Subgroups — C₂×#14583701 #19149345

Order 5504.

1512 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	C₂×#13063371	C₂	1	2	yes	maximal, complemented
2	#14583701	C₂	1	16	yes	maximal, direct product factor
3	C₂×#13063352	C₂	1	4	yes	maximal, complemented
4	C₂×#13063350	C₂	1	2	yes	maximal, complemented
5	C₂×#13063348	C₂	1	2	yes	maximal, complemented
6	C₂²×D₃₄₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₈×D₄₃	C₂	1	2	yes	maximal, complemented
8	C₂×#13063331	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#13063371	—	2	8	no	—
11	#13063371	C₂²	1	16	yes	complemented
12	C₂×#10621902	C₂²	1	4	yes	complemented
13	C₂×#10621898	—	2	2	no	—
14	C₂×#10621898	C₂²	1	1	yes	characteristic
15	C₂²×D₁₇₂	—	2	2	no	—
16	C₂²×D₁₇₂	C₂²	1	2	yes	complemented
17	C₂×D₄×D₄₃	—	2	4	no	—
18	C₂×D₄×D₄₃	C₂²	1	4	yes	complemented
19	#13063352	C₂²	1	32	yes	complemented
20	#13063350	C₂²	1	16	yes	complemented
21	#13063348	C₂²	1	16	yes	complemented
22	C₂×#10621884	C₂²	1	4	yes	complemented
23	C₂×D₃₄₄	C₂²	1	2	yes	complemented
24	C₂×D₃₄₄	C₂²	1	2	yes	complemented
25	C₂×D₃₄₄	C₂²	1	8	yes	complemented
26	C₂×#10621882	C₂²	1	4	yes	complemented
27	D₈×D₄₃	C₂²	1	16	yes	complemented
28	#13063331	C₂²	1	8	yes	complemented
29	C₂×#10621877	C₂²	1	1	yes	characteristic, complemented
30	C₂×#10621872	C₂²	1	2	yes	complemented
31	C₂×C₈×D₄₃	C₂²	1	2	yes	complemented
32	C₂×#41×C₄₃	C₂²	1	2	yes	complemented
33	#135×C₄₃	C₂²	1	8	yes	complemented
34	C₂×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic, complemented
35	C₂×SD₁₆×C₄₃	C₂²	1	2	yes	complemented
36	C₂×D₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
37	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
38	C₂³×D₄₃	—	2	2	no	—
39	C₂³×D₄₃	—	2	4	no	—
40	#41×C₄₃	—	2	4	no	—
41	#41×C₄₃	C₂³	1	8	yes	complemented
42	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
43	C₂×D₄×C₄₃	—	2	2	no	—
44	C₂×D₄×C₄₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₄₃	—	2	2	no	—
46	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₄₃	—	2	4	no	—
48	C₂×C₂²⋊D₄₃	D₄	1	2	yes	—
49	#10621902	—	2	8	no	—
50	#10621902	C₂³	1	16	yes	complemented
51	D₄×D₄₃	—	2	32	no	—
52	D₄×D₄₃	—	2	8	no	—
53	D₄×D₄₃	C₂³	1	16	yes	complemented
54	#10621898	—	2	16	no	—
55	#10621898	D₄	1	4	yes	—
56	#10621898	C₂³	1	4	yes	—
57	C₂×D₁₇₂	—	2	2	no	—
58	C₂×D₁₇₂	—	2	4	no	—
59	C₂×D₁₇₂	—	2	16	no	—
60	C₂×D₁₇₂	—	2	4	no	—
61	C₂×D₁₇₂	D₄	1	1	yes	characteristic
62	C₂×D₁₇₂	C₂³	1	4	yes	—
63	C₂×D₁₇₂	C₂³	1	8	yes	complemented
64	C₂×C₄×D₄₃	—	2	4	no	—
65	C₂×C₄×D₄₃	C₂³	1	2	yes	—
66	C₂×Dic₈₆	D₄	1	1	yes	characteristic
67	Q₁₆×C₄₃	C₂³	1	4	yes	complemented
