Subgroups — C₂×C₄₃×#1620 #24938495

Order 11008.

1190 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₃×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₃×#493	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₃×#460	C₂	1	2	yes	maximal, complemented
4	C₄₃×#1620	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₃×#331	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₃×Q₈⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₂²×SD₁₆×C₄₃	—	2	2	no	—
8	C₂²×D₈×C₄₃	—	2	2	no	—
9	C₄₃×#559	—	2	2	no	—
10	C₄₃×#559	—	2	4	no	—
11	C₄₃×#559	C₂²	1	4	yes	complemented
12	C₂×#127×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×#127×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×#124×C₄₃	—	2	2	no	—
15	C₂×#123×C₄₃	—	2	2	no	—
16	C₂×C₄×Q₈×C₄₃	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₄₃	—	2	2	no	—
18	C₄₃×#493	—	2	2	no	—
19	C₄₃×#493	C₂²	1	4	yes	—
20	C₄₃×#460	—	2	4	no	—
21	C₄₃×#460	C₂²	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
23	C₂×#102×C₄₃	—	2	2	no	—
24	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#331	C₂²	1	4	yes	complemented
26	C₄₃×Q₈⋊C₈	C₂²	1	8	yes	complemented
27	C₂²×Q₈×C₄₃	D₄	1	2	yes	—
28	C₂²×D₄×C₄₃	—	2	2	no	—
29	C₂²×D₄×C₄₃	—	2	1	no	—
30	C₂²×D₄×C₄₃	—	2	2	no	—
31	C₂²×D₄×C₄₃	D₄	1	2	yes	—
32	C₂³×C₄×C₄₃	—	2	2	no	—
33	C₂×SD₁₆×C₄₃	—	2	8	no	—
34	C₂×SD₁₆×C₄₃	—	4	8	no	—
35	C₂×D₈×C₄₃	—	2	8	no	—
36	C₂×D₈×C₄₃	—	4	8	no	—
37	C₂²×C₈×C₄₃	—	2	1	no	—
38	C₂²×C₈×C₄₃	—	2	2	no	—
39	#127×C₄₃	—	2	2	no	—
40	#127×C₄₃	—	2	2	no	—
41	#127×C₄₃	—	2	1	no	—
42	#127×C₄₃	D₄	1	2	yes	—
43	#127×C₄₃	C₂³	1	2	yes	—
44	#127×C₄₃	C₂³	1	2	yes	—
45	#124×C₄₃	—	2	8	no	—
46	#124×C₄₃	—	2	8	no	—
47	#123×C₄₃	—	2	8	no	—
48	#123×C₄₃	—	2	8	no	—
49	C₄×Q₈×C₄₃	—	2	2	no	—
50	C₄×Q₈×C₄₃	—	2	2	no	—
51	C₄×Q₈×C₄₃	D₄	1	4	yes	—
52	C₄×Q₈×C₄₃	C₂³	1	4	yes	—
53	C₄×D₄×C₄₃	—	2	8	no	—
54	C₄×D₄×C₄₃	—	2	8	no	—
55	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
57	C₂×C₂²⋊C₄×C₄₃	—	2	4	no	—
58	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
59	C₂×C₄²×C₄₃	—	2	2	no	—
60	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₄₃	—	2	2	no	—
62	C₄⋊C₈×C₄₃	C₂³	1	4	yes	—
63	#102×C₄₃	—	2	8	no	—
64	C₄×C₈×C₄₃	—	2	1	no	—
65	C₄×C₈×C₄₃	C₂³	1	2	yes	—
66	C₂⁴×C₄₃	—	2	2	no	—
67	C₂⁴×C₄₃	—	4	1	no	—
68	C₂×Q₈×C₄₃	—	2	2	no	—
69	C₂×Q₈×C₄₃	—	2	4	no	—
70	C₂×Q₈×C₄₃	—	2	4	no	—
71	C₂×Q₈×C₄₃	C₂×D₄	1	4	yes	—
72	C₂×D₄×C₄₃	—	2	4	no	—
73	C₂×D₄×C₄₃	—	2	2	no	—
74	C₂×D₄×C₄₃	—	2	4	no	—
75	C₂×D₄×C₄₃	—	2	4	no	—
76	C₂×D₄×C₄₃	—	2	2	no	—
77	C₂×D₄×C₄₃	—	4	1	no	—
78	C₂×D₄×C₄₃	—	4	2	no	—
79	C₂×D₄×C₄₃	—	4	4	no	—
