Subgroups — #24992459

Order 11008.

478 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144404	C₂	1	1	yes	maximal, characteristic
2	#19143164	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582648	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132983	C₂	1	1	yes	maximal, characteristic
5	#19132491	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132469	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063333	—	2	1	no	—
9	C₄×#10621898	C₂²	1	1	yes	characteristic
10	#14583191	—	2	1	no	—
11	#14583186	—	2	1	no	—
12	#14583181	C₄	1	2	yes	—
13	#14583176	C₄	1	2	yes	—
14	#14582939	—	2	1	no	—
15	#14582927	—	2	1	no	—
16	#14582924	—	2	2	no	—
17	#14582923	C₂²	1	1	yes	characteristic
18	#14582921	—	2	1	no	—
19	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×D₄₃	—	2	1	no	—
22	#14582898	—	2	1	no	—
23	#14582648	—	2	1	no	—
24	#14582648	—	2	1	no	—
25	#14582631	—	2	1	no	—
26	#14582628	—	2	1	no	—
27	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#431	C₂²	1	1	yes	characteristic
29	C₄₃×#423	C₂²	1	1	yes	characteristic
30	C₄₃×#332	—	2	2	no	—
31	C₂×#10621898	D₄	1	1	yes	characteristic
32	#13063333	—	4	2	no	—
33	#13063333	—	4	2	no	—
34	C₂×#10621877	—	2	1	no	—
35	C₂×#10621872	—	2	2	no	—
36	C₂²×C₄₃⋊C₈	—	2	1	no	—
37	C₂²×C₄₃⋊C₈	—	2	1	no	—
38	C₂×C₈×D₄₃	—	2	2	no	—
39	C₄×Dic₈₆	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₈₆	D₄	1	1	yes	characteristic
41	C₄×D₁₇₂	C₂×C₄	1	1	yes	characteristic
42	C₄×D₁₇₂	D₄	1	1	yes	characteristic
43	C₄×C₂²⋊D₄₃	—	2	1	no	—
44	C₄×C₂²⋊D₄₃	—	2	1	no	—
45	#13063284	D₄	1	1	yes	characteristic
46	#13063282	—	2	1	no	—
47	C₄²×D₄₃	—	2	1	no	—
48	#13063258	—	2	2	no	—
49	#13063256	—	2	2	no	—
50	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
51	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
52	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
53	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
54	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
55	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
56	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
57	Dic₄₃⋊C₈	—	2	2	no	—
58	Dic₄₃⋊C₈	—	2	2	no	—
59	#13063248	—	2	2	no	—
60	#13063245	—	2	2	no	—
61	C₂²×Q₈×C₄₃	—	2	1	no	—
62	C₂×M₁₆×C₄₃	—	2	1	no	—
63	C₄⋊Q₈×C₄₃	—	2	1	no	—
64	C₄⋊Q₈×C₄₃	—	2	1	no	—
65	C₄⋊Q₈×C₄₃	—	2	1	no	—
66	C₄⋊Q₈×C₄₃	C₂×C₄	1	2	yes	—
67	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
68	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
69	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₄₃	—	2	1	no	—
71	C₄⋊C₈×C₄₃	—	2	1	no	—
72	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
73	C₂²⋊C₈×C₄₃	—	2	1	no	—
74	C₄×C₈×C₄₃	—	2	1	no	—
75	C₂×Q₈×C₄₃	—	2	2	no	—
76	C₂×Q₈×C₄₃	—	4	1	no	—
77	C₂×Q₈×C₄₃	—	8	2	no	—
78	C₂²×C₄×C₄₃	—	4	1	no	—
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
82	C₂×C₂²⋊D₄₃	—	2	1	no	—
83	#10621898	—	2	1	no	—
