Subgroups — #19147517

Order 5504.

360 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583625	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic
3	#14583308	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583307	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583303	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583201	C₂	1	1	yes	maximal, characteristic
7	#14583199	C₂	1	1	yes	maximal, characteristic
8	#14583194	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583192	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583182	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583177	C₂	1	1	yes	maximal, characteristic
12	#14583027	C₂	1	1	yes	maximal, characteristic, complemented
13	#14583000	C₂	1	1	yes	maximal, characteristic
14	SD₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#440	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₈₆	—	2	1	no	—
17	#13063325	C₂²	1	1	yes	characteristic
18	#13063324	—	2	1	no	—
19	#13063324	C₂²	1	1	yes	characteristic
20	#13063324	C₂²	1	1	yes	characteristic
21	#13063324	C₂²	1	1	yes	characteristic
22	#13063317	—	2	1	no	—
23	#13063317	C₂²	1	1	yes	characteristic
24	#13063316	—	2	1	no	—
25	#13063316	C₂²	1	1	yes	characteristic
26	#13063304	—	2	1	no	—
27	#13063302	—	2	1	no	—
28	#13063302	—	2	1	no	—
29	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
30	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
31	C₄×Dic₈₆	—	2	1	no	—
32	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
33	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
34	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
35	#13063259	C₂²	1	1	yes	characteristic
36	#13063259	C₂²	1	1	yes	characteristic
37	#13063258	C₂²	1	1	yes	characteristic
38	#13063258	C₂²	1	1	yes	characteristic
39	#13063258	C₂²	1	1	yes	characteristic
40	#13063257	C₂²	1	1	yes	characteristic
41	#13063257	C₂²	1	1	yes	characteristic
42	#13063256	C₂²	1	1	yes	characteristic
43	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
46	#13063249	C₂²	1	1	yes	characteristic
47	#13063249	C₂²	1	1	yes	characteristic
48	#13063246	C₂²	1	1	yes	characteristic
49	#13063246	C₂²	1	1	yes	characteristic
50	#13063235	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
55	#107×C₄₃	C₂²	1	1	yes	characteristic
56	#103×C₄₃	C₂²	1	1	yes	characteristic
57	#102×C₄₃	C₂²	1	1	yes	characteristic
58	#97×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	—	2	1	no	—
62	C₂²×Dic₄₃	—	2	1	no	—
63	C₂²×Dic₄₃	—	2	1	no	—
64	C₂×Dic₈₆	—	2	1	no	—
65	C₂×Dic₈₆	—	2	1	no	—
66	C₂×Dic₈₆	—	2	1	no	—
67	C₂×Dic₈₆	—	2	1	no	—
68	C₂×Dic₈₆	—	4	4	no	—
69	C₂×Dic₈₆	D₄	1	1	yes	characteristic
70	SD₁₆×C₄₃	Q₈	1	4	yes	complemented
71	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₄₃	—	2	1	no	—
74	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₄₃	—	2	1	no	—
77	C₄²×C₄₃	—	2	1	no	—
78	C₄²×C₄₃	C₂³	1	1	yes	characteristic
79	C₂²⋊Dic₄₃	—	2	1	no	—
80	C₂²⋊Dic₄₃	—	2	1	no	—
81	C₄⋊Dic₄₃	—	2	1	no	—
82	C₄⋊Dic₄₃	—	2	1	no	—
83	C₄⋊Dic₄₃	—	2	1	no	—
84	C₄⋊Dic₄₃	—	2	1	no	—
85	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
86	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
90	Dic₄₃⋊C₄	—	2	1	no	—
