Subgroups — C₂×#14583707 #19149352

Order 5504.

1460 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063371	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583707	C₂	1	16	yes	maximal, direct product factor
3	C₂×#13063356	C₂	1	3	yes	maximal, complemented
4	C₂×#13063350	C₂	1	3	yes	maximal, complemented
5	C₂×#13063345	C₂	1	3	yes	maximal, complemented
6	C₂²×#10621884	C₂	1	3	yes	maximal, complemented
7	C₂×#13063332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×2₋¹⁺⁴×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#13063371	—	2	4	no	—
10	#13063371	C₂²	1	8	yes	complemented
11	C₂×#10621902	—	2	1	no	—
12	C₂×#10621898	C₂²	1	3	yes	—
13	C₂²×D₁₇₂	C₂²	1	3	yes	—
14	C₂×D₄×D₄₃	—	2	3	no	—
15	#13063356	C₂²	1	24	yes	complemented
16	#13063350	C₂²	1	24	yes	complemented
17	#13063345	C₂²	1	24	yes	complemented
18	C₂×#10621885	C₂²	1	3	yes	complemented
19	C₂×#10621883	C₂²	1	1	yes	characteristic, complemented
20	C₂×#10621884	C₂²	1	3	yes	complemented
21	C₂×#10621884	C₂²	1	6	yes	complemented
22	C₂×#10621884	C₂²	1	24	yes	complemented
23	C₂×#10621882	C₂²	1	3	yes	complemented
24	#13063332	C₂²	1	8	yes	complemented
25	C₂×#10621877	C₂²	1	3	yes	complemented
26	C₂²×C₄₃⋊C₈	C₂²	1	3	yes	complemented
27	2₋¹⁺⁴×C₄₃	—	2	4	no	—
28	2₋¹⁺⁴×C₄₃	C₂²	1	8	yes	complemented
29	C₂×#41×C₄₃	—	2	3	no	—
30	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×#41×C₄₃	C₂²	1	3	yes	complemented
32	C₂²×Q₈×C₄₃	—	2	1	no	—
33	C₂²×Q₈×C₄₃	C₂²	1	3	yes	complemented
34	C₂³×D₄₃	—	2	3	no	—
35	#41×C₄₃	—	2	24	no	—
36	#41×C₄₃	—	2	6	no	—
37	#41×C₄₃	D₄	1	4	yes	—
38	#41×C₄₃	C₂³	1	4	yes	—
39	#41×C₄₃	C₂³	1	12	yes	complemented
40	C₂×Q₈×C₄₃	—	2	8	no	—
41	C₂×Q₈×C₄₃	—	2	3	no	—
42	C₂×Q₈×C₄₃	—	2	6	no	—
43	C₂×Q₈×C₄₃	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₄₃	C₂³	1	3	yes	—
45	C₂×Q₈×C₄₃	C₂³	1	12	yes	complemented
46	C₂×D₄×C₄₃	—	2	3	no	—
47	C₂×D₄×C₄₃	D₄	1	3	yes	—
48	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	—	2	3	no	—
50	C₂²×C₄×C₄₃	—	2	3	no	—
51	C₂²×C₄×C₄₃	C₂³	1	3	yes	—
52	C₂×C₂²⋊D₄₃	—	2	3	no	—
53	#10621902	—	2	8	no	—
54	D₄×D₄₃	—	2	24	no	—
55	#10621898	—	2	6	no	—
56	#10621898	C₂³	1	12	yes	—
57	C₂×D₁₇₂	—	2	3	no	—
58	C₂×D₁₇₂	—	2	6	no	—
59	C₂×D₁₇₂	C₂³	1	3	yes	—
60	C₂×D₁₇₂	C₂³	1	12	yes	—
