Subgroups — C₄₃×#28392 #24948409

Order 11008.

930 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₃×#479	C₂	1	2	yes	maximal, complemented
2	C₂×C₄₃×#460	C₂	1	2	yes	maximal
3	C₄₃×#2641	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#2327	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#2092	C₂	1	2	yes	maximal, complemented
6	C₄₃×#2013	C₂	1	2	yes	maximal, complemented
7	C₄₃×#1973	C₂	1	2	yes	maximal, complemented
8	C₄₃×#1968	C₂	1	2	yes	maximal, complemented
9	C₄₃×#1768	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₄₃	C₂²	1	2	yes	—
11	C₂²×SD₁₆×C₄₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×#127×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×#125×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₄₃	C₂²	1	2	yes	—
16	C₄₃×#479	—	2	8	no	—
17	C₄₃×#460	—	2	8	no	—
18	C₂×#107×C₄₃	C₂²	1	2	yes	—
19	C₂×#106×C₄₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
21	C₂×#103×C₄₃	C₂²	1	2	yes	—
22	C₂×#103×C₄₃	C₂²	1	2	yes	—
23	C₂×#102×C₄₃	C₂²	1	2	yes	—
24	C₂×#102×C₄₃	C₂²	1	2	yes	—
25	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₄₃×#394	—	2	4	no	—
28	C₄₃×#391	C₂²	1	2	yes	—
29	C₄₃×#390	C₂²	1	2	yes	—
30	C₄₃×#390	C₂²	1	2	yes	—
31	C₄₃×#387	—	2	4	no	—
32	C₄₃×#382	C₂²	1	1	yes	characteristic
33	C₄₃×#376	C₂²	1	1	yes	characteristic
34	C₄₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
36	C₂²×D₄×C₄₃	—	2	2	no	—
37	C₂²×D₄×C₄₃	C₂³	1	2	yes	—
38	C₂×SD₁₆×C₄₃	—	2	4	no	—
39	C₂×SD₁₆×C₄₃	—	2	4	no	—
40	C₂×SD₁₆×C₄₃	—	2	4	no	—
41	C₂×SD₁₆×C₄₃	—	2	4	no	—
42	C₂×SD₁₆×C₄₃	—	4	2	no	—
43	C₂×SD₁₆×C₄₃	—	4	2	no	—
44	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
45	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₄₃	—	2	4	no	—
47	#127×C₄₃	—	2	4	no	—
48	#125×C₄₃	—	2	4	no	—
49	C₄×Q₈×C₄₃	—	2	4	no	—
50	C₄×Q₈×C₄₃	—	2	4	no	—
51	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄×C₄₃	—	2	4	no	—
57	C₂×C₂²⋊C₄×C₄₃	—	2	4	no	—
58	C₂×C₄²×C₄₃	—	2	2	no	—
59	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
62	#107×C₄₃	—	2	4	no	—
63	#106×C₄₃	—	2	4	no	—
64	C₄⋊C₈×C₄₃	D₄	1	8	yes	—
65	#103×C₄₃	—	2	4	no	—
66	#103×C₄₃	—	2	4	no	—
67	#102×C₄₃	—	2	4	no	—
68	#102×C₄₃	—	2	4	no	—
69	#97×C₄₃	—	4	1	no	—
70	C₄×C₈×C₄₃	—	4	1	no	—
71	#95×C₄₃	—	2	2	no	—
72	#95×C₄₃	—	2	4	no	—
73	#95×C₄₃	D₄	1	4	yes	—
74	C₂⁴×C₄₃	—	2	2	no	—
75	C₂×Q₈×C₄₃	—	2	2	no	—
76	C₂×Q₈×C₄₃	—	2	2	no	—
77	C₂×Q₈×C₄₃	—	2	2	no	—
78	C₂×Q₈×C₄₃	—	2	2	no	—
79	C₂×Q₈×C₄₃	—	2	2	no	—
80	C₂×Q₈×C₄₃	#41	1	4	yes	—
81	C₂×D₄×C₄₃	—	2	2	no	—
82	C₂×D₄×C₄₃	—	2	2	no	—
