Subgroups — #24992108

Order 11008.

546 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143676	C₂	1	1	yes	maximal, characteristic, complemented
2	#19132986	C₂	1	2	yes	maximal
3	#19132919	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132424	C₂	1	2	yes	maximal, complemented
5	C₄₃×#2326	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₄₃⋊C₈	—	2	4	no	—
7	#14582952	—	2	2	no	—
8	#14582952	C₂²	1	2	yes	—
9	#14582952	C₂²	1	2	yes	—
10	D₄⋊(C₄₃⋊C₈)	—	2	4	no	—
11	#14582625	—	2	2	no	—
12	C₂×#124×C₄₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#391	C₄	1	1	yes	characteristic
16	C₄₃×#390	C₄	1	1	yes	characteristic
17	C₄₃×#388	—	2	2	no	—
18	C₄₃×#385	—	2	2	no	—
19	C₄₃×#384	C₄	1	1	yes	characteristic
20	C₄₃×#376	C₂²	1	1	yes	characteristic
21	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
26	C₄×C₄₃⋊C₈	—	2	4	no	—
27	C₄×C₄₃⋊C₈	—	2	2	no	—
28	C₄×C₄₃⋊C₈	D₄	1	4	yes	—
29	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₃	—	2	1	no	—
32	#124×C₄₃	—	2	2	no	—
33	#124×C₄₃	C₈	1	4	yes	complemented
34	C₄×Q₈×C₄₃	—	2	2	no	—
35	C₄×Q₈×C₄₃	—	2	2	no	—
36	C₄×D₄×C₄₃	—	2	2	no	—
37	C₄×D₄×C₄₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
45	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	D₄	1	2	yes	—
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	#95×C₄₃	—	2	2	no	—
50	#95×C₄₃	—	2	1	no	—
51	#95×C₄₃	—	2	1	no	—
52	C₂⁴×C₄₃	—	2	1	no	—
53	C₂×Q₈×C₄₃	—	2	2	no	—
54	C₂×Q₈×C₄₃	—	2	2	no	—
55	C₂×Q₈×C₄₃	C₂×C₈	1	2	yes	—
56	C₂×Q₈×C₄₃	M₁₆	1	2	yes	—
57	C₂×D₄×C₄₃	—	2	1	no	—
58	C₂×D₄×C₄₃	—	2	2	no	—
59	C₂×D₄×C₄₃	—	2	2	no	—
60	C₂×D₄×C₄₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₄₃	—	2	1	no	—
62	C₂²×C₄×C₄₃	—	2	1	no	—
63	C₂²×C₄×C₄₃	—	2	1	no	—
64	C₂²×C₄×C₄₃	—	4	2	no	—
65	C₂²×C₄×C₄₃	—	8	2	no	—
66	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₄₃	—	2	2	no	—
72	C₄⋊C₄×C₄₃	—	2	4	no	—
73	C₄⋊C₄×C₄₃	—	2	2	no	—
74	C₄⋊C₄×C₄₃	—	8	2	no	—
75	C₂²⋊C₄×C₄₃	—	2	8	no	—
76	C₂²⋊C₄×C₄₃	—	4	1	no	—
77	C₂²⋊C₄×C₄₃	—	4	1	no	—
78	C₂²⋊C₄×C₄₃	—	8	2	no	—
79	C₄²×C₄₃	—	2	2	no	—
80	C₄²×C₄₃	—	2	2	no	—
81	C₄²×C₄₃	—	2	1	no	—
82	C₄²×C₄₃	—	2	2	no	—
83	C₄²×C₄₃	C₂×C₈	1	2	yes	—
84	C₄²×C₄₃	C₂×D₄	1	2	yes	—
85	C₄²×C₄₃	#41	1	2	yes	—
86	C₂×C₄₃⋊C₈	—	2	4	no	—
87	C₂×C₄₃⋊C₈	—	4	2	no	—
88	C₂³×C₄₃	—	2	2	no	—
89	C₂³×C₄₃	—	4	2	no	—
90	C₂³×C₄₃	—	4	2	no	—
91	C₂³×C₄₃	—	4	1	no	—
92	C₂³×C₄₃	—	4	1	no	—
93	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	Q₈×C₄₃	—	2	4	no	—
95	Q₈×C₄₃	—	2	4	no	—
96	D₄×C₄₃	—	2	4	no	—
97	D₄×C₄₃	—	4	2	no	—
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	—	4	1	no	—
103	C₂×C₄×C₄₃	—	4	1	no	—
104	C₂×C₄×C₄₃	—	4	1	no	—
105	C₂×C₄×C₄₃	—	4	1	no	—
