Subgroups — C₄₃×#10802 #24930819

Order 11008.

802 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×C₄×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₃×#343	C₂	1	2	yes	maximal
3	C₂×C₄₃×#343	C₂	1	2	yes	maximal
4	C₄₃×#1728	C₂	1	2	yes	maximal
5	C₂²×M₁₆×C₄₃	C₂²	1	4	yes	—
6	C₂²×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₄₃	C₄	1	2	yes	—
9	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊C₄×C₄₃	—	2	2	no	—
11	C₄×C₄⋊C₄×C₄₃	—	2	2	no	—
12	C₂×#95×C₄₃	C₄	1	2	yes	—
13	C₄₃×#343	—	2	4	no	—
14	C₄₃×#343	C₄	1	8	yes	complemented
15	C₄₃×#336	—	2	2	no	—
16	C₄₃×#336	—	2	2	no	—
17	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₄₃	—	2	4	no	—
23	C₂×M₁₆×C₄₃	—	2	4	no	—
24	C₂×M₁₆×C₄₃	—	2	4	no	—
25	C₂×M₁₆×C₄₃	—	2	4	no	—
26	C₂×M₁₆×C₄₃	—	2	4	no	—
27	C₂×M₁₆×C₄₃	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₄₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₄₃	Q₈	1	2	yes	—
36	C₂×C₄²×C₄₃	—	2	2	no	—
37	C₂×C₄²×C₄₃	—	2	2	no	—
38	C₂×C₄²×C₄₃	—	2	2	no	—
39	C₂×C₄²×C₄₃	—	2	2	no	—
40	C₂×C₄²×C₄₃	—	4	4	no	—
41	C₂×C₄²×C₄₃	D₄	1	2	yes	—
42	C₂×C₄²×C₄₃	D₄	1	2	yes	—
43	#95×C₄₃	—	2	4	no	—
44	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₃	—	2	2	no	—
46	C₂²×C₄×C₄₃	—	2	2	no	—
47	C₂²×C₄×C₄₃	—	2	2	no	—
48	C₂²×C₄×C₄₃	—	2	2	no	—
49	C₂²×C₄×C₄₃	—	2	1	no	—
50	C₂²×C₄×C₄₃	—	2	1	no	—
51	C₂²×C₄×C₄₃	—	2	1	no	—
52	C₂²×C₄×C₄₃	—	2	1	no	—
53	C₂²×C₄×C₄₃	—	2	2	no	—
54	C₂²×C₄×C₄₃	—	2	2	no	—
55	C₂²×C₄×C₄₃	—	4	1	no	—
56	C₂²×C₄×C₄₃	—	4	1	no	—
57	C₂²×C₄×C₄₃	—	4	2	no	—
58	C₂²×C₄×C₄₃	—	4	2	no	—
59	C₂²×C₄×C₄₃	—	4	2	no	—
60	C₂²×C₄×C₄₃	—	4	2	no	—
61	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
62	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	#41	1	2	yes	—
71	C₂²×C₄×C₄₃	#41	1	2	yes	—
72	C₂²×C₄×C₄₃	#41	1	2	yes	—
73	C₂²×C₄×C₄₃	#41	1	2	yes	—
74	M₁₆×C₄₃	—	2	8	no	—
75	M₁₆×C₄₃	—	2	8	no	—
76	M₁₆×C₄₃	—	4	4	no	—
77	M₁₆×C₄₃	—	4	4	no	—
78	C₂×C₈×C₄₃	—	2	8	no	—
79	C₂×C₈×C₄₃	—	4	4	no	—
80	C₂×C₈×C₄₃	—	4	4	no	—
81	C₄⋊C₄×C₄₃	—	2	8	no	—
82	C₄⋊C₄×C₄₃	—	2	4	no	—
83	C₄⋊C₄×C₄₃	—	2	4	no	—
84	C₄²×C₄₃	—	2	4	no	—
85	C₄²×C₄₃	—	2	4	no	—
86	C₄²×C₄₃	—	4	4	no	—
87	C₄²×C₄₃	—	4	4	no	—
88	C₂³×C₄₃	—	2	1	no	—
89	C₂³×C₄₃	—	2	1	no	—
90	C₂³×C₄₃	—	2	1	no	—
91	C₂³×C₄₃	—	2	1	no	—
92	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
