Subgroups — #24995122

Order 11008.

318 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19135540	C₂	1	1	yes	maximal, characteristic
2	#19135371	C₂	1	1	yes	maximal, characteristic
3	#19133050	C₂	1	1	yes	maximal, characteristic
4	#19132885	C₂	1	1	yes	maximal, characteristic
5	#19132497	C₂	1	1	yes	maximal, characteristic
6	#19132434	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1542	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063261	—	2	1	no	—
9	#14582923	C₂²	1	1	yes	characteristic
10	C₄×#10621877	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
13	#14582664	—	2	1	no	—
14	#14582658	—	2	1	no	—
15	#14582658	—	2	1	no	—
16	Q₈⋊(C₄₃⋊C₈)	—	2	1	no	—
17	Q₈⋊(C₄₃⋊C₈)	—	2	1	no	—
18	#14582621	—	2	1	no	—
19	#14582620	—	2	1	no	—
20	C₄₃×#533	C₂²	1	1	yes	characteristic
21	C₄₃×#431	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄₃×#333	C₄	1	1	yes	characteristic, complemented
24	C₄₃×#333	C₄	1	1	yes	characteristic, complemented
25	C₄₃×#332	C₄	1	1	yes	characteristic, complemented
26	C₄₃×#332	C₄	1	1	yes	characteristic, complemented
27	C₂×#10621877	—	2	1	no	—
28	C₂×#10621877	D₄	1	1	yes	characteristic
29	C₂×#10621877	D₄	1	1	yes	characteristic
30	C₂²×C₄₃⋊C₈	—	2	1	no	—
31	C₂×C₄⋊Dic₄₃	—	2	1	no	—
32	C₂×C₄×Dic₄₃	—	2	1	no	—
33	#13063261	—	4	2	no	—
34	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
35	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
36	#13063236	—	2	1	no	—
37	C₄×C₄₃⋊C₈	—	2	1	no	—
38	C₄×C₄₃⋊C₈	—	2	1	no	—
39	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
40	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
41	C₄₃⋊#95	—	2	1	no	—
42	C₂²×C₈×C₄₃	—	2	1	no	—
43	C₄⋊Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₄⋊Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
45	#125×C₄₃	—	2	1	no	—
46	C₂²⋊Q₈×C₄₃	—	2	1	no	—
47	C₂²⋊Q₈×C₄₃	—	2	1	no	—
48	C₄×Q₈×C₄₃	—	2	1	no	—
49	C₄×Q₈×C₄₃	—	2	1	no	—
50	#117×C₄₃	—	2	1	no	—
51	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₄₃	—	2	1	no	—
53	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
55	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
57	C₂²⋊C₈×C₄₃	—	2	1	no	—
58	C₄×C₈×C₄₃	—	2	1	no	—
59	C₂×Q₈×C₄₃	—	2	1	no	—
60	C₂×Q₈×C₄₃	—	2	1	no	—
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₂²×Dic₄₃	—	4	1	no	—
65	C₂×C₈×C₄₃	—	2	1	no	—
66	C₂×C₈×C₄₃	—	2	1	no	—
67	C₂×C₈×C₄₃	—	2	1	no	—
68	C₂×C₈×C₄₃	—	2	1	no	—
69	C₂×C₈×C₄₃	—	4	1	no	—
70	C₂×C₈×C₄₃	—	4	1	no	—
71	C₄⋊C₄×C₄₃	—	2	1	no	—
72	C₄⋊C₄×C₄₃	—	2	1	no	—
73	C₄⋊C₄×C₄₃	—	2	1	no	—
74	C₄⋊C₄×C₄₃	—	2	1	no	—
75	C₄⋊C₄×C₄₃	—	4	1	no	—
76	C₄⋊C₄×C₄₃	—	4	1	no	—
77	C₂²⋊C₄×C₄₃	—	4	1	no	—
78	C₄²×C₄₃	—	4	1	no	—
79	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₄₃	#41	1	1	yes	characteristic
82	C₄²×C₄₃	#41	1	1	yes	characteristic
83	C₄⋊Dic₄₃	—	2	2	no	—
84	C₄⋊Dic₄₃	—	4	1	no	—
85	C₄×Dic₄₃	—	4	1	no	—
86	C₄×Dic₄₃	—	4	1	no	—
87	#10621877	—	2	1	no	—
88	#10621877	—	2	1	no	—
89	#10621877	—	2	1	no	—
90	#10621877	—	2	1	no	—
91	#10621877	—	4	1	no	—
92	#10621877	—	4	1	no	—
93	C₂×C₄₃⋊C₈	—	2	1	no	—
94	C₂×C₄₃⋊C₈	—	2	1	no	—
95	C₂×C₄₃⋊C₈	—	2	1	no	—
96	C₂×C₄₃⋊C₈	—	2	1	no	—
97	C₂×C₄₃⋊C₈	—	4	2	no	—
98	C₂×C₄₃⋊C₈	—	4	1	no	—
99	C₂³×C₄₃	#124	1	1	yes	characteristic
