Subgroups — C₄₃×#20524 #24940541

Order 11008.

1314 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₃×#407	C₂	1	2	yes	maximal
2	C₂×C₄×C₂²⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#1753	C₂	1	8	yes	maximal, complemented
4	C₄₃×#1749	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₈×C₄₃	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₄₃	C₂²	1	4	yes	—
8	C₂²×C₂²⋊C₄×C₄₃	C₄	1	2	yes	—
9	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
11	C₄₃×#407	C₄	1	16	yes	complemented
12	C₂×C₂²⋊C₈×C₄₃	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₈×C₄₃	C₂²	1	4	yes	—
14	C₂×#97×C₄₃	C₂²	1	4	yes	complemented
15	C₂×C₄×C₈×C₄₃	C₂²	1	4	yes	complemented
16	C₄×C₂²⋊C₄×C₄₃	C₄	1	4	yes	—
17	C₄×C₂²⋊C₄×C₄₃	C₂²	1	4	yes	—
18	C₂×#95×C₄₃	C₄	1	2	yes	—
19	C₄₃×#336	C₂²	1	4	yes	complemented
20	C₄₃×#336	C₂²	1	4	yes	complemented
21	C₂⁵×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₃	—	4	2	no	—
23	C₂³×C₄×C₄₃	—	4	2	no	—
24	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₄₃	C₂×C₄	1	8	yes	—
28	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
31	C₂×M₁₆×C₄₃	—	4	4	no	—
32	C₂×M₁₆×C₄₃	—	4	4	no	—
33	C₂×M₁₆×C₄₃	C₂×C₄	1	16	yes	complemented
34	C₂²×C₈×C₄₃	C₂³	1	8	yes	—
35	C₂×C₂²⋊C₄×C₄₃	—	2	8	no	—
36	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	8	yes	—
37	C₂×C₄²×C₄₃	—	4	2	no	—
38	C₂×C₄²×C₄₃	—	4	2	no	—
39	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
43	C₂²⋊C₈×C₄₃	C₂×C₄	1	16	yes	—
44	C₂²⋊C₈×C₄₃	C₂×C₄	1	16	yes	complemented
45	#97×C₄₃	—	4	4	no	—
46	C₄×C₈×C₄₃	—	4	4	no	—
47	#95×C₄₃	C₂×C₄	1	8	yes	—
48	C₂⁴×C₄₃	—	2	4	no	—
49	C₂⁴×C₄₃	—	4	1	no	—
50	C₂⁴×C₄₃	—	4	1	no	—
51	C₂⁴×C₄₃	—	4	2	no	—
52	C₂⁴×C₄₃	C₄²	1	4	yes	—
53	C₂⁴×C₄₃	C₂²×C₄	1	2	yes	—
54	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	—	2	2	no	—
56	C₂²×C₄×C₄₃	—	2	2	no	—
57	C₂²×C₄×C₄₃	—	2	4	no	—
58	C₂²×C₄×C₄₃	—	2	4	no	—
59	C₂²×C₄×C₄₃	—	2	4	no	—
60	C₂²×C₄×C₄₃	—	4	2	no	—
61	C₂²×C₄×C₄₃	—	4	2	no	—
62	C₂²×C₄×C₄₃	—	4	1	no	—
63	C₂²×C₄×C₄₃	—	4	1	no	—
64	C₂²×C₄×C₄₃	—	4	2	no	—
65	C₂²×C₄×C₄₃	—	4	2	no	—
66	C₂²×C₄×C₄₃	—	4	2	no	—
67	C₂²×C₄×C₄₃	—	4	2	no	—
68	C₂²×C₄×C₄₃	—	4	16	no	—
69	C₂²×C₄×C₄₃	C₄²	1	8	yes	—
70	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
71	C₂²×C₄×C₄₃	M₁₆	1	8	yes	complemented
72	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₄₃	C₂²×C₄	1	4	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	4	yes	—
