Subgroups — C₄₃×#29624 #24949641

Order 11008.

1388 conjugacy classes of subgroups.

The subgroup lattice has 360 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, complemented
2	C₂×C₄₃×#448	C₂	1	2	yes	maximal
3	C₄₃×#2854	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#2043	C₂	1	4	yes	maximal, complemented
5	C₄₃×#2013	C₂	1	4	yes	maximal, complemented
6	C₄₃×#1817	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1787	C₂	1	2	yes	maximal, complemented
8	C₂³×Q₈×C₄₃	C₂²	1	2	yes	—
9	C₂²×SD₁₆×C₄₃	C₂²	1	4	yes	—
10	C₂×C₄⋊Q₈×C₄₃	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₄₃	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₄₃	C₂²	1	4	yes	—
13	C₂×#123×C₄₃	—	2	1	no	—
14	C₂×#121×C₄₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₄₃	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₄₃	—	2	1	no	—
17	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#448	—	2	8	no	—
19	C₂×#103×C₄₃	C₂²	1	4	yes	—
20	C₂×#103×C₄₃	C₂²	1	8	yes	—
21	C₂×C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
22	C₄₃×#390	C₂²	1	2	yes	—
23	C₄₃×#389	C₂²	1	2	yes	—
24	C₄₃×#388	—	2	2	no	—
25	C₄₃×#386	—	2	1	no	—
26	C₄₃×#380	C₂²	1	1	yes	characteristic
27	C₄₃×#336	C₂²	1	4	yes	—
28	C₂²×Q₈×C₄₃	—	2	4	no	—
29	C₂²×Q₈×C₄₃	—	2	2	no	—
30	C₂²×Q₈×C₄₃	—	4	8	no	—
31	C₂²×Q₈×C₄₃	D₄	1	8	yes	—
32	C₂²×Q₈×C₄₃	C₂³	1	4	yes	—
33	C₂²×D₄×C₄₃	—	2	1	no	—
34	C₂²×D₄×C₄₃	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₄₃	—	2	2	no	—
36	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
39	C₂×SD₁₆×C₄₃	—	2	8	no	—
40	C₂×SD₁₆×C₄₃	—	4	8	no	—
41	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
42	C₄⋊Q₈×C₄₃	—	2	16	no	—
43	C₄⋊Q₈×C₄₃	—	2	16	no	—
44	#123×C₄₃	—	4	4	no	—
45	#121×C₄₃	—	2	4	no	—
46	C₂×C₄⋊C₄×C₄₃	—	2	8	no	—
47	C₂×C₄⋊C₄×C₄₃	—	2	8	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	C₂³	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
53	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
54	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
55	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
56	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
57	C₂×C₄²×C₄₃	—	2	4	no	—
58	C₂×C₄²×C₄₃	D₄	1	2	yes	—
59	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
60	#103×C₄₃	—	2	8	no	—
61	#103×C₄₃	—	4	8	no	—
62	C₂²⋊C₈×C₄₃	—	2	4	no	—
63	#95×C₄₃	—	2	1	no	—
64	#95×C₄₃	—	2	2	no	—
65	C₂⁴×C₄₃	—	2	1	no	—
66	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂×Q₈×C₄₃	—	2	8	no	—
68	C₂×Q₈×C₄₃	—	2	4	no	—
69	C₂×Q₈×C₄₃	—	2	8	no	—
70	C₂×Q₈×C₄₃	—	2	8	no	—
71	C₂×Q₈×C₄₃	—	4	16	no	—
72	C₂×Q₈×C₄₃	—	4	8	no	—
73	C₂×Q₈×C₄₃	—	4	16	no	—
74	C₂×Q₈×C₄₃	C₂×D₄	1	8	yes	—
75	C₂×D₄×C₄₃	—	2	2	no	—
76	C₂×D₄×C₄₃	—	2	4	no	—
77	C₂×D₄×C₄₃	—	4	2	no	—
78	C₂×D₄×C₄₃	—	4	2	no	—