84	#10621898	—	2	1	no	—
85	#10621898	SD₁₆	1	2	yes	—
86	#10621898	Q₁₆	1	2	yes	—
87	C₂×D₁₇₂	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×D₄₃	—	2	1	no	—
89	C₂×C₄×D₄₃	—	2	1	no	—
90	C₂×C₄×D₄₃	—	2	1	no	—
91	C₂×Dic₈₆	C₂²⋊C₄	1	1	yes	characteristic
92	M₁₆×C₄₃	—	4	1	no	—
93	M₁₆×C₄₃	—	4	1	no	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₂×C₈×C₄₃	—	4	1	no	—
96	C₄⋊C₄×C₄₃	—	2	2	no	—
97	C₄⋊C₄×C₄₃	—	2	2	no	—
98	C₄⋊C₄×C₄₃	—	4	1	no	—
99	C₄⋊C₄×C₄₃	—	4	1	no	—
100	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₂²⋊Dic₄₃	—	2	1	no	—
105	(C₂×D₄₃)⋊C₄	—	2	1	no	—
106	(C₂×D₄₃)⋊C₄	—	2	1	no	—
107	C₄⋊Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
108	C₄⋊Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
109	Dic₄₃⋊C₄	—	2	1	no	—
110	Dic₄₃⋊C₄	—	2	1	no	—
111	C₄×Dic₄₃	—	2	1	no	—
112	C₄×Dic₄₃	—	2	1	no	—
113	C₄×Dic₄₃	—	2	1	no	—
114	#10621877	—	4	1	no	—
115	#10621877	—	4	1	no	—
116	C₂×C₄₃⋊C₈	—	2	2	no	—
117	C₂×C₄₃⋊C₈	—	2	1	no	—
118	C₂×C₄₃⋊C₈	—	2	1	no	—
119	C₂×C₄₃⋊C₈	—	4	1	no	—
120	C₂×C₄₃⋊C₈	—	4	1	no	—
121	C₂×C₄₃⋊C₈	—	4	1	no	—
122	C₂×C₄₃⋊C₈	—	8	1	no	—
123	#10621872	—	4	4	no	—
124	C₈×D₄₃	—	4	4	no	—
125	C₂³×C₄₃	#123	1	1	yes	characteristic
126	C₂²×D₄₃	—	2	1	no	—
127	Q₈×C₄₃	—	8	2	no	—
128	Q₈×C₄₃	—	8	2	no	—
129	C₂×C₄×C₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	—	4	1	no	—
132	C₂×C₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	—	8	2	no	—
134	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
142	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
144	C₂²⋊D₄₃	—	2	2	no	—
145	C₂²⋊D₄₃	—	2	2	no	—
146	C₂×Dic₄₃	—	2	1	no	—
147	C₂×Dic₄₃	—	2	1	no	—
148	C₂×Dic₄₃	—	2	1	no	—
149	D₁₇₂	—	2	1	no	—
150	D₁₇₂	#103	1	2	yes	—
151	C₄×D₄₃	—	2	2	no	—
152	C₄×D₄₃	—	2	2	no	—
153	C₄×D₄₃	—	4	1	no	—
154	C₄×D₄₃	—	4	1	no	—
155	C₄×D₄₃	—	8	1	no	—
156	Dic₈₆	—	2	1	no	—
157	Dic₈₆	#103	1	2	yes	—
158	C₈×C₄₃	—	4	2	no	—
159	C₈×C₄₃	—	8	1	no	—
160	C₄₃⋊C₈	—	4	2	no	—
161	C₄₃⋊C₈	—	4	1	no	—
162	C₄₃⋊C₈	—	4	1	no	—
163	C₄₃⋊C₈	—	8	1	no	—
164	#9169	—	43	1	no	maximal, Sylow
165	C₂²×C₄₃	—	2	1	no	—
166	C₂²×C₄₃	—	2	1	no	—
167	C₂²×C₄₃	#386	1	1	yes	characteristic
168	C₂²×C₄₃	#483	1	1	yes	characteristic
169	C₂²×C₄₃	#484	1	1	yes	characteristic
170	C₂×D₄₃	—	2	2	no	—
171	C₂×D₄₃	—	4	1	no	—
172	C₂×D₄₃	—	4	1	no	—
173	C₄×C₄₃	—	2	1	no	—
174	C₄×C₄₃	—	2	1	no	—
175	C₄×C₄₃	—	2	1	no	—
176	C₄×C₄₃	—	8	2	no	—
177	C₄×C₄₃	C₂×#103	1	1	yes	characteristic
178	C₄×C₄₃	#417	1	1	yes	characteristic
179	C₄×C₄₃	#450	1	1	yes	characteristic
180	C₄×C₄₃	#451	1	1	yes	characteristic
181	C₄×C₄₃	#471	1	1	yes	characteristic
182	C₄×C₄₃	#473	1	1	yes	characteristic
183	Dic₄₃	—	2	2	no	—
184	Dic₄₃	—	4	1	no	—
185	Dic₄₃	—	4	1	no	—
186	#3079	—	43	1	no	—
187	#2906	—	43	1	no	—
188	#1549	D₄₃	1	1	yes	characteristic, complemented
189	C₂×#333	—	43	1	no	—
190	#1477	—	43	1	no	—
191	#1301	—	43	1	no	—
192	#1292	—	43	1	no	—
193	C₂×C₄₃	—	2	1	no	—