106	C₂×C₄×C₄₃	—	4	2	no	—
107	C₂×C₄×C₄₃	—	4	1	no	—
108	C₂×C₄×C₄₃	—	4	2	no	—
109	C₂×C₄×C₄₃	—	4	1	no	—
110	C₂×C₄×C₄₃	—	4	1	no	—
111	C₂×C₄×C₄₃	—	4	1	no	—
112	C₂×C₄×C₄₃	—	8	4	no	—
113	C₂×C₄×C₄₃	C₂²⋊C₈	1	4	yes	—
114	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
116	C₂×C₄×C₄₃	#121	1	2	yes	—
117	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#131	1	2	yes	—
121	C₄₃⋊C₈	—	4	4	no	—
122	C₄₃⋊C₈	—	8	2	no	—
123	#9102	—	43	1	no	maximal, Sylow
124	C₂²×C₄₃	—	2	1	no	—
125	C₂²×C₄₃	—	2	1	no	—
126	C₂²×C₄₃	—	4	4	no	—
127	C₂²×C₄₃	—	4	2	no	—
128	C₂²×C₄₃	—	8	2	no	—
129	C₂²×C₄₃	—	8	1	no	—
130	C₂²×C₄₃	—	8	1	no	—
131	C₂²×C₄₃	#386	1	1	yes	characteristic
132	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₂²×C₄₃	#408	1	1	yes	characteristic
134	C₄×C₄₃	—	2	4	no	—
135	C₄×C₄₃	—	2	2	no	—
136	C₄×C₄₃	—	2	1	no	—
137	C₄×C₄₃	—	2	1	no	—
138	C₄×C₄₃	—	4	2	no	—
139	C₄×C₄₃	—	4	2	no	—
140	C₄×C₄₃	—	8	1	no	—
141	C₄×C₄₃	—	8	1	no	—
142	C₄×C₄₃	C₈×D₄	1	2	yes	—
143	C₄×C₄₃	#435	1	2	yes	—
144	#2982	—	43	1	no	—
145	#2326	D₄₃	1	1	yes	characteristic
146	#1478	—	43	2	no	—
147	#1461	—	43	1	no	—
148	#1274	—	43	2	no	—
149	C₂×C₄₃	—	2	1	no	—
150	C₂×C₄₃	—	2	1	no	—
151	C₂×C₄₃	—	8	2	no	—
152	C₂×C₄₃	#1869	1	1	yes	characteristic
153	C₂×C₄₃	#1910	1	1	yes	characteristic
154	C₂×C₄₃	#1912	1	1	yes	characteristic
155	C₂×#124	Dic₄₃	1	1	yes	characteristic
156	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
157	C₂×C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
158	C₈×Q₈	—	86	4	no	—
159	#431	—	43	2	no	—
160	#431	—	43	2	no	—
161	#431	—	86	2	no	—
162	#391	Dic₄₃	1	1	yes	characteristic
163	#390	Dic₄₃	1	1	yes	characteristic
164	#388	—	2	2	no	—
165	#385	—	2	2	no	—
166	#384	Dic₄₃	1	1	yes	characteristic
167	#376	C₂×D₄₃	1	1	yes	characteristic
168	D₄⋊C₈	—	86	4	no	—
169	#324	—	86	2	no	—
170	C₄₃	#9102	1	1	yes	characteristic, complemented, Sylow
171	C₂²×Q₈	C₂×Dic₄₃	1	1	yes	characteristic
172	C₂²×D₄	C₂×Dic₄₃	1	1	yes	characteristic
173	C₂³×C₄	—	2	1	no	—
174	#124	—	2	2	no	—
175	#124	C₄₃⋊C₈	1	4	yes	complemented
176	C₄×Q₈	—	2	2	no	—
177	C₄×Q₈	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂²⋊D₄₃	1	2	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₂²⋊C₄	—	2	1	no	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
188	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
190	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
191	C₄⋊C₈	—	86	4	no	—
192	C₄⋊C₈	—	86	2	no	—
193	C₄⋊C₈	—	86	2	no	—
194	C₂²⋊C₈	—	86	2	no	—
195	C₂²⋊C₈	—	86	2	no	—
196	C₄×C₈	—	43	4	no	—
197	C₄×C₈	—	86	4	no	—
198	C₄×C₈	—	86	2	no	—
199	#95	—	2	2	no	—
200	#95	—	2	1	no	—
201	#95	—	2	1	no	—