79	C₂×D₄×C₄₃	—	4	2	no	—
80	C₂²×C₄×C₄₃	—	2	4	no	—
81	C₂²×C₄×C₄₃	—	2	4	no	—
82	C₂²×C₄×C₄₃	—	2	2	no	—
83	C₂²×C₄×C₄₃	—	2	2	no	—
84	C₂²×C₄×C₄₃	—	2	8	no	—
85	C₂²×C₄×C₄₃	—	2	2	no	—
86	C₂²×C₄×C₄₃	—	2	4	no	—
87	C₂²×C₄×C₄₃	—	2	2	no	—
88	C₂²×C₄×C₄₃	—	2	1	no	—
89	C₂²×C₄×C₄₃	—	4	8	no	—
90	C₂²×C₄×C₄₃	SD₁₆	1	4	yes	complemented
91	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
92	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
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	16	no	—
97	C₂×C₈×C₄₃	—	2	4	no	—
98	C₂×C₈×C₄₃	—	2	8	no	—
99	C₄⋊C₄×C₄₃	—	2	16	no	—
100	C₄⋊C₄×C₄₃	—	2	16	no	—
101	C₄⋊C₄×C₄₃	—	2	2	no	—
102	C₄⋊C₄×C₄₃	—	2	4	no	—
103	C₄⋊C₄×C₄₃	—	4	4	no	—
104	C₂²⋊C₄×C₄₃	—	4	2	no	—
105	C₂²⋊C₄×C₄₃	—	4	4	no	—
106	C₂²⋊C₄×C₄₃	—	4	4	no	—
107	C₂²⋊C₄×C₄₃	—	4	2	no	—
108	C₄²×C₄₃	—	2	4	no	—
109	C₄²×C₄₃	—	2	4	no	—
110	C₂³×C₄₃	—	2	2	no	—
111	C₂³×C₄₃	—	2	2	no	—
112	C₂³×C₄₃	—	4	1	no	—
113	C₂³×C₄₃	—	4	2	no	—
114	C₂³×C₄₃	—	4	1	no	—
115	C₂³×C₄₃	—	4	2	no	—
116	C₂³×C₄₃	—	4	1	no	—
117	C₂³×C₄₃	#120	1	2	yes	—
118	C₂³×C₄₃	C₂×SD₁₆	1	2	yes	—
119	C₂³×C₄₃	#137	1	2	yes	—
120	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
121	Q₈×C₄₃	—	2	8	no	—
122	Q₈×C₄₃	—	2	8	no	—
123	Q₈×C₄₃	—	4	16	no	—
124	Q₈×C₄₃	—	4	8	no	—
125	D₄×C₄₃	—	4	4	no	—
126	D₄×C₄₃	—	8	4	no	—
127	C₂×C₄×C₄₃	—	2	8	no	—
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	2	2	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	2	2	no	—
132	C₂×C₄×C₄₃	—	2	2	no	—
133	C₂×C₄×C₄₃	—	2	8	no	—
134	C₂×C₄×C₄₃	—	2	8	no	—
135	C₂×C₄×C₄₃	—	2	8	no	—
136	C₂×C₄×C₄₃	—	4	16	no	—
137	C₂×C₄×C₄₃	—	4	8	no	—
138	C₂×C₄×C₄₃	—	4	1	no	—
139	C₂×C₄×C₄₃	—	4	2	no	—
140	C₂×C₄×C₄₃	—	4	2	no	—
141	C₂×C₄×C₄₃	—	4	1	no	—
142	C₂×C₄×C₄₃	—	4	2	no	—
143	C₂×C₄×C₄₃	—	4	1	no	—
144	C₂×C₄×C₄₃	#120	1	2	yes	—
145	C₂×C₄×C₄₃	C₂×SD₁₆	1	4	yes	—
146	C₂×C₄×C₄₃	#137	1	4	yes	—
147	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
148	C₂×C₄×C₄₃	H(5,𝔽₂)	1	4	yes	—
149	C₈×C₄₃	—	4	4	no	—
150	#29624	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂²×C₄₃	—	2	2	no	—
152	C₂²×C₄₃	—	2	4	no	—
153	C₂²×C₄₃	—	2	4	no	—
154	C₂²×C₄₃	—	2	2	no	—
155	C₂²×C₄₃	—	8	1	no	—
156	C₂²×C₄₃	—	8	2	no	—
157	C₂²×C₄₃	—	8	2	no	—
158	C₂²×C₄₃	—	8	1	no	—
159	C₂²×C₄₃	—	8	1	no	—
160	C₂²×C₄₃	#448	1	4	yes	—
161	C₂²×C₄₃	C₂×#120	1	1	yes	characteristic
162	C₂²×C₄₃	#535	1	2	yes	—
163	C₂²×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
164	C₂²×C₄₃	C₂×#137	1	2	yes	—
165	C₂²×C₄₃	C₂×#137	1	1	yes	characteristic
166	C₄×C₄₃	—	2	2	no	—
167	C₄×C₄₃	—	2	4	no	—
168	C₄×C₄₃	—	2	4	no	—
169	C₄×C₄₃	—	4	8	no	—
170	C₄×C₄₃	—	8	2	no	—
171	C₄×C₄₃	—	8	1	no	—
172	C₄×C₄₃	#534	1	4	yes	—
173	C₂²×C₄⋊Q₈	C₂×C₄₃	1	1	yes	characteristic, complemented
174	C₂×#448	C₂×C₄₃	1	2	yes	—
