Subgroups — C₄₃×#20694 #24940711

Order 11008.

1430 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₃×#407	C₂	1	2	yes	maximal
3	C₄₃×#1822	C₂	1	4	yes	maximal, complemented
4	C₄₃×#1821	C₂	1	4	yes	maximal, complemented
5	C₄₃×#1742	C₂	1	4	yes	maximal, complemented
6	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₄₃	C₂²	1	4	yes	—
8	C₂×#117×C₄₃	—	2	4	no	—
9	C₂×#117×C₄₃	—	2	4	no	—
10	C₂×#117×C₄₃	C₄	1	2	yes	—
11	C₂×#117×C₄₃	C₄	1	2	yes	—
12	C₂×#117×C₄₃	C₂²	1	2	yes	—
13	C₂×#117×C₄₃	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
15	C₂²×C₂²⋊C₄×C₄₃	C₄	1	2	yes	—
16	C₂²×C₄²×C₄₃	C₄	1	2	yes	—
17	C₄₃×#407	—	2	4	no	—
18	C₄₃×#407	C₂²	1	4	yes	—
19	C₄₃×#407	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₈×C₄₃	—	2	4	no	—
21	C₄₃×#343	C₂²	1	16	yes	complemented
22	C₂⁵×C₄₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₃	—	2	8	no	—
24	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₃	C₂×C₄	1	4	yes	—
28	C₂³×C₄×C₄₃	D₄	1	4	yes	—
29	C₂³×C₄×C₄₃	D₄	1	4	yes	—
30	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₄₃	—	2	4	no	—
34	C₂×M₁₆×C₄₃	—	2	16	no	—
35	C₂×M₁₆×C₄₃	C₂³	1	8	yes	—
36	C₂²×C₈×C₄₃	—	2	4	no	—
37	#117×C₄₃	—	2	8	no	—
38	#117×C₄₃	—	2	4	no	—
39	#117×C₄₃	—	2	4	no	—
40	#117×C₄₃	—	2	8	no	—
41	#117×C₄₃	—	2	4	no	—
42	#117×C₄₃	—	2	4	no	—
43	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
44	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
45	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄×C₄₃	—	2	8	no	—
48	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
49	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₄₃	—	2	4	no	—
51	C₂×C₄²×C₄₃	—	2	4	no	—
52	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
54	C₂²⋊C₈×C₄₃	—	2	8	no	—
55	C₂²⋊C₈×C₄₃	—	2	8	no	—
56	C₂⁴×C₄₃	—	2	4	no	—
57	C₂⁴×C₄₃	—	4	4	no	—
58	C₂⁴×C₄₃	C₂²⋊C₄	1	4	yes	—
59	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
60	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	—	2	4	no	—
63	C₂²×C₄×C₄₃	—	2	4	no	—
64	C₂²×C₄×C₄₃	—	2	16	no	—
65	C₂²×C₄×C₄₃	—	2	16	no	—
66	C₂²×C₄×C₄₃	—	2	4	no	—
67	C₂²×C₄×C₄₃	—	2	4	no	—
68	C₂²×C₄×C₄₃	—	2	4	no	—
69	C₂²×C₄×C₄₃	—	2	4	no	—
70	C₂²×C₄×C₄₃	—	2	4	no	—
71	C₂²×C₄×C₄₃	—	4	8	no	—
72	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
75	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
80	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
82	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₄₃	#41	1	8	yes	—
86	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
87	M₁₆×C₄₃	—	2	16	no	—
88	M₁₆×C₄₃	—	4	8	no	—
89	C₂×C₈×C₄₃	—	2	8	no	—
90	C₂×C₈×C₄₃	—	4	8	no	—
91	C₄⋊C₄×C₄₃	—	2	8	no	—
92	C₄⋊C₄×C₄₃	—	2	8	no	—
93	C₂²⋊C₄×C₄₃	—	2	8	no	—
94	C₂²⋊C₄×C₄₃	—	2	8	no	—
95	C₄²×C₄₃	—	2	8	no	—
96	C₄²×C₄₃	—	2	4	no	—
97	C₄²×C₄₃	—	2	4	no	—
98	C₂³×C₄₃	—	2	2	no	—
99	C₂³×C₄₃	—	2	4	no	—
100	C₂³×C₄₃	—	2	2	no	—
101	C₂³×C₄₃	—	2	8	no	—
102	C₂³×C₄₃	—	2	8	no	—
103	C₂³×C₄₃	—	4	4	no	—
104	C₂³×C₄₃	—	4	2	no	—
105	C₂³×C₄₃	—	4	2	no	—
106	C₂³×C₄₃	—	4	16	no	—
107	C₂³×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂³×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
109	C₂³×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂³×C₄₃	#117	1	4	yes	—
111	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	8	no	—
115	C₂×C₄×C₄₃	—	2	8	no	—
