Subgroups — #19147382

Order 5504.

384 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583631	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583260	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583258	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583244	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583190	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583187	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583182	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583116	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583092	C₂	1	1	yes	maximal, characteristic
11	#14583028	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583011	C₂	1	1	yes	maximal, characteristic, complemented
13	#14583009	C₂	1	1	yes	maximal, characteristic, complemented
14	SD₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#463	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₈₆	—	2	1	no	—
17	C₂×#10621882	C₂²	1	1	yes	characteristic
18	#13063325	C₂²	1	1	yes	characteristic
19	#13063324	C₂²	1	1	yes	characteristic
20	#13063324	C₂²	1	1	yes	characteristic
21	#13063321	C₂²	1	1	yes	characteristic
22	#13063314	C₂²	1	1	yes	characteristic
23	#13063311	—	2	1	no	—
24	#13063309	—	2	1	no	—
25	#13063307	—	2	1	no	—
26	#13063304	—	2	1	no	—
27	#13063302	—	2	1	no	—
28	#13063302	—	2	1	no	—
29	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
30	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
31	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
32	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
33	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
34	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
35	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
36	#13063257	C₂²	1	1	yes	characteristic
37	#13063257	C₂²	1	1	yes	characteristic
38	#13063256	C₂²	1	1	yes	characteristic
39	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
41	#13063249	C₂²	1	1	yes	characteristic
42	#13063249	C₂²	1	1	yes	characteristic
43	#13063247	C₂²	1	1	yes	characteristic
44	#13063247	C₂²	1	1	yes	characteristic
45	#13063246	C₂²	1	1	yes	characteristic
46	#13063246	C₂²	1	1	yes	characteristic
47	#13063246	C₂²	1	1	yes	characteristic
48	#13063245	C₂²	1	1	yes	characteristic
49	#13063236	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	C₂×D₈×C₄₃	C₂²	1	1	yes	characteristic
54	#124×C₄₃	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
57	#103×C₄₃	C₂²	1	1	yes	characteristic
58	#102×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	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	—	2	1	no	—
63	C₂×C₂²⋊D₄₃	—	2	1	no	—
64	C₂×C₂²⋊D₄₃	—	2	1	no	—
65	C₂²×Dic₄₃	—	2	1	no	—
66	C₂²×Dic₄₃	—	2	1	no	—
67	C₂²×Dic₄₃	—	2	1	no	—
68	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₄₃	—	2	1	no	—
70	C₂×Dic₈₆	—	2	1	no	—
71	C₂×Dic₈₆	—	2	1	no	—
72	C₂×Dic₈₆	—	2	1	no	—
73	C₂×Dic₈₆	—	4	4	no	—
74	C₂×Dic₈₆	D₄	1	1	yes	characteristic
75	SD₁₆×C₄₃	—	2	2	no	—
76	D₈×C₄₃	—	2	2	no	—
77	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
80	C₂²⋊C₄×C₄₃	—	2	1	no	—
81	C₂²⋊C₄×C₄₃	—	2	1	no	—
82	C₂²⋊C₄×C₄₃	—	2	1	no	—
83	C₄²×C₄₃	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₄₃	—	2	1	no	—
85	C₂²⋊Dic₄₃	—	2	1	no	—
86	C₂²⋊Dic₄₃	—	2	1	no	—
87	#10621882	—	2	2	no	—
88	(C₂×D₄₃)⋊C₄	—	2	1	no	—
89	(C₂×D₄₃)⋊C₄	—	2	1	no	—
90	(C₂×D₄₃)⋊C₄	—	2	1	no	—
91	C₄⋊Dic₄₃	—	2	1	no	—
92	C₄⋊Dic₄₃	—	2	1	no	—