100	Q₈×C₄₃	—	4	1	no	—
101	Q₈×C₄₃	—	4	1	no	—
102	Q₈×C₄₃	—	8	1	no	—
103	C₂×C₄×C₄₃	—	2	1	no	—
104	C₂×C₄×C₄₃	—	2	1	no	—
105	C₂×C₄×C₄₃	—	4	1	no	—
106	C₂×C₄×C₄₃	—	4	1	no	—
107	C₂×C₄×C₄₃	—	4	1	no	—
108	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
113	C₂×C₄×C₄₃	#126	1	1	yes	characteristic
114	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
116	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
118	C₂×Dic₄₃	—	4	2	no	—
119	C₂×Dic₄₃	—	8	1	no	—
120	C₂×Dic₄₃	—	8	1	no	—
121	C₈×C₄₃	—	4	1	no	—
122	C₈×C₄₃	—	4	1	no	—
123	C₈×C₄₃	—	4	1	no	—
124	C₈×C₄₃	—	4	1	no	—
125	C₄₃⋊C₈	—	2	2	no	—
126	C₄₃⋊C₈	—	4	1	no	—
127	C₄₃⋊C₈	—	4	1	no	—
128	C₄₃⋊C₈	—	4	1	no	—
129	C₄₃⋊C₈	—	8	1	no	—
130	#9591	—	43	1	no	maximal, Sylow
131	C₂²×C₄₃	—	2	1	no	—
132	C₂²×C₄₃	—	2	1	no	—
133	C₂²×C₄₃	#388	1	1	yes	characteristic
134	C₂²×C₄₃	#489	1	1	yes	characteristic
135	C₂²×C₄₃	#491	1	1	yes	characteristic
136	C₄×C₄₃	—	2	1	no	—
137	C₄×C₄₃	—	2	1	no	—
138	C₄×C₄₃	—	2	1	no	—
139	C₄×C₄₃	—	8	1	no	—
140	C₄×C₄₃	—	8	1	no	—
141	C₄×C₄₃	#440	1	1	yes	characteristic
142	C₄×C₄₃	#441	1	1	yes	characteristic
143	C₄×C₄₃	#443	1	1	yes	characteristic
144	C₄×C₄₃	#444	1	1	yes	characteristic
145	C₄×C₄₃	#468	1	1	yes	characteristic
146	C₄×C₄₃	#470	1	1	yes	characteristic
147	Dic₄₃	—	8	2	no	—
148	#1986	—	43	1	no	—
149	#1950	—	43	1	no	—
150	#1542	D₄₃	1	1	yes	characteristic
151	#1493	—	43	1	no	—
152	#1451	—	43	1	no	—
153	#1303	—	43	1	no	—
154	#1279	—	43	1	no	—
155	C₂×C₄₃	—	2	1	no	—
156	C₂×C₄₃	—	2	1	no	—
157	C₂×C₄₃	#1929	1	1	yes	characteristic
158	C₂×C₄₃	#1932	1	1	yes	characteristic
159	C₂×C₄₃	#1935	1	1	yes	characteristic
160	#533	C₂×D₄₃	1	1	yes	characteristic
161	#431	C₂×D₄₃	1	1	yes	characteristic
162	C₂×#108	—	86	1	no	—
163	#423	—	43	1	no	—
164	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
165	C₄×M₁₆	—	43	1	no	—
166	C₂×C₄×C₈	—	43	1	no	—
167	C₄×C₄⋊C₄	—	43	1	no	—
168	#341	—	86	1	no	—
169	#339	—	86	1	no	—
170	#339	—	86	1	no	—
171	#333	Dic₄₃	1	1	yes	characteristic, complemented
172	#333	Dic₄₃	1	1	yes	characteristic, complemented
173	#332	Dic₄₃	1	1	yes	characteristic, complemented
174	#332	Dic₄₃	1	1	yes	characteristic, complemented
175	Q₈⋊C₈	—	86	1	no	—
176	Q₈⋊C₈	—	86	1	no	—
177	#322	—	86	1	no	—
178	#322	—	86	1	no	—
179	C₄₃	#9591	1	1	yes	characteristic, complemented, Sylow
180	C₂×M₁₆	—	43	1	no	—
181	C₂×M₁₆	—	43	1	no	—
182	C₂×M₁₆	—	86	1	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	86	1	no	—
185	C₄⋊Q₈	C₂×Dic₄₃	1	1	yes	characteristic
186	C₄⋊Q₈	C₂×Dic₄₃	1	1	yes	characteristic
187	#125	—	2	1	no	—
188	C₂²⋊Q₈	—	2	1	no	—
189	C₂²⋊Q₈	—	2	1	no	—
190	C₄×Q₈	—	2	1	no	—
191	C₄×Q₈	—	2	1	no	—
192	#117	—	2	1	no	—
193	C₂×C₄⋊C₄	—	86	1	no	—
194	C₂×C₄²	—	86	1	no	—
195	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
196	#108	—	172	2	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	86	1	no	—
199	C₄⋊C₈	—	86	1	no	—
200	C₄⋊C₈	C₂×Dic₄₃	1	1	yes	characteristic
201	C₄⋊C₈	C₂×Dic₄₃	1	1	yes	characteristic
202	C₄⋊C₈	C₂×Dic₄₃	1	1	yes	characteristic
203	C₄⋊C₈	C₂×Dic₄₃	1	1	yes	characteristic
204	C₂²⋊C₈	—	2	1	no	—
205	#97	—	86	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	43	1	no	—
208	C₄×C₈	—	43	1	no	—
209	C₄×C₈	—	86	1	no	—
210	C₄×C₈	—	86	1	no	—