116	C₂×C₄×C₄₃	—	2	8	no	—
117	C₂×C₄×C₄₃	—	2	4	no	—
118	C₂×C₄×C₄₃	—	2	4	no	—
119	C₂×C₄×C₄₃	—	2	4	no	—
120	C₂×C₄×C₄₃	—	2	4	no	—
121	C₂×C₄×C₄₃	—	2	8	no	—
122	C₂×C₄×C₄₃	—	4	8	no	—
123	C₂×C₄×C₄₃	—	4	8	no	—
124	C₂×C₄×C₄₃	—	4	8	no	—
125	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
126	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
127	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	#117	1	4	yes	—
130	C₂×C₄×C₄₃	#123	1	8	yes	—
131	C₂×C₄×C₄₃	#123	1	8	yes	—
132	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
133	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
134	C₈×C₄₃	—	4	8	no	—
135	#20694	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₄₃	—	2	8	no	—
137	C₂²×C₄₃	—	2	4	no	—
138	C₂²×C₄₃	—	2	2	no	—
139	C₂²×C₄₃	—	2	2	no	—
140	C₂²×C₄₃	—	2	2	no	—
141	C₂²×C₄₃	—	2	4	no	—
142	C₂²×C₄₃	—	2	2	no	—
143	C₂²×C₄₃	—	4	4	no	—
144	C₂²×C₄₃	—	4	16	no	—
145	C₂²×C₄₃	—	4	4	no	—
146	C₂²×C₄₃	#381	1	4	yes	—
147	C₂²×C₄₃	#413	1	4	yes	—
148	C₂²×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄₃	C₂×#117	1	2	yes	—
150	C₄×C₄₃	—	2	2	no	—
151	C₄×C₄₃	—	2	2	no	—
152	C₄×C₄₃	—	4	8	no	—
153	C₄×C₄₃	#381	1	2	yes	—
154	C₄×C₄₃	#381	1	2	yes	—
155	C₄×C₄₃	C₂×#123	1	4	yes	—
156	C₂²×#117	C₂×C₄₃	1	1	yes	characteristic
157	C₂×#407	C₂×C₄₃	1	2	yes	—
158	#1822	C₂×C₄₃	1	4	yes	complemented
159	#1821	C₂×C₄₃	1	4	yes	complemented
160	#1742	C₂×C₄₃	1	4	yes	complemented
161	C₂×C₄₃	—	2	2	no	—
162	C₂×C₄₃	—	2	4	no	—
163	C₂×C₄₃	—	2	4	no	—
164	C₂×C₄₃	—	4	2	no	—
165	C₂×C₄₃	C₂×#381	1	1	yes	characteristic
166	C₂×C₄₃	C₂×#413	1	2	yes	—
167	C₂⁴×C₄	C₂²×C₄₃	1	1	yes	characteristic
168	C₂²×M₁₆	C₂²×C₄₃	1	4	yes	—
169	C₂×#117	—	2	4	no	—
170	C₂×#117	—	2	4	no	—
171	C₂×#117	C₄×C₄₃	1	2	yes	—
172	C₂×#117	C₄×C₄₃	1	2	yes	—
173	C₂×#117	C₂²×C₄₃	1	2	yes	—
174	C₂×#117	C₂²×C₄₃	1	2	yes	—
175	C₂²×C₄⋊C₄	C₂²×C₄₃	1	2	yes	—
176	C₂²×C₂²⋊C₄	C₄×C₄₃	1	2	yes	—
177	C₂²×C₄²	C₄×C₄₃	1	2	yes	—
178	#407	—	2	4	no	—
179	#407	C₂²×C₄₃	1	4	yes	—
180	#407	C₂²×C₄₃	1	4	yes	—
181	C₂×C₂²⋊C₈	—	2	4	no	—
182	#343	C₂²×C₄₃	1	16	yes	complemented
183	C₄₃	#20694	1	1	yes	characteristic, direct product factor, Sylow
184	C₂⁵	C₂×C₄×C₄₃	1	1	yes	characteristic
185	C₂³×C₄	—	2	8	no	—
186	C₂³×C₄	C₂×C₄×C₄₃	1	1	yes	characteristic
187	C₂³×C₄	C₂×C₄×C₄₃	1	1	yes	characteristic
188	C₂³×C₄	C₂×C₄×C₄₃	1	1	yes	characteristic
189	C₂³×C₄	C₂×C₄×C₄₃	1	4	yes	—
190	C₂³×C₄	D₄×C₄₃	1	4	yes	—
191	C₂³×C₄	D₄×C₄₃	1	4	yes	—
192	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
193	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
194	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
195	C₂×M₁₆	—	2	4	no	—
196	C₂×M₁₆	—	2	16	no	—
197	C₂×M₁₆	C₂³×C₄₃	1	8	yes	—
198	C₂²×C₈	—	2	4	no	—
199	#117	—	2	8	no	—
200	#117	—	2	4	no	—
201	#117	—	2	4	no	—
202	#117	—	2	8	no	—
203	#117	—	2	4	no	—
204	#117	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	C₂×C₄×C₄₃	1	4	yes	—
208	C₂×C₄⋊C₄	C₂³×C₄₃	1	4	yes	—
209	C₂×C₂²⋊C₄	—	2	8	no	—
210	C₂×C₂²⋊C₄	C₂×C₄×C₄₃	1	4	yes	—
211	C₂×C₂²⋊C₄	C₂×C₄×C₄₃	1	4	yes	—
212	C₂×C₄²	—	2	4	no	—
213	C₂×C₄²	—	2	4	no	—
214	C₂×C₄²	C₂×C₄×C₄₃	1	4	yes	—
215	C₂×C₄²	C₂×C₄×C₄₃	1	4	yes	—
216	C₂²⋊C₈	—	2	8	no	—
217	C₂²⋊C₈	—	2	8	no	—
218	C₂⁴	—	2	4	no	—
219	C₂⁴	—	4	4	no	—
220	C₂⁴	C₂²⋊C₄×C₄₃	1	4	yes	—
221	C₂⁴	C₂²×C₄×C₄₃	1	1	yes	characteristic
