Subgroups — #25096969

Order 11008.

608 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143021	C₂	1	1	yes	maximal, characteristic
2	#19134742	C₂	1	2	yes	maximal, complemented
3	#19134718	C₂	1	2	yes	maximal, complemented
4	#19134696	C₂	1	2	yes	maximal
5	#19134645	C₂	1	1	yes	maximal, characteristic
6	#19134481	C₂	1	2	yes	maximal, complemented
7	#19134397	C₂	1	2	yes	maximal
8	#19134349	C₂	1	2	yes	maximal, complemented
9	C₄₃×#1812	C₂	1	1	yes	maximal, characteristic
10	C₂×#13063332	C₂²	1	1	yes	characteristic
11	#41×Dic₄₃	C₂²	1	1	yes	characteristic
12	C₂×#13063261	C₂²	1	2	yes	—
13	#14582883	C₂²	1	2	yes	—
14	#14582857	C₄	1	2	yes	complemented
15	#14582857	C₂²	1	1	yes	characteristic
16	#14582855	C₂²	1	4	yes	—
17	#14582854	C₄	1	2	yes	complemented
18	#14582854	C₂²	1	1	yes	characteristic
19	M₁₆×Dic₄₃	C₄	1	2	yes	complemented
20	M₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
21	C₂×#13063235	C₄	1	2	yes	complemented
22	C₂×#13063235	C₂²	1	1	yes	characteristic
23	#14582669	C₂²	1	4	yes	—
24	#14582658	C₂²	1	4	yes	—
25	C₄₃⋊#338	C₂²	1	2	yes	—
26	#14582655	C₂²	1	4	yes	—
27	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#427	C₂²	1	2	yes	—
29	C₄₃×#410	C₂²	1	2	yes	—
30	C₄₃×H(3,ℤ/4)	C₂²	1	2	yes	—
31	#13063332	—	2	4	no	—
32	C₂×#10621877	C₂×C₄	1	2	yes	—
33	C₂×#10621877	C₂³	1	4	yes	—
34	C₂×#10621877	C₂³	1	1	yes	characteristic
35	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
37	Q₈×Dic₄₃	—	2	1	no	—
38	D₄×Dic₄₃	—	2	2	no	—
39	D₄×Dic₄₃	—	2	1	no	—
40	#13063284	C₂×C₄	1	2	yes	—
41	#13063284	C₂³	1	1	yes	characteristic
42	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
44	#13063261	—	2	2	no	—
45	#13063261	—	2	2	no	—
46	#13063244	—	2	2	no	—
47	#13063242	—	2	2	no	—
48	#13063236	—	2	4	no	—
49	#13063235	C₂×C₄	1	1	yes	characteristic
50	#13063235	C₂×C₄	1	2	yes	—
51	#13063235	C₂×C₄	1	2	yes	—
52	#13063235	C₂×C₄	1	2	yes	—
53	#13063235	C₂×C₄	1	1	yes	characteristic
54	#13063235	C₂×C₄	1	1	yes	characteristic
55	#13063235	C₂×C₄	1	1	yes	characteristic
56	C₄×C₄₃⋊C₈	—	2	4	no	—
57	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
58	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
59	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
60	C₂×#41×C₄₃	C₂³	1	1	yes	characteristic
61	#131×C₄₃	—	2	2	no	—
62	#131×C₄₃	—	2	2	no	—
63	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
64	C₂×M₁₆×C₄₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
66	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
67	#117×C₄₃	C₂³	1	4	yes	—
