Subgroups — #19143476

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	#14583379	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583009	C₂	1	2	yes	maximal, complemented
4	#14583004	C₂	1	2	yes	maximal, complemented
5	C₄×#10621884	C₂	1	2	yes	maximal, complemented
6	C₄×#10621883	C₂	1	2	yes	maximal, complemented
7	C₂×#13063256	C₂	1	1	yes	maximal, characteristic, complemented
8	#14582900	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582898	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582855	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#520	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	C₂×#10621877	C₂²	1	1	yes	characteristic
19	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×Dic₈₆	C₂²	1	2	yes	—
21	C₄×D₁₇₂	C₂²	1	2	yes	—
22	(C₂²⋊D₄₃)⋊C₄	—	2	2	no	—
23	#13063283	—	2	1	no	—
24	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×D₄₃	—	2	1	no	—
26	#13063256	C₂²	1	2	yes	—
27	#13063256	C₂²	1	2	yes	—
28	#13063249	C₂²	1	2	yes	—
29	#13063248	C₂²	1	2	yes	—
30	#13063246	C₂²	1	2	yes	—
31	#13063245	C₂²	1	2	yes	—
32	#13063236	C₂²	1	2	yes	—
33	C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
34	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₄₃	C₂²	1	2	yes	—
36	C₄×D₄×C₄₃	—	2	2	no	—
37	C₄×D₄×C₄₃	C₂²	1	2	yes	—
38	#117×C₄₃	—	2	1	no	—
39	#117×C₄₃	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
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	2	no	—
49	C₂²×C₄×C₄₃	—	2	1	no	—
50	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₄₃	—	2	1	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	C₄⋊C₄×C₄₃	—	2	1	no	—
60	C₄⋊C₄×C₄₃	—	2	2	no	—
61	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
64	C₂²⋊C₄×C₄₃	—	2	2	no	—
65	C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
66	C₄²×C₄₃	—	2	2	no	—
67	C₄²×C₄₃	C₂³	1	2	yes	—
68	#10621885	C₂×C₄	1	4	yes	complemented
69	#10621884	C₂×C₄	1	4	yes	complemented
70	#10621883	C₂×C₄	1	4	yes	complemented
71	#10621882	C₂×C₄	1	4	yes	complemented
72	(C₂×D₄₃)⋊C₄	—	2	2	no	—
73	C₄⋊Dic₄₃	—	2	1	no	—
74	C₄⋊Dic₄₃	C₂³	1	2	yes	—
75	Dic₄₃⋊C₄	—	2	2	no	—
76	C₄×Dic₄₃	—	2	2	no	—
77	#10621877	—	2	2	no	—
78	C₂×C₄₃⋊C₈	C₂×C₄	1	4	yes	complemented
79	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
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	2	no	—
94	C₂×C₄×C₄₃	—	8	1	no	—
95	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₃	#41	1	2	yes	—
100	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₄₃	—	2	4	no	—
102	C₂×Dic₄₃	—	2	1	no	—
103	C₂×Dic₄₃	—	2	2	no	—
104	C₂×Dic₄₃	—	8	1	no	—
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	C₄₃⋊C₈	—	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	—	2	2	no	—
126	C₄×C₄₃	—	2	2	no	—
127	C₄×C₄₃	—	4	2	no	—
128	C₄×C₄₃	C₄×D₄	1	2	yes	—
129	C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
130	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
131	Dic₄₃	—	2	2	no	—
132	Dic₄₃	—	4	2	no	—
133	#2952	—	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	#577	1	1	yes	characteristic
138	C₂×C₄₃	#577	1	1	yes	characteristic
139	D₄₃	—	4	2	no	—
140	C₂×#135	—	43	1	no	—
141	#520	—	43	1	no	—
142	#520	D₄₃	1	1	yes	characteristic, complemented
143	#442	—	43	2	no	—
144	#441	—	43	2	no	—
145	C₄×SD₁₆	—	43	2	no	—
146	C₄×SD₁₆	—	43	2	no	—
147	C₂×#106	—	43	1	no	—
148	#417	—	43	1	no	—