61	C₂×C₄×D₄₃	—	2	3	no	—
62	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
63	#10621885	C₂³	1	12	yes	complemented
64	#10621884	C₂³	1	12	yes	complemented
65	#10621884	C₂³	1	24	yes	complemented
66	#10621883	C₂³	1	4	yes	complemented
67	#10621882	C₂³	1	12	yes	complemented
68	#10621877	C₂³	1	12	yes	complemented
69	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₄₃⋊C₈	C₂³	1	3	yes	—
71	C₂×C₄₃⋊C₈	C₂³	1	12	yes	complemented
72	C₂³×C₄₃	—	2	1	no	—
73	C₂³×C₄₃	C₂×D₄	1	3	yes	—
74	C₂²×D₄₃	—	2	3	no	—
75	C₂²×D₄₃	—	2	12	no	—
76	C₂²×D₄₃	—	8	6	no	—
77	Q₈×C₄₃	—	2	3	no	—
78	Q₈×C₄₃	—	2	12	no	—
79	Q₈×C₄₃	C₂×D₄	1	4	yes	—
80	Q₈×C₄₃	C₂⁴	1	6	yes	—
81	D₄×C₄₃	—	2	12	no	—
82	D₄×C₄₃	—	2	1	no	—
83	D₄×C₄₃	C₂×D₄	1	12	yes	—
84	D₄×C₄₃	C₂⁴	1	2	yes	—
85	C₂×C₄×C₄₃	—	2	3	no	—
86	C₂×C₄×C₄₃	—	2	12	no	—
87	C₂×C₄×C₄₃	—	2	12	no	—
88	C₂×C₄×C₄₃	C₂×D₄	1	3	yes	—
89	C₂×C₄×C₄₃	C₂×D₄	1	6	yes	—
90	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₂⁴	1	6	yes	—
92	C₂²⋊D₄₃	—	2	12	no	—
93	C₂×Dic₄₃	—	2	1	no	—
94	D₁₇₂	—	2	3	no	—
95	D₁₇₂	—	2	12	no	—
96	D₁₇₂	C₂⁴	1	6	yes	—
97	C₄×D₄₃	—	2	12	no	—
98	Dic₈₆	—	2	1	no	—
99	Dic₈₆	C₂⁴	1	2	yes	—
100	C₄₃⋊C₈	C₂⁴	1	2	yes	—
101	C₄₃⋊C₈	C₂⁴	1	6	yes	—
102	C₂²×C₄₃	—	2	6	no	—
103	C₂²×C₄₃	—	2	2	no	—
104	C₂²×C₄₃	—	4	2	no	—
105	C₂²×C₄₃	C₂²×D₄	1	6	yes	—
106	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×D₄₃	—	2	6	no	—
108	C₂×D₄₃	—	4	6	no	—
109	C₂×D₄₃	—	8	12	no	—
110	C₄×C₄₃	—	2	6	no	—
111	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
112	C₄×C₄₃	C₂²×D₄	1	6	yes	—
113	C₄×C₄₃	C₂⁵	1	1	yes	characteristic
114	Dic₄₃	—	2	2	no	—
115	C₂×#577	—	43	1	no	maximal, Sylow
116	C₂×C₄₃	—	2	6	no	—
117	C₂×C₄₃	—	4	2	no	—
118	C₂×C₄₃	#577	1	2	yes	complemented
119	C₂×C₄₃	C₂³×D₄	1	1	yes	characteristic
120	D₄₃	—	4	6	no	—
121	C₂×2₋¹⁺⁴	D₄₃	1	1	yes	characteristic, complemented
122	C₂×H(5,𝔽₂)	—	43	1	no	—
123	#577	—	43	16	no	—
124	C₂×#137	—	43	3	no	—
125	C₂×AGL(1,ℤ/8)	—	43	3	no	—
126	C₂×#135	—	43	3	no	—
127	C₂²×SD₁₆	—	43	3	no	—
128	C₂×#131	—	43	1	no	—
129	C₄₃	C₂×#577	1	1	yes	characteristic, complemented, Sylow
130	2₋¹⁺⁴	—	2	4	no	—
131	2₋¹⁺⁴	C₂×D₄₃	1	8	yes	complemented
132	H(5,𝔽₂)	—	43	8	no	—
133	H(5,𝔽₂)	—	86	4	no	—
134	C₂×#41	—	2	3	no	—
135	C₂×#41	—	43	3	no	—
136	C₂×#41	—	86	1	no	—
