Subgroups — C₄₃×#29724 #24949741

Order 11008.

922 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₃×#451	C₂	1	2	yes	maximal
2	C₄₃×#2621	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#2310	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#2074	C₂	1	2	yes	maximal, complemented
5	C₄₃×#2056	C₂	1	4	yes	maximal, complemented
6	C₄₃×#2016	C₂	1	2	yes	maximal, complemented
7	C₄₃×#1873	C₂	1	2	yes	maximal
8	C₄₃×#1798	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₄₃	C₂²	1	1	yes	characteristic
10	C₂²×Q₁₆×C₄₃	C₂²	1	4	yes	—
11	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Q₈×C₄₃	C₂²	1	2	yes	—
13	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#451	—	2	8	no	—
16	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
17	C₂×#103×C₄₃	C₂²	1	4	yes	—
18	C₂×#103×C₄₃	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
20	C₄₃×#398	—	2	2	no	—
21	C₄₃×#397	—	2	2	no	—
22	C₄₃×#391	—	2	2	no	—
23	C₄₃×#391	C₂²	1	2	yes	—
24	C₄₃×#391	C₂²	1	2	yes	—
25	C₄₃×#391	C₂²	1	2	yes	—
26	C₄₃×#382	C₂²	1	1	yes	characteristic
27	C₄₃×#381	—	2	2	no	—
28	C₄₃×#340	C₂²	1	4	yes	—
29	C₄₃×#336	C₂²	1	2	yes	—
30	C₂²×Q₈×C₄₃	—	2	1	no	—
31	C₂²×Q₈×C₄₃	—	2	2	no	—
32	C₂²×Q₈×C₄₃	—	4	2	no	—
33	C₂²×Q₈×C₄₃	—	4	2	no	—
34	C₂²×Q₈×C₄₃	D₄	1	4	yes	—
35	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
36	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
37	C₂³×C₄×C₄₃	—	2	1	no	—
38	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
39	C₂×Q₁₆×C₄₃	—	2	8	no	—
40	C₂×Q₁₆×C₄₃	—	4	4	no	—
41	C₂×Q₁₆×C₄₃	—	4	4	no	—
42	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
43	C₄⋊Q₈×C₄₃	—	4	2	no	—
44	C₂²⋊Q₈×C₄₃	—	2	4	no	—
45	C₂²⋊Q₈×C₄₃	—	2	4	no	—
46	#117×C₄₃	—	4	2	no	—
47	#117×C₄₃	—	4	2	no	—
48	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
53	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
54	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
55	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
56	C₂×C₄²×C₄₃	—	2	1	no	—
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₄₃	—	4	2	no	—
60	#103×C₄₃	—	2	8	no	—
61	#103×C₄₃	—	4	4	no	—
62	C₂²⋊C₈×C₄₃	—	2	4	no	—
63	#95×C₄₃	—	2	4	no	—
64	#95×C₄₃	—	2	4	no	—
65	#95×C₄₃	—	2	2	no	—
66	#95×C₄₃	—	2	4	no	—
67	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂×Q₈×C₄₃	—	2	4	no	—
69	C₂×Q₈×C₄₃	—	2	2	no	—
70	C₂×Q₈×C₄₃	—	2	2	no	—
71	C₂×Q₈×C₄₃	—	2	2	no	—
72	C₂×Q₈×C₄₃	—	2	4	no	—
73	C₂×Q₈×C₄₃	—	2	4	no	—
74	C₂×Q₈×C₄₃	—	2	4	no	—
75	C₂×Q₈×C₄₃	—	2	4	no	—
76	C₂×Q₈×C₄₃	—	4	8	no	—
77	C₂×Q₈×C₄₃	—	4	2	no	—
78	C₂×Q₈×C₄₃	—	4	2	no	—
79	C₂×Q₈×C₄₃	—	4	8	no	—
80	C₂×Q₈×C₄₃	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₄₃	—	2	2	no	—
82	C₂²×C₄×C₄₃	—	2	2	no	—
83	C₂²×C₄×C₄₃	—	2	4	no	—
84	C₂²×C₄×C₄₃	—	2	2	no	—
85	C₂²×C₄×C₄₃	—	2	2	no	—
86	C₂²×C₄×C₄₃	—	2	2	no	—
87	C₂²×C₄×C₄₃	—	2	2	no	—
88	C₂²×C₄×C₄₃	—	4	2	no	—
89	C₂²×C₄×C₄₃	—	4	2	no	—
90	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₃	#41	1	2	yes	—
95	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
96	Q₁₆×C₄₃	—	4	16	no	—
97	C₂×C₈×C₄₃	—	2	4	no	—
98	C₂×C₈×C₄₃	—	2	4	no	—
99	C₂×C₈×C₄₃	—	2	4	no	—
100	C₄⋊C₄×C₄₃	—	2	2	no	—
101	C₄⋊C₄×C₄₃	—	2	2	no	—
102	C₄⋊C₄×C₄₃	—	2	4	no	—