222	C₂⁴	C₂²×C₄×C₄₃	1	1	yes	characteristic
223	C₂⁴	C₂²×C₄×C₄₃	1	1	yes	characteristic
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	16	no	—
228	C₂²×C₄	—	2	16	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	4	8	no	—
234	C₂²×C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
235	C₂²×C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
236	C₂²×C₄	C₂²⋊C₄×C₄₃	1	4	yes	—
237	C₂²×C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
238	C₂²×C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
241	C₂²×C₄	C₂²×C₄×C₄₃	1	8	yes	—
242	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₄₃	1	4	yes	—
244	C₂²×C₄	C₂×D₄×C₄₃	1	4	yes	—
245	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
247	C₂²×C₄	#41×C₄₃	1	8	yes	—
248	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
249	M₁₆	—	2	16	no	—
250	M₁₆	—	4	8	no	—
251	C₂×C₈	—	2	8	no	—
252	C₂×C₈	—	4	8	no	—
253	C₄⋊C₄	—	2	8	no	—
254	C₄⋊C₄	—	2	8	no	—
255	C₂²⋊C₄	—	2	8	no	—
256	C₂²⋊C₄	—	2	8	no	—
257	C₄²	—	2	8	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	4	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	4	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	8	no	—
264	C₂³	—	2	8	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	4	2	no	—
267	C₂³	—	4	2	no	—
268	C₂³	—	4	16	no	—
269	C₂³	C₂×C₂²⋊C₄×C₄₃	1	2	yes	—
270	C₂³	C₂×C₂²⋊C₄×C₄₃	1	2	yes	—
271	C₂³	C₂×C₂²⋊C₄×C₄₃	1	2	yes	—
272	C₂³	#117×C₄₃	1	4	yes	—
273	C₂³	C₂³×C₄×C₄₃	1	1	yes	characteristic
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	4	8	no	—
286	C₂×C₄	—	4	8	no	—
287	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	2	yes	—
288	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	2	yes	—
289	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
290	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
291	C₂×C₄	#117×C₄₃	1	4	yes	—
292	C₂×C₄	#123×C₄₃	1	8	yes	—
293	C₂×C₄	#123×C₄₃	1	8	yes	—
294	C₂×C₄	C₂²×D₄×C₄₃	1	2	yes	—
295	C₂×C₄	C₂×#41×C₄₃	1	4	yes	—
296	C₈	—	4	8	no	—
297	C₂²	—	2	8	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	4	16	no	—
306	C₂²	—	4	4	no	—
307	C₂²	C₄₃×#381	1	4	yes	—
308	C₂²	C₄₃×#413	1	4	yes	—
309	C₂²	C₂²×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
310	C₂²	C₂×#117×C₄₃	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	4	8	no	—
314	C₄	C₄₃×#381	1	2	yes	—
315	C₄	C₄₃×#381	1	2	yes	—
316	C₄	C₂×#123×C₄₃	1	4	yes	—
317	C₂	—	2	2	no	—
318	C₂	—	2	4	no	—
319	C₂	—	2	4	no	—
320	C₂	—	4	2	no	—
321	C₂	C₂×C₄₃×#381	1	1	yes	characteristic
322	C₂	C₂×C₄₃×#413	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₄₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₃
socle	C₂²×C₄₃	C₂²×C₂²⋊C₄
prime radical	C₂⁵×C₄₃	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₃ ◁ C₂⁴×C₄×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₃×#1822	C₂	semidirect
4. C₄₃×#1821	C₂	semidirect
5. C₄₃×#1742	C₂	semidirect
21. C₄₃×#343	C₂²	semidirect
135. #20694	C₄₃	direct (N × H)
158. #1822	C₂×C₄₃	semidirect
159. #1821	C₂×C₄₃	semidirect
160. #1742	C₂×C₄₃	semidirect
182. #343	C₂²×C₄₃	semidirect
183. C₄₃	#20694	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#117×C₄₃	2
C₂×C₄₃×#407	2
C₄₃×#1822	2
C₄₃×#1821	2
C₄₃×#1742	2
#20694	43

Subgroups — C43×#20694 #24940711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#20694 #24940711