Subgroups — #19143442

Order 5504.

528 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×D₄₃	C₂	1	1	yes	maximal, characteristic
2	#14583371	C₂	1	2	yes	maximal, complemented
3	#14583009	C₂	1	2	yes	maximal, complemented
4	#14583008	C₂	1	2	yes	maximal, complemented
5	C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic, complemented
6	D₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#14582981	C₂	1	1	yes	maximal, characteristic, complemented
8	#14582936	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582902	C₂	1	2	yes	maximal, complemented
10	#14582853	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₄₃	C₂²	1	2	yes	—
13	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
14	C₂×#10621882	C₂²	1	2	yes	—
15	D₈×D₄₃	C₄	1	8	yes	complemented
16	C₂×#10621872	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₄₃	C₂²	1	1	yes	characteristic
18	C₄×D₁₇₂	C₂²	1	2	yes	complemented
19	D₄×Dic₄₃	C₂²	1	2	yes	complemented
20	#13063287	C₂²	1	2	yes	complemented
21	C₄×C₂²⋊D₄₃	—	2	2	no	—
22	(C₂²⋊D₄₃)⋊C₄	—	2	2	no	—
23	#13063283	C₂²	1	2	yes	—
24	#13063282	C₂²	1	1	yes	characteristic
25	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
26	C₂²⋊C₄×D₄₃	—	2	2	no	—
27	#13063259	C₂²	1	1	yes	characteristic, complemented
28	#13063256	C₂²	1	2	yes	complemented
29	#13063247	C₂²	1	2	yes	complemented
30	#13063246	C₂²	1	2	yes	complemented
31	#13063245	C₂²	1	2	yes	complemented
32	#13063236	C₂²	1	1	yes	characteristic, complemented
33	#13063235	C₂²	1	1	yes	characteristic, complemented
34	C₈×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₄₃	C₂²	1	2	yes	complemented
37	#107×C₄₃	C₂²	1	1	yes	characteristic, complemented
38	#102×C₄₃	C₂²	1	2	yes	complemented
39	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
40	C₂³×D₄₃	—	2	2	no	—
41	C₂×D₄×C₄₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₄₃	—	2	2	no	—
43	C₂×C₂²⋊D₄₃	—	2	2	no	—
44	C₂²×Dic₄₃	—	2	2	no	—
45	D₄×D₄₃	—	2	4	no	—
46	D₄×D₄₃	C₂×C₄	1	8	yes	complemented
47	C₂×D₁₇₂	C₂³	1	2	yes	—
48	C₂×C₄×D₄₃	—	2	2	no	—
49	C₂×C₄×D₄₃	—	2	2	no	—
50	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
51	D₈×C₄₃	C₂×C₄	1	4	yes	complemented
52	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₄₃	—	2	2	no	—
56	C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₄₃	—	2	2	no	—
58	#10621882	C₂×C₄	1	8	yes	complemented
59	(C₂×D₄₃)⋊C₄	—	2	2	no	—
60	(C₂×D₄₃)⋊C₄	—	2	2	no	—
61	(C₂×D₄₃)⋊C₄	—	2	2	no	—
62	(C₂×D₄₃)⋊C₄	D₄	1	1	yes	characteristic
63	(C₂×D₄₃)⋊C₄	D₄	1	1	yes	characteristic
64	C₄⋊Dic₄₃	C₂³	1	2	yes	—
65	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
66	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
67	C₄×Dic₄₃	—	2	2	no	—
68	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
69	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
71	D₃₄₄	C₂×C₄	1	4	yes	complemented
72	#10621872	—	2	2	no	—
73	C₈×D₄₃	C₂×C₄	1	4	yes	complemented
74	C₂³×C₄₃	—	2	2	no	—
75	C₂²×D₄₃	—	2	2	no	—
76	C₂²×D₄₃	—	2	8	no	—
77	C₂²×D₄₃	—	8	2	no	—
78	C₂²×D₄₃	—	8	2	no	—
