Subgroups — C₄₃×#7575 #24927592

Order 11008.

668 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×#391	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#1797	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#1272	C₂	1	4	yes	maximal, complemented
5	C₂³×Q₈×C₄₃	C₄	1	1	yes	characteristic
6	C₂²×C₄⋊C₄×C₄₃	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₄₃	—	2	1	no	—
9	C₂×C₂²⋊C₈×C₄₃	—	2	1	no	—
10	C₂×C₂²⋊C₈×C₄₃	—	2	1	no	—
11	C₂×C₄×C₈×C₄₃	—	2	1	no	—
12	C₄₃×#391	—	2	1	no	—
13	C₄₃×#391	—	2	1	no	—
14	C₄₃×#391	C₄	1	2	yes	—
15	C₄₃×#391	C₂²	1	2	yes	—
16	C₂×#95×C₄₃	—	2	1	no	—
17	C₂×#95×C₄₃	—	2	1	no	—
18	C₄₃×#336	C₂²	1	2	yes	—
19	C₄₃×#336	C₂²	1	2	yes	—
20	C₂²×Q₈×C₄₃	—	2	1	no	—
21	C₂²×Q₈×C₄₃	—	2	2	no	—
22	C₂²×Q₈×C₄₃	—	4	2	no	—
23	C₂²×Q₈×C₄₃	—	4	2	no	—
24	C₂²×Q₈×C₄₃	C₈	1	4	yes	complemented
25	C₂²×Q₈×C₄₃	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₄₃	—	2	1	no	—
27	C₂³×C₄×C₄₃	—	2	1	no	—
28	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₃	—	2	2	no	—
32	C₂²×C₈×C₄₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₄₃	—	4	2	no	—
35	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₄₃	—	4	2	no	—
37	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₄₃	—	4	2	no	—
42	C₂²⋊C₈×C₄₃	—	2	4	no	—
43	C₂²⋊C₈×C₄₃	—	4	2	no	—
44	C₄×C₈×C₄₃	—	4	2	no	—
45	#95×C₄₃	—	2	2	no	—
46	#95×C₄₃	—	2	2	no	—
47	#95×C₄₃	—	2	2	no	—
48	#95×C₄₃	—	2	2	no	—
49	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×Q₈×C₄₃	—	2	4	no	—
51	C₂×Q₈×C₄₃	—	2	2	no	—
52	C₂×Q₈×C₄₃	—	4	8	no	—
53	C₂×Q₈×C₄₃	—	4	8	no	—
54	C₂×Q₈×C₄₃	—	4	2	no	—
55	C₂×Q₈×C₄₃	—	4	2	no	—
56	C₂×Q₈×C₄₃	—	4	2	no	—
57	C₂×Q₈×C₄₃	—	4	2	no	—
58	C₂×Q₈×C₄₃	C₂×C₈	1	4	yes	—
59	C₂²×C₄×C₄₃	—	2	2	no	—
60	C₂²×C₄×C₄₃	—	2	2	no	—
61	C₂²×C₄×C₄₃	—	2	2	no	—
62	C₂²×C₄×C₄₃	—	2	4	no	—
63	C₂²×C₄×C₄₃	—	2	2	no	—
64	C₂²×C₄×C₄₃	—	2	2	no	—
65	C₂²×C₄×C₄₃	—	2	2	no	—
66	C₂²×C₄×C₄₃	—	4	1	no	—
67	C₂²×C₄×C₄₃	—	4	1	no	—
68	C₂²×C₄×C₄₃	—	4	1	no	—
69	C₂²×C₄×C₄₃	—	4	2	no	—
70	C₂²×C₄×C₄₃	—	4	2	no	—
71	C₂²×C₄×C₄₃	—	4	2	no	—
72	C₂²×C₄×C₄₃	—	4	2	no	—
73	C₂²×C₄×C₄₃	—	4	2	no	—
74	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
78	C₂²×C₄×C₄₃	M₁₆	1	2	yes	—
79	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₄₃	Q₁₆	1	2	yes	—
81	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₃	—	2	4	no	—
85	C₂×C₈×C₄₃	—	4	2	no	—
86	C₂×C₈×C₄₃	—	4	2	no	—
87	C₂×C₈×C₄₃	—	4	2	no	—
88	C₂×C₈×C₄₃	—	4	2	no	—
89	C₂×C₈×C₄₃	—	4	2	no	—
90	C₄⋊C₄×C₄₃	—	4	2	no	—
91	C₄⋊C₄×C₄₃	—	4	2	no	—
92	C₄²×C₄₃	—	2	2	no	—
93	C₄²×C₄₃	—	2	2	no	—
94	C₄²×C₄₃	—	4	1	no	—
95	C₄²×C₄₃	—	4	1	no	—
96	C₂³×C₄₃	—	2	1	no	—
97	C₂³×C₄₃	—	2	1	no	—
98	C₂³×C₄₃	—	2	1	no	—
99	C₂³×C₄₃	—	2	1	no	—
100	C₂³×C₄₃	C₂²⋊C₈	1	2	yes	—
101	C₂³×C₄₃	#103	1	2	yes	—
102	C₂³×C₄₃	#104	1	2	yes	—
103	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	Q₈×C₄₃	—	4	4	no	—
105	Q₈×C₄₃	—	4	4	no	—
106	Q₈×C₄₃	—	4	8	no	—
