Subgroups — C₄₃×#10816 #24930833

Order 11008.

722 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#1742	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#1740	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₃×#339	C₂	1	2	yes	maximal
5	C₂×C₄₃×H(3,ℤ/4)	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
7	C₂×#117×C₄₃	C₂²	1	2	yes	—
8	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
9	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₄₃	—	2	2	no	—
12	C₄×C₂²⋊C₄×C₄₃	—	2	2	no	—
13	C₄₃×#376	C₄	1	2	yes	complemented
14	C₄³×C₄₃	C₄	1	2	yes	complemented
15	C₄₃×#343	—	2	2	no	—
16	C₄₃×#342	—	2	2	no	—
17	C₄₃×#339	—	2	4	no	—
18	C₄₃×H(3,ℤ/4)	C₄	1	8	yes	complemented
19	C₂³×C₄×C₄₃	—	2	1	no	—
20	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₄₃	—	2	2	no	—
22	C₂×M₁₆×C₄₃	—	2	2	no	—
23	C₂×M₁₆×C₄₃	—	2	2	no	—
24	C₂×M₁₆×C₄₃	—	2	2	no	—
25	C₂×M₁₆×C₄₃	—	2	2	no	—
26	C₂×M₁₆×C₄₃	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₄₃	—	2	2	no	—
28	#117×C₄₃	—	2	2	no	—
29	#117×C₄₃	—	2	2	no	—
30	#117×C₄₃	—	2	2	no	—
31	#117×C₄₃	—	2	2	no	—
32	#117×C₄₃	—	2	2	no	—
33	#117×C₄₃	—	2	2	no	—
34	#117×C₄₃	C₂×C₄	1	4	yes	—
35	#117×C₄₃	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
40	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
41	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
42	C₂×C₄²×C₄₃	—	2	1	no	—
43	C₂×C₄²×C₄₃	—	2	1	no	—
44	C₂×C₄²×C₄₃	—	2	1	no	—
45	C₂×C₄²×C₄₃	—	2	2	no	—
46	C₂×C₄²×C₄₃	—	2	4	no	—
47	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	D₄	1	2	yes	—
52	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
53	#95×C₄₃	—	2	4	no	—
54	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	—	2	1	no	—
56	C₂²×C₄×C₄₃	—	2	1	no	—
57	C₂²×C₄×C₄₃	—	2	2	no	—
58	C₂²×C₄×C₄₃	—	2	2	no	—
59	C₂²×C₄×C₄₃	—	2	2	no	—
60	C₂²×C₄×C₄₃	—	4	4	no	—
61	C₂²×C₄×C₄₃	—	4	2	no	—
62	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
63	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	#41	1	2	yes	—
69	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	#41	1	2	yes	—
72	C₂²×C₄×C₄₃	#41	1	2	yes	—
73	M₁₆×C₄₃	—	2	4	no	—
74	M₁₆×C₄₃	—	2	4	no	—
75	M₁₆×C₄₃	—	4	2	no	—
76	M₁₆×C₄₃	—	4	2	no	—
77	C₂×C₈×C₄₃	—	2	4	no	—
78	C₂×C₈×C₄₃	—	4	2	no	—
79	C₂×C₈×C₄₃	—	4	2	no	—
80	C₄⋊C₄×C₄₃	—	2	4	no	—
81	C₄⋊C₄×C₄₃	—	2	4	no	—
82	C₄⋊C₄×C₄₃	—	2	4	no	—
83	C₄⋊C₄×C₄₃	—	4	2	no	—
84	C₂²⋊C₄×C₄₃	—	2	4	no	—
85	C₂²⋊C₄×C₄₃	—	2	4	no	—
86	C₂²⋊C₄×C₄₃	—	2	4	no	—
87	C₂²⋊C₄×C₄₃	—	4	2	no	—
88	C₄²×C₄₃	—	2	1	no	—
89	C₄²×C₄₃	—	2	1	no	—
90	C₄²×C₄₃	—	2	2	no	—
91	C₄²×C₄₃	—	2	2	no	—
92	C₄²×C₄₃	—	2	2	no	—
93	C₄²×C₄₃	—	2	2	no	—
94	C₄²×C₄₃	—	2	2	no	—
95	C₄²×C₄₃	—	2	4	no	—
96	C₄²×C₄₃	—	4	8	no	—
97	C₄²×C₄₃	—	4	2	no	—
98	C₄²×C₄₃	C₄²	1	4	yes	—
99	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
100	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
101	C₂³×C₄₃	—	4	2	no	—
102	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₄₃	C₄×D₄	1	2	yes	—
104	C₂³×C₄₃	#121	1	1	yes	characteristic
105	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₄₃	#123	1	2	yes	—
107	C₂×C₄×C₄₃	—	2	4	no	—
108	C₂×C₄×C₄₃	—	2	2	no	—