175	#2854	C₂×C₄₃	1	1	yes	characteristic
176	#2043	C₂×C₄₃	1	4	yes	complemented
177	#2013	C₂×C₄₃	1	4	yes	complemented
178	#1817	C₂×C₄₃	1	1	yes	characteristic, complemented
179	#1787	C₂×C₄₃	1	2	yes	complemented
180	C₂×C₄₃	—	2	4	no	—
181	C₂×C₄₃	—	16	1	no	—
182	C₂×C₄₃	#2613	1	1	yes	characteristic
183	C₂×C₄₃	C₂×#448	1	1	yes	characteristic
184	C₂×C₄₃	#3004	1	1	yes	characteristic
185	C₂×C₄₃	#3206	1	2	yes	—
186	C₂×C₄₃	#3218	1	2	yes	—
187	C₂³×Q₈	C₂²×C₄₃	1	2	yes	—
188	C₂²×SD₁₆	C₂²×C₄₃	1	4	yes	—
189	C₂×C₄⋊Q₈	—	2	8	no	—
190	C₂×C₄⋊Q₈	—	2	2	no	—
191	C₂×C₄⋊Q₈	C₂²×C₄₃	1	4	yes	—
192	C₂×#123	—	2	1	no	—
193	C₂×#121	C₂²×C₄₃	1	1	yes	characteristic
194	C₂²×C₄⋊C₄	—	2	1	no	—
195	C₂²×C₄⋊C₄	—	2	1	no	—
196	C₂²×C₄²	C₂²×C₄₃	1	1	yes	characteristic
197	#448	—	2	8	no	—
198	C₂×#103	C₂²×C₄₃	1	4	yes	—
199	C₂×#103	C₂²×C₄₃	1	8	yes	—
200	C₂×C₂²⋊C₈	C₂²×C₄₃	1	2	yes	—
201	#390	C₂²×C₄₃	1	2	yes	—
202	#389	C₂²×C₄₃	1	2	yes	—
203	#388	—	2	2	no	—
204	#386	—	2	1	no	—
205	#380	C₂²×C₄₃	1	1	yes	characteristic
206	#336	C₂²×C₄₃	1	4	yes	—
207	C₄₃	#29624	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²×Q₈	—	2	4	no	—
209	C₂²×Q₈	—	2	2	no	—
210	C₂²×Q₈	—	4	8	no	—
211	C₂²×Q₈	D₄×C₄₃	1	8	yes	—
212	C₂²×Q₈	C₂³×C₄₃	1	4	yes	—
213	C₂²×D₄	—	2	1	no	—
214	C₂²×D₄	C₂³×C₄₃	1	1	yes	characteristic
215	C₂³×C₄	—	2	2	no	—
216	C₂³×C₄	D₄×C₄₃	1	1	yes	characteristic
217	C₂³×C₄	D₄×C₄₃	1	1	yes	characteristic
218	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
219	C₂×SD₁₆	—	2	8	no	—
220	C₂×SD₁₆	—	4	8	no	—
221	C₂²×C₈	C₂³×C₄₃	1	4	yes	—
222	C₄⋊Q₈	—	2	16	no	—
223	C₄⋊Q₈	—	2	16	no	—
224	#123	—	4	4	no	—
225	#121	—	2	4	no	—
226	C₂×C₄⋊C₄	—	2	8	no	—
227	C₂×C₄⋊C₄	—	2	8	no	—
228	C₂×C₄⋊C₄	—	2	4	no	—
229	C₂×C₄⋊C₄	—	2	4	no	—
230	C₂×C₄⋊C₄	—	2	2	no	—
231	C₂×C₄⋊C₄	C₂³×C₄₃	1	1	yes	characteristic
232	C₂×C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
233	C₂×C₂²⋊C₄	—	2	1	no	—
234	C₂×C₂²⋊C₄	—	2	1	no	—
235	C₂×C₂²⋊C₄	—	2	2	no	—
236	C₂×C₂²⋊C₄	—	2	2	no	—
237	C₂×C₄²	—	2	4	no	—
238	C₂×C₄²	D₄×C₄₃	1	2	yes	—
239	C₂×C₄²	C₂³×C₄₃	1	2	yes	—
240	#103	—	2	8	no	—
241	#103	—	4	8	no	—
242	C₂²⋊C₈	—	2	4	no	—
243	#95	—	2	1	no	—
244	#95	—	2	2	no	—
245	C₂⁴	—	2	1	no	—
246	C₂⁴	C₂×D₄×C₄₃	1	1	yes	characteristic
247	C₂×Q₈	—	2	8	no	—
248	C₂×Q₈	—	2	4	no	—
249	C₂×Q₈	—	2	8	no	—
250	C₂×Q₈	—	2	8	no	—
251	C₂×Q₈	—	4	16	no	—
252	C₂×Q₈	—	4	8	no	—
253	C₂×Q₈	—	4	16	no	—
254	C₂×Q₈	C₂×D₄×C₄₃	1	8	yes	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	4	no	—
264	C₂²×C₄	—	2	4	no	—
265	C₂²×C₄	—	2	8	no	—
266	C₂²×C₄	—	2	4	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	4	8	no	—
270	C₂²×C₄	SD₁₆×C₄₃	1	4	yes	complemented
271	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₄₃	1	4	yes	—
273	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