93	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂³×C₄₃	#117	1	1	yes	characteristic
96	C₂³×C₄₃	#117	1	1	yes	characteristic
97	C₂³×C₄₃	#117	1	1	yes	characteristic
98	C₂³×C₄₃	#117	1	1	yes	characteristic
99	C₂×C₄×C₄₃	—	2	4	no	—
100	C₂×C₄×C₄₃	—	2	4	no	—
101	C₂×C₄×C₄₃	—	2	1	no	—
102	C₂×C₄×C₄₃	—	2	1	no	—
103	C₂×C₄×C₄₃	—	2	1	no	—
104	C₂×C₄×C₄₃	—	2	1	no	—
105	C₂×C₄×C₄₃	—	2	4	no	—
106	C₂×C₄×C₄₃	—	2	4	no	—
107	C₂×C₄×C₄₃	—	2	2	no	—
108	C₂×C₄×C₄₃	—	2	2	no	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	4	4	no	—
111	C₂×C₄×C₄₃	—	4	2	no	—
112	C₂×C₄×C₄₃	—	4	2	no	—
113	C₂×C₄×C₄₃	—	4	4	no	—
114	C₂×C₄×C₄₃	—	4	2	no	—
115	C₂×C₄×C₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	—	4	2	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	—	4	4	no	—
119	C₂×C₄×C₄₃	—	4	4	no	—
120	C₂×C₄×C₄₃	#100	1	2	yes	—
121	C₂×C₄×C₄₃	#101	1	2	yes	—
122	C₂×C₄×C₄₃	#117	1	4	yes	—
123	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
124	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₄₃	#121	1	2	yes	—
127	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₄₃	#123	1	4	yes	—
129	C₂×C₄×C₄₃	#124	1	2	yes	—
130	C₂×C₄×C₄₃	#125	1	2	yes	—
131	C₂×C₄×C₄₃	#126	1	4	yes	—
132	C₈×C₄₃	—	4	8	no	—
133	#10802	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₄₃	—	2	1	no	—
135	C₂²×C₄₃	—	2	1	no	—
136	C₂²×C₄₃	—	2	1	no	—
137	C₂²×C₄₃	—	2	1	no	—
138	C₂²×C₄₃	—	2	1	no	—
139	C₂²×C₄₃	—	2	1	no	—
140	C₂²×C₄₃	—	2	1	no	—
141	C₂²×C₄₃	—	2	1	no	—
142	C₂²×C₄₃	—	2	1	no	—
143	C₂²×C₄₃	—	2	1	no	—
144	C₂²×C₄₃	—	2	1	no	—
145	C₂²×C₄₃	—	2	1	no	—
146	C₂²×C₄₃	#376	1	1	yes	characteristic
147	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₄₃	#380	1	1	yes	characteristic
150	C₂²×C₄₃	#381	1	1	yes	characteristic
151	C₂²×C₄₃	#382	1	1	yes	characteristic
152	C₂²×C₄₃	#383	1	1	yes	characteristic
153	C₂²×C₄₃	C₂×#100	1	1	yes	characteristic
154	C₂²×C₄₃	C₂×#101	1	1	yes	characteristic
155	C₂²×C₄₃	#413	1	1	yes	characteristic
156	C₂²×C₄₃	#413	1	1	yes	characteristic
157	C₄×C₄₃	—	2	2	no	—
158	C₄×C₄₃	—	2	2	no	—
159	C₄×C₄₃	—	4	4	no	—
160	C₄×C₄₃	—	4	2	no	—
161	C₄×C₄₃	—	4	4	no	—
162	C₄×C₄₃	#387	1	4	yes	—
163	C₄×C₄₃	#388	1	4	yes	—
164	C₂×C₄×C₄⋊C₄	C₂×C₄₃	1	1	yes	characteristic
165	C₂×#343	C₂×C₄₃	1	2	yes	—
166	C₂×#343	C₂×C₄₃	1	2	yes	—