137	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
138	C₂×#41	C₂×D₄₃	1	3	yes	complemented
139	C₂²×Q₈	—	2	1	no	—
140	C₂²×Q₈	C₂×D₄₃	1	3	yes	complemented
141	C₂²×D₄	—	43	3	no	—
142	C₂²×D₄	—	86	3	no	—
143	#137	—	43	24	no	—
144	AGL(1,ℤ/8)	—	43	24	no	—
145	#135	—	43	24	no	—
146	C₂×Q₁₆	—	43	3	no	—
147	C₂×SD₁₆	—	43	3	no	—
148	C₂×SD₁₆	—	43	6	no	—
149	C₂×SD₁₆	—	43	24	no	—
150	C₂×SD₁₆	—	43	1	no	—
151	C₂×D₈	—	43	3	no	—
152	#131	—	43	8	no	—
153	C₂×M₁₆	—	43	3	no	—
154	C₂²×C₈	—	43	3	no	—
155	C₂⁴	—	86	3	no	—
156	#41	—	2	24	no	—
157	#41	—	2	6	no	—
158	#41	—	43	12	no	—
159	#41	—	86	8	no	—
160	#41	—	86	6	no	—
161	#41	C₂²⋊D₄₃	1	4	yes	—
162	#41	C₂²×D₄₃	1	4	yes	—
163	#41	C₂²×D₄₃	1	12	yes	complemented
164	C₂×Q₈	—	2	8	no	—
165	C₂×Q₈	—	2	3	no	—
166	C₂×Q₈	—	2	6	no	—
167	C₂×Q₈	—	43	1	no	—
168	C₂×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
169	C₂×Q₈	C₂²×D₄₃	1	3	yes	—
170	C₂×Q₈	C₂²×D₄₃	1	12	yes	complemented
171	C₂×D₄	—	2	3	no	—
172	C₂×D₄	—	43	3	no	—
173	C₂×D₄	—	43	12	no	—
174	C₂×D₄	—	86	3	no	—
175	C₂×D₄	—	86	24	no	—
176	C₂×D₄	—	86	3	no	—
177	C₂×D₄	—	86	6	no	—
178	C₂×D₄	C₂²⋊D₄₃	1	3	yes	—
179	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
180	C₂²×C₄	—	2	3	no	—
181	C₂²×C₄	—	2	3	no	—
182	C₂²×C₄	—	86	3	no	—
183	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
184	Q₁₆	—	43	12	no	—
185	SD₁₆	—	43	12	no	—
186	SD₁₆	—	43	24	no	—
187	SD₁₆	—	43	4	no	—
188	D₈	—	43	12	no	—
189	M₁₆	—	43	12	no	—
190	C₂×C₈	—	43	1	no	—
191	C₂×C₈	—	43	3	no	—
192	C₂×C₈	—	43	12	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	86	12	no	—
195	C₂³	—	86	3	no	—
196	C₂³	—	344	6	no	—
197	C₂³	C₂×C₂²⋊D₄₃	1	3	yes	—
198	Q₈	—	2	3	no	—
199	Q₈	—	2	12	no	—
200	Q₈	—	43	2	no	—
201	Q₈	—	86	1	no	—
202	Q₈	C₂×C₂²⋊D₄₃	1	4	yes	—
203	Q₈	C₂³×D₄₃	1	6	yes	—
204	D₄	—	2	12	no	—
205	D₄	—	2	1	no	—
206	D₄	—	43	6	no	—
207	D₄	—	86	12	no	—
208	D₄	—	86	12	no	—
209	D₄	—	86	3	no	—
210	D₄	C₂×C₂²⋊D₄₃	1	12	yes	—
211	D₄	C₂³×D₄₃	1	2	yes	—
212	C₂×C₄	—	2	12	no	—
213	C₂×C₄	—	2	3	no	—
214	C₂×C₄	—	2	12	no	—
215	C₂×C₄	—	86	1	no	—
216	C₂×C₄	—	86	12	no	—
217	C₂×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
218	C₂×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
219	C₂×C₄	C₂³×D₄₃	1	6	yes	—
