Subgroups — #25013147

Order 11008.

420 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108×D₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582721	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136205	C₂	1	1	yes	maximal, characteristic
4	#19134617	C₂	1	1	yes	maximal, characteristic
5	#19132695	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132694	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₄₃	—	2	1	no	—
9	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
10	C₂×#13063265	C₂²	1	1	yes	characteristic
11	C₂×#13063261	C₂²	1	1	yes	characteristic
12	C₂×#13063260	—	2	1	no	—
13	#108×D₄₃	—	2	2	no	—
14	#108×D₄₃	—	2	2	no	—
15	C₂×#13063257	C₂²	1	1	yes	characteristic
16	#14582756	—	2	1	no	—
17	#14582721	—	2	2	no	—
18	(C₂×D₄₃)⋊C₁₆	C₄	1	2	yes	complemented
19	(C₂×D₄₃)⋊C₁₆	C₄	1	2	yes	complemented
20	#14582661	—	2	1	no	—
21	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#365	—	2	2	no	—
25	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
26	C₂×#10621877	—	2	1	no	—
27	C₂×#10621872	—	4	1	no	—
28	M₁₆×D₄₃	—	2	4	no	—
29	M₁₆×D₄₃	—	4	2	no	—
30	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
31	C₂×C₈×D₄₃	—	2	1	no	—
32	C₂×C₈×D₄₃	—	2	1	no	—
33	C₂×C₈×D₄₃	—	2	1	no	—
34	C₂×C₈×D₄₃	—	2	1	no	—
35	C₂×C₈×D₄₃	—	4	1	no	—
36	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₄₃	Q₈	1	1	yes	characteristic
39	C₂×C₈×D₄₃	Q₈	1	1	yes	characteristic
40	C₂×C₄⋊Dic₄₃	—	2	1	no	—
41	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
42	#13063265	—	2	1	no	—
43	#13063265	—	2	1	no	—
44	C₂×C₄₃⋊C₁₆	C₂×C₄	1	2	yes	—
45	#13063261	—	2	1	no	—
46	#13063261	D₄	1	2	yes	—
47	#13063260	—	2	4	no	—
48	#13063257	—	2	1	no	—
49	#13063257	—	2	1	no	—
50	C₄₃⋊#95	—	4	1	no	—
51	C₂×M₁₆×C₄₃	—	2	1	no	—
52	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
54	M₃₂×C₄₃	—	2	1	no	—
55	M₃₂×C₄₃	—	2	1	no	—
56	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
57	#108×C₄₃	—	2	1	no	—
58	#108×C₄₃	D₄	1	2	yes	—
59	#106×C₄₃	—	2	1	no	—
60	#106×C₄₃	—	2	1	no	—
61	C₂³×D₄₃	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	—	4	1	no	—
63	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×Dic₄₃	—	4	1	no	—
65	C₂²×Dic₄₃	SD₁₆	1	1	yes	characteristic
66	C₂×C₄×D₄₃	—	2	1	no	—
67	C₂×C₄×D₄₃	—	2	1	no	—
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×C₄×D₄₃	—	2	1	no	—
70	C₂×C₄×D₄₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×D₄₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×D₄₃	SD₁₆	1	1	yes	characteristic
73	C₂×C₄×D₄₃	SD₁₆	1	1	yes	characteristic
74	M₁₆×C₄₃	—	2	2	no	—
75	M₁₆×C₄₃	—	4	1	no	—
76	C₂×C₈×C₄₃	—	4	1	no	—
77	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
82	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