107	C₂×C₄×C₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	1	no	—
109	C₂×C₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	4	no	—
116	C₂×C₄×C₄₃	—	2	4	no	—
117	C₂×C₄×C₄₃	—	2	4	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	4	2	no	—
122	C₂×C₄×C₄₃	—	4	2	no	—
123	C₂×C₄×C₄₃	—	4	4	no	—
124	C₂×C₄×C₄₃	—	4	8	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	2	no	—
130	C₂×C₄×C₄₃	—	8	4	no	—
131	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
132	C₂×C₄×C₄₃	#99	1	2	yes	—
133	C₂×C₄×C₄₃	#100	1	2	yes	—
134	C₂×C₄×C₄₃	#103	1	2	yes	—
135	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#137	1	2	yes	—
140	C₈×C₄₃	—	4	4	no	—
141	C₈×C₄₃	—	8	2	no	—
142	#7575	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂²×C₄₃	—	2	2	no	—
144	C₂²×C₄₃	—	2	2	no	—
145	C₂²×C₄₃	—	2	2	no	—
146	C₂²×C₄₃	—	2	2	no	—
147	C₂²×C₄₃	—	2	2	no	—
148	C₂²×C₄₃	—	2	2	no	—
149	C₂²×C₄₃	Q₈⋊C₈	1	4	yes	—
150	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
151	C₂²×C₄₃	C₂×#99	1	1	yes	characteristic
152	C₂²×C₄₃	C₂×#100	1	1	yes	characteristic
153	C₂²×C₄₃	C₂×#103	1	1	yes	characteristic
154	C₂²×C₄₃	#419	1	1	yes	characteristic
155	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
156	C₂²×C₄₃	#421	1	1	yes	characteristic
157	C₄×C₄₃	—	2	1	no	—
158	C₄×C₄₃	—	2	1	no	—
159	C₄×C₄₃	—	2	2	no	—
160	C₄×C₄₃	—	2	1	no	—
161	C₄×C₄₃	—	2	2	no	—
162	C₄×C₄₃	—	2	1	no	—
163	C₄×C₄₃	—	4	4	no	—
164	C₄×C₄₃	—	4	1	no	—
165	C₄×C₄₃	—	8	2	no	—
166	C₄×C₄₃	#323	1	4	yes	—
167	C₂×#391	C₂×C₄₃	1	1	yes	characteristic
168	#1797	C₂×C₄₃	1	1	yes	characteristic
169	C₄×C₂²⋊C₈	C₂×C₄₃	1	1	yes	characteristic
170	#1272	C₂×C₄₃	1	4	yes	complemented
171	C₂×C₄₃	—	2	4	no	—
172	C₂×C₄₃	C₂×#323	1	1	yes	characteristic
173	C₂×C₄₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
174	C₂×C₄₃	#1479	1	1	yes	characteristic
175	C₂×C₄₃	#1508	1	1	yes	characteristic
176	C₂×C₄₃	#1520	1	1	yes	characteristic
177	C₂×C₄₃	#1527	1	1	yes	characteristic
178	C₂×C₄₃	#1548	1	1	yes	characteristic
179	C₂³×Q₈	C₄×C₄₃	1	1	yes	characteristic
180	C₂²×C₄⋊C₄	C₄×C₄₃	1	1	yes	characteristic
181	C₂²×C₄²	C₂²×C₄₃	1	1	yes	characteristic
182	C₂×C₄⋊C₈	—	2	1	no	—
183	C₂×C₂²⋊C₈	—	2	1	no	—
184	C₂×C₂²⋊C₈	—	2	1	no	—
185	C₂×C₄×C₈	—	2	1	no	—
186	#391	—	2	1	no	—
187	#391	—	2	1	no	—
188	#391	C₄×C₄₃	1	2	yes	—
189	#391	C₂²×C₄₃	1	2	yes	—
190	C₂×#95	—	2	1	no	—
191	C₂×#95	—	2	1	no	—
192	#336	C₂²×C₄₃	1	2	yes	—
193	#336	C₂²×C₄₃	1	2	yes	—
194	C₄₃	#7575	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²×Q₈	—	2	1	no	—
196	C₂²×Q₈	—	2	2	no	—
197	C₂²×Q₈	—	4	2	no	—
198	C₂²×Q₈	—	4	2	no	—
199	C₂²×Q₈	C₈×C₄₃	1	4	yes	complemented
200	C₂²×Q₈	C₂×C₄×C₄₃	1	2	yes	—
201	C₂³×C₄	—	2	1	no	—
202	C₂³×C₄	—	2	1	no	—
203	C₂³×C₄	C₂×C₄×C₄₃	1	1	yes	characteristic
204	C₂³×C₄	D₄×C₄₃	1	1	yes	characteristic
205	C₂³×C₄	D₄×C₄₃	1	1	yes	characteristic
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	4	2	no	—