79	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
80	D₄×C₄₃	—	2	2	no	—
81	D₄×C₄₃	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	—	8	2	no	—
84	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
87	C₂²⋊D₄₃	—	2	8	no	—
88	C₂×Dic₄₃	—	2	2	no	—
89	C₂×Dic₄₃	—	8	2	no	—
90	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
93	D₁₇₂	—	2	2	no	—
94	D₁₇₂	C₂²×C₄	1	4	yes	—
95	C₄×D₄₃	—	8	2	no	—
96	C₄×D₄₃	—	8	2	no	—
97	C₄×D₄₃	C₂²×C₄	1	2	yes	—
98	C₄×D₄₃	#41	1	2	yes	—
99	C₈×C₄₃	—	2	1	no	—
100	C₈×C₄₃	C₂²×C₄	1	2	yes	—
101	C₄₃⋊C₈	—	2	1	no	—
102	C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
103	C₂²×C₄₃	—	2	4	no	—
104	C₂²×C₄₃	—	4	2	no	—
105	C₂²×C₄₃	—	4	2	no	—
106	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×D₄₃	—	2	1	no	—
108	C₂×D₄₃	—	2	1	no	—
109	C₂×D₄₃	—	2	4	no	—
110	C₂×D₄₃	—	4	2	no	—
111	C₂×D₄₃	—	4	2	no	—
112	C₂×D₄₃	—	8	8	no	—
113	C₂×D₄₃	C₄×D₄	1	2	yes	—
114	C₄×C₄₃	—	2	1	no	—
115	C₄×C₄₃	—	2	1	no	—
116	C₄×C₄₃	—	4	2	no	—
117	C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
118	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
119	Dic₄₃	—	2	1	no	—
120	Dic₄₃	—	2	1	no	—
121	Dic₄₃	—	4	2	no	—
122	Dic₄₃	C₄×D₄	1	2	yes	—
123	#2948	—	43	1	no	maximal, Sylow
124	C₂×C₄₃	—	4	4	no	—
125	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
126	C₂×C₄₃	#576	1	1	yes	characteristic
127	C₂×C₄₃	#576	1	1	yes	characteristic
128	D₄₃	—	2	2	no	—
129	D₄₃	—	4	4	no	—
130	C₂²×D₈	—	43	1	no	—
131	#518	—	43	2	no	—
132	#442	—	43	2	no	—
133	#442	—	43	2	no	—
134	C₄×D₈	—	43	1	no	—
135	C₄×D₈	—	43	1	no	—
136	C₄×D₈	—	43	1	no	—
137	C₄×D₈	D₄₃	1	1	yes	characteristic, complemented
138	#427	—	43	1	no	—
139	#418	—	43	2	no	—
140	#405	—	43	1	no	—
141	C₄₃	#2948	1	1	yes	characteristic, complemented, Sylow
142	C₂²×D₄	—	43	2	no	—
143	C₂×D₈	—	43	2	no	—
144	C₂×D₈	—	43	8	no	—
145	C₂×D₈	—	43	1	no	—
146	C₂×D₈	C₂×D₄₃	1	1	yes	characteristic
147	C₂×M₁₆	—	43	1	no	—
148	C₂²×C₈	—	43	1	no	—
149	C₄×D₄	—	43	2	no	—
150	C₄×D₄	—	43	2	no	—
151	C₄×D₄	—	43	2	no	—
152	C₄×D₄	—	86	2	no	—
153	C₄×D₄	—	86	2	no	—
154	C₄×D₄	C₂×D₄₃	1	2	yes	complemented
155	#117	—	43	1	no	—
156	#117	—	43	2	no	—
157	C₂×C₂²⋊C₄	—	86	2	no	—
158	C₂×C₂²⋊C₄	—	86	2	no	—
159	#107	—	43	1	no	—
160	#107	C₂×D₄₃	1	1	yes	characteristic, complemented
161	#106	—	43	2	no	—
162	#102	—	43	2	no	—
163	#102	—	43	2	no	—
164	#102	—	43	2	no	—
165	#102	C₂×D₄₃	1	2	yes	complemented
166	#97	—	43	1	no	—
167	#97	—	43	1	no	—
168	C₄×C₈	—	43	1	no	—
169	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic, complemented
170	C₂⁴	—	86	2	no	—
171	C₂×D₄	—	43	8	no	—
172	C₂×D₄	—	43	2	no	—
173	C₂×D₄	—	86	4	no	—
174	C₂×D₄	—	86	2	no	—
175	C₂×D₄	C₂²×D₄₃	1	2	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	43	1	no	—
178	C₂²×C₄	—	86	2	no	—
179	C₂²×C₄	—	86	2	no	—
180	C₂²×C₄	—	86	2	no	—
181	D₈	—	43	8	no	—
182	D₈	—	43	4	no	—
183	D₈	C₄×D₄₃	1	4	yes	complemented
184	M₁₆	—	86	2	no	—
185	C₂×C₈	—	43	1	no	—
186	C₂×C₈	—	43	4	no	—
187	C₂×C₈	—	43	1	no	—
188	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
189	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
