Subgroups — #19143466

Order 5504.

484 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063345	C₂	1	1	yes	maximal, characteristic
2	#14583382	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583380	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583002	C₂	1	2	yes	maximal, complemented
5	#14582996	C₂	1	2	yes	maximal, complemented
6	#14582991	C₂	1	2	yes	maximal, complemented
7	#14582979	C₂	1	2	yes	maximal, complemented
8	C₂×#13063258	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582901	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582857	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#417	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621898	C₂²	1	1	yes	characteristic
13	#13063345	C₄	1	8	yes	complemented
14	C₂×#10621885	C₂²	1	1	yes	characteristic
15	C₂×#10621883	C₂²	1	1	yes	characteristic
16	C₂×#10621884	C₂²	1	1	yes	characteristic
17	C₂×#10621882	C₂²	1	1	yes	characteristic
18	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
19	Q₈×Dic₄₃	C₂²	1	2	yes	—
20	#13063290	C₂²	1	2	yes	—
21	D₄×Dic₄₃	—	2	2	no	—
22	D₄×Dic₄₃	C₂²	1	2	yes	—
23	#13063287	C₂²	1	2	yes	—
24	C₄×C₂²⋊D₄₃	—	2	2	no	—
25	#13063284	—	2	1	no	—
26	#13063284	C₂²	1	1	yes	characteristic
27	#13063283	—	2	1	no	—
28	C₂×C₄⋊Dic₄₃	—	2	1	no	—
29	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×D₄₃	—	2	1	no	—
31	#13063258	C₂²	1	2	yes	—
32	#13063258	C₂²	1	2	yes	—
33	#13063250	C₂²	1	2	yes	—
34	#13063247	C₂²	1	2	yes	—
35	#13063235	C₂²	1	2	yes	—
36	C₈×Dic₄₃	C₂²	1	2	yes	—
37	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
38	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
40	#103×C₄₃	C₂²	1	2	yes	—
41	#102×C₄₃	C₂²	1	2	yes	—
42	#41×C₄₃	—	2	2	no	—
43	#41×C₄₃	C₂×C₄	1	4	yes	complemented
44	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₄₃	—	2	1	no	—
46	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	—	2	1	no	—
48	C₂²×C₄×C₄₃	—	2	1	no	—
49	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₄₃	—	2	1	no	—
51	C₂²×Dic₄₃	—	2	2	no	—
52	C₂²×Dic₄₃	—	2	1	no	—
53	#10621898	—	2	2	no	—
54	#10621898	C₂×C₄	1	4	yes	complemented
55	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₄₃	—	2	1	no	—
57	C₂×C₄×D₄₃	—	2	2	no	—
58	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
59	M₁₆×C₄₃	—	2	2	no	—
60	C₂×C₈×C₄₃	C₂³	1	2	yes	—
61	C₄⋊C₄×C₄₃	—	2	1	no	—
62	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
63	C₂²⋊Dic₄₃	—	2	2	no	—
64	C₂²⋊Dic₄₃	D₄	1	2	yes	—
65	#10621885	C₂×C₄	1	4	yes	complemented
66	#10621884	C₂×C₄	1	4	yes	complemented
67	#10621883	C₂×C₄	1	4	yes	complemented
68	#10621882	C₂×C₄	1	4	yes	complemented
69	(C₂×D₄₃)⋊C₄	—	2	2	no	—
70	C₄⋊Dic₄₃	—	2	1	no	—
71	C₄⋊Dic₄₃	—	2	2	no	—
72	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
73	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
74	C₄⋊Dic₄₃	C₂³	1	2	yes	—
75	Dic₄₃⋊C₄	—	2	2	no	—
76	C₄×Dic₄₃	—	2	2	no	—
77	C₄×Dic₄₃	—	2	2	no	—
78	C₄×Dic₄₃	C₂³	1	2	yes	—
79	C₂×C₄₃⋊C₈	C₂×C₄	1	4	yes	complemented
80	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₂³×C₄₃	—	2	1	no	—
