Subgroups — #19147999

Order 5504.

996 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583530	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583529	C₂	1	4	yes	maximal, complemented
3	#14583527	C₂	1	2	yes	maximal, complemented
4	#14583526	C₂	1	4	yes	maximal, complemented
5	#14583524	C₂	1	4	yes	maximal
6	#14583455	C₂	1	4	yes	maximal, complemented
7	#14583454	C₂	1	4	yes	maximal, complemented
8	#14583451	C₂	1	4	yes	maximal, complemented
9	C₂×#13063283	C₂	1	1	yes	maximal, characteristic
10	#117×D₄₃	C₂	1	2	yes	maximal
11	C₄₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₄₃	C₂²	1	1	yes	characteristic
13	#13063325	C₂²	1	4	yes	complemented
14	#13063324	C₂²	1	4	yes	complemented
15	#13063324	C₂²	1	4	yes	complemented
16	#13063324	C₂²	1	4	yes	complemented
17	#13063323	C₂²	1	4	yes	complemented
18	#13063317	C₂²	1	4	yes	complemented
19	#13063316	C₂²	1	4	yes	complemented
20	#13063316	C₂²	1	4	yes	complemented
21	#13063315	C₂²	1	8	yes	complemented
22	#13063305	C₂²	1	8	yes	—
23	#13063304	C₂²	1	4	yes	—
24	#13063303	C₂²	1	8	yes	—
25	#13063303	C₂²	1	8	yes	—
26	#13063302	C₂²	1	8	yes	—
27	#13063301	C₂²	1	8	yes	—
28	Q₈×Dic₄₃	C₂²	1	4	yes	complemented
29	#13063290	C₂²	1	8	yes	complemented
30	#13063290	C₂²	1	4	yes	complemented
31	#13063284	C₂²	1	2	yes	—
32	#13063283	C₂²	1	2	yes	—
33	#13063283	C₂²	1	2	yes	—
34	#13063283	C₂²	1	4	yes	—
35	#13063283	C₂²	1	4	yes	—
36	#13063282	C₂²	1	4	yes	—
37	#13063281	C₂²	1	4	yes	—
38	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
39	C₄⋊C₄×D₄₃	C₂²	1	2	yes	—
40	C₄⋊C₄×D₄₃	C₂²	1	2	yes	—
41	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
42	C₂²⋊C₄×D₄₃	C₂²	1	4	yes	—
43	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
44	C₄²×D₄₃	C₂²	1	4	yes	—
45	C₄⋊Q₈×C₄₃	C₂²	1	4	yes	complemented
46	#125×C₄₃	C₂²	1	4	yes	complemented
47	C₂²⋊Q₈×C₄₃	C₂²	1	4	yes	—
48	#117×C₄₃	C₂²	1	2	yes	—
49	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
50	C₂³×D₄₃	C₂³	1	1	yes	characteristic
51	C₂×Q₈×C₄₃	C₂³	1	4	yes	—
52	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
54	C₂²×Dic₄₃	C₂³	1	2	yes	—
55	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₄₃	Q₈	1	8	yes	complemented
57	C₂×C₄×D₄₃	C₂³	1	4	yes	—
58	C₂×C₄×D₄₃	C₂³	1	4	yes	—
59	C₂×C₄×D₄₃	C₂³	1	2	yes	—
60	C₂×C₄×D₄₃	C₂³	1	2	yes	—
61	C₂×Dic₈₆	C₂³	1	8	yes	—
62	C₂×Dic₈₆	C₂³	1	4	yes	—
