Subgroups — #25114202

Order 11008.

692 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143255	C₂	1	2	yes	maximal
2	C₂×#14582937	C₂	1	1	yes	maximal, characteristic
3	#19135346	C₂	1	2	yes	maximal, complemented
4	#19135193	C₂	1	4	yes	maximal, complemented
5	#19134692	C₂	1	2	yes	maximal
6	#19134482	C₂	1	2	yes	maximal, complemented
7	#19134371	C₂	1	1	yes	maximal, characteristic
8	C₄₃×#1810	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063284	C₂²	1	2	yes	—
10	#14582957	—	2	4	no	—
11	#14582953	—	2	2	no	—
12	#14582949	—	2	2	no	—
13	#14582937	C₄	1	1	yes	characteristic, complemented
14	#14582937	C₄	1	1	yes	characteristic, complemented
15	#14582937	C₄	1	2	yes	complemented
16	#14582937	C₂²	1	1	yes	characteristic
17	#14582937	C₂²	1	1	yes	characteristic
18	#14582937	C₂²	1	2	yes	—
19	C₂×#13063259	C₄	1	1	yes	characteristic, complemented
20	C₂×#13063259	C₄	1	1	yes	characteristic, complemented
21	C₂×#13063257	C₄	1	2	yes	complemented
22	#14582927	C₂²	1	4	yes	—
23	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
24	#14582857	C₂²	1	4	yes	—
25	C₂×#13063235	C₂²	1	2	yes	—
26	C₄×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
27	#14582660	C₂²	1	4	yes	—
28	#14582660	C₂²	1	4	yes	—
29	#14582653	C₂²	1	2	yes	—
30	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×#108×C₄₃	C₂²	1	2	yes	—
32	C₄₃×#407	C₂²	1	2	yes	—
33	C₄₃×#341	C₂²	1	2	yes	—
34	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
35	#13063284	—	2	2	no	—
36	#13063284	—	2	2	no	—
37	#13063284	C₂×C₄	1	2	yes	—
38	#13063284	C₂×C₄	1	2	yes	—
39	#13063284	C₂³	1	2	yes	—
40	#13063284	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
44	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
45	C₂×C₄×Dic₄₃	—	2	2	no	—
46	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
47	#13063259	C₂×C₄	1	2	yes	—
48	#13063259	C₂×C₄	1	2	yes	—
49	#13063259	C₂×C₄	1	1	yes	characteristic
50	#13063259	C₂×C₄	1	1	yes	characteristic
51	#13063259	C₂×C₄	1	1	yes	characteristic
52	#13063259	C₂×C₄	1	1	yes	characteristic
53	#13063257	C₂×C₄	1	2	yes	—
54	#13063257	C₂×C₄	1	2	yes	—
55	#13063257	C₂×C₄	1	2	yes	—
56	#13063257	C₂×C₄	1	2	yes	—
57	Dic₄₃⋊C₈	—	2	4	no	—
58	Dic₄₃⋊C₈	—	2	4	no	—
59	C₂²⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
60	C₂²⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
61	#13063235	—	2	4	no	—
62	#13063235	—	2	4	no	—
63	C₈×Dic₄₃	—	2	4	no	—
64	C₄₃⋊#95	—	2	1	no	—
65	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
66	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
67	C₂²×C₈×C₄₃	—	2	4	no	—
68	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
71	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	#108×C₄₃	—	2	4	no	—
75	C₂²⋊C₈×C₄₃	—	2	2	no	—
76	C₂²⋊C₈×C₄₃	—	2	2	no	—