79	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
80	M₁₆×C₄₃	—	4	16	no	—
81	C₂×C₈×C₄₃	—	2	8	no	—
82	C₂×C₈×C₄₃	—	2	8	no	—
83	C₂×C₈×C₄₃	C₂²×C₄	1	16	yes	—
84	C₂²⋊C₄×C₄₃	—	2	16	no	—
85	C₄²×C₄₃	—	2	4	no	—
86	C₄²×C₄₃	—	2	4	no	—
87	C₄²×C₄₃	—	4	2	no	—
88	C₄²×C₄₃	—	4	2	no	—
89	C₂³×C₄₃	—	2	1	no	—
90	C₂³×C₄₃	—	2	1	no	—
91	C₂³×C₄₃	—	2	2	no	—
92	C₂³×C₄₃	—	2	2	no	—
93	C₂³×C₄₃	—	2	2	no	—
94	C₂³×C₄₃	—	2	4	no	—
95	C₂³×C₄₃	—	2	4	no	—
96	C₂³×C₄₃	—	2	8	no	—
97	C₂³×C₄₃	—	4	4	no	—
98	C₂³×C₄₃	—	4	4	no	—
99	C₂³×C₄₃	—	4	8	no	—
100	C₂³×C₄₃	—	4	8	no	—
101	C₂³×C₄₃	C₂×C₄²	1	4	yes	—
102	C₂³×C₄₃	C₂×C₄²	1	2	yes	—
103	C₂³×C₄₃	C₂×M₁₆	1	4	yes	—
104	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	—	2	4	no	—
106	C₂×C₄×C₄₃	—	2	1	no	—
107	C₂×C₄×C₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	2	no	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	2	4	no	—
111	C₂×C₄×C₄₃	—	2	8	no	—
112	C₂×C₄×C₄₃	—	2	8	no	—
113	C₂×C₄×C₄₃	—	2	8	no	—
114	C₂×C₄×C₄₃	—	2	4	no	—
115	C₂×C₄×C₄₃	—	2	4	no	—
116	C₂×C₄×C₄₃	—	2	4	no	—
117	C₂×C₄×C₄₃	—	4	4	no	—
118	C₂×C₄×C₄₃	—	4	8	no	—
119	C₂×C₄×C₄₃	—	4	8	no	—
120	C₂×C₄×C₄₃	—	4	8	no	—
121	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
123	C₂×C₄×C₄₃	C₂×M₁₆	1	8	yes	—
124	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
125	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
126	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
127	C₈×C₄₃	—	2	16	no	—
128	#20524	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂²×C₄₃	—	2	2	no	—
130	C₂²×C₄₃	—	2	4	no	—
131	C₂²×C₄₃	—	2	4	no	—
132	C₂²×C₄₃	—	2	2	no	—
133	C₂²×C₄₃	—	2	4	no	—
134	C₂²×C₄₃	—	2	4	no	—
135	C₂²×C₄₃	—	2	4	no	—
136	C₂²×C₄₃	—	4	16	no	—
137	C₂²×C₄₃	—	4	4	no	—
138	C₂²×C₄₃	—	4	4	no	—
139	C₂²×C₄₃	C₄×M₁₆	1	4	yes	—
140	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
141	C₂²×C₄₃	#518	1	1	yes	characteristic
142	C₂²×C₄₃	#518	1	1	yes	characteristic
143	C₂²×C₄₃	C₂²×M₁₆	1	2	yes	—
144	C₂²×C₄₃	#568	1	2	yes	—
145	C₄×C₄₃	—	2	8	no	—
146	C₄×C₄₃	—	2	2	no	—
147	C₄×C₄₃	—	2	4	no	—
148	C₄×C₄₃	—	4	4	no	—
149	C₄×C₄₃	#518	1	4	yes	—
150	C₂×#407	C₂×C₄₃	1	2	yes	—
151	C₂×C₄×C₂²⋊C₄	C₂×C₄₃	1	1	yes	characteristic
152	#1753	C₂×C₄₃	1	8	yes	complemented
153	#1749	C₂×C₄₃	1	2	yes	complemented
154	C₄×C₂²⋊C₈	C₂×C₄₃	1	2	yes	complemented
155	C₂×C₄₃	—	2	4	no	—
156	C₂×C₄₃	—	4	4	no	—
157	C₂×C₄₃	#2273	1	1	yes	characteristic
158	C₂×C₄₃	C₂×C₄×M₁₆	1	1	yes	characteristic
159	C₂×C₄₃	#2873	1	1	yes	characteristic
