Subgroups — #24990908

Order 11008.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143164	C₂	1	1	yes	maximal, characteristic
2	#19134612	C₂	1	1	yes	maximal, characteristic
3	#19133367	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132978	C₂	1	1	yes	maximal, characteristic
5	#19132496	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132435	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063333	—	2	1	no	—
9	C₄×#10621898	C₂²	1	1	yes	characteristic
10	#14582952	C₂²	1	1	yes	characteristic
11	#14582927	—	2	1	no	—
12	#14582924	—	2	2	no	—
13	#14582921	—	2	1	no	—
14	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×D₄₃	—	2	1	no	—
16	C₂×#13063251	—	2	1	no	—
17	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
18	#14582660	—	2	1	no	—
19	#14582649	—	2	1	no	—
20	#14582649	—	2	1	no	—
21	#14582631	C₄	1	1	yes	characteristic, complemented
22	#14582631	C₄	1	1	yes	characteristic, complemented
23	#14582628	C₄	1	1	yes	characteristic, complemented
24	#14582628	C₄	1	1	yes	characteristic, complemented
25	C₄₃×#423	C₂²	1	1	yes	characteristic
26	C₄₃×#423	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#334	—	2	2	no	—
29	C₂×#10621898	D₄	1	1	yes	characteristic
30	#13063333	—	4	2	no	—
31	#13063333	—	4	2	no	—
32	C₂×#10621872	—	2	2	no	—
33	C₂²×C₄₃⋊C₈	—	2	1	no	—
34	C₂×C₈×D₄₃	—	2	2	no	—
35	C₄×Dic₈₆	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₈₆	Q₈	1	1	yes	characteristic
37	C₄×D₁₇₂	C₂×C₄	1	1	yes	characteristic
38	C₄×D₁₇₂	Q₈	1	1	yes	characteristic
39	C₄×C₂²⋊D₄₃	—	2	1	no	—
40	C₄×C₂²⋊D₄₃	—	2	1	no	—
41	#13063284	D₄	1	1	yes	characteristic
42	#13063282	—	2	1	no	—
43	C₂×C₄⋊Dic₄₃	—	2	1	no	—
44	C₂×C₄×Dic₄₃	—	2	1	no	—
45	C₄²×D₄₃	—	2	1	no	—
46	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
47	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
48	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
49	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
50	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
51	Dic₄₃⋊C₈	—	2	2	no	—
52	Dic₄₃⋊C₈	—	2	2	no	—
53	#13063251	—	4	2	no	—
54	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
55	C₄×C₄₃⋊C₈	—	2	1	no	—
56	C₄₃⋊#95	—	2	1	no	—
57	C₂×M₁₆×C₄₃	—	2	1	no	—
58	C₂×M₁₆×C₄₃	—	2	1	no	—
59	C₂²×C₈×C₄₃	—	2	1	no	—
60	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₄₃	—	2	1	no	—
62	C₄⋊C₈×C₄₃	—	2	1	no	—
63	C₄⋊C₈×C₄₃	—	2	1	no	—
64	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
65	C₄×C₈×C₄₃	—	2	1	no	—
66	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
71	C₂×C₂²⋊D₄₃	—	2	1	no	—
72	C₂²×Dic₄₃	—	4	1	no	—
73	#10621898	—	2	1	no	—
74	#10621898	—	2	1	no	—
75	#10621898	SD₁₆	1	2	yes	—
76	#10621898	SD₁₆	1	2	yes	—
77	C₂×D₁₇₂	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×D₄₃	—	2	1	no	—