109	C₂×C₄×C₄₃	—	2	4	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	2	4	no	—
112	C₂×C₄×C₄₃	—	4	4	no	—
113	C₂×C₄×C₄₃	—	4	4	no	—
114	C₂×C₄×C₄₃	—	4	4	no	—
115	C₂×C₄×C₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	—	4	4	no	—
117	C₂×C₄×C₄₃	—	4	4	no	—
118	C₂×C₄×C₄₃	—	8	4	no	—
119	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#117	1	2	yes	—
123	C₂×C₄×C₄₃	#117	1	2	yes	—
124	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
125	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
126	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₄₃	#123	1	2	yes	—
133	C₂×C₄×C₄₃	#124	1	2	yes	—
134	C₂×C₄×C₄₃	#125	1	2	yes	—
135	C₂×C₄×C₄₃	#126	1	2	yes	—
136	C₂×C₄×C₄₃	#126	1	2	yes	—
137	C₈×C₄₃	—	4	4	no	—
138	#10816	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂²×C₄₃	—	2	2	no	—
140	C₂²×C₄₃	—	2	4	no	—
141	C₂²×C₄₃	—	4	2	no	—
142	C₂²×C₄₃	—	4	1	no	—
143	C₂²×C₄₃	—	4	1	no	—
144	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄₃	#380	1	1	yes	characteristic
146	C₂²×C₄₃	#381	1	1	yes	characteristic
147	C₂²×C₄₃	#387	1	2	yes	—
148	C₂²×C₄₃	#388	1	2	yes	—
149	C₄×C₄₃	—	2	4	no	—
150	C₄×C₄₃	—	4	4	no	—
151	C₄×C₄₃	—	4	2	no	—
152	C₄×C₄₃	—	4	4	no	—
153	C₄×C₄₃	—	8	4	no	—
154	C₄×C₄₃	#376	1	1	yes	characteristic
155	C₄×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
156	C₄×C₄₃	#382	1	1	yes	characteristic
157	C₄×C₄₃	#383	1	1	yes	characteristic
158	C₄×C₄₃	#387	1	2	yes	—
159	C₄×C₄₃	#388	1	2	yes	—
160	C₄×#117	C₂×C₄₃	1	1	yes	characteristic
161	#1742	C₂×C₄₃	1	1	yes	characteristic
162	#1740	C₂×C₄₃	1	1	yes	characteristic
163	C₂×#339	C₂×C₄₃	1	2	yes	—
164	C₂×H(3,ℤ/4)	C₂×C₄₃	1	2	yes	—
165	C₂×C₄₃	—	2	1	no	—
166	C₂×C₄₃	—	2	1	no	—
167	C₂×C₄₃	—	2	2	no	—
168	C₂×C₄₃	—	2	2	no	—
169	C₂×C₄₃	#1434	1	1	yes	characteristic
170	C₂×C₄₃	#1827	1	2	yes	—
171	C₂²×M₁₆	C₂²×C₄₃	1	2	yes	—
172	C₂×#117	C₂²×C₄₃	1	2	yes	—
173	C₂×#117	C₂²×C₄₃	1	1	yes	characteristic
174	C₂×#117	C₂²×C₄₃	1	1	yes	characteristic
175	C₂²×C₄²	C₂²×C₄₃	1	1	yes	characteristic
176	C₄×C₄⋊C₄	—	2	2	no	—
177	C₄×C₂²⋊C₄	—	2	2	no	—
178	#376	C₄×C₄₃	1	2	yes	complemented
179	C₄³	C₄×C₄₃	1	2	yes	complemented
180	#343	—	2	2	no	—
181	#342	—	2	2	no	—
182	#339	—	2	4	no	—
183	H(3,ℤ/4)	C₄×C₄₃	1	8	yes	complemented
184	C₄₃	#10816	1	1	yes	characteristic, direct product factor, Sylow
185	C₂³×C₄	—	2	1	no	—
186	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	C₂×C₄×C₄₃	1	4	yes	—
193	C₂²×C₈	—	2	2	no	—
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	C₂×C₄×C₄₃	1	4	yes	—
201	#117	C₂×C₄×C₄₃	1	4	yes	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	D₄×C₄₃	1	2	yes	—
205	C₂×C₂²⋊C₄	—	2	1	no	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₂²⋊C₄	D₄×C₄₃	1	2	yes	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	—	2	1	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	—	2	4	no	—
213	C₂×C₄²	C₂×C₄×C₄₃	1	2	yes	—
214	C₂×C₄²	C₂×C₄×C₄₃	1	2	yes	—
215	C₂×C₄²	C₂×C₄×C₄₃	1	1	yes	characteristic
216	C₂×C₄²	C₂×C₄×C₄₃	1	1	yes	characteristic
217	C₂×C₄²	D₄×C₄₃	1	2	yes	—
218	C₂×C₄²	Q₈×C₄₃	1	2	yes	—
219	#95	—	2	4	no	—
220	C₂⁴	C₂×D₄×C₄₃	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	C₂²⋊C₄×C₄₃	1	4	yes	—