77	C₂⁴×C₄₃	#41	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	—	2	2	no	—
79	C₂²×C₄×C₄₃	—	2	2	no	—
80	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
85	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
86	C₂²×Dic₄₃	—	2	2	no	—
87	C₂²×Dic₄₃	—	2	2	no	—
88	C₂²×Dic₄₃	—	2	2	no	—
89	M₁₆×C₄₃	—	2	4	no	—
90	M₁₆×C₄₃	—	2	4	no	—
91	C₂×C₈×C₄₃	—	2	2	no	—
92	C₂×C₈×C₄₃	—	2	2	no	—
93	C₂×C₈×C₄₃	—	2	2	no	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₂×C₈×C₄₃	—	2	2	no	—
96	C₂×C₈×C₄₃	—	2	4	no	—
97	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₄₃	#41	1	4	yes	—
102	C₂²⋊Dic₄₃	—	2	2	no	—
103	C₂²⋊Dic₄₃	—	2	2	no	—
104	C₂²⋊Dic₄₃	—	2	2	no	—
105	C₂²⋊Dic₄₃	—	2	2	no	—
106	C₂²⋊Dic₄₃	—	4	2	no	—
107	C₄⋊Dic₄₃	—	2	2	no	—
108	C₄⋊Dic₄₃	—	2	2	no	—
109	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
110	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
111	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
112	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
113	C₄×Dic₄₃	—	2	2	no	—
114	C₄×Dic₄₃	—	2	2	no	—
115	C₄×Dic₄₃	—	2	2	no	—
116	C₄×Dic₄₃	—	2	2	no	—
117	C₄×Dic₄₃	—	2	2	no	—
118	C₄×Dic₄₃	—	2	2	no	—
119	C₂×C₄₃⋊C₈	—	2	2	no	—
120	C₂×C₄₃⋊C₈	—	2	2	no	—
121	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
122	C₂³×C₄₃	—	2	2	no	—
123	C₂³×C₄₃	—	2	2	no	—
124	C₂³×C₄₃	—	4	1	no	—
125	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
126	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
127	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	—	2	1	no	—
132	C₂×C₄×C₄₃	—	2	1	no	—
133	C₂×C₄×C₄₃	—	2	2	no	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
140	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
141	C₂×Dic₄₃	—	2	2	no	—
142	C₂×Dic₄₃	—	2	2	no	—
143	C₂×Dic₄₃	—	2	2	no	—
144	C₂×Dic₄₃	—	2	2	no	—
145	C₂×Dic₄₃	—	4	4	no	—
146	C₂×Dic₄₃	—	4	2	no	—
147	C₂×Dic₄₃	—	4	2	no	—
148	C₂×Dic₄₃	—	4	2	no	—
149	C₂×Dic₄₃	—	8	2	no	—
150	C₈×C₄₃	—	2	1	no	—
151	C₈×C₄₃	—	2	1	no	—
152	C₈×C₄₃	—	4	4	no	—
153	C₈×C₄₃	#117	1	2	yes	—
154	C₈×C₄₃	#117	1	2	yes	—
155	C₄₃⋊C₈	—	2	4	no	—
156	#22607	—	43	1	no	maximal, Sylow
157	C₂²×C₄₃	—	2	1	no	—
158	C₂²×C₄₃	—	2	1	no	—
159	C₂²×C₄₃	—	2	1	no	—
160	C₂²×C₄₃	—	2	1	no	—
161	C₂²×C₄₃	—	2	1	no	—
162	C₂²×C₄₃	—	2	1	no	—
163	C₂²×C₄₃	—	2	1	no	—
164	C₂²×C₄₃	—	4	4	no	—
165	C₂²×C₄₃	#444	1	2	yes	—
166	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₄₃	#552	1	2	yes	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	2	1	no	—
170	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
171	C₄×C₄₃	#519	1	1	yes	characteristic
172	C₄×C₄₃	#547	1	2	yes	—
173	Dic₄₃	—	4	2	no	—
174	Dic₄₃	—	4	2	no	—
175	Dic₄₃	—	4	2	no	—
176	Dic₄₃	—	4	2	no	—
177	Dic₄₃	—	8	2	no	—
178	#2924	—	43	2	no	—
179	C₂×#427	—	43	1	no	—
180	#1946	—	43	2	no	—
181	#1922	—	43	4	no	—
182	#1826	—	43	2	no	—
