Subgroups — C₄₃×#3091 #19131908

Order 5504.

420 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₃×#525	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₃×#485	C₂	1	2	yes	maximal, complemented
4	C₄₃×#484	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#483	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₃×#482	C₂	1	2	yes	maximal, complemented
7	C₄₃×#481	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₃×#480	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#427	C₂	1	2	yes	maximal, complemented
10	C₄₃×#416	C₂	1	2	yes	maximal, complemented
11	C₂×C₂²⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
15	#123×C₄₃	—	2	1	no	—
16	C₂²⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
18	#121×C₄₃	C₂²	1	1	yes	characteristic
19	#121×C₄₃	C₂²	1	1	yes	characteristic
20	#120×C₄₃	—	2	1	no	—
21	C₄×D₄×C₄₃	—	2	1	no	—
22	C₄×D₄×C₄₃	—	2	1	no	—
23	#117×C₄₃	—	2	1	no	—
24	#117×C₄₃	—	2	1	no	—
25	#117×C₄₃	C₂²	1	2	yes	—
26	#117×C₄₃	C₂²	1	2	yes	—
27	#117×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
31	C₂×C₄²×C₄₃	—	2	1	no	—
32	#107×C₄₃	C₂²	1	2	yes	complemented
33	#107×C₄₃	C₂²	1	2	yes	complemented
34	#106×C₄₃	C₂²	1	2	yes	complemented
35	#106×C₄₃	C₂²	1	2	yes	complemented
36	#103×C₄₃	C₂²	1	2	yes	complemented
37	#103×C₄₃	C₂²	1	2	yes	complemented
38	#102×C₄₃	C₂²	1	2	yes	complemented
39	#102×C₄₃	C₂²	1	2	yes	complemented
40	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
42	C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
43	C₂⁴×C₄₃	D₄	1	1	yes	characteristic
44	#41×C₄₃	—	2	4	no	—
45	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₄₃	—	2	1	no	—
47	C₂×D₄×C₄₃	—	2	1	no	—
48	C₂×D₄×C₄₃	—	2	1	no	—
49	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	—	2	4	no	—
51	C₂²×C₄×C₄₃	—	2	1	no	—
52	C₂²×C₄×C₄₃	—	2	1	no	—
53	C₂²×C₄×C₄₃	—	2	1	no	—
54	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₄₃	—	2	4	no	—
61	C₂×C₈×C₄₃	C₂³	1	2	yes	—
62	C₂×C₈×C₄₃	C₂³	1	2	yes	—
63	C₄⋊C₄×C₄₃	—	2	1	no	—
64	C₄⋊C₄×C₄₃	—	2	1	no	—
65	C₄⋊C₄×C₄₃	—	2	1	no	—
66	C₄⋊C₄×C₄₃	—	2	1	no	—
67	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
68	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
69	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₄₃	—	2	1	no	—
72	C₂²⋊C₄×C₄₃	—	2	1	no	—
73	C₂²⋊C₄×C₄₃	—	2	2	no	—
74	C₂²⋊C₄×C₄₃	—	2	2	no	—
75	C₂²⋊C₄×C₄₃	—	2	1	no	—
76	C₂²⋊C₄×C₄₃	—	2	1	no	—
77	C₂²⋊C₄×C₄₃	—	2	1	no	—
78	C₄²×C₄₃	—	2	2	no	—
79	C₄²×C₄₃	—	2	2	no	—
80	C₄²×C₄₃	—	2	1	no	—
81	C₂³×C₄₃	—	2	2	no	—
82	C₂³×C₄₃	—	2	2	no	—
83	C₂³×C₄₃	—	2	1	no	—
84	C₂³×C₄₃	—	4	1	no	—
85	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
88	Q₈×C₄₃	—	2	2	no	—
89	D₄×C₄₃	—	2	2	no	—
90	D₄×C₄₃	—	4	2	no	—
91	D₄×C₄₃	—	4	2	no	—
92	D₄×C₄₃	—	4	2	no	—
93	C₂×C₄×C₄₃	—	2	2	no	—
94	C₂×C₄×C₄₃	—	2	2	no	—
95	C₂×C₄×C₄₃	—	2	2	no	—
96	C₂×C₄×C₄₃	—	2	2	no	—
97	C₂×C₄×C₄₃	—	2	2	no	—
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	1	no	—
101	C₂×C₄×C₄₃	—	2	1	no	—
102	C₂×C₄×C₄₃	—	2	1	no	—
103	C₂×C₄×C₄₃	—	4	2	no	—
104	C₂×C₄×C₄₃	—	4	2	no	—
105	C₂×C₄×C₄₃	—	4	2	no	—