63	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
64	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
65	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
66	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
67	C₄⋊C₄×C₄₃	C₂³	1	8	yes	—
68	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
69	C₄²×C₄₃	C₂³	1	4	yes	—
70	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
71	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
72	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
73	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
74	C₄⋊Dic₄₃	C₂³	1	2	yes	—
75	C₄⋊Dic₄₃	C₂³	1	2	yes	—
76	C₄⋊Dic₄₃	C₂³	1	4	yes	—
77	C₄⋊Dic₄₃	C₂³	1	8	yes	—
78	C₄⋊Dic₄₃	C₂³	1	4	yes	—
79	Dic₄₃⋊C₄	C₂³	1	4	yes	—
80	Dic₄₃⋊C₄	C₂³	1	4	yes	—
81	Dic₄₃⋊C₄	C₂³	1	8	yes	—
82	Dic₄₃⋊C₄	C₂³	1	8	yes	—
83	Dic₄₃⋊C₄	C₂³	1	8	yes	—
84	C₄×Dic₄₃	C₂³	1	4	yes	—
85	C₄×Dic₄₃	C₂³	1	4	yes	—
86	C₄×Dic₄₃	C₂³	1	4	yes	—
87	C₄×Dic₄₃	C₂³	1	4	yes	—
88	C₄×Dic₄₃	C₂³	1	4	yes	—
89	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
90	C₂²×D₄₃	—	4	1	no	—
91	C₂²×D₄₃	—	4	1	no	—
92	C₂²×D₄₃	C₂×Q₈	1	4	yes	—
93	C₂²×D₄₃	C₂⁴	1	2	yes	—
94	Q₈×C₄₃	—	4	4	no	—
95	C₂×C₄×C₄₃	—	4	2	no	—
96	C₂×C₄×C₄₃	C₂×Q₈	1	4	yes	—
97	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
98	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
99	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
100	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
101	C₂×Dic₄₃	—	4	2	no	—
102	C₂×Dic₄₃	C₂×Q₈	1	4	yes	—
103	C₂×Dic₄₃	C₂⁴	1	4	yes	—
104	C₂×Dic₄₃	C₂⁴	1	4	yes	—
105	C₂×Dic₄₃	C₂⁴	1	4	yes	—
106	C₂×Dic₄₃	C₂⁴	1	1	yes	characteristic
107	C₂×Dic₄₃	C₂⁴	1	1	yes	characteristic
108	C₄×D₄₃	—	4	4	no	—
109	C₄×D₄₃	—	4	4	no	—
110	C₄×D₄₃	C₂×Q₈	1	8	yes	—
111	C₄×D₄₃	C₂×Q₈	1	8	yes	—
112	Dic₈₆	—	4	4	no	—
113	Dic₈₆	—	4	8	no	—
114	C₂²×C₄₃	—	2	1	no	—
115	C₂²×C₄₃	—	2	1	no	—
116	C₂²×C₄₃	C₂²×Q₈	1	2	yes	—
117	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
118	C₂×D₄₃	—	2	2	no	—
119	C₂×D₄₃	—	2	2	no	—
120	C₂×D₄₃	—	4	4	no	—
121	C₂×D₄₃	C₂²×Q₈	1	4	yes	—
122	C₄×C₄₃	—	2	4	no	—
123	C₄×C₄₃	—	2	4	no	—
124	C₄×C₄₃	—	2	4	no	—
125	C₄×C₄₃	C₂²×Q₈	1	4	yes	—
126	Dic₄₃	—	2	4	no	—
127	Dic₄₃	—	2	4	no	—
128	Dic₄₃	—	2	4	no	—
129	Dic₄₃	C₂²×Q₈	1	2	yes	—
130	Dic₄₃	C₂²×Q₈	1	2	yes	—
131	#3458	—	43	1	no	maximal, Sylow
132	C₂×C₄₃	—	2	2	no	—
133	C₂×C₄₃	C₂³×Q₈	1	1	yes	characteristic
