Subgroups — #19148285

Order 5504.

1328 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583576	C₂	1	8	yes	maximal, complemented
2	#14583540	C₂	1	8	yes	maximal, complemented
3	#14583489	C₂	1	4	yes	maximal, complemented
4	#14583461	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583412	C₂	1	4	yes	maximal, complemented
6	C₂×#13063295	C₂	1	2	yes	maximal, complemented
7	#14583372	C₂	1	2	yes	maximal, complemented
8	C₂×C₂²⋊C₄×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#543	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₄₃	C₂²	1	1	yes	characteristic
11	C₂×#10621902	C₂²	1	4	yes	complemented
12	C₂²×D₁₇₂	C₂²	1	4	yes	complemented
13	C₂²×C₂²⋊D₄₃	C₂²	1	2	yes	—
14	C₂²×C₄×D₄₃	C₂²	1	2	yes	complemented
15	#13063320	C₂²	1	8	yes	complemented
16	#13063314	C₂²	1	8	yes	complemented
17	#13063310	C₂²	1	8	yes	complemented
18	#13063310	C₂²	1	8	yes	complemented
19	#13063309	C₂²	1	4	yes	complemented
20	#13063301	C₂²	1	4	yes	complemented
21	#13063300	C₂²	1	8	yes	complemented
22	#13063300	C₂²	1	4	yes	complemented
23	#13063299	C₂²	1	4	yes	complemented
24	#13063295	C₂²	1	8	yes	—
25	#13063295	C₂²	1	4	yes	complemented
26	#13063295	C₂²	1	4	yes	complemented
27	#13063295	C₂²	1	4	yes	complemented
28	#13063287	C₂²	1	8	yes	complemented
29	C₄×C₂²⋊D₄₃	C₂²	1	8	yes	complemented
30	(C₂²⋊D₄₃)⋊C₄	C₂²	1	8	yes	complemented
31	#13063283	C₂²	1	4	yes	complemented
32	#13063281	C₂²	1	4	yes	complemented
33	C₂×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
34	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	complemented
35	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	complemented
36	C₂²⋊C₄×D₄₃	C₂²	1	8	yes	complemented
37	C₂²⋊C₄×D₄₃	C₂²	1	4	yes	complemented
38	C₂²⋊Q₈×C₄₃	C₂²	1	8	yes	complemented
39	C₂²⋊Q₈×C₄₃	C₂²	1	4	yes	complemented
40	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
42	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
43	C₂³×D₄₃	—	2	4	no	—
44	C₂³×D₄₃	—	4	4	no	—
45	C₂³×D₄₃	C₂³	1	1	yes	characteristic
46	C₂³×D₄₃	C₂³	1	4	yes	—
47	C₂³×D₄₃	C₂³	1	2	yes	—
48	C₂³×D₄₃	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₄₃	C₂³	1	4	yes	complemented
50	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
51	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
52	C₂×C₂²⋊D₄₃	—	2	8	no	—
53	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
54	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
55	C₂²×Dic₄₃	C₂³	1	2	yes	—
56	C₂²×Dic₄₃	C₂³	1	2	yes	—
57	#10621902	—	4	8	no	—
58	C₂×D₁₇₂	—	4	8	no	—
59	C₂×D₁₇₂	C₂³	1	8	yes	—
60	C₂×D₁₇₂	C₂³	1	8	yes	—
61	C₂×D₁₇₂	C₂³	1	4	yes	complemented
62	C₂×C₄×D₄₃	—	4	4	no	—
63	C₂×C₄×D₄₃	C₂³	1	4	yes	—
64	C₂×C₄×D₄₃	C₂³	1	4	yes	—