106	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#41	1	4	yes	—
110	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
111	C₈×C₄₃	—	2	2	no	—
112	C₈×C₄₃	—	2	2	no	—
113	C₂²×C₄₃	—	2	1	no	—
114	C₂²×C₄₃	—	2	1	no	—
115	C₂²×C₄₃	—	2	1	no	—
116	C₂²×C₄₃	—	2	1	no	—
117	C₂²×C₄₃	—	2	1	no	—
118	C₂²×C₄₃	—	2	1	no	—
119	C₂²×C₄₃	—	2	1	no	—
120	C₂²×C₄₃	—	4	4	no	—
121	C₂²×C₄₃	—	4	1	no	—
122	C₂²×C₄₃	—	4	1	no	—
123	C₂²×C₄₃	—	4	1	no	—
124	C₂²×C₄₃	#123	1	2	yes	—
125	C₂²×C₄₃	#135	1	2	yes	—
126	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
127	C₄×C₄₃	—	2	1	no	—
128	C₄×C₄₃	—	2	1	no	—
129	C₄×C₄₃	—	4	1	no	—
130	C₄×C₄₃	—	4	2	no	—
131	C₄×C₄₃	—	4	1	no	—
132	C₄×C₄₃	—	4	2	no	—
133	C₄×C₄₃	—	4	1	no	—
134	C₄×C₄₃	#123	1	2	yes	—
135	C₄×C₄₃	C₂×#41	1	2	yes	—
136	#3091	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂×C₄₃	—	2	2	no	—
138	C₂×C₄₃	—	2	2	no	—
139	C₂×C₄₃	—	4	1	no	—
140	C₂×C₄₃	—	8	1	no	—
141	C₂×C₄₃	C₂×#123	1	1	yes	characteristic
142	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
143	C₂×C₄₃	#575	1	1	yes	characteristic
144	#525	C₂×C₄₃	1	1	yes	characteristic
145	C₂×#117	C₂×C₄₃	1	1	yes	characteristic, complemented
146	#485	C₂×C₄₃	1	2	yes	complemented
147	#484	C₂×C₄₃	1	1	yes	characteristic, complemented
148	#483	C₂×C₄₃	1	1	yes	characteristic, complemented
149	#482	C₂×C₄₃	1	2	yes	complemented
150	#481	C₂×C₄₃	1	1	yes	characteristic, complemented
151	#480	C₂×C₄₃	1	1	yes	characteristic, complemented
152	#427	C₂×C₄₃	1	2	yes	complemented
153	#416	C₂×C₄₃	1	2	yes	complemented
154	C₂×C₂²⋊C₈	C₂×C₄₃	1	1	yes	characteristic, complemented
155	C₄₃	#3091	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×#41	C₂²×C₄₃	1	1	yes	characteristic
157	C₂³×C₄	C₂²×C₄₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²×C₄₃	1	2	yes	—
159	#123	—	2	1	no	—
160	C₂²⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
161	C₂²⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
162	#121	C₂²×C₄₃	1	1	yes	characteristic
163	#121	C₂²×C₄₃	1	1	yes	characteristic
164	#120	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	#117	—	2	1	no	—
168	#117	—	2	1	no	—
169	#117	C₂²×C₄₃	1	2	yes	—
170	#117	C₂²×C₄₃	1	2	yes	—
171	#117	C₂²×C₄₃	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂²×C₄₃	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂²×C₄₃	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	—	2	1	no	—
175	C₂×C₄²	—	2	1	no	—
176	#107	C₂²×C₄₃	1	2	yes	complemented
177	#107	C₂²×C₄₃	1	2	yes	complemented
178	#106	C₂²×C₄₃	1	2	yes	complemented
179	#106	C₂²×C₄₃	1	2	yes	complemented
180	#103	C₂²×C₄₃	1	2	yes	complemented
181	#103	C₂²×C₄₃	1	2	yes	complemented
182	#102	C₂²×C₄₃	1	2	yes	complemented
183	#102	C₂²×C₄₃	1	2	yes	complemented
184	C₂²⋊C₈	C₂²×C₄₃	1	1	yes	characteristic
185	C₂²⋊C₈	C₂²×C₄₃	1	1	yes	characteristic
186	C₂²⋊C₈	C₂²×C₄₃	1	2	yes	—
187	C₂⁴	D₄×C₄₃	1	1	yes	characteristic
188	#41	—	2	4	no	—
189	C₂×Q₈	C₂³×C₄₃	1	1	yes	characteristic
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	C₂³×C₄₃	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	D₄×C₄₃	1	1	yes	characteristic
