Subgroups — C₂×#16810742 #20283608

Order 6784.

464 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133237	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133225	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133224	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810742	C₂	1	8	yes	maximal, direct product factor
5	C₂×#13133152	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13133145	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133128	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#104×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11719945	C₂²	1	1	yes	characteristic
10	#13133237	C₄	1	4	yes	complemented
11	#13133237	C₂²	1	4	yes	—
12	C₂×#11719925	C₄	1	1	yes	characteristic, complemented
13	C₂×#11719923	C₄	1	1	yes	characteristic, complemented
14	C₂×#11719924	C₄	1	1	yes	characteristic, complemented
15	C₂×#11719922	C₄	1	1	yes	characteristic, complemented
16	#13133225	—	2	2	no	—
17	#13133225	C₂²	1	4	yes	complemented
18	#13133224	—	2	2	no	—
19	#13133224	C₂²	1	4	yes	complemented
20	C₂×#11719917	—	2	1	no	—
21	C₂×#11719917	C₂²	1	1	yes	characteristic
22	C₂×#11719912	—	2	1	no	—
23	C₂²×#88322	—	2	1	no	—
24	C₂²×#88322	C₂²	1	1	yes	characteristic
25	C₂²×#88322	C₂²	1	1	yes	characteristic
26	C₂×C₈×D₅₃	—	2	1	no	—
27	#13133152	C₂²	1	4	yes	complemented
28	#13133145	C₂²	1	4	yes	complemented
29	#13133128	C₂²	1	4	yes	complemented
30	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
33	#104×C₅₃	C₂²	1	4	yes	complemented
34	#41×C₅₃	—	2	1	no	—
35	#41×C₅₃	—	2	1	no	—
36	#41×C₅₃	C₂×C₄	1	2	yes	—
37	#41×C₅₃	C₂³	1	2	yes	—
38	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×D₄×C₅₃	—	2	1	no	—
40	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	—	2	1	no	—
42	C₂²×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₅₃	—	2	1	no	—
45	#11719945	—	2	1	no	—
46	#11719945	—	2	1	no	—
47	#11719945	C₂×C₄	1	2	yes	—
48	#11719945	C₂³	1	2	yes	—
49	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₅₃	—	2	1	no	—
51	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
52	M₁₆×C₅₃	—	2	1	no	—
53	M₁₆×C₅₃	C₂³	1	2	yes	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₄²×C₅₃	—	2	1	no	—
56	C₄²×C₅₃	C₂³	1	2	yes	—
57	#11719925	C₂×C₄	1	4	yes	complemented
58	#11719924	C₂×C₄	1	4	yes	complemented
59	#11719923	C₂×C₄	1	4	yes	complemented
60	#11719922	C₂×C₄	1	4	yes	complemented
61	#11719917	—	2	1	no	—
62	#11719917	—	2	4	no	—
63	#11719917	C₂³	1	2	yes	—
64	C₂×#88322	—	2	1	no	—
65	C₂×#88322	—	2	4	no	—
66	C₂×#88322	C₂×C₄	1	1	yes	characteristic
67	C₂×#88322	C₂×C₄	1	1	yes	characteristic
68	C₂×#88322	C₂×C₄	1	2	yes	—
69	C₂×#88322	D₄	1	1	yes	characteristic
70	C₂×#88322	D₄	1	1	yes	characteristic
71	C₂×#88322	D₄	1	2	yes	—
72	C₂×#88322	C₂³	1	2	yes	—
73	C₂×#88322	C₂³	1	2	yes	—
74	#11719912	—	2	4	no	—
75	C₈×D₅₃	—	2	4	no	—
76	C₂³×C₅₃	—	2	1	no	—
77	C₂³×C₅₃	#41	1	1	yes	characteristic
78	C₂²×D₅₃	—	2	1	no	—
79	Q₈×C₅₃	—	2	1	no	—
80	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
81	D₄×C₅₃	—	2	2	no	—
82	D₄×C₅₃	—	2	2	no	—
83	D₄×C₅₃	—	2	1	no	—
84	D₄×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₅₃	—	2	2	no	—
97	C₂²⋊D₅₃	—	2	2	no	—
98	C₂×Dic₅₃	—	2	1	no	—
99	D₂₁₂	—	2	1	no	—
100	D₂₁₂	C₂²×C₄	1	2	yes	—
101	C₄×D₅₃	—	2	2	no	—
102	C₄×D₅₃	—	2	2	no	—
103	Dic₁₀₆	—	2	1	no	—
104	Dic₁₀₆	C₂²×C₄	1	2	yes	—
105	C₈×C₅₃	—	2	2	no	—
106	#88322	—	2	2	no	—
107	#88322	C₂²×C₄	1	2	yes	—
108	#88322	C₂²×C₄	1	2	yes	—
109	#88322	C₂×D₄	1	2	yes	—
110	#88322	C₂×D₄	1	2	yes	—
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	4	2	no	—
114	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
117	C₂×D₅₃	—	2	2	no	—
118	C₂×D₅₃	—	4	2	no	—
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	4	2	no	—
121	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
124	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	Dic₅₃	—	2	2	no	—
126	C₂×#444	—	53	1	no	maximal, Sylow
127	C₂×C₅₃	—	2	1	no	—
128	C₂×C₅₃	—	2	1	no	—
129	C₂×C₅₃	—	4	2	no	—
130	C₂×C₅₃	#444	1	2	yes	complemented
131	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
132	D₅₃	—	4	2	no	—
133	C₂×#135	—	53	1	no	—
134	C₂×#131	—	53	1	no	—
135	C₂×#131	—	53	1	no	—
136	#444	—	53	8	no	—
137	C₂×#108	—	53	1	no	—
138	C₂×#104	—	53	1	no	—
139	C₂×#104	D₅₃	1	1	yes	characteristic, complemented
140	C₂×#97	—	53	1	no	—
141	C₅₃	C₂×#444	1	1	yes	characteristic, complemented, Sylow
142	C₂×#41	—	53	1	no	—
143	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
144	#135	—	53	4	no	—
145	#135	—	53	4	no	—
146	C₂×Q₁₆	—	53	1	no	—
147	C₂×SD₁₆	—	53	1	no	—
148	C₂×SD₁₆	—	53	1	no	—
149	C₂×D₈	—	53	1	no	—
150	#131	—	53	4	no	—
151	#131	—	53	4	no	—
152	#131	—	106	2	no	—
153	#131	—	106	2	no	—
154	C₂×M₁₆	—	53	1	no	—
155	C₂×M₁₆	—	106	1	no	—
156	C₂×M₁₆	—	106	1	no	—
157	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
158	C₂²×C₈	—	53	1	no	—
159	C₂²×C₈	—	53	1	no	—
160	C₂²×C₈	—	106	1	no	—
161	C₂²×C₈	—	106	1	no	—
162	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
163	#108	—	53	4	no	—
164	#104	—	53	4	no	—
165	#104	C₂×D₅₃	1	4	yes	complemented
166	#97	—	53	4	no	—
167	#41	—	2	1	no	—
168	#41	—	2	1	no	—
169	#41	—	53	2	no	—
170	#41	—	53	2	no	—
171	#41	—	106	1	no	—
172	#41	—	106	1	no	—
173	#41	C₄×D₅₃	1	2	yes	—
174	#41	C₂²×D₅₃	1	2	yes	—
175	C₂×Q₈	—	53	1	no	—
176	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	—	53	1	no	—
179	C₂×D₄	—	106	1	no	—
180	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	106	1	no	—
184	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
185	Q₁₆	—	53	4	no	—
186	SD₁₆	—	53	4	no	—
187	SD₁₆	—	53	4	no	—
188	D₈	—	53	4	no	—
189	M₁₆	—	2	1	no	—
190	M₁₆	—	53	2	no	—
191	M₁₆	—	106	4	no	—
192	M₁₆	—	106	4	no	—
193	M₁₆	—	106	1	no	—
194	M₁₆	C₂²×D₅₃	1	2	yes	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	53	1	no	—
197	C₂×C₈	—	53	1	no	—
198	C₂×C₈	—	53	2	no	—
199	C₂×C₈	—	53	2	no	—
200	C₂×C₈	—	53	1	no	—
201	C₂×C₈	—	53	1	no	—
202	C₂×C₈	—	53	2	no	—
203	C₂×C₈	—	53	2	no	—
204	C₂×C₈	—	106	4	no	—
205	C₂×C₈	—	106	4	no	—
206	C₂×C₈	—	106	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	C₂²×D₅₃	1	2	yes	—
209	C₂³	—	2	1	no	—
210	C₂³	—	106	1	no	—
211	C₂³	#11719947	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	53	2	no	—
214	Q₈	—	106	1	no	—
215	Q₈	C₂×C₄×D₅₃	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	53	2	no	—
220	D₄	—	106	2	no	—
221	D₄	—	106	2	no	—
222	D₄	—	106	1	no	—
223	D₄	C₂×C₄×D₅₃	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	106	1	no	—
230	C₂×C₄	—	106	2	no	—
231	C₂×C₄	—	106	2	no	—
232	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
234	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
235	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
236	C₂×C₄	#11719947	1	1	yes	characteristic
237	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	53	2	no	—
240	C₈	—	53	2	no	—
241	C₈	—	53	2	no	—
242	C₈	—	53	2	no	—
243	C₈	—	106	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	106	2	no	—
248	C₂²	—	212	2	no	—
249	C₂²	#13133177	1	1	yes	characteristic
250	C₂²	#13133177	1	1	yes	characteristic
251	C₂²	C₂×#11719947	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	4	2	no	—
254	C₄	—	106	2	no	—
255	C₄	#13133177	1	1	yes	characteristic
256	C₄	#13133177	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
258	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	4	2	no	—
262	C₂	—	212	2	no	—
263	C₂	#16810742	1	2	yes	direct product factor
264	C₂	C₂×#13133177	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#104×C₅₃	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	C₂×#104	D₅₃
hypocenter	C₅₃	C₂×#444
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#104

