Subgroups — #25094845

Order 11008.

990 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582956	C₂	1	1	yes	maximal, characteristic
2	#19136993	C₂	1	1	yes	maximal, characteristic
3	#19134462	C₂	1	2	yes	maximal, complemented
4	C₈×C₄⋊Dic₄₃	C₂	1	2	yes	maximal, complemented
5	#19134371	C₂	1	2	yes	maximal
6	C₈×C₂²⋊Dic₄₃	C₂	1	2	yes	maximal
7	#19134346	C₂	1	2	yes	maximal, complemented
8	#19134281	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
10	#14582956	C₄	1	8	yes	complemented
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
13	C₂×#13063236	C₂²	1	2	yes	—
14	C₂×#13063235	C₂²	1	4	yes	—
15	C₂×C₈×Dic₄₃	C₂²	1	4	yes	—
16	#14582767	C₄	1	2	yes	—
17	#14582765	C₄	1	2	yes	—
18	#14582762	C₄	1	2	yes	—
19	#14582759	C₄	1	2	yes	—
20	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
21	C₄×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
22	#14582745	C₂²	1	2	yes	—
23	#14582653	C₂²	1	4	yes	—
24	#14582653	C₂²	1	4	yes	—
25	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₃	C₄	1	8	yes	complemented
29	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊Dic₄₃	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₄₃	C₂³	1	4	yes	—
37	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
38	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
39	#13063236	C₂×C₄	1	8	yes	—
40	#13063235	—	2	8	no	—
41	C₈×Dic₄₃	—	2	8	no	—
42	C₄₃⋊#95	C₂×C₄	1	4	yes	—
43	C₄₃⋊#95	C₂×C₄	1	4	yes	—
44	C₄₃⋊#95	C₂×C₄	1	4	yes	—
45	C₄₃⋊#95	C₂×C₄	1	4	yes	—
46	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₃	—	2	4	no	—
50	C₂²×C₈×C₄₃	—	2	4	no	—
51	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
60	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
61	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
64	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
65	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
66	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
67	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	—	2	4	no	—
69	C₂²×C₄×C₄₃	—	2	2	no	—
70	C₂²×C₄×C₄₃	—	2	2	no	—
71	C₂²×C₄×C₄₃	—	2	4	no	—
72	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
73	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
74	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
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	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₄₃	C₂²×C₄	1	8	yes	—
83	C₂×C₈×C₄₃	—	2	2	no	—
84	C₂×C₈×C₄₃	—	2	2	no	—
85	C₂×C₈×C₄₃	—	2	8	no	—
86	C₂×C₈×C₄₃	—	2	8	no	—
87	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₄₃	#41	1	4	yes	—
92	C₂×C₈×C₄₃	#41	1	4	yes	—
93	C₄²×C₄₃	C₄²	1	4	yes	—
94	C₄²×C₄₃	C₄²	1	4	yes	—
95	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
97	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
99	C₂²⋊Dic₄₃	—	2	8	no	—
100	C₄⋊Dic₄₃	—	2	8	no	—
101	C₄×Dic₄₃	—	2	4	no	—
102	C₄×Dic₄₃	—	2	4	no	—
103	C₂×C₄₃⋊C₈	—	2	4	no	—
104	C₂×C₄₃⋊C₈	—	2	4	no	—
105	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
106	C₂³×C₄₃	—	2	2	no	—
107	C₂³×C₄₃	—	2	2	no	—
108	C₂³×C₄₃	C₂×C₄²	1	2	yes	—
109	C₂³×C₄₃	#131	1	4	yes	—
110	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	2	8	no	—
116	C₂×C₄×C₄₃	—	2	2	no	—
117	C₂×C₄×C₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	—	2	4	no	—
119	C₂×C₄×C₄₃	—	2	8	no	—
120	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
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	#131	1	4	yes	—
126	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
129	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
130	C₂×Dic₄₃	—	2	8	no	—
131	C₂×Dic₄₃	—	2	8	no	—
132	C₂×Dic₄₃	—	2	8	no	—
133	C₈×C₄₃	—	2	2	no	—
134	C₈×C₄₃	—	2	2	no	—
135	C₈×C₄₃	#117	1	4	yes	—
136	C₈×C₄₃	#117	1	4	yes	—
137	C₄₃⋊C₈	—	2	8	no	—
138	#20476	—	43	1	no	maximal, Sylow
139	C₂²×C₄₃	—	2	2	no	—
140	C₂²×C₄₃	—	2	8	no	—
141	C₂²×C₄₃	—	2	2	no	—
142	C₂²×C₄₃	#405	1	4	yes	—
143	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
144	C₂²×C₄₃	C₄×#41	1	2	yes	—
145	C₂²×C₄₃	C₂×#131	1	2	yes	—
146	C₂²×C₄₃	C₂×#131	1	2	yes	—
147	C₄×C₄₃	—	2	2	no	—
148	C₄×C₄₃	—	2	2	no	—
149	C₄×C₄₃	—	2	2	no	—
150	C₄×C₄₃	—	2	2	no	—
151	C₄×C₄₃	#405	1	4	yes	—
152	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
153	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
154	C₄×C₄₃	C₄×#41	1	2	yes	—
155	Dic₄₃	—	4	8	no	—
156	C₂×#432	—	43	1	no	—
157	C₂²×C₄×C₈	D₄₃	1	1	yes	characteristic
158	#2283	—	43	1	no	—
159	#1770	—	43	2	no	—
160	C₈×C₄⋊C₄	—	43	2	no	—
161	#1752	—	43	2	no	—
162	C₈×C₂²⋊C₄	—	43	2	no	—
163	#1746	—	43	2	no	—
164	#1726	—	43	2	no	—
165	C₂×C₄₃	—	2	4	no	—
166	C₂×C₄₃	C₄×#117	1	1	yes	characteristic
167	C₂×C₄₃	C₂×#405	1	1	yes	characteristic
168	C₂×C₄₃	C₂×#405	1	1	yes	characteristic
169	C₂×C₄₃	#2966	1	4	yes	—
170	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
171	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
172	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
173	#432	—	43	8	no	—
174	C₂×C₄⋊C₈	—	43	2	no	—
