Subgroups — #25095356

Order 11008.

922 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582952	C₂	1	1	yes	maximal, characteristic
2	#19137002	C₂	1	1	yes	maximal, characteristic
3	#19134465	C₂	1	2	yes	maximal, complemented
4	#19134462	C₂	1	2	yes	maximal, complemented
5	#19134460	C₂	1	2	yes	maximal, complemented
6	C₈×C₄⋊Dic₄₃	C₂	1	2	yes	maximal, complemented
7	#19134363	C₂	1	4	yes	maximal
8	C₂×C₄₃×#431	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063284	C₂²	1	2	yes	—
10	#14582952	C₄	1	8	yes	complemented
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
14	C₂×#13063235	C₂²	1	4	yes	—
15	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
16	C₂×C₈×Dic₄₃	C₂²	1	4	yes	—
17	#14582769	C₄	1	2	yes	—
18	#14582765	C₄	1	2	yes	—
19	#14582759	C₄	1	2	yes	—
20	#14582759	C₄	1	2	yes	—
21	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
22	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
23	#14582653	C₂²	1	4	yes	—
24	#14582653	C₂²	1	4	yes	—
25	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×#431	C₄	1	8	yes	complemented
27	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
31	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
32	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
33	#13063284	—	4	2	no	—
34	#13063284	—	4	2	no	—
35	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊Dic₄₃	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₄₃	C₂³	1	4	yes	—
40	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
41	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
42	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
43	#13063235	—	4	4	no	—
44	C₄×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
45	C₈×Dic₄₃	—	4	4	no	—
46	C₄₃⋊#95	C₂×C₄	1	4	yes	—
47	C₄₃⋊#95	C₂×C₄	1	4	yes	—
48	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
56	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₄₃	C₂×C₄	1	4	yes	—
60	C₄⋊C₈×C₄₃	C₂×C₄	1	4	yes	—
61	C₂²⋊C₈×C₄₃	C₂×C₄	1	8	yes	—
62	C₄×C₈×C₄₃	C₂×C₄	1	8	yes	—
63	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	—	2	4	no	—
65	C₂²×C₄×C₄₃	—	2	4	no	—
66	C₂²×C₄×C₄₃	—	2	1	no	—
67	C₂²×C₄×C₄₃	—	2	1	no	—
68	C₂²×C₄×C₄₃	—	2	1	no	—
69	C₂²×C₄×C₄₃	—	2	1	no	—
70	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
71	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
72	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₄₃	C₂²×C₄	1	8	yes	—
81	C₂×C₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	—	2	2	no	—
83	C₂×C₈×C₄₃	—	2	2	no	—
84	C₂×C₈×C₄₃	—	2	2	no	—
85	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
87	C₄²×C₄₃	C₄²	1	4	yes	—
88	C₄²×C₄₃	C₄²	1	4	yes	—
89	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
91	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
92	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
93	C₂²⋊Dic₄₃	—	4	4	no	—
94	C₄⋊Dic₄₃	—	2	8	no	—
95	C₄⋊Dic₄₃	—	2	8	no	—
96	C₄⋊Dic₄₃	—	4	4	no	—
97	C₄×Dic₄₃	—	2	4	no	—
98	C₄×Dic₄₃	—	2	4	no	—
99	C₂×C₄₃⋊C₈	—	2	2	no	—
100	C₂×C₄₃⋊C₈	—	2	2	no	—
101	C₂×C₄₃⋊C₈	—	2	2	no	—
102	C₂×C₄₃⋊C₈	—	2	2	no	—
103	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
104	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
105	C₂³×C₄₃	—	2	1	no	—
106	C₂³×C₄₃	—	2	1	no	—
107	C₂³×C₄₃	—	2	2	no	—
108	C₂³×C₄₃	—	2	2	no	—
109	C₂³×C₄₃	C₂×C₄²	1	2	yes	—
