Subgroups — #19143408

Order 5504.

548 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063350	C₂	1	1	yes	maximal, characteristic
2	#41×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13063287	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583008	C₂	1	2	yes	maximal, complemented
5	#14583002	C₂	1	2	yes	maximal, complemented
6	#14582984	C₂	1	2	yes	maximal, complemented
7	#14582979	C₂	1	2	yes	maximal, complemented
8	#14582939	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582905	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#417	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₄₃	—	2	1	no	—
14	#13063350	C₄	1	8	yes	complemented
15	C₂×#10621884	C₂²	1	2	yes	—
16	C₂×#10621882	C₂²	1	2	yes	—
17	C₂×#10621877	C₂²	1	1	yes	characteristic
18	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
19	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
20	D₄×Dic₄₃	—	2	1	no	—
21	D₄×Dic₄₃	—	2	1	no	—
22	D₄×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
23	D₄×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
24	#13063287	—	2	1	no	—
25	#13063287	—	2	1	no	—
26	#13063287	C₂²	1	2	yes	—
27	#13063287	C₂²	1	2	yes	—
28	#13063284	—	2	1	no	—
29	#13063284	C₂²	1	1	yes	characteristic
30	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₄₃	—	2	1	no	—
32	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
33	#13063258	C₂²	1	2	yes	complemented
34	#13063256	C₂²	1	2	yes	complemented
35	#13063247	C₂²	1	2	yes	—
36	#13063247	C₂²	1	2	yes	—
37	#13063235	C₂²	1	2	yes	complemented
38	C₈×Dic₄₃	C₂²	1	2	yes	complemented
39	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
42	#103×C₄₃	C₂²	1	2	yes	complemented
43	#102×C₄₃	C₂²	1	2	yes	complemented
44	C₂³×D₄₃	—	2	1	no	—
45	#41×C₄₃	—	2	2	no	—
46	#41×C₄₃	C₂×C₄	1	4	yes	complemented
47	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₄₃	—	2	1	no	—
49	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	—	2	1	no	—
51	C₂²×C₄×C₄₃	—	2	1	no	—
52	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
53	C₂²×Dic₄₃	—	2	1	no	—
54	C₂²×Dic₄₃	—	2	1	no	—
55	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
56	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
57	C₂×D₁₇₂	—	2	1	no	—
58	C₂×D₁₇₂	—	2	2	no	—
59	C₂×D₁₇₂	C₂×C₄	1	4	yes	complemented
60	C₂×D₁₇₂	C₂³	1	2	yes	—
61	C₂×C₄×D₄₃	—	2	2	no	—
62	C₂×C₄×D₄₃	—	2	2	no	—
63	C₂×C₄×D₄₃	—	4	4	no	—
64	M₁₆×C₄₃	—	2	2	no	—
65	C₂×C₈×C₄₃	C₂³	1	2	yes	—
66	C₄⋊C₄×C₄₃	—	2	1	no	—
67	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
68	C₂²⋊Dic₄₃	—	2	1	no	—
69	C₂²⋊Dic₄₃	—	2	1	no	—
70	C₂²⋊Dic₄₃	—	2	1	no	—
71	#10621884	C₂×C₄	1	8	yes	complemented
72	#10621882	C₂×C₄	1	8	yes	complemented
73	(C₂×D₄₃)⋊C₄	—	2	2	no	—
74	(C₂×D₄₃)⋊C₄	—	2	2	no	—
75	C₄⋊Dic₄₃	—	2	1	no	—
76	C₄⋊Dic₄₃	—	2	1	no	—
77	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
79	C₄×Dic₄₃	—	2	1	no	—
80	C₄×Dic₄₃	—	2	1	no	—
81	C₄×Dic₄₃	—	2	1	no	—
82	C₄×Dic₄₃	D₄	1	1	yes	characteristic
83	C₄×Dic₄₃	D₄	1	1	yes	characteristic
84	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
85	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
86	#10621877	C₂×C₄	1	4	yes	complemented
87	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
88	C₂³×C₄₃	—	2	1	no	—
89	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₄₃	—	2	4	no	—
91	C₂²×D₄₃	—	2	2	no	—
92	C₂²×D₄₃	—	4	2	no	—
93	C₂²×D₄₃	—	4	2	no	—
94	Q₈×C₄₃	—	2	1	no	—
95	Q₈×C₄₃	C₂²×C₄	1	2	yes	complemented
96	D₄×C₄₃	—	2	4	no	—
