Subgroups — #19147256

Order 5504.

362 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583632	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583608	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583254	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583252	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583244	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583202	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583199	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583193	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583114	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583112	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583092	C₂	1	1	yes	maximal, characteristic
12	C₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582990	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582979	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×D₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621883	C₂²	1	1	yes	characteristic
17	C₂×#10621882	C₂²	1	1	yes	characteristic
18	#13063324	C₂²	1	1	yes	characteristic
19	#13063321	C₂²	1	1	yes	characteristic
20	#13063317	C₂²	1	1	yes	characteristic
21	#13063316	C₂²	1	1	yes	characteristic
22	#13063313	C₂²	1	1	yes	characteristic
23	#13063312	C₂²	1	1	yes	characteristic
24	#13063311	—	2	1	no	—
25	#13063308	—	2	1	no	—
26	#13063307	—	2	1	no	—
27	#13063304	—	2	1	no	—
28	#13063302	—	2	1	no	—
29	#13063302	—	2	1	no	—
30	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
31	#13063290	C₂²	1	1	yes	characteristic
32	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
33	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
34	#13063287	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊D₄₃	—	2	1	no	—
36	C₂×Dic₄₃⋊C₄	—	2	1	no	—
37	#13063258	C₂²	1	1	yes	characteristic
38	#13063256	C₂²	1	1	yes	characteristic
39	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
40	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
41	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
42	#13063250	C₂²	1	1	yes	characteristic
43	#13063250	C₂²	1	1	yes	characteristic
44	#13063248	C₂²	1	1	yes	characteristic
45	#13063247	C₂²	1	1	yes	characteristic
46	#13063247	C₂²	1	1	yes	characteristic
47	#13063246	C₂²	1	1	yes	characteristic
48	#13063246	C₂²	1	1	yes	characteristic
49	#13063245	C₂²	1	1	yes	characteristic
50	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₄₃	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
55	#107×C₄₃	C₂²	1	1	yes	characteristic
56	#102×C₄₃	C₂²	1	1	yes	characteristic
57	#102×C₄₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	—	2	1	no	—
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	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×Dic₈₆	—	2	1	no	—
70	C₂×Dic₈₆	D₄	1	1	yes	characteristic
71	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
72	D₈×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₄₃	—	2	1	no	—
78	C₂²⋊C₄×C₄₃	—	2	1	no	—
79	C₄²×C₄₃	C₂³	1	1	yes	characteristic
80	C₂²⋊Dic₄₃	—	2	1	no	—
81	C₂²⋊Dic₄₃	—	2	1	no	—
82	C₂²⋊Dic₄₃	—	2	1	no	—
83	C₂²⋊Dic₄₃	—	2	1	no	—
84	#10621883	—	2	2	no	—
85	#10621882	—	2	2	no	—
86	(C₂×D₄₃)⋊C₄	—	2	1	no	—
87	(C₂×D₄₃)⋊C₄	—	2	1	no	—
88	(C₂×D₄₃)⋊C₄	—	2	1	no	—
89	C₄⋊Dic₄₃	—	2	1	no	—
90	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
