Subgroups — #25012747

Order 11008.

396 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582723	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582721	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134630	C₂	1	1	yes	maximal, characteristic
5	#19134617	C₂	1	1	yes	maximal, characteristic
6	#19132694	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
9	C₂×#13063265	C₂²	1	1	yes	characteristic
10	C₂×#13063264	C₄	1	2	yes	complemented
11	M₃₂×D₄₃	—	2	2	no	—
12	M₃₂×D₄₃	—	2	2	no	—
13	C₂×C₁₆×D₄₃	C₄	1	2	yes	complemented
14	C₂×#13063261	C₂²	1	1	yes	characteristic
15	C₂×#13063257	C₂²	1	1	yes	characteristic
16	#14582862	—	2	1	no	—
17	#14582756	—	2	1	no	—
18	#14582723	—	2	2	no	—
19	#14582721	—	2	2	no	—
20	#14582663	—	2	1	no	—
21	#14582661	—	2	1	no	—
22	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
26	C₂×#10621877	—	2	1	no	—
27	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₈×D₄₃	—	2	1	no	—
29	C₂×C₈×D₄₃	—	2	1	no	—
30	C₂×C₈×D₄₃	—	2	1	no	—
31	C₂×C₈×D₄₃	—	2	1	no	—
32	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₄₃	Q₈	1	1	yes	characteristic
35	C₂×C₈×D₄₃	Q₈	1	1	yes	characteristic
36	C₂×C₄⋊Dic₄₃	—	2	1	no	—
37	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
38	#13063265	D₄	1	2	yes	—
39	#13063265	D₄	1	2	yes	—
40	#13063264	—	2	4	no	—
41	C₂×C₄₃⋊C₁₆	C₂×C₄	1	2	yes	—
42	C₁₆×D₄₃	—	2	4	no	—
43	#13063261	—	2	1	no	—
44	#13063261	—	2	1	no	—
45	#13063257	—	2	1	no	—
46	#13063257	—	2	1	no	—
47	(C₂×D₄₃)⋊C₈	—	4	1	no	—
48	(C₂×D₄₃)⋊C₈	—	4	1	no	—
49	C₄₃⋊#95	—	4	1	no	—
50	C₂×M₁₆×C₄₃	—	2	1	no	—
51	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
53	M₃₂×C₄₃	—	2	1	no	—
54	M₃₂×C₄₃	—	2	1	no	—
55	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
56	#108×C₄₃	—	2	1	no	—
57	#108×C₄₃	—	2	1	no	—
58	#106×C₄₃	—	2	1	no	—
59	#106×C₄₃	—	2	1	no	—
60	C₂³×D₄₃	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	—	4	1	no	—
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₄₃	—	4	1	no	—
64	C₂²×Dic₄₃	SD₁₆	1	1	yes	characteristic
65	C₂×C₄×D₄₃	—	2	1	no	—
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	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×D₄₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×D₄₃	SD₁₆	1	1	yes	characteristic
72	C₂×C₄×D₄₃	SD₁₆	1	1	yes	characteristic
73	M₁₆×C₄₃	—	4	1	no	—
74	M₁₆×C₄₃	—	4	1	no	—
75	C₂×C₈×C₄₃	—	4	1	no	—
76	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
81	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
82	C₄⋊C₄×C₄₃	—	4	1	no	—
83	C₄⋊C₄×C₄₃	—	4	1	no	—
84	(C₂×D₄₃)⋊C₄	—	8	2	no	—
85	C₄⋊Dic₄₃	—	4	1	no	—
86	C₄⋊Dic₄₃	—	4	1	no	—
87	#10621877	—	4	1	no	—
88	#10621877	—	4	1	no	—
89	C₂×C₄₃⋊C₈	—	2	1	no	—
90	C₂×C₄₃⋊C₈	—	2	1	no	—
91	C₂×C₄₃⋊C₈	—	4	1	no	—
92	C₂×C₄₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
94	C₈×D₄₃	—	2	1	no	—
95	C₈×D₄₃	—	2	1	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	C₁₆×C₄₃	—	2	2	no	—
103	C₄₃⋊C₁₆	C₂²⋊C₄	1	4	yes	complemented
104	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
105	C₂²×D₄₃	—	2	1	no	—
106	C₂²×D₄₃	—	2	1	no	—
107	C₂²×D₄₃	—	2	2	no	—
108	C₂²×D₄₃	—	4	1	no	—
109	C₂²×D₄₃	#106	1	1	yes	characteristic
110	C₂²×D₄₃	#106	1	1	yes	characteristic
111	C₂×C₄×C₄₃	—	8	1	no	—
112	C₂×C₄×C₄₃	—	8	1	no	—
113	C₂×C₄×C₄₃	—	8	1	no	—
114	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
119	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
