Subgroups — #25081845

Order 11008.

642 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145743	C₂	1	1	yes	maximal, characteristic, complemented
2	#19144388	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583190	C₂	1	1	yes	maximal, characteristic
4	#19143618	C₂	1	1	yes	maximal, characteristic
5	#19134045	C₂	1	4	yes	maximal, complemented
6	#19133993	C₂	1	4	yes	maximal, complemented
7	#19133369	C₂	1	1	yes	maximal, characteristic, complemented
8	#19133275	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#1542	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13063325	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
12	#14583302	C₂²	1	2	yes	complemented
13	#14583249	C₂²	1	2	yes	complemented
14	#14583232	—	2	2	no	—
15	#14583196	—	2	2	no	—
16	#14583194	—	2	2	no	—
17	#14583190	—	2	2	no	—
18	#14583190	C₂²	1	4	yes	—
19	#14583186	C₂²	1	2	yes	—
20	#14583176	C₂²	1	2	yes	—
21	C₈×D₁₇₂	—	2	2	no	—
22	#14582964	C₂²	1	4	yes	—
23	#14582952	C₂²	1	1	yes	characteristic
24	#14582942	—	2	1	no	—
25	C₂×#13063257	—	2	1	no	—
26	#14582923	C₂²	1	1	yes	characteristic
27	#14582903	—	2	1	no	—
28	C₂×#13063247	—	2	1	no	—
29	C₂²⋊C₈×D₄₃	—	2	2	no	—
30	#14582651	C₂²	1	4	yes	complemented
31	#14582647	C₂²	1	4	yes	complemented
32	C₄₃×#533	C₂²	1	1	yes	characteristic
33	C₄₃×#431	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
35	C₄₃×#333	C₂²	1	2	yes	complemented
36	C₄₃×#332	C₂²	1	2	yes	complemented
37	C₂²×Dic₈₆	—	2	1	no	—
38	C₂²×D₁₇₂	D₄	1	1	yes	characteristic
39	C₂²×C₄×D₄₃	—	2	1	no	—
40	C₂×#10621877	—	2	1	no	—
41	C₂×C₈×D₄₃	—	2	4	no	—
42	#13063325	—	2	1	no	—
43	#13063325	—	2	1	no	—
44	#13063325	—	2	1	no	—
45	#13063325	C₂³	1	2	yes	—
46	C₄×D₁₇₂	—	2	1	no	—
47	C₄×D₁₇₂	—	2	1	no	—
48	C₄×D₁₇₂	D₄	1	2	yes	—
49	C₄×D₁₇₂	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₄₃	—	2	1	no	—
51	C₂×C₄⋊Dic₄₃	—	2	1	no	—
52	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
53	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
54	#13063258	—	2	2	no	—
55	#13063258	—	2	4	no	—
56	#13063257	—	2	4	no	—
57	#13063256	—	2	2	no	—
58	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
59	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
60	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
61	#13063249	—	2	4	no	—
62	#13063247	—	2	4	no	—
63	#13063245	—	2	2	no	—
64	#13063245	—	2	2	no	—
65	C₂²⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
66	(C₂×D₄₃)⋊C₈	—	2	4	no	—
67	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
68	C₂²×C₈×C₄₃	—	2	1	no	—
69	C₄⋊Q₈×C₄₃	C₂³	1	1	yes	characteristic
70	C₄⋊Q₈×C₄₃	C₂³	1	1	yes	characteristic
71	#125×C₄₃	—	2	1	no	—
72	C₂²⋊Q₈×C₄₃	—	2	1	no	—
73	C₂²⋊Q₈×C₄₃	—	2	1	no	—
74	C₄×Q₈×C₄₃	—	2	1	no	—
75	C₄×Q₈×C₄₃	—	2	1	no	—
76	#117×C₄₃	—	2	1	no	—
77	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₄₃	—	2	1	no	—
79	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