68	SD₁₆×C₄₃	C₂³	1	8	yes	complemented
69	D₈×C₄₃	C₂³	1	4	yes	complemented
70	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₄₃	C₂³	1	4	yes	complemented
73	#10621884	C₂³	1	16	yes	complemented
74	#10621882	C₂³	1	16	yes	complemented
75	#10621877	C₂³	1	4	yes	complemented
76	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
77	D₃₄₄	C₂³	1	8	yes	complemented
78	D₃₄₄	C₂³	1	8	yes	complemented
79	#10621872	C₂³	1	8	yes	complemented
80	C₈×D₄₃	C₂³	1	8	yes	complemented
81	C₂³×C₄₃	—	2	2	no	—
82	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×D₄₃	—	2	4	no	—
84	C₂²×D₄₃	—	2	8	no	—
85	C₂²×D₄₃	—	2	16	no	—
86	C₂²×D₄₃	—	8	8	no	—
87	C₂²×D₄₃	—	8	4	no	—
88	C₂²×D₄₃	C₂×D₄	1	2	yes	—
89	Q₈×C₄₃	—	2	2	no	—
90	Q₈×C₄₃	C₂⁴	1	4	yes	—
91	D₄×C₄₃	—	2	8	no	—
92	D₄×C₄₃	—	2	2	no	—
93	D₄×C₄₃	C₂⁴	1	4	yes	—
94	C₂×C₄×C₄₃	—	2	2	no	—
95	C₂×C₄×C₄₃	—	2	8	no	—
96	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
100	C₂²⋊D₄₃	—	2	16	no	—
101	C₂²⋊D₄₃	C₂×D₄	1	8	yes	—
102	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
103	D₁₇₂	—	2	16	no	—
104	D₁₇₂	—	2	8	no	—
105	D₁₇₂	—	2	4	no	—
106	D₁₇₂	C₂×D₄	1	4	yes	—
107	D₁₇₂	C₂⁴	1	8	yes	—
108	C₄×D₄₃	—	2	16	no	—
109	C₄×D₄₃	C₂×D₄	1	4	yes	—
110	C₄×D₄₃	C₂⁴	1	4	yes	—
111	Dic₈₆	C₂×D₄	1	4	yes	—
112	C₈×C₄₃	C₂⁴	1	2	yes	—
113	C₈×C₄₃	C₂⁴	1	2	yes	—
114	C₄₃⋊C₈	C₂⁴	1	4	yes	—
115	C₂²×C₄₃	—	2	2	no	—
116	C₂²×C₄₃	—	2	4	no	—
117	C₂²×C₄₃	—	4	4	no	—
118	C₂²×C₄₃	C₂²×D₄	1	2	yes	—
119	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₄₃	—	2	4	no	—
121	C₂×D₄₃	—	2	8	no	—
122	C₂×D₄₃	—	4	8	no	—
123	C₂×D₄₃	—	8	16	no	—
124	C₂×D₄₃	—	8	8	no	—
125	C₂×D₄₃	C₂²×D₄	1	4	yes	—
126	C₄×C₄₃	—	2	4	no	—
127	C₄×C₄₃	C₂²×D₄	1	2	yes	—
128	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
129	C₄×C₄₃	C₂⁵	1	1	yes	characteristic
130	Dic₄₃	C₂²×D₄	1	4	yes	—
131	C₂×#576	—	43	1	no	maximal, Sylow
132	C₂×C₄₃	—	2	2	no	—
133	C₂×C₄₃	—	4	4	no	—
134	C₂×C₄₃	#576	1	2	yes	complemented
135	C₂×C₄₃	C₂³×D₄	1	1	yes	characteristic
136	D₄₃	—	2	4	no	—
137	D₄₃	—	4	8	no	—
138	C₂×H(5,𝔽₂)	—	43	2	no	—
139	#576	—	43	16	no	—
140	C₂×AGL(1,ℤ/8)	—	43	2	no	—
141	C₂×AGL(1,ℤ/8)	—	43	4	no	—
142	C₂×#135	—	43	2	no	—
143	C₂×#135	D₄₃	1	1	yes	characteristic, complemented
144	C₂²×D₈	—	43	2	no	—
145	C₂²×D₈	—	43	1	no	—
146	C₂×#131	—	43	1	no	—
147	C₄₃	C₂×#576	1	1	yes	characteristic, complemented, Sylow
148	H(5,𝔽₂)	—	43	16	no	—
149	H(5,𝔽₂)	—	86	8	no	—
150	C₂×#41	—	43	1	no	—
151	C₂×#41	—	43	4	no	—
152	C₂×#41	—	86	2	no	—
153	C₂×#41	C₂×D₄₃	1	2	yes	complemented
154	C₂²×D₄	—	43	4	no	—
155	C₂²×D₄	—	43	2	no	—
156	C₂²×D₄	—	86	2	no	—
157	C₂²×D₄	—	86	4	no	—
158	AGL(1,ℤ/8)	—	43	16	no	—
159	AGL(1,ℤ/8)	—	43	32	no	—