120	C₂×Dic₄₃	—	2	1	no	—
121	C₂×Dic₄₃	—	2	1	no	—
122	C₂×Dic₄₃	—	8	1	no	—
123	C₂×Dic₄₃	—	8	1	no	—
124	C₂×Dic₄₃	—	8	1	no	—
125	C₂×Dic₄₃	#106	1	1	yes	characteristic
126	C₂×Dic₄₃	#106	1	1	yes	characteristic
127	C₄×D₄₃	—	2	1	no	—
128	C₄×D₄₃	—	2	1	no	—
129	C₄×D₄₃	—	2	1	no	—
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	C₈×C₄₃	—	8	1	no	—
136	C₈×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₈×C₄₃	#117	1	1	yes	characteristic
138	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
139	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
140	C₄₃⋊C₈	—	2	1	no	—
141	C₄₃⋊C₈	—	2	1	no	—
142	C₄₃⋊C₈	—	8	1	no	—
143	#12831	—	43	1	no	maximal, Sylow
144	C₂²×C₄₃	—	2	1	no	—
145	C₂²×C₄₃	—	2	1	no	—
146	C₂²×C₄₃	C₂×#106	1	1	yes	characteristic
147	C₂²×C₄₃	#465	1	1	yes	characteristic
148	C₂²×C₄₃	#465	1	1	yes	characteristic
149	C₂×D₄₃	—	2	1	no	—
150	C₂×D₄₃	—	2	1	no	—
151	C₂×D₄₃	—	2	1	no	—
152	C₂×D₄₃	—	2	1	no	—
153	C₂×D₄₃	—	2	1	no	—
154	C₂×D₄₃	—	4	2	no	—
155	C₂×D₄₃	—	4	2	no	—
156	C₂×D₄₃	#365	1	2	yes	complemented
157	C₄×C₄₃	—	16	1	no	—
158	C₄×C₄₃	#380	1	1	yes	characteristic
159	C₄×C₄₃	#427	1	1	yes	characteristic
160	C₄×C₄₃	#465	1	1	yes	characteristic
161	C₄×C₄₃	#465	1	1	yes	characteristic
162	Dic₄₃	—	2	1	no	—
163	Dic₄₃	—	2	1	no	—
164	Dic₄₃	—	16	1	no	—
165	C₂²×M₃₂	—	43	1	no	—
166	C₂×#365	—	43	1	no	—
167	C₂×#365	—	43	1	no	—
168	#1810	—	43	1	no	—
169	#1807	—	43	1	no	—
170	#1383	—	43	1	no	—
171	#1383	D₄₃	1	1	yes	characteristic, complemented
172	C₂×C₄₃	—	2	1	no	—
173	C₂×C₄₃	—	2	1	no	—
174	C₂×C₄₃	#1807	1	1	yes	characteristic
175	C₂×C₄₃	C₂×#365	1	1	yes	characteristic
176	C₂×C₄₃	#2155	1	1	yes	characteristic
177	D₄₃	—	2	2	no	—
178	D₄₃	—	4	1	no	—
179	C₂³×C₈	—	43	1	no	—
180	C₂×M₃₂	—	43	1	no	—
181	C₂×M₃₂	—	43	2	no	—
182	C₂×M₃₂	—	86	2	no	—
183	C₂×M₃₂	—	86	2	no	—
184	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
185	C₂²×C₁₆	—	43	2	no	—
186	C₂×#108	—	43	1	no	—
187	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
188	C₂×#106	—	43	1	no	—
189	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
190	#407	—	86	1	no	—
191	#380	—	86	1	no	—
192	#365	—	86	2	no	—
193	#365	—	86	2	no	—
194	#341	—	86	1	no	—
195	#340	—	86	1	no	—
196	C₄₃	#12831	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₄	—	43	1	no	—
198	C₂×M₁₆	—	2	1	no	—
199	C₂×M₁₆	—	86	1	no	—
200	C₂²×C₈	—	43	1	no	—
201	C₂²×C₈	—	43	1	no	—
202	C₂²×C₈	—	43	1	no	—
203	C₂²×C₈	—	43	1	no	—
204	C₂²×C₈	—	43	1	no	—
205	C₂²×C₈	—	86	1	no	—
206	C₂²×C₈	—	86	1	no	—
207	C₂²×C₈	—	86	1	no	—
208	C₂²×C₈	—	86	1	no	—
209	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	86	1	no	—
212	C₂×C₂²⋊C₄	—	172	1	no	—
213	M₃₂	—	2	1	no	—
214	M₃₂	—	2	1	no	—
215	M₃₂	—	43	2	no	—
216	M₃₂	—	43	2	no	—
217	M₃₂	—	86	4	no	—
218	C₂×C₁₆	—	43	2	no	—
219	C₂×C₁₆	—	86	4	no	—
220	C₂×C₁₆	C₄×D₄₃	1	2	yes	—
221	#108	—	2	1	no	—
222	#108	—	2	1	no	—
223	#108	—	86	1	no	—
224	#108	—	86	1	no	—
225	#106	—	2	1	no	—
226	#106	—	2	1	no	—
227	#106	—	86	1	no	—
228	#106	—	86	1	no	—
229	C₂²⋊C₈	—	172	1	no	—
230	C₂²⋊C₈	—	172	1	no	—
231	#95	—	172	1	no	—
232	C₂⁴	—	43	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	43	1	no	—
235	C₂²×C₄	—	43	1	no	—
236	C₂²×C₄	—	43	1	no	—
237	C₂²×C₄	—	43	1	no	—
238	C₂²×C₄	—	43	1	no	—
239	C₂²×C₄	—	86	1	no	—
240	C₂²×C₄	—	86	1	no	—
241	C₂²×C₄	—	86	1	no	—
242	C₂²×C₄	—	86	1	no	—
243	C₂²×C₄	—	172	1	no	—
