Subgroups — #24993101

Order 11008.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145522	C₂	1	1	yes	maximal, characteristic
2	#19143338	C₂	1	1	yes	maximal, characteristic
3	#19135583	C₂	1	1	yes	maximal, characteristic, complemented
4	#19133273	C₂	1	1	yes	maximal, characteristic
5	C₂×#14582647	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132444	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063331	—	2	1	no	—
9	#14583449	C₂²	1	1	yes	characteristic
10	C₂×#13063325	C₂²	1	1	yes	characteristic
11	#14583272	—	2	1	no	—
12	#14583250	—	2	1	no	—
13	#14583105	—	2	2	no	—
14	#14583040	C₄	1	4	yes	—
15	#14582976	—	2	1	no	—
16	#14582969	—	2	2	no	—
17	#14582964	—	2	1	no	—
18	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
19	#14582865	—	2	1	no	—
20	C₄×#10621877	C₂²	1	1	yes	characteristic
21	#14582667	—	2	1	no	—
22	#14582667	—	2	1	no	—
23	#14582647	—	2	1	no	—
24	#14582647	—	2	1	no	—
25	#14582646	—	2	1	no	—
26	#14582642	—	2	1	no	—
27	C₄₃×#431	C₂²	1	1	yes	characteristic
28	C₄₃×#423	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×#10621898	—	2	1	no	—
31	C₂²×Dic₈₆	C₂×C₄	1	2	yes	—
32	C₂×#10621873	—	2	4	no	—
33	#13063331	—	4	2	no	—
34	#13063331	—	4	2	no	—
35	C₂×#10621877	D₄	1	1	yes	characteristic
36	C₂×#10621877	D₄	1	1	yes	characteristic
37	C₂×#10621872	—	2	2	no	—
38	C₂²×C₄₃⋊C₈	—	2	1	no	—
39	C₂×C₈×D₄₃	—	2	2	no	—
40	#13063325	—	2	1	no	—
41	#13063325	—	2	1	no	—
42	#13063325	—	2	1	no	—
43	#13063325	—	2	1	no	—
44	#13063325	D₄	1	1	yes	characteristic
45	#13063322	D₄	1	1	yes	characteristic
46	#13063313	—	2	1	no	—
47	#13063299	—	2	1	no	—
48	#13063299	C₂×C₄	1	2	yes	—
49	C₄×D₁₇₂	—	2	1	no	—
50	C₄×D₁₇₂	D₄	1	2	yes	—
51	C₂×C₄⋊Dic₄₃	—	2	1	no	—
52	C₂×C₄⋊Dic₄₃	—	2	1	no	—
53	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
54	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
55	#13063250	—	2	4	no	—
56	(C₂×D₄₃)⋊C₈	—	2	2	no	—
57	(C₂×D₄₃)⋊C₈	—	2	2	no	—
58	#13063236	—	2	1	no	—
59	C₄×C₄₃⋊C₈	—	2	1	no	—
60	C₂×M₁₆×C₄₃	—	2	1	no	—
61	C₂²×C₈×C₄₃	—	2	1	no	—
62	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₄₃	—	2	1	no	—
64	C₄⋊C₈×C₄₃	—	2	1	no	—
65	C₄⋊C₈×C₄₃	—	2	1	no	—
66	C₄⋊C₈×C₄₃	—	2	1	no	—
67	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
68	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
69	C₂²⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₄₃	—	2	1	no	—
71	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
74	C₂×C₂²⋊D₄₃	—	2	2	no	—
75	C₂²×Dic₄₃	—	2	2	no	—
76	#10621898	—	4	1	no	—
77	#10621898	—	4	1	no	—
78	#10621898	—	4	1	no	—
79	#10621898	—	4	1	no	—
80	C₂×D₁₇₂	—	2	1	no	—
81	C₂×D₁₇₂	—	2	1	no	—