91	Dic₄₃⋊C₄	—	2	1	no	—
92	Dic₄₃⋊C₄	—	2	1	no	—
93	Dic₄₃⋊C₄	—	2	1	no	—
94	Dic₄₃⋊C₄	—	2	1	no	—
95	Dic₄₃⋊C₄	—	2	1	no	—
96	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
97	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
98	C₄×Dic₄₃	—	2	1	no	—
99	C₄×Dic₄₃	—	2	1	no	—
100	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
101	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
102	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
104	C₂³×C₄₃	—	2	1	no	—
105	Q₈×C₄₃	—	2	1	no	—
106	Q₈×C₄₃	C₂×Q₈	1	2	yes	complemented
107	D₄×C₄₃	—	2	1	no	—
108	D₄×C₄₃	C₂×Q₈	1	2	yes	—
109	C₂×C₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	4	2	no	—
113	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₄₃	—	2	1	no	—
117	C₂×Dic₄₃	—	2	1	no	—
118	C₂×Dic₄₃	—	2	1	no	—
119	C₂×Dic₄₃	—	2	1	no	—
120	C₂×Dic₄₃	—	4	2	no	—
121	C₂×Dic₄₃	—	4	2	no	—
122	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
126	Dic₈₆	—	2	2	no	—
127	Dic₈₆	—	4	2	no	—
128	Dic₈₆	—	4	2	no	—
129	Dic₈₆	—	4	2	no	—
130	Dic₈₆	—	4	2	no	—
131	C₈×C₄₃	—	2	1	no	—
132	C₈×C₄₃	C₂×Q₈	1	2	yes	complemented
133	C₄₃⋊C₈	—	2	1	no	—
134	C₄₃⋊C₈	—	2	1	no	—
135	C₂²×C₄₃	—	2	2	no	—
136	C₂²×C₄₃	—	4	1	no	—
137	C₂²×C₄₃	—	4	1	no	—
138	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₄₃	—	2	2	no	—
140	C₄×C₄₃	—	2	1	no	—
141	C₄×C₄₃	—	2	1	no	—
142	C₄×C₄₃	—	4	1	no	—
143	C₄×C₄₃	—	4	1	no	—
144	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
145	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
146	Dic₄₃	—	2	1	no	—
147	Dic₄₃	—	2	1	no	—
148	Dic₄₃	—	2	1	no	—
149	Dic₄₃	—	4	1	no	—
150	Dic₄₃	—	4	1	no	—
151	Dic₄₃	—	4	1	no	—
152	Dic₄₃	—	4	1	no	—
153	Dic₄₃	AGL(1,ℤ/8)	1	2	yes	—
154	#3386	—	43	1	no	maximal, Sylow
155	C₂×C₄₃	—	4	2	no	—
156	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
157	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂×C₄₃	#578	1	1	yes	characteristic
159	#557	—	43	1	no	—
160	D₄×Q₈	—	43	1	no	—
161	#501	—	43	1	no	—
162	#501	—	43	1	no	—
163	#500	—	43	1	no	—
164	#478	—	43	1	no	—
165	#478	—	43	1	no	—
166	#477	—	43	1	no	—
167	#477	—	43	1	no	—
168	#475	—	43	1	no	—
169	#474	—	43	1	no	—
170	#446	—	43	1	no	—
171	#440	—	43	1	no	—
172	#440	D₄₃	1	1	yes	characteristic
173	C₄×SD₁₆	—	43	1	no	—
174	C₄₃	#3386	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₈	—	86	1	no	—
176	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
177	C₄⋊Q₈	—	43	1	no	—
178	C₄⋊Q₈	—	43	1	no	—
179	C₄⋊Q₈	—	43	1	no	—
180	C₄⋊Q₈	—	43	1	no	—
181	C₄⋊Q₈	—	86	1	no	—
182	#125	—	43	1	no	—
183	#125	—	43	1	no	—
184	#125	—	86	1	no	—
185	#125	—	86	1	no	—
186	C₂²⋊Q₈	—	86	1	no	—
187	C₂²⋊Q₈	—	86	1	no	—
188	C₂²⋊Q₈	—	86	1	no	—
189	C₄×Q₈	—	43	1	no	—
190	C₄×Q₈	—	43	1	no	—
191	C₄×Q₈	—	43	1	no	—
192	C₄×Q₈	—	86	1	no	—
193	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
194	C₄×D₄	—	43	1	no	—
195	C₄×D₄	—	43	1	no	—
196	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
197	#107	—	43	1	no	—
198	#107	—	43	1	no	—
199	#107	—	43	1	no	—
200	#107	—	43	1	no	—
201	#107	—	43	1	no	—
202	#107	C₂×D₄₃	1	1	yes	characteristic
203	#106	—	43	1	no	—
204	#106	—	43	1	no	—
205	#106	—	43	1	no	—
206	C₄⋊C₈	—	43	1	no	—
207	C₄⋊C₈	—	43	1	no	—
