Subgroups — C₄₃×#25722 #24945739

Order 11008.

1546 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₃×#494	C₂	1	2	yes	maximal, complemented
3	C₂×C₄₃×#451	C₂	1	4	yes	maximal
4	C₄₃×#2857	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#2016	C₂	1	4	yes	maximal, complemented
6	C₄₃×#1956	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂³×Q₈×C₄₃	C₂²	1	2	yes	—
9	C₂²×Q₁₆×C₄₃	C₂²	1	8	yes	—
10	C₂×C₄⋊Q₈×C₄₃	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₄₃	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊Q₈×C₄₃	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₄₃	—	2	2	no	—
17	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#494	—	2	8	no	—
19	C₄₃×#451	—	2	16	no	—
20	C₂×#103×C₄₃	C₂²	1	8	yes	—
21	C₂×C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
23	C₄₃×#391	C₂²	1	2	yes	—
24	C₄₃×#389	—	2	2	no	—
25	C₄₃×#384	C₂²	1	1	yes	characteristic
26	C₄₃×#380	—	2	2	no	—
27	C₄₃×#336	C₂²	1	2	yes	—
28	C₂²×Q₈×C₄₃	—	2	8	no	—
29	C₂²×Q₈×C₄₃	—	2	2	no	—
30	C₂²×Q₈×C₄₃	—	2	8	no	—
31	C₂²×Q₈×C₄₃	—	4	4	no	—
32	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
33	C₂²×Q₈×C₄₃	C₂³	1	4	yes	—
34	C₂³×C₄×C₄₃	—	2	2	no	—
35	C₂³×C₄×C₄₃	D₄	1	2	yes	—
36	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
37	C₂×Q₁₆×C₄₃	—	2	16	no	—
38	C₂×Q₁₆×C₄₃	—	2	16	no	—
39	C₂×Q₁₆×C₄₃	—	4	8	no	—
40	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
41	C₄⋊Q₈×C₄₃	—	2	16	no	—
42	C₄⋊Q₈×C₄₃	—	2	8	no	—
43	C₄⋊Q₈×C₄₃	—	2	4	no	—
44	C₄⋊Q₈×C₄₃	—	2	4	no	—
45	C₄⋊Q₈×C₄₃	D₄	1	8	yes	—
46	C₂²⋊Q₈×C₄₃	—	2	8	no	—
47	C₂×C₄⋊C₄×C₄₃	—	2	16	no	—
48	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
49	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
50	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
54	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
55	C₂×C₄²×C₄₃	—	2	4	no	—
56	C₂×C₄²×C₄₃	D₄	1	2	yes	—
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	#103×C₄₃	—	2	16	no	—
60	C₂²⋊C₈×C₄₃	D₄	1	8	yes	—
61	C₄×C₈×C₄₃	—	2	4	no	—
62	#95×C₄₃	—	2	4	no	—
63	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂×Q₈×C₄₃	—	2	8	no	—
65	C₂×Q₈×C₄₃	—	2	4	no	—
66	C₂×Q₈×C₄₃	—	2	8	no	—
67	C₂×Q₈×C₄₃	—	2	4	no	—
68	C₂×Q₈×C₄₃	—	2	4	no	—
69	C₂×Q₈×C₄₃	—	2	8	no	—
70	C₂×Q₈×C₄₃	—	2	4	no	—
71	C₂×Q₈×C₄₃	—	4	32	no	—
72	C₂×Q₈×C₄₃	—	4	4	no	—
73	C₂×Q₈×C₄₃	—	4	4	no	—
74	C₂×Q₈×C₄₃	C₂×D₄	1	8	yes	—
75	C₂²×C₄×C₄₃	—	2	4	no	—
76	C₂²×C₄×C₄₃	—	2	4	no	—
77	C₂²×C₄×C₄₃	—	2	2	no	—
78	C₂²×C₄×C₄₃	—	2	2	no	—
