Subgroups — #25251847

Order 11008.

526 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145324	C₂	1	1	yes	maximal, characteristic, complemented
2	#19143506	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136815	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136810	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136772	C₂	1	2	yes	maximal, complemented
6	#19136665	C₂	1	2	yes	maximal, complemented
7	#19136548	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136543	C₂	1	1	yes	maximal, characteristic
9	#19136495	C₂	1	2	yes	maximal, complemented
10	#19136488	C₂	1	2	yes	maximal, complemented
11	C₄₃×#2165	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583699	C₂²	1	1	yes	characteristic
13	#14583665	—	2	1	no	—
14	#14583433	—	2	1	no	—
15	#14583342	C₂²	1	2	yes	—
16	#14583337	C₂²	1	2	yes	—
17	#14583301	C₂²	1	1	yes	characteristic, complemented
18	#14583283	C₂²	1	1	yes	characteristic, complemented
19	#14583254	—	2	1	no	—
20	#14583157	C₂²	1	1	yes	characteristic
21	#14583152	C₂²	1	1	yes	characteristic
22	#14583145	—	2	1	no	—
23	#14583022	—	2	1	no	—
24	#14583022	C₂²	1	2	yes	—
25	#14583017	—	2	1	no	—
26	#14583017	C₂²	1	2	yes	—
27	#14583012	C₂²	1	1	yes	characteristic, complemented
28	#14583007	C₂²	1	1	yes	characteristic, complemented
29	#14583001	—	2	1	no	—
30	#14582948	C₂²	1	1	yes	characteristic
31	C₂×#13063253	—	2	1	no	—
32	#14582722	C₂²	1	2	yes	complemented
33	#14582722	C₂²	1	2	yes	complemented
34	#14582716	C₂²	1	2	yes	complemented
35	#14582713	C₂²	1	2	yes	complemented
36	#14582710	C₂²	1	2	yes	—
37	#14582710	C₂²	1	2	yes	—
38	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
39	C₄₃×#364	C₂²	1	2	yes	complemented
40	C₄₃×#349	C₂²	1	1	yes	characteristic
41	C₄₃×#349	C₂²	1	1	yes	characteristic
42	C₄₃×#347	C₂²	1	2	yes	complemented
43	C₂×#10621898	—	2	1	no	—
44	#13063358	—	2	2	no	—
45	#13063353	—	2	2	no	—
46	#13063347	—	2	1	no	—
47	#13063347	C₂³	1	2	yes	—
48	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
49	C₂×#10621873	—	2	1	no	—
50	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
51	#13063333	—	4	2	no	—
52	#13063331	—	2	2	no	—
53	#13063331	—	2	2	no	—
54	C₂×#10621877	—	2	1	no	—
55	M₁₆×D₄₃	—	4	2	no	—
56	#13063319	—	4	1	no	—
57	#13063317	—	2	1	no	—
58	#13063313	—	2	1	no	—
59	#13063308	—	4	1	no	—
60	#13063304	—	2	1	no	—
61	#13063299	—	2	1	no	—
62	C₄×Dic₈₆	—	2	1	no	—
63	C₄×D₁₇₂	—	2	1	no	—
64	C₄×C₂²⋊D₄₃	—	4	1	no	—
65	#13063282	—	4	1	no	—