91	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
93	Dic₄₃⋊C₄	—	2	1	no	—
94	Dic₄₃⋊C₄	—	2	1	no	—
95	Dic₄₃⋊C₄	—	2	1	no	—
96	Dic₄₃⋊C₄	—	2	1	no	—
97	Dic₄₃⋊C₄	—	2	1	no	—
98	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
99	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
102	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
103	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
105	C₂³×C₄₃	—	2	1	no	—
106	C₂³×C₄₃	—	2	1	no	—
107	C₂²×D₄₃	—	2	1	no	—
108	D₄×C₄₃	—	2	1	no	—
109	D₄×C₄₃	—	2	1	no	—
110	D₄×C₄₃	—	2	1	no	—
111	D₄×C₄₃	#41	1	2	yes	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	8	1	no	—
115	C₂×C₄×C₄₃	—	8	1	no	—
116	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₄₃	—	4	2	no	—
120	C₂²⋊D₄₃	—	8	1	no	—
121	C₂×Dic₄₃	—	2	1	no	—
122	C₂×Dic₄₃	—	2	1	no	—
123	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	—	4	2	no	—
125	C₂×Dic₄₃	—	8	1	no	—
126	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
130	D₁₇₂	—	2	1	no	—
131	D₁₇₂	—	2	1	no	—
132	C₄×D₄₃	—	8	1	no	—
133	Dic₈₆	—	2	2	no	—
134	Dic₈₆	—	2	1	no	—
135	Dic₈₆	—	2	1	no	—
136	Dic₈₆	—	8	1	no	—
137	C₈×C₄₃	—	2	1	no	—
138	C₈×C₄₃	—	2	1	no	—
139	C₄₃⋊C₈	—	2	1	no	—
140	C₄₃⋊C₈	—	2	1	no	—
141	C₂²×C₄₃	—	2	2	no	—
142	C₂²×C₄₃	—	4	1	no	—
143	C₂²×C₄₃	—	4	1	no	—
144	C₂²×C₄₃	—	4	1	no	—
145	C₂²×C₄₃	—	4	1	no	—
146	C₂²×C₄₃	—	4	1	no	—
147	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₄₃	—	4	1	no	—
149	C₂×D₄₃	—	4	1	no	—
150	C₂×D₄₃	—	4	1	no	—
151	C₄×C₄₃	—	2	1	no	—
152	C₄×C₄₃	—	2	1	no	—
153	C₄×C₄₃	—	4	1	no	—
154	C₄×C₄₃	—	4	1	no	—
155	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
156	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
157	Dic₄₃	—	2	1	no	—
158	Dic₄₃	—	2	1	no	—
159	Dic₄₃	—	2	1	no	—
160	Dic₄₃	—	4	1	no	—
161	Dic₄₃	—	4	1	no	—
162	Dic₄₃	—	4	1	no	—
163	Dic₄₃	#135	1	2	yes	—
164	#3361	—	43	1	no	maximal, Sylow
165	C₂×C₄₃	—	4	2	no	—
166	C₂×C₄₃	—	8	1	no	—
167	C₂×C₄₃	#548	1	1	yes	characteristic
168	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
169	C₂×C₄₃	#576	1	1	yes	characteristic
170	D₄₃	—	8	1	no	—
171	#559	—	43	1	no	—
172	#554	—	43	1	no	—
173	#488	—	43	1	no	—
174	#488	—	43	1	no	—
175	#486	—	43	1	no	—
176	#478	—	43	1	no	—
177	#478	—	43	1	no	—
178	#477	—	43	1	no	—
179	#463	—	43	1	no	—
180	#463	—	43	1	no	—
181	#459	—	43	1	no	—
182	C₈×Q₈	—	43	1	no	—
183	C₄×SD₁₆	—	43	1	no	—
184	C₄×D₈	—	43	1	no	—
185	C₄×D₈	D₄₃	1	1	yes	characteristic, complemented
186	C₄₃	#3361	1	1	yes	characteristic, complemented, Sylow
187	C₂×SD₁₆	—	43	1	no	—
188	C₂×D₈	—	43	1	no	—
189	C₂×D₈	C₂×D₄₃	1	1	yes	characteristic
190	C₄⋊Q₈	—	43	1	no	—
191	#127	—	43	1	no	—
192	#125	—	43	1	no	—
193	#125	—	43	1	no	—
194	#124	—	43	1	no	—
195	#124	—	43	1	no	—
196	#124	—	86	1	no	—
197	#123	—	86	1	no	—
198	#123	—	86	1	no	—
199	C₂²⋊Q₈	—	86	1	no	—
200	C₂²⋊Q₈	—	86	1	no	—
201	C₂²⋊Q₈	—	86	1	no	—
202	C₄×Q₈	—	43	1	no	—
203	C₄×Q₈	—	43	1	no	—
204	C₄×D₄	—	43	1	no	—
205	C₄×D₄	—	43	1	no	—
206	C₄×D₄	—	43	1	no	—
207	C₄×D₄	—	86	1	no	—
208	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
209	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
210	C₂×C₄⋊C₄	—	86	1	no	—
211	#107	—	43	1	no	—
212	#107	C₂×D₄₃	1	1	yes	characteristic
213	#106	—	43	1	no	—
214	C₄⋊C₈	—	43	1	no	—
215	C₄⋊C₈	—	43	1	no	—
216	C₄⋊C₈	—	43	1	no	—