160	#135	—	43	16	no	—
161	#135	C₂×D₄₃	1	8	yes	complemented
162	C₂×Q₁₆	C₂×D₄₃	1	1	yes	characteristic, complemented
163	C₂×SD₁₆	—	43	4	no	—
164	C₂×SD₁₆	C₂×D₄₃	1	2	yes	complemented
165	C₂×D₈	—	43	4	no	—
166	C₂×D₈	—	43	16	no	—
167	C₂×D₈	—	43	2	no	—
168	C₂×D₈	—	43	2	no	—
169	C₂×D₈	—	43	8	no	—
170	C₂×D₈	C₂×D₄₃	1	1	yes	characteristic, complemented
171	#131	—	43	8	no	—
172	C₂×M₁₆	—	43	1	no	—
173	C₂×M₁₆	—	43	2	no	—
174	C₂²×C₈	—	43	2	no	—
175	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic, complemented
176	C₂⁴	—	86	4	no	—
177	C₂⁴	—	86	2	no	—
178	#41	—	2	4	no	—
179	#41	—	43	4	no	—
180	#41	—	43	4	no	—
181	#41	—	43	16	no	—
182	#41	—	86	16	no	—
183	#41	—	86	8	no	—
184	#41	C₂²×D₄₃	1	8	yes	complemented
185	C₂×Q₈	—	43	1	no	—
186	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	43	2	no	—
189	C₂×D₄	—	43	16	no	—
190	C₂×D₄	—	43	1	no	—
191	C₂×D₄	—	43	4	no	—
192	C₂×D₄	—	43	8	no	—
193	C₂×D₄	—	86	16	no	—
194	C₂×D₄	—	86	4	no	—
195	C₂×D₄	—	86	8	no	—
196	C₂×D₄	—	86	4	no	—
197	C₂×D₄	—	86	32	no	—
198	C₂×D₄	—	86	2	no	—
199	C₂×D₄	—	86	4	no	—
200	C₂×D₄	C₂²×D₄₃	1	2	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	43	2	no	—
203	C₂²×C₄	—	86	4	no	—
204	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
205	Q₁₆	C₂²×D₄₃	1	4	yes	complemented
206	SD₁₆	—	43	16	no	—
207	SD₁₆	C₂²×D₄₃	1	8	yes	complemented
208	D₈	—	43	16	no	—
209	D₈	—	43	8	no	—
210	D₈	—	43	8	no	—
211	D₈	C₂²×D₄₃	1	4	yes	complemented
212	M₁₆	—	43	4	no	—
213	M₁₆	—	43	8	no	—
214	C₂×C₈	—	43	2	no	—
215	C₂×C₈	—	43	8	no	—
216	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
217	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
218	C₂×C₈	C₂²×D₄₃	1	4	yes	complemented
219	C₂³	—	2	2	no	—
220	C₂³	—	43	2	no	—
221	C₂³	—	86	16	no	—
222	C₂³	—	86	8	no	—
223	C₂³	—	86	4	no	—
224	C₂³	—	344	4	no	—
225	C₂³	—	344	8	no	—
226	C₂³	D₄×D₄₃	1	1	yes	characteristic
227	Q₈	—	2	2	no	—
228	Q₈	—	43	4	no	—
229	Q₈	C₂³×D₄₃	1	4	yes	—
230	D₄	—	2	8	no	—
231	D₄	—	2	2	no	—
232	D₄	—	43	8	no	—
233	D₄	—	43	4	no	—
234	D₄	—	43	8	no	—
235	D₄	—	86	16	no	—
236	D₄	—	86	16	no	—
237	D₄	—	86	8	no	—
238	D₄	—	86	4	no	—
239	D₄	C₂³×D₄₃	1	4	yes	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	43	2	no	—
243	C₂×C₄	—	43	4	no	—
244	C₂×C₄	—	43	4	no	—
245	C₂×C₄	—	86	16	no	—
246	C₂×C₄	D₄×D₄₃	1	2	yes	—
247	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
248	C₂×C₄	C₂³×D₄₃	1	2	yes	—
249	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
250	C₈	—	43	4	no	—
251	C₈	C₂³×D₄₃	1	2	yes	—
252	C₈	C₂³×D₄₃	1	2	yes	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	43	4	no	—
257	C₂²	—	86	4	no	—
258	C₂²	—	86	8	no	—
