Subgroups — #25094834

Order 11008.

906 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582960	C₂	1	1	yes	maximal, characteristic
2	#19137093	C₂	1	1	yes	maximal, characteristic
3	#19134467	C₂	1	2	yes	maximal, complemented
4	#19134462	C₂	1	2	yes	maximal, complemented
5	#19134376	C₂	1	2	yes	maximal
6	#19134371	C₂	1	2	yes	maximal
7	#19134348	C₂	1	1	yes	maximal, characteristic, complemented
8	#19134343	C₂	1	1	yes	maximal, characteristic, complemented
9	#19134281	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13063235	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄₃×#405	C₂	1	1	yes	maximal, characteristic
12	#14582960	C₄	1	8	yes	complemented
13	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
15	C₂×#13063236	C₂²	1	1	yes	characteristic
16	C₂×#13063235	C₂²	1	2	yes	—
17	C₂×#13063235	C₂²	1	2	yes	—
18	C₂×#13063235	C₂²	1	2	yes	—
19	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
21	#14582792	C₄	1	4	yes	—
22	#14582785	C₄	1	4	yes	—
23	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
24	C₄×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
25	#14582745	C₂²	1	1	yes	characteristic
26	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
27	#14582653	C₂²	1	2	yes	—
28	#14582653	C₂²	1	2	yes	—
29	#14582653	C₂²	1	2	yes	—
30	#14582653	C₂²	1	2	yes	—
31	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
32	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
34	C₄₃×#405	C₄	1	8	yes	complemented
35	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
39	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
40	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊Dic₄₃	C₂×C₄	1	4	yes	—
43	C₂×C₄×Dic₄₃	C₂×C₄	1	4	yes	—
44	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
45	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
46	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
47	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
48	#13063236	C₂×C₄	1	4	yes	—
49	#13063235	—	2	4	no	—
50	#13063235	—	2	4	no	—
51	#13063235	—	2	4	no	—
52	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
53	C₈×Dic₄₃	—	2	4	no	—
54	C₄₃⋊#95	C₂×C₄	1	4	yes	—
55	C₄₃⋊#95	C₂×C₄	1	4	yes	—
56	C₄₃⋊#95	C₂×C₄	1	4	yes	—
57	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₄₃	—	2	4	no	—
59	C₂×M₁₆×C₄₃	C₂×C₄	1	4	yes	—
60	C₂²×C₈×C₄₃	—	2	4	no	—
61	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
62	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
66	#117×C₄₃	C₂×C₄	1	4	yes	—
67	#117×C₄₃	C₂×C₄	1	4	yes	—
68	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
69	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
70	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
72	#97×C₄₃	C₂×C₄	1	4	yes	—
73	#97×C₄₃	C₂×C₄	1	4	yes	—
74	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
75	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
76	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	—	2	2	no	—
78	C₂²×C₄×C₄₃	—	2	2	no	—
79	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
86	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
87	M₁₆×C₄₃	—	2	8	no	—
88	C₂×C₈×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	—	2	8	no	—
91	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₄₃	#41	1	4	yes	—
96	C₂×C₈×C₄₃	#41	1	4	yes	—
97	C₄⋊C₄×C₄₃	C₄²	1	8	yes	—
98	C₂²⋊C₄×C₄₃	C₄²	1	8	yes	—
99	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
101	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
103	C₂²⋊Dic₄₃	—	4	4	no	—
104	C₄⋊Dic₄₃	—	4	4	no	—
105	C₄×Dic₄₃	—	2	2	no	—
106	C₄×Dic₄₃	—	2	2	no	—
107	C₄×Dic₄₃	—	2	2	no	—
