Subgroups — #19145891

Order 5504.

320 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583527	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583516	C₂	1	1	yes	maximal, characteristic
3	#14583307	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583296	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583257	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583252	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583240	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583230	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583215	C₂	1	1	yes	maximal, characteristic
10	#14583205	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583201	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583187	C₂	1	1	yes	maximal, characteristic, complemented
13	#14583183	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582957	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#482	C₂	1	1	yes	maximal, characteristic
16	#13063325	—	2	1	no	—
17	#13063324	C₂²	1	1	yes	characteristic
18	#13063324	C₂²	1	1	yes	characteristic
19	#13063324	C₂²	1	1	yes	characteristic
20	#13063318	—	2	1	no	—
21	#13063317	—	2	1	no	—
22	#13063316	C₂²	1	1	yes	characteristic
23	#13063316	C₂²	1	1	yes	characteristic
24	#13063312	—	2	1	no	—
25	#13063312	C₂²	1	1	yes	characteristic
26	#13063306	—	2	1	no	—
27	#13063304	C₂²	1	1	yes	characteristic
28	#13063302	—	2	1	no	—
29	#13063302	—	2	1	no	—
30	#13063302	—	2	1	no	—
31	#13063290	C₂²	1	1	yes	characteristic
32	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
33	#13063284	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
35	#13063258	C₂²	1	1	yes	characteristic
36	#13063257	C₂²	1	1	yes	characteristic
37	#13063257	C₂²	1	1	yes	characteristic
38	#13063256	C₂²	1	1	yes	characteristic
39	#13063256	C₂²	1	1	yes	characteristic
40	#13063256	C₂²	1	1	yes	characteristic
41	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
43	#13063250	C₂²	1	1	yes	characteristic
44	#13063250	C₂²	1	1	yes	characteristic
45	#13063248	C₂²	1	1	yes	characteristic
46	#13063248	C₂²	1	1	yes	characteristic
47	#13063246	C₂²	1	1	yes	characteristic
48	#13063246	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
50	#13063235	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	#121×C₄₃	C₂²	1	1	yes	characteristic
53	#117×C₄₃	C₂²	1	1	yes	characteristic
54	#107×C₄₃	C₂²	1	1	yes	characteristic
55	#106×C₄₃	C₂²	1	1	yes	characteristic
56	#102×C₄₃	C₂²	1	1	yes	characteristic
57	#102×C₄₃	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₄₃	—	2	1	no	—
60	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
73	C₂²⋊C₄×C₄₃	—	2	1	no	—
74	C₂²⋊C₄×C₄₃	—	2	1	no	—
75	C₄²×C₄₃	—	2	1	no	—
76	C₂²⋊Dic₄₃	—	2	1	no	—
77	C₂²⋊Dic₄₃	—	2	1	no	—
78	C₂²⋊Dic₄₃	—	2	1	no	—
79	C₂²⋊Dic₄₃	—	2	1	no	—
80	C₂²⋊Dic₄₃	D₄	1	1	yes	characteristic
81	C₂²⋊Dic₄₃	D₄	1	1	yes	characteristic
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	D₄	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	—	2	1	no	—
97	C₄×Dic₄₃	—	2	1	no	—
98	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
99	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
100	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
102	C₂³×C₄₃	—	2	1	no	—
103	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
104	D₄×C₄₃	—	2	1	no	—
105	D₄×C₄₃	—	2	1	no	—
106	D₄×C₄₃	—	8	1	no	—
107	C₂×C₄×C₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
114	C₂×Dic₄₃	—	2	1	no	—
115	C₂×Dic₄₃	—	2	1	no	—
116	C₂×Dic₄₃	—	2	1	no	—
117	C₂×Dic₄₃	—	2	1	no	—
118	C₂×Dic₄₃	—	8	1	no	—
119	C₂×Dic₄₃	—	8	1	no	—
120	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
124	Dic₈₆	—	2	1	no	—
125	Dic₈₆	—	2	1	no	—
126	Dic₈₆	—	8	1	no	—
127	Dic₈₆	—	8	1	no	—
128	Dic₈₆	—	8	1	no	—
129	C₈×C₄₃	—	2	1	no	—
130	C₈×C₄₃	—	2	1	no	—
131	C₄₃⋊C₈	—	2	1	no	—
132	C₄₃⋊C₈	—	2	1	no	—
133	C₂²×C₄₃	—	2	1	no	—
134	C₂²×C₄₃	—	2	1	no	—
135	C₂²×C₄₃	—	2	1	no	—
136	C₂²×C₄₃	—	4	1	no	—
137	C₂²×C₄₃	—	4	1	no	—
138	C₂²×C₄₃	—	4	1	no	—
139	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₄₃	—	2	1	no	—
141	C₄×C₄₃	—	4	1	no	—
142	C₄×C₄₃	—	4	1	no	—
143	C₄×C₄₃	—	4	1	no	—
144	C₄×C₄₃	2₋¹⁺⁴	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	—	2	1	no	—
150	Dic₄₃	—	4	1	no	—
151	Dic₄₃	—	4	1	no	—
