Subgroups — C₄₃×#8174 #24928191

Order 11008.

698 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₄×C₄₃	C₂	1	1	yes	maximal, characteristic
2	C₄₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄₃×D₄⋊C₈	C₂	1	2	yes	maximal
5	C₄₃×#1279	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄×C₄₃	C₄	1	2	yes	—
7	C₂×C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₄₃	—	2	1	no	—
9	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
10	C₂×#97×C₄₃	—	2	1	no	—
11	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
12	C₄×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
13	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
14	C₄×C₂²⋊C₄×C₄₃	—	2	1	no	—
15	C₄×C₂²⋊C₄×C₄₃	—	2	2	no	—
16	C₄³×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#336	—	2	2	no	—
18	C₄₃×#336	—	2	1	no	—
19	C₄₃×D₄⋊C₈	C₄	1	8	yes	complemented
20	C₂²×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₄₃	—	2	2	no	—
22	C₂³×C₄×C₄₃	—	2	1	no	—
23	C₂²×C₈×C₄₃	—	2	2	no	—
24	C₂²×C₈×C₄₃	—	2	2	no	—
25	C₄×D₄×C₄₃	—	2	4	no	—
26	C₄×D₄×C₄₃	—	2	4	no	—
27	C₄×D₄×C₄₃	—	2	2	no	—
28	C₄×D₄×C₄₃	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
32	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
33	C₂×C₄²×C₄₃	—	2	2	no	—
34	C₂×C₄²×C₄₃	—	2	1	no	—
35	C₂×C₄²×C₄₃	—	4	2	no	—
36	C₂×C₄²×C₄₃	—	4	2	no	—
37	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₃	D₄	1	2	yes	—
40	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₄₃	—	2	2	no	—
43	C₄⋊C₈×C₄₃	C₂×C₄	1	4	yes	complemented
44	#97×C₄₃	—	4	2	no	—
45	C₄×C₈×C₄₃	—	2	2	no	—
46	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	complemented
47	#95×C₄₃	—	2	2	no	—
48	#95×C₄₃	—	2	1	no	—
49	C₂⁴×C₄₃	—	2	1	no	—
50	C₂×D₄×C₄₃	—	2	1	no	—
51	C₂×D₄×C₄₃	—	2	2	no	—
52	C₂×D₄×C₄₃	C₄²	1	2	yes	—
53	C₂×D₄×C₄₃	M₁₆	1	4	yes	—
54	C₂²×C₄×C₄₃	—	2	8	no	—
55	C₂²×C₄×C₄₃	—	2	4	no	—
56	C₂²×C₄×C₄₃	—	2	1	no	—
57	C₂²×C₄×C₄₃	—	2	2	no	—
58	C₂²×C₄×C₄₃	—	4	4	no	—
59	C₂²×C₄×C₄₃	—	4	4	no	—
60	C₂²×C₄×C₄₃	—	4	1	no	—
61	C₂²×C₄×C₄₃	—	4	1	no	—
62	C₂²×C₄×C₄₃	—	4	2	no	—
63	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₄₃	—	2	4	no	—
69	C₂×C₈×C₄₃	—	2	4	no	—
70	C₂×C₈×C₄₃	—	4	4	no	—
71	C₂×C₈×C₄₃	—	4	2	no	—
72	C₂×C₈×C₄₃	—	4	2	no	—
73	C₄⋊C₄×C₄₃	—	2	1	no	—
74	C₄⋊C₄×C₄₃	—	2	2	no	—
75	C₄⋊C₄×C₄₃	C₄²	1	2	yes	—
76	C₄⋊C₄×C₄₃	M₁₆	1	4	yes	—
77	C₂²⋊C₄×C₄₃	—	2	8	no	—
78	C₂²⋊C₄×C₄₃	—	2	4	no	—
79	C₄²×C₄₃	—	2	2	no	—
80	C₄²×C₄₃	—	2	2	no	—
81	C₄²×C₄₃	—	2	2	no	—
82	C₄²×C₄₃	—	4	2	no	—
83	C₄²×C₄₃	—	4	4	no	—
84	C₄²×C₄₃	—	4	1	no	—
85	C₄²×C₄₃	—	4	1	no	—
86	C₄²×C₄₃	D₈	1	2	yes	—
87	C₄²×C₄₃	SD₁₆	1	2	yes	—
88	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₄₃	#41	1	2	yes	—
90	C₂³×C₄₃	—	2	2	no	—
91	C₂³×C₄₃	—	2	4	no	—
92	C₂³×C₄₃	—	4	1	no	—
93	C₂³×C₄₃	—	4	1	no	—
94	C₂³×C₄₃	—	4	2	no	—
95	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	D₄×C₄₃	—	2	2	no	—
97	D₄×C₄₃	—	2	4	no	—
98	D₄×C₄₃	#97	1	4	yes	complemented
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	—	2	2	no	—