134	C₂×C₄₃	#585	1	1	yes	characteristic
135	C₂×C₄₃	#585	1	1	yes	characteristic
136	D₄₃	—	2	4	no	—
137	#544	—	43	4	no	—
138	#544	—	43	4	no	—
139	#544	—	43	2	no	—
140	#544	—	43	1	no	—
141	#544	D₄₃	1	1	yes	characteristic, complemented
142	#543	—	43	4	no	—
143	#533	—	43	4	no	—
144	#533	—	43	4	no	—
145	#533	—	43	4	no	—
146	C₂×#117	—	43	2	no	—
147	C₂×#117	—	43	1	no	—
148	C₄₃	#3458	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₄	—	43	1	no	—
150	C₄⋊Q₈	—	43	4	no	—
151	C₄⋊Q₈	—	43	4	no	—
152	C₄⋊Q₈	—	43	4	no	—
153	C₄⋊Q₈	—	43	4	no	—
154	C₄⋊Q₈	—	43	4	no	—
155	C₄⋊Q₈	C₂×D₄₃	1	4	yes	complemented
156	#125	—	43	8	no	—
157	#125	—	43	4	no	—
158	#125	—	43	4	no	—
159	#125	—	43	4	no	—
160	#125	C₂×D₄₃	1	4	yes	complemented
161	C₂²⋊Q₈	—	43	8	no	—
162	C₂²⋊Q₈	—	43	4	no	—
163	C₂²⋊Q₈	—	43	8	no	—
164	C₂²⋊Q₈	—	43	8	no	—
165	C₂²⋊Q₈	—	43	8	no	—
166	C₂²⋊Q₈	—	43	8	no	—
167	C₂²⋊Q₈	C₂×D₄₃	1	4	yes	—
168	C₄×Q₈	—	43	4	no	—
169	C₄×Q₈	—	43	8	no	—
170	C₄×Q₈	—	43	4	no	—
171	#117	—	43	2	no	—
172	#117	—	43	2	no	—
173	#117	—	43	2	no	—
174	#117	—	43	4	no	—
175	#117	—	43	4	no	—
176	#117	—	43	4	no	—
177	#117	—	43	4	no	—
178	#117	C₂×D₄₃	1	2	yes	—
179	C₂×C₄⋊C₄	—	43	1	no	—
180	C₂×C₄⋊C₄	—	43	2	no	—
181	C₂×C₄⋊C₄	—	43	2	no	—
182	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
183	C₂×C₂²⋊C₄	—	43	2	no	—
184	C₂×C₂²⋊C₄	—	43	4	no	—
185	C₂×C₄²	—	43	2	no	—
186	C₂×C₄²	—	43	4	no	—
187	C₂⁴	—	43	1	no	—
188	C₂×Q₈	—	43	4	no	—
189	C₂×Q₈	—	43	8	no	—
190	C₂×Q₈	C₂²×D₄₃	1	4	yes	—
191	C₂²×C₄	—	43	1	no	—
192	C₂²×C₄	—	43	8	no	—
193	C₂²×C₄	—	43	2	no	—
194	C₂²×C₄	—	43	2	no	—
195	C₂²×C₄	—	43	2	no	—
196	C₂²×C₄	—	43	4	no	—
197	C₂²×C₄	—	43	4	no	—
198	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
199	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
200	C₄⋊C₄	—	43	4	no	—
201	C₄⋊C₄	—	43	4	no	—
202	C₄⋊C₄	—	43	4	no	—
203	C₄⋊C₄	—	43	2	no	—
204	C₄⋊C₄	—	43	2	no	—
205	C₄⋊C₄	—	43	8	no	—
206	C₄⋊C₄	—	43	8	no	—
207	C₄⋊C₄	—	43	8	no	—
208	C₄⋊C₄	—	43	4	no	—
209	C₄⋊C₄	—	43	8	no	—
210	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
211	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
212	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
213	C₄⋊C₄	C₂²×D₄₃	1	8	yes	—
214	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
215	C₂²⋊C₄	—	43	4	no	—
216	C₂²⋊C₄	—	43	8	no	—
217	C₂²⋊C₄	—	43	4	no	—
218	C₂²⋊C₄	—	43	4	no	—