66	Dic₃₄₄	—	2	2	no	—
67	#13063270	—	2	2	no	—
68	#13063270	—	2	2	no	—
69	D₆₈₈	—	2	2	no	—
70	#13063261	—	2	1	no	—
71	#13063261	C₂³	1	2	yes	—
72	#13063259	—	2	1	no	—
73	#13063257	C₂³	1	1	yes	characteristic
74	#13063257	C₂³	1	1	yes	characteristic
75	#13063253	—	2	2	no	—
76	#13063253	—	2	2	no	—
77	#13063250	—	2	1	no	—
78	#13063250	—	2	1	no	—
79	#13063247	—	2	1	no	—
80	#13063247	—	2	1	no	—
81	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
82	C₂×M₁₆×C₄₃	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
84	M₃₂×C₄₃	C₂³	1	2	yes	—
85	M₃₂×C₄₃	C₂³	1	2	yes	—
86	#97×C₄₃	D₄	1	1	yes	characteristic
87	#97×C₄₃	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
89	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
90	C₂²×C₄×C₄₃	—	2	1	no	—
91	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₄₃	—	4	1	no	—
93	C₂×C₂²⋊D₄₃	—	4	1	no	—
94	#10621898	—	2	2	no	—
95	#10621898	—	2	2	no	—
96	#10621898	—	4	1	no	—
97	#10621898	—	4	1	no	—
98	C₂×D₁₇₂	—	2	1	no	—
99	C₂×D₁₇₂	—	2	1	no	—
100	C₂×C₄×D₄₃	—	4	1	no	—
101	C₂×C₄×D₄₃	—	4	1	no	—
102	C₂×Dic₈₆	—	2	1	no	—
103	C₂×Dic₈₆	—	2	1	no	—
104	M₁₆×C₄₃	—	2	2	no	—
105	M₁₆×C₄₃	—	2	2	no	—
106	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
110	C₄²×C₄₃	—	2	1	no	—
111	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
113	C₂²⋊Dic₄₃	—	4	1	no	—
114	(C₂×D₄₃)⋊C₄	—	4	1	no	—
115	(C₂×D₄₃)⋊C₄	—	4	1	no	—
116	(C₂×D₄₃)⋊C₄	—	4	1	no	—
117	(C₂×D₄₃)⋊C₄	—	4	1	no	—
118	C₄⋊Dic₄₃	—	2	1	no	—
119	C₄⋊Dic₄₃	—	2	1	no	—
120	Dic₄₃⋊C₄	—	4	1	no	—
121	Dic₄₃⋊C₄	—	4	1	no	—
122	Dic₄₃⋊C₄	—	4	1	no	—
123	Dic₄₃⋊C₄	—	4	1	no	—
124	C₄×Dic₄₃	—	4	1	no	—
125	#10621877	—	2	2	no	—
126	#10621877	—	4	1	no	—
127	C₂×C₄₃⋊C₈	—	4	1	no	—
128	Dic₁₇₂	—	2	1	no	—
129	Dic₁₇₂	—	2	2	no	—
130	Dic₁₇₂	#41	1	2	yes	—
131	D₃₄₄	—	2	1	no	—
132	D₃₄₄	—	2	2	no	—
133	D₃₄₄	#41	1	2	yes	—
134	#10621873	—	2	2	no	—
135	#10621873	—	2	2	no	—
136	#10621873	—	4	2	no	—
137	#10621872	—	4	2	no	—
138	#10621872	—	4	2	no	—
139	C₈×D₄₃	—	4	2	no	—
140	C₈×D₄₃	—	4	2	no	—
141	C₁₆×C₄₃	—	2	2	no	—
142	C₁₆×C₄₃	—	2	2	no	—
143	C₂³×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂²×D₄₃	—	4	1	no	—
145	C₂²×D₄₃	—	4	1	no	—
146	C₂×C₄×C₄₃	—	2	1	no	—
147	C₂×C₄×C₄₃	—	2	1	no	—
148	C₂×C₄×C₄₃	—	2	1	no	—
149	C₂×C₄×C₄₃	—	2	1	no	—
150	C₂×C₄×C₄₃	—	8	1	no	—
151	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
153	C₂²⋊D₄₃	—	4	2	no	—
154	C₂²⋊D₄₃	—	4	2	no	—
155	C₂²⋊D₄₃	—	8	2	no	—
156	C₂×Dic₄₃	—	4	1	no	—
157	C₂×Dic₄₃	—	4	1	no	—