108	C₄×Dic₄₃	—	2	2	no	—
109	C₄×Dic₄₃	—	4	4	no	—
110	C₂×C₄₃⋊C₈	—	2	2	no	—
111	C₂×C₄₃⋊C₈	—	2	2	no	—
112	C₂×C₄₃⋊C₈	—	2	2	no	—
113	C₂×C₄₃⋊C₈	—	2	2	no	—
114	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
115	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
116	C₂³×C₄₃	—	2	2	no	—
117	C₂³×C₄₃	—	2	4	no	—
118	C₂³×C₄₃	C₂×C₄²	1	2	yes	—
119	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	—	2	2	no	—
121	C₂×C₄×C₄₃	—	2	4	no	—
122	C₂×C₄×C₄₃	—	2	4	no	—
123	C₂×C₄×C₄₃	—	2	2	no	—
124	C₂×C₄×C₄₃	—	2	2	no	—
125	C₂×C₄×C₄₃	—	2	4	no	—
126	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
127	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
128	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
132	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
133	C₂×Dic₄₃	—	2	4	no	—
134	C₂×Dic₄₃	—	2	4	no	—
135	C₂×Dic₄₃	—	2	4	no	—
136	C₂×Dic₄₃	—	2	4	no	—
137	C₂×Dic₄₃	—	2	4	no	—
138	C₂×Dic₄₃	—	2	4	no	—
139	C₈×C₄₃	—	2	2	no	—
140	C₈×C₄₃	—	2	2	no	—
141	C₈×C₄₃	#117	1	4	yes	—
142	C₈×C₄₃	#117	1	4	yes	—
143	C₄₃⋊C₈	—	2	4	no	—
144	C₄₃⋊C₈	—	2	4	no	—
145	#20475	—	43	1	no	maximal, Sylow
146	C₂²×C₄₃	—	2	4	no	—
147	C₂²×C₄₃	—	2	2	no	—
148	C₂²×C₄₃	—	4	4	no	—
149	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
150	C₂²×C₄₃	C₄×#41	1	2	yes	—
151	C₂²×C₄₃	#568	1	4	yes	—
152	C₄×C₄₃	—	2	4	no	—
153	C₄×C₄₃	—	2	4	no	—
154	C₄×C₄₃	—	2	2	no	—
155	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
156	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
157	C₄×C₄₃	C₄×#41	1	2	yes	—
158	Dic₄₃	—	4	4	no	—
159	Dic₄₃	—	4	4	no	—
160	C₂×#433	—	43	1	no	—
161	C₂×#405	D₄₃	1	1	yes	characteristic
162	#2298	—	43	1	no	—
163	#1771	—	43	2	no	—
164	#1770	—	43	2	no	—
165	#1753	—	43	2	no	—
166	#1752	—	43	2	no	—
167	#1746	—	43	1	no	—
168	#1745	—	43	1	no	—
169	#1726	—	43	1	no	—
170	C₄×#97	—	43	1	no	—
171	C₂×C₄₃	—	4	2	no	—
172	C₂×C₄₃	C₄×#117	1	1	yes	characteristic
173	C₂×C₄₃	#2875	1	2	yes	—
174	C₂×C₄₃	#2967	1	4	yes	—
175	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
176	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
177	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
178	#433	—	43	8	no	—
179	C₂×C₄⋊C₈	—	43	2	no	—
180	C₂×C₂²⋊C₈	—	43	2	no	—
181	#405	Dic₄₃	1	8	yes	complemented
182	C₂×#97	—	43	1	no	—
183	C₂×#97	—	43	2	no	—
184	C₂×#97	—	43	2	no	—
185	C₂×#97	—	43	2	no	—
186	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
187	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
188	C₂×C₄×C₈	—	43	1	no	—
189	C₂×C₄×C₈	—	43	2	no	—
190	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
191	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
192	#388	—	43	4	no	—
193	#387	—	43	4	no	—
194	C₄×C₄⋊C₄	—	43	2	no	—
195	C₄×C₂²⋊C₄	—	43	2	no	—
196	#376	—	43	1	no	—
197	C₄³	—	43	1	no	—
198	#336	—	43	2	no	—
199	#336	—	43	2	no	—
200	#336	—	43	2	no	—
201	#336	—	43	2	no	—
202	C₄₃	#20475	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
204	C₂×M₁₆	—	2	4	no	—
205	C₂×M₁₆	C₂×Dic₄₃	1	4	yes	—
206	C₂²×C₈	—	2	4	no	—
207	C₂²×C₈	—	43	2	no	—
208	C₂²×C₈	—	43	2	no	—
209	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
210	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
211	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
214	#117	C₄×D₄₃	1	4	yes	—
215	#117	C₂×Dic₄₃	1	4	yes	—
216	C₂×C₄⋊C₄	—	43	4	no	—
217	C₂×C₄⋊C₄	C₄×D₄₃	1	2	yes	—
218	C₂×C₂²⋊C₄	—	43	4	no	—
219	C₂×C₂²⋊C₄	C₄×D₄₃	1	2	yes	—
220	C₂×C₄²	—	43	2	no	—
221	C₂×C₄²	—	43	2	no	—
222	C₂×C₄²	—	43	4	no	—
223	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
224	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
