Subgroups — #25105378

Order 11008.

750 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137344	C₂	1	2	yes	maximal
2	#19135569	C₂	1	2	yes	maximal
3	#19135334	C₂	1	2	yes	maximal
4	#19135329	C₂	1	2	yes	maximal
5	#19134759	C₂	1	1	yes	maximal, characteristic
6	#19134494	C₂	1	2	yes	maximal
7	#107×Dic₄₃	C₂	1	2	yes	maximal
8	#19134281	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₃×#500	C₂	1	1	yes	maximal, characteristic
10	C₂×#13063259	C₂²	1	2	yes	—
11	C₂×#13063257	C₂²	1	2	yes	—
12	C₂×#13063236	C₂²	1	1	yes	characteristic
13	C₂×#13063235	C₂²	1	1	yes	characteristic
14	C₂×#13063235	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
17	#14582806	—	2	4	no	—
18	#14582798	C₂²	1	4	yes	—
19	#14582785	—	2	4	no	—
20	#14582765	C₂²	1	2	yes	—
21	C₄⋊C₄×Dic₄₃	C₂²	1	4	yes	—
22	#14582745	C₂²	1	1	yes	characteristic
23	#14582661	C₂²	1	4	yes	—
24	#14582661	C₂²	1	4	yes	—
25	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
26	C₄₃×#500	C₄	1	8	yes	complemented
27	C₂×#107×C₄₃	C₂²	1	2	yes	—
28	C₂×#107×C₄₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₄₃	—	2	4	no	—
33	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
34	C₂×C₄×Dic₄₃	—	2	4	no	—
35	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
36	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
37	#13063259	—	2	4	no	—
38	#13063257	—	2	4	no	—
39	#13063236	—	2	2	no	—
40	#13063235	—	2	2	no	—
41	#13063235	Q₈	1	4	yes	complemented
42	C₈×Dic₄₃	—	2	2	no	—
43	C₈×Dic₄₃	Q₈	1	4	yes	complemented
44	C₄₃⋊#95	—	2	2	no	—
45	C₄₃⋊#95	—	2	2	no	—
46	C₄₃⋊#95	—	2	4	no	—
47	C₄₃⋊#95	—	2	4	no	—
48	C₄₃⋊#95	D₄	1	2	yes	—
49	C₄₃⋊#95	D₄	1	2	yes	—
50	C₂²×Q₈×C₄₃	—	2	2	no	—
51	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₄₃	—	2	4	no	—
54	C₄⋊Q₈×C₄₃	C₂×C₄	1	8	yes	—
55	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	#107×C₄₃	C₂×C₄	1	8	yes	—
59	#107×C₄₃	C₂×C₄	1	8	yes	—
60	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
61	C₂×Q₈×C₄₃	—	2	8	no	—
62	C₂×Q₈×C₄₃	—	8	4	no	—
63	C₂²×C₄×C₄₃	—	2	4	no	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₃	—	2	4	no	—
68	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
69	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₄₃	#41	1	2	yes	—
75	C₂×C₈×C₄₃	#41	1	2	yes	—
76	C₄⋊C₄×C₄₃	—	2	8	no	—
77	C₄⋊C₄×C₄₃	—	2	4	no	—
78	C₄⋊C₄×C₄₃	C₂²×C₄	1	8	yes	—
79	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₄₃	#41	1	2	yes	—
81	C₄⋊Dic₄₃	—	2	4	no	—
82	C₄⋊Dic₄₃	—	2	4	no	—
83	C₄⋊Dic₄₃	—	8	4	no	—
84	C₄×Dic₄₃	—	2	1	no	—
85	C₄×Dic₄₃	—	2	1	no	—
86	C₄×Dic₄₃	—	8	4	no	—
87	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
88	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
89	C₂×C₄₃⋊C₈	—	2	1	no	—
90	C₂×C₄₃⋊C₈	—	2	1	no	—
91	C₂×C₄₃⋊C₈	—	2	1	no	—
92	C₂×C₄₃⋊C₈	—	2	1	no	—
93	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
94	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
95	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₄₃	—	8	8	no	—
97	C₂×C₄×C₄₃	—	2	2	no	—