80	C₂×D₄×C₄₃	—	4	4	no	—
81	C₂×D₄×C₄₃	—	4	4	no	—
82	C₂×D₄×C₄₃	—	4	4	no	—
83	C₂×D₄×C₄₃	—	4	4	no	—
84	C₂×D₄×C₄₃	—	4	4	no	—
85	C₂×D₄×C₄₃	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₄₃	—	2	8	no	—
87	C₂²×C₄×C₄₃	—	2	2	no	—
88	C₂²×C₄×C₄₃	—	2	2	no	—
89	C₂²×C₄×C₄₃	—	8	4	no	—
90	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
92	SD₁₆×C₄₃	—	4	16	no	—
93	D₈×C₄₃	—	4	16	no	—
94	C₂×C₈×C₄₃	—	2	4	no	—
95	C₂×C₈×C₄₃	—	2	2	no	—
96	C₂×C₈×C₄₃	—	2	4	no	—
97	C₂×C₈×C₄₃	—	4	4	no	—
98	C₄⋊C₄×C₄₃	—	2	4	no	—
99	C₄⋊C₄×C₄₃	—	2	2	no	—
100	C₄⋊C₄×C₄₃	—	2	4	no	—
101	C₄⋊C₄×C₄₃	C₂×D₄	1	4	yes	—
102	C₂²⋊C₄×C₄₃	—	2	16	no	—
103	C₂²⋊C₄×C₄₃	—	2	8	no	—
104	C₄²×C₄₃	—	2	4	no	—
105	C₄²×C₄₃	—	2	4	no	—
106	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₄₃	C₂×D₄	1	2	yes	—
108	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
109	C₂³×C₄₃	—	2	4	no	—
110	C₂³×C₄₃	—	4	8	no	—
111	C₂³×C₄₃	—	4	4	no	—
112	C₂³×C₄₃	—	4	2	no	—
113	C₂³×C₄₃	—	8	4	no	—
114	C₂³×C₄₃	—	8	2	no	—
115	C₂³×C₄₃	#120	1	1	yes	characteristic
116	Q₈×C₄₃	—	2	4	no	—
117	Q₈×C₄₃	—	2	4	no	—
118	Q₈×C₄₃	—	4	4	no	—
119	D₄×C₄₃	—	4	4	no	—
120	D₄×C₄₃	—	4	4	no	—
121	D₄×C₄₃	—	4	4	no	—
122	D₄×C₄₃	—	4	4	no	—
123	D₄×C₄₃	—	4	4	no	—
124	D₄×C₄₃	—	4	4	no	—
125	D₄×C₄₃	—	4	8	no	—
126	D₄×C₄₃	—	8	2	no	—
127	D₄×C₄₃	—	8	2	no	—
128	D₄×C₄₃	—	8	4	no	—
129	C₂×C₄×C₄₃	—	2	2	no	—
130	C₂×C₄×C₄₃	—	2	4	no	—
131	C₂×C₄×C₄₃	—	2	4	no	—
132	C₂×C₄×C₄₃	—	4	4	no	—
133	C₂×C₄×C₄₃	—	4	4	no	—
134	C₂×C₄×C₄₃	—	8	8	no	—
135	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
136	C₂×C₄×C₄₃	#120	1	2	yes	—
137	C₂×C₄×C₄₃	#135	1	4	yes	—
138	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
139	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
141	C₈×C₄₃	—	2	4	no	—
142	C₈×C₄₃	—	4	4	no	—
143	C₂×#1620	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₄₃	—	4	8	no	—
145	C₂²×C₄₃	—	4	4	no	—
146	C₂²×C₄₃	—	8	8	no	—
147	C₂²×C₄₃	—	8	4	no	—
148	C₂²×C₄₃	—	8	2	no	—
149	C₂²×C₄₃	—	16	4	no	—
150	C₂²×C₄₃	#449	1	4	yes	—
151	C₂²×C₄₃	#453	1	2	yes	—
152	C₂²×C₄₃	C₂×#120	1	1	yes	characteristic
153	C₄×C₄₃	—	2	1	no	—
154	C₄×C₄₃	—	2	1	no	—
155	C₄×C₄₃	—	4	4	no	—
156	C₄×C₄₃	—	4	4	no	—
157	C₄×C₄₃	#449	1	4	yes	—
158	C₄×C₄₃	C₂×#135	1	2	yes	—
159	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
160	C₂×#559	C₂×C₄₃	1	1	yes	characteristic, complemented
161	C₂×#493	C₂×C₄₃	1	1	yes	characteristic
162	C₂×#460	C₂×C₄₃	1	2	yes	complemented
163	#1620	C₂×C₄₃	1	8	yes	direct product factor
164	C₂×#331	C₂×C₄₃	1	1	yes	characteristic, complemented
165	C₂×Q₈⋊C₈	C₂×C₄₃	1	2	yes	complemented
166	C₂×C₄₃	—	8	4	no	—
