Subgroups — #25253010

Order 11008.

512 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145245	C₂	1	1	yes	maximal, characteristic, complemented
2	#19136752	C₂	1	2	yes	maximal, complemented
3	#19136749	C₂	1	2	yes	maximal, complemented
4	#19136696	C₂	1	2	yes	maximal, complemented
5	#19136679	C₂	1	2	yes	maximal, complemented
6	#19136630	C₂	1	2	yes	maximal, complemented
7	#19136627	C₂	1	2	yes	maximal, complemented
8	#19136354	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#3176	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583504	—	2	1	no	—
11	C₂×#13063266	C₂²	1	2	yes	—
12	C₂×#13063266	C₂²	1	2	yes	—
13	C₂×#13063265	C₂²	1	1	yes	characteristic
14	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	#14583301	C₂²	1	1	yes	characteristic
16	#14583276	C₂²	1	1	yes	characteristic
17	#14583264	—	2	1	no	—
18	#14583152	—	2	1	no	—
19	#14583135	C₂²	1	2	yes	—
20	#14583012	—	2	1	no	—
21	C₄×D₃₄₄	C₂²	1	2	yes	—
22	#14582726	C₂²	1	2	yes	—
23	#14582724	C₂²	1	2	yes	—
24	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
25	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
26	#14582700	C₂²	1	2	yes	—
27	#14582700	C₂²	1	2	yes	—
28	#14582699	C₂²	1	2	yes	—
29	#14582699	C₂²	1	2	yes	—
30	C₄₃×#540	—	2	1	no	—
31	C₄₃×#499	C₂²	1	1	yes	characteristic
32	C₄₃×#493	C₂²	1	1	yes	characteristic
33	C₄₃×#490	—	2	1	no	—
34	C₄₃×#469	—	2	1	no	—
35	C₄₃×#466	C₂²	1	2	yes	—
36	C₄₃×#442	—	2	1	no	—
37	C₄×D₈×C₄₃	C₂²	1	2	yes	—
38	C₄₃×#405	C₂²	1	1	yes	characteristic
39	C₂×D₃₄₄	—	2	1	no	—
40	C₂×D₃₄₄	C₂³	1	2	yes	—
41	#13063322	—	2	1	no	—
42	#13063317	—	2	1	no	—
43	#13063310	—	4	2	no	—
44	#13063299	—	2	2	no	—
45	#13063298	—	4	2	no	—
46	C₄×D₁₇₂	—	2	2	no	—
47	#13063266	—	4	2	no	—
48	#13063266	—	4	2	no	—
49	#13063265	—	4	1	no	—
50	C₂×C₄₃⋊C₁₆	—	4	1	no	—
51	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
52	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
53	#13063259	C₂³	1	2	yes	—
54	#13063257	—	2	1	no	—
55	#13063247	—	2	2	no	—
56	#13063247	—	2	2	no	—
57	C₂×D₈×C₄₃	—	2	1	no	—
58	C₂×D₈×C₄₃	C₂³	1	2	yes	—
59	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	#127×C₄₃	—	2	1	no	—
62	#125×C₄₃	—	2	1	no	—
63	#123×C₄₃	—	4	2	no	—
64	#121×C₄₃	—	2	2	no	—
65	#121×C₄₃	—	4	2	no	—
66	C₄×D₄×C₄₃	—	2	2	no	—
67	#117×C₄₃	D₄	1	1	yes	characteristic
68	#107×C₄₃	C₂³	1	2	yes	—
69	#106×C₄₃	—	2	1	no	—
70	#102×C₄₃	—	2	2	no	—
71	#102×C₄₃	—	2	2	no	—
72	#97×C₄₃	D₄	1	1	yes	characteristic
73	#97×C₄₃	D₄	1	1	yes	characteristic
74	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
76	C₂×D₄×C₄₃	—	2	2	no	—
77	C₂×D₄×C₄₃	—	4	2	no	—
78	C₂×D₄×C₄₃	—	4	2	no	—
79	C₂²×C₄×C₄₃	—	4	2	no	—
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₄₃	—	4	2	no	—
82	C₂×D₁₇₂	—	2	2	no	—
83	C₂×D₁₇₂	—	4	2	no	—
84	C₂×C₄×D₄₃	—	4	2	no	—
85	D₈×C₄₃	—	2	2	no	—
86	D₈×C₄₃	—	2	2	no	—
87	D₈×C₄₃	—	4	2	no	—
88	M₁₆×C₄₃	—	4	1	no	—
89	C₂×C₈×C₄₃	—	2	1	no	—
90	C₂×C₈×C₄₃	—	2	1	no	—
91	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₄₃	—	2	2	no	—
96	C₄⋊C₄×C₄₃	—	4	2	no	—
97	C₄⋊C₄×C₄₃	D₈	1	2	yes	—
98	C₂²⋊C₄×C₄₃	—	4	2	no	—
99	C₂²⋊C₄×C₄₃	—	4	2	no	—