103	C₂×C₄×C₄₃	—	2	2	no	—
104	C₂×C₄×C₄₃	—	2	4	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	2	no	—
107	C₂×C₄×C₄₃	—	4	16	no	—
108	C₂×C₄×C₄₃	—	4	2	no	—
109	C₂×C₄×C₄₃	—	4	1	no	—
110	C₂×C₄×C₄₃	—	4	1	no	—
111	C₂×C₄×C₄₃	—	4	2	no	—
112	C₂×C₄×C₄₃	—	4	4	no	—
113	C₂×C₄×C₄₃	—	4	4	no	—
114	C₂×C₄×C₄₃	#102	1	4	yes	—
115	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
120	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
121	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₄₃	#135	1	2	yes	—
124	C₈×C₄₃	—	4	4	no	—
125	C₈×C₄₃	—	4	4	no	—
126	#8174	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₄₃	—	2	4	no	—
128	C₂²×C₄₃	—	4	4	no	—
129	C₂²×C₄₃	—	4	4	no	—
130	C₂²×C₄₃	—	4	2	no	—
131	C₂²×C₄₃	—	4	2	no	—
132	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₄₃	#407	1	2	yes	—
134	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
135	C₂²×C₄₃	#416	1	1	yes	characteristic
136	C₂²×C₄₃	#421	1	2	yes	—
137	C₄×C₄₃	—	2	4	no	—
138	C₄×C₄₃	—	2	2	no	—
139	C₄×C₄₃	—	4	4	no	—
140	C₄×C₄₃	—	4	2	no	—
141	C₄×C₄₃	—	4	2	no	—
142	C₄×C₄₃	C₂×#97	1	1	yes	characteristic
143	C₄×C₄₃	C₄×M₁₆	1	1	yes	characteristic
144	C₄×C₄₃	#432	1	2	yes	—
145	C₄×C₄₃	C₄×D₈	1	2	yes	—
146	C₄×C₄₃	C₄×SD₁₆	1	2	yes	—
147	C₄²×D₄	C₂×C₄₃	1	1	yes	characteristic
148	#1767	C₂×C₄₃	1	1	yes	characteristic, complemented
149	#1744	C₂×C₄₃	1	1	yes	characteristic, complemented
150	C₂×D₄⋊C₈	C₂×C₄₃	1	2	yes	—
151	#1279	C₂×C₄₃	1	2	yes	complemented
152	C₂×C₄₃	—	4	4	no	—
153	C₂×C₄₃	#1486	1	2	yes	—
154	C₂×C₄₃	#1488	1	2	yes	—
155	C₂×C₄₃	#1749	1	1	yes	characteristic
156	C₂×C₄₃	#1759	1	1	yes	characteristic
157	C₂×C₄₃	C₄×#102	1	1	yes	characteristic
158	C₂×C₄×D₄	C₄×C₄₃	1	2	yes	—
159	C₂×C₄×D₄	C₂²×C₄₃	1	1	yes	characteristic
160	C₂²×C₄²	—	2	1	no	—
161	C₂×C₄⋊C₈	C₂²×C₄₃	1	2	yes	—
162	C₂×#97	—	2	1	no	—
163	C₂×C₄×C₈	C₂²×C₄₃	1	2	yes	—
164	C₄×C₄⋊C₄	C₄×C₄₃	1	2	yes	—
165	C₄×C₄⋊C₄	C₂²×C₄₃	1	1	yes	characteristic
166	C₄×C₂²⋊C₄	—	2	1	no	—
167	C₄×C₂²⋊C₄	—	2	2	no	—
168	C₄³	C₂²×C₄₃	1	1	yes	characteristic
169	#336	—	2	2	no	—
170	#336	—	2	1	no	—
171	D₄⋊C₈	C₄×C₄₃	1	8	yes	complemented
172	C₄₃	#8174	1	1	yes	characteristic, direct product factor, Sylow
173	C₂²×D₄	C₂×C₄×C₄₃	1	1	yes	characteristic
174	C₂³×C₄	—	2	2	no	—
175	C₂³×C₄	—	2	1	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₄×D₄	—	2	4	no	—
179	C₄×D₄	—	2	4	no	—
180	C₄×D₄	—	2	2	no	—
181	C₄×D₄	C₂×C₄×C₄₃	1	4	yes	—
182	C₂×C₄⋊C₄	C₂×C₄×C₄₃	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×C₄×C₄₃	1	2	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₂²⋊C₄	—	2	1	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	1	no	—
188	C₂×C₄²	—	4	2	no	—
189	C₂×C₄²	—	4	2	no	—
190	C₂×C₄²	C₂×C₄×C₄₃	1	2	yes	—
191	C₂×C₄²	D₄×C₄₃	1	1	yes	characteristic
192	C₂×C₄²	D₄×C₄₃	1	2	yes	—
193	C₂×C₄²	D₄×C₄₃	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	C₂×C₄×C₄₃	1	4	yes	complemented
197	#97	—	4	2	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	C₂×C₄×C₄₃	1	4	yes	complemented