79	C₂×C₄×D₄₃	—	2	1	no	—
80	C₂×C₄×D₄₃	—	2	1	no	—
81	C₂×Dic₈₆	C₄⋊C₄	1	1	yes	characteristic
82	M₁₆×C₄₃	—	4	1	no	—
83	M₁₆×C₄₃	—	4	1	no	—
84	M₁₆×C₄₃	—	4	1	no	—
85	M₁₆×C₄₃	—	4	1	no	—
86	C₂×C₈×C₄₃	—	2	1	no	—
87	C₂×C₈×C₄₃	—	2	1	no	—
88	C₂×C₈×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	—	4	2	no	—
90	C₂×C₈×C₄₃	—	4	1	no	—
91	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
95	C₂²⋊Dic₄₃	—	2	1	no	—
96	(C₂×D₄₃)⋊C₄	—	2	1	no	—
97	(C₂×D₄₃)⋊C₄	—	2	1	no	—
98	C₄⋊Dic₄₃	—	2	2	no	—
99	C₄⋊Dic₄₃	—	4	1	no	—
100	C₄⋊Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄⋊Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
102	Dic₄₃⋊C₄	—	2	1	no	—
103	Dic₄₃⋊C₄	—	2	1	no	—
104	C₄×Dic₄₃	—	2	1	no	—
105	C₄×Dic₄₃	—	2	1	no	—
106	C₄×Dic₄₃	—	2	1	no	—
107	C₄×Dic₄₃	—	4	1	no	—
108	C₄×Dic₄₃	—	4	1	no	—
109	C₂×C₄₃⋊C₈	—	2	1	no	—
110	C₂×C₄₃⋊C₈	—	2	1	no	—
111	C₂×C₄₃⋊C₈	—	2	1	no	—
112	C₂×C₄₃⋊C₈	—	2	1	no	—
113	C₂×C₄₃⋊C₈	—	4	1	no	—
114	C₂×C₄₃⋊C₈	—	4	1	no	—
115	#10621872	—	4	4	no	—
116	C₈×D₄₃	—	4	4	no	—
117	C₂³×C₄₃	#123	1	1	yes	characteristic
118	C₂²×D₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	4	1	no	—
122	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
132	C₂²⋊D₄₃	—	2	2	no	—
133	C₂²⋊D₄₃	—	2	2	no	—
134	C₂×Dic₄₃	—	2	1	no	—
135	C₂×Dic₄₃	—	2	1	no	—
136	C₂×Dic₄₃	—	2	1	no	—
137	C₂×Dic₄₃	—	4	2	no	—
138	C₂×Dic₄₃	—	8	1	no	—
139	C₂×Dic₄₃	—	8	1	no	—
140	D₁₇₂	—	2	1	no	—
141	D₁₇₂	#106	1	2	yes	—
142	C₄×D₄₃	—	2	2	no	—
143	C₄×D₄₃	—	2	2	no	—
144	C₄×D₄₃	—	4	1	no	—
145	C₄×D₄₃	—	4	1	no	—
146	C₄×D₄₃	—	8	1	no	—
147	Dic₈₆	—	2	1	no	—
148	Dic₈₆	#106	1	2	yes	—
149	C₈×C₄₃	—	2	2	no	—
150	C₈×C₄₃	—	4	1	no	—
151	C₈×C₄₃	—	4	2	no	—
152	C₈×C₄₃	—	8	1	no	—
153	C₄₃⋊C₈	—	4	1	no	—
154	C₄₃⋊C₈	—	4	1	no	—
155	C₄₃⋊C₈	—	4	1	no	—
156	C₄₃⋊C₈	—	4	1	no	—
157	#8904	—	43	1	no	maximal, Sylow
158	C₂²×C₄₃	—	2	1	no	—
159	C₂²×C₄₃	—	2	1	no	—
160	C₂²×C₄₃	#385	1	1	yes	characteristic
161	C₂²×C₄₃	#481	1	1	yes	characteristic
162	C₂²×C₄₃	#483	1	1	yes	characteristic
163	C₂×D₄₃	—	2	2	no	—
164	C₂×D₄₃	—	4	1	no	—
165	C₂×D₄₃	—	4	1	no	—
166	C₄×C₄₃	—	2	1	no	—
167	C₄×C₄₃	—	2	1	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	C₂×#106	1	1	yes	characteristic
170	C₄×C₄₃	#428	1	1	yes	characteristic
171	C₄×C₄₃	#443	1	1	yes	characteristic
172	C₄×C₄₃	#444	1	1	yes	characteristic
173	C₄×C₄₃	#448	1	1	yes	characteristic
174	C₄×C₄₃	#450	1	1	yes	characteristic
175	Dic₄₃	—	2	2	no	—
176	Dic₄₃	—	4	1	no	—
177	Dic₄₃	—	4	1	no	—
178	Dic₄₃	—	8	2	no	—
179	#2906	—	43	1	no	—
180	#1806	—	43	1	no	—
181	#1570	—	43	1	no	—
182	#1566	D₄₃	1	1	yes	characteristic, complemented
183	#1476	—	43	1	no	—
184	#1303	—	43	1	no	—
185	#1279	—	43	1	no	—
186	C₂×C₄₃	—	2	1	no	—
187	C₂×C₄₃	—	2	1	no	—
188	C₂×C₄₃	#1853	1	1	yes	characteristic