83	C₂×D₄×C₄₃	—	2	2	no	—
84	C₂×D₄×C₄₃	—	2	2	no	—
85	C₂×D₄×C₄₃	—	2	2	no	—
86	C₂×D₄×C₄₃	—	2	2	no	—
87	C₂×D₄×C₄₃	—	4	4	no	—
88	C₂×D₄×C₄₃	—	8	4	no	—
89	C₂²×C₄×C₄₃	—	2	2	no	—
90	C₂²×C₄×C₄₃	—	2	2	no	—
91	C₂²×C₄×C₄₃	—	2	2	no	—
92	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
96	SD₁₆×C₄₃	—	4	8	no	—
97	SD₁₆×C₄₃	—	4	8	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	—	2	2	no	—
100	C₂×C₈×C₄₃	—	2	2	no	—
101	C₂×C₈×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
103	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
104	C₄⋊C₄×C₄₃	—	2	8	no	—
105	C₄⋊C₄×C₄₃	—	2	2	no	—
106	C₄⋊C₄×C₄₃	—	2	2	no	—
107	C₄⋊C₄×C₄₃	—	2	2	no	—
108	C₄⋊C₄×C₄₃	—	2	2	no	—
109	C₄⋊C₄×C₄₃	—	4	4	no	—
110	C₄⋊C₄×C₄₃	—	4	4	no	—
111	C₂²⋊C₄×C₄₃	—	8	4	no	—
112	C₂²⋊C₄×C₄₃	—	8	4	no	—
113	C₄²×C₄₃	—	2	2	no	—
114	C₄²×C₄₃	—	2	1	no	—
115	C₄²×C₄₃	—	2	1	no	—
116	C₄²×C₄₃	—	4	4	no	—
117	C₄²×C₄₃	C₂×D₄	1	4	yes	—
118	C₂³×C₄₃	—	4	2	no	—
119	C₂³×C₄₃	—	4	2	no	—
120	C₂³×C₄₃	—	4	2	no	—
121	C₂³×C₄₃	—	4	4	no	—
122	C₂³×C₄₃	—	4	4	no	—
123	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
124	Q₈×C₄₃	—	2	4	no	—
125	Q₈×C₄₃	—	2	4	no	—
126	Q₈×C₄₃	—	4	2	no	—
127	Q₈×C₄₃	—	4	2	no	—
128	D₄×C₄₃	—	4	2	no	—
129	D₄×C₄₃	—	4	2	no	—
130	D₄×C₄₃	—	4	2	no	—
131	D₄×C₄₃	—	4	2	no	—
132	D₄×C₄₃	—	8	8	no	—
133	C₂×C₄×C₄₃	—	2	4	no	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	2	4	no	—
136	C₂×C₄×C₄₃	—	2	2	no	—
137	C₂×C₄×C₄₃	—	2	4	no	—
138	C₂×C₄×C₄₃	—	2	4	no	—
139	C₂×C₄×C₄₃	—	4	2	no	—
140	C₂×C₄×C₄₃	—	4	2	no	—
141	C₂×C₄×C₄₃	—	4	2	no	—
142	C₂×C₄×C₄₃	—	4	2	no	—
143	C₂×C₄×C₄₃	—	4	4	no	—
144	C₂×C₄×C₄₃	—	4	4	no	—
145	C₂×C₄×C₄₃	#127	1	4	yes	—
146	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
147	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
148	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
150	C₈×C₄₃	—	2	4	no	—
151	C₈×C₄₃	—	2	4	no	—
152	#28392	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂²×C₄₃	—	8	4	no	—
154	C₂²×C₄₃	—	8	2	no	—
155	C₂²×C₄₃	—	8	2	no	—
156	C₂²×C₄₃	—	8	4	no	—
157	C₂²×C₄₃	—	8	2	no	—
158	C₂²×C₄₃	#525	1	1	yes	characteristic
159	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
160	C₂²×C₄₃	#537	1	1	yes	characteristic
161	C₂²×C₄₃	#577	1	4	yes	—
162	C₄×C₄₃	—	2	4	no	—
163	C₄×C₄₃	—	4	2	no	—
164	C₄×C₄₃	—	4	2	no	—
165	C₄×C₄₃	—	4	4	no	—
166	C₄×C₄₃	—	8	2	no	—
167	C₄×C₄₃	—	8	2	no	—
168	C₄×C₄₃	#546	1	2	yes	—
169	C₄×C₄₃	#548	1	2	yes	—
170	C₂×#479	C₂×C₄₃	1	2	yes	complemented
171	C₂×#460	C₂×C₄₃	1	2	yes	—
172	#2641	C₂×C₄₃	1	1	yes	characteristic
