Subgroups — C₂×#14583735 #19149403

Order 5504.

5956 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583735	C₂	1	32	yes	maximal, direct product factor
2	C₂×#13063373	C₂	1	2	yes	maximal, complemented
3	C₂×#13063372	C₂	1	4	yes	maximal, complemented
4	C₂×#13063371	C₂	1	4	yes	maximal, complemented
5	C₂×#13063370	C₂	1	6	yes	maximal, complemented
6	C₂²×#10621898	C₂	1	6	yes	maximal, complemented
7	C₂×#41×D₄₃	C₂	1	8	yes	maximal, complemented
8	C₂²×#41×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#13063373	C₂²	1	32	yes	complemented
10	#13063372	C₂²	1	64	yes	complemented
11	#13063371	C₂²	1	64	yes	complemented
12	#13063370	C₂²	1	96	yes	complemented
13	C₂×#10621902	C₂²	1	8	yes	complemented
14	C₂×#10621898	C₂²	1	12	yes	complemented
15	C₂×#10621898	C₂²	1	4	yes	complemented
16	C₂×#10621898	C₂²	1	12	yes	complemented
17	C₂×#10621898	C₂²	1	12	yes	complemented
18	C₂×#10621898	C₂²	1	96	yes	complemented
19	C₂×#10621900	C₂²	1	24	yes	complemented
20	#41×D₄₃	C₂²	1	128	yes	complemented
21	C₂²×Dic₈₆	C₂²	1	6	yes	complemented
22	C₂×Q₈×D₄₃	C₂²	1	8	yes	complemented
23	C₂²×D₁₇₂	C₂²	1	6	yes	complemented
24	C₂²×C₂²⋊D₄₃	C₂²	1	12	yes	complemented
25	C₂×D₄×D₄₃	C₂²	1	24	yes	complemented
26	C₂²×C₄×D₄₃	C₂²	1	12	yes	complemented
27	C₂×#41×C₄₃	C₂²	1	4	yes	complemented
28	C₂×#41×C₄₃	C₂²	1	4	yes	complemented
29	C₂×#41×C₄₃	C₂²	1	16	yes	complemented
30	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
31	C₂²×D₄×C₄₃	C₂²	1	3	yes	complemented
32	C₂³×C₄×C₄₃	C₂²	1	3	yes	complemented
33	C₂⁴×C₄₃	C₂³	1	3	yes	—
34	C₂³×D₄₃	C₂³	1	12	yes	—
35	#41×C₄₃	C₂³	1	32	yes	complemented
36	#41×C₄₃	C₂³	1	32	yes	complemented
37	C₂×Q₈×C₄₃	C₂³	1	4	yes	—
38	C₂×Q₈×C₄₃	C₂³	1	8	yes	complemented
39	C₂×D₄×C₄₃	C₂³	1	12	yes	—
40	C₂×D₄×C₄₃	C₂³	1	24	yes	complemented
41	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂³	1	3	yes	—
43	C₂²×C₄×C₄₃	C₂³	1	6	yes	—
44	C₂²×C₄×C₄₃	C₂³	1	6	yes	—
45	C₂²×C₄×C₄₃	C₂³	1	24	yes	complemented
46	C₂×C₂²⋊D₄₃	C₂³	1	24	yes	—
47	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
48	C₂×C₂²⋊D₄₃	C₂³	1	96	yes	complemented
49	C₂²×Dic₄₃	C₂³	1	12	yes	—
50	#10621902	C₂³	1	64	yes	complemented
51	Q₈×D₄₃	C₂³	1	64	yes	complemented
52	#10621900	C₂³	1	192	yes	complemented
53	D₄×D₄₃	C₂³	1	192	yes	complemented
54	#10621898	C₂³	1	96	yes	complemented
55	#10621898	C₂³	1	32	yes	complemented
56	#10621898	C₂³	1	96	yes	complemented