158	C₂×Dic₄₃	—	4	1	no	—
159	C₂×Dic₄₃	—	4	1	no	—
160	D₁₇₂	—	2	2	no	—
161	D₁₇₂	—	4	1	no	—
162	D₁₇₂	—	4	1	no	—
163	D₁₇₂	—	4	1	no	—
164	C₄×D₄₃	—	4	2	no	—
165	C₄×D₄₃	—	4	2	no	—
166	C₄×D₄₃	—	8	2	no	—
167	Dic₈₆	—	2	2	no	—
168	Dic₈₆	—	4	1	no	—
169	Dic₈₆	—	4	1	no	—
170	Dic₈₆	—	4	1	no	—
171	C₈×C₄₃	—	2	1	no	—
172	C₈×C₄₃	—	2	1	no	—
173	C₈×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
174	C₈×C₄₃	C₂×#41	1	2	yes	—
175	C₄₃⋊C₈	—	4	2	no	—
176	#33690	—	43	1	no	maximal, Sylow
177	C₂²×C₄₃	—	2	1	no	—
178	C₂²×C₄₃	—	4	1	no	—
179	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₂×D₄₃	—	4	2	no	—
181	C₂×D₄₃	—	8	1	no	—
182	C₂×D₄₃	—	8	1	no	—
183	C₂×D₄₃	—	8	1	no	—
184	C₂×D₄₃	—	16	1	no	—
185	C₄×C₄₃	—	2	1	no	—
186	C₄×C₄₃	—	4	1	no	—
187	C₄×C₄₃	—	4	1	no	—
188	C₄×C₄₃	#525	1	1	yes	characteristic
189	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	Dic₄₃	—	4	2	no	—
191	Dic₄₃	—	8	1	no	—
192	Dic₄₃	—	8	1	no	—
193	Dic₄₃	—	8	1	no	—
194	#3183	—	43	1	no	—
195	#2956	—	43	1	no	—
196	#2239	—	43	1	no	—
197	#2238	—	43	1	no	—
198	#2231	—	43	2	no	—
199	#2212	—	43	2	no	—
200	#2191	—	43	1	no	—
201	#2190	—	43	1	no	—
202	AGL(1,ℤ/16)	—	43	2	no	—
203	#2180	—	43	2	no	—
204	#2165	D₄₃	1	1	yes	characteristic, complemented
205	C₂×C₄₃	—	2	1	no	—
206	C₂×C₄₃	—	4	1	no	—
207	C₂×C₄₃	#3109	1	1	yes	characteristic
208	D₄₃	—	8	2	no	—
209	D₄₃	—	16	1	no	—
210	#575	—	43	1	no	—
211	#568	—	86	1	no	—
212	#529	—	86	1	no	—
213	#510	—	43	2	no	—
214	#509	—	43	2	no	—
215	#499	—	43	1	no	—
216	#495	—	43	1	no	—
217	#488	—	86	1	no	—
218	#470	—	43	1	no	—
219	#469	—	43	1	no	—
220	#468	—	86	1	no	—
221	#444	—	43	2	no	—
222	#444	—	86	1	no	—
223	#443	—	43	2	no	—
224	#443	—	86	1	no	—
225	#442	—	43	1	no	—
226	#441	—	43	1	no	—
227	#440	—	86	1	no	—
228	#430	—	43	1	no	—
229	C₂×#104	—	86	1	no	—
230	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
231	#365	—	43	2	no	—
232	#365	—	43	2	no	—
233	#364	C₂×D₄₃	1	2	yes	complemented
234	#362	—	43	2	no	—
235	#361	—	43	2	no	—
236	#360	—	43	2	no	—
237	#360	—	43	2	no	—
238	#349	C₂×D₄₃	1	1	yes	characteristic
239	#349	C₂×D₄₃	1	1	yes	characteristic
240	#347	C₂×D₄₃	1	2	yes	complemented
241	C₄₃	#33690	1	1	yes	characteristic, complemented, Sylow
242	C₂×#41	—	86	1	no	—
243	#137	—	86	2	no	—
244	AGL(1,ℤ/8)	—	86	2	no	—
245	#135	—	43	2	no	—
246	#135	—	86	1	no	—
247	C₂×Q₁₆	—	43	1	no	—
248	C₂×SD₁₆	—	86	1	no	—
249	C₂×D₈	—	43	1	no	—
250	#131	—	86	2	no	—
251	#131	—	86	2	no	—
252	#131	—	172	2	no	—
253	C₂×M₁₆	—	86	1	no	—