189	C₂×C₄₃	#1861	1	1	yes	characteristic
190	C₂×C₄₃	#1866	1	1	yes	characteristic
191	D₄₃	—	4	2	no	—
192	C₂×#131	—	86	1	no	—
193	C₄×#41	—	43	1	no	—
194	#431	—	43	1	no	—
195	#424	—	86	1	no	—
196	#424	—	86	2	no	—
197	#423	—	86	1	no	—
198	#423	C₂×D₄₃	1	1	yes	characteristic
199	#423	C₂×D₄₃	1	1	yes	characteristic
200	C₂×C₄⋊C₈	—	43	1	no	—
201	C₂×C₄⋊C₈	—	86	1	no	—
202	C₂×#104	—	86	1	no	—
203	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
204	C₄×C₄⋊C₄	—	43	1	no	—
205	#339	—	86	1	no	—
206	#334	—	2	2	no	—
207	#334	—	86	1	no	—
208	#334	—	86	1	no	—
209	Q₈⋊C₈	—	43	1	no	—
210	Q₈⋊C₈	—	43	1	no	—
211	D₄⋊C₈	—	43	1	no	—
212	D₄⋊C₈	—	43	1	no	—
213	C₄₃	#8904	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	43	1	no	—
215	#131	—	172	2	no	—
216	#131	—	172	2	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	—	86	2	no	—
220	C₂²×C₈	—	2	1	no	—
221	C₂²×C₈	—	86	2	no	—
222	C₂²×C₈	—	86	1	no	—
223	C₄×Q₈	—	43	1	no	—
224	C₄×Q₈	—	43	1	no	—
225	C₄×D₄	—	43	1	no	—
226	C₄×D₄	—	43	1	no	—
227	C₄×D₄	—	86	1	no	—
228	C₄×D₄	—	86	1	no	—
229	#117	—	43	1	no	—
230	#117	—	86	1	no	—
231	C₂×C₄⋊C₄	—	86	1	no	—
232	C₂×C₄²	—	86	1	no	—
233	C₂×C₄²	—	86	1	no	—
234	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	43	1	no	—
239	C₄⋊C₈	—	43	1	no	—
240	C₄⋊C₈	—	43	1	no	—
241	C₄⋊C₈	—	43	1	no	—
242	C₄⋊C₈	—	86	2	no	—
243	C₄⋊C₈	—	86	2	no	—
244	C₄⋊C₈	—	86	1	no	—
245	C₄⋊C₈	C₂²⋊D₄₃	1	2	yes	—
246	#104	—	172	2	no	—
247	C₂²⋊C₈	—	86	1	no	—
248	C₄×C₈	—	2	1	no	—
249	C₄×C₈	—	86	1	no	—
250	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
251	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
252	#95	—	86	1	no	—
253	#41	—	43	2	no	—
254	#41	—	43	2	no	—
255	#41	—	86	1	no	—
256	#41	—	86	1	no	—
257	C₂×Q₈	—	43	1	no	—
258	C₂×D₄	—	43	1	no	—
259	C₂×D₄	—	86	1	no	—
260	C₂²×C₄	—	86	1	no	—
261	C₂²×C₄	—	86	1	no	—
262	C₂²×C₄	—	86	1	no	—
263	C₂²×C₄	—	172	1	no	—
264	C₂²×C₄	#10621898	1	1	yes	characteristic
265	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
266	C₂²×C₄	#10621902	1	1	yes	characteristic
267	M₁₆	—	4	1	no	—
268	M₁₆	—	4	1	no	—
269	M₁₆	—	4	1	no	—
270	M₁₆	—	4	1	no	—
271	M₁₆	—	172	4	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	4	2	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	86	1	no	—
278	C₂×C₈	—	86	1	no	—
279	C₂×C₈	—	86	1	no	—
280	C₂×C₈	—	86	1	no	—
281	C₂×C₈	—	172	4	no	—
282	C₂×C₈	—	172	1	no	—
283	C₂×C₈	—	172	1	no	—
284	C₄⋊C₄	—	43	1	no	—
285	C₄⋊C₄	—	43	1	no	—
286	C₄⋊C₄	—	86	2	no	—
287	C₄⋊C₄	—	86	1	no	—
288	C₄⋊C₄	—	86	1	no	—
289	C₄⋊C₄	—	172	1	no	—
290	C₂²⋊C₄	—	86	1	no	—
291	C₂²⋊C₄	—	86	1	no	—
292	C₂²⋊C₄	—	86	1	no	—
293	C₄²	—	86	1	no	—
294	C₄²	—	86	1	no	—
295	C₄²	—	86	1	no	—
296	C₄²	—	172	1	no	—
297	C₄²	—	172	1	no	—
298	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
299	C₄²	D₄×D₄₃	1	1	yes	characteristic