210	C₂×C₄⋊C₄	C₂×C₄×C₄₃	1	2	yes	—
211	C₂×C₄²	—	4	2	no	—
212	C₂×C₄²	C₂×C₄×C₄₃	1	1	yes	characteristic
213	C₂×C₄²	D₄×C₄₃	1	1	yes	characteristic
214	C₂×C₄²	D₄×C₄₃	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
216	C₄⋊C₈	—	4	2	no	—
217	C₂²⋊C₈	—	2	4	no	—
218	C₂²⋊C₈	—	4	2	no	—
219	C₄×C₈	—	4	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	C₂⁴	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	8	no	—
228	C₂×Q₈	—	4	8	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂×Q₈	—	4	2	no	—
232	C₂×Q₈	—	4	2	no	—
233	C₂×Q₈	C₂×C₈×C₄₃	1	4	yes	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₈×C₄₃	1	2	yes	—
253	C₂²×C₄	M₁₆×C₄₃	1	2	yes	—
254	C₂²×C₄	SD₁₆×C₄₃	1	2	yes	—
255	C₂²×C₄	Q₁₆×C₄₃	1	2	yes	—
256	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	4	1	no	—
270	C₄²	—	4	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	C₂²⋊C₈×C₄₃	1	2	yes	—
276	C₂³	#103×C₄₃	1	2	yes	—
277	C₂³	#104×C₄₃	1	2	yes	—
278	C₂³	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
279	Q₈	—	4	4	no	—
280	Q₈	—	4	4	no	—
281	Q₈	—	4	8	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	8	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	4	no	—
306	C₂×C₄	C₂²⋊C₈×C₄₃	1	2	yes	—
307	C₂×C₄	#99×C₄₃	1	2	yes	—
308	C₂×C₄	#100×C₄₃	1	2	yes	—
309	C₂×C₄	#103×C₄₃	1	2	yes	—
310	C₂×C₄	C₂²×C₈×C₄₃	1	1	yes	characteristic
311	C₂×C₄	C₂×M₁₆×C₄₃	1	1	yes	characteristic
312	C₂×C₄	C₂×SD₁₆×C₄₃	1	1	yes	characteristic
313	C₂×C₄	C₂×Q₁₆×C₄₃	1	1	yes	characteristic
314	C₂×C₄	#137×C₄₃	1	2	yes	—
315	C₈	—	4	4	no	—
316	C₈	—	8	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	C₄₃×Q₈⋊C₈	1	4	yes	—
324	C₂²	C₂×C₂²⋊C₈×C₄₃	1	1	yes	characteristic
325	C₂²	C₂×#99×C₄₃	1	1	yes	characteristic
326	C₂²	C₂×#100×C₄₃	1	1	yes	characteristic
327	C₂²	C₂×#103×C₄₃	1	1	yes	characteristic
328	C₂²	C₄₃×#419	1	1	yes	characteristic
329	C₂²	C₂×#104×C₄₃	1	1	yes	characteristic
330	C₂²	C₄₃×#421	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	1	no	—
337	C₄	—	4	4	no	—
338	C₄	—	4	1	no	—
339	C₄	—	8	2	no	—
340	C₄	C₄₃×#323	1	4	yes	—
341	C₂	—	2	4	no	—
342	C₂	C₂×C₄₃×#323	1	1	yes	characteristic
343	C₂	C₂×C₄₃×Q₈⋊C₈	1	1	yes	characteristic
344	C₂	C₄₃×#1479	1	1	yes	characteristic
345	C₂	C₄₃×#1508	1	1	yes	characteristic
346	C₂	C₄₃×#1520	1	1	yes	characteristic
347	C₂	C₄₃×#1527	1	1	yes	characteristic
348	C₂	C₄₃×#1548	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₄₃
Frattini, cosocle	C₂×C₄²	C₂³×C₄₃
prime radical	C₂⁴×C₄₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
4. C₄₃×#1272	C₂	semidirect
24. C₂²×Q₈×C₄₃	C₈	semidirect
142. #7575	C₄₃	direct (N × H)
170. #1272	C₂×C₄₃	semidirect
194. C₄₃	#7575	direct (N × H)
199. C₂²×Q₈	C₈×C₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₃×#391	2
C₄₃×#1797	2
C₄×C₂²⋊C₈×C₄₃	2
C₄₃×#1272	2
#7575	43

Subgroups — C43×#7575 #24927592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#7575 #24927592