83	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₄₃	—	2	1	no	—
85	Q₈×C₄₃	—	2	1	no	—
86	Q₈×C₄₃	C₂²×C₄	1	2	yes	—
87	D₄×C₄₃	—	2	4	no	—
88	D₄×C₄₃	—	2	1	no	—
89	D₄×C₄₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₄₃	—	2	1	no	—
91	C₂×C₄×C₄₃	—	2	4	no	—
92	C₂×C₄×C₄₃	—	2	2	no	—
93	C₂×C₄×C₄₃	—	8	1	no	—
94	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₃	#41	1	2	yes	—
97	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₄₃	—	2	4	no	—
99	C₂×Dic₄₃	—	2	1	no	—
100	C₂×Dic₄₃	—	2	2	no	—
101	C₂×Dic₄₃	—	8	2	no	—
102	C₂×Dic₄₃	—	8	1	no	—
103	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
104	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
105	D₁₇₂	—	2	1	no	—
106	D₁₇₂	C₂²×C₄	1	2	yes	—
107	C₄×D₄₃	—	2	4	no	—
108	C₄×D₄₃	—	8	2	no	—
109	Dic₈₆	—	2	1	no	—
110	Dic₈₆	C₂²×C₄	1	2	yes	—
111	C₈×C₄₃	—	2	2	no	—
112	C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
113	C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
114	C₂²×C₄₃	—	2	1	no	—
115	C₂²×C₄₃	—	2	1	no	—
116	C₂²×C₄₃	—	2	2	no	—
117	C₂²×C₄₃	—	4	1	no	—
118	C₂²×C₄₃	—	4	1	no	—
119	C₂²×C₄₃	C₄×D₄	1	2	yes	—
120	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₄₃	—	2	2	no	—
122	C₂×D₄₃	—	4	1	no	—
123	C₂×D₄₃	—	4	1	no	—
124	C₄×C₄₃	—	2	2	no	—
125	C₄×C₄₃	—	4	2	no	—
126	C₄×C₄₃	C₄×D₄	1	2	yes	—
127	C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
129	Dic₄₃	—	2	2	no	—
130	Dic₄₃	—	2	2	no	—
131	Dic₄₃	—	2	2	no	—
132	Dic₄₃	—	4	2	no	—
133	#2951	—	43	1	no	maximal, Sylow
134	C₂×C₄₃	—	2	2	no	—
135	C₂×C₄₃	—	4	2	no	—
136	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂×C₄₃	#576	1	1	yes	characteristic
138	C₂×C₄₃	#578	1	1	yes	characteristic
139	D₄₃	—	4	2	no	—
140	C₂×#135	—	43	1	no	—
141	#520	—	43	1	no	—
142	#520	—	43	1	no	—
143	#440	—	43	2	no	—
144	#440	—	43	2	no	—
145	C₄×Q₁₆	—	43	2	no	—
146	C₄×D₈	—	43	2	no	—
147	C₂×#107	—	43	1	no	—
148	#417	—	43	1	no	—
149	#417	D₄₃	1	1	yes	characteristic, complemented
150	#405	—	43	1	no	—
151	C₄₃	#2951	1	1	yes	characteristic, complemented, Sylow
152	C₂×#41	—	43	1	no	—
153	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
154	#135	—	43	8	no	—
155	C₂×Q₁₆	—	43	1	no	—
156	C₂×SD₁₆	—	43	1	no	—
157	C₂×SD₁₆	—	43	1	no	—
158	C₂×D₈	—	43	1	no	—
159	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
160	C₂²×C₈	—	43	1	no	—
161	C₄×Q₈	—	43	2	no	—
162	C₄×Q₈	—	43	2	no	—
163	C₄×D₄	—	43	2	no	—
164	C₄×D₄	—	43	2	no	—
165	C₄×D₄	—	86	2	no	—
166	C₄×D₄	—	86	2	no	—
167	#117	—	43	1	no	—
168	#117	—	86	1	no	—
169	#117	—	86	1	no	—
170	C₂×C₄⋊C₄	—	43	1	no	—
171	C₂×C₄⋊C₄	—	86	1	no	—
172	C₂×C₄⋊C₄	—	86	1	no	—
173	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
174	#107	—	43	2	no	—
175	#107	—	43	2	no	—
176	#103	—	43	2	no	—
177	#103	C₂×D₄₃	1	2	yes	—
178	#102	—	43	2	no	—
179	#102	C₂×D₄₃	1	2	yes	—
180	#97	—	43	2	no	—
181	C₄×C₈	—	43	2	no	—
182	#41	—	2	2	no	—
183	#41	—	43	4	no	—
184	#41	—	86	2	no	—
185	#41	C₄×D₄₃	1	4	yes	complemented
186	C₂×Q₈	—	43	1	no	—
187	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	43	1	no	—
190	C₂×D₄	—	86	1	no	—