110	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	2	2	no	—
117	C₂×C₄×C₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	—	2	2	no	—
119	C₂×C₄×C₄₃	—	2	8	no	—
120	C₂×C₄×C₄₃	—	2	8	no	—
121	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₄₃	#131	1	4	yes	—
127	C₂×C₄×C₄₃	#131	1	4	yes	—
128	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
134	C₂×Dic₄₃	—	2	8	no	—
135	C₂×Dic₄₃	—	2	8	no	—
136	C₂×Dic₄₃	—	2	8	no	—
137	C₈×C₄₃	—	2	4	no	—
138	C₈×C₄₃	—	2	4	no	—
139	C₄₃⋊C₈	—	2	4	no	—
140	C₄₃⋊C₈	—	2	4	no	—
141	#20538	—	43	1	no	maximal, Sylow
142	C₂²×C₄₃	—	2	2	no	—
143	C₂²×C₄₃	—	2	2	no	—
144	C₂²×C₄₃	—	2	2	no	—
145	C₂²×C₄₃	—	4	2	no	—
146	C₂²×C₄₃	—	4	2	no	—
147	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
148	C₂²×C₄₃	#518	1	1	yes	characteristic
149	C₂²×C₄₃	#519	1	1	yes	characteristic
150	C₂²×C₄₃	C₂×#131	1	2	yes	—
151	C₂²×C₄₃	C₂×#131	1	2	yes	—
152	C₄×C₄₃	—	2	2	no	—
153	C₄×C₄₃	—	2	2	no	—
154	C₄×C₄₃	—	2	2	no	—
155	C₄×C₄₃	#405	1	4	yes	—
156	C₄×C₄₃	#405	1	4	yes	—
157	C₄×C₄₃	#520	1	1	yes	characteristic
158	C₄×C₄₃	#520	1	1	yes	characteristic
159	C₄×C₄₃	#520	1	1	yes	characteristic
160	C₄×C₄₃	#520	1	1	yes	characteristic
161	Dic₄₃	—	4	8	no	—
162	C₂×#431	—	43	1	no	—
163	C₂×#431	D₄₃	1	1	yes	characteristic
164	#2285	—	43	1	no	—
165	#1770	—	43	2	no	—
166	#1770	—	43	2	no	—
167	C₈×C₄⋊C₄	—	43	2	no	—
168	C₈×C₄⋊C₄	—	43	2	no	—
169	#1750	—	43	4	no	—
170	C₂×C₄₃	—	4	2	no	—
171	C₂×C₄₃	#2274	1	1	yes	characteristic
172	C₂×C₄₃	C₂×#405	1	1	yes	characteristic
173	C₂×C₄₃	C₂×#405	1	1	yes	characteristic
174	C₂×C₄₃	#2979	1	2	yes	—
175	C₂×C₄₃	#2986	1	2	yes	—
176	C₂×#117	—	43	2	no	—
177	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
178	#431	—	43	8	no	—
179	#431	Dic₄₃	1	8	yes	complemented
180	C₂×C₄⋊C₈	—	43	1	no	—
181	C₂×C₄⋊C₈	—	43	1	no	—
182	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
183	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
184	C₂×C₂²⋊C₈	—	43	2	no	—
185	C₂×C₂²⋊C₈	C₂×D₄₃	1	2	yes	—
186	C₂×#97	—	43	4	no	—
187	C₂×C₄×C₈	—	43	2	no	—
188	C₂×C₄×C₈	—	43	4	no	—
189	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
190	#384	—	43	2	no	—
191	#382	—	43	2	no	—
192	#380	—	43	2	no	—
193	#380	—	43	2	no	—
194	C₄×C₄⋊C₄	—	43	2	no	—
195	C₄×C₄⋊C₄	—	43	2	no	—
196	#336	—	43	4	no	—
197	#336	—	43	4	no	—
198	C₄₃	#20538	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
200	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
201	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
202	C₂²×C₈	—	43	2	no	—
203	C₂²×C₈	—	43	2	no	—
204	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
205	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
206	#117	—	172	2	no	—
207	#117	—	172	2	no	—
208	C₂×C₄⋊C₄	—	43	4	no	—
209	C₂×C₄⋊C₄	—	43	4	no	—
210	C₂×C₄⋊C₄	—	43	4	no	—
211	C₂×C₂²⋊C₄	—	43	4	no	—
212	C₂×C₄²	—	43	4	no	—
213	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
214	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
215	C₂×C₄²	C₄×D₄₃	1	4	yes	—
216	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
217	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
218	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
219	C₄⋊C₈	—	43	4	no	—
220	C₄⋊C₈	—	43	4	no	—
221	C₄⋊C₈	C₂×Dic₄₃	1	4	yes	—
222	C₄⋊C₈	C₂×Dic₄₃	1	4	yes	—
223	C₂²⋊C₈	—	43	8	no	—
224	C₂²⋊C₈	C₂×Dic₄₃	1	8	yes	—
