₆	D₅₃	1	1	yes	characteristic
137	C₂×#117	—	53	1	no	—
138	#344	—	53	4	no	—
139	C₅₃	C₂×#344	1	1	yes	characteristic, complemented, Sylow
140	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
141	C₂×M₁₆	—	2	1	no	—
142	C₂×M₁₆	—	2	1	no	—
143	C₂×M₁₆	—	2	1	no	—
144	C₂×M₁₆	—	53	2	no	—
145	C₂×M₁₆	—	53	2	no	—
146	C₂×M₁₆	—	106	1	no	—
147	C₂×M₁₆	—	106	1	no	—
148	C₂×M₁₆	—	106	1	no	—
149	C₂×M₁₆	—	106	1	no	—
150	C₂×M₁₆	Dic₅₃	1	2	yes	complemented
151	C₂×M₁₆	Dic₅₃	1	2	yes	complemented
152	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	—	53	2	no	—
155	#117	—	53	2	no	—
156	#117	—	53	2	no	—
157	#117	—	106	1	no	—
158	#117	—	106	1	no	—
159	C₂×C₄⋊C₄	—	106	1	no	—
160	C₂×C₂²⋊C₄	—	53	2	no	—
161	C₂×C₄²	—	106	1	no	—
162	C₂⁴	C₄×D₅₃	1	1	yes	characteristic
163	C₂²×C₄	—	106	2	no	—
164	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
165	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
166	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
167	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
168	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
169	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
170	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
171	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
174	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
176	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
178	M₁₆	—	2	2	no	—
179	M₁₆	—	2	2	no	—
180	M₁₆	—	2	2	no	—
181	M₁₆	—	106	2	no	—
182	M₁₆	—	106	2	no	—
183	M₁₆	—	106	2	no	—
184	M₁₆	—	106	2	no	—
185	M₁₆	C₂×Dic₅₃	1	4	yes	complemented
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	53	4	no	—
190	C₂×C₈	—	106	2	no	—
191	C₂×C₈	—	106	2	no	—
192	C₄⋊C₄	—	106	2	no	—
193	C₄⋊C₄	—	106	2	no	—
194	C₂²⋊C₄	—	53	4	no	—
195	C₂²⋊C₄	—	106	2	no	—
196	C₄²	—	106	2	no	—
197	C₄²	—	106	2	no	—
198	C₂³	—	4	2	no	—
199	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
200	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
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	C₂×C₄×D₅₃	1	1	yes	characteristic
206	C₂×C₄	—	106	4	no	—
207	C₂×C₄	—	212	4	no	—
208	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
209	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
210	C₂×C₄	#11719919	1	1	yes	characteristic
211	C₂×C₄	#11719919	1	1	yes	characteristic
212	C₂×C₄	#11719919	1	1	yes	characteristic
213	C₂×C₄	#11719919	1	1	yes	characteristic
214	C₂×C₄	#11719919	1	1	yes	characteristic
215	C₂×C₄	#11719919	1	1	yes	characteristic
216	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
217	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
218	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
219	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
220	C₂×C₄	#11719921	1	1	yes	characteristic
221	C₂×C₄	#11719921	1	1	yes	characteristic
222	C₂×C₄	#11719921	1	1	yes	characteristic
223	C₂×C₄	#11719921	1	1	yes	characteristic
224	C₂×C₄	#11719921	1	1	yes	characteristic
225	C₂×C₄	#11719921	1	1	yes	characteristic
226	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
227	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
228	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
229	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
231	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
232	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
233	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
236	C₈	—	2	4	no	—
237	C₈	—	106	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	#13133124	1	1	yes	characteristic
246	C₂²	#13133124	1	1	yes	characteristic
247	C₂²	#13133124	1	1	yes	characteristic
248	C₂²	#13133124	1	1	yes	characteristic
249	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
250	C₂²	C₂×#11719921	1	1	yes	characteristic
251	C₂²	C₂×#11719919	1	1	yes	characteristic
252	C₄	—	212	4	no	—
253	C₄	#13133124	1	1	yes	characteristic
254	C₄	#13133124	1	1	yes	characteristic
255	C₄	#13133124	1	1	yes	characteristic
256	C₄	#13133124	1	1	yes	characteristic
257	C₄	C₂×#11719921	1	1	yes	characteristic
258	C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
259	C₄	C₂×#11719919	1	1	yes	characteristic
260	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	#16810395	1	2	yes	direct product factor
268	C₂	C₂×#13133124	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#11719921
commutator	C₂²×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂²×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	#95
hypercenter	C₂²×M₁₆	D₅₃
hypocenter	C₅₃	C₂×#344
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #16810395	C₂	direct (N × H)
9. C₂×#11719917	C₄	semidirect
11. C₂²×#88322	C₄	semidirect
23. C₂×M₁₆×C₅₃	C₄	semidirect
24. C₂×M₁₆×C₅₃	C₄	semidirect
46. M₁₆×C₅₃	C₂×C₄	semidirect
62. C₂×#88322	C₂×C₄	semidirect
133. C₂×C₅₃	#344	semidirect
139. C₅₃	C₂×#344	semidirect
150. C₂×M₁₆	Dic₅₃	semidirect
151. C₂×M₁₆	Dic₅₃	semidirect
185. M₁₆	C₂×Dic₅₃	semidirect
267. C₂	#16810395	direct (N × H)

Subgroups — C2×#16810395 #20274458

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810395 #20274458