167	C₂×C₄₃	—	16	2	no	—
168	C₂×C₄₃	#1620	1	4	yes	direct product factor
169	C₂×C₄₃	C₂×#449	1	2	yes	—
170	C₂×C₄₃	C₂×#453	1	1	yes	characteristic
171	C₂²×SD₁₆	—	2	2	no	—
172	C₂²×D₈	—	2	2	no	—
173	#559	—	2	2	no	—
174	#559	—	2	4	no	—
175	#559	C₂²×C₄₃	1	4	yes	complemented
176	C₂×#127	C₂²×C₄₃	1	1	yes	characteristic
177	C₂×#127	C₂²×C₄₃	1	1	yes	characteristic
178	C₂×#124	—	2	2	no	—
179	C₂×#123	—	2	2	no	—
180	C₂×C₄×Q₈	C₂²×C₄₃	1	2	yes	—
181	C₂×C₄×D₄	—	2	2	no	—
182	#493	—	2	2	no	—
183	#493	C₂²×C₄₃	1	4	yes	—
184	#460	—	2	4	no	—
185	#460	C₂²×C₄₃	1	8	yes	complemented
186	C₂×C₄⋊C₈	C₂²×C₄₃	1	2	yes	—
187	C₂×#102	—	2	2	no	—
188	C₂×C₄×C₈	C₂²×C₄₃	1	1	yes	characteristic
189	#331	C₂²×C₄₃	1	4	yes	complemented
190	Q₈⋊C₈	C₂²×C₄₃	1	8	yes	complemented
191	C₄₃	C₂×#1620	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²×Q₈	D₄×C₄₃	1	2	yes	—
193	C₂²×D₄	—	2	2	no	—
194	C₂²×D₄	—	2	1	no	—
195	C₂²×D₄	—	2	2	no	—
196	C₂²×D₄	D₄×C₄₃	1	2	yes	—
197	C₂³×C₄	—	2	2	no	—
198	C₂×SD₁₆	—	2	8	no	—
199	C₂×SD₁₆	—	4	8	no	—
200	C₂×D₈	—	2	8	no	—
201	C₂×D₈	—	4	8	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	2	2	no	—
204	#127	—	2	2	no	—
205	#127	—	2	2	no	—
206	#127	—	2	1	no	—
207	#127	D₄×C₄₃	1	2	yes	—
208	#127	C₂³×C₄₃	1	2	yes	—
209	#127	C₂³×C₄₃	1	2	yes	—
210	#124	—	2	8	no	—
211	#124	—	2	8	no	—
212	#123	—	2	8	no	—
213	#123	—	2	8	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×Q₈	D₄×C₄₃	1	4	yes	—
217	C₄×Q₈	C₂³×C₄₃	1	4	yes	—
218	C₄×D₄	—	2	8	no	—
219	C₄×D₄	—	2	8	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	D₄×C₄₃	1	2	yes	—
222	C₂×C₂²⋊C₄	—	2	4	no	—
223	C₂×C₂²⋊C₄	—	2	2	no	—
224	C₂×C₄²	—	2	2	no	—
225	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	C₂³×C₄₃	1	4	yes	—
228	#102	—	2	8	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	C₂³×C₄₃	1	2	yes	—
231	C₂⁴	—	2	2	no	—
232	C₂⁴	—	4	1	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	C₂×D₄×C₄₃	1	4	yes	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	4	no	—
243	C₂×D₄	—	4	4	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	4	no	—
247	C₂×D₄	—	4	4	no	—
248	C₂×D₄	—	4	4	no	—
249	C₂×D₄	—	4	4	no	—
250	C₂×D₄	C₂×D₄×C₄₃	1	4	yes	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	8	no	—
254	C₂²×C₄	—	8	4	no	—
255	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
257	SD₁₆	—	4	16	no	—
258	D₈	—	4	16	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	4	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	C₂×D₄×C₄₃	1	4	yes	—
267	C₂²⋊C₄	—	2	16	no	—
268	C₂²⋊C₄	—	2	8	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	4	no	—