79	C₂²×C₄×C₄₃	—	2	8	no	—
80	C₂²×C₄×C₄₃	—	2	4	no	—
81	C₂²×C₄×C₄₃	—	2	2	no	—
82	C₂²×C₄×C₄₃	—	2	2	no	—
83	C₂²×C₄×C₄₃	—	4	4	no	—
84	C₂²×C₄×C₄₃	—	4	4	no	—
85	C₂²×C₄×C₄₃	Q₁₆	1	8	yes	complemented
86	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
91	Q₁₆×C₄₃	—	4	32	no	—
92	C₂×C₈×C₄₃	—	2	4	no	—
93	C₂×C₈×C₄₃	—	2	4	no	—
94	C₂×C₈×C₄₃	C₂×D₄	1	8	yes	—
95	C₄⋊C₄×C₄₃	—	2	16	no	—
96	C₄⋊C₄×C₄₃	—	2	16	no	—
97	C₄⋊C₄×C₄₃	—	2	4	no	—
98	C₄⋊C₄×C₄₃	—	4	4	no	—
99	C₄⋊C₄×C₄₃	—	4	4	no	—
100	C₄⋊C₄×C₄₃	—	4	4	no	—
101	C₂²⋊C₄×C₄₃	—	4	4	no	—
102	C₄²×C₄₃	—	2	4	no	—
103	C₄²×C₄₃	—	2	2	no	—
104	C₄²×C₄₃	C₂×D₄	1	4	yes	—
105	C₂³×C₄₃	—	2	1	no	—
106	C₂³×C₄₃	—	2	1	no	—
107	C₂³×C₄₃	—	2	2	no	—
108	C₂³×C₄₃	#127	1	2	yes	—
109	C₂³×C₄₃	C₂×Q₁₆	1	4	yes	—
110	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₄₃	—	2	16	no	—
112	Q₈×C₄₃	—	4	16	no	—
113	Q₈×C₄₃	—	4	8	no	—
114	Q₈×C₄₃	—	4	8	no	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	2	no	—
119	C₂×C₄×C₄₃	—	2	8	no	—
120	C₂×C₄×C₄₃	—	2	16	no	—
121	C₂×C₄×C₄₃	—	2	8	no	—
122	C₂×C₄×C₄₃	—	4	16	no	—
123	C₂×C₄×C₄₃	—	4	4	no	—
124	C₂×C₄×C₄₃	—	4	4	no	—
125	C₂×C₄×C₄₃	—	4	2	no	—
126	C₂×C₄×C₄₃	—	4	2	no	—
127	C₂×C₄×C₄₃	—	4	2	no	—
128	C₂×C₄×C₄₃	—	4	2	no	—
129	C₂×C₄×C₄₃	—	4	4	no	—
130	C₂×C₄×C₄₃	#120	1	4	yes	—
131	C₂×C₄×C₄₃	#127	1	2	yes	—
132	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
133	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
134	C₂×C₄×C₄₃	#137	1	4	yes	—
135	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
136	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
137	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
138	C₈×C₄₃	—	2	8	no	—
139	#25722	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₄₃	—	2	2	no	—
141	C₂²×C₄₃	—	2	2	no	—
142	C₂²×C₄₃	—	2	4	no	—
143	C₂²×C₄₃	—	2	4	no	—
144	C₂²×C₄₃	#494	1	4	yes	complemented
145	C₂²×C₄₃	C₂×#120	1	1	yes	characteristic
146	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
147	C₂²×C₄₃	#535	1	1	yes	characteristic
148	C₂²×C₄₃	C₂²×Q₁₆	1	2	yes	—
149	C₂²×C₄₃	C₂×#137	1	2	yes	—
150	C₄×C₄₃	—	2	2	no	—
151	C₄×C₄₃	—	2	4	no	—
152	C₄×C₄₃	—	4	8	no	—
153	C₄×C₄₃	—	8	2	no	—
154	C₄×C₄₃	—	8	2	no	—
155	C₄×C₄₃	#451	1	8	yes	complemented
156	C₄×C₄₃	D₄²	1	4	yes	—
157	C₂²×C₄⋊Q₈	C₂×C₄₃	1	1	yes	characteristic
158	C₂×#494	C₂×C₄₃	1	2	yes	complemented
159	C₂×#451	C₂×C₄₃	1	4	yes	—
160	#2857	C₂×C₄₃	1	1	yes	characteristic
161	#2016	C₂×C₄₃	1	4	yes	complemented
162	#1956	C₂×C₄₃	1	2	yes	complemented