83	C₄⋊C₄×C₄₃	—	4	1	no	—
84	C₄⋊C₄×C₄₃	—	4	1	no	—
85	(C₂×D₄₃)⋊C₄	—	8	2	no	—
86	C₄⋊Dic₄₃	—	4	1	no	—
87	C₄⋊Dic₄₃	—	4	1	no	—
88	#10621877	—	2	2	no	—
89	#10621877	—	4	1	no	—
90	C₂×C₄₃⋊C₈	—	2	1	no	—
91	C₂×C₄₃⋊C₈	—	2	1	no	—
92	C₂×C₄₃⋊C₈	—	4	1	no	—
93	C₂×C₄₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
95	#10621872	—	4	4	no	—
96	C₈×D₄₃	—	2	1	no	—
97	C₈×D₄₃	—	2	1	no	—
98	C₈×D₄₃	—	2	1	no	—
99	C₈×D₄₃	—	2	1	no	—
100	C₈×D₄₃	—	2	1	no	—
101	C₈×D₄₃	—	2	1	no	—
102	C₈×D₄₃	—	2	1	no	—
103	C₈×D₄₃	—	2	1	no	—
104	C₈×D₄₃	—	4	4	no	—
105	C₁₆×C₄₃	—	2	2	no	—
106	C₄₃⋊C₁₆	—	2	2	no	—
107	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
108	C₂²×D₄₃	—	2	1	no	—
109	C₂²×D₄₃	—	2	1	no	—
110	C₂²×D₄₃	—	2	2	no	—
111	C₂²×D₄₃	—	4	1	no	—
112	C₂²×D₄₃	#106	1	1	yes	characteristic
113	C₂²×D₄₃	#106	1	1	yes	characteristic
114	C₂×C₄×C₄₃	—	8	1	no	—
115	C₂×C₄×C₄₃	—	8	1	no	—
116	C₂×C₄×C₄₃	—	8	1	no	—
117	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
119	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
123	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	8	1	no	—
126	C₂×Dic₄₃	—	8	1	no	—
127	C₂×Dic₄₃	—	8	1	no	—
128	C₂×Dic₄₃	#106	1	1	yes	characteristic
129	C₂×Dic₄₃	#106	1	1	yes	characteristic
130	C₄×D₄₃	—	2	1	no	—
131	C₄×D₄₃	—	2	1	no	—
132	C₄×D₄₃	—	2	1	no	—
133	C₄×D₄₃	—	2	1	no	—
134	C₄×D₄₃	—	2	1	no	—
135	C₄×D₄₃	—	2	1	no	—
136	C₄×D₄₃	—	2	1	no	—
137	C₄×D₄₃	—	2	1	no	—
138	C₈×C₄₃	—	4	2	no	—
139	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
140	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
141	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
142	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
143	C₄₃⋊C₈	—	2	1	no	—
144	C₄₃⋊C₈	—	2	1	no	—
145	C₄₃⋊C₈	—	4	2	no	—
146	#12928	—	43	1	no	maximal, Sylow
147	C₂²×C₄₃	—	2	1	no	—
148	C₂²×C₄₃	—	2	1	no	—
149	C₂²×C₄₃	C₂×#106	1	1	yes	characteristic
150	C₂²×C₄₃	#448	1	1	yes	characteristic
151	C₂²×C₄₃	#481	1	1	yes	characteristic
152	C₂×D₄₃	—	2	1	no	—
153	C₂×D₄₃	—	2	1	no	—
154	C₂×D₄₃	—	2	1	no	—
155	C₂×D₄₃	—	2	1	no	—
156	C₂×D₄₃	—	2	1	no	—
157	C₂×D₄₃	—	4	2	no	—
158	C₂×D₄₃	—	4	2	no	—
159	C₂×D₄₃	#365	1	2	yes	complemented
160	C₄×C₄₃	—	16	1	no	—
161	C₄×C₄₃	#385	1	1	yes	characteristic
162	C₄×C₄₃	#428	1	1	yes	characteristic
163	C₄×C₄₃	#450	1	1	yes	characteristic
164	C₄×C₄₃	#483	1	1	yes	characteristic
165	Dic₄₃	—	2	1	no	—
166	Dic₄₃	—	2	1	no	—
167	Dic₄₃	—	16	1	no	—
168	C₂²×#108	—	43	1	no	—
169	C₂×#365	—	43	1	no	—
170	C₂×#365	D₄₃	1	1	yes	characteristic, complemented
171	#2112	—	43	1	no	—
172	#1807	—	43	1	no	—
173	#1383	—	43	1	no	—
174	#1383	—	43	1	no	—
175	C₂×C₄₃	—	2	1	no	—
176	C₂×C₄₃	—	2	1	no	—
177	C₂×C₄₃	#1853	1	1	yes	characteristic
178	C₂×C₄₃	C₂×#365	1	1	yes	characteristic