82	C₂×D₁₇₂	C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×D₄₃	—	2	2	no	—
84	C₂×Dic₈₆	—	2	1	no	—
85	C₂×Dic₈₆	—	2	2	no	—
86	C₂×Dic₈₆	—	2	2	no	—
87	C₂×Dic₈₆	—	8	2	no	—
88	M₁₆×C₄₃	—	4	1	no	—
89	M₁₆×C₄₃	—	4	1	no	—
90	C₂×C₈×C₄₃	—	2	2	no	—
91	C₂×C₈×C₄₃	—	2	1	no	—
92	C₂×C₈×C₄₃	—	2	1	no	—
93	C₂×C₈×C₄₃	—	4	1	no	—
94	C₂×C₈×C₄₃	—	4	1	no	—
95	C₂×C₈×C₄₃	—	4	1	no	—
96	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₄₃	#41	1	1	yes	characteristic
100	(C₂×D₄₃)⋊C₄	—	2	2	no	—
101	(C₂×D₄₃)⋊C₄	—	2	2	no	—
102	C₄⋊Dic₄₃	—	2	1	no	—
103	C₄⋊Dic₄₃	—	4	1	no	—
104	C₄⋊Dic₄₃	—	4	1	no	—
105	C₄⋊Dic₄₃	—	4	1	no	—
106	C₄⋊Dic₄₃	—	4	1	no	—
107	C₄⋊Dic₄₃	C₂²⋊C₄	1	2	yes	—
108	#10621877	—	2	1	no	—
109	#10621877	—	2	1	no	—
110	#10621877	—	2	1	no	—
111	#10621877	SD₁₆	1	2	yes	complemented
112	C₂×C₄₃⋊C₈	—	2	1	no	—
113	C₂×C₄₃⋊C₈	—	2	1	no	—
114	C₂×C₄₃⋊C₈	—	4	1	no	—
115	C₂×C₄₃⋊C₈	—	8	1	no	—
116	#10621873	—	8	4	no	—
117	#10621872	—	4	4	no	—
118	C₈×D₄₃	—	4	4	no	—
119	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
120	C₂²×D₄₃	—	2	2	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	4	1	no	—
124	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
127	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
132	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
134	C₂²⋊D₄₃	—	4	2	no	—
135	C₂²⋊D₄₃	—	4	2	no	—
136	C₂×Dic₄₃	—	2	2	no	—
137	C₂×Dic₄₃	—	4	2	no	—
138	C₂×Dic₄₃	—	8	2	no	—
139	D₁₇₂	—	4	2	no	—
140	D₁₇₂	—	4	1	no	—
141	D₁₇₂	—	4	1	no	—
142	D₁₇₂	—	4	1	no	—
143	D₁₇₂	—	4	1	no	—
144	C₄×D₄₃	—	4	2	no	—
145	C₄×D₄₃	—	4	2	no	—
146	Dic₈₆	—	4	1	no	—
147	Dic₈₆	—	4	1	no	—
148	Dic₈₆	—	4	4	no	—
149	Dic₈₆	—	8	2	no	—
150	C₈×C₄₃	—	4	2	no	—
151	C₈×C₄₃	—	4	1	no	—
152	C₈×C₄₃	—	4	1	no	—
153	C₈×C₄₃	—	8	1	no	—
154	C₄₃⋊C₈	—	2	2	no	—
155	C₄₃⋊C₈	—	4	1	no	—
156	C₄₃⋊C₈	—	8	1	no	—
157	#9267	—	43	1	no	maximal, Sylow
158	C₂²×C₄₃	—	2	1	no	—
159	C₂²×C₄₃	—	2	1	no	—
160	C₂²×C₄₃	#386	1	1	yes	characteristic
161	C₂²×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
162	C₂²×C₄₃	#440	1	1	yes	characteristic
163	C₂×D₄₃	—	4	2	no	—
164	C₂×D₄₃	—	4	2	no	—
165	C₂×D₄₃	—	4	2	no	—
166	C₄×C₄₃	—	2	1	no	—
167	C₄×C₄₃	—	2	1	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	#450	1	1	yes	characteristic
170	C₄×C₄₃	#452	1	1	yes	characteristic
171	C₄×C₄₃	#461	1	1	yes	characteristic
172	C₄×C₄₃	#463	1	1	yes	characteristic
173	C₄×C₄₃	#471	1	1	yes	characteristic
174	C₄×C₄₃	#473	1	1	yes	characteristic
175	Dic₄₃	—	4	2	no	—
176	Dic₄₃	—	8	2	no	—
177	#3206	—	43	1	no	—
178	#2935	—	43	1	no	—
179	#1997	—	43	1	no	—
180	#1547	—	43	1	no	—
181	C₂×#333	—	43	1	no	—
182	#1301	D₄₃	1	1	yes	characteristic, complemented
183	#1283	—	43	1	no	—
184	C₂×C₄₃	—	2	1	no	—
185	C₂×C₄₃	—	2	1	no	—