103	C₄⋊C₄×C₄₃	—	4	2	no	—
104	C₄⋊C₄×C₄₃	—	4	4	no	—
105	C₄⋊C₄×C₄₃	—	4	4	no	—
106	C₂²⋊C₄×C₄₃	—	4	2	no	—
107	C₂²⋊C₄×C₄₃	—	4	2	no	—
108	C₂²⋊C₄×C₄₃	—	4	4	no	—
109	C₄²×C₄₃	—	2	2	no	—
110	C₄²×C₄₃	—	2	2	no	—
111	C₄²×C₄₃	—	4	2	no	—
112	C₂³×C₄₃	—	2	1	no	—
113	C₂³×C₄₃	—	2	1	no	—
114	C₂³×C₄₃	—	2	4	no	—
115	C₂³×C₄₃	#121	1	2	yes	—
116	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
117	Q₈×C₄₃	—	2	8	no	—
118	Q₈×C₄₃	—	4	8	no	—
119	Q₈×C₄₃	—	4	4	no	—
120	Q₈×C₄₃	—	4	4	no	—
121	Q₈×C₄₃	—	4	4	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	2	1	no	—
124	C₂×C₄×C₄₃	—	2	4	no	—
125	C₂×C₄×C₄₃	—	2	4	no	—
126	C₂×C₄×C₄₃	—	2	4	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	2	4	no	—
130	C₂×C₄×C₄₃	—	2	4	no	—
131	C₂×C₄×C₄₃	—	4	8	no	—
132	C₂×C₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	—	4	2	no	—
134	C₂×C₄×C₄₃	—	4	2	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	—	4	2	no	—
137	C₂×C₄×C₄₃	—	4	2	no	—
138	C₂×C₄×C₄₃	—	4	2	no	—
139	C₂×C₄×C₄₃	—	4	4	no	—
140	C₂×C₄×C₄₃	—	4	4	no	—
141	C₂×C₄×C₄₃	#121	1	2	yes	—
142	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
145	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
146	C₈×C₄₃	—	4	4	no	—
147	#29724	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₄₃	—	2	2	no	—
149	C₂²×C₄₃	—	2	4	no	—
150	C₂²×C₄₃	—	4	2	no	—
151	C₂²×C₄₃	—	4	2	no	—
152	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
153	C₂²×C₄₃	#534	1	1	yes	characteristic
154	C₂²×C₄₃	#535	1	1	yes	characteristic
155	C₂²×C₄₃	#578	1	4	yes	—
156	C₄×C₄₃	—	2	2	no	—
157	C₄×C₄₃	—	4	4	no	—
158	C₄×C₄₃	—	4	2	no	—
159	C₄×C₄₃	—	4	2	no	—
160	C₄×C₄₃	—	8	2	no	—
161	C₄×C₄₃	—	8	2	no	—
162	C₄×C₄₃	—	8	2	no	—
163	C₄×C₄₃	#534	1	2	yes	—
164	C₄×C₄₃	#538	1	2	yes	—
165	C₂×#451	C₂×C₄₃	1	2	yes	—
166	#2621	C₂×C₄₃	1	1	yes	characteristic
167	#2310	C₂×C₄₃	1	1	yes	characteristic
168	#2074	C₂×C₄₃	1	2	yes	complemented
169	#2056	C₂×C₄₃	1	4	yes	complemented
170	#2016	C₂×C₄₃	1	2	yes	complemented
171	#1873	C₂×C₄₃	1	2	yes	—
172	#1798	C₂×C₄₃	1	1	yes	characteristic
173	C₂×C₄₃	—	4	2	no	—
174	C₂×C₄₃	#2617	1	1	yes	characteristic
175	C₂×C₄₃	#3044	1	2	yes	—
176	C₂×C₄₃	#3202	1	2	yes	—
177	C₂×C₄₃	#3223	1	2	yes	—
178	C₂³×Q₈	C₂²×C₄₃	1	1	yes	characteristic
179	C₂²×Q₁₆	C₂²×C₄₃	1	4	yes	—
180	C₂×C₄⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
181	C₂×C₂²⋊Q₈	C₂²×C₄₃	1	2	yes	—
182	C₂×#117	C₂²×C₄₃	1	1	yes	characteristic
183	C₂×#117	C₂²×C₄₃	1	1	yes	characteristic
184	#451	—	2	8	no	—
185	C₂×C₄⋊C₈	C₂²×C₄₃	1	2	yes	—
186	C₂×#103	C₂²×C₄₃	1	4	yes	—
187	C₂×#103	C₂²×C₄₃	1	4	yes	—
188	C₂×C₂²⋊C₈	C₂²×C₄₃	1	2	yes	—
189	#398	—	2	2	no	—
190	#397	—	2	2	no	—
191	#391	—	2	2	no	—
192	#391	C₂²×C₄₃	1	2	yes	—
193	#391	C₂²×C₄₃	1	2	yes	—
194	#391	C₂²×C₄₃	1	2	yes	—
195	#382	C₂²×C₄₃	1	1	yes	characteristic
196	#381	—	2	2	no	—
197	#340	C₂²×C₄₃	1	4	yes	—
198	#336	C₂²×C₄₃	1	2	yes	—
199	C₄₃	#29724	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²×Q₈	—	2	1	no	—
201	C₂²×Q₈	—	2	2	no	—
202	C₂²×Q₈	—	4	2	no	—
203	C₂²×Q₈	—	4	2	no	—
204	C₂²×Q₈	D₄×C₄₃	1	4	yes	—
205	C₂²×Q₈	C₂³×C₄₃	1	2	yes	—
206	C₂²×Q₈	C₂³×C₄₃	1	2	yes	—
207	C₂³×C₄	—	2	1	no	—
208	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