254	C₂×M₁₆	—	172	2	no	—
255	C₂×M₁₆	D₁₇₂	1	1	yes	characteristic
256	C₂×M₁₆	C₂²×D₄₃	1	1	yes	characteristic
257	#126	—	172	1	no	—
258	#125	—	86	1	no	—
259	#124	—	86	1	no	—
260	#123	—	172	1	no	—
261	C₂²⋊Q₈	—	86	1	no	—
262	#121	—	86	1	no	—
263	C₄×Q₈	—	86	1	no	—
264	C₄×D₄	—	86	1	no	—
265	C₄×D₄	—	172	1	no	—
266	#117	—	172	1	no	—
267	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
268	Q₃₂	—	86	2	no	—
269	SD₃₂	—	86	2	no	—
270	SD₃₂	—	86	2	no	—
271	D₁₆	—	86	2	no	—
272	M₃₂	C₂²×D₄₃	1	2	yes	—
273	M₃₂	C₂²×D₄₃	1	2	yes	—
274	#108	—	43	2	no	—
275	#108	—	86	1	no	—
276	#107	—	86	1	no	—
277	#106	—	43	1	no	—
278	#106	—	43	1	no	—
279	#104	—	86	2	no	—
280	#104	—	86	2	no	—
281	#103	—	86	1	no	—
282	#103	—	86	1	no	—
283	#102	—	86	1	no	—
284	#102	—	86	1	no	—
285	#97	D₁₇₂	1	1	yes	characteristic
286	#97	C₂²×D₄₃	1	1	yes	characteristic
287	C₄×C₈	D₁₇₂	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
289	#41	—	86	2	no	—
290	#41	—	86	2	no	—
291	#41	—	172	1	no	—
292	#41	—	172	1	no	—
293	C₂×Q₈	—	86	1	no	—
294	C₂×Q₈	—	86	1	no	—
295	C₂×D₄	—	86	1	no	—
296	C₂×D₄	—	86	1	no	—
297	C₂×D₄	—	172	1	no	—
298	C₂×D₄	—	172	1	no	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	172	1	no	—
301	C₂²×C₄	—	172	1	no	—
302	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
303	Q₁₆	—	43	2	no	—
304	Q₁₆	—	86	1	no	—
305	Q₁₆	—	86	2	no	—
306	SD₁₆	—	86	2	no	—
307	SD₁₆	—	86	2	no	—
308	SD₁₆	—	172	2	no	—
309	D₈	—	43	2	no	—
310	D₈	—	86	1	no	—
311	D₈	—	86	2	no	—
312	M₁₆	—	2	2	no	—
313	M₁₆	—	2	2	no	—
314	M₁₆	—	86	2	no	—
315	M₁₆	—	172	2	no	—
316	M₁₆	—	172	1	no	—
317	M₁₆	—	172	2	no	—
318	C₂×C₈	—	172	2	no	—
319	C₂×C₈	—	172	1	no	—
320	C₂×C₈	—	172	2	no	—
321	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
322	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
323	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
324	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
325	C₄⋊C₄	—	86	1	no	—
326	C₄⋊C₄	—	86	1	no	—
327	C₄⋊C₄	—	172	1	no	—
328	C₄⋊C₄	—	172	1	no	—
329	C₄⋊C₄	—	172	1	no	—
330	C₄⋊C₄	—	172	1	no	—
331	C₂²⋊C₄	—	172	1	no	—
332	C₂²⋊C₄	—	172	1	no	—
333	C₂²⋊C₄	—	172	1	no	—
334	C₂²⋊C₄	—	172	1	no	—
335	C₂²⋊C₄	—	172	1	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	172	1	no	—
338	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
339	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
340	C₁₆	—	2	2	no	—
341	C₁₆	—	2	2	no	—
342	C₂³	—	172	1	no	—
343	C₂³	—	172	1	no	—