179	C₂×C₄₃	#2155	1	1	yes	characteristic
180	D₄₃	—	2	2	no	—
181	D₄₃	—	4	1	no	—
182	C₂²×M₁₆	—	86	1	no	—
183	C₂³×C₈	—	43	1	no	—
184	C₂×M₃₂	—	43	1	no	—
185	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
186	C₂×#108	—	43	1	no	—
187	C₂×#108	—	86	2	no	—
188	C₂×#108	—	86	1	no	—
189	C₂×#108	—	86	2	no	—
190	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
191	C₂×#106	—	43	1	no	—
192	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
193	#380	—	86	1	no	—
194	#365	—	2	2	no	—
195	#365	—	86	2	no	—
196	C₂²⋊C₁₆	—	43	2	no	—
197	C₂²⋊C₁₆	—	43	2	no	—
198	#340	—	86	1	no	—
199	C₄₃	#12928	1	1	yes	characteristic, complemented, Sylow
200	C₂³×C₄	—	43	1	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	—	86	4	no	—
203	C₂×M₁₆	—	86	1	no	—
204	C₂×M₁₆	—	172	1	no	—
205	C₂×M₁₆	—	172	2	no	—
206	C₂²×C₈	—	43	1	no	—
207	C₂²×C₈	—	43	1	no	—
208	C₂²×C₈	—	43	1	no	—
209	C₂²×C₈	—	43	1	no	—
210	C₂²×C₈	—	43	1	no	—
211	C₂²×C₈	—	86	1	no	—
212	C₂²×C₈	—	86	1	no	—
213	C₂²×C₈	—	86	1	no	—
214	C₂²×C₈	—	86	1	no	—
215	C₂²×C₈	—	172	1	no	—
216	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	86	1	no	—
219	C₂×C₂²⋊C₄	—	172	1	no	—
220	M₃₂	—	2	1	no	—
221	M₃₂	—	2	1	no	—
222	M₃₂	—	86	1	no	—
223	M₃₂	—	86	1	no	—
224	C₂×C₁₆	—	43	2	no	—
225	C₂×C₁₆	C₄×D₄₃	1	2	yes	—
226	#108	—	2	1	no	—
227	#108	—	43	2	no	—
228	#108	—	86	4	no	—
229	#108	—	86	1	no	—
230	#108	C₂²⋊D₄₃	1	2	yes	—
231	#106	—	2	1	no	—
232	#106	—	2	1	no	—
233	#106	—	86	1	no	—
234	#106	—	86	1	no	—
235	#95	—	172	1	no	—
236	C₂⁴	—	43	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	43	1	no	—
239	C₂²×C₄	—	43	1	no	—
240	C₂²×C₄	—	43	1	no	—
241	C₂²×C₄	—	43	1	no	—
242	C₂²×C₄	—	43	1	no	—
243	C₂²×C₄	—	86	1	no	—
244	C₂²×C₄	—	86	1	no	—
245	C₂²×C₄	—	86	1	no	—
246	C₂²×C₄	—	86	1	no	—
247	C₂²×C₄	—	172	1	no	—
248	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
249	M₁₆	—	2	2	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	86	2	no	—
252	M₁₆	—	172	4	no	—
253	M₁₆	—	172	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	43	1	no	—
256	C₂×C₈	—	43	1	no	—
257	C₂×C₈	—	86	1	no	—
258	C₂×C₈	—	86	1	no	—
259	C₂×C₈	—	86	1	no	—
260	C₂×C₈	—	86	1	no	—
261	C₂×C₈	—	86	1	no	—
262	C₂×C₈	—	86	1	no	—
263	C₂×C₈	—	86	1	no	—
264	C₂×C₈	—	86	1	no	—
265	C₂×C₈	—	86	1	no	—
266	C₂×C₈	—	86	1	no	—
267	C₂×C₈	—	172	4	no	—
268	C₂×C₈	—	172	1	no	—
269	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
270	C₂×C₈	#10621898	1	1	yes	characteristic
271	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
272	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
273	C₂×C₈	#10621902	1	1	yes	characteristic
274	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	172	1	no	—
278	C₄⋊C₄	—	172	1	no	—