211	#95	—	86	1	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×Q₈	—	2	1	no	—
214	C₂²×C₄	—	172	1	no	—
215	C₂²×C₄	#10621898	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
217	C₂²×C₄	#10621900	1	1	yes	characteristic
218	M₁₆	—	86	1	no	—
219	M₁₆	—	86	1	no	—
220	M₁₆	—	86	1	no	—
221	M₁₆	—	86	1	no	—
222	M₁₆	—	172	1	no	—
223	M₁₆	—	172	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	86	1	no	—
231	C₂×C₈	—	86	1	no	—
232	C₂×C₈	—	86	1	no	—
233	C₂×C₈	—	86	1	no	—
234	C₂×C₈	—	172	2	no	—
235	C₂×C₈	—	172	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	86	2	no	—
243	C₄⋊C₄	—	172	1	no	—
244	C₂²⋊C₄	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	172	1	no	—
247	C₄²	—	172	1	no	—
248	C₄²	#10621898	1	1	yes	characteristic
249	C₄²	D₄×D₄₃	1	1	yes	characteristic
250	C₄²	#10621900	1	1	yes	characteristic
251	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
252	C₂³	#13063311	1	1	yes	characteristic
253	Q₈	—	4	1	no	—
254	Q₈	—	4	1	no	—
255	Q₈	—	8	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	344	1	no	—
262	C₂×C₄	—	344	1	no	—
263	C₂×C₄	—	172	2	no	—
264	C₂×C₄	#13063284	1	1	yes	characteristic
265	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
266	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
267	C₂×C₄	#13063298	1	1	yes	characteristic
268	C₂×C₄	#13063307	1	1	yes	characteristic
269	C₂×C₄	#13063318	1	1	yes	characteristic
270	C₂×C₄	#13063352	1	1	yes	characteristic
271	C₂×C₄	#13063354	1	1	yes	characteristic
272	C₂×C₄	#13063357	1	1	yes	characteristic
273	C₂×C₄	#13063357	1	1	yes	characteristic
274	C₈	—	4	1	no	—
275	C₈	—	4	1	no	—
276	C₈	—	4	1	no	—
277	C₈	—	4	1	no	—
278	C₈	—	86	2	no	—
279	C₈	—	344	1	no	—
280	C₈	—	172	1	no	—
281	C₈	—	172	1	no	—
282	C₈	—	172	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	#14582792	1	1	yes	characteristic
286	C₂²	#14583260	1	1	yes	characteristic
287	C₂²	#14583266	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	8	1	no	—
292	C₄	—	8	1	no	—
293	C₄	—	344	2	no	—
294	C₄	#14583000	1	1	yes	characteristic
295	C₄	#14583006	1	1	yes	characteristic
296	C₄	#14583016	1	1	yes	characteristic
297	C₄	#14583021	1	1	yes	characteristic
298	C₄	#14583148	1	1	yes	characteristic
299	C₄	#14583156	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	#19135245	1	1	yes	characteristic
303	C₂	#19135266	1	1	yes	characteristic
304	C₂	#19135287	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063298
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1542	C₂
socle	C₂²×C₄₃	#388
prime radical	C₂³×C₄₃	#124
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1542	D₄₃
hypocenter	C₄₃	#9591
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1542

Complements

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

N	H ≅ G/N	type
23. C₄₃×#333	C₄	semidirect
24. C₄₃×#333	C₄	semidirect
25. C₄₃×#332	C₄	semidirect
26. C₄₃×#332	C₄	semidirect
171. #333	Dic₄₃	semidirect
172. #333	Dic₄₃	semidirect
173. #332	Dic₄₃	semidirect
174. #332	Dic₄₃	semidirect
179. C₄₃	#9591	semidirect

Maximal subgroups

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

subgroup	index
#19135540	2
#19135371	2
#19133050	2
#19132885	2
#19132497	2
#19132434	2
C₄₃×#1542	2
#9591	43