194	C₂×C₄₃	—	2	1	no	—
195	C₂×C₄₃	#1872	1	1	yes	characteristic
196	C₂×C₄₃	#1903	1	1	yes	characteristic
197	C₂×C₄₃	#1908	1	1	yes	characteristic
198	D₄₃	—	4	2	no	—
199	C₂×#131	—	86	1	no	—
200	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
201	C₄×#41	—	43	1	no	—
202	#477	—	86	1	no	—
203	#476	—	86	1	no	—
204	#475	—	43	2	no	—
205	#474	—	43	2	no	—
206	#431	C₂×D₄₃	1	1	yes	characteristic
207	#428	—	86	1	no	—
208	#424	—	86	1	no	—
209	#424	—	86	2	no	—
210	#423	—	43	1	no	—
211	#423	—	86	1	no	—
212	#423	C₂×D₄₃	1	1	yes	characteristic
213	C₂×C₄⋊C₈	—	43	1	no	—
214	C₂×C₄⋊C₈	—	43	1	no	—
215	C₂×C₄⋊C₈	—	86	1	no	—
216	#417	—	86	1	no	—
217	#333	—	86	1	no	—
218	#333	—	86	1	no	—
219	#332	—	2	2	no	—
220	Q₈⋊C₈	—	86	1	no	—
221	D₄⋊C₈	—	86	1	no	—
222	C₄₃	#9169	1	1	yes	characteristic, complemented, Sylow
223	C₂×#41	—	43	1	no	—
224	C₂²×Q₈	—	2	1	no	—
225	#131	—	172	2	no	—
226	#131	—	172	2	no	—
227	C₂×M₁₆	—	2	1	no	—
228	C₂×M₁₆	—	86	2	no	—
229	C₂×M₁₆	—	86	1	no	—
230	C₂²×C₈	—	86	2	no	—
231	C₂²×C₈	—	86	1	no	—
232	C₂²×C₈	—	86	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	—	2	1	no	—
236	C₄⋊Q₈	C₄×D₄₃	1	2	yes	—
237	C₄×Q₈	—	43	1	no	—
238	C₄×Q₈	—	43	1	no	—
239	C₄×D₄	—	43	1	no	—
240	C₄×D₄	—	43	1	no	—
241	C₄×D₄	—	86	1	no	—
242	C₄×D₄	—	86	1	no	—
243	#117	—	43	1	no	—
244	#117	—	86	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄²	—	86	1	no	—
248	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
249	#107	—	86	2	no	—
250	#106	—	86	2	no	—
251	C₄⋊C₈	—	2	1	no	—
252	C₄⋊C₈	—	2	1	no	—
253	C₄⋊C₈	—	43	2	no	—
254	C₄⋊C₈	—	43	1	no	—
255	C₄⋊C₈	—	43	1	no	—
256	C₄⋊C₈	—	86	2	no	—
257	C₄⋊C₈	—	86	2	no	—
258	C₄⋊C₈	—	86	1	no	—
259	C₄⋊C₈	—	86	1	no	—
260	C₄⋊C₈	—	86	1	no	—
261	C₄⋊C₈	—	86	1	no	—
262	C₄⋊C₈	C₂²⋊D₄₃	1	2	yes	—
263	#103	—	86	2	no	—
264	#102	—	86	2	no	—
265	C₂²⋊C₈	—	2	1	no	—
266	C₄×C₈	—	2	1	no	—
267	#41	—	43	2	no	—
268	#41	—	43	2	no	—
269	#41	—	86	1	no	—
270	#41	—	86	1	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	4	1	no	—
273	C₂×Q₈	—	8	2	no	—
274	C₂×Q₈	—	43	1	no	—
275	C₂×D₄	—	43	1	no	—
276	C₂×D₄	—	86	1	no	—
277	C₂²×C₄	—	4	1	no	—
278	C₂²×C₄	—	86	1	no	—
279	C₂²×C₄	—	86	1	no	—
280	C₂²×C₄	—	86	1	no	—
281	C₂²×C₄	#10621898	1	1	yes	characteristic
282	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
283	C₂²×C₄	#10621900	1	1	yes	characteristic
284	M₁₆	—	4	1	no	—
285	M₁₆	—	4	1	no	—
286	M₁₆	—	172	4	no	—
287	M₁₆	—	172	1	no	—
288	M₁₆	—	172	1	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	4	1	no	—
291	C₂×C₈	—	86	2	no	—
292	C₂×C₈	—	86	1	no	—
293	C₂×C₈	—	86	1	no	—
294	C₂×C₈	—	344	1	no	—
295	C₂×C₈	—	172	4	no	—
296	C₂×C₈	—	172	1	no	—
297	C₂×C₈	—	172	1	no	—
298	C₂×C₈	—	172	1	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	43	1	no	—
304	C₄⋊C₄	—	43	1	no	—
305	C₄⋊C₄	—	86	1	no	—
306	C₄⋊C₄	—	86	1	no	—
307	C₂²⋊C₄	—	86	1	no	—
308	C₂²⋊C₄	—	86	1	no	—
309	C₂²⋊C₄	—	86	1	no	—