93	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
94	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
97	Dic₄₃⋊C₄	—	2	1	no	—
98	Dic₄₃⋊C₄	—	2	1	no	—
99	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
100	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
101	C₄×Dic₄₃	—	2	1	no	—
102	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
103	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
104	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
106	C₂³×C₄₃	—	2	1	no	—
107	C₂³×C₄₃	—	2	1	no	—
108	C₂²×D₄₃	—	2	1	no	—
109	Q₈×C₄₃	—	2	1	no	—
110	Q₈×C₄₃	—	2	1	no	—
111	D₄×C₄₃	—	2	1	no	—
112	D₄×C₄₃	—	2	1	no	—
113	D₄×C₄₃	—	2	1	no	—
114	D₄×C₄₃	#41	1	2	yes	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₄₃	—	4	2	no	—
122	C₂²⋊D₄₃	—	8	1	no	—
123	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	2	1	no	—
126	C₂×Dic₄₃	—	4	2	no	—
127	C₂×Dic₄₃	—	4	2	no	—
128	C₂×Dic₄₃	—	8	1	no	—
129	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
133	D₁₇₂	—	2	1	no	—
134	D₁₇₂	—	2	1	no	—
135	C₄×D₄₃	—	8	1	no	—
136	Dic₈₆	—	2	2	no	—
137	Dic₈₆	—	4	2	no	—
138	Dic₈₆	—	4	2	no	—
139	Dic₈₆	—	4	2	no	—
140	C₈×C₄₃	—	2	1	no	—
141	C₈×C₄₃	—	2	1	no	—
142	C₄₃⋊C₈	—	2	1	no	—
143	C₄₃⋊C₈	—	2	1	no	—
144	C₂²×C₄₃	—	2	2	no	—
145	C₂²×C₄₃	—	4	1	no	—
146	C₂²×C₄₃	—	4	1	no	—
147	C₂²×C₄₃	—	4	1	no	—
148	C₂²×C₄₃	—	4	1	no	—
149	C₂²×C₄₃	—	4	1	no	—
150	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₄₃	—	4	1	no	—
152	C₂×D₄₃	—	4	1	no	—
153	C₂×D₄₃	—	4	1	no	—
154	C₄×C₄₃	—	2	1	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	—	4	1	no	—
157	C₄×C₄₃	—	4	1	no	—
158	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
159	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₄₃	—	2	1	no	—
161	Dic₄₃	—	2	1	no	—
162	Dic₄₃	—	2	1	no	—
163	Dic₄₃	—	4	1	no	—
164	Dic₄₃	—	4	1	no	—
165	Dic₄₃	—	4	1	no	—
166	Dic₄₃	AGL(1,ℤ/8)	1	2	yes	—
167	#3373	—	43	1	no	maximal, Sylow
168	C₂×C₄₃	—	4	2	no	—
169	C₂×C₄₃	—	8	1	no	—
170	C₂×C₄₃	#548	1	1	yes	characteristic
171	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₂×C₄₃	#577	1	1	yes	characteristic
173	D₄₃	—	8	1	no	—
174	#559	—	43	1	no	—
175	D₄×Q₈	—	43	1	no	—
176	#489	—	43	1	no	—
177	#489	—	43	1	no	—
178	#486	—	43	1	no	—
179	#476	—	43	1	no	—
180	#476	—	43	1	no	—
181	#475	—	43	1	no	—
182	#463	—	43	1	no	—
183	#463	D₄₃	1	1	yes	characteristic, complemented
184	#459	—	43	1	no	—
185	#446	—	43	1	no	—
186	#442	—	43	1	no	—
187	#442	—	43	1	no	—
188	C₄×SD₁₆	—	43	1	no	—
189	C₄₃	#3373	1	1	yes	characteristic, complemented, Sylow
190	C₂²×Q₈	—	86	1	no	—
191	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
192	C₂×D₈	—	43	1	no	—
193	C₂×D₈	C₂×D₄₃	1	1	yes	characteristic
194	C₄⋊Q₈	—	43	1	no	—
195	C₄⋊Q₈	—	43	1	no	—
196	C₄⋊Q₈	—	43	1	no	—
197	#127	—	43	1	no	—
198	#124	—	43	1	no	—
199	#124	—	86	1	no	—
200	#124	C₂×D₄₃	1	1	yes	characteristic
201	#123	—	86	1	no	—
202	#123	—	86	1	no	—
203	C₂²⋊Q₈	—	86	1	no	—
204	C₂²⋊Q₈	—	86	1	no	—
205	C₂²⋊Q₈	—	86	1	no	—
206	C₄×Q₈	—	43	1	no	—
207	C₄×Q₈	—	43	1	no	—
208	C₄×D₄	—	43	1	no	—
209	C₄×D₄	—	43	1	no	—
210	C₄×D₄	—	43	1	no	—
211	C₄×D₄	—	43	1	no	—
212	C₄×D₄	—	86	1	no	—
213	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
214	#106	—	43	1	no	—
215	#106	—	43	1	no	—
216	#106	—	43	1	no	—
217	C₄⋊C₈	—	43	1	no	—
218	C₄⋊C₈	—	43	1	no	—
219	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
220	#103	—	43	1	no	—
221	#103	—	43	1	no	—