80	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
81	C₂²⋊C₈×C₄₃	—	2	1	no	—
82	C₄×C₈×C₄₃	—	2	1	no	—
83	C₂³×D₄₃	—	2	1	no	—
84	C₂×Q₈×C₄₃	—	2	1	no	—
85	C₂×Q₈×C₄₃	—	2	1	no	—
86	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
89	C₂²×Dic₄₃	—	2	1	no	—
90	C₂²×Dic₄₃	—	4	1	no	—
91	C₂×D₁₇₂	—	2	1	no	—
92	C₂×D₁₇₂	—	2	2	no	—
93	C₂×D₁₇₂	Q₁₆	1	4	yes	—
94	C₂×D₁₇₂	C₂×D₄	1	2	yes	—
95	C₂×C₄×D₄₃	—	2	2	no	—
96	C₂×C₄×D₄₃	—	2	2	no	—
97	C₂×C₄×D₄₃	—	8	2	no	—
98	C₂×Dic₈₆	—	2	2	no	—
99	C₂×Dic₈₆	—	4	1	no	—
100	C₂×Dic₈₆	—	8	2	no	—
101	C₂×C₈×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	—	2	2	no	—
103	C₂×C₈×C₄₃	—	4	2	no	—
104	C₄⋊C₄×C₄₃	—	2	1	no	—
105	C₄⋊C₄×C₄₃	—	2	1	no	—
106	C₄⋊C₄×C₄₃	—	2	1	no	—
107	C₄⋊C₄×C₄₃	—	2	1	no	—
108	C₄⋊C₄×C₄₃	—	4	1	no	—
109	C₄⋊C₄×C₄₃	—	4	1	no	—
110	C₂²⋊C₄×C₄₃	—	4	1	no	—
111	C₄²×C₄₃	—	4	1	no	—
112	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₄₃	#41	1	1	yes	characteristic
115	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
116	(C₂×D₄₃)⋊C₄	—	2	2	no	—
117	(C₂×D₄₃)⋊C₄	—	2	2	no	—
118	C₄⋊Dic₄₃	—	2	1	no	—
119	C₄⋊Dic₄₃	—	2	2	no	—
120	C₄⋊Dic₄₃	—	2	2	no	—
121	C₄⋊Dic₄₃	—	4	1	no	—
122	C₄⋊Dic₄₃	—	4	1	no	—
123	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
124	#10621877	—	4	2	no	—
125	C₂×C₄₃⋊C₈	—	2	2	no	—
126	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
127	C₈×D₄₃	—	8	4	no	—
128	C₂³×C₄₃	#121	1	1	yes	characteristic
129	C₂²×D₄₃	—	2	2	no	—
130	C₂²×D₄₃	—	2	4	no	—
131	C₂²×D₄₃	—	8	1	no	—
132	C₂²×D₄₃	—	8	1	no	—
133	Q₈×C₄₃	—	4	2	no	—
134	Q₈×C₄₃	—	8	1	no	—
135	C₂×C₄×C₄₃	—	2	2	no	—
136	C₂×C₄×C₄₃	—	4	1	no	—
137	C₂×C₄×C₄₃	—	4	1	no	—
138	C₂×C₄×C₄₃	—	4	1	no	—
139	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
143	C₂×C₄×C₄₃	#135	1	2	yes	—
144	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
147	C₂×Dic₄₃	—	2	2	no	—
148	C₂×Dic₄₃	—	4	2	no	—
149	C₂×Dic₄₃	—	8	1	no	—
150	C₂×Dic₄₃	—	8	2	no	—
151	D₁₇₂	—	2	2	no	—
152	D₁₇₂	—	2	4	no	—
153	D₁₇₂	C₂×Q₁₆	1	4	yes	—
154	C₄×D₄₃	—	4	4	no	—
155	C₄×D₄₃	—	8	2	no	—
156	Dic₈₆	—	8	2	no	—
157	Dic₈₆	—	8	2	no	—
158	C₈×C₄₃	—	4	2	no	—
159	C₈×C₄₃	—	4	2	no	—
160	C₄₃⋊C₈	—	2	4	no	—
161	C₄₃⋊C₈	—	4	2	no	—
162	#19393	—	43	1	no	maximal, Sylow
163	C₂²×C₄₃	—	2	1	no	—
164	C₂²×C₄₃	—	2	1	no	—
165	C₂²×C₄₃	#467	1	2	yes	—
166	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
167	C₂×D₄₃	—	2	4	no	—
168	C₂×D₄₃	—	4	2	no	—
169	C₂×D₄₃	—	4	2	no	—
170	C₂×D₄₃	—	8	4	no	—
171	C₄×C₄₃	—	2	1	no	—
172	C₄×C₄₃	—	2	1	no	—
173	C₄×C₄₃	—	2	1	no	—
174	C₄×C₄₃	—	8	1	no	—
175	C₄×C₄₃	—	8	1	no	—
176	C₄×C₄₃	#467	1	2	yes	—
177	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
178	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
179	C₄×C₄₃	#577	1	2	yes	—
180	Dic₄₃	—	4	2	no	—
181	Dic₄₃	—	8	2	no	—
182	#3230	—	43	1	no	—
183	#3077	—	43	1	no	—
184	C₂×#476	—	43	1	no	—
185	#2972	—	43	1	no	—
186	#1683	—	43	4	no	—
187	#1675	—	43	4	no	—
188	#1571	—	43	1	no	—
189	#1548	—	43	1	no	—