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	2	8	no	—
100	C₂×C₄×C₄₃	—	4	4	no	—
101	C₂×C₄×C₄₃	—	4	4	no	—
102	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
110	C₂×Dic₄₃	—	2	2	no	—
111	C₂×Dic₄₃	—	2	2	no	—
112	C₂×Dic₄₃	—	2	2	no	—
113	C₂×Dic₄₃	—	2	2	no	—
114	C₂×Dic₄₃	—	2	2	no	—
115	C₂×Dic₄₃	—	4	4	no	—
116	C₂×Dic₄₃	—	4	4	no	—
117	C₂×Dic₄₃	—	4	4	no	—
118	C₂×Dic₄₃	C₄⋊Q₈	1	4	yes	—
119	C₈×C₄₃	C₄×Q₈	1	4	yes	—
120	C₈×C₄₃	C₄×Q₈	1	4	yes	—
121	C₄₃⋊C₈	—	2	2	no	—
122	C₄₃⋊C₈	—	2	2	no	—
123	#21725	—	43	1	no	maximal, Sylow
124	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
126	C₂²×C₄₃	#532	1	1	yes	characteristic
127	C₂²×C₄₃	#576	1	2	yes	—
128	C₂²×C₄₃	#578	1	2	yes	—
129	C₄×C₄₃	—	2	2	no	—
130	C₄×C₄₃	—	2	2	no	—
131	C₄×C₄₃	—	4	8	no	—
132	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
134	C₄×C₄₃	#533	1	1	yes	characteristic
135	C₄×C₄₃	#533	1	1	yes	characteristic
136	Dic₄₃	—	2	4	no	—
137	Dic₄₃	—	4	2	no	—
138	Dic₄₃	—	8	4	no	—
139	C₂×#500	D₄₃	1	1	yes	characteristic
140	#2331	—	43	2	no	—
141	#1993	—	43	2	no	—
142	#1943	—	43	2	no	—
143	#1942	—	43	2	no	—
144	#1846	—	43	1	no	—
145	#1776	—	43	2	no	—
146	C₄×#107	—	43	2	no	—
147	#1726	—	43	1	no	—
148	C₂×C₄₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
149	C₂×C₄₃	#2948	1	1	yes	characteristic
150	C₂×C₄₃	#2950	1	1	yes	characteristic
151	C₂×C₄₃	#3164	1	2	yes	—
152	C₂×C₄₃	#3175	1	2	yes	—
153	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
154	#500	Dic₄₃	1	8	yes	complemented
155	C₂×#107	—	43	2	no	—
156	C₂×#107	C₂×D₄₃	1	2	yes	—
157	C₂×#107	C₂×D₄₃	1	2	yes	—
158	C₂×#106	—	43	2	no	—
159	C₂×#97	—	43	1	no	—
160	C₂×#97	—	43	1	no	—
161	C₂×#97	—	43	1	no	—
162	C₂×C₄×C₈	—	43	1	no	—
163	C₂×C₄×C₈	—	43	1	no	—
164	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
165	#391	—	86	4	no	—
166	#389	—	43	4	no	—
167	#387	—	86	4	no	—
168	#382	—	43	2	no	—
169	C₄×C₄⋊C₄	—	43	4	no	—
170	#376	—	43	1	no	—
171	#340	—	43	4	no	—
172	#340	—	43	4	no	—
173	C₄₃	#21725	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₈	—	2	2	no	—
175	C₂²×C₈	—	43	1	no	—
176	C₂²×C₈	—	43	1	no	—
177	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
179	C₄⋊Q₈	—	2	4	no	—
180	C₄⋊Q₈	C₂×Dic₄₃	1	8	yes	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	43	4	no	—
183	C₂×C₄⋊C₄	—	86	4	no	—
184	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
185	C₂×C₄²	—	43	1	no	—
186	C₂×C₄²	—	43	1	no	—
187	C₂×C₄²	—	86	4	no	—
188	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
189	#107	—	86	4	no	—
190	#107	C₂×Dic₄₃	1	8	yes	—
191	#107	C₂×Dic₄₃	1	8	yes	—
192	#106	—	86	4	no	—
193	#97	—	43	4	no	—
194	#97	—	86	2	no	—
195	#97	—	86	2	no	—
196	C₄×C₈	—	43	4	no	—
197	C₄×C₈	—	86	2	no	—
198	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
199	#95	—	43	2	no	—
200	#95	—	43	2	no	—
201	#95	—	86	4	no	—
202	#95	—	86	4	no	—
203	#95	—	86	2	no	—
204	#95	—	86	2	no	—
205	C₂×Q₈	—	2	8	no	—
206	C₂×Q₈	—	8	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	43	2	no	—
209	C₂²×C₄	—	43	2	no	—
210	C₂²×C₄	—	86	4	no	—