200	#95	—	2	2	no	—
201	#95	—	2	1	no	—
202	C₂⁴	—	2	1	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	C₄²×C₄₃	1	2	yes	—
206	C₂×D₄	M₁₆×C₄₃	1	4	yes	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
217	C₂²×C₄	C₂²⋊C₄×C₄₃	1	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	4	4	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	C₄²×C₄₃	1	2	yes	—
229	C₄⋊C₄	M₁₆×C₄₃	1	4	yes	—
230	C₂²⋊C₄	—	2	8	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	4	2	no	—
236	C₄²	—	4	4	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	D₈×C₄₃	1	2	yes	—
240	C₄²	SD₁₆×C₄₃	1	2	yes	—
241	C₄²	C₂²×C₄×C₄₃	1	2	yes	—
242	C₄²	#41×C₄₃	1	2	yes	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	4	no	—
245	C₂³	—	4	1	no	—
246	C₂³	—	4	1	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
249	D₄	—	2	2	no	—
250	D₄	—	2	4	no	—
251	D₄	#97×C₄₃	1	4	yes	complemented
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	16	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	#102×C₄₃	1	4	yes	—
268	C₂×C₄	C₂×C₄²×C₄₃	1	1	yes	characteristic
269	C₂×C₄	#117×C₄₃	1	1	yes	characteristic
270	C₂×C₄	C₄×D₄×C₄₃	1	2	yes	—
271	C₂×C₄	C₄×D₄×C₄₃	1	2	yes	—
272	C₂×C₄	C₂×M₁₆×C₄₃	1	2	yes	—
273	C₂×C₄	C₂×M₁₆×C₄₃	1	2	yes	—
274	C₂×C₄	C₂×D₈×C₄₃	1	1	yes	characteristic
275	C₂×C₄	C₂×SD₁₆×C₄₃	1	1	yes	characteristic
276	C₂×C₄	#135×C₄₃	1	2	yes	—
277	C₈	—	4	4	no	—
278	C₈	—	4	4	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	4	4	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	C₄×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
285	C₂²	C₄₃×#407	1	2	yes	—
286	C₂²	C₂×#102×C₄₃	1	1	yes	characteristic
287	C₂²	C₄₃×#416	1	1	yes	characteristic
288	C₂²	C₄₃×#421	1	2	yes	—
289	C₄	—	2	4	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	4	no	—
292	C₄	—	4	2	no	—
293	C₄	—	4	2	no	—
294	C₄	C₂×#97×C₄₃	1	1	yes	characteristic
295	C₄	C₄×M₁₆×C₄₃	1	1	yes	characteristic
296	C₄	C₄₃×#432	1	2	yes	—
297	C₄	C₄×D₈×C₄₃	1	2	yes	—
298	C₄	C₄×SD₁₆×C₄₃	1	2	yes	—
299	C₂	—	4	4	no	—
300	C₂	C₄₃×#1486	1	2	yes	—
301	C₂	C₄₃×#1488	1	2	yes	—
302	C₂	C₄₃×#1749	1	1	yes	characteristic
303	C₂	C₄₃×#1759	1	1	yes	characteristic
304	C₂	C₄×#102×C₄₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₄₃×#1767	C₂	semidirect
3. C₄₃×#1744	C₂	semidirect
5. C₄₃×#1279	C₂	semidirect
19. C₄₃×D₄⋊C₈	C₄	semidirect
43. C₄⋊C₈×C₄₃	C₂×C₄	semidirect
46. C₄×C₈×C₄₃	C₂×C₄	semidirect
98. D₄×C₄₃	#97	semidirect
126. #8174	C₄₃	direct (N × H)
148. #1767	C₂×C₄₃	semidirect
149. #1744	C₂×C₄₃	semidirect
151. #1279	C₂×C₄₃	semidirect
171. D₄⋊C₈	C₄×C₄₃	semidirect
172. C₄₃	#8174	direct (N × H)
196. C₄⋊C₈	C₂×C₄×C₄₃	semidirect
199. C₄×C₈	C₂×C₄×C₄₃	semidirect
251. D₄	#97×C₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₄×C₄₃	2
C₄₃×#1767	2
C₄₃×#1744	2
C₂×C₄₃×D₄⋊C₈	2
C₄₃×#1279	2
#8174	43

Subgroups — C43×#8174 #24928191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#8174 #24928191