100	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
101	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
103	(C₂×D₄₃)⋊C₄	—	4	2	no	—
104	(C₂×D₄₃)⋊C₄	—	4	2	no	—
105	C₄⋊Dic₄₃	—	2	2	no	—
106	Dic₄₃⋊C₄	—	4	2	no	—
107	D₃₄₄	—	2	2	no	—
108	D₃₄₄	—	2	2	no	—
109	D₃₄₄	—	4	2	no	—
110	C₄₃⋊C₁₆	—	2	2	no	—
111	C₄₃⋊C₁₆	—	2	2	no	—
112	C₂³×C₄₃	—	4	2	no	—
113	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
114	C₂²×D₄₃	—	4	2	no	—
115	D₄×C₄₃	—	4	2	no	—
116	D₄×C₄₃	—	4	2	no	—
117	D₄×C₄₃	—	16	2	no	—
118	D₄×C₄₃	—	16	2	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	4	2	no	—
122	C₂×C₄×C₄₃	—	16	2	no	—
123	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
127	C₂²⋊D₄₃	—	16	2	no	—
128	C₂×Dic₄₃	—	4	2	no	—
129	D₁₇₂	—	4	2	no	—
130	D₁₇₂	—	4	2	no	—
131	D₁₇₂	—	16	2	no	—
132	C₄×D₄₃	—	16	2	no	—
133	C₈×C₄₃	—	2	1	no	—
134	C₈×C₄₃	—	2	1	no	—
135	C₈×C₄₃	—	2	1	no	—
136	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
137	C₈×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
138	#33825	—	43	1	no	maximal, Sylow
139	C₂²×C₄₃	—	2	2	no	—
140	C₂²×C₄₃	—	2	1	no	—
141	C₂²×C₄₃	—	8	2	no	—
142	C₂²×C₄₃	—	8	2	no	—
143	C₂²×C₄₃	—	8	2	no	—
144	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₄₃	—	8	2	no	—
146	C₂×D₄₃	—	8	2	no	—
147	C₂×D₄₃	—	8	2	no	—
148	C₄×C₄₃	—	2	2	no	—
149	C₄×C₄₃	—	2	2	no	—
150	C₄×C₄₃	—	2	1	no	—
151	C₄×C₄₃	—	8	2	no	—
152	C₄×C₄₃	#537	1	1	yes	characteristic
153	C₄×C₄₃	#575	1	1	yes	characteristic
154	Dic₄₃	—	8	2	no	—
155	#3176	—	43	1	no	—
156	#3176	D₄₃	1	1	yes	characteristic, complemented
157	#2228	—	43	2	no	—
158	#2228	—	43	2	no	—
159	#2218	—	43	2	no	—
160	#2215	—	43	2	no	—
161	#2206	—	43	2	no	—
162	#2206	—	43	2	no	—
163	#2151	—	43	1	no	—
164	C₂×C₄₃	—	4	1	no	—
165	C₂×C₄₃	—	16	2	no	—
166	C₂×C₄₃	#3245	1	1	yes	characteristic
167	C₂×C₄₃	#3415	1	2	yes	—
168	D₄₃	—	16	2	no	—
169	#540	—	2	1	no	—
170	#540	—	86	1	no	—
171	C₂×D₁₆	—	43	2	no	—
172	C₂×D₁₆	—	43	2	no	—
173	C₂×M₃₂	—	43	1	no	—
174	C₂²×C₁₆	—	43	1	no	—
175	#499	—	43	1	no	—
176	#499	C₂×D₄₃	1	1	yes	characteristic
177	#493	—	43	1	no	—
178	#493	C₂×D₄₃	1	1	yes	characteristic
179	#490	—	2	1	no	—
180	#490	—	86	1	no	—
181	#469	—	2	1	no	—
182	#469	—	86	1	no	—
183	#466	—	43	2	no	—
184	#466	C₂×D₄₃	1	2	yes	—
185	#442	—	2	1	no	—
186	#442	—	86	1	no	—
187	C₄×D₈	—	43	2	no	—
188	C₄×D₈	C₂×D₄₃	1	2	yes	—
189	#405	C₂×D₄₃	1	1	yes	characteristic
190	#367	—	43	2	no	—
191	#366	—	43	2	no	—
192	C₄⋊C₁₆	—	43	2	no	—
193	C₄⋊C₁₆	—	43	2	no	—
194	#357	—	43	2	no	—
195	#357	—	43	2	no	—
196	#357	—	43	2	no	—
197	#357	—	43	2	no	—
198	C₄₃	#33825	1	1	yes	characteristic, complemented, Sylow
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	—	43	2	no	—
201	C₂×D₈	—	86	1	no	—
202	C₂×D₈	C₂²×D₄₃	1	2	yes	—
203	C₂×M₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
205	#127	—	2	1	no	—
206	#127	—	86	1	no	—
207	#125	—	2	1	no	—
208	#125	—	86	1	no	—
209	#123	—	4	2	no	—
210	#123	—	172	2	no	—
211	#121	—	2	2	no	—
212	#121	—	4	2	no	—
213	#121	—	86	2	no	—
214	#121	—	172	2	no	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	—	86	2	no	—