202	C₂⁴	—	2	1	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	C₂×C₄₃⋊C₈	1	2	yes	—
206	C₂×Q₈	#10621877	1	2	yes	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₂×C₄₃⋊C₈	1	2	yes	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	8	2	no	—
216	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
217	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
221	C₂×C₈	—	86	4	no	—
222	C₂×C₈	—	172	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	8	2	no	—
227	C₂²⋊C₄	—	2	8	no	—
228	C₂²⋊C₄	—	4	1	no	—
229	C₂²⋊C₄	—	4	1	no	—
230	C₂²⋊C₄	—	8	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	2	no	—
235	C₄²	C₂×C₄₃⋊C₈	1	2	yes	—
236	C₄²	D₄×D₄₃	1	2	yes	—
237	C₄²	#10621900	1	2	yes	—
238	C₂³	—	2	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	4	1	no	—
242	C₂³	—	4	1	no	—
243	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
244	Q₈	—	2	4	no	—
245	Q₈	—	2	4	no	—
246	D₄	—	2	4	no	—
247	D₄	—	4	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	8	4	no	—
263	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	4	yes	—
264	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
265	C₂×C₄	#13063293	1	1	yes	characteristic
266	C₂×C₄	#13063297	1	2	yes	—
267	C₂×C₄	#13063306	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
269	C₂×C₄	C₂×#10621877	1	1	yes	characteristic
270	C₂×C₄	#13063332	1	2	yes	—
271	C₈	—	344	2	no	—
272	C₈	—	172	4	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	4	4	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	8	1	no	—
279	C₂²	—	8	1	no	—
280	C₂²	#14582779	1	1	yes	characteristic
281	C₂²	C₂×C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
282	C₂²	#14582866	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	8	1	no	—
290	C₄	—	8	1	no	—
291	C₄	D₄×C₄₃⋊C₈	1	2	yes	—
292	C₄	#14582971	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	8	2	no	—
296	C₂	#19134888	1	1	yes	characteristic
297	C₂	#19135123	1	1	yes	characteristic
298	C₂	#19135133	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14582779
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#2326	C₂
socle	C₂²×C₄₃	#386
prime radical	C₂²×D₄×C₄₃	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#2326	D₄₃
hypocenter	C₄₃	#9102
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2326

Complements

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

N	H ≅ G/N	type
1. #19143676	C₂	semidirect
3. #19132919	C₂	semidirect
4. #19132424	C₂	semidirect
33. #124×C₄₃	C₈	semidirect
170. C₄₃	#9102	semidirect
175. #124	C₄₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#19143676	2
#19132986	2
#19132919	2
#19132424	2
C₄₃×#2326	2
#9102	43