Subgroups — C₂×C₄₃×#1573 #24937821

Order 11008.

726 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₃×#431	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#1573	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₃×#335	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₃×#334	C₂	1	2	yes	maximal, complemented
7	C₂³×C₈×C₄₃	—	2	1	no	—
8	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
9	C₄₃×#432	—	2	2	no	—
10	C₄₃×#432	C₂²	1	4	yes	—
11	C₄₃×#431	C₄	1	4	yes	—
12	C₄₃×#431	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₄₃	—	2	2	no	—
14	C₂×C₄⋊C₈×C₄₃	—	2	1	no	—
15	C₂×C₄⋊C₈×C₄₃	C₄	1	2	yes	—
16	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₄₃	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₈×C₄₃	—	2	1	no	—
21	C₂×C₂²⋊C₈×C₄₃	C₄	1	2	yes	—
22	C₂×#97×C₄₃	—	2	1	no	—
23	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
24	C₄₃×#335	C₂²	1	8	yes	complemented
25	C₄₃×#334	C₂²	1	8	yes	complemented
26	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₃	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₄₃	—	2	4	no	—
30	C₂²×C₈×C₄₃	—	2	2	no	—
31	C₂²×C₈×C₄₃	—	2	1	no	—
32	C₂²×C₈×C₄₃	—	2	1	no	—
33	C₂²×C₈×C₄₃	—	4	4	no	—
34	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₄₃	D₄	1	2	yes	—
38	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₃	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
41	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₄₃	—	2	4	no	—
44	C₄⋊C₈×C₄₃	—	2	2	no	—
45	C₄⋊C₈×C₄₃	—	2	1	no	—
46	C₄⋊C₈×C₄₃	—	2	1	no	—
47	C₄⋊C₈×C₄₃	C₂×C₄	1	8	yes	—
48	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
49	C₄⋊C₈×C₄₃	C₂³	1	4	yes	—
50	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
51	C₂²⋊C₈×C₄₃	—	2	4	no	—
52	C₂²⋊C₈×C₄₃	C₂×C₄	1	8	yes	—
53	#97×C₄₃	—	2	4	no	—
54	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
55	C₄×C₈×C₄₃	C₂³	1	4	yes	—
56	C₂⁴×C₄₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	—	2	4	no	—
58	C₂²×C₄×C₄₃	—	2	2	no	—
59	C₂²×C₄×C₄₃	—	2	2	no	—
60	C₂²×C₄×C₄₃	—	2	4	no	—
61	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₈×C₄₃	—	2	8	no	—
74	C₂×C₈×C₄₃	—	2	2	no	—
75	C₂×C₈×C₄₃	—	2	2	no	—
76	C₂×C₈×C₄₃	—	2	4	no	—
77	C₂×C₈×C₄₃	—	4	8	no	—
78	C₂×C₈×C₄₃	—	4	2	no	—
79	C₂×C₈×C₄₃	—	4	4	no	—
80	C₂×C₈×C₄₃	—	4	2	no	—
81	C₂×C₈×C₄₃	C₂²×C₄	1	8	yes	—
82	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
85	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
87	C₄²×C₄₃	C₂×D₄	1	2	yes	—
88	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
90	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
92	C₂³×C₄₃	—	2	2	no	—
93	C₂³×C₄₃	—	2	2	no	—
94	C₂³×C₄₃	#108	1	4	yes	—
95	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₃	—	2	2	no	—
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	2	4	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	4	no	—
102	C₂×C₄×C₄₃	—	2	2	no	—
103	C₂×C₄×C₄₃	—	2	4	no	—
104	C₂×C₄×C₄₃	—	2	2	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	4	no	—