191	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	86	2	no	—
195	C₂²×C₄	—	86	2	no	—
196	C₂²×C₄	—	86	1	no	—
197	C₂²×C₄	—	86	1	no	—
198	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
199	Q₁₆	—	43	4	no	—
200	SD₁₆	—	43	4	no	—
201	SD₁₆	—	43	4	no	—
202	D₈	—	43	4	no	—
203	M₁₆	—	2	2	no	—
204	C₂×C₈	—	43	4	no	—
205	C₂×C₈	—	43	1	no	—
206	C₂×C₈	—	43	1	no	—
207	C₂×C₈	C₂²×D₄₃	1	2	yes	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	43	1	no	—
210	C₄⋊C₄	—	43	1	no	—
211	C₄⋊C₄	—	43	2	no	—
212	C₄⋊C₄	—	86	2	no	—
213	C₄⋊C₄	—	86	2	no	—
214	C₄⋊C₄	—	86	1	no	—
215	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
216	C₂²⋊C₄	—	43	2	no	—
217	C₂²⋊C₄	—	86	2	no	—
218	C₂²⋊C₄	—	86	2	no	—
219	C₄²	—	43	2	no	—
220	C₄²	—	86	2	no	—
221	C₄²	—	86	2	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	86	1	no	—
224	C₂³	D₄×D₄₃	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	—	43	2	no	—
227	Q₈	—	86	1	no	—
228	Q₈	C₂×C₄×D₄₃	1	2	yes	—
229	D₄	—	2	4	no	—
230	D₄	—	2	1	no	—
231	D₄	—	43	2	no	—
232	D₄	—	86	4	no	—
233	D₄	—	86	1	no	—
234	D₄	C₂×C₄×D₄₃	1	2	yes	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	8	1	no	—
239	C₂×C₄	—	43	2	no	—
240	C₂×C₄	—	43	2	no	—
241	C₂×C₄	—	86	1	no	—
242	C₂×C₄	—	86	4	no	—
243	C₂×C₄	—	86	2	no	—
244	C₂×C₄	—	344	2	no	—
245	C₂×C₄	—	344	2	no	—
246	C₂×C₄	—	344	1	no	—
247	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
248	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
249	C₂×C₄	#10621900	1	2	yes	—
250	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	43	2	no	—
253	C₈	—	43	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	86	2	no	—
260	C₂²	—	172	1	no	—
261	C₂²	—	172	1	no	—
262	C₂²	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
263	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	86	2	no	—
267	C₄	—	86	2	no	—
268	C₄	—	86	2	no	—
269	C₄	—	172	2	no	—
270	C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
271	C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
272	C₄	C₂×#10621900	1	1	yes	characteristic
273	C₂	—	2	2	no	—
274	C₂	—	4	2	no	—
275	C₂	—	172	2	no	—
276	C₂	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
277	C₂	#14583703	1	1	yes	characteristic
278	C₂	#14583715	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₄×C₄₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#417	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#417	D₄₃
hypocenter	C₄₃	#2951
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
2. #14583382	C₂	semidirect
3. #14583380	C₂	semidirect
4. #14583002	C₂	semidirect
5. #14582996	C₂	semidirect
6. #14582991	C₂	semidirect
7. #14582979	C₂	semidirect
8. C₂×#13063258	C₂	semidirect
9. #14582901	C₂	semidirect
10. #14582857	C₂	semidirect
11. C₄₃×#417	C₂	semidirect
13. #13063345	C₄	semidirect
43. #41×C₄₃	C₂×C₄	semidirect
54. #10621898	C₂×C₄	semidirect
65. #10621885	C₂×C₄	semidirect
66. #10621884	C₂×C₄	semidirect
67. #10621883	C₂×C₄	semidirect
68. #10621882	C₂×C₄	semidirect
79. C₂×C₄₃⋊C₈	C₂×C₄	semidirect
149. #417	D₄₃	semidirect
151. C₄₃	#2951	semidirect
185. #41	C₄×D₄₃	semidirect