199	C₂²×C₄	D₄×C₄₃	1	1	yes	characteristic
200	C₂²×C₄	D₄×C₄₃	1	1	yes	characteristic
201	C₂²×C₄	C₂³×C₄₃	1	1	yes	characteristic
202	C₂²×C₄	C₂³×C₄₃	1	1	yes	characteristic
203	C₂²×C₄	C₂³×C₄₃	1	1	yes	characteristic
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	C₂³×C₄₃	1	2	yes	—
206	C₂×C₈	C₂³×C₄₃	1	2	yes	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
212	C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
213	C₄⋊C₄	C₂³×C₄₃	1	1	yes	characteristic
214	C₄⋊C₄	C₂³×C₄₃	1	1	yes	characteristic
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	1	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	4	1	no	—
229	C₂³	C₂×D₄×C₄₃	1	1	yes	characteristic
230	C₂³	C₂×D₄×C₄₃	1	1	yes	characteristic
231	C₂³	C₂×D₄×C₄₃	1	1	yes	characteristic
232	Q₈	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
251	C₂×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
252	C₂×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
253	C₂×C₄	#41×C₄₃	1	4	yes	—
254	C₂×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	4	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	#123×C₄₃	1	2	yes	—
269	C₂²	#135×C₄₃	1	2	yes	—
270	C₂²	C₂²×D₄×C₄₃	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	2	no	—
277	C₄	—	4	1	no	—
278	C₄	#123×C₄₃	1	2	yes	—
279	C₄	C₂×#41×C₄₃	1	2	yes	—
280	C₂	—	2	2	no	—
281	C₂	—	2	2	no	—
282	C₂	—	4	1	no	—
283	C₂	—	8	1	no	—
284	C₂	C₂×#123×C₄₃	1	1	yes	characteristic
285	C₂	C₂×#135×C₄₃	1	1	yes	characteristic
286	C₂	C₄₃×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₄₃
socle	C₂²×C₄₃	C₂²×D₄
prime radical	C₄₃×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₃ ◁ C₂³×C₄×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#117×C₄₃	C₂	semidirect
3. C₄₃×#485	C₂	semidirect
4. C₄₃×#484	C₂	semidirect
5. C₄₃×#483	C₂	semidirect
6. C₄₃×#482	C₂	semidirect
7. C₄₃×#481	C₂	semidirect
8. C₄₃×#480	C₂	semidirect
9. C₄₃×#427	C₂	semidirect
10. C₄₃×#416	C₂	semidirect
11. C₂×C₂²⋊C₈×C₄₃	C₂	semidirect
32. #107×C₄₃	C₂²	semidirect
33. #107×C₄₃	C₂²	semidirect
34. #106×C₄₃	C₂²	semidirect
35. #106×C₄₃	C₂²	semidirect
36. #103×C₄₃	C₂²	semidirect
37. #103×C₄₃	C₂²	semidirect
38. #102×C₄₃	C₂²	semidirect
39. #102×C₄₃	C₂²	semidirect
136. #3091	C₄₃	direct (N × H)
145. C₂×#117	C₂×C₄₃	semidirect
146. #485	C₂×C₄₃	semidirect
147. #484	C₂×C₄₃	semidirect
148. #483	C₂×C₄₃	semidirect
149. #482	C₂×C₄₃	semidirect
150. #481	C₂×C₄₃	semidirect
151. #480	C₂×C₄₃	semidirect
152. #427	C₂×C₄₃	semidirect
153. #416	C₂×C₄₃	semidirect
154. C₂×C₂²⋊C₈	C₂×C₄₃	semidirect
155. C₄₃	#3091	direct (N × H)
176. #107	C₂²×C₄₃	semidirect
177. #107	C₂²×C₄₃	semidirect
178. #106	C₂²×C₄₃	semidirect
179. #106	C₂²×C₄₃	semidirect
180. #103	C₂²×C₄₃	semidirect
181. #103	C₂²×C₄₃	semidirect
182. #102	C₂²×C₄₃	semidirect
183. #102	C₂²×C₄₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄₃×#525	2
C₂×#117×C₄₃	2
C₄₃×#485	2
C₄₃×#484	2
C₄₃×#483	2
C₄₃×#482	2
C₄₃×#481	2
C₄₃×#480	2
C₄₃×#427	2
C₄₃×#416	2
C₂×C₂²⋊C₈×C₄₃	2
#3091	43

Subgroups — C43×#3091 #19131908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#3091 #19131908