183	#1810	D₄₃	1	1	yes	characteristic
184	#1773	—	43	2	no	—
185	#1752	—	43	1	no	—
186	C₂×C₄₃	—	2	2	no	—
187	C₂×C₄₃	—	2	2	no	—
188	C₂×C₄₃	—	4	1	no	—
189	C₂×C₄₃	#2344	1	1	yes	characteristic
190	C₂×C₄₃	C₂×#444	1	1	yes	characteristic
191	C₂×C₄₃	#2957	1	1	yes	characteristic
192	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
193	C₂×#117	—	43	2	no	—
194	#433	—	86	4	no	—
195	#432	—	86	2	no	—
196	#431	—	86	2	no	—
197	C₂×#108	C₂×D₄₃	1	2	yes	—
198	#427	—	43	2	no	—
199	#427	—	43	1	no	—
200	#427	—	43	1	no	—
201	#427	—	43	1	no	—
202	#427	—	43	1	no	—
203	#427	—	43	2	no	—
204	C₂×#107	—	43	1	no	—
205	C₂×#107	—	43	1	no	—
206	C₂×#106	—	43	2	no	—
207	#424	—	43	4	no	—
208	#407	C₂×D₄₃	1	2	yes	—
209	C₂×C₂²⋊C₈	—	43	1	no	—
210	#405	—	43	4	no	—
211	C₂×#97	—	43	2	no	—
212	C₄×C₂²⋊C₄	—	43	1	no	—
213	#341	C₂×D₄₃	1	2	yes	—
214	#339	—	43	4	no	—
215	#339	—	43	4	no	—
216	#336	—	43	2	no	—
217	C₄₃	#22607	1	1	yes	characteristic, complemented, Sylow
218	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
219	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
220	C₂²×C₈	—	2	4	no	—
221	C₂²×C₈	—	43	2	no	—
222	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
223	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
224	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
225	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
227	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
228	#117	—	43	2	no	—
229	#117	—	43	2	no	—
230	#117	—	43	2	no	—
231	#117	—	43	2	no	—
232	#117	—	86	2	no	—
233	#117	—	86	2	no	—
234	C₂×C₄⋊C₄	—	43	2	no	—
235	C₂×C₄⋊C₄	—	43	2	no	—
236	C₂×C₂²⋊C₄	—	86	2	no	—
237	C₂×C₂²⋊C₄	—	86	1	no	—
238	C₂×C₄²	—	43	2	no	—
239	C₂×C₄²	—	86	2	no	—
240	#108	—	2	4	no	—
241	#107	—	43	2	no	—
242	#107	—	43	2	no	—
243	#107	—	43	1	no	—
244	#107	—	43	1	no	—
245	#107	—	43	1	no	—
246	#107	—	43	1	no	—
247	#106	—	43	2	no	—
248	#106	—	43	2	no	—
249	#106	—	43	2	no	—
250	#106	—	43	2	no	—
251	C₄⋊C₈	—	86	4	no	—
252	C₄⋊C₈	—	86	4	no	—
253	C₂²⋊C₈	—	2	2	no	—
254	C₂²⋊C₈	—	2	2	no	—
255	C₂²⋊C₈	—	43	2	no	—
256	C₂²⋊C₈	—	43	2	no	—
257	#97	—	86	4	no	—
258	#97	—	86	4	no	—
259	C₄×C₈	—	86	4	no	—
260	#95	—	86	1	no	—
261	C₂⁴	#10621902	1	1	yes	characteristic
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	86	2	no	—
265	C₂²×C₄	—	86	2	no	—
266	C₂²×C₄	—	86	2	no	—
267	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
269	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
270	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
271	C₂²×C₄	#10621902	1	1	yes	characteristic
272	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
273	M₁₆	—	2	4	no	—
274	M₁₆	—	2	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	43	4	no	—
282	C₂×C₈	—	86	2	no	—
283	C₂×C₈	—	86	2	no	—
284	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
