Subgroups — #19146382

Order 5504.

454 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583716	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583706	C₂	1	2	yes	maximal, complemented
3	#14583704	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583283	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583174	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583169	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583163	C₂	1	2	yes	maximal, complemented
8	#14583078	C₂	1	2	yes	maximal, complemented
9	#14583073	C₂	1	2	yes	maximal, complemented
10	#14583017	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
12	#13063372	C₂²	1	2	yes	complemented
13	#13063371	C₂²	1	2	yes	—
14	#13063358	—	2	2	no	—
15	#13063358	C₂²	1	2	yes	complemented
16	#13063357	—	2	2	no	—
17	#13063355	C₂²	1	2	yes	—
18	#13063353	—	2	2	no	—
19	#13063353	C₂²	1	2	yes	—
20	#13063350	C₂²	1	2	yes	complemented
21	#13063349	—	2	2	no	—
22	#13063347	—	2	1	no	—
23	#13063347	—	2	1	no	—
24	#13063347	C₂²	1	1	yes	characteristic
25	#13063346	—	2	2	no	—
26	C₂×Dic₁₇₂	—	2	1	no	—
27	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic, complemented
28	C₂×#10621873	C₂²	1	2	yes	complemented
29	SD₁₆×D₄₃	—	2	2	no	—
30	C₂×D₃₄₄	—	2	1	no	—
31	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
32	#13063331	C₂²	1	2	yes	complemented
33	#13063313	C₂²	1	2	yes	complemented
34	#13063261	C₂²	1	1	yes	characteristic, complemented
35	#13063253	C₂²	1	2	yes	complemented
36	#13063243	C₂²	1	2	yes	complemented
37	#13063241	C₂²	1	2	yes	complemented
38	#135×C₄₃	C₂²	1	1	yes	characteristic
39	#131×C₄₃	C₂²	1	1	yes	characteristic
40	#131×C₄₃	C₂²	1	1	yes	characteristic, complemented
41	#108×C₄₃	C₂²	1	1	yes	characteristic, complemented
42	#104×C₄₃	C₂²	1	1	yes	characteristic
43	#104×C₄₃	C₂²	1	1	yes	characteristic, complemented
44	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
45	#41×C₄₃	C₂³	1	1	yes	characteristic
46	#41×C₄₃	C₂³	1	1	yes	characteristic
47	#10621902	—	2	2	no	—
48	Q₈×D₄₃	—	2	2	no	—
49	#10621900	—	2	2	no	—
50	#10621900	—	4	2	no	—
51	D₄×D₄₃	—	2	2	no	—
52	D₄×D₄₃	—	4	2	no	—
53	#10621898	—	2	2	no	—
54	#10621898	—	2	2	no	—
55	#10621898	C₂³	1	2	yes	—
56	C₂×D₁₇₂	—	2	2	no	—
57	C₂×D₁₇₂	C₂³	1	2	yes	—
58	C₂×Dic₈₆	—	2	2	no	—
59	C₂×Dic₈₆	C₂³	1	2	yes	—
60	Q₁₆×C₄₃	D₄	1	1	yes	characteristic
61	SD₁₆×C₄₃	D₄	1	1	yes	characteristic
62	SD₁₆×C₄₃	D₄	1	1	yes	characteristic
63	D₈×C₄₃	D₄	1	1	yes	characteristic
64	M₁₆×C₄₃	—	2	1	no	—
65	M₁₆×C₄₃	—	2	1	no	—
66	M₁₆×C₄₃	C₂³	1	1	yes	characteristic
67	M₁₆×C₄₃	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₄₃	—	2	1	no	—
69	C₂×C₈×C₄₃	—	2	1	no	—
70	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₄²×C₄₃	C₂³	1	1	yes	characteristic
73	#10621885	—	2	2	no	—
74	#10621884	—	2	2	no	—
75	#10621883	—	2	2	no	—
76	#10621882	—	2	2	no	—
77	(C₂×D₄₃)⋊C₄	—	4	2	no	—
78	#10621877	C₂³	1	2	yes	—
79	Dic₁₇₂	—	2	2	no	—
80	Dic₁₇₂	—	2	1	no	—
81	Dic₁₇₂	—	2	1	no	—
82	Dic₁₇₂	D₄	1	1	yes	characteristic
83	D₃₄₄	—	2	2	no	—
84	D₃₄₄	—	2	1	no	—
85	D₃₄₄	—	2	1	no	—
86	D₃₄₄	D₄	1	1	yes	characteristic
87	#10621873	—	2	2	no	—
88	#10621873	—	2	2	no	—
89	#10621873	—	2	2	no	—
90	#10621873	—	2	2	no	—
91	#10621873	D₄	1	2	yes	—
92	#10621872	—	2	2	no	—
93	C₈×D₄₃	—	2	2	no	—