65	C₂×C₄×D₄₃	C₂³	1	8	yes	complemented
66	C₄⋊C₄×C₄₃	C₂³	1	4	yes	complemented
67	C₄⋊C₄×C₄₃	C₂³	1	8	yes	complemented
68	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
69	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
70	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
71	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
72	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
73	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
74	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
75	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	complemented
76	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	complemented
77	C₄⋊Dic₄₃	C₂³	1	4	yes	complemented
78	Dic₄₃⋊C₄	C₂³	1	8	yes	complemented
79	C₄×Dic₄₃	C₂³	1	8	yes	complemented
80	C₂³×C₄₃	—	4	2	no	—
81	C₂³×C₄₃	#41	1	4	yes	complemented
82	C₂³×C₄₃	C₂⁴	1	2	yes	—
83	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
84	C₂²×D₄₃	—	2	2	no	—
85	C₂²×D₄₃	—	2	4	no	—
86	C₂²×D₄₃	—	2	8	no	—
87	C₂²×D₄₃	—	2	2	no	—
88	C₂²×D₄₃	—	2	4	no	—
89	C₂²×D₄₃	—	2	8	no	—
90	C₂²×D₄₃	—	4	4	no	—
91	C₂²×D₄₃	—	4	2	no	—
92	C₂²×D₄₃	—	4	4	no	—
93	C₂²×D₄₃	—	4	16	no	—
94	C₂²×D₄₃	C₂⁴	1	2	yes	—
95	C₂²×D₄₃	C₂⁴	1	4	yes	—
96	C₂²×D₄₃	C₂⁴	1	2	yes	—
97	Q₈×C₄₃	—	4	4	no	—
98	C₂×C₄×C₄₃	—	4	2	no	—
99	C₂×C₄×C₄₃	—	4	4	no	—
100	C₂×C₄×C₄₃	C₂⁴	1	8	yes	—
101	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
102	C₂²⋊D₄₃	—	2	8	no	—
103	C₂²⋊D₄₃	—	2	8	no	—
104	C₂×Dic₄₃	—	2	4	no	—
105	C₂×Dic₄₃	—	4	2	no	—
106	C₂×Dic₄₃	C₂⁴	1	4	yes	—
107	C₂×Dic₄₃	C₂⁴	1	4	yes	—
108	D₁₇₂	—	2	16	no	—
109	D₁₇₂	—	4	8	no	—
110	D₁₇₂	—	4	4	no	—
111	C₄×D₄₃	—	2	8	no	—
112	C₄×D₄₃	—	4	4	no	—
113	C₄×D₄₃	—	4	8	no	—
114	C₂²×C₄₃	—	2	4	no	—
115	C₂²×C₄₃	—	2	2	no	—
116	C₂²×C₄₃	—	4	4	no	—
117	C₂²×C₄₃	C₂×#41	1	4	yes	—
118	C₂²×C₄₃	C₂×#41	1	2	yes	—
119	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
120	C₂×D₄₃	—	2	8	no	—
121	C₂×D₄₃	—	2	2	no	—
122	C₂×D₄₃	—	2	2	no	—
123	C₂×D₄₃	—	2	2	no	—
124	C₂×D₄₃	—	2	4	no	—
125	C₂×D₄₃	—	2	4	no	—
126	C₂×D₄₃	—	2	8	no	—
127	C₂×D₄₃	—	4	16	no	—
128	C₂×D₄₃	—	4	4	no	—
129	C₂×D₄₃	—	8	4	no	—
130	C₂×D₄₃	H(5,𝔽₂)	1	4	yes	—
131	C₄×C₄₃	—	2	4	no	—
132	C₄×C₄₃	—	2	8	no	—
133	Dic₄₃	—	2	4	no	—
134	Dic₄₃	—	2	4	no	—
135	#3490	—	43	1	no	maximal, Sylow
136	C₂×C₄₃	—	2	2	no	—
137	C₂×C₄₃	—	2	4	no	—
138	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
139	C₂×C₄₃	C₂×H(5,𝔽₂)	1	2	yes	—
140	D₄₃	—	2	4	no	—
141	D₄₃	—	4	2	no	—
142	D₄₃	—	4	4	no	—
143	#551	—	43	8	no	—
144	#546	—	43	8	no	—
145	#543	D₄₃	1	1	yes	characteristic, complemented
146	#538	—	43	4	no	—
147	#534	—	43	1	no	—
148	#525	—	43	4	no	—
149	C₂×#120	—	43	2	no	—
150	#518	—	43	2	no	—
151	C₂²×C₂²⋊C₄	—	43	1	no	—