271	C₄²	C₂×D₄×C₄₃	1	1	yes	characteristic
272	C₄²	C₂×D₄×C₄₃	1	2	yes	—
273	C₄²	C₂⁴×C₄₃	1	1	yes	characteristic
274	C₂³	—	2	4	no	—
275	C₂³	—	4	8	no	—
276	C₂³	—	4	4	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	8	4	no	—
279	C₂³	—	8	2	no	—
280	C₂³	#120×C₄₃	1	1	yes	characteristic
281	Q₈	—	2	4	no	—
282	Q₈	—	2	4	no	—
283	Q₈	—	4	4	no	—
284	D₄	—	4	4	no	—
285	D₄	—	4	4	no	—
286	D₄	—	4	4	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	8	no	—
291	D₄	—	8	2	no	—
292	D₄	—	8	2	no	—
293	D₄	—	8	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	8	8	no	—
300	C₂×C₄	#120×C₄₃	1	1	yes	characteristic
301	C₂×C₄	#120×C₄₃	1	2	yes	—
302	C₂×C₄	#135×C₄₃	1	4	yes	—
303	C₂×C₄	AGL(1,ℤ/8)×C₄₃	1	4	yes	—
304	C₂×C₄	C₂²×D₄×C₄₃	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₄₃	1	2	yes	—
306	C₈	—	2	4	no	—
307	C₈	—	4	4	no	—
308	C₂²	—	4	8	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	8	8	no	—
311	C₂²	—	8	4	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	16	4	no	—
314	C₂²	C₄₃×#449	1	4	yes	—
315	C₂²	C₄₃×#453	1	2	yes	—
316	C₂²	C₂×#120×C₄₃	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	4	no	—
320	C₄	—	4	4	no	—
321	C₄	C₄₃×#449	1	4	yes	—
322	C₄	C₂×#135×C₄₃	1	2	yes	—
323	C₄	C₂×AGL(1,ℤ/8)×C₄₃	1	2	yes	—
324	C₂	—	8	4	no	—
325	C₂	—	16	2	no	—
326	C₂	C₄₃×#1620	1	4	yes	direct product factor
327	C₂	C₂×C₄₃×#449	1	2	yes	—
328	C₂	C₂×C₄₃×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₃	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₃
prime radical	C₂×C₄₃×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₃ ◁ C₂×C₄²×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₃×#559	C₂	semidirect
3. C₂×C₄₃×#460	C₂	semidirect
4. C₄₃×#1620	C₂	direct (N × H)
5. C₂×C₄₃×#331	C₂	semidirect
6. C₂×C₄₃×Q₈⋊C₈	C₂	semidirect
11. C₄₃×#559	C₂²	semidirect
21. C₄₃×#460	C₂²	semidirect
25. C₄₃×#331	C₂²	semidirect
26. C₄₃×Q₈⋊C₈	C₂²	semidirect
143. C₂×#1620	C₄₃	direct (N × H)
160. C₂×#559	C₂×C₄₃	semidirect
162. C₂×#460	C₂×C₄₃	semidirect
163. #1620	C₂×C₄₃	direct (N × H)
164. C₂×#331	C₂×C₄₃	semidirect
165. C₂×Q₈⋊C₈	C₂×C₄₃	semidirect
168. C₂×C₄₃	#1620	direct (N × H)
175. #559	C₂²×C₄₃	semidirect
185. #460	C₂²×C₄₃	semidirect
189. #331	C₂²×C₄₃	semidirect
190. Q₈⋊C₈	C₂²×C₄₃	semidirect
191. C₄₃	C₂×#1620	direct (N × H)
326. C₂	C₄₃×#1620	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×C₄₃×#559	2
C₂×C₄₃×#493	2
C₂×C₄₃×#460	2
C₄₃×#1620	2
C₂×C₄₃×#331	2
C₂×C₄₃×Q₈⋊C₈	2
C₂×#1620	43

Subgroups — C2×C43×#1620 #24938495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₃×#1620 #24938495