94	C₂²×D₄₃	—	2	2	no	—
95	C₂²×D₄₃	—	4	2	no	—
96	Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
97	Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
98	D₄×C₄₃	—	2	1	no	—
99	D₄×C₄₃	—	2	1	no	—
100	D₄×C₄₃	C₂×D₄	1	1	yes	characteristic
101	D₄×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₃	—	2	1	no	—
103	C₂×C₄×C₄₃	—	2	1	no	—
104	C₂×C₄×C₄₃	—	2	1	no	—
105	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₄₃	—	2	2	no	—
107	C₂²⋊D₄₃	—	4	2	no	—
108	C₂²⋊D₄₃	—	4	2	no	—
109	C₂×Dic₄₃	—	2	2	no	—
110	C₂×Dic₄₃	—	4	2	no	—
111	D₁₇₂	—	2	2	no	—
112	D₁₇₂	—	2	2	no	—
113	D₁₇₂	—	4	2	no	—
114	D₁₇₂	C₂×D₄	1	2	yes	—
115	C₄×D₄₃	—	2	2	no	—
116	C₄×D₄₃	—	4	2	no	—
117	C₄×D₄₃	—	4	2	no	—
118	Dic₈₆	—	2	2	no	—
119	Dic₈₆	—	2	2	no	—
120	Dic₈₆	—	4	2	no	—
121	Dic₈₆	C₂×D₄	1	2	yes	—
122	C₈×C₄₃	—	2	1	no	—
123	C₈×C₄₃	—	2	1	no	—
124	C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
125	C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
126	C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
127	C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
128	C₄₃⋊C₈	—	2	2	no	—
129	C₂²×C₄₃	—	2	1	no	—
130	C₂²×C₄₃	—	2	1	no	—
131	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×D₄₃	—	2	2	no	—
133	C₂×D₄₃	—	4	2	no	—
134	C₂×D₄₃	—	8	2	no	—
135	C₂×D₄₃	—	16	2	no	—
136	C₄×C₄₃	—	2	1	no	—
137	C₄×C₄₃	—	2	1	no	—
138	C₄×C₄₃	—	4	1	no	—
139	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
141	Dic₄₃	—	2	2	no	—
142	Dic₄₃	—	4	2	no	—
143	#3290	—	43	1	no	maximal, Sylow
144	C₂×C₄₃	—	2	1	no	—
145	C₂×C₄₃	—	4	1	no	—
146	C₂×C₄₃	—	4	1	no	—
147	C₂×C₄₃	D₄²	1	1	yes	characteristic
148	D₄₃	—	4	2	no	—
149	D₄₃	—	8	2	no	—
150	#578	—	43	1	no	—
151	#577	—	43	2	no	—
152	#576	—	43	1	no	—
153	#495	—	43	1	no	—
154	#473	—	43	1	no	—
155	#472	—	43	1	no	—
156	#471	—	43	2	no	—
157	#455	—	43	2	no	—
158	#454	—	43	2	no	—
159	#443	—	43	1	no	—
160	#443	D₄₃	1	1	yes	characteristic, complemented
161	C₄₃	#3290	1	1	yes	characteristic, complemented, Sylow
162	2₋¹⁺⁴	—	43	2	no	—
163	H(5,𝔽₂)	—	43	2	no	—
164	#137	—	43	2	no	—
165	#137	—	43	2	no	—
166	#137	—	86	2	no	—
167	#137	—	86	2	no	—
168	AGL(1,ℤ/8)	—	43	2	no	—
169	AGL(1,ℤ/8)	—	43	2	no	—
170	AGL(1,ℤ/8)	—	86	2	no	—
171	AGL(1,ℤ/8)	—	86	2	no	—
172	#135	—	43	1	no	—
173	#135	—	86	1	no	—
174	#135	—	86	1	no	—
175	#135	—	86	2	no	—
176	#135	C₂×D₄₃	1	1	yes	characteristic
177	C₂×Q₁₆	—	43	1	no	—
178	C₂×Q₁₆	—	86	1	no	—
179	C₂×SD₁₆	—	43	2	no	—
180	C₂×SD₁₆	—	86	2	no	—
181	C₂×D₈	—	43	1	no	—
182	C₂×D₈	—	86	1	no	—
183	#131	—	43	2	no	—
184	#131	C₂×D₄₃	1	1	yes	characteristic
185	#131	C₂×D₄₃	1	1	yes	characteristic, complemented
186	#124	—	43	2	no	—
187	#108	—	43	1	no	—
188	#108	C₂×D₄₃	1	1	yes	characteristic, complemented
189	#104	—	43	2	no	—
190	#104	C₂×D₄₃	1	1	yes	characteristic
191	#104	C₂×D₄₃	1	1	yes	characteristic, complemented
192	#101	—	43	2	no	—
193	#100	—	43	2	no	—
194	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic, complemented
195	#41	—	43	2	no	—
196	#41	—	86	2	no	—
197	#41	—	86	2	no	—
198	#41	—	86	2	no	—
199	#41	—	86	2	no	—
200	#41	—	172	2	no	—
201	#41	C₂²×D₄₃	1	1	yes	characteristic
202	#41	C₂²×D₄₃	1	1	yes	characteristic
203	C₂×Q₈	—	43	2	no	—
204	C₂×Q₈	—	86	2	no	—
205	C₂×Q₈	—	86	2	no	—
206	C₂×D₄	—	43	2	no	—
207	C₂×D₄	—	86	2	no	—