152	C₄₃	#3490	1	1	yes	characteristic, complemented, Sylow
153	C₂⁵	—	43	1	no	—
154	C₂×#41	—	43	4	no	—
155	C₂²×D₄	—	43	2	no	—
156	C₂²×D₄	—	43	4	no	—
157	C₂³×C₄	—	43	2	no	—
158	#126	—	43	8	no	—
159	#124	—	43	8	no	—
160	#123	—	43	8	no	—
161	#123	—	43	8	no	—
162	#123	—	43	4	no	—
163	C₂²⋊Q₈	—	43	4	no	—
164	C₂²⋊Q₈	C₂×D₄₃	1	4	yes	complemented
165	C₂²⋊Q₈	C₂×D₄₃	1	8	yes	complemented
166	#121	—	43	4	no	—
167	#121	—	43	4	no	—
168	#121	—	43	8	no	—
169	#120	—	43	4	no	—
170	#120	—	43	4	no	—
171	#120	—	43	8	no	—
172	#120	—	43	4	no	—
173	C₄×D₄	—	43	8	no	—
174	C₄×D₄	—	43	8	no	—
175	C₄×D₄	—	43	8	no	—
176	#117	—	43	4	no	—
177	#117	—	43	4	no	—
178	C₂×C₂²⋊C₄	—	43	1	no	—
179	C₂×C₂²⋊C₄	—	43	2	no	—
180	C₂×C₂²⋊C₄	—	43	4	no	—
181	C₂×C₂²⋊C₄	—	43	8	no	—
182	C₂×C₂²⋊C₄	—	43	4	no	—
183	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
184	C₂×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
185	C₂⁴	—	43	1	no	—
186	C₂⁴	—	43	4	no	—
187	C₂⁴	—	43	1	no	—
188	C₂⁴	—	43	2	no	—
189	C₂⁴	—	86	4	no	—
190	C₂⁴	—	172	4	no	—
191	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
192	#41	—	172	8	no	—
193	C₂×Q₈	C₂²×D₄₃	1	4	yes	complemented
194	C₂×D₄	—	43	4	no	—
195	C₂×D₄	—	43	4	no	—
196	C₂×D₄	—	43	8	no	—
197	C₂×D₄	—	43	4	no	—
198	C₂×D₄	—	43	8	no	—
199	C₂×D₄	—	86	8	no	—
200	C₂×D₄	—	172	8	no	—
201	C₂²×C₄	—	43	2	no	—
202	C₂²×C₄	—	43	2	no	—
203	C₂²×C₄	—	43	4	no	—
204	C₂²×C₄	—	43	4	no	—
205	C₂²×C₄	—	43	8	no	—
206	C₂²×C₄	—	172	4	no	—
207	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
208	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
209	C₄⋊C₄	—	43	8	no	—
210	C₄⋊C₄	—	43	4	no	—
211	C₄⋊C₄	C₂²×D₄₃	1	4	yes	complemented
212	C₄⋊C₄	C₂²×D₄₃	1	8	yes	complemented
213	C₂²⋊C₄	—	43	4	no	—
214	C₂²⋊C₄	—	43	8	no	—
215	C₂²⋊C₄	—	43	8	no	—
216	C₂²⋊C₄	—	43	8	no	—
217	C₂²⋊C₄	—	43	8	no	—
218	C₂²⋊C₄	—	43	8	no	—
219	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
220	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
221	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
222	C₄²	—	43	8	no	—
223	C₂³	—	4	2	no	—
224	C₂³	—	43	2	no	—
225	C₂³	—	43	2	no	—
226	C₂³	—	43	4	no	—
227	C₂³	—	86	4	no	—
228	C₂³	—	86	8	no	—
229	C₂³	—	86	2	no	—
230	C₂³	—	86	4	no	—
231	C₂³	—	86	8	no	—
232	C₂³	—	86	2	no	—
233	C₂³	—	172	4	no	—
234	C₂³	—	172	2	no	—
235	C₂³	—	172	4	no	—
236	C₂³	—	172	16	no	—
237	C₂³	#10621902	1	4	yes	complemented
238	C₂³	C₂³×D₄₃	1	2	yes	—
239	C₂³	C₂³×D₄₃	1	1	yes	characteristic
240	Q₈	—	4	4	no	—
241	D₄	—	86	16	no	—
242	D₄	—	86	8	no	—
243	D₄	—	86	8	no	—
244	D₄	—	172	8	no	—
245	D₄	—	172	4	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	43	4	no	—
249	C₂×C₄	—	43	4	no	—
250	C₂×C₄	—	86	8	no	—
251	C₂×C₄	—	86	4	no	—