107	C₂×C₄×C₄₃	—	2	4	no	—
108	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	#131	1	4	yes	—
115	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₄₃	#137	1	2	yes	—
117	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
120	C₈×C₄₃	—	2	8	no	—
121	C₈×C₄₃	—	4	2	no	—
122	C₈×C₄₃	—	4	4	no	—
123	C₈×C₄₃	—	4	2	no	—
124	C₂×#1573	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₄₃	—	2	8	no	—
126	C₂²×C₄₃	—	2	2	no	—
127	C₂²×C₄₃	—	2	2	no	—
128	C₂²×C₄₃	#424	1	2	yes	—
129	C₂²×C₄₃	#428	1	2	yes	—
130	C₂²×C₄₃	C₂×#108	1	4	yes	—
131	C₂²×C₄₃	C₂×#108	1	2	yes	—
132	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₄₃	—	2	2	no	—
134	C₄×C₄₃	—	2	2	no	—
135	C₄×C₄₃	—	2	1	no	—
136	C₄×C₄₃	—	2	2	no	—
137	C₄×C₄₃	—	2	1	no	—
138	C₄×C₄₃	#424	1	2	yes	—
139	C₄×C₄₃	#428	1	2	yes	—
140	C₄×C₄₃	C₂×#131	1	2	yes	—
141	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
143	C₂×#432	C₂×C₄₃	1	1	yes	characteristic
144	C₂×#431	C₂×C₄₃	1	1	yes	characteristic
145	C₂²×C₄⋊C₈	C₂×C₄₃	1	1	yes	characteristic
146	#1573	C₂×C₄₃	1	8	yes	direct product factor
147	C₂×#335	C₂×C₄₃	1	2	yes	complemented
148	C₂×#334	C₂×C₄₃	1	2	yes	complemented
149	C₂×C₄₃	—	2	4	no	—
150	C₂×C₄₃	#1573	1	4	yes	direct product factor
151	C₂×C₄₃	C₂×#424	1	1	yes	characteristic
152	C₂×C₄₃	C₂×#428	1	1	yes	characteristic
153	C₂×C₄₃	C₂²×#108	1	1	yes	characteristic
154	C₂³×C₈	—	2	1	no	—
155	C₂²×C₄²	C₂²×C₄₃	1	1	yes	characteristic
156	#432	—	2	2	no	—
157	#432	C₂²×C₄₃	1	4	yes	—
158	#431	C₄×C₄₃	1	4	yes	—
159	#431	C₂²×C₄₃	1	4	yes	—
160	C₂×C₄⋊C₈	—	2	1	no	—
161	C₂×C₄⋊C₈	—	2	2	no	—
162	C₂×C₄⋊C₈	C₄×C₄₃	1	2	yes	—
163	C₂×C₄⋊C₈	C₂²×C₄₃	1	1	yes	characteristic
164	C₂×C₄⋊C₈	C₂²×C₄₃	1	2	yes	—
165	C₂×C₄⋊C₈	C₂²×C₄₃	1	1	yes	characteristic
166	C₂×C₄⋊C₈	C₂²×C₄₃	1	4	yes	—
167	C₂×C₂²⋊C₈	—	2	1	no	—
168	C₂×C₂²⋊C₈	C₄×C₄₃	1	2	yes	—
169	C₂×#97	—	2	1	no	—
170	C₂×C₄×C₈	C₂²×C₄₃	1	2	yes	—
171	#335	C₂²×C₄₃	1	8	yes	complemented
172	#334	C₂²×C₄₃	1	8	yes	complemented
173	C₄₃	C₂×#1573	1	1	yes	characteristic, direct product factor, Sylow
174	C₂³×C₄	C₂×C₄×C₄₃	1	1	yes	characteristic
175	C₂³×C₄	D₄×C₄₃	1	1	yes	characteristic
176	C₂³×C₄	Q₈×C₄₃	1	1	yes	characteristic
177	C₂²×C₈	—	2	4	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	4	4	no	—
182	C₂²×C₈	C₂×C₄×C₄₃	1	4	yes	—
183	C₂×C₄²	C₂×C₄×C₄₃	1	1	yes	characteristic
184	C₂×C₄²	C₂×C₄×C₄₃	1	2	yes	—
185	C₂×C₄²	D₄×C₄₃	1	1	yes	characteristic
186	C₂×C₄²	D₄×C₄₃	1	2	yes	—
187	C₂×C₄²	Q₈×C₄₃	1	1	yes	characteristic
188	C₂×C₄²	Q₈×C₄₃	1	2	yes	—
189	C₂×C₄²	C₂³×C₄₃	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₄₃	1	2	yes	—
191	C₄⋊C₈	—	2	4	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	C₂×C₄×C₄₃	1	8	yes	—
196	C₄⋊C₈	C₂³×C₄₃	1	2	yes	—
197	C₄⋊C₈	C₂³×C₄₃	1	4	yes	—
198	C₄⋊C₈	C₂³×C₄₃	1	2	yes	—
199	C₂²⋊C₈	—	2	4	no	—
200	C₂²⋊C₈	C₂×C₄×C₄₃	1	8	yes	—
201	#97	—	2	4	no	—
202	C₄×C₈	C₂×C₄×C₄₃	1	4	yes	—
203	C₄×C₈	C₂³×C₄₃	1	4	yes	—
204	C₂⁴	C₄⋊C₄×C₄₃	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₄⋊C₄×C₄₃	1	2	yes	—