229	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²×C₄×C₄₃	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×Q₈×C₄₃	1	1	yes	characteristic
234	C₂²×C₄	#41×C₄₃	1	1	yes	characteristic
235	C₂²×C₄	#41×C₄₃	1	1	yes	characteristic
236	C₂²×C₄	#41×C₄₃	1	2	yes	—
237	C₂²×C₄	#41×C₄₃	1	2	yes	—
238	C₂²×C₄	#41×C₄₃	1	2	yes	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	2	4	no	—
241	M₁₆	—	4	2	no	—
242	M₁₆	—	4	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	4	8	no	—
263	C₄²	—	4	2	no	—
264	C₄²	C₄²×C₄₃	1	4	yes	—
265	C₄²	C₄⋊C₄×C₄₃	1	2	yes	—
266	C₄²	C₄⋊C₄×C₄₃	1	2	yes	—
267	C₂³	—	4	2	no	—
268	C₂³	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
269	C₂³	C₄×D₄×C₄₃	1	2	yes	—
270	C₂³	#121×C₄₃	1	1	yes	characteristic
271	C₂³	C₂²⋊Q₈×C₄₃	1	1	yes	characteristic
272	C₂³	#123×C₄₃	1	2	yes	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	8	4	no	—
285	C₂×C₄	C₂×C₄²×C₄₃	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄⋊C₄×C₄₃	1	1	yes	characteristic
287	C₂×C₄	#117×C₄₃	1	1	yes	characteristic
288	C₂×C₄	#117×C₄₃	1	1	yes	characteristic
289	C₂×C₄	#117×C₄₃	1	2	yes	—
290	C₂×C₄	#117×C₄₃	1	2	yes	—
291	C₂×C₄	#117×C₄₃	1	1	yes	characteristic
292	C₂×C₄	#117×C₄₃	1	1	yes	characteristic
293	C₂×C₄	C₄×D₄×C₄₃	1	2	yes	—
294	C₂×C₄	C₄×D₄×C₄₃	1	2	yes	—
295	C₂×C₄	C₄×Q₈×C₄₃	1	2	yes	—
296	C₂×C₄	#121×C₄₃	1	1	yes	characteristic
297	C₂×C₄	C₂²⋊Q₈×C₄₃	1	1	yes	characteristic
298	C₂×C₄	#123×C₄₃	1	2	yes	—
299	C₂×C₄	#124×C₄₃	1	2	yes	—
300	C₂×C₄	#125×C₄₃	1	2	yes	—
301	C₂×C₄	#126×C₄₃	1	2	yes	—
302	C₂×C₄	#126×C₄₃	1	2	yes	—
303	C₈	—	4	4	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₄×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
310	C₂²	C₄₃×#380	1	1	yes	characteristic
311	C₂²	C₄₃×#381	1	1	yes	characteristic
312	C₂²	C₄₃×#387	1	2	yes	—
313	C₂²	C₄₃×#388	1	2	yes	—
314	C₄	—	2	4	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	4	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	4	no	—
319	C₄	C₄₃×#376	1	1	yes	characteristic
320	C₄	C₄×C₄⋊C₄×C₄₃	1	1	yes	characteristic
321	C₄	C₄₃×#382	1	1	yes	characteristic
322	C₄	C₄₃×#383	1	1	yes	characteristic
323	C₄	C₄₃×#387	1	2	yes	—
324	C₄	C₄₃×#388	1	2	yes	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	2	2	no	—
328	C₂	—	2	2	no	—
329	C₂	C₄₃×#1434	1	1	yes	characteristic
330	C₂	C₄₃×#1827	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₃	#117
commutator	C₂×C₄	C₂×C₄²×C₄₃
Frattini, cosocle	C₂³×C₄	C₂³×C₄₃
socle	C₂²×C₄₃	#381
prime radical	C₂⁴×C₄₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₃ ◁ C₂×#117×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄₃×#376	C₄	semidirect
14. C₄³×C₄₃	C₄	semidirect
18. C₄₃×H(3,ℤ/4)	C₄	semidirect
138. #10816	C₄₃	direct (N × H)
178. #376	C₄×C₄₃	semidirect
179. C₄³	C₄×C₄₃	semidirect
183. H(3,ℤ/4)	C₄×C₄₃	semidirect
184. C₄₃	#10816	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#117×C₄₃	2
C₄₃×#1742	2
C₄₃×#1740	2
C₂×C₄₃×#339	2
C₂×C₄₃×H(3,ℤ/4)	2
#10816	43

Subgroups — C43×#10816 #24930833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#10816 #24930833