173	#2327	C₂×C₄₃	1	1	yes	characteristic, complemented
174	#2092	C₂×C₄₃	1	2	yes	complemented
175	#2013	C₂×C₄₃	1	2	yes	complemented
176	#1973	C₂×C₄₃	1	2	yes	complemented
177	#1968	C₂×C₄₃	1	2	yes	complemented
178	#1768	C₂×C₄₃	1	1	yes	characteristic, complemented
179	C₂×C₄₃	—	16	2	no	—
180	C₂×C₄₃	#2540	1	1	yes	characteristic
181	C₂×C₄₃	#3126	1	2	yes	—
182	C₂×C₄₃	#3154	1	2	yes	—
183	C₂×C₄₃	#3260	1	2	yes	—
184	C₂²×SD₁₆	C₂²×C₄₃	1	2	yes	—
185	C₂²×SD₁₆	C₂²×C₄₃	1	2	yes	—
186	C₂×C₄⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
187	C₂×#127	C₂²×C₄₃	1	1	yes	characteristic
188	C₂×#125	C₂²×C₄₃	1	1	yes	characteristic
189	C₂×C₄×Q₈	C₂²×C₄₃	1	2	yes	—
190	#479	—	2	8	no	—
191	#460	—	2	8	no	—
192	C₂×#107	C₂²×C₄₃	1	2	yes	—
193	C₂×#106	C₂²×C₄₃	1	2	yes	—
194	C₂×C₄⋊C₈	C₂²×C₄₃	1	2	yes	—
195	C₂×#103	C₂²×C₄₃	1	2	yes	—
196	C₂×#103	C₂²×C₄₃	1	2	yes	—
197	C₂×#102	C₂²×C₄₃	1	2	yes	—
198	C₂×#102	C₂²×C₄₃	1	2	yes	—
199	C₂×#97	C₂²×C₄₃	1	1	yes	characteristic
200	C₂×C₄×C₈	C₂²×C₄₃	1	1	yes	characteristic
201	#394	—	2	4	no	—
202	#391	C₂²×C₄₃	1	2	yes	—
203	#390	C₂²×C₄₃	1	2	yes	—
204	#390	C₂²×C₄₃	1	2	yes	—
205	#387	—	2	4	no	—
206	#382	C₂²×C₄₃	1	1	yes	characteristic
207	#376	C₂²×C₄₃	1	1	yes	characteristic
208	#336	C₂²×C₄₃	1	2	yes	—
209	C₄₃	#28392	1	1	yes	characteristic, direct product factor, Sylow
210	C₂²×Q₈	C₂³×C₄₃	1	2	yes	—
211	C₂²×D₄	—	2	2	no	—
212	C₂²×D₄	C₂³×C₄₃	1	2	yes	—
213	C₂×SD₁₆	—	2	4	no	—
214	C₂×SD₁₆	—	2	4	no	—
215	C₂×SD₁₆	—	2	4	no	—
216	C₂×SD₁₆	—	2	4	no	—
217	C₂×SD₁₆	—	4	2	no	—
218	C₂×SD₁₆	—	4	2	no	—
219	C₂²×C₈	C₂³×C₄₃	1	2	yes	—
220	C₂²×C₈	C₂³×C₄₃	1	2	yes	—
221	C₄⋊Q₈	—	2	4	no	—
222	#127	—	2	4	no	—
223	#125	—	2	4	no	—
224	C₄×Q₈	—	2	4	no	—
225	C₄×Q₈	—	2	4	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
230	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
231	C₂×C₂²⋊C₄	—	2	4	no	—
232	C₂×C₂²⋊C₄	—	2	4	no	—
233	C₂×C₄²	—	2	2	no	—
234	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
236	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
237	#107	—	2	4	no	—
238	#106	—	2	4	no	—
239	C₄⋊C₈	D₄×C₄₃	1	8	yes	—
240	#103	—	2	4	no	—
241	#103	—	2	4	no	—
242	#102	—	2	4	no	—
243	#102	—	2	4	no	—
244	#97	—	4	1	no	—
245	C₄×C₈	—	4	1	no	—
246	#95	—	2	2	no	—
247	#95	—	2	4	no	—
248	#95	D₄×C₄₃	1	4	yes	—
249	C₂⁴	—	2	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	#41×C₄₃	1	4	yes	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	4	4	no	—
263	C₂×D₄	—	8	4	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
268	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