259	C₂²	—	344	8	no	—
260	C₂²	—	344	16	no	—
261	C₂²	—	172	8	no	—
262	C₂²	C₂×D₄×D₄₃	1	2	yes	—
263	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	43	4	no	—
266	C₄	C₂×D₄×D₄₃	1	2	yes	—
267	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
268	C₄	C₂⁴×D₄₃	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	4	4	no	—
271	C₂	—	86	4	no	—
272	C₂	—	172	8	no	—
273	C₂	#14583701	1	2	yes	direct product factor
274	C₂	C₂²×D₄×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₄×C₄₃	C₂⁵
Frattini	C₄	C₂⁴×D₄₃
Fitting	C₂×#135×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₂×#135	D₄₃
hypocenter	C₄₃	C₂×#576
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#13063371	C₂	semidirect
2. #14583701	C₂	direct (N × H)
3. C₂×#13063352	C₂	semidirect
4. C₂×#13063350	C₂	semidirect
5. C₂×#13063348	C₂	semidirect
6. C₂²×D₃₄₄	C₂	semidirect
7. C₂×D₈×D₄₃	C₂	semidirect
8. C₂×#13063331	C₂	semidirect
9. C₂×#135×C₄₃	C₂	semidirect
11. #13063371	C₂²	semidirect
12. C₂×#10621902	C₂²	semidirect
16. C₂²×D₁₇₂	C₂²	semidirect
18. C₂×D₄×D₄₃	C₂²	semidirect
19. #13063352	C₂²	semidirect
20. #13063350	C₂²	semidirect
21. #13063348	C₂²	semidirect
22. C₂×#10621884	C₂²	semidirect
23. C₂×D₃₄₄	C₂²	semidirect
24. C₂×D₃₄₄	C₂²	semidirect
25. C₂×D₃₄₄	C₂²	semidirect
26. C₂×#10621882	C₂²	semidirect
27. D₈×D₄₃	C₂²	semidirect
28. #13063331	C₂²	semidirect
29. C₂×#10621877	C₂²	semidirect
30. C₂×#10621872	C₂²	semidirect
31. C₂×C₈×D₄₃	C₂²	semidirect
32. C₂×#41×C₄₃	C₂²	semidirect
33. #135×C₄₃	C₂²	semidirect
34. C₂×Q₁₆×C₄₃	C₂²	semidirect
35. C₂×SD₁₆×C₄₃	C₂²	semidirect
36. C₂×D₈×C₄₃	C₂²	semidirect
37. C₂²×C₈×C₄₃	C₂²	semidirect
41. #41×C₄₃	C₂³	semidirect
50. #10621902	C₂³	semidirect
53. D₄×D₄₃	C₂³	semidirect
63. C₂×D₁₇₂	C₂³	semidirect
67. Q₁₆×C₄₃	C₂³	semidirect
68. SD₁₆×C₄₃	C₂³	semidirect
69. D₈×C₄₃	C₂³	semidirect
72. C₂×C₈×C₄₃	C₂³	semidirect
73. #10621884	C₂³	semidirect
74. #10621882	C₂³	semidirect
75. #10621877	C₂³	semidirect
77. D₃₄₄	C₂³	semidirect
78. D₃₄₄	C₂³	semidirect
79. #10621872	C₂³	semidirect
80. C₈×D₄₃	C₂³	semidirect
134. C₂×C₄₃	#576	semidirect
143. C₂×#135	D₄₃	semidirect
147. C₄₃	C₂×#576	semidirect
153. C₂×#41	C₂×D₄₃	semidirect
161. #135	C₂×D₄₃	semidirect
162. C₂×Q₁₆	C₂×D₄₃	semidirect
164. C₂×SD₁₆	C₂×D₄₃	semidirect
170. C₂×D₈	C₂×D₄₃	semidirect
175. C₂²×C₈	C₂×D₄₃	semidirect
184. #41	C₂²×D₄₃	semidirect
205. Q₁₆	C₂²×D₄₃	semidirect
207. SD₁₆	C₂²×D₄₃	semidirect
211. D₈	C₂²×D₄₃	semidirect
218. C₂×C₈	C₂²×D₄₃	semidirect
273. C₂	#14583701	direct (N × H)

Subgroups — C2×#14583701 #19149345

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14583701 #19149345