57	#10621898	C₂³	1	96	yes	complemented
58	C₂×D₁₇₂	C₂³	1	12	yes	—
59	C₂×D₁₇₂	C₂³	1	4	yes	—
60	C₂×D₁₇₂	C₂³	1	48	yes	complemented
61	C₂×C₄×D₄₃	C₂³	1	8	yes	—
62	C₂×C₄×D₄₃	C₂³	1	24	yes	—
63	C₂×C₄×D₄₃	C₂³	1	96	yes	complemented
64	C₂×Dic₈₆	C₂³	1	12	yes	—
65	C₂×Dic₈₆	C₂³	1	4	yes	—
66	C₂×Dic₈₆	C₂³	1	48	yes	complemented
67	C₂³×C₄₃	—	4	6	no	—
68	C₂³×C₄₃	C₂⁴	1	6	yes	—
69	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
70	C₂³×C₄₃	C₂⁴	1	12	yes	—
71	C₂²×D₄₃	—	4	24	no	—
72	C₂²×D₄₃	C₂⁴	1	8	yes	—
73	C₂²×D₄₃	C₂⁴	1	48	yes	—
74	Q₈×C₄₃	C₂⁴	1	16	yes	—
75	D₄×C₄₃	C₂⁴	1	48	yes	—
76	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
77	C₂×C₄×C₄₃	C₂⁴	1	6	yes	—
78	C₂×C₄×C₄₃	C₂⁴	1	24	yes	—
79	C₂×C₄×C₄₃	C₂⁴	1	24	yes	—
80	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
81	C₂×C₄×C₄₃	C₂⁴	1	12	yes	—
82	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₄₃	C₂⁴	1	32	yes	—
84	C₂²⋊D₄₃	C₂⁴	1	96	yes	—
85	C₂×Dic₄₃	C₂⁴	1	8	yes	—
86	C₂×Dic₄₃	C₂⁴	1	48	yes	—
87	D₁₇₂	C₂⁴	1	16	yes	—
88	D₁₇₂	C₂⁴	1	48	yes	—
89	C₄×D₄₃	C₂⁴	1	32	yes	—
90	C₄×D₄₃	C₂⁴	1	96	yes	—
91	Dic₈₆	C₂⁴	1	16	yes	—
92	Dic₈₆	C₂⁴	1	48	yes	—
93	C₂²×C₄₃	—	2	12	no	—
94	C₂²×C₄₃	—	2	2	no	—
95	C₂²×C₄₃	—	4	12	no	—
96	C₂²×C₄₃	C₂⁵	1	12	yes	—
97	C₂²×C₄₃	C₂⁵	1	2	yes	—
98	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
99	C₂×D₄₃	—	2	16	no	—
100	C₂×D₄₃	—	4	48	no	—
101	C₂×D₄₃	C₂⁵	1	16	yes	—
102	C₄×C₄₃	C₂⁵	1	2	yes	—
103	C₄×C₄₃	C₂⁵	1	12	yes	—
104	C₄×C₄₃	C₂⁵	1	2	yes	—
105	Dic₄₃	C₂⁵	1	16	yes	—
106	C₂×#585	—	43	1	no	maximal, Sylow
107	C₂×C₄₃	—	2	12	no	—
108	C₂×C₄₃	—	2	2	no	—
109	C₂×C₄₃	#585	1	2	yes	complemented
110	C₂×C₄₃	C₂⁶	1	1	yes	characteristic
111	D₄₃	—	2	16	no	—
112	#585	—	43	32	no	—
113	C₂×2₋¹⁺⁴	—	43	4	no	—
114	C₂×2₋¹⁺⁴	—	43	2	no	—
115	C₂×H(5,𝔽₂)	—	43	4	no	—
116	C₂×H(5,𝔽₂)	—	43	6	no	—
117	C₂²×#41	—	43	8	no	—
118	C₂²×#41	—	43	6	no	—
119	C₂²×#41	D₄₃	1	1	yes	characteristic, complemented
120	C₄₃	C₂×#585	1	1	yes	characteristic, complemented, Sylow
121	2₋¹⁺⁴	—	43	64	no	—
122	2₋¹⁺⁴	—	43	32	no	—
123	H(5,𝔽₂)	—	43	64	no	—
124	H(5,𝔽₂)	—	43	96	no	—
125	C₂×#41	—	43	128	no	—
126	C₂×#41	—	43	8	no	—
127	C₂×#41	—	43	24	no	—
128	C₂×#41	—	43	12	no	—
129	C₂×#41	—	43	4	no	—
130	C₂×#41	—	43	12	no	—
131	C₂×#41	—	43	12	no	—
132	C₂×#41	—	43	96	no	—
133	C₂×#41	C₂×D₄₃	1	4	yes	complemented
134	C₂×#41	C₂×D₄₃	1	4	yes	complemented
135	C₂×#41	C₂×D₄₃	1	16	yes	complemented