222	#103	C₂×D₄₃	1	1	yes	characteristic
223	#102	—	43	1	no	—
224	#102	—	43	1	no	—
225	#102	—	43	1	no	—
226	#102	—	43	1	no	—
227	#102	—	43	1	no	—
228	#102	—	43	1	no	—
229	#102	C₂×D₄₃	1	1	yes	characteristic
230	#97	—	43	1	no	—
231	#97	—	43	1	no	—
232	C₄×C₈	—	43	1	no	—
233	C₂×Q₈	—	43	1	no	—
234	C₂×Q₈	—	86	1	no	—
235	C₂×Q₈	—	86	1	no	—
236	C₂×Q₈	—	86	1	no	—
237	C₂×Q₈	—	172	4	no	—
238	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
239	C₂×D₄	—	43	1	no	—
240	C₂×D₄	—	86	1	no	—
241	C₂×D₄	—	86	1	no	—
242	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
243	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	86	1	no	—
246	C₂²×C₄	—	86	1	no	—
247	C₂²×C₄	—	86	1	no	—
248	C₂²×C₄	—	86	1	no	—
249	SD₁₆	—	2	2	no	—
250	D₈	—	2	2	no	—
251	D₈	—	86	2	no	—
252	C₂×C₈	—	43	1	no	—
253	C₂×C₈	—	43	1	no	—
254	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
256	C₄⋊C₄	—	43	1	no	—
257	C₄⋊C₄	—	43	1	no	—
258	C₄⋊C₄	—	43	1	no	—
259	C₄⋊C₄	—	43	1	no	—
260	C₄⋊C₄	—	43	1	no	—
261	C₄⋊C₄	—	43	1	no	—
262	C₄⋊C₄	—	86	1	no	—
263	C₄⋊C₄	—	86	1	no	—
264	C₄⋊C₄	—	86	1	no	—
265	C₄⋊C₄	—	86	1	no	—
266	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	86	1	no	—
271	C₂²⋊C₄	—	86	1	no	—
272	C₂²⋊C₄	—	86	1	no	—
273	C₂²⋊C₄	—	86	1	no	—
274	C₂²⋊C₄	—	86	1	no	—
275	C₂²⋊C₄	—	86	1	no	—
276	C₄²	—	43	1	no	—
277	C₄²	—	43	1	no	—
278	C₄²	—	86	1	no	—
279	C₄²	C₂²×D₄₃	1	1	yes	characteristic
280	C₂³	—	2	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	86	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	86	2	no	—
286	Q₈	—	172	2	no	—
287	Q₈	—	172	2	no	—
288	Q₈	—	172	2	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	86	1	no	—
293	D₄	—	86	1	no	—
294	D₄	—	344	1	no	—
295	D₄	—	172	2	no	—
296	D₄	#10621900	1	2	yes	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	43	1	no	—
301	C₂×C₄	—	43	1	no	—
302	C₂×C₄	—	43	1	no	—
303	C₂×C₄	—	43	1	no	—
304	C₂×C₄	—	86	1	no	—
305	C₂×C₄	—	86	1	no	—
306	C₂×C₄	—	86	1	no	—
307	C₂×C₄	—	344	1	no	—
308	C₂×C₄	—	344	1	no	—
309	C₂×C₄	—	172	2	no	—
310	C₂×C₄	—	172	2	no	—
311	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
312	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
313	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	86	1	no	—
317	C₈	—	86	1	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	172	1	no	—
325	C₂²	—	172	1	no	—
326	C₂²	—	172	1	no	—
327	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	43	2	no	—
333	C₄	—	86	1	no	—
334	C₄	—	86	1	no	—
335	C₄	—	86	1	no	—
336	C₄	—	172	1	no	—
337	C₄	—	172	1	no	—
338	C₄	—	172	1	no	—
339	C₄	C₂×#10621900	1	1	yes	characteristic
340	C₄	#13063372	1	1	yes	characteristic
341	C₂	—	4	2	no	—
342	C₂	—	8	1	no	—
343	C₂	—	344	1	no	—
344	C₂	#14583555	1	1	yes	characteristic
345	C₂	AGL(1,ℤ/8)×D₄₃	1	1	yes	characteristic
346	C₂	#14583706	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#463	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#463	D₄₃
hypocenter	C₄₃	#3373
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #14583631	C₂	semidirect
2. D₄×Dic₈₆	C₂	semidirect
3. #14583260	C₂	semidirect
4. #14583258	C₂	semidirect
5. #14583244	C₂	semidirect
6. #14583190	C₂	semidirect
7. #14583187	C₂	semidirect
8. #14583182	C₂	semidirect
9. #14583116	C₂	semidirect
11. #14583028	C₂	semidirect
12. #14583011	C₂	semidirect
13. #14583009	C₂	semidirect
14. SD₁₆×Dic₄₃	C₂	semidirect
15. C₄₃×#463	C₂	semidirect
183. #463	D₄₃	semidirect
189. C₄₃	#3373	semidirect