186	C₂×C₄₃	#1877	1	1	yes	characteristic
187	C₂×C₄₃	#1908	1	1	yes	characteristic
188	C₂×C₄₃	#1914	1	1	yes	characteristic
189	D₄₃	—	8	2	no	—
190	C₂×#131	—	86	1	no	—
191	#532	—	43	1	no	—
192	C₂×C₄⋊Q₈	—	43	1	no	—
193	#492	—	86	1	no	—
194	#487	—	86	1	no	—
195	#461	—	86	2	no	—
196	#448	—	43	4	no	—
197	#436	—	86	1	no	—
198	#435	—	86	2	no	—
199	C₈×D₄	—	86	1	no	—
200	#431	C₂×D₄₃	1	1	yes	characteristic
201	#423	C₂×D₄₃	1	1	yes	characteristic
202	C₂×C₄⋊C₈	—	43	1	no	—
203	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
204	#408	—	86	1	no	—
205	C₄×M₁₆	—	43	1	no	—
206	#343	—	86	1	no	—
207	#343	—	86	1	no	—
208	#333	—	86	1	no	—
209	#333	—	86	1	no	—
210	#332	—	86	1	no	—
211	#331	—	86	1	no	—
212	C₄₃	#9267	1	1	yes	characteristic, complemented, Sylow
213	C₂×#41	—	86	1	no	—
214	C₂²×Q₈	—	43	2	no	—
215	C₂×SD₁₆	—	86	4	no	—
216	#131	—	172	2	no	—
217	#131	—	172	2	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	—	43	1	no	—
220	C₂×M₁₆	—	43	1	no	—
221	C₂×M₁₆	—	86	2	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	—	86	2	no	—
224	C₂²×C₈	—	86	1	no	—
225	C₄⋊Q₈	—	43	1	no	—
226	C₄⋊Q₈	—	86	1	no	—
227	C₄⋊Q₈	—	86	1	no	—
228	C₄⋊Q₈	—	86	1	no	—
229	C₄⋊Q₈	—	86	1	no	—
230	#127	—	43	1	no	—
231	#124	—	86	1	no	—
232	#121	—	43	2	no	—
233	#121	—	86	1	no	—
234	C₄×D₄	—	43	2	no	—
235	C₄×D₄	—	86	1	no	—
236	C₂×C₄⋊C₄	—	86	1	no	—
237	C₂×C₄⋊C₄	—	86	1	no	—
238	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	86	1	no	—
244	C₄⋊C₈	—	86	1	no	—
245	C₄⋊C₈	D₁₇₂	1	1	yes	characteristic
246	C₄⋊C₈	D₁₇₂	1	1	yes	characteristic
247	#103	—	86	4	no	—
248	C₂²⋊C₈	—	86	2	no	—
249	C₂²⋊C₈	—	86	2	no	—
250	C₂²⋊C₈	C₄×D₄₃	1	2	yes	—
251	#97	—	86	1	no	—
252	C₄×C₈	—	2	1	no	—
253	C₄×C₈	—	86	1	no	—
254	#41	—	172	1	no	—
255	#41	—	172	1	no	—
256	#41	—	172	1	no	—
257	#41	—	172	1	no	—
258	C₂×Q₈	—	86	1	no	—
259	C₂×Q₈	—	86	2	no	—
260	C₂×Q₈	—	86	2	no	—
261	C₂×Q₈	—	344	2	no	—
262	C₂×D₄	—	43	2	no	—
263	C₂×D₄	—	86	2	no	—
264	C₂×D₄	—	86	1	no	—
265	C₂×D₄	—	86	1	no	—
266	C₂²×C₄	—	86	2	no	—
267	C₂²×C₄	—	86	2	no	—
268	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
269	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
270	C₂²×C₄	#10621902	1	1	yes	characteristic
271	SD₁₆	—	344	4	no	—
272	M₁₆	—	4	1	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	43	2	no	—
275	M₁₆	—	86	1	no	—
276	M₁₆	—	86	1	no	—
277	M₁₆	—	86	1	no	—
278	M₁₆	—	172	4	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	86	1	no	—
286	C₂×C₈	—	86	1	no	—
287	C₂×C₈	—	344	1	no	—
288	C₂×C₈	—	172	4	no	—
289	C₂×C₈	—	172	1	no	—
290	C₄⋊C₄	—	43	2	no	—
291	C₄⋊C₄	—	86	1	no	—
292	C₄⋊C₄	—	172	1	no	—
293	C₄⋊C₄	—	172	1	no	—
294	C₄⋊C₄	—	172	1	no	—
295	C₄⋊C₄	—	172	1	no	—