136	C₂²×Q₈	—	43	8	no	—
137	C₂²×Q₈	—	43	6	no	—
138	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic, complemented
139	C₂²×D₄	—	43	12	no	—
140	C₂²×D₄	—	43	24	no	—
141	C₂²×D₄	—	43	6	no	—
142	C₂²×D₄	C₂×D₄₃	1	3	yes	complemented
143	C₂³×C₄	—	43	12	no	—
144	C₂³×C₄	C₂×D₄₃	1	3	yes	complemented
145	C₂⁴	—	43	12	no	—
146	C₂⁴	C₂²×D₄₃	1	3	yes	—
147	#41	—	43	64	no	—
148	#41	—	43	192	no	—
149	#41	—	43	96	no	—
150	#41	—	43	32	no	—
151	#41	—	43	96	no	—
152	#41	—	43	96	no	—
153	#41	C₂²×D₄₃	1	32	yes	complemented
154	#41	C₂²×D₄₃	1	32	yes	complemented
155	C₂×Q₈	—	43	64	no	—
156	C₂×Q₈	—	43	12	no	—
157	C₂×Q₈	—	43	4	no	—
158	C₂×Q₈	—	43	48	no	—
159	C₂×Q₈	C₂²×D₄₃	1	4	yes	—
160	C₂×Q₈	C₂²×D₄₃	1	8	yes	complemented
161	C₂×D₄	—	43	24	no	—
162	C₂×D₄	—	43	8	no	—
163	C₂×D₄	—	43	96	no	—
164	C₂×D₄	—	43	192	no	—
165	C₂×D₄	—	43	12	no	—
166	C₂×D₄	—	43	4	no	—
167	C₂×D₄	—	43	48	no	—
168	C₂×D₄	C₂²×D₄₃	1	12	yes	—
169	C₂×D₄	C₂²×D₄₃	1	24	yes	complemented
170	C₂²×C₄	—	43	12	no	—
171	C₂²×C₄	—	43	24	no	—
172	C₂²×C₄	—	43	96	no	—
173	C₂²×C₄	—	43	8	no	—
174	C₂²×C₄	C₂²×D₄₃	1	6	yes	—
175	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
176	C₂²×C₄	C₂²×D₄₃	1	6	yes	—
177	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₄₃	1	24	yes	complemented
179	C₂³	—	4	6	no	—
180	C₂³	—	43	8	no	—
181	C₂³	—	43	48	no	—
182	C₂³	—	172	24	no	—
183	C₂³	C₂³×D₄₃	1	6	yes	—
184	C₂³	C₂³×D₄₃	1	1	yes	characteristic
185	C₂³	C₂³×D₄₃	1	12	yes	—
186	Q₈	—	43	48	no	—
187	Q₈	—	43	16	no	—
188	Q₈	C₂³×D₄₃	1	16	yes	—
189	D₄	—	43	96	no	—
190	D₄	—	43	32	no	—
191	D₄	—	43	48	no	—
192	D₄	—	43	16	no	—
193	D₄	C₂³×D₄₃	1	48	yes	—
194	C₂×C₄	—	43	8	no	—
195	C₂×C₄	—	43	48	no	—
196	C₂×C₄	—	43	32	no	—
197	C₂×C₄	—	43	96	no	—
198	C₂×C₄	C₂³×D₄₃	1	6	yes	—
199	C₂×C₄	C₂³×D₄₃	1	24	yes	—
200	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₄₃	1	4	yes	—
202	C₂×C₄	C₂³×D₄₃	1	24	yes	—
203	C₂×C₄	C₂³×D₄₃	1	12	yes	—
204	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
205	C₂²	—	2	2	no	—
206	C₂²	—	2	12	no	—
207	C₂²	—	4	12	no	—
208	C₂²	—	43	16	no	—
209	C₂²	—	86	16	no	—
210	C₂²	—	172	48	no	—
211	C₂²	C₂⁴×D₄₃	1	12	yes	—
212	C₂²	C₂⁴×D₄₃	1	2	yes	—
213	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
214	C₄	—	43	16	no	—
215	C₄	C₂⁴×D₄₃	1	2	yes	—
216	C₄	C₂⁴×D₄₃	1	12	yes	—
217	C₄	C₂⁴×D₄₃	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	2	12	no	—