220	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
221	C₈	—	43	2	no	—
222	C₈	—	43	6	no	—
223	C₂²	—	2	6	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	86	6	no	—
227	C₂²	—	344	12	no	—
228	C₂²	—	172	6	no	—
229	C₂²	C₂²×C₂²⋊D₄₃	1	6	yes	—
230	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
231	C₄	—	2	6	no	—
232	C₄	—	86	2	no	—
233	C₄	C₂²×C₂²⋊D₄₃	1	6	yes	—
234	C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
235	C₄	C₂⁴×D₄₃	1	1	yes	characteristic
236	C₂	—	2	6	no	—
237	C₂	—	4	2	no	—
238	C₂	—	172	6	no	—
239	C₂	#14583707	1	2	yes	direct product factor
240	C₂	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₃
commutator	C₄×C₄₃	C₂⁵
Frattini	C₄	C₂⁴×D₄₃
Fitting	C₂×2₋¹⁺⁴×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₂×2₋¹⁺⁴	D₄₃
hypocenter	C₄₃	C₂×#577
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂×#13063371	C₂	semidirect
2. #14583707	C₂	direct (N × H)
3. C₂×#13063356	C₂	semidirect
4. C₂×#13063350	C₂	semidirect
5. C₂×#13063345	C₂	semidirect
6. C₂²×#10621884	C₂	semidirect
7. C₂×#13063332	C₂	semidirect
8. C₂×2₋¹⁺⁴×C₄₃	C₂	semidirect
10. #13063371	C₂²	semidirect
15. #13063356	C₂²	semidirect
16. #13063350	C₂²	semidirect
17. #13063345	C₂²	semidirect
18. C₂×#10621885	C₂²	semidirect
19. C₂×#10621883	C₂²	semidirect
20. C₂×#10621884	C₂²	semidirect
21. C₂×#10621884	C₂²	semidirect
22. C₂×#10621884	C₂²	semidirect
23. C₂×#10621882	C₂²	semidirect
24. #13063332	C₂²	semidirect
25. C₂×#10621877	C₂²	semidirect
26. C₂²×C₄₃⋊C₈	C₂²	semidirect
28. 2₋¹⁺⁴×C₄₃	C₂²	semidirect
31. C₂×#41×C₄₃	C₂²	semidirect
33. C₂²×Q₈×C₄₃	C₂²	semidirect
39. #41×C₄₃	C₂³	semidirect
45. C₂×Q₈×C₄₃	C₂³	semidirect
63. #10621885	C₂³	semidirect
64. #10621884	C₂³	semidirect
65. #10621884	C₂³	semidirect
66. #10621883	C₂³	semidirect
67. #10621882	C₂³	semidirect
68. #10621877	C₂³	semidirect
71. C₂×C₄₃⋊C₈	C₂³	semidirect
118. C₂×C₄₃	#577	semidirect
121. C₂×2₋¹⁺⁴	D₄₃	semidirect
129. C₄₃	C₂×#577	semidirect
131. 2₋¹⁺⁴	C₂×D₄₃	semidirect
138. C₂×#41	C₂×D₄₃	semidirect
140. C₂²×Q₈	C₂×D₄₃	semidirect
163. #41	C₂²×D₄₃	semidirect
170. C₂×Q₈	C₂²×D₄₃	semidirect
239. C₂	#14583707	direct (N × H)

Subgroups — C2×#14583707 #19149352

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14583707 #19149352