160	C₂×C₄₃	#2973	1	2	yes	—
161	C₂×C₄₃	#2975	1	2	yes	—
162	C₂⁴×C₄	C₂²×C₄₃	1	1	yes	characteristic
163	C₂²×M₁₆	C₂²×C₄₃	1	4	yes	—
164	C₂²×C₂²⋊C₄	C₄×C₄₃	1	2	yes	—
165	C₂²×C₄²	C₂²×C₄₃	1	1	yes	characteristic
166	C₂²×C₄²	C₂²×C₄₃	1	1	yes	characteristic
167	#407	C₄×C₄₃	1	16	yes	complemented
168	C₂×C₂²⋊C₈	C₂²×C₄₃	1	4	yes	—
169	C₂×C₂²⋊C₈	C₂²×C₄₃	1	4	yes	—
170	C₂×#97	C₂²×C₄₃	1	4	yes	complemented
171	C₂×C₄×C₈	C₂²×C₄₃	1	4	yes	complemented
172	C₄×C₂²⋊C₄	C₄×C₄₃	1	4	yes	—
173	C₄×C₂²⋊C₄	C₂²×C₄₃	1	4	yes	—
174	C₂×#95	C₄×C₄₃	1	2	yes	—
175	#336	C₂²×C₄₃	1	4	yes	complemented
176	#336	C₂²×C₄₃	1	4	yes	complemented
177	C₄₃	#20524	1	1	yes	characteristic, direct product factor, Sylow
178	C₂⁵	C₂×C₄×C₄₃	1	1	yes	characteristic
179	C₂³×C₄	—	4	2	no	—
180	C₂³×C₄	—	4	2	no	—
181	C₂³×C₄	C₂×C₄×C₄₃	1	2	yes	—
182	C₂³×C₄	C₂×C₄×C₄₃	1	1	yes	characteristic
183	C₂³×C₄	C₂×C₄×C₄₃	1	2	yes	—
184	C₂³×C₄	C₂×C₄×C₄₃	1	8	yes	—
185	C₂³×C₄	C₂×C₄×C₄₃	1	2	yes	—
186	C₂³×C₄	C₂³×C₄₃	1	2	yes	—
187	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
188	C₂×M₁₆	—	4	4	no	—
189	C₂×M₁₆	—	4	4	no	—
190	C₂×M₁₆	C₂×C₄×C₄₃	1	16	yes	complemented
191	C₂²×C₈	C₂³×C₄₃	1	8	yes	—
192	C₂×C₂²⋊C₄	—	2	8	no	—
193	C₂×C₂²⋊C₄	C₂×C₄×C₄₃	1	8	yes	—
194	C₂×C₄²	—	4	2	no	—
195	C₂×C₄²	—	4	2	no	—
196	C₂×C₄²	C₂×C₄×C₄₃	1	2	yes	—
197	C₂×C₄²	C₂×C₄×C₄₃	1	2	yes	—
198	C₂×C₄²	C₂³×C₄₃	1	2	yes	—
199	C₂×C₄²	C₂³×C₄₃	1	2	yes	—
200	C₂²⋊C₈	C₂×C₄×C₄₃	1	16	yes	—
201	C₂²⋊C₈	C₂×C₄×C₄₃	1	16	yes	complemented
202	#97	—	4	4	no	—
203	C₄×C₈	—	4	4	no	—
204	#95	C₂×C₄×C₄₃	1	8	yes	—
205	C₂⁴	—	2	4	no	—
206	C₂⁴	—	4	1	no	—
207	C₂⁴	—	4	1	no	—
208	C₂⁴	—	4	2	no	—
209	C₂⁴	C₄²×C₄₃	1	4	yes	—
210	C₂⁴	C₂²×C₄×C₄₃	1	2	yes	—
211	C₂⁴	C₂²×C₄×C₄₃	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	4	16	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₄²×C₄₃	1	8	yes	—
227	C₂²×C₄	C₄²×C₄₃	1	4	yes	—
228	C₂²×C₄	M₁₆×C₄₃	1	8	yes	complemented
229	C₂²×C₄	C₂²×C₄×C₄₃	1	4	yes	—
230	C₂²×C₄	C₂²×C₄×C₄₃	1	2	yes	—
231	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₄₃	1	4	yes	—
233	C₂²×C₄	C₂²×C₄×C₄₃	1	4	yes	—
234	C₂²×C₄	C₂²×C₄×C₄₃	1	4	yes	—
235	C₂²×C₄	C₂²×C₄×C₄₃	1	4	yes	—
236	C₂²×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
237	M₁₆	—	4	16	no	—
238	C₂×C₈	—	2	8	no	—
239	C₂×C₈	—	2	8	no	—
240	C₂×C₈	C₂²×C₄×C₄₃	1	16	yes	—
241	C₂²⋊C₄	—	2	16	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	4	2	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	2	8	no	—