190	C₄⋊C₄	—	43	1	no	—
191	C₄⋊C₄	—	43	1	no	—
192	C₄⋊C₄	—	43	2	no	—
193	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	43	1	no	—
196	C₂²⋊C₄	—	43	1	no	—
197	C₂²⋊C₄	—	86	2	no	—
198	C₂²⋊C₄	—	86	2	no	—
199	C₂²⋊C₄	—	86	2	no	—
200	C₂²⋊C₄	—	86	2	no	—
201	C₄²	—	43	1	no	—
202	C₄²	—	86	2	no	—
203	C₄²	C₂²×D₄₃	1	1	yes	characteristic
204	C₂³	—	2	2	no	—
205	C₂³	—	43	1	no	—
206	C₂³	—	86	8	no	—
207	C₂³	—	86	2	no	—
208	C₂³	—	344	2	no	—
209	C₂³	—	344	2	no	—
210	D₄	—	2	2	no	—
211	D₄	—	43	4	no	—
212	D₄	—	86	8	no	—
213	D₄	—	86	2	no	—
214	D₄	C₂×C₄×D₄₃	1	4	yes	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	43	1	no	—
218	C₂×C₄	—	43	2	no	—
219	C₂×C₄	—	43	2	no	—
220	C₂×C₄	—	43	1	no	—
221	C₂×C₄	—	43	1	no	—
222	C₂×C₄	—	86	2	no	—
223	C₂×C₄	—	344	2	no	—
224	C₂×C₄	—	344	2	no	—
225	C₂×C₄	—	344	2	no	—
226	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
227	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
228	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	43	2	no	—
231	C₈	—	86	1	no	—
232	C₈	C₂×C₄×D₄₃	1	2	yes	—
233	C₂²	—	2	4	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	43	2	no	—
237	C₂²	—	86	1	no	—
238	C₂²	—	86	1	no	—
239	C₂²	—	86	4	no	—
240	C₂²	—	344	8	no	—
241	C₂²	—	172	2	no	—
242	C₂²	—	172	2	no	—
243	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	4	2	no	—
247	C₄	—	43	2	no	—
248	C₄	—	86	1	no	—
249	C₄	—	86	1	no	—
250	C₄	—	172	2	no	—
251	C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
252	C₄	#41×D₄₃	1	1	yes	characteristic
253	C₂	—	4	4	no	—
254	C₂	—	86	2	no	—
255	C₂	—	172	4	no	—
256	C₂	C₄×D₄×D₄₃	1	1	yes	characteristic
257	C₂	#14583700	1	1	yes	characteristic
258	C₂	#14583701	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₄×C₄₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄×D₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₄×D₈	D₄₃
hypocenter	C₄₃	#2948
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #14583371	C₂	semidirect
3. #14583009	C₂	semidirect
4. #14583008	C₂	semidirect
5. C₄×D₃₄₄	C₂	semidirect
6. D₈×Dic₄₃	C₂	semidirect
7. #14582981	C₂	semidirect
8. #14582936	C₂	semidirect
9. #14582902	C₂	semidirect
10. #14582853	C₂	semidirect
11. C₄×D₈×C₄₃	C₂	semidirect
15. D₈×D₄₃	C₄	semidirect
18. C₄×D₁₇₂	C₂²	semidirect
19. D₄×Dic₄₃	C₂²	semidirect
20. #13063287	C₂²	semidirect
27. #13063259	C₂²	semidirect
28. #13063256	C₂²	semidirect
29. #13063247	C₂²	semidirect
30. #13063246	C₂²	semidirect
31. #13063245	C₂²	semidirect
32. #13063236	C₂²	semidirect
33. #13063235	C₂²	semidirect
34. C₈×Dic₄₃	C₂²	semidirect
36. C₄×D₄×C₄₃	C₂²	semidirect
37. #107×C₄₃	C₂²	semidirect
38. #102×C₄₃	C₂²	semidirect
39. C₄×C₈×C₄₃	C₂²	semidirect
46. D₄×D₄₃	C₂×C₄	semidirect
51. D₈×C₄₃	C₂×C₄	semidirect
58. #10621882	C₂×C₄	semidirect
71. D₃₄₄	C₂×C₄	semidirect
73. C₈×D₄₃	C₂×C₄	semidirect
137. C₄×D₈	D₄₃	semidirect
141. C₄₃	#2948	semidirect
154. C₄×D₄	C₂×D₄₃	semidirect
160. #107	C₂×D₄₃	semidirect
165. #102	C₂×D₄₃	semidirect
169. C₄×C₈	C₂×D₄₃	semidirect
183. D₈	C₄×D₄₃	semidirect