163	C₄×C₂²⋊C₈	C₂×C₄₃	1	1	yes	characteristic
164	C₂×C₄₃	—	2	4	no	—
165	C₂×C₄₃	#2433	1	1	yes	characteristic
166	C₂×C₄₃	C₂×#451	1	2	yes	—
167	C₂×C₄₃	C₂×#494	1	1	yes	characteristic
168	C₂×C₄₃	#3152	1	1	yes	characteristic
169	C₂×C₄₃	#3209	1	2	yes	—
170	C₂³×Q₈	C₂²×C₄₃	1	2	yes	—
171	C₂²×Q₁₆	C₂²×C₄₃	1	8	yes	—
172	C₂×C₄⋊Q₈	—	2	8	no	—
173	C₂×C₄⋊Q₈	—	2	2	no	—
174	C₂×C₄⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
175	C₂×C₄⋊Q₈	C₂²×C₄₃	1	2	yes	—
176	C₂×C₄⋊Q₈	C₂²×C₄₃	1	2	yes	—
177	C₂×C₂²⋊Q₈	C₂²×C₄₃	1	2	yes	—
178	C₂²×C₄⋊C₄	—	2	2	no	—
179	C₂²×C₄²	C₂²×C₄₃	1	1	yes	characteristic
180	#494	—	2	8	no	—
181	#451	—	2	16	no	—
182	C₂×#103	C₂²×C₄₃	1	8	yes	—
183	C₂×C₂²⋊C₈	C₂²×C₄₃	1	2	yes	—
184	C₂×C₄×C₈	C₂²×C₄₃	1	2	yes	—
185	#391	C₂²×C₄₃	1	2	yes	—
186	#389	—	2	2	no	—
187	#384	C₂²×C₄₃	1	1	yes	characteristic
188	#380	—	2	2	no	—
189	#336	C₂²×C₄₃	1	2	yes	—
190	C₄₃	#25722	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²×Q₈	—	2	8	no	—
192	C₂²×Q₈	—	2	2	no	—
193	C₂²×Q₈	—	2	8	no	—
194	C₂²×Q₈	—	4	4	no	—
195	C₂²×Q₈	C₂³×C₄₃	1	2	yes	—
196	C₂²×Q₈	C₂³×C₄₃	1	4	yes	—
197	C₂³×C₄	—	2	2	no	—
198	C₂³×C₄	D₄×C₄₃	1	2	yes	—
199	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
200	C₂×Q₁₆	—	2	16	no	—
201	C₂×Q₁₆	—	2	16	no	—
202	C₂×Q₁₆	—	4	8	no	—
203	C₂²×C₈	C₂³×C₄₃	1	4	yes	—
204	C₄⋊Q₈	—	2	16	no	—
205	C₄⋊Q₈	—	2	8	no	—
206	C₄⋊Q₈	—	2	4	no	—
207	C₄⋊Q₈	—	2	4	no	—
208	C₄⋊Q₈	D₄×C₄₃	1	8	yes	—
209	C₂²⋊Q₈	—	2	8	no	—
210	C₂×C₄⋊C₄	—	2	16	no	—
211	C₂×C₄⋊C₄	—	2	4	no	—
212	C₂×C₄⋊C₄	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
217	C₂×C₂²⋊C₄	—	2	2	no	—
218	C₂×C₄²	—	2	4	no	—
219	C₂×C₄²	D₄×C₄₃	1	2	yes	—
220	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
222	#103	—	2	16	no	—
223	C₂²⋊C₈	D₄×C₄₃	1	8	yes	—
224	C₄×C₈	—	2	4	no	—
225	#95	—	2	4	no	—
226	C₂⁴	C₂×D₄×C₄₃	1	1	yes	characteristic
227	C₂×Q₈	—	2	8	no	—
228	C₂×Q₈	—	2	4	no	—
229	C₂×Q₈	—	2	8	no	—
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	2	8	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	4	32	no	—
235	C₂×Q₈	—	4	4	no	—
236	C₂×Q₈	—	4	4	no	—
237	C₂×Q₈	C₂×D₄×C₄₃	1	8	yes	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	—	4	4	no	—
248	C₂²×C₄	Q₁₆×C₄₃	1	8	yes	complemented
249	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₄₃	1	4	yes	—
251	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
253	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
254	Q₁₆	—	4	32	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	C₂×D₄×C₄₃	1	8	yes	—