210	C₂²×C₄	C₄⋊C₄×C₄₃	1	1	yes	characteristic
211	C₂²×C₄	C₄⋊C₄×C₄₃	1	1	yes	characteristic
212	C₂²×C₄	C₄⋊C₄×C₄₃	1	2	yes	—
213	C₂²×C₄	C₄⋊C₄×C₄₃	1	1	yes	characteristic
214	C₂²×C₄	C₄⋊C₄×C₄₃	1	2	yes	—
215	C₂²×C₄	C₂²×C₄×C₄₃	1	2	yes	—
216	C₂²×C₄	C₂²×C₄×C₄₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×Q₈×C₄₃	1	2	yes	—
220	C₂²×C₄	C₂×Q₈×C₄₃	1	1	yes	characteristic
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	4	8	no	—
226	C₂×C₈	—	4	4	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	C₂²×C₄×C₄₃	1	8	yes	—
230	C₄²	C₄⋊C₄×C₄₃	1	2	yes	—
231	C₄²	C₄⋊C₄×C₄₃	1	2	yes	—
232	C₄²	C₄⋊C₄×C₄₃	1	2	yes	—
233	C₄²	C₂²×C₄×C₄₃	1	1	yes	characteristic
234	C₄²	C₂²×C₄×C₄₃	1	2	yes	—
235	C₄²	C₂×D₄×C₄₃	1	2	yes	—
236	C₄²	C₂×D₄×C₄₃	1	1	yes	characteristic
237	C₄²	C₂×Q₈×C₄₃	1	2	yes	—
238	C₄²	C₂×Q₈×C₄₃	1	1	yes	characteristic
239	C₄²	C₂⁴×C₄₃	1	1	yes	characteristic
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	#108×C₄₃	1	4	yes	—
243	C₂³	C₂×C₄⋊C₄×C₄₃	1	2	yes	—
244	C₂³	C₂×C₄⋊C₄×C₄₃	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	C₂×C₄⋊C₄×C₄₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊C₄×C₄₃	1	2	yes	—
258	C₂×C₄	C₂×C₄⋊C₄×C₄₃	1	2	yes	—
259	C₂×C₄	C₂×C₄⋊C₄×C₄₃	1	2	yes	—
260	C₂×C₄	C₂×C₄⋊C₄×C₄₃	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊C₄×C₄₃	1	1	yes	characteristic
262	C₂×C₄	#131×C₄₃	1	4	yes	—
263	C₂×C₄	AGL(1,ℤ/8)×C₄₃	1	2	yes	—
264	C₂×C₄	#137×C₄₃	1	2	yes	—
265	C₂×C₄	C₂³×C₄×C₄₃	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₄₃	1	1	yes	characteristic
267	C₂×C₄	C₂²×Q₈×C₄₃	1	1	yes	characteristic
268	C₈	—	2	8	no	—
269	C₈	—	4	4	no	—
270	C₈	—	4	2	no	—
271	C₈	—	4	2	no	—
272	C₂²	—	2	8	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	C₄₃×#424	1	2	yes	—
276	C₂²	C₄₃×#428	1	2	yes	—
277	C₂²	C₂×#108×C₄₃	1	4	yes	—
278	C₂²	C₂×#108×C₄₃	1	2	yes	—
279	C₂²	C₂²×C₄⋊C₄×C₄₃	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	1	no	—
285	C₄	C₄₃×#424	1	2	yes	—
286	C₄	C₄₃×#428	1	2	yes	—
287	C₄	C₂×#131×C₄₃	1	2	yes	—
288	C₄	C₂×AGL(1,ℤ/8)×C₄₃	1	1	yes	characteristic
289	C₄	C₂×#137×C₄₃	1	1	yes	characteristic
290	C₂	—	2	4	no	—
291	C₂	C₄₃×#1573	1	4	yes	direct product factor
292	C₂	C₂×C₄₃×#424	1	1	yes	characteristic
293	C₂	C₂×C₄₃×#428	1	1	yes	characteristic
294	C₂	C₂²×#108×C₄₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₄₃×#1573	C₂	direct (N × H)
5. C₂×C₄₃×#335	C₂	semidirect
6. C₂×C₄₃×#334	C₂	semidirect
24. C₄₃×#335	C₂²	semidirect
25. C₄₃×#334	C₂²	semidirect
124. C₂×#1573	C₄₃	direct (N × H)
146. #1573	C₂×C₄₃	direct (N × H)
147. C₂×#335	C₂×C₄₃	semidirect
148. C₂×#334	C₂×C₄₃	semidirect
150. C₂×C₄₃	#1573	direct (N × H)
171. #335	C₂²×C₄₃	semidirect
172. #334	C₂²×C₄₃	semidirect
173. C₄₃	C₂×#1573	direct (N × H)
291. C₂	C₄₃×#1573	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×#432	2
C₂×C₄₃×#431	2
C₂²×C₄⋊C₈×C₄₃	2
C₄₃×#1573	2
C₂×C₄₃×#335	2
C₂×C₄₃×#334	2
C₂×#1573	43

Subgroups — C2×C43×#1573 #24937821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₃×#1573 #24937821