209	C₂×Q₁₆	—	2	8	no	—
210	C₂×Q₁₆	—	4	4	no	—
211	C₂×Q₁₆	—	4	4	no	—
212	C₂²×C₈	C₂³×C₄₃	1	4	yes	—
213	C₄⋊Q₈	—	4	2	no	—
214	C₂²⋊Q₈	—	2	4	no	—
215	C₂²⋊Q₈	—	2	4	no	—
216	#117	—	4	2	no	—
217	#117	—	4	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	D₄×C₄₃	1	2	yes	—
221	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
222	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
223	C₂×C₂²⋊C₄	—	2	1	no	—
224	C₂×C₂²⋊C₄	—	2	2	no	—
225	C₂×C₂²⋊C₄	D₄×C₄₃	1	2	yes	—
226	C₂×C₄²	—	2	1	no	—
227	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
228	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
229	C₄⋊C₈	—	4	2	no	—
230	#103	—	2	8	no	—
231	#103	—	4	4	no	—
232	C₂²⋊C₈	—	2	4	no	—
233	#95	—	2	4	no	—
234	#95	—	2	4	no	—
235	#95	—	2	2	no	—
236	#95	—	2	4	no	—
237	C₂⁴	C₂×D₄×C₄₃	1	1	yes	characteristic
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	4	8	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×Q₈	—	4	2	no	—
249	C₂×Q₈	—	4	8	no	—
250	C₂×Q₈	C₂×D₄×C₄₃	1	4	yes	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
261	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
262	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
264	C₂²×C₄	#41×C₄₃	1	2	yes	—
265	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
266	Q₁₆	—	4	16	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	2	4	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	4	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	4	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	4	2	no	—
282	C₂³	—	2	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	4	no	—
285	C₂³	#121×C₄₃	1	2	yes	—
286	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
287	Q₈	—	2	8	no	—
288	Q₈	—	4	8	no	—
289	Q₈	—	4	4	no	—
290	Q₈	—	4	4	no	—
291	Q₈	—	4	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	4	8	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	#121×C₄₃	1	2	yes	—
312	C₂×C₄	C₂²×D₄×C₄₃	1	1	yes	characteristic
313	C₂×C₄	C₂×#41×C₄₃	1	1	yes	characteristic
314	C₂×C₄	H(5,𝔽₂)×C₄₃	1	2	yes	—
315	C₂×C₄	H(5,𝔽₂)×C₄₃	1	2	yes	—
316	C₈	—	4	4	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	C₂×#121×C₄₃	1	1	yes	characteristic
322	C₂²	C₄₃×#534	1	1	yes	characteristic
323	C₂²	C₄₃×#535	1	1	yes	characteristic
324	C₂²	C₄₃×#578	1	4	yes	—
325	C₄	—	2	2	no	—
326	C₄	—	4	4	no	—
327	C₄	—	4	2	no	—
328	C₄	—	4	2	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	C₄₃×#534	1	2	yes	—
333	C₄	C₄₃×#538	1	2	yes	—
334	C₂	—	4	2	no	—
335	C₂	C₄₃×#2617	1	1	yes	characteristic
336	C₂	C₄₃×#3044	1	2	yes	—
337	C₂	C₄₃×#3202	1	2	yes	—
338	C₂	C₄₃×#3223	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 ◁ C₂×#117×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₃×#2074	C₂	semidirect
5. C₄₃×#2056	C₂	semidirect
6. C₄₃×#2016	C₂	semidirect
147. #29724	C₄₃	direct (N × H)
168. #2074	C₂×C₄₃	semidirect
169. #2056	C₂×C₄₃	semidirect
170. #2016	C₂×C₄₃	semidirect
199. C₄₃	#29724	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×#451	2
C₄₃×#2621	2
C₄₃×#2310	2
C₄₃×#2074	2
C₄₃×#2056	2
C₄₃×#2016	2
C₄₃×#1873	2
C₄₃×#1798	2
#29724	43

Subgroups — C43×#29724 #24949741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#29724 #24949741