344	C₂³	#13063353	1	1	yes	characteristic
345	Q₈	—	86	2	no	—
346	Q₈	—	172	1	no	—
347	Q₈	—	172	1	no	—
348	Q₈	—	172	1	no	—
349	D₄	—	86	2	no	—
350	D₄	—	344	2	no	—
351	D₄	—	172	1	no	—
352	D₄	—	172	2	no	—
353	D₄	—	172	2	no	—
354	D₄	—	172	1	no	—
355	D₄	—	172	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	8	1	no	—
361	C₂×C₄	—	344	2	no	—
362	C₂×C₄	—	172	2	no	—
363	C₂×C₄	—	172	1	no	—
364	C₂×C₄	—	172	1	no	—
365	C₂×C₄	—	172	1	no	—
366	C₂×C₄	—	172	1	no	—
367	C₂×C₄	—	172	2	no	—
368	C₂×C₄	#13063353	1	1	yes	characteristic
369	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	172	2	no	—
373	C₈	#13063353	1	2	yes	—
374	C₈	#41×D₄₃	1	2	yes	—
375	C₂²	—	2	1	no	—
376	C₂²	—	4	1	no	—
377	C₂²	—	344	1	no	—
378	C₂²	—	344	1	no	—
379	C₂²	—	344	1	no	—
380	C₂²	—	172	2	no	—
381	C₂²	—	688	1	no	—
382	C₂²	C₂×#13063353	1	1	yes	characteristic
383	C₄	—	2	1	no	—
384	C₄	—	4	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	344	1	no	—
387	C₄	—	344	1	no	—
388	C₄	—	344	1	no	—
389	C₄	—	172	2	no	—
390	C₄	#14583411	1	1	yes	characteristic
391	C₄	C₂×#13063353	1	1	yes	characteristic
392	C₂	—	2	1	no	—
393	C₂	—	4	1	no	—
394	C₂	—	344	2	no	—
395	C₂	—	688	1	no	—
396	C₂	#19144687	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19144687
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2165	C₂
socle	C₂×C₄₃	#3109
prime radical	#19136543	C₂
hypercenter	#2165	D₄₃
hypocenter	C₄₃	#33690
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #2165

Complements

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

N	H ≅ G/N	type
1. #19145324	C₂	semidirect
2. #19143506	C₂	semidirect
3. #19136815	C₂	semidirect
4. #19136810	C₂	semidirect
5. #19136772	C₂	semidirect
6. #19136665	C₂	semidirect
7. #19136548	C₂	semidirect
9. #19136495	C₂	semidirect
10. #19136488	C₂	semidirect
11. C₄₃×#2165	C₂	semidirect
17. #14583301	C₂²	semidirect
18. #14583283	C₂²	semidirect
27. #14583012	C₂²	semidirect
28. #14583007	C₂²	semidirect
32. #14582722	C₂²	semidirect
33. #14582722	C₂²	semidirect
34. #14582716	C₂²	semidirect
35. #14582713	C₂²	semidirect
39. C₄₃×#364	C₂²	semidirect
42. C₄₃×#347	C₂²	semidirect
204. #2165	D₄₃	semidirect
233. #364	C₂×D₄₃	semidirect
240. #347	C₂×D₄₃	semidirect
241. C₄₃	#33690	semidirect

Maximal subgroups

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

subgroup	index
#19145324	2
#19143506	2
#19136815	2
#19136810	2
#19136772	2
#19136665	2
#19136548	2
#19136543	2
#19136495	2
#19136488	2
C₄₃×#2165	2
#33690	43