258	C₄⋊C₄	—	2	16	no	—
259	C₄⋊C₄	—	2	16	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	4	4	no	—
263	C₄⋊C₄	—	4	4	no	—
264	C₂²⋊C₄	—	4	4	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	C₂×D₄×C₄₃	1	4	yes	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	2	no	—
271	C₂³	#127×C₄₃	1	2	yes	—
272	C₂³	C₂×Q₁₆×C₄₃	1	4	yes	—
273	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
274	Q₈	—	2	16	no	—
275	Q₈	—	4	16	no	—
276	Q₈	—	4	8	no	—
277	Q₈	—	4	8	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	16	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	4	16	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	#120×C₄₃	1	4	yes	—
294	C₂×C₄	#127×C₄₃	1	2	yes	—
295	C₂×C₄	C₂×Q₁₆×C₄₃	1	4	yes	—
296	C₂×C₄	C₂×Q₁₆×C₄₃	1	4	yes	—
297	C₂×C₄	#137×C₄₃	1	4	yes	—
298	C₂×C₄	C₂²×D₄×C₄₃	1	2	yes	—
299	C₂×C₄	C₂²×D₄×C₄₃	1	2	yes	—
300	C₂×C₄	H(5,𝔽₂)×C₄₃	1	2	yes	—
301	C₈	—	2	8	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	4	no	—
305	C₂²	—	2	4	no	—
306	C₂²	C₄₃×#494	1	4	yes	complemented
307	C₂²	C₂×#120×C₄₃	1	1	yes	characteristic
308	C₂²	C₂×#127×C₄₃	1	1	yes	characteristic
309	C₂²	C₄₃×#535	1	1	yes	characteristic
310	C₂²	C₂²×Q₁₆×C₄₃	1	2	yes	—
311	C₂²	C₂×#137×C₄₃	1	2	yes	—
312	C₄	—	2	2	no	—
313	C₄	—	2	4	no	—
314	C₄	—	4	8	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	C₄₃×#451	1	8	yes	complemented
318	C₄	D₄²×C₄₃	1	4	yes	—
319	C₂	—	2	4	no	—
320	C₂	C₄₃×#2433	1	1	yes	characteristic
321	C₂	C₂×C₄₃×#451	1	2	yes	—
322	C₂	C₂×C₄₃×#494	1	1	yes	characteristic
323	C₂	C₄₃×#3152	1	1	yes	characteristic
324	C₂	C₄₃×#3209	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₃
prime radical	C₂⁴×C₄₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₃ ◁ C₂²×C₄²×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₃×#494	C₂	semidirect
5. C₄₃×#2016	C₂	semidirect
6. C₄₃×#1956	C₂	semidirect
85. C₂²×C₄×C₄₃	Q₁₆	semidirect
139. #25722	C₄₃	direct (N × H)
144. C₂²×C₄₃	#494	semidirect
155. C₄×C₄₃	#451	semidirect
158. C₂×#494	C₂×C₄₃	semidirect
161. #2016	C₂×C₄₃	semidirect
162. #1956	C₂×C₄₃	semidirect
190. C₄₃	#25722	direct (N × H)
248. C₂²×C₄	Q₁₆×C₄₃	semidirect
306. C₂²	C₄₃×#494	semidirect
317. C₄	C₄₃×#451	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₄₃	2
C₂×C₄₃×#494	2
C₂×C₄₃×#451	2
C₄₃×#2857	2
C₄₃×#2016	2
C₄₃×#1956	2
C₄×C₂²⋊C₈×C₄₃	2
#25722	43

Subgroups — C43×#25722 #24945739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#25722 #24945739