217	#117	C₂²⋊D₄₃	1	1	yes	characteristic
218	D₁₆	—	172	2	no	—
219	D₁₆	—	172	2	no	—
220	M₃₂	—	172	1	no	—
221	C₂×C₁₆	—	43	2	no	—
222	C₂×C₁₆	—	43	2	no	—
223	C₂×C₁₆	—	172	1	no	—
224	#107	—	43	2	no	—
225	#107	C₂²×D₄₃	1	2	yes	—
226	#106	—	2	1	no	—
227	#106	—	86	1	no	—
228	#102	—	2	2	no	—
229	#102	—	2	2	no	—
230	#102	—	86	2	no	—
231	#102	—	86	2	no	—
232	#97	C₂²⋊D₄₃	1	1	yes	characteristic
233	#97	C₂²⋊D₄₃	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	—	86	2	no	—
240	C₂×D₄	—	172	2	no	—
241	C₂×D₄	—	172	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	172	2	no	—
244	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
245	D₈	—	2	2	no	—
246	D₈	—	2	2	no	—
247	D₈	—	4	2	no	—
248	D₈	—	86	2	no	—
249	D₈	—	86	2	no	—
250	D₈	—	172	2	no	—
251	M₁₆	—	4	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
255	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
256	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
257	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	86	2	no	—
261	C₄⋊C₄	—	172	2	no	—
262	C₄⋊C₄	#10621882	1	2	yes	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	172	2	no	—
266	C₂²⋊C₄	—	172	2	no	—
267	C₂²⋊C₄	#10621882	1	2	yes	—
268	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
270	C₁₆	—	86	2	no	—
271	C₁₆	—	86	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	172	2	no	—
274	C₂³	C₂×#10621882	1	1	yes	characteristic
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	16	2	no	—
278	D₄	—	16	2	no	—
279	D₄	—	172	2	no	—
280	D₄	—	172	2	no	—
281	D₄	—	688	2	no	—
282	D₄	—	688	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	16	2	no	—
287	C₂×C₄	—	172	2	no	—
288	C₂×C₄	—	688	2	no	—
289	C₂×C₄	C₂×#10621882	1	2	yes	—
290	C₂×C₄	C₂×#10621882	1	2	yes	—
291	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	#13063371	1	1	yes	characteristic
297	C₈	#13063372	1	1	yes	characteristic
298	C₂²	—	2	1	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	344	2	no	—
304	C₂²	—	344	2	no	—
305	C₂²	—	344	2	no	—
306	C₂²	C₂²×#10621882	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	344	2	no	—
312	C₄	#14583477	1	1	yes	characteristic
313	C₄	#14583697	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	16	2	no	—
316	C₂	—	688	2	no	—
317	C₂	#19145902	1	1	yes	characteristic
318	C₂	#19147734	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#3176	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	#3176	D₄₃
hypocenter	C₄₃	#33825
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3176

Complements

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

N	H ≅ G/N	type
1. #19145245	C₂	semidirect
2. #19136752	C₂	semidirect
3. #19136749	C₂	semidirect
4. #19136696	C₂	semidirect
5. #19136679	C₂	semidirect
6. #19136630	C₂	semidirect
7. #19136627	C₂	semidirect
8. #19136354	C₂	semidirect
9. C₄₃×#3176	C₂	semidirect
156. #3176	D₄₃	semidirect
198. C₄₃	#33825	semidirect

Maximal subgroups

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

subgroup	index
#19145245	2
#19136752	2
#19136749	2
#19136696	2
#19136679	2
#19136630	2
#19136627	2
#19136354	2
C₄₃×#3176	2
#33825	43