68	#107×C₄₃	—	2	2	no	—
69	#106×C₄₃	—	2	2	no	—
70	#99×C₄₃	—	2	4	no	—
71	#41×C₄₃	—	2	2	no	—
72	#41×C₄₃	—	2	2	no	—
73	C₂×Q₈×C₄₃	#41	1	1	yes	characteristic
74	C₂×D₄×C₄₃	#41	1	1	yes	characteristic
75	C₂×D₄×C₄₃	#41	1	2	yes	—
76	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₄₃	—	2	2	no	—
79	C₂²×Dic₄₃	—	2	1	no	—
80	M₁₆×C₄₃	—	2	2	no	—
81	M₁₆×C₄₃	—	2	4	no	—
82	C₂×C₈×C₄₃	—	2	2	no	—
83	C₂×C₈×C₄₃	—	2	2	no	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	#41	1	2	yes	—
88	C₂×C₈×C₄₃	#41	1	2	yes	—
89	C₄⋊C₄×C₄₃	—	2	4	no	—
90	C₂²⋊C₄×C₄₃	—	2	4	no	—
91	C₄²×C₄₃	—	2	4	no	—
92	C₂²⋊Dic₄₃	—	2	2	no	—
93	C₂²⋊Dic₄₃	—	2	1	no	—
94	C₄⋊Dic₄₃	—	2	2	no	—
95	C₄⋊Dic₄₃	—	2	1	no	—
96	C₄×Dic₄₃	—	2	2	no	—
97	C₄×Dic₄₃	—	2	1	no	—
98	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
99	C₄×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
100	C₄×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
101	#10621877	—	2	2	no	—
102	#10621877	—	2	4	no	—
103	#10621877	—	2	4	no	—
104	#10621877	—	2	4	no	—
105	C₂×C₄₃⋊C₈	—	2	4	no	—
106	C₂×C₄₃⋊C₈	—	2	4	no	—
107	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
108	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄₃⋊C₈	#41	1	4	yes	—
111	C₂³×C₄₃	#117	1	2	yes	—
112	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
113	Q₈×C₄₃	—	2	2	no	—
114	D₄×C₄₃	—	2	4	no	—
115	D₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	2	2	no	—
117	C₂×C₄×C₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	—	4	4	no	—
119	C₂×C₄×C₄₃	#117	1	2	yes	—
120	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
122	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
123	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	2	2	no	—
126	C₂×Dic₄₃	—	2	1	no	—
127	C₂×Dic₄₃	—	8	1	no	—
128	C₂×Dic₄₃	—	8	2	no	—
129	C₈×C₄₃	—	2	2	no	—
130	C₈×C₄₃	#117	1	1	yes	characteristic
131	C₈×C₄₃	#117	1	1	yes	characteristic
132	C₈×C₄₃	#117	1	2	yes	—
133	C₄₃⋊C₈	—	2	2	no	—
134	C₄₃⋊C₈	—	4	4	no	—
135	C₄₃⋊C₈	#117	1	2	yes	—
136	C₄₃⋊C₈	#117	1	2	yes	—
137	#20788	—	43	1	no	maximal, Sylow
138	C₂²×C₄₃	—	2	2	no	—
139	C₂²×C₄₃	—	4	1	no	—
140	C₂²×C₄₃	—	4	2	no	—
141	C₂²×C₄₃	C₂×#117	1	1	yes	characteristic
142	C₂²×C₄₃	#529	1	2	yes	—
143	C₄×C₄₃	—	2	2	no	—
144	C₄×C₄₃	—	8	4	no	—
145	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
146	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
147	C₄×C₄₃	#529	1	2	yes	—
148	Dic₄₃	—	4	2	no	—
149	Dic₄₃	—	4	1	no	—
150	Dic₄₃	—	4	1	no	—
151	#2875	—	43	1	no	—
152	#1841	—	43	2	no	—
153	#1833	—	43	2	no	—
154	#1827	—	43	2	no	—