310	C₄²	—	86	1	no	—
311	C₄²	—	86	1	no	—
312	C₄²	—	86	1	no	—
313	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
314	C₄²	D₄×D₄₃	1	1	yes	characteristic
315	C₄²	D₄×D₄₃	1	1	yes	characteristic
316	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
317	C₂³	—	86	1	no	—
318	C₂³	#13063307	1	1	yes	characteristic
319	Q₈	—	8	2	no	—
320	Q₈	—	8	2	no	—
321	Q₈	—	43	2	no	—
322	Q₈	—	86	1	no	—
323	D₄	—	43	2	no	—
324	D₄	—	86	2	no	—
325	D₄	—	86	2	no	—
326	D₄	—	86	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	8	2	no	—
332	C₂×C₄	—	86	1	no	—
333	C₂×C₄	—	86	2	no	—
334	C₂×C₄	—	86	2	no	—
335	C₂×C₄	—	86	1	no	—
336	C₂×C₄	—	86	1	no	—
337	C₂×C₄	—	344	1	no	—
338	C₂×C₄	—	172	1	no	—
339	C₂×C₄	—	172	1	no	—
340	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
341	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
342	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
343	C₂×C₄	#13063293	1	1	yes	characteristic
344	C₂×C₄	#13063296	1	1	yes	characteristic
345	C₂×C₄	#13063298	1	1	yes	characteristic
346	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
347	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
348	C₂×C₄	#13063352	1	1	yes	characteristic
349	C₂×C₄	#13063357	1	1	yes	characteristic
350	C₈	—	4	2	no	—
351	C₈	—	8	1	no	—
352	C₈	—	344	1	no	—
353	C₈	—	172	1	no	—
354	C₈	—	172	1	no	—
355	C₈	—	172	2	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	86	2	no	—
359	C₂²	—	172	1	no	—
360	C₂²	—	172	1	no	—
361	C₂²	#14582775	1	1	yes	characteristic
362	C₂²	#14583228	1	1	yes	characteristic
363	C₂²	#14583233	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	8	2	no	—
368	C₄	—	86	2	no	—
369	C₄	—	172	1	no	—
370	C₄	—	172	1	no	—
371	C₄	#103×D₄₃	1	1	yes	characteristic
372	C₄	#14582899	1	1	yes	characteristic
373	C₄	#14583048	1	1	yes	characteristic
374	C₄	#14583053	1	1	yes	characteristic
375	C₄	#14583165	1	1	yes	characteristic
376	C₄	#14583173	1	1	yes	characteristic
377	C₂	—	2	1	no	—
378	C₂	—	2	1	no	—
379	C₂	—	172	2	no	—
380	C₂	#19134899	1	1	yes	characteristic
381	C₂	#19135072	1	1	yes	characteristic
382	C₂	#19135107	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14582775
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1549	C₂
socle	C₂²×C₄₃	#386
prime radical	C₂×#10621898	D₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1549	D₄₃
hypocenter	C₄₃	#9169
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1549

Complements

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

N	H ≅ G/N	type
3. C₂×#14582648	C₂	semidirect
5. #19132491	C₂	semidirect
6. #19132469	C₂	semidirect
7. C₄₃×#1549	C₂	semidirect
188. #1549	D₄₃	semidirect
222. C₄₃	#9169	semidirect

Maximal subgroups

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

subgroup	index
#19144404	2
#19143164	2
C₂×#14582648	2
#19132983	2
#19132491	2
#19132469	2
C₄₃×#1549	2
#9169	43