167	#1728	C₂×C₄₃	1	2	yes	—
168	C₂×C₄₃	—	2	1	no	—
169	C₂×C₄₃	—	2	1	no	—
170	C₂×C₄₃	—	2	1	no	—
171	C₂×C₄₃	—	2	1	no	—
172	C₂×C₄₃	#1434	1	1	yes	characteristic
173	C₂×C₄₃	C₄×#100	1	1	yes	characteristic
174	C₂×C₄₃	C₄×#101	1	1	yes	characteristic
175	C₂×C₄₃	#1800	1	1	yes	characteristic
176	C₂×C₄₃	#1801	1	1	yes	characteristic
177	C₂×C₄₃	#1821	1	1	yes	characteristic
178	C₂×C₄₃	#1822	1	1	yes	characteristic
179	C₂²×M₁₆	C₂²×C₄₃	1	4	yes	—
180	C₂²×C₄⋊C₄	C₂²×C₄₃	1	1	yes	characteristic
181	C₂²×C₄⋊C₄	C₂²×C₄₃	1	1	yes	characteristic
182	C₂²×C₄²	C₄×C₄₃	1	2	yes	—
183	C₂²×C₄²	C₂²×C₄₃	1	1	yes	characteristic
184	C₄×C₄⋊C₄	—	2	2	no	—
185	C₄×C₄⋊C₄	—	2	2	no	—
186	C₂×#95	C₄×C₄₃	1	2	yes	—
187	#343	—	2	4	no	—
188	#343	C₄×C₄₃	1	8	yes	complemented
189	#336	—	2	2	no	—
190	#336	—	2	2	no	—
191	C₄₃	#10802	1	1	yes	characteristic, direct product factor, Sylow
192	C₂³×C₄	C₂×C₄×C₄₃	1	1	yes	characteristic
193	C₂³×C₄	C₂×C₄×C₄₃	1	2	yes	—
194	C₂³×C₄	C₂×C₄×C₄₃	1	1	yes	characteristic
195	C₂³×C₄	C₂×C₄×C₄₃	1	2	yes	—
196	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	—	2	4	no	—
200	C₂×M₁₆	—	2	4	no	—
201	C₂×M₁₆	—	2	4	no	—
202	C₂×M₁₆	C₂×C₄×C₄₃	1	8	yes	—
203	C₂²×C₈	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	C₂×C₄×C₄₃	1	4	yes	—
209	C₂×C₄⋊C₄	D₄×C₄₃	1	2	yes	—
210	C₂×C₄⋊C₄	Q₈×C₄₃	1	2	yes	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	4	4	no	—
216	C₂×C₄²	D₄×C₄₃	1	2	yes	—
217	C₂×C₄²	D₄×C₄₃	1	2	yes	—
218	#95	—	2	4	no	—
219	C₂⁴	C₂²×C₄×C₄₃	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	C₄²×C₄₃	1	4	yes	—
237	C₂²×C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
238	C₂²×C₄	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊C₄×C₄₃	1	1	yes	characteristic
240	C₂²×C₄	C₄⋊C₄×C₄₃	1	4	yes	—
241	C₂²×C₄	C₂²×C₄×C₄₃	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
244	C₂²×C₄	C₂×Q₈×C₄₃	1	1	yes	characteristic
245	C₂²×C₄	#41×C₄₃	1	2	yes	—
246	C₂²×C₄	#41×C₄₃	1	2	yes	—
247	C₂²×C₄	#41×C₄₃	1	2	yes	—
248	C₂²×C₄	#41×C₄₃	1	2	yes	—
249	M₁₆	—	2	8	no	—
250	M₁₆	—	2	8	no	—
251	M₁₆	—	4	4	no	—
252	M₁₆	—	4	4	no	—
253	C₂×C₈	—	2	8	no	—
254	C₂×C₈	—	4	4	no	—
255	C₂×C₈	—	4	4	no	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	4	4	no	—
262	C₄²	—	4	4	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	C₂×C₄²×C₄₃	1	1	yes	characteristic