252	C₂×C₄	—	172	4	no	—
253	C₂×C₄	—	172	8	no	—
254	C₂×C₄	—	172	2	no	—
255	C₂×C₄	C₂³×D₄₃	1	8	yes	—
256	C₂×C₄	C₂³×D₄₃	1	4	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	4	4	no	—
260	C₂²	—	43	4	no	—
261	C₂²	—	86	8	no	—
262	C₂²	—	86	2	no	—
263	C₂²	—	86	2	no	—
264	C₂²	—	86	4	no	—
265	C₂²	—	86	2	no	—
266	C₂²	—	86	8	no	—
267	C₂²	—	86	4	no	—
268	C₂²	—	344	4	no	—
269	C₂²	—	172	16	no	—
270	C₂²	—	172	4	no	—
271	C₂²	C₂×#10621902	1	4	yes	—
272	C₂²	C₂×#10621902	1	2	yes	—
273	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
274	C₄	—	2	4	no	—
275	C₄	—	2	8	no	—
276	C₄	—	86	4	no	—
277	C₄	—	86	4	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	4	no	—
280	C₂	—	86	4	no	—
281	C₂	—	172	4	no	—
282	C₂	—	172	2	no	—
283	C₂	C₂²×#10621902	1	1	yes	characteristic
284	C₂	H(5,𝔽₂)×D₄₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₄₃×#543	C₂
hypercenter	#543	D₄₃
hypocenter	C₄₃	#3490
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #543

Complements

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

N	H ≅ G/N	type
1. #14583576	C₂	semidirect
2. #14583540	C₂	semidirect
3. #14583489	C₂	semidirect
4. #14583461	C₂	semidirect
5. #14583412	C₂	semidirect
6. C₂×#13063295	C₂	semidirect
7. #14583372	C₂	semidirect
8. C₂×C₂²⋊C₄×D₄₃	C₂	semidirect
9. C₄₃×#543	C₂	semidirect
11. C₂×#10621902	C₂²	semidirect
12. C₂²×D₁₇₂	C₂²	semidirect
14. C₂²×C₄×D₄₃	C₂²	semidirect
15. #13063320	C₂²	semidirect
16. #13063314	C₂²	semidirect
17. #13063310	C₂²	semidirect
18. #13063310	C₂²	semidirect
19. #13063309	C₂²	semidirect
20. #13063301	C₂²	semidirect
21. #13063300	C₂²	semidirect
22. #13063300	C₂²	semidirect
23. #13063299	C₂²	semidirect
25. #13063295	C₂²	semidirect
26. #13063295	C₂²	semidirect
27. #13063295	C₂²	semidirect
28. #13063287	C₂²	semidirect
29. C₄×C₂²⋊D₄₃	C₂²	semidirect
30. (C₂²⋊D₄₃)⋊C₄	C₂²	semidirect
31. #13063283	C₂²	semidirect
32. #13063281	C₂²	semidirect
34. C₂×(C₂×D₄₃)⋊C₄	C₂²	semidirect
35. C₂×(C₂×D₄₃)⋊C₄	C₂²	semidirect
36. C₂²⋊C₄×D₄₃	C₂²	semidirect
37. C₂²⋊C₄×D₄₃	C₂²	semidirect
38. C₂²⋊Q₈×C₄₃	C₂²	semidirect
39. C₂²⋊Q₈×C₄₃	C₂²	semidirect
49. C₂×Q₈×C₄₃	C₂³	semidirect
61. C₂×D₁₇₂	C₂³	semidirect
65. C₂×C₄×D₄₃	C₂³	semidirect
66. C₄⋊C₄×C₄₃	C₂³	semidirect
67. C₄⋊C₄×C₄₃	C₂³	semidirect
75. (C₂×D₄₃)⋊C₄	C₂³	semidirect
76. (C₂×D₄₃)⋊C₄	C₂³	semidirect
77. C₄⋊Dic₄₃	C₂³	semidirect
78. Dic₄₃⋊C₄	C₂³	semidirect
79. C₄×Dic₄₃	C₂³	semidirect
81. C₂³×C₄₃	#41	semidirect
145. #543	D₄₃	semidirect
152. C₄₃	#3490	semidirect
164. C₂²⋊Q₈	C₂×D₄₃	semidirect
165. C₂²⋊Q₈	C₂×D₄₃	semidirect
193. C₂×Q₈	C₂²×D₄₃	semidirect
211. C₄⋊C₄	C₂²×D₄₃	semidirect
212. C₄⋊C₄	C₂²×D₄₃	semidirect
237. C₂³	#10621902	semidirect