285	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
286	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
287	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
288	C₂×C₈	#10621900	1	4	yes	—
289	C₄⋊C₄	—	43	2	no	—
290	C₄⋊C₄	—	43	2	no	—
291	C₄⋊C₄	—	43	2	no	—
292	C₄⋊C₄	—	43	2	no	—
293	C₄⋊C₄	—	86	2	no	—
294	C₄⋊C₄	—	86	2	no	—
295	C₂²⋊C₄	—	86	2	no	—
296	C₂²⋊C₄	—	86	2	no	—
297	C₂²⋊C₄	—	86	2	no	—
298	C₂²⋊C₄	—	86	2	no	—
299	C₂²⋊C₄	—	172	2	no	—
300	C₄²	—	86	2	no	—
301	C₄²	—	86	2	no	—
302	C₄²	—	86	2	no	—
303	C₄²	—	86	2	no	—
304	C₄²	—	86	2	no	—
305	C₄²	—	86	2	no	—
306	C₂³	—	2	2	no	—
307	C₂³	—	2	2	no	—
308	C₂³	—	4	1	no	—
309	C₂³	#13063287	1	1	yes	characteristic
310	C₂³	#13063287	1	1	yes	characteristic
311	C₂³	C₂×#10621902	1	1	yes	characteristic
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	86	2	no	—
320	C₂×C₄	—	86	2	no	—
321	C₂×C₄	—	86	2	no	—
322	C₂×C₄	—	86	2	no	—
323	C₂×C₄	—	344	2	no	—
324	C₂×C₄	—	172	2	no	—
325	C₂×C₄	—	172	2	no	—
326	C₂×C₄	—	172	2	no	—
327	C₂×C₄	—	172	4	no	—
328	C₂×C₄	#13063287	1	1	yes	characteristic
329	C₂×C₄	#13063287	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
331	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
332	C₂×C₄	C₂×#10621900	1	2	yes	—
333	C₂×C₄	#13063372	1	2	yes	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	4	4	no	—
337	C₈	—	86	4	no	—
338	C₈	#13063281	1	2	yes	—
339	C₈	#13063281	1	2	yes	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	#14583020	1	2	yes	—
349	C₂²	C₂×#13063287	1	1	yes	characteristic
350	C₂²	#14583591	1	2	yes	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	344	2	no	—
354	C₄	—	172	2	no	—
355	C₄	—	172	2	no	—
356	C₄	—	172	2	no	—
357	C₄	—	172	2	no	—
358	C₄	C₂×#13063281	1	1	yes	characteristic
359	C₄	#14583376	1	1	yes	characteristic
360	C₄	#14583549	1	2	yes	—
361	C₂	—	2	2	no	—
362	C₂	—	2	2	no	—
363	C₂	—	4	1	no	—
364	C₂	#19137466	1	1	yes	characteristic
365	C₂	C₂×#14583020	1	1	yes	characteristic
366	C₂	#19143511	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1810	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1810	D₄₃
hypocenter	C₄₃	#22607
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1810

Complements

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

N	H ≅ G/N	type
3. #19135346	C₂	semidirect
4. #19135193	C₂	semidirect
6. #19134482	C₂	semidirect
13. #14582937	C₄	semidirect
14. #14582937	C₄	semidirect
15. #14582937	C₄	semidirect
19. C₂×#13063259	C₄	semidirect
20. C₂×#13063259	C₄	semidirect
21. C₂×#13063257	C₄	semidirect
217. C₄₃	#22607	semidirect

Maximal subgroups

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

subgroup	index
#19143255	2
C₂×#14582937	2
#19135346	2
#19135193	2
#19134692	2
#19134482	2
#19134371	2
C₄₃×#1810	2
#22607	43