211	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
212	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
213	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
214	C₂×C₈	—	43	2	no	—
215	C₂×C₈	—	43	2	no	—
216	C₂×C₈	—	86	1	no	—
217	C₂×C₈	—	86	1	no	—
218	C₂×C₈	—	86	1	no	—
219	C₂×C₈	—	86	1	no	—
220	C₂×C₈	Q₈×D₄₃	1	2	yes	—
221	C₂×C₈	Q₈×D₄₃	1	2	yes	—
222	C₂×C₈	#10621902	1	2	yes	—
223	C₂×C₈	#10621902	1	2	yes	—
224	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
225	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
226	C₄⋊C₄	—	2	8	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	86	4	no	—
229	C₄⋊C₄	—	86	4	no	—
230	C₄⋊C₄	—	344	4	no	—
231	C₄⋊C₄	C₂²×Dic₄₃	1	8	yes	—
232	C₄²	—	43	2	no	—
233	C₄²	—	43	2	no	—
234	C₄²	—	86	1	no	—
235	C₄²	—	86	1	no	—
236	C₄²	—	344	4	no	—
237	C₄²	#10621900	1	2	yes	—
238	C₄²	C₂²×Dic₄₃	1	2	yes	—
239	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
240	Q₈	—	8	8	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	43	4	no	—
247	C₂×C₄	—	86	2	no	—
248	C₂×C₄	—	86	2	no	—
249	C₂×C₄	—	86	2	no	—
250	C₂×C₄	—	86	2	no	—
251	C₂×C₄	—	86	2	no	—
252	C₂×C₄	—	172	4	no	—
253	C₂×C₄	—	172	4	no	—
254	C₂×C₄	—	172	4	no	—
255	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
256	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
257	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
258	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
259	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
260	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
261	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
262	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
263	C₈	—	86	2	no	—
264	C₈	—	86	2	no	—
265	C₈	Q₈×Dic₄₃	1	4	yes	—
266	C₈	Q₈×Dic₄₃	1	4	yes	—
267	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
268	C₂²	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
269	C₂²	#14583450	1	1	yes	characteristic
270	C₂²	#14583700	1	2	yes	—
271	C₂²	#14583712	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	8	no	—
275	C₄	—	86	4	no	—
276	C₄	—	344	4	no	—
277	C₄	—	172	2	no	—
278	C₄	C₂×Q₈×Dic₄₃	1	1	yes	characteristic
279	C₄	C₂×Q₈×Dic₄₃	1	1	yes	characteristic
280	C₄	#14583454	1	1	yes	characteristic
281	C₄	#14583454	1	1	yes	characteristic
282	C₂	C₄⋊Q₈×Dic₄₃	1	1	yes	characteristic
283	C₂	#19143447	1	1	yes	characteristic
284	C₂	#19143463	1	1	yes	characteristic
285	C₂	#19145128	1	2	yes	—
286	C₂	#19145237	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#500	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#500	D₄₃
hypocenter	C₄₃	#21725
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
26. C₄₃×#500	C₄	semidirect
41. #13063235	Q₈	semidirect
43. C₈×Dic₄₃	Q₈	semidirect
154. #500	Dic₄₃	semidirect
173. C₄₃	#21725	semidirect

Maximal subgroups

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

subgroup	index
#19137344	2
#19135569	2
#19135334	2
#19135329	2
#19134759	2
#19134494	2
#107×Dic₄₃	2
#19134281	2
C₂×C₄₃×#500	2
#21725	43