244	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	1	no	—
247	M₁₆	—	172	1	no	—
248	M₁₆	—	172	1	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	43	1	no	—
251	C₂×C₈	—	43	1	no	—
252	C₂×C₈	—	86	1	no	—
253	C₂×C₈	—	86	1	no	—
254	C₂×C₈	—	86	1	no	—
255	C₂×C₈	—	86	1	no	—
256	C₂×C₈	—	86	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	—	172	1	no	—
263	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
264	C₂×C₈	#10621898	1	1	yes	characteristic
265	C₂×C₈	#10621898	1	1	yes	characteristic
266	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
267	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
268	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	172	1	no	—
272	C₄⋊C₄	—	172	1	no	—
273	C₂²⋊C₄	—	344	2	no	—
274	C₁₆	—	2	2	no	—
275	C₁₆	—	43	4	no	—
276	C₂³	—	43	1	no	—
277	C₂³	—	43	1	no	—
278	C₂³	—	86	2	no	—
279	C₂³	—	86	1	no	—
280	C₂³	—	86	1	no	—
281	C₂³	—	172	1	no	—
282	C₂³	SD₁₆×D₄₃	1	1	yes	characteristic
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	43	1	no	—
287	C₂×C₄	—	43	1	no	—
288	C₂×C₄	—	86	1	no	—
289	C₂×C₄	—	86	1	no	—
290	C₂×C₄	—	86	1	no	—
291	C₂×C₄	—	86	1	no	—
292	C₂×C₄	—	86	1	no	—
293	C₂×C₄	—	86	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	—	344	1	no	—
299	C₂×C₄	—	344	1	no	—
300	C₂×C₄	—	344	1	no	—
301	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
302	C₂×C₄	#13063298	1	1	yes	characteristic
303	C₂×C₄	#13063298	1	1	yes	characteristic
304	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
305	C₂×C₄	#13063344	1	1	yes	characteristic
306	C₂×C₄	#13063348	1	1	yes	characteristic
307	C₈	—	8	1	no	—
308	C₈	—	86	1	no	—
309	C₈	—	86	1	no	—
310	C₈	—	344	1	no	—
311	C₈	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
312	C₈	#13063282	1	1	yes	characteristic
313	C₈	#13063303	1	1	yes	characteristic
314	C₈	#13063303	1	1	yes	characteristic
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	43	2	no	—
318	C₂²	—	86	1	no	—
319	C₂²	—	86	1	no	—
320	C₂²	—	86	1	no	—
321	C₂²	—	86	1	no	—
322	C₂²	—	86	1	no	—
323	C₂²	—	172	2	no	—
324	C₂²	—	172	2	no	—
325	C₂²	#106×D₄₃	1	1	yes	characteristic
326	C₂²	#14583127	1	1	yes	characteristic
327	C₂²	#14583131	1	1	yes	characteristic
328	C₄	—	16	1	no	—
329	C₄	—	86	1	no	—
330	C₄	—	86	1	no	—
331	C₄	—	688	1	no	—
332	C₄	#14582758	1	1	yes	characteristic
333	C₄	#14582936	1	1	yes	characteristic
334	C₄	#14583127	1	1	yes	characteristic
335	C₄	#14583131	1	1	yes	characteristic
336	C₂	—	2	1	no	—
337	C₂	—	2	1	no	—
338	C₂	—	86	2	no	—
339	C₂	—	172	1	no	—
340	C₂	#19134619	1	1	yes	characteristic
341	C₂	#365×D₄₃	1	1	yes	characteristic
342	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	C₄₃×#1383	C₂
socle	C₂²×C₄₃	C₂×#106
prime radical	C₂³×D₄₃	SD₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1383	D₄₃
hypocenter	C₄₃	#12831
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1383

Complements

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

N	H ≅ G/N	type
2. C₂×#14582723	C₂	semidirect
3. C₂×#14582721	C₂	semidirect
6. #19132694	C₂	semidirect
7. C₄₃×#1383	C₂	semidirect
10. C₂×#13063264	C₄	semidirect
13. C₂×C₁₆×D₄₃	C₄	semidirect
103. C₄₃⋊C₁₆	C₂²⋊C₄	semidirect
156. C₂×D₄₃	#365	semidirect
171. #1383	D₄₃	semidirect
196. C₄₃	#12831	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₄₃	2
C₂×#14582723	2
C₂×#14582721	2
#19134630	2
#19134617	2
#19132694	2
C₄₃×#1383	2
#12831	43