97	D₄×C₄₃	—	2	1	no	—
98	D₄×C₄₃	C₂²×C₄	1	2	yes	complemented
99	C₂×C₄×C₄₃	—	2	1	no	—
100	C₂×C₄×C₄₃	—	2	4	no	—
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	—	4	2	no	—
103	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	#41	1	2	yes	—
106	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
107	C₂×Dic₄₃	—	2	2	no	—
108	C₂×Dic₄₃	—	2	1	no	—
109	C₂×Dic₄₃	—	2	1	no	—
110	C₂×Dic₄₃	—	4	1	no	—
111	C₂×Dic₄₃	—	4	2	no	—
112	C₂×Dic₄₃	—	4	2	no	—
113	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
114	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
117	D₁₇₂	—	2	4	no	—
118	D₁₇₂	—	2	2	no	—
119	D₁₇₂	C₂²×C₄	1	4	yes	—
120	C₄×D₄₃	—	4	4	no	—
121	C₄×D₄₃	—	4	4	no	—
122	C₈×C₄₃	—	2	2	no	—
123	C₄₃⋊C₈	C₂²×C₄	1	4	yes	complemented
124	C₂²×C₄₃	—	2	1	no	—
125	C₂²×C₄₃	—	2	1	no	—
126	C₂²×C₄₃	—	2	2	no	—
127	C₂²×C₄₃	—	4	1	no	—
128	C₂²×C₄₃	—	4	1	no	—
129	C₂²×C₄₃	C₄×D₄	1	2	yes	—
130	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×D₄₃	—	2	4	no	—
132	C₂×D₄₃	—	4	8	no	—
133	C₂×D₄₃	—	4	2	no	—
134	C₂×D₄₃	—	4	2	no	—
135	C₄×C₄₃	—	2	2	no	—
136	C₄×C₄₃	—	4	2	no	—
137	C₄×C₄₃	C₄×D₄	1	2	yes	—
138	C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
139	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
140	Dic₄₃	—	2	1	no	—
141	Dic₄₃	—	2	1	no	—
142	Dic₄₃	—	2	1	no	—
143	Dic₄₃	—	4	1	no	—
144	Dic₄₃	—	4	1	no	—
145	Dic₄₃	AGL(1,ℤ/8)	1	2	yes	complemented
146	C₄×AGL(1,ℤ/8)	—	43	1	no	maximal, Sylow
147	C₂×C₄₃	—	2	2	no	—
148	C₂×C₄₃	—	4	2	no	—
149	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
150	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₄₃	#575	1	1	yes	characteristic
152	D₄₃	—	4	4	no	—
153	C₂×AGL(1,ℤ/8)	—	43	1	no	—
154	C₄×#41	—	43	1	no	—
155	C₂×C₄×D₄	—	43	1	no	—
156	#442	—	43	2	no	—
157	#440	—	43	2	no	—
158	C₄×SD₁₆	—	43	2	no	—
159	C₄×D₈	—	43	2	no	—
160	#428	—	43	1	no	—
161	#418	—	43	1	no	—
162	#417	D₄₃	1	1	yes	characteristic, complemented
163	C₄×M₁₆	—	43	1	no	—
164	C₄₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
165	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
166	C₂²×D₄	—	43	1	no	—
167	C₂³×C₄	—	86	1	no	—
168	AGL(1,ℤ/8)	—	43	8	no	—
169	C₂×SD₁₆	—	43	2	no	—
170	C₂×D₈	—	43	2	no	—
171	C₂×M₁₆	—	43	1	no	—
172	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
173	C₄×Q₈	—	43	1	no	—
174	C₄×Q₈	—	43	1	no	—
175	C₄×D₄	—	43	1	no	—
176	C₄×D₄	—	43	1	no	—
177	C₄×D₄	—	43	2	no	—
178	C₄×D₄	—	43	2	no	—
179	C₄×D₄	—	86	1	no	—
180	C₄×D₄	—	86	1	no	—
181	C₄×D₄	—	86	1	no	—
182	C₄×D₄	—	86	1	no	—
183	#117	—	43	1	no	—
184	#117	—	86	1	no	—
185	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	—	86	1	no	—
187	C₂×C₄²	—	43	1	no	—
188	C₂×C₄²	—	86	1	no	—
189	#107	—	43	2	no	—
190	#106	—	43	2	no	—
191	#103	C₂×D₄₃	1	2	yes	complemented
192	#102	—	43	2	no	—
193	#102	—	43	2	no	—
194	#102	C₂×D₄₃	1	2	yes	complemented
195	#97	—	43	2	no	—
196	C₄×C₈	—	43	2	no	—
197	C₂⁴	—	86	1	no	—
198	#41	—	2	2	no	—
199	#41	C₄×D₄₃	1	4	yes	complemented
200	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	43	4	no	—
203	C₂×D₄	—	43	2	no	—
204	C₂×D₄	—	86	1	no	—
205	C₂×D₄	—	86	2	no	—
206	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	43	1	no	—
210	C₂²×C₄	—	43	1	no	—
211	C₂²×C₄	—	86	2	no	—
212	C₂²×C₄	—	86	2	no	—
213	C₂²×C₄	—	86	1	no	—
214	C₂²×C₄	—	86	1	no	—
215	C₂²×C₄	—	172	4	no	—
216	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
217	SD₁₆	—	43	8	no	—
218	D₈	—	43	8	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	43	4	no	—
221	C₂×C₈	—	43	2	no	—