190	#1542	D₄₃	1	1	yes	characteristic, complemented
191	C₂×C₄₃	—	2	2	no	—
192	C₂×C₄₃	#3042	1	1	yes	characteristic
193	C₂×C₄₃	C₂×#467	1	1	yes	characteristic
194	C₂×C₄₃	#3068	1	1	yes	characteristic
195	D₄₃	—	4	4	no	—
196	#533	C₂×D₄₃	1	1	yes	characteristic
197	C₂×C₄⋊Q₈	—	43	1	no	—
198	C₂×C₄×D₄	—	43	1	no	—
199	#500	—	43	2	no	—
200	#487	—	43	2	no	—
201	#484	—	86	2	no	—
202	#478	—	86	2	no	—
203	#477	—	86	2	no	—
204	#476	—	43	2	no	—
205	#476	—	43	4	no	—
206	#476	—	86	2	no	—
207	#474	—	43	2	no	—
208	C₈×D₄	—	43	4	no	—
209	C₈×D₄	—	86	2	no	—
210	#431	—	43	1	no	—
211	#431	C₂×D₄₃	1	1	yes	characteristic
212	#428	—	86	1	no	—
213	C₂×#106	—	86	1	no	—
214	#423	—	43	1	no	—
215	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
216	#418	—	86	1	no	—
217	C₂×#102	—	86	1	no	—
218	C₂×C₂²⋊C₈	—	86	2	no	—
219	#335	—	43	4	no	—
220	#333	—	43	4	no	—
221	#333	C₂×D₄₃	1	2	yes	complemented
222	#332	C₂×D₄₃	1	2	yes	complemented
223	C₄₃	#19393	1	1	yes	characteristic, complemented, Sylow
224	C₂²×Q₈	—	86	1	no	—
225	C₂²×D₄	—	43	1	no	—
226	C₂³×C₄	—	86	1	no	—
227	C₂×M₁₆	—	86	1	no	—
228	C₂²×C₈	—	2	1	no	—
229	C₂²×C₈	—	86	4	no	—
230	C₄⋊Q₈	—	43	2	no	—
231	C₄⋊Q₈	—	86	1	no	—
232	C₄⋊Q₈	—	86	1	no	—
233	C₄⋊Q₈	—	86	1	no	—
234	C₄⋊Q₈	C₂²×D₄₃	1	1	yes	characteristic
235	C₄⋊Q₈	C₂²×D₄₃	1	1	yes	characteristic
236	#125	—	2	1	no	—
237	C₂²⋊Q₈	—	2	1	no	—
238	C₂²⋊Q₈	—	2	1	no	—
239	C₄×Q₈	—	2	1	no	—
240	C₄×Q₈	—	2	1	no	—
241	C₄×D₄	—	43	2	no	—
242	C₄×D₄	—	43	2	no	—
243	C₄×D₄	—	86	1	no	—
244	C₄×D₄	—	86	1	no	—
245	#117	—	2	1	no	—
246	C₂×C₄⋊C₄	—	43	1	no	—
247	C₂×C₄⋊C₄	—	86	1	no	—
248	C₂×C₄⋊C₄	—	86	1	no	—
249	C₂×C₂²⋊C₄	—	86	1	no	—
250	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
251	#107	—	86	2	no	—
252	#107	—	86	4	no	—
253	#106	—	86	2	no	—
254	#106	—	86	4	no	—
255	C₄⋊C₈	—	2	1	no	—
256	C₄⋊C₈	—	43	2	no	—
257	C₄⋊C₈	—	43	2	no	—
258	C₄⋊C₈	—	86	1	no	—
259	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
260	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
261	#103	—	86	4	no	—
262	#102	—	86	2	no	—
263	#102	—	86	2	no	—
264	#102	—	86	4	no	—
265	C₂²⋊C₈	—	2	1	no	—
266	C₂²⋊C₈	—	43	2	no	—
267	C₂²⋊C₈	—	86	4	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	—	43	2	no	—
270	C₂⁴	—	86	1	no	—
271	C₂×Q₈	—	2	1	no	—
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	86	2	no	—
274	C₂×Q₈	—	344	2	no	—
275	C₂×Q₈	—	172	1	no	—
276	C₂×D₄	—	43	4	no	—
277	C₂×D₄	—	43	2	no	—
278	C₂×D₄	—	86	1	no	—
279	C₂×D₄	—	86	2	no	—
280	C₂²×C₄	—	86	1	no	—
281	C₂²×C₄	—	86	2	no	—
282	C₂²×C₄	—	86	2	no	—
283	C₂²×C₄	—	344	2	no	—
284	C₂²×C₄	—	172	1	no	—
285	C₂²×C₄	#10621898	1	1	yes	characteristic
286	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
287	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
288	M₁₆	—	172	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	4	2	no	—
292	C₂×C₈	—	43	4	no	—
293	C₂×C₈	—	86	2	no	—
294	C₂×C₈	—	344	4	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	43	2	no	—