274	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
275	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
276	SD₁₆	—	4	16	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	2	8	no	—
279	C₄⋊C₄	—	2	16	no	—
280	C₄⋊C₄	—	2	16	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	4	4	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	4	no	—
287	C₂²⋊C₄	—	4	4	no	—
288	C₄²	—	2	4	no	—
289	C₄²	—	2	4	no	—
290	C₂³	—	2	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	1	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	1	no	—
297	C₂³	#120×C₄₃	1	2	yes	—
298	C₂³	C₂×SD₁₆×C₄₃	1	2	yes	—
299	C₂³	#137×C₄₃	1	2	yes	—
300	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
301	Q₈	—	2	8	no	—
302	Q₈	—	2	8	no	—
303	Q₈	—	4	16	no	—
304	Q₈	—	4	8	no	—
305	D₄	—	4	4	no	—
306	D₄	—	8	4	no	—
307	C₂×C₄	—	2	8	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	8	no	—
314	C₂×C₄	—	2	8	no	—
315	C₂×C₄	—	2	8	no	—
316	C₂×C₄	—	4	16	no	—
317	C₂×C₄	—	4	8	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	#120×C₄₃	1	2	yes	—
325	C₂×C₄	C₂×SD₁₆×C₄₃	1	4	yes	—
326	C₂×C₄	#137×C₄₃	1	4	yes	—
327	C₂×C₄	C₂²×D₄×C₄₃	1	2	yes	—
328	C₂×C₄	H(5,𝔽₂)×C₄₃	1	4	yes	—
329	C₈	—	4	4	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	4	no	—
332	C₂²	—	2	4	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	8	1	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	C₄₃×#448	1	4	yes	—
340	C₂²	C₂×#120×C₄₃	1	1	yes	characteristic
341	C₂²	C₄₃×#535	1	2	yes	—
342	C₂²	C₂²×SD₁₆×C₄₃	1	1	yes	characteristic
343	C₂²	C₂×#137×C₄₃	1	2	yes	—
344	C₂²	C₂×#137×C₄₃	1	1	yes	characteristic
345	C₄	—	2	2	no	—
346	C₄	—	2	4	no	—
347	C₄	—	2	4	no	—
348	C₄	—	4	8	no	—
349	C₄	—	8	2	no	—
350	C₄	—	8	1	no	—
351	C₄	C₄₃×#534	1	4	yes	—
352	C₂	—	2	4	no	—
353	C₂	—	16	1	no	—
354	C₂	C₄₃×#2613	1	1	yes	characteristic
355	C₂	C₂×C₄₃×#448	1	1	yes	characteristic
356	C₂	C₄₃×#3004	1	1	yes	characteristic
357	C₂	C₄₃×#3206	1	2	yes	—
358	C₂	C₄₃×#3218	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₃
prime radical	C₂×#121×C₄₃	C₂²
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
1. C₂²×C₄⋊Q₈×C₄₃	C₂	semidirect
4. C₄₃×#2043	C₂	semidirect
5. C₄₃×#2013	C₂	semidirect
6. C₄₃×#1817	C₂	semidirect
7. C₄₃×#1787	C₂	semidirect
90. C₂²×C₄×C₄₃	SD₁₆	semidirect
150. #29624	C₄₃	direct (N × H)
173. C₂²×C₄⋊Q₈	C₂×C₄₃	semidirect
176. #2043	C₂×C₄₃	semidirect
177. #2013	C₂×C₄₃	semidirect
178. #1817	C₂×C₄₃	semidirect
179. #1787	C₂×C₄₃	semidirect
207. C₄₃	#29624	direct (N × H)
270. C₂²×C₄	SD₁₆×C₄₃	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×#448	2
C₄₃×#2854	2
C₄₃×#2043	2
C₄₃×#2013	2
C₄₃×#1817	2
C₄₃×#1787	2
#29624	43

Subgroups — C43×#29624 #24949641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#29624 #24949641