152	Dic₄₃	—	4	1	no	—
153	Dic₄₃	—	4	1	no	—
154	#3244	—	43	1	no	maximal, Sylow
155	C₂×C₄₃	—	4	1	no	—
156	C₂×C₄₃	—	8	1	no	—
157	C₂×C₄₃	#536	1	1	yes	characteristic
158	C₂×C₄₃	#577	1	1	yes	characteristic
159	C₂×C₄₃	#577	1	1	yes	characteristic
160	#544	—	43	1	no	—
161	#542	—	43	1	no	—
162	#501	—	43	1	no	—
163	#498	—	43	1	no	—
164	#489	—	43	1	no	—
165	#488	—	43	1	no	—
166	#485	—	43	1	no	—
167	#483	—	43	1	no	—
168	#482	D₄₃	1	1	yes	characteristic
169	#481	—	43	1	no	—
170	#479	—	43	1	no	—
171	#478	—	43	1	no	—
172	#476	—	43	1	no	—
173	#475	—	43	1	no	—
174	#433	—	43	1	no	—
175	C₄₃	#3244	1	1	yes	characteristic, complemented, Sylow
176	C₄⋊Q₈	—	43	1	no	—
177	C₄⋊Q₈	—	43	1	no	—
178	C₄⋊Q₈	—	43	1	no	—
179	C₄⋊Q₈	—	86	1	no	—
180	#126	—	86	1	no	—
181	#125	—	43	1	no	—
182	#125	—	43	1	no	—
183	#125	—	86	1	no	—
184	#124	—	43	1	no	—
185	#124	—	86	1	no	—
186	#123	—	86	1	no	—
187	C₂²⋊Q₈	—	43	1	no	—
188	C₂²⋊Q₈	—	86	1	no	—
189	C₂²⋊Q₈	—	86	1	no	—
190	C₂²⋊Q₈	—	86	1	no	—
191	#121	C₂×D₄₃	1	1	yes	characteristic
192	C₄×Q₈	—	43	1	no	—
193	C₄×D₄	—	43	1	no	—
194	#117	—	43	1	no	—
195	#117	C₂×D₄₃	1	1	yes	characteristic
196	C₂×C₄⋊C₄	—	43	1	no	—
197	#107	—	43	1	no	—
198	#107	C₂×D₄₃	1	1	yes	characteristic
199	#106	—	43	1	no	—
200	#106	—	43	1	no	—
201	#106	—	43	1	no	—
202	#106	—	43	1	no	—
203	#106	—	43	1	no	—
204	#106	C₂×D₄₃	1	1	yes	characteristic
205	C₄⋊C₈	—	43	1	no	—
206	C₄⋊C₈	—	43	1	no	—
207	#103	—	43	1	no	—
208	#103	—	43	1	no	—
209	#103	—	43	1	no	—
210	#103	—	43	1	no	—
211	#102	—	43	1	no	—
212	#102	—	43	1	no	—
213	#102	C₂×D₄₃	1	1	yes	characteristic
214	#102	C₂×D₄₃	1	1	yes	characteristic
215	C₂²⋊C₈	—	43	1	no	—
216	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
217	#97	—	43	1	no	—
218	C₄×C₈	—	43	1	no	—
219	C₂×Q₈	—	43	1	no	—
220	C₂×Q₈	—	86	1	no	—
221	C₂×Q₈	—	86	1	no	—
222	C₂×Q₈	—	86	1	no	—
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
225	C₂²×C₄	—	86	1	no	—
226	C₂²×C₄	—	86	1	no	—
227	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
228	C₂×C₈	—	43	1	no	—
229	C₂×C₈	—	43	1	no	—
230	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
232	C₄⋊C₄	—	43	1	no	—
233	C₄⋊C₄	—	43	1	no	—
234	C₄⋊C₄	—	43	1	no	—
235	C₄⋊C₄	—	43	1	no	—
236	C₄⋊C₄	—	43	1	no	—
237	C₄⋊C₄	—	86	1	no	—
238	C₄⋊C₄	—	86	1	no	—
239	C₄⋊C₄	—	86	1	no	—
240	C₄⋊C₄	—	86	1	no	—
241	C₄⋊C₄	—	86	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	C₂²×D₄₃	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	43	1	no	—
253	C₂²⋊C₄	—	43	1	no	—
254	C₂²⋊C₄	—	86	1	no	—
255	C₂²⋊C₄	—	86	1	no	—
256	C₂²⋊C₄	—	86	1	no	—
257	C₂²⋊C₄	—	86	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	43	1	no	—
260	C₄²	—	43	1	no	—
261	C₄²	—	86	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	D₄×D₄₃	1	1	yes	characteristic
264	Q₈	—	86	1	no	—
265	Q₈	—	86	1	no	—
266	Q₈	—	344	1	no	—
267	Q₈	—	344	1	no	—
268	Q₈	—	344	1	no	—
269	D₄	—	2	1	no	—
270	D₄	—	2	1	no	—
271	D₄	—	8	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	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	—	344	1	no	—
286	C₂×C₄	—	344	1	no	—
287	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	86	1	no	—
292	C₈	—	86	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	86	1	no	—
305	C₄	—	86	1	no	—
306	C₄	—	86	1	no	—
307	C₄	—	86	1	no	—
308	C₄	—	172	1	no	—
309	C₄	—	172	1	no	—
310	C₄	—	172	1	no	—
311	C₄	—	172	1	no	—
312	C₄	#13063372	1	1	yes	characteristic
313	C₄	#13063372	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	8	1	no	—
316	C₂	#14583471	1	1	yes	characteristic
317	C₂	#14583706	1	1	yes	characteristic
318	C₂	#14583709	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#482	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#482	D₄₃
hypocenter	C₄₃	#3244
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #482

Complements

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

N	H ≅ G/N	type
1. #14583527	C₂	semidirect
3. #14583307	C₂	semidirect
4. #14583296	C₂	semidirect
5. #14583257	C₂	semidirect
6. #14583252	C₂	semidirect
7. #14583240	C₂	semidirect
8. #14583230	C₂	semidirect
10. #14583205	C₂	semidirect
11. #14583201	C₂	semidirect
12. #14583187	C₂	semidirect
13. #14583183	C₂	semidirect
14. #14582957	C₂	semidirect
175. C₄₃	#3244	semidirect