217	#103	—	43	1	no	—
218	#103	—	43	1	no	—
219	#103	—	43	1	no	—
220	#102	—	43	1	no	—
221	#102	—	43	1	no	—
222	#102	—	43	1	no	—
223	#102	—	43	1	no	—
224	#102	—	43	1	no	—
225	#102	C₂×D₄₃	1	1	yes	characteristic
226	#102	C₂×D₄₃	1	1	yes	characteristic
227	C₄×C₈	—	43	1	no	—
228	C₄×C₈	—	43	1	no	—
229	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
230	C₂×Q₈	—	43	1	no	—
231	C₂×Q₈	—	43	1	no	—
232	C₂×Q₈	—	86	1	no	—
233	C₂×D₄	—	43	1	no	—
234	C₂×D₄	—	86	1	no	—
235	C₂×D₄	—	86	1	no	—
236	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
237	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	86	1	no	—
241	C₂²×C₄	—	86	1	no	—
242	C₂²×C₄	—	86	1	no	—
243	SD₁₆	—	86	2	no	—
244	D₈	—	2	2	no	—
245	D₈	—	86	2	no	—
246	C₂×C₈	—	43	1	no	—
247	C₂×C₈	—	43	1	no	—
248	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
250	C₄⋊C₄	—	43	1	no	—
251	C₄⋊C₄	—	43	1	no	—
252	C₄⋊C₄	—	43	1	no	—
253	C₄⋊C₄	—	43	1	no	—
254	C₄⋊C₄	—	43	1	no	—
255	C₄⋊C₄	—	86	1	no	—
256	C₄⋊C₄	—	86	1	no	—
257	C₄⋊C₄	—	86	1	no	—
258	C₄⋊C₄	—	86	1	no	—
259	C₄⋊C₄	—	86	1	no	—
260	C₄⋊C₄	—	86	1	no	—
261	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	86	1	no	—
266	C₂²⋊C₄	—	86	1	no	—
267	C₂²⋊C₄	—	86	1	no	—
268	C₂²⋊C₄	—	86	1	no	—
269	C₂²⋊C₄	—	86	1	no	—
270	C₂²⋊C₄	—	86	1	no	—
271	C₂²⋊C₄	—	86	1	no	—
272	C₄²	—	43	1	no	—
273	C₄²	—	43	1	no	—
274	C₄²	—	86	1	no	—
275	C₄²	C₂²×D₄₃	1	1	yes	characteristic
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	86	1	no	—
279	Q₈	—	86	2	no	—
280	Q₈	—	86	1	no	—
281	Q₈	—	86	1	no	—
282	Q₈	—	344	1	no	—
283	D₄	—	2	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	86	1	no	—
287	D₄	—	86	1	no	—
288	D₄	—	344	1	no	—
289	D₄	—	172	2	no	—
290	D₄	#10621898	1	2	yes	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	—	43	1	no	—
296	C₂×C₄	—	43	1	no	—
297	C₂×C₄	—	43	1	no	—
298	C₂×C₄	—	43	1	no	—
299	C₂×C₄	—	86	1	no	—
300	C₂×C₄	—	86	1	no	—
301	C₂×C₄	—	86	1	no	—
302	C₂×C₄	—	344	1	no	—
303	C₂×C₄	—	344	1	no	—
304	C₂×C₄	—	172	2	no	—
305	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
306	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	86	1	no	—
311	C₈	—	86	1	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	172	1	no	—
319	C₂²	—	172	1	no	—
320	C₂²	—	172	1	no	—
321	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	43	2	no	—
327	C₄	—	86	1	no	—
328	C₄	—	86	1	no	—
329	C₄	—	86	1	no	—
330	C₄	—	172	1	no	—
331	C₄	—	172	1	no	—
332	C₄	—	172	1	no	—
333	C₄	C₂×#10621898	1	1	yes	characteristic
334	C₄	#13063372	1	1	yes	characteristic
335	C₂	—	4	2	no	—
336	C₂	—	8	1	no	—
337	C₂	—	344	1	no	—
338	C₂	#14583558	1	1	yes	characteristic
339	C₂	#135×D₄₃	1	1	yes	characteristic
340	C₂	#14583700	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄×D₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₄×D₈	D₄₃
hypocenter	C₄₃	#3361
LCS	G ▷ C₂×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
1. #14583632	C₂	semidirect
2. #14583608	C₂	semidirect
3. #14583254	C₂	semidirect
4. #14583252	C₂	semidirect
5. #14583244	C₂	semidirect
6. #14583202	C₂	semidirect
7. #14583199	C₂	semidirect
8. #14583193	C₂	semidirect
9. #14583114	C₂	semidirect
10. #14583112	C₂	semidirect
12. C₈×Dic₈₆	C₂	semidirect
13. #14582990	C₂	semidirect
14. #14582979	C₂	semidirect
15. C₄×D₈×C₄₃	C₂	semidirect
185. C₄×D₈	D₄₃	semidirect
186. C₄₃	#3361	semidirect