279	C₂²⋊C₄	—	344	2	no	—
280	C₁₆	—	2	2	no	—
281	C₁₆	—	86	2	no	—
282	C₂³	—	43	1	no	—
283	C₂³	—	43	1	no	—
284	C₂³	—	86	2	no	—
285	C₂³	—	86	1	no	—
286	C₂³	—	86	1	no	—
287	C₂³	—	172	1	no	—
288	C₂³	SD₁₆×D₄₃	1	1	yes	characteristic
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	43	1	no	—
293	C₂×C₄	—	43	1	no	—
294	C₂×C₄	—	86	1	no	—
295	C₂×C₄	—	86	1	no	—
296	C₂×C₄	—	86	1	no	—
297	C₂×C₄	—	86	1	no	—
298	C₂×C₄	—	86	1	no	—
299	C₂×C₄	—	86	1	no	—
300	C₂×C₄	—	86	1	no	—
301	C₂×C₄	—	86	1	no	—
302	C₂×C₄	—	86	1	no	—
303	C₂×C₄	—	86	1	no	—
304	C₂×C₄	—	344	1	no	—
305	C₂×C₄	—	344	1	no	—
306	C₂×C₄	—	344	1	no	—
307	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
308	C₂×C₄	#13063293	1	1	yes	characteristic
309	C₂×C₄	#13063310	1	1	yes	characteristic
310	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
311	C₂×C₄	#13063352	1	1	yes	characteristic
312	C₂×C₄	#13063354	1	1	yes	characteristic
313	C₈	—	4	2	no	—
314	C₈	—	86	1	no	—
315	C₈	—	86	1	no	—
316	C₈	—	172	2	no	—
317	C₈	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
318	C₈	#13063287	1	1	yes	characteristic
319	C₈	#13063301	1	1	yes	characteristic
320	C₈	#13063303	1	1	yes	characteristic
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	43	2	no	—
324	C₂²	—	86	1	no	—
325	C₂²	—	86	1	no	—
326	C₂²	—	86	1	no	—
327	C₂²	—	86	1	no	—
328	C₂²	—	86	1	no	—
329	C₂²	—	172	2	no	—
330	C₂²	—	172	2	no	—
331	C₂²	#106×D₄₃	1	1	yes	characteristic
332	C₂²	#14583036	1	1	yes	characteristic
333	C₂²	#14583219	1	1	yes	characteristic
334	C₄	—	16	1	no	—
335	C₄	—	86	1	no	—
336	C₄	—	86	1	no	—
337	C₄	—	688	1	no	—
338	C₄	#14582770	1	1	yes	characteristic
339	C₄	#14582938	1	1	yes	characteristic
340	C₄	#14583048	1	1	yes	characteristic
341	C₄	#14583231	1	1	yes	characteristic
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	—	86	2	no	—
345	C₂	—	172	1	no	—
346	C₂	#19134786	1	1	yes	characteristic
347	C₂	#365×D₄₃	1	1	yes	characteristic
348	C₂	#19136366	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#106×D₄₃
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#365	C₂
socle	C₂²×C₄₃	C₂×#106
prime radical	C₂³×D₄₃	SD₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#365	D₄₃
hypocenter	C₄₃	#12928
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
2. C₂×#14582721	C₂	semidirect
5. #19132695	C₂	semidirect
6. #19132694	C₂	semidirect
7. C₂×C₄₃×#365	C₂	semidirect
18. (C₂×D₄₃)⋊C₁₆	C₄	semidirect
19. (C₂×D₄₃)⋊C₁₆	C₄	semidirect
159. C₂×D₄₃	#365	semidirect
170. C₂×#365	D₄₃	semidirect
199. C₄₃	#12928	semidirect

Maximal subgroups

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

subgroup	index
C₂×#108×D₄₃	2
C₂×#14582721	2
#19136205	2
#19134617	2
#19132695	2
#19132694	2
C₂×C₄₃×#365	2
#12928	43