302	C₄⋊C₄	—	86	1	no	—
303	C₄⋊C₄	—	86	2	no	—
304	C₄⋊C₄	—	86	2	no	—
305	C₄⋊C₄	—	172	1	no	—
306	C₄⋊C₄	—	172	1	no	—
307	C₂²⋊C₄	—	4	1	no	—
308	C₂²⋊C₄	—	86	2	no	—
309	C₂²⋊C₄	—	86	2	no	—
310	C₄²	—	4	1	no	—
311	C₄²	#10621898	1	1	yes	characteristic
312	C₄²	D₄×D₄₃	1	1	yes	characteristic
313	C₄²	D₄×D₄₃	1	1	yes	characteristic
314	C₄²	C₂³×D₄₃	1	1	yes	characteristic
315	C₂³	—	86	4	no	—
316	C₂³	—	86	2	no	—
317	C₂³	—	344	1	no	—
318	C₂³	—	344	1	no	—
319	C₂³	#13063298	1	1	yes	characteristic
320	Q₈	—	4	2	no	—
321	Q₈	—	8	1	no	—
322	Q₈	—	344	2	no	—
323	Q₈	—	344	2	no	—
324	D₄	—	43	4	no	—
325	D₄	—	86	4	no	—
326	D₄	—	86	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	86	2	no	—
332	C₂×C₄	—	344	2	no	—
333	C₂×C₄	—	344	2	no	—
334	C₂×C₄	—	344	1	no	—
335	C₂×C₄	—	172	2	no	—
336	C₂×C₄	—	172	4	no	—
337	C₂×C₄	#13063298	1	1	yes	characteristic
338	C₂×C₄	#13063298	1	1	yes	characteristic
339	C₂×C₄	#13063298	1	1	yes	characteristic
340	C₂×C₄	Q₁₆×D₄₃	1	2	yes	—
341	C₂×C₄	#13063346	1	2	yes	—
342	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
344	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
345	C₈	—	4	2	no	—
346	C₈	—	4	2	no	—
347	C₈	—	86	4	no	—
348	C₈	—	172	2	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	86	4	no	—
352	C₂²	—	344	4	no	—
353	C₂²	—	172	2	no	—
354	C₂²	—	172	2	no	—
355	C₂²	#14583139	1	2	yes	—
356	C₂²	C₂×#13063298	1	1	yes	characteristic
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	8	1	no	—
361	C₄	—	8	1	no	—
362	C₄	—	344	2	no	—
363	C₄	—	172	2	no	—
364	C₄	#14583139	1	2	yes	—
365	C₄	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
366	C₄	C₂×#13063346	1	1	yes	characteristic
367	C₄	#14583706	1	2	yes	—
368	C₂	—	2	2	no	—
369	C₂	—	172	4	no	—
370	C₂	#19144110	1	1	yes	characteristic
371	C₂	C₂×#14583139	1	1	yes	characteristic
372	C₂	#19144326	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063298
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1542	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	C₂²×D₁₇₂	D₄
hypercenter	#1542	D₄₃
hypocenter	C₄₃	#19393
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1542

Complements

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

N	H ≅ G/N	type
1. #19145743	C₂	semidirect
5. #19134045	C₂	semidirect
6. #19133993	C₂	semidirect
7. #19133369	C₂	semidirect
8. #19133275	C₂	semidirect
9. C₄₃×#1542	C₂	semidirect
12. #14583302	C₂²	semidirect
13. #14583249	C₂²	semidirect
30. #14582651	C₂²	semidirect
31. #14582647	C₂²	semidirect
35. C₄₃×#333	C₂²	semidirect
36. C₄₃×#332	C₂²	semidirect
190. #1542	D₄₃	semidirect
221. #333	C₂×D₄₃	semidirect
222. #332	C₂×D₄₃	semidirect
223. C₄₃	#19393	semidirect

Maximal subgroups

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

subgroup	index
#19145743	2
#19144388	2
C₂×#14583190	2
#19143618	2
#19134045	2
#19133993	2
#19133369	2
#19133275	2
C₄₃×#1542	2
#19393	43