300	C₄²	Q₈×D₄₃	1	1	yes	characteristic
301	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
302	C₂³	—	86	1	no	—
303	C₂³	#13063310	1	1	yes	characteristic
304	Q₈	—	43	2	no	—
305	Q₈	—	86	1	no	—
306	D₄	—	43	2	no	—
307	D₄	—	86	2	no	—
308	D₄	—	86	2	no	—
309	D₄	—	86	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	86	1	no	—
314	C₂×C₄	—	86	2	no	—
315	C₂×C₄	—	86	2	no	—
316	C₂×C₄	—	86	1	no	—
317	C₂×C₄	—	86	1	no	—
318	C₂×C₄	—	344	1	no	—
319	C₂×C₄	—	344	1	no	—
320	C₂×C₄	—	344	1	no	—
321	C₂×C₄	—	172	2	no	—
322	C₂×C₄	—	172	1	no	—
323	C₂×C₄	—	172	1	no	—
324	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
325	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
326	C₂×C₄	#13063287	1	1	yes	characteristic
327	C₂×C₄	#13063293	1	1	yes	characteristic
328	C₂×C₄	#13063301	1	1	yes	characteristic
329	C₂×C₄	#13063303	1	1	yes	characteristic
330	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
331	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
332	C₂×C₄	#13063352	1	1	yes	characteristic
333	C₂×C₄	#13063354	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	2	no	—
337	C₈	—	8	1	no	—
338	C₈	—	172	1	no	—
339	C₈	—	172	1	no	—
340	C₈	—	172	1	no	—
341	C₈	—	172	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	86	2	no	—
345	C₂²	—	172	1	no	—
346	C₂²	—	172	1	no	—
347	C₂²	#14582770	1	1	yes	characteristic
348	C₂²	#14583219	1	1	yes	characteristic
349	C₂²	#14583231	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	86	2	no	—
354	C₄	—	344	2	no	—
355	C₄	—	172	1	no	—
356	C₄	—	172	1	no	—
357	C₄	#106×D₄₃	1	1	yes	characteristic
358	C₄	#14582938	1	1	yes	characteristic
359	C₄	#14583015	1	1	yes	characteristic
360	C₄	#14583020	1	1	yes	characteristic
361	C₄	#14583036	1	1	yes	characteristic
362	C₄	#14583048	1	1	yes	characteristic
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	172	2	no	—
366	C₂	#19134786	1	1	yes	characteristic
367	C₂	#19134834	1	1	yes	characteristic
368	C₂	#19134867	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063293
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1566	C₂
socle	C₂²×C₄₃	#385
prime radical	C₂×#10621898	D₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1566	D₄₃
hypocenter	C₄₃	#8904
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
3. #19133367	C₂	semidirect
5. #19132496	C₂	semidirect
6. #19132435	C₂	semidirect
7. C₄₃×#1566	C₂	semidirect
21. #14582631	C₄	semidirect
22. #14582631	C₄	semidirect
23. #14582628	C₄	semidirect
24. #14582628	C₄	semidirect
182. #1566	D₄₃	semidirect
213. C₄₃	#8904	semidirect

Maximal subgroups

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

subgroup	index
#19143164	2
#19134612	2
#19133367	2
#19132978	2
#19132496	2
#19132435	2
C₄₃×#1566	2
#8904	43