254	C₂³	—	4	4	no	—
255	C₂³	—	4	4	no	—
256	C₂³	—	4	8	no	—
257	C₂³	—	4	8	no	—
258	C₂³	C₂×C₄²×C₄₃	1	4	yes	—
259	C₂³	C₂×C₄²×C₄₃	1	2	yes	—
260	C₂³	C₂×M₁₆×C₄₃	1	4	yes	—
261	C₂³	C₂³×C₄×C₄₃	1	1	yes	characteristic
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	8	no	—
276	C₂×C₄	—	4	8	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	C₂×C₄²×C₄₃	1	4	yes	—
279	C₂×C₄	C₂×C₄²×C₄₃	1	2	yes	—
280	C₂×C₄	C₂×M₁₆×C₄₃	1	8	yes	—
281	C₂×C₄	C₂³×C₄×C₄₃	1	2	yes	—
282	C₂×C₄	H(5,𝔽₂)×C₄₃	1	2	yes	—
283	C₂×C₄	H(5,𝔽₂)×C₄₃	1	2	yes	—
284	C₈	—	2	16	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	4	16	no	—
293	C₂²	—	4	4	no	—
294	C₂²	—	4	4	no	—
295	C₂²	C₄×M₁₆×C₄₃	1	4	yes	—
296	C₂²	C₂²×C₄²×C₄₃	1	1	yes	characteristic
297	C₂²	C₄₃×#518	1	1	yes	characteristic
298	C₂²	C₄₃×#518	1	1	yes	characteristic
299	C₂²	C₂²×M₁₆×C₄₃	1	2	yes	—
300	C₂²	C₄₃×#568	1	2	yes	—
301	C₄	—	2	8	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	4	no	—
304	C₄	—	4	4	no	—
305	C₄	C₄₃×#518	1	4	yes	—
306	C₂	—	2	4	no	—
307	C₂	—	4	4	no	—
308	C₂	C₄₃×#2273	1	1	yes	characteristic
309	C₂	C₂×C₄×M₁₆×C₄₃	1	1	yes	characteristic
310	C₂	C₄₃×#2873	1	1	yes	characteristic
311	C₂	C₄₃×#2973	1	2	yes	—
312	C₂	C₄₃×#2975	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₄₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₃
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁵×C₄₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₃×#1753	C₂	semidirect
4. C₄₃×#1749	C₂	semidirect
5. C₄×C₂²⋊C₈×C₄₃	C₂	semidirect
11. C₄₃×#407	C₄	semidirect
14. C₂×#97×C₄₃	C₂²	semidirect
15. C₂×C₄×C₈×C₄₃	C₂²	semidirect
19. C₄₃×#336	C₂²	semidirect
20. C₄₃×#336	C₂²	semidirect
33. C₂×M₁₆×C₄₃	C₂×C₄	semidirect
44. C₂²⋊C₈×C₄₃	C₂×C₄	semidirect
71. C₂²×C₄×C₄₃	M₁₆	semidirect
128. #20524	C₄₃	direct (N × H)
152. #1753	C₂×C₄₃	semidirect
153. #1749	C₂×C₄₃	semidirect
154. C₄×C₂²⋊C₈	C₂×C₄₃	semidirect
167. #407	C₄×C₄₃	semidirect
170. C₂×#97	C₂²×C₄₃	semidirect
171. C₂×C₄×C₈	C₂²×C₄₃	semidirect
175. #336	C₂²×C₄₃	semidirect
176. #336	C₂²×C₄₃	semidirect
177. C₄₃	#20524	direct (N × H)
190. C₂×M₁₆	C₂×C₄×C₄₃	semidirect
201. C₂²⋊C₈	C₂×C₄×C₄₃	semidirect
228. C₂²×C₄	M₁₆×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×#407	2
C₂×C₄×C₂²⋊C₄×C₄₃	2
C₄₃×#1753	2
C₄₃×#1749	2
C₄×C₂²⋊C₈×C₄₃	2
#20524	43

Subgroups — C43×#20524 #24940541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#20524 #24940541