208	C₂×D₄	—	86	2	no	—
209	C₂×D₄	—	172	2	no	—
210	Q₁₆	—	43	1	no	—
211	Q₁₆	—	86	2	no	—
212	Q₁₆	—	86	2	no	—
213	Q₁₆	—	86	1	no	—
214	Q₁₆	—	86	1	no	—
215	Q₁₆	D₁₇₂	1	1	yes	characteristic
216	SD₁₆	—	43	2	no	—
217	SD₁₆	—	86	2	no	—
218	SD₁₆	—	86	2	no	—
219	SD₁₆	—	86	2	no	—
220	SD₁₆	—	86	2	no	—
221	SD₁₆	—	86	2	no	—
222	SD₁₆	—	86	2	no	—
223	SD₁₆	D₁₇₂	1	1	yes	characteristic
224	SD₁₆	D₁₇₂	1	1	yes	characteristic
225	D₈	—	43	1	no	—
226	D₈	—	86	2	no	—
227	D₈	—	86	2	no	—
228	D₈	—	86	1	no	—
229	D₈	—	86	1	no	—
230	D₈	D₁₇₂	1	1	yes	characteristic
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	1	no	—
233	M₁₆	—	43	2	no	—
234	M₁₆	—	86	2	no	—
235	M₁₆	C₂²×D₄₃	1	1	yes	characteristic
236	M₁₆	C₂²×D₄₃	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	86	2	no	—
240	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
242	C₂²⋊C₄	—	172	2	no	—
243	C₄²	C₂²×D₄₃	1	1	yes	characteristic
244	C₂³	—	86	2	no	—
245	C₂³	—	172	2	no	—
246	Q₈	—	43	2	no	—
247	Q₈	—	86	2	no	—
248	Q₈	—	86	2	no	—
249	Q₈	—	172	2	no	—
250	Q₈	C₂×D₁₇₂	1	1	yes	characteristic
251	Q₈	C₂×D₁₇₂	1	1	yes	characteristic
252	D₄	—	2	1	no	—
253	D₄	—	2	1	no	—
254	D₄	—	43	2	no	—
255	D₄	—	86	2	no	—
256	D₄	—	86	2	no	—
257	D₄	—	86	2	no	—
258	D₄	—	172	2	no	—
259	D₄	—	172	2	no	—
260	D₄	—	172	2	no	—
261	D₄	C₂×D₁₇₂	1	1	yes	characteristic
262	D₄	C₂×D₁₇₂	1	1	yes	characteristic
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	86	2	no	—
267	C₂×C₄	—	86	2	no	—
268	C₂×C₄	—	172	2	no	—
269	C₂×C₄	—	172	2	no	—
270	C₂×C₄	—	172	2	no	—
271	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	86	2	no	—
275	C₈	C₂×D₁₇₂	1	1	yes	characteristic
276	C₈	C₂×D₁₇₂	1	1	yes	characteristic
277	C₈	D₄×D₄₃	1	1	yes	characteristic
278	C₈	D₄×D₄₃	1	1	yes	characteristic
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	86	2	no	—
282	C₂²	—	344	2	no	—
283	C₂²	—	172	2	no	—
284	C₂²	—	688	2	no	—
285	C₂²	#13063371	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	86	2	no	—
290	C₄	—	172	2	no	—
291	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
292	C₄	C₂²×D₁₇₂	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	1	no	—
296	C₂	—	344	2	no	—
297	C₂	—	172	2	no	—
298	C₂	D₄×D₁₇₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#443	C₂
socle	C₂×C₄₃	D₄²
hypercenter	#443	D₄₃
hypocenter	C₄₃	#3290
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #14583716	C₂	semidirect
2. #14583706	C₂	semidirect
3. #14583704	C₂	semidirect
4. #14583283	C₂	semidirect
5. #14583174	C₂	semidirect
6. #14583169	C₂	semidirect
7. #14583163	C₂	semidirect
8. #14583078	C₂	semidirect
9. #14583073	C₂	semidirect
10. #14583017	C₂	semidirect
11. C₄₃×#443	C₂	semidirect
12. #13063372	C₂²	semidirect
15. #13063358	C₂²	semidirect
20. #13063350	C₂²	semidirect
27. C₂×Dic₁₇₂	C₂²	semidirect
28. C₂×#10621873	C₂²	semidirect
32. #13063331	C₂²	semidirect
33. #13063313	C₂²	semidirect
34. #13063261	C₂²	semidirect
35. #13063253	C₂²	semidirect
36. #13063243	C₂²	semidirect
37. #13063241	C₂²	semidirect
40. #131×C₄₃	C₂²	semidirect
41. #108×C₄₃	C₂²	semidirect
43. #104×C₄₃	C₂²	semidirect
44. C₄×C₈×C₄₃	C₂²	semidirect
160. #443	D₄₃	semidirect
161. C₄₃	#3290	semidirect
185. #131	C₂×D₄₃	semidirect
188. #108	C₂×D₄₃	semidirect
191. #104	C₂×D₄₃	semidirect
194. C₄×C₈	C₂×D₄₃	semidirect