225	C₄⋊C₈	—	43	8	no	—
226	C₂²⋊C₈	—	43	8	no	—
227	#97	—	43	4	no	—
228	#97	—	86	4	no	—
229	#97	—	86	4	no	—
230	#97	—	86	4	no	—
231	#97	C₂×Dic₄₃	1	4	yes	—
232	#97	C₂×Dic₄₃	1	4	yes	—
233	C₄×C₈	—	43	4	no	—
234	C₄×C₈	—	86	4	no	—
235	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
236	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
237	#95	—	43	4	no	—
238	#95	—	43	4	no	—
239	#95	—	43	4	no	—
240	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	43	4	no	—
244	C₂²×C₄	—	43	4	no	—
245	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
247	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
248	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
249	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
250	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
251	M₁₆	—	2	8	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	8	no	—
255	C₂×C₈	—	43	4	no	—
256	C₂×C₈	—	43	4	no	—
257	C₂×C₈	—	86	2	no	—
258	C₂×C₈	—	86	2	no	—
259	C₂×C₈	—	86	2	no	—
260	C₂×C₈	—	86	2	no	—
261	C₂×C₈	#10621898	1	4	yes	—
262	C₂×C₈	#10621898	1	4	yes	—
263	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
264	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
265	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
266	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
267	C₄⋊C₄	—	172	4	no	—
268	C₄⋊C₄	C₄×Dic₄₃	1	8	yes	—
269	C₂²⋊C₄	—	172	4	no	—
270	C₂²⋊C₄	C₄×Dic₄₃	1	8	yes	—
271	C₄²	—	86	2	no	—
272	C₄²	—	86	2	no	—
273	C₄²	—	86	2	no	—
274	C₄²	—	86	2	no	—
275	C₄²	—	172	4	no	—
276	C₄²	C₂×C₄×D₄₃	1	2	yes	—
277	C₄²	C₂×C₄×D₄₃	1	2	yes	—
278	C₄²	C₂²×Dic₄₃	1	2	yes	—
279	C₄²	C₂²×Dic₄₃	1	2	yes	—
280	C₂³	—	2	2	no	—
281	C₂³	—	2	4	no	—
282	C₂³	C₂×C₄×Dic₄₃	1	2	yes	—
283	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	86	4	no	—
291	C₂×C₄	—	86	4	no	—
292	C₂×C₄	—	86	4	no	—
293	C₂×C₄	—	86	4	no	—
294	C₂×C₄	—	86	4	no	—
295	C₂×C₄	—	86	4	no	—
296	C₂×C₄	C₂×C₄×Dic₄₃	1	4	yes	—
297	C₂×C₄	C₂×C₄×Dic₄₃	1	4	yes	—
298	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
299	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
300	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
301	C₂×C₄	C₂×#10621898	1	2	yes	—
302	C₂×C₄	C₂×#10621898	1	2	yes	—
303	C₈	—	2	2	no	—
304	C₈	—	2	2	no	—
305	C₈	—	86	4	no	—
306	C₈	—	86	4	no	—
307	C₈	#13063284	1	4	yes	—
308	C₈	#13063284	1	4	yes	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	4	4	no	—
312	C₂²	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
313	C₂²	C₄×#10621898	1	2	yes	—
314	C₂²	#14583664	1	4	yes	—
315	C₄	—	2	4	no	—
316	C₄	—	2	4	no	—
317	C₄	—	2	2	no	—
318	C₄	—	172	4	no	—
319	C₄	—	172	4	no	—
320	C₄	C₂×#13063284	1	1	yes	characteristic
321	C₄	C₂×#13063284	1	1	yes	characteristic
322	C₄	C₄×#10621898	1	2	yes	—
323	C₂	—	4	2	no	—
324	C₂	C₄×#13063284	1	1	yes	characteristic
325	C₂	#19143021	1	2	yes	—
326	C₂	#19143576	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Fitting	C₂×C₄₃×#405	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#405	D₄₃
hypocenter	C₄₃	#20475
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #19134467	C₂	semidirect
4. #19134462	C₂	semidirect
7. #19134348	C₂	semidirect
8. #19134343	C₂	semidirect
9. #19134281	C₂	semidirect
10. C₄×#13063235	C₂	semidirect
12. #14582960	C₄	semidirect
34. C₄₃×#405	C₄	semidirect
181. #405	Dic₄₃	semidirect
202. C₄₃	#20475	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582960	2
#19137093	2
#19134467	2
#19134462	2
#19134376	2
#19134371	2
#19134348	2
#19134343	2
#19134281	2
C₄×#13063235	2
C₂×C₄₃×#405	2
#20475	43