269	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
270	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
271	SD₁₆	—	4	8	no	—
272	SD₁₆	—	4	8	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	C₂×D₄×C₄₃	1	4	yes	—
278	C₂×C₈	C₂×D₄×C₄₃	1	4	yes	—
279	C₄⋊C₄	—	2	8	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	4	no	—
285	C₄⋊C₄	—	4	4	no	—
286	C₂²⋊C₄	—	8	4	no	—
287	C₂²⋊C₄	—	8	4	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	4	4	no	—
292	C₄²	C₂×D₄×C₄₃	1	4	yes	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	4	no	—
297	C₂³	—	4	4	no	—
298	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
299	Q₈	—	2	4	no	—
300	Q₈	—	2	4	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	8	8	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	#127×C₄₃	1	4	yes	—
321	C₂×C₄	C₂²×D₄×C₄₃	1	2	yes	—
322	C₂×C₄	C₂×#41×C₄₃	1	2	yes	—
323	C₂×C₄	H(5,𝔽₂)×C₄₃	1	1	yes	characteristic
324	C₂×C₄	2₋¹⁺⁴×C₄₃	1	1	yes	characteristic
325	C₈	—	2	4	no	—
326	C₈	—	2	4	no	—
327	C₂²	—	8	4	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	4	no	—
331	C₂²	—	8	2	no	—
332	C₂²	C₄₃×#525	1	1	yes	characteristic
333	C₂²	C₂×#127×C₄₃	1	1	yes	characteristic
334	C₂²	C₄₃×#537	1	1	yes	characteristic
335	C₂²	C₄₃×#577	1	4	yes	—
336	C₄	—	2	4	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	2	no	—
339	C₄	—	4	4	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	C₄₃×#546	1	2	yes	—
343	C₄	C₄₃×#548	1	2	yes	—
344	C₂	—	16	2	no	—
345	C₂	C₄₃×#2540	1	1	yes	characteristic
346	C₂	C₄₃×#3126	1	2	yes	—
347	C₂	C₄₃×#3154	1	2	yes	—
348	C₂	C₄₃×#3260	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₃
prime radical	C₂×#127×C₄₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₃ ◁ C₄₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₃×#479	C₂	semidirect
4. C₄₃×#2327	C₂	semidirect
5. C₄₃×#2092	C₂	semidirect
6. C₄₃×#2013	C₂	semidirect
7. C₄₃×#1973	C₂	semidirect
8. C₄₃×#1968	C₂	semidirect
9. C₄₃×#1768	C₂	semidirect
152. #28392	C₄₃	direct (N × H)
170. C₂×#479	C₂×C₄₃	semidirect
173. #2327	C₂×C₄₃	semidirect
174. #2092	C₂×C₄₃	semidirect
175. #2013	C₂×C₄₃	semidirect
176. #1973	C₂×C₄₃	semidirect
177. #1968	C₂×C₄₃	semidirect
178. #1768	C₂×C₄₃	semidirect
209. C₄₃	#28392	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₃×#479	2
C₂×C₄₃×#460	2
C₄₃×#2641	2
C₄₃×#2327	2
C₄₃×#2092	2
C₄₃×#2013	2
C₄₃×#1973	2
C₄₃×#1968	2
C₄₃×#1768	2
#28392	43

Subgroups — C43×#28392 #24948409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#28392 #24948409