Complements

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

N	H ≅ G/N	type
2. C₂×#13133225	C₂	semidirect
3. C₂×#13133224	C₂	semidirect
4. #16810742	C₂	direct (N × H)
5. C₂×#13133152	C₂	semidirect
6. C₂×#13133145	C₂	semidirect
7. C₂×#13133128	C₂	semidirect
8. C₂×#104×C₅₃	C₂	semidirect
10. #13133237	C₄	semidirect
12. C₂×#11719925	C₄	semidirect
13. C₂×#11719923	C₄	semidirect
14. C₂×#11719924	C₄	semidirect
15. C₂×#11719922	C₄	semidirect
17. #13133225	C₂²	semidirect
19. #13133224	C₂²	semidirect
27. #13133152	C₂²	semidirect
28. #13133145	C₂²	semidirect
29. #13133128	C₂²	semidirect
33. #104×C₅₃	C₂²	semidirect
57. #11719925	C₂×C₄	semidirect
58. #11719924	C₂×C₄	semidirect
59. #11719923	C₂×C₄	semidirect
60. #11719922	C₂×C₄	semidirect
130. C₂×C₅₃	#444	semidirect
139. C₂×#104	D₅₃	semidirect
141. C₅₃	C₂×#444	semidirect
165. #104	C₂×D₅₃	semidirect
263. C₂	#16810742	direct (N × H)

Subgroups — C2×#16810742 #20283608

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810742 #20283608