220	C₂	—	86	16	no	—
221	C₂	#14583735	1	2	yes	direct product factor
222	C₂	C₂⁵×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₄₃
commutator	C₂×C₄₃	C₂⁶
Frattini	C₂	C₂⁵×D₄₃
Fitting	C₂²×#41×C₄₃	C₂
socle	C₂²×C₄₃	C₂⁵
hypercenter	C₂²×#41	D₄₃
hypocenter	C₄₃	C₂×#585
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. #14583735	C₂	direct (N × H)
2. C₂×#13063373	C₂	semidirect
3. C₂×#13063372	C₂	semidirect
4. C₂×#13063371	C₂	semidirect
5. C₂×#13063370	C₂	semidirect
6. C₂²×#10621898	C₂	semidirect
7. C₂×#41×D₄₃	C₂	semidirect
8. C₂²×#41×C₄₃	C₂	semidirect
9. #13063373	C₂²	semidirect
10. #13063372	C₂²	semidirect
11. #13063371	C₂²	semidirect
12. #13063370	C₂²	semidirect
13. C₂×#10621902	C₂²	semidirect
14. C₂×#10621898	C₂²	semidirect
15. C₂×#10621898	C₂²	semidirect
16. C₂×#10621898	C₂²	semidirect
17. C₂×#10621898	C₂²	semidirect
18. C₂×#10621898	C₂²	semidirect
19. C₂×#10621900	C₂²	semidirect
20. #41×D₄₃	C₂²	semidirect
21. C₂²×Dic₈₆	C₂²	semidirect
22. C₂×Q₈×D₄₃	C₂²	semidirect
23. C₂²×D₁₇₂	C₂²	semidirect
24. C₂²×C₂²⋊D₄₃	C₂²	semidirect
25. C₂×D₄×D₄₃	C₂²	semidirect
26. C₂²×C₄×D₄₃	C₂²	semidirect
27. C₂×#41×C₄₃	C₂²	semidirect
28. C₂×#41×C₄₃	C₂²	semidirect
29. C₂×#41×C₄₃	C₂²	semidirect
30. C₂²×Q₈×C₄₃	C₂²	semidirect
31. C₂²×D₄×C₄₃	C₂²	semidirect
32. C₂³×C₄×C₄₃	C₂²	semidirect
35. #41×C₄₃	C₂³	semidirect
36. #41×C₄₃	C₂³	semidirect
38. C₂×Q₈×C₄₃	C₂³	semidirect
40. C₂×D₄×C₄₃	C₂³	semidirect
45. C₂²×C₄×C₄₃	C₂³	semidirect
48. C₂×C₂²⋊D₄₃	C₂³	semidirect
50. #10621902	C₂³	semidirect
51. Q₈×D₄₃	C₂³	semidirect
52. #10621900	C₂³	semidirect
53. D₄×D₄₃	C₂³	semidirect
54. #10621898	C₂³	semidirect
55. #10621898	C₂³	semidirect
56. #10621898	C₂³	semidirect
57. #10621898	C₂³	semidirect
60. C₂×D₁₇₂	C₂³	semidirect
63. C₂×C₄×D₄₃	C₂³	semidirect
66. C₂×Dic₈₆	C₂³	semidirect
109. C₂×C₄₃	#585	semidirect
119. C₂²×#41	D₄₃	semidirect
120. C₄₃	C₂×#585	semidirect
133. C₂×#41	C₂×D₄₃	semidirect
134. C₂×#41	C₂×D₄₃	semidirect
135. C₂×#41	C₂×D₄₃	semidirect
138. C₂²×Q₈	C₂×D₄₃	semidirect
142. C₂²×D₄	C₂×D₄₃	semidirect
144. C₂³×C₄	C₂×D₄₃	semidirect
153. #41	C₂²×D₄₃	semidirect
154. #41	C₂²×D₄₃	semidirect
160. C₂×Q₈	C₂²×D₄₃	semidirect
169. C₂×D₄	C₂²×D₄₃	semidirect
178. C₂²×C₄	C₂²×D₄₃	semidirect
221. C₂	#14583735	direct (N × H)

Subgroups — C2×#14583735 #19149403

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14583735 #19149403