219	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
220	C₄²	—	43	4	no	—
221	C₄²	—	43	4	no	—
222	C₄²	—	43	4	no	—
223	C₄²	—	43	4	no	—
224	C₄²	—	43	4	no	—
225	C₄²	C₂²×D₄₃	1	4	yes	—
226	C₂³	—	43	2	no	—
227	C₂³	—	43	4	no	—
228	C₂³	—	172	1	no	—
229	C₂³	—	172	1	no	—
230	C₂³	C₂³×D₄₃	1	1	yes	characteristic
231	Q₈	—	4	4	no	—
232	Q₈	—	172	4	no	—
233	Q₈	—	172	8	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	43	4	no	—
236	C₂×C₄	—	43	1	no	—
237	C₂×C₄	—	43	1	no	—
238	C₂×C₄	—	43	8	no	—
239	C₂×C₄	—	43	8	no	—
240	C₂×C₄	—	43	4	no	—
241	C₂×C₄	—	43	4	no	—
242	C₂×C₄	—	43	4	no	—
243	C₂×C₄	—	172	4	no	—
244	C₂×C₄	—	172	4	no	—
245	C₂×C₄	—	172	2	no	—
246	C₂×C₄	Q₈×D₄₃	1	4	yes	—
247	C₂×C₄	C₂³×D₄₃	1	4	yes	—
248	C₂×C₄	C₂³×D₄₃	1	4	yes	—
249	C₂×C₄	C₂³×D₄₃	1	4	yes	—
250	C₂×C₄	C₂³×D₄₃	1	2	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	43	4	no	—
254	C₂²	—	86	2	no	—
255	C₂²	—	86	2	no	—
256	C₂²	—	172	4	no	—
257	C₂²	C₂×Q₈×D₄₃	1	2	yes	—
258	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
259	C₄	—	2	4	no	—
260	C₄	—	2	4	no	—
261	C₄	—	2	4	no	—
262	C₄	—	43	2	no	—
263	C₄	—	43	2	no	—
264	C₄	—	86	4	no	—
265	C₄	—	86	4	no	—
266	C₄	—	86	4	no	—
267	C₄	C₂×Q₈×D₄₃	1	4	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	86	4	no	—
270	C₂	C₂²×Q₈×D₄₃	1	1	yes	characteristic
271	C₂	#14583734	1	1	yes	characteristic
272	C₂	#14583736	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₄₃×#544	C₂
hypercenter	#544	D₄₃
hypocenter	C₄₃	#3458
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #544

Complements

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

N	H ≅ G/N	type
1. #14583530	C₂	semidirect
2. #14583529	C₂	semidirect
3. #14583527	C₂	semidirect
4. #14583526	C₂	semidirect
6. #14583455	C₂	semidirect
7. #14583454	C₂	semidirect
8. #14583451	C₂	semidirect
11. C₄₃×#544	C₂	semidirect
13. #13063325	C₂²	semidirect
14. #13063324	C₂²	semidirect
15. #13063324	C₂²	semidirect
16. #13063324	C₂²	semidirect
17. #13063323	C₂²	semidirect
18. #13063317	C₂²	semidirect
19. #13063316	C₂²	semidirect
20. #13063316	C₂²	semidirect
21. #13063315	C₂²	semidirect
28. Q₈×Dic₄₃	C₂²	semidirect
29. #13063290	C₂²	semidirect
30. #13063290	C₂²	semidirect
45. C₄⋊Q₈×C₄₃	C₂²	semidirect
46. #125×C₄₃	C₂²	semidirect
56. C₂×C₄×D₄₃	Q₈	semidirect
141. #544	D₄₃	semidirect
148. C₄₃	#3458	semidirect
155. C₄⋊Q₈	C₂×D₄₃	semidirect
160. #125	C₂×D₄₃	semidirect