175	C₂×C₂²⋊C₈	—	43	2	no	—
176	C₂×#97	—	43	2	no	—
177	C₂×#97	—	43	4	no	—
178	C₂×C₄×C₈	—	43	4	no	—
179	C₂×C₄×C₈	Dic₄₃	1	8	yes	complemented
180	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
182	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
184	#383	—	43	2	no	—
185	#382	—	43	2	no	—
186	#381	—	43	2	no	—
187	#380	—	43	2	no	—
188	C₄×C₄⋊C₄	—	43	2	no	—
189	C₄×C₂²⋊C₄	—	43	2	no	—
190	#376	—	43	2	no	—
191	#336	—	43	4	no	—
192	#336	—	43	4	no	—
193	C₄₃	#20476	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
195	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
196	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
197	C₂²×C₈	—	2	4	no	—
198	C₂²×C₈	—	2	4	no	—
199	C₂²×C₈	—	43	4	no	—
200	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
201	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
202	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
205	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
206	C₂×C₄⋊C₄	—	43	4	no	—
207	C₂×C₂²⋊C₄	—	43	4	no	—
208	C₂×C₄²	—	43	4	no	—
209	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
211	C₂×C₄²	C₄×D₄₃	1	4	yes	—
212	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
213	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
214	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
215	C₄⋊C₈	—	43	8	no	—
216	C₂²⋊C₈	—	43	8	no	—
217	#97	—	43	8	no	—
218	#97	—	86	8	no	—
219	C₄×C₈	—	86	8	no	—
220	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
221	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
222	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
223	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
224	#95	—	43	4	no	—
225	#95	—	43	4	no	—
226	#95	—	43	4	no	—
227	#95	—	43	4	no	—
228	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	43	8	no	—
234	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
235	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
236	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
241	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
242	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
243	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	8	no	—
247	C₂×C₈	—	2	8	no	—
248	C₂×C₈	—	43	8	no	—
249	C₂×C₈	—	86	4	no	—
250	C₂×C₈	—	86	4	no	—
251	C₂×C₈	#10621898	1	4	yes	—
252	C₂×C₈	#10621898	1	4	yes	—
253	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
254	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
255	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
256	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
257	C₄⋊C₄	—	86	8	no	—
258	C₂²⋊C₄	—	86	8	no	—
259	C₄²	—	86	4	no	—
260	C₄²	—	86	4	no	—
261	C₄²	C₄×Dic₄₃	1	4	yes	—
262	C₄²	C₄×Dic₄₃	1	4	yes	—
263	C₄²	C₂×C₄×D₄₃	1	2	yes	—
264	C₄²	C₂×C₄×D₄₃	1	2	yes	—
265	C₄²	C₂²×Dic₄₃	1	2	yes	—
266	C₄²	C₂²×Dic₄₃	1	2	yes	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	C₂×C₄×Dic₄₃	1	2	yes	—
270	C₂³	#13063331	1	4	yes	—
271	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	86	8	no	—
282	C₂×C₄	—	86	8	no	—
283	C₂×C₄	—	86	8	no	—
284	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
285	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
286	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
287	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
288	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
289	C₂×C₄	#13063331	1	4	yes	—
290	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
291	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
292	C₂×C₄	C₂×#10621898	1	2	yes	—
293	C₂×C₄	C₂×#10621898	1	2	yes	—
294	C₈	—	2	2	no	—
295	C₈	—	2	2	no	—
296	C₈	—	86	8	no	—
297	C₈	#13063284	1	4	yes	—
298	C₈	#13063284	1	4	yes	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	8	no	—
301	C₂²	—	2	2	no	—
302	C₂²	#14582854	1	4	yes	—
303	C₂²	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
304	C₂²	C₄×#10621898	1	2	yes	—
305	C₂²	C₂×#13063331	1	2	yes	—
306	C₂²	C₂×#13063331	1	2	yes	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	172	8	no	—
312	C₄	#14582854	1	4	yes	—
313	C₄	C₂×#13063284	1	1	yes	characteristic
314	C₄	C₂×#13063284	1	1	yes	characteristic
315	C₄	C₄×#10621898	1	2	yes	—
316	C₂	—	2	4	no	—
317	C₂	C₄×#13063284	1	1	yes	characteristic
318	C₂	C₂×#14582854	1	1	yes	characteristic
319	C₂	C₂×#14582854	1	1	yes	characteristic
320	C₂	#19143570	1	4	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂²×C₄×C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₄₃
hypocenter	C₄₃	#20476
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #19134462	C₂	semidirect
4. C₈×C₄⋊Dic₄₃	C₂	semidirect
7. #19134346	C₂	semidirect
8. #19134281	C₂	semidirect
10. #14582956	C₄	semidirect
28. C₂×C₄×C₈×C₄₃	C₄	semidirect
179. C₂×C₄×C₈	Dic₄₃	semidirect
193. C₄₃	#20476	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582956	2
#19136993	2
#19134462	2
C₈×C₄⋊Dic₄₃	2
#19134371	2
C₈×C₂²⋊Dic₄₃	2
#19134346	2
#19134281	2
C₂²×C₄×C₈×C₄₃	2
#20476	43