222	C₂×C₈	C₂²×D₄₃	1	2	yes	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	43	1	no	—
225	C₄⋊C₄	—	43	1	no	—
226	C₄⋊C₄	—	86	1	no	—
227	C₄⋊C₄	—	86	1	no	—
228	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
229	C₂²⋊C₄	—	86	2	no	—
230	C₂²⋊C₄	—	86	2	no	—
231	C₂²⋊C₄	—	86	1	no	—
232	C₂²⋊C₄	—	86	1	no	—
233	C₂²⋊C₄	—	86	1	no	—
234	C₄²	—	43	1	no	—
235	C₄²	—	43	1	no	—
236	C₄²	—	43	1	no	—
237	C₄²	—	43	1	no	—
238	C₄²	—	86	1	no	—
239	C₄²	—	86	1	no	—
240	C₄²	—	86	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	86	4	no	—
243	C₂³	—	86	2	no	—
244	C₂³	—	172	2	no	—
245	C₂³	—	172	2	no	—
246	C₂³	D₄×D₄₃	1	1	yes	characteristic
247	Q₈	—	2	1	no	—
248	Q₈	C₂×C₄×D₄₃	1	2	yes	complemented
249	D₄	—	2	4	no	—
250	D₄	—	2	1	no	—
251	D₄	—	43	4	no	—
252	D₄	—	86	4	no	—
253	D₄	—	86	2	no	—
254	D₄	C₂×C₄×D₄₃	1	2	yes	complemented
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	43	1	no	—
260	C₂×C₄	—	43	1	no	—
261	C₂×C₄	—	43	1	no	—
262	C₂×C₄	—	43	1	no	—
263	C₂×C₄	—	86	2	no	—
264	C₂×C₄	—	86	1	no	—
265	C₂×C₄	—	86	1	no	—
266	C₂×C₄	—	172	4	no	—
267	C₂×C₄	—	172	4	no	—
268	C₂×C₄	—	172	2	no	—
269	C₂×C₄	—	172	1	no	—
270	C₂×C₄	—	172	2	no	—
271	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
272	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
273	C₂×C₄	#10621900	1	2	yes	—
274	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	43	4	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	86	4	no	—
283	C₂²	—	172	8	no	—
284	C₂²	—	172	2	no	—
285	C₂²	—	172	2	no	—
286	C₂²	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
287	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	4	2	no	—
290	C₄	—	43	2	no	—
291	C₄	—	86	1	no	—
292	C₄	—	86	1	no	—
293	C₄	—	86	1	no	—
294	C₄	—	172	1	no	—
295	C₄	—	172	1	no	—
296	C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
297	C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
298	C₄	C₂×#10621900	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	4	2	no	—
301	C₂	—	172	4	no	—
302	C₂	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
303	C₂	AGL(1,ℤ/8)×D₄₃	1	1	yes	characteristic
304	C₂	#14583698	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₄×C₄₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#417	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#417	D₄₃
hypocenter	C₄₃	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
2. #41×Dic₄₃	C₂	semidirect
3. C₂×#13063287	C₂	semidirect
4. #14583008	C₂	semidirect
5. #14583002	C₂	semidirect
6. #14582984	C₂	semidirect
7. #14582979	C₂	semidirect
8. #14582939	C₂	semidirect
9. #14582905	C₂	semidirect
10. M₁₆×Dic₄₃	C₂	semidirect
11. C₄₃×#417	C₂	semidirect
14. #13063350	C₄	semidirect
18. Q₈×Dic₄₃	C₂²	semidirect
19. Q₈×Dic₄₃	C₂²	semidirect
22. D₄×Dic₄₃	C₂²	semidirect
23. D₄×Dic₄₃	C₂²	semidirect
33. #13063258	C₂²	semidirect
34. #13063256	C₂²	semidirect
37. #13063235	C₂²	semidirect
38. C₈×Dic₄₃	C₂²	semidirect
42. #103×C₄₃	C₂²	semidirect
43. #102×C₄₃	C₂²	semidirect
46. #41×C₄₃	C₂×C₄	semidirect
59. C₂×D₁₇₂	C₂×C₄	semidirect
71. #10621884	C₂×C₄	semidirect
72. #10621882	C₂×C₄	semidirect
86. #10621877	C₂×C₄	semidirect
95. Q₈×C₄₃	C₂²×C₄	semidirect
98. D₄×C₄₃	C₂²×C₄	semidirect
123. C₄₃⋊C₈	C₂²×C₄	semidirect
145. Dic₄₃	AGL(1,ℤ/8)	semidirect
162. #417	D₄₃	semidirect
164. C₄₃	C₄×AGL(1,ℤ/8)	semidirect
191. #103	C₂×D₄₃	semidirect
194. #102	C₂×D₄₃	semidirect
199. #41	C₄×D₄₃	semidirect
248. Q₈	C₂×C₄×D₄₃	semidirect
254. D₄	C₂×C₄×D₄₃	semidirect