208	C₄⋊C₈	—	43	1	no	—
209	#103	—	43	1	no	—
210	#103	—	43	1	no	—
211	#103	C₂×D₄₃	1	1	yes	characteristic
212	#102	—	43	1	no	—
213	#102	—	43	1	no	—
214	#102	C₂×D₄₃	1	1	yes	characteristic
215	#97	—	43	1	no	—
216	#97	C₂×D₄₃	1	1	yes	characteristic
217	C₄×C₈	—	43	1	no	—
218	C₂×Q₈	—	43	1	no	—
219	C₂×Q₈	—	86	1	no	—
220	C₂×Q₈	—	86	1	no	—
221	C₂×Q₈	—	86	1	no	—
222	C₂×Q₈	—	86	1	no	—
223	C₂×Q₈	—	172	4	no	—
224	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
225	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	86	1	no	—
228	C₂²×C₄	—	86	1	no	—
229	SD₁₆	Dic₈₆	1	4	yes	complemented
230	C₂×C₈	—	43	1	no	—
231	C₂×C₈	—	43	1	no	—
232	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
233	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	43	1	no	—
236	C₄⋊C₄	—	43	1	no	—
237	C₄⋊C₄	—	43	1	no	—
238	C₄⋊C₄	—	43	1	no	—
239	C₄⋊C₄	—	43	1	no	—
240	C₄⋊C₄	—	43	1	no	—
241	C₄⋊C₄	—	43	1	no	—
242	C₄⋊C₄	—	86	1	no	—
243	C₄⋊C₄	—	86	1	no	—
244	C₄⋊C₄	—	86	1	no	—
245	C₄⋊C₄	—	86	1	no	—
246	C₄⋊C₄	—	86	1	no	—
247	C₄⋊C₄	—	86	1	no	—
248	C₄⋊C₄	—	86	1	no	—
249	C₄⋊C₄	—	86	1	no	—
250	C₄⋊C₄	—	86	1	no	—
251	C₄⋊C₄	—	86	1	no	—
252	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
253	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	86	1	no	—
256	C₂²⋊C₄	—	86	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	43	1	no	—
259	C₄²	—	43	1	no	—
260	C₄²	—	86	1	no	—
261	C₄²	—	86	1	no	—
262	C₄²	C₂²×D₄₃	1	1	yes	characteristic
263	C₂³	—	2	1	no	—
264	Q₈	—	2	1	no	—
265	Q₈	—	86	2	no	—
266	Q₈	—	172	2	no	—
267	Q₈	—	172	2	no	—
268	Q₈	—	172	2	no	—
269	Q₈	—	172	2	no	—
270	Q₈	C₂×Dic₈₆	1	2	yes	complemented
271	D₄	—	2	1	no	—
272	D₄	C₂×Dic₈₆	1	2	yes	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	43	1	no	—
278	C₂×C₄	—	43	1	no	—
279	C₂×C₄	—	43	1	no	—
280	C₂×C₄	—	43	1	no	—
281	C₂×C₄	—	86	1	no	—
282	C₂×C₄	—	86	1	no	—
283	C₂×C₄	—	86	1	no	—
284	C₂×C₄	—	86	1	no	—
285	C₂×C₄	—	172	2	no	—
286	C₂×C₄	—	172	2	no	—
287	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
288	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	86	1	no	—
292	C₈	—	86	1	no	—
293	C₈	C₂×Dic₈₆	1	2	yes	complemented
294	C₂²	—	2	2	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	43	2	no	—
304	C₄	—	86	1	no	—
305	C₄	—	86	1	no	—
306	C₄	—	86	1	no	—
307	C₄	—	172	1	no	—
308	C₄	—	172	1	no	—
309	C₄	—	172	1	no	—
310	C₄	—	172	1	no	—
311	C₄	C₂²×Dic₈₆	1	1	yes	characteristic
312	C₄	#13063372	1	1	yes	characteristic
313	C₂	—	4	2	no	—
314	C₂	D₄×Dic₈₆	1	1	yes	characteristic
315	C₂	AGL(1,ℤ/8)×D₄₃	1	1	yes	characteristic
316	C₂	#14583715	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#440	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#440	D₄₃
hypocenter	C₄₃	#3386
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #14583625	C₂	semidirect
3. #14583308	C₂	semidirect
4. #14583307	C₂	semidirect
5. #14583303	C₂	semidirect
8. #14583194	C₂	semidirect
9. #14583192	C₂	semidirect
10. #14583182	C₂	semidirect
12. #14583027	C₂	semidirect
70. SD₁₆×C₄₃	Q₈	semidirect
106. Q₈×C₄₃	C₂×Q₈	semidirect
132. C₈×C₄₃	C₂×Q₈	semidirect
174. C₄₃	#3386	semidirect
229. SD₁₆	Dic₈₆	semidirect
270. Q₈	C₂×Dic₈₆	semidirect
293. C₈	C₂×Dic₈₆	semidirect