296	C₂²⋊C₄	—	86	2	no	—
297	C₂²⋊C₄	—	86	2	no	—
298	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
299	C₄²	#10621898	1	1	yes	characteristic
300	C₄²	D₄×D₄₃	1	1	yes	characteristic
301	C₄²	D₄×D₄₃	1	1	yes	characteristic
302	C₂³	—	86	2	no	—
303	C₂³	#13063287	1	1	yes	characteristic
304	Q₈	—	344	2	no	—
305	Q₈	—	172	4	no	—
306	Q₈	—	172	1	no	—
307	Q₈	—	172	1	no	—
308	D₄	—	172	2	no	—
309	D₄	—	172	2	no	—
310	D₄	—	172	2	no	—
311	D₄	—	172	1	no	—
312	D₄	—	172	1	no	—
313	D₄	—	172	1	no	—
314	D₄	—	172	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	86	2	no	—
319	C₂×C₄	—	344	2	no	—
320	C₂×C₄	—	172	2	no	—
321	C₂×C₄	—	172	2	no	—
322	C₂×C₄	—	172	2	no	—
323	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
324	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
325	C₂×C₄	#13063295	1	1	yes	characteristic
326	C₂×C₄	#13063298	1	1	yes	characteristic
327	C₂×C₄	#13063300	1	1	yes	characteristic
328	C₂×C₄	#13063310	1	1	yes	characteristic
329	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
330	C₂×C₄	#13063346	1	1	yes	characteristic
331	C₂×C₄	#13063349	1	1	yes	characteristic
332	C₂×C₄	#13063357	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	8	1	no	—
337	C₈	—	86	2	no	—
338	C₈	—	344	1	no	—
339	C₈	—	172	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	172	2	no	—
343	C₂²	—	172	2	no	—
344	C₂²	—	172	2	no	—
345	C₂²	#14582777	1	1	yes	characteristic
346	C₂²	#14582986	1	1	yes	characteristic
347	C₂²	#14582998	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	344	2	no	—
352	C₄	—	172	2	no	—
353	C₄	#14583051	1	1	yes	characteristic
354	C₄	#14583064	1	1	yes	characteristic
355	C₄	#14583103	1	1	yes	characteristic
356	C₄	#14583113	1	1	yes	characteristic
357	C₄	#14583163	1	1	yes	characteristic
358	C₄	#14583175	1	1	yes	characteristic
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	344	2	no	—
362	C₂	#19134930	1	1	yes	characteristic
363	C₂	#19135109	1	1	yes	characteristic
364	C₂	#19135141	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14582777
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1301	C₂
socle	C₂²×C₄₃	#386
prime radical	#14583449	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1301	D₄₃
hypocenter	C₄₃	#9267
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
3. #19135583	C₂	semidirect
5. C₂×#14582647	C₂	semidirect
6. #19132444	C₂	semidirect
7. C₄₃×#1301	C₂	semidirect
111. #10621877	SD₁₆	semidirect
182. #1301	D₄₃	semidirect
212. C₄₃	#9267	semidirect

Maximal subgroups

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

subgroup	index
#19145522	2
#19143338	2
#19135583	2
#19133273	2
C₂×#14582647	2
#19132444	2
C₄₃×#1301	2
#9267	43