149	#417	—	43	1	no	—
150	#405	—	43	1	no	—
151	C₄₃	#2952	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	—	43	1	no	—
160	C₂²×C₈	—	43	1	no	—
161	C₄×Q₈	—	43	2	no	—
162	C₄×Q₈	C₂×D₄₃	1	2	yes	—
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	—	43	2	no	—
165	C₄×D₄	—	86	2	no	—
166	C₄×D₄	C₂×D₄₃	1	2	yes	—
167	#117	—	2	1	no	—
168	#117	—	86	1	no	—
169	#117	C₂×D₄₃	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	43	1	no	—
172	C₂×C₄⋊C₄	—	86	1	no	—
173	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
174	#106	—	43	2	no	—
175	#106	—	43	2	no	—
176	#103	—	43	2	no	—
177	#103	—	43	2	no	—
178	#102	—	43	2	no	—
179	#102	—	43	2	no	—
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	—	2	2	no	—
195	C₂²×C₄	—	86	1	no	—
196	C₂²×C₄	—	86	1	no	—
197	C₂²×C₄	—	86	2	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	—	86	2	no	—
204	C₂×C₈	—	43	4	no	—
205	C₂×C₈	—	43	1	no	—
206	C₂×C₈	—	43	1	no	—
207	C₂×C₈	—	43	2	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	43	2	no	—
211	C₄⋊C₄	—	86	1	no	—
212	C₄⋊C₄	—	86	2	no	—
213	C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
214	C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
215	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	86	2	no	—
218	C₂²⋊C₄	C₂²⋊D₄₃	1	2	yes	—
219	C₄²	—	2	2	no	—
220	C₄²	—	86	2	no	—
221	C₄²	C₂²×D₄₃	1	2	yes	—
222	C₂³	—	2	1	no	—
223	C₂³	—	86	1	no	—
224	C₂³	C₂×C₂²⋊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	2	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	—	86	1	no	—
241	C₂×C₄	—	86	4	no	—
242	C₂×C₄	—	86	2	no	—
243	C₂×C₄	—	344	2	no	—
244	C₂×C₄	—	344	1	no	—
245	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
246	C₂×C₄	#10621898	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
248	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
249	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
251	C₈	—	43	2	no	—
252	C₈	—	43	2	no	—
253	C₈	—	86	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	C₄×C₂²⋊D₄₃	1	2	yes	—
263	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	86	2	no	—
269	C₄	—	172	2	no	—
270	C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
271	C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
272	C₄	C₂×#10621898	1	1	yes	characteristic
273	C₂	—	2	2	no	—
274	C₂	—	4	2	no	—
275	C₂	—	172	2	no	—
276	C₂	C₂×C₄×C₂²⋊D₄₃	1	1	yes	characteristic
277	C₂	#14583707	1	1	yes	characteristic
278	C₂	#14583710	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₃
commutator	C₄×C₄₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#520	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#520	D₄₃
hypocenter	C₄₃	#2952
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
2. #14583379	C₂	semidirect
3. #14583009	C₂	semidirect
4. #14583004	C₂	semidirect
5. C₄×#10621884	C₂	semidirect
6. C₄×#10621883	C₂	semidirect
7. C₂×#13063256	C₂	semidirect
8. #14582900	C₂	semidirect
9. #14582898	C₂	semidirect
10. #14582855	C₂	semidirect
11. C₄₃×#520	C₂	semidirect
13. #13063345	C₄	semidirect
43. #41×C₄₃	C₂×C₄	semidirect
54. #10621898	C₂×C₄	semidirect
68. #10621885	C₂×C₄	semidirect
69. #10621884	C₂×C₄	semidirect
70. #10621883	C₂×C₄	semidirect
71. #10621882	C₂×C₄	semidirect
78. C₂×C₄₃⋊C₈	C₂×C₄	semidirect
142. #520	D₄₃	semidirect
151. C₄₃	#2952	semidirect
185. #41	C₄×D₄₃	semidirect