225	#97	—	172	4	no	—
226	C₄×C₈	—	43	8	no	—
227	C₄×C₈	—	172	4	no	—
228	C₄×C₈	C₂×Dic₄₃	1	8	yes	—
229	#95	—	43	4	no	—
230	#95	—	43	4	no	—
231	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	43	8	no	—
239	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
240	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
241	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
245	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
247	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
248	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	43	4	no	—
254	C₂×C₈	—	43	4	no	—
255	C₂×C₈	—	86	2	no	—
256	C₂×C₈	—	86	2	no	—
257	C₂×C₈	—	86	2	no	—
258	C₂×C₈	—	86	2	no	—
259	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
260	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
261	C₄⋊C₄	—	86	8	no	—
262	C₄⋊C₄	—	86	8	no	—
263	C₄⋊C₄	—	172	4	no	—
264	C₂²⋊C₄	—	172	4	no	—
265	C₄²	—	86	4	no	—
266	C₄²	—	86	4	no	—
267	C₄²	C₄×Dic₄₃	1	4	yes	—
268	C₄²	C₄×Dic₄₃	1	4	yes	—
269	C₄²	C₂×C₄×D₄₃	1	2	yes	—
270	C₄²	C₂×C₄×D₄₃	1	2	yes	—
271	C₄²	C₂²×Dic₄₃	1	2	yes	—
272	C₄²	C₂²×Dic₄₃	1	2	yes	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	C₂×C₄×Dic₄₃	1	2	yes	—
278	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	86	8	no	—
290	C₂×C₄	—	86	8	no	—
291	C₂×C₄	—	86	8	no	—
292	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
293	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
294	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
295	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
296	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
297	C₂×C₄	#13063331	1	4	yes	—
298	C₂×C₄	#13063333	1	4	yes	—
299	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
300	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
301	C₂×C₄	#13063371	1	1	yes	characteristic
302	C₂×C₄	#13063371	1	1	yes	characteristic
303	C₂×C₄	#13063372	1	1	yes	characteristic
304	C₂×C₄	#13063372	1	1	yes	characteristic
305	C₈	—	2	4	no	—
306	C₈	—	2	4	no	—
307	C₈	—	86	4	no	—
308	C₈	—	86	4	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
315	C₂²	#14583373	1	1	yes	characteristic
316	C₂²	#14583376	1	1	yes	characteristic
317	C₂²	C₂×#13063331	1	2	yes	—
318	C₂²	C₂×#13063333	1	2	yes	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	172	8	no	—
323	C₄	#14582854	1	4	yes	—
324	C₄	#14582857	1	4	yes	—
325	C₄	#14583379	1	1	yes	characteristic
326	C₄	#14583379	1	1	yes	characteristic
327	C₄	#14583382	1	1	yes	characteristic
328	C₄	#14583382	1	1	yes	characteristic
329	C₂	—	4	2	no	—
330	C₂	#19136951	1	1	yes	characteristic
331	C₂	C₂×#14582854	1	1	yes	characteristic
332	C₂	C₂×#14582857	1	1	yes	characteristic
333	C₂	#19143661	1	2	yes	—
334	C₂	#19143696	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Fitting	C₂×C₄₃×#431	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#431	D₄₃
hypocenter	C₄₃	#20538
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #19134465	C₂	semidirect
4. #19134462	C₂	semidirect
5. #19134460	C₂	semidirect
6. C₈×C₄⋊Dic₄₃	C₂	semidirect
10. #14582952	C₄	semidirect
26. C₄₃×#431	C₄	semidirect
179. #431	Dic₄₃	semidirect
198. C₄₃	#20538	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582952	2
#19137002	2
#19134465	2
#19134462	2
#19134460	2
C₈×C₄⋊Dic₄₃	2
#19134363	2
C₂×C₄₃×#431	2
#20538	43