268	C₂³	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
269	C₂³	C₂×C₄⋊C₄×C₄₃	1	1	yes	characteristic
270	C₂³	#117×C₄₃	1	1	yes	characteristic
271	C₂³	#117×C₄₃	1	1	yes	characteristic
272	C₂³	#117×C₄₃	1	1	yes	characteristic
273	C₂³	#117×C₄₃	1	1	yes	characteristic
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	#100×C₄₃	1	2	yes	—
296	C₂×C₄	#101×C₄₃	1	2	yes	—
297	C₂×C₄	#117×C₄₃	1	4	yes	—
298	C₂×C₄	C₄×D₄×C₄₃	1	4	yes	—
299	C₂×C₄	C₄×D₄×C₄₃	1	2	yes	—
300	C₂×C₄	C₄×Q₈×C₄₃	1	2	yes	—
301	C₂×C₄	#121×C₄₃	1	2	yes	—
302	C₂×C₄	C₂²⋊Q₈×C₄₃	1	2	yes	—
303	C₂×C₄	#123×C₄₃	1	4	yes	—
304	C₂×C₄	#124×C₄₃	1	2	yes	—
305	C₂×C₄	#125×C₄₃	1	2	yes	—
306	C₂×C₄	#126×C₄₃	1	4	yes	—
307	C₈	—	4	8	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	C₄₃×#376	1	1	yes	characteristic
321	C₂²	C₄×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
322	C₂²	C₄×C₄⋊C₄×C₄₃	1	1	yes	characteristic
323	C₂²	C₄₃×#380	1	1	yes	characteristic
324	C₂²	C₄₃×#381	1	1	yes	characteristic
325	C₂²	C₄₃×#382	1	1	yes	characteristic
326	C₂²	C₄₃×#383	1	1	yes	characteristic
327	C₂²	C₂×#100×C₄₃	1	1	yes	characteristic
328	C₂²	C₂×#101×C₄₃	1	1	yes	characteristic
329	C₂²	C₄₃×#413	1	1	yes	characteristic
330	C₂²	C₄₃×#413	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	2	no	—
334	C₄	—	4	4	no	—
335	C₄	—	4	4	no	—
336	C₄	C₄₃×#387	1	4	yes	—
337	C₄	C₄₃×#388	1	4	yes	—
338	C₂	—	2	1	no	—
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	2	1	no	—
342	C₂	C₄₃×#1434	1	1	yes	characteristic
343	C₂	C₄×#100×C₄₃	1	1	yes	characteristic
344	C₂	C₄×#101×C₄₃	1	1	yes	characteristic
345	C₂	C₄₃×#1800	1	1	yes	characteristic
346	C₂	C₄₃×#1801	1	1	yes	characteristic
347	C₂	C₄₃×#1821	1	1	yes	characteristic
348	C₂	C₄₃×#1822	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₄₃
Frattini, cosocle	C₂³×C₄	C₂³×C₄₃
prime radical	C₂⁴×C₄₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ 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
14. C₄₃×#343	C₄	semidirect
133. #10802	C₄₃	direct (N × H)
188. #343	C₄×C₄₃	semidirect
191. C₄₃	#10802	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×C₄×C₄⋊C₄×C₄₃	2
C₂×C₄₃×#343	2
C₄₃×#1728	2
#10802	43

Subgroups — C43×#10802 #24930819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#10802 #24930819