155	#1813	—	43	1	no	—
156	#1812	D₄₃	1	1	yes	characteristic
157	#1773	—	43	2	no	—
158	#1757	—	43	2	no	—
159	#1747	—	43	2	no	—
160	C₂×C₄₃	—	2	1	no	—
161	C₂×C₄₃	—	2	2	no	—
162	C₂×C₄₃	—	4	2	no	—
163	C₂×C₄₃	#2283	1	1	yes	characteristic
164	C₂×#131	—	43	1	no	—
165	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
166	C₄×#41	—	43	1	no	—
167	C₂×#108	—	43	2	no	—
168	#427	C₂×D₄₃	1	2	yes	—
169	#413	—	43	2	no	—
170	#410	C₂×D₄₃	1	2	yes	—
171	#405	—	43	1	no	—
172	#405	—	43	2	no	—
173	#405	—	43	1	no	—
174	#405	—	43	2	no	—
175	#405	—	43	4	no	—
176	C₄×M₁₆	—	43	1	no	—
177	C₄×M₁₆	—	43	2	no	—
178	C₂×#97	—	43	1	no	—
179	C₂×#97	—	43	2	no	—
180	#344	—	43	4	no	—
181	#339	—	43	4	no	—
182	#338	—	43	2	no	—
183	H(3,ℤ/4)	—	43	4	no	—
184	H(3,ℤ/4)	C₂×D₄₃	1	2	yes	—
185	C₄₃	#20788	1	1	yes	characteristic, complemented, Sylow
186	C₂×#41	C₂²×D₄₃	1	1	yes	characteristic
187	#131	—	2	2	no	—
188	#131	—	2	2	no	—
189	#131	—	86	4	no	—
190	C₂×M₁₆	—	43	2	no	—
191	C₂×M₁₆	—	43	1	no	—
192	C₂×M₁₆	—	43	4	no	—
193	C₂×M₁₆	C₄×D₄₃	1	2	yes	—
194	C₂×M₁₆	C₂²×D₄₃	1	1	yes	characteristic
195	C₂²×C₈	—	43	1	no	—
196	C₂²×C₈	—	43	2	no	—
197	C₂²×C₈	C₄×D₄₃	1	2	yes	—
198	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
199	C₄×Q₈	—	86	1	no	—
200	C₄×D₄	—	86	2	no	—
201	C₄×D₄	—	86	1	no	—
202	#117	—	43	2	no	—
203	#117	—	43	1	no	—
204	#117	C₂²×D₄₃	1	4	yes	—
205	C₂×C₄²	—	43	2	no	—
206	C₂×C₄²	—	43	1	no	—
207	#108	—	86	2	no	—
208	#108	—	86	2	no	—
209	#107	—	2	2	no	—
210	#106	—	2	2	no	—
211	#101	—	86	2	no	—
212	#100	—	86	2	no	—
213	#99	—	2	4	no	—
214	#97	—	43	1	no	—
215	#97	—	43	1	no	—
216	#97	—	43	2	no	—
217	#97	—	43	1	no	—
218	#97	—	43	2	no	—
219	#97	—	43	1	no	—
220	#97	—	43	2	no	—
221	#97	—	86	4	no	—
222	C₄×C₈	—	43	2	no	—
223	C₄×C₈	—	43	2	no	—
224	C₄×C₈	—	43	2	no	—
225	C₄×C₈	—	86	4	no	—
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	C₂×Q₈	#10621902	1	1	yes	characteristic
229	C₂×D₄	#10621900	1	2	yes	—
230	C₂×D₄	#10621902	1	1	yes	characteristic
231	C₂²×C₄	—	86	2	no	—
232	C₂²×C₄	—	86	1	no	—
233	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
234	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	86	4	no	—
238	M₁₆	—	86	2	no	—
239	M₁₆	—	86	4	no	—
240	M₁₆	—	86	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	43	1	no	—
244	C₂×C₈	—	43	1	no	—
245	C₂×C₈	—	43	4	no	—
246	C₂×C₈	—	43	2	no	—
247	C₂×C₈	—	86	4	no	—
248	C₂×C₈	—	86	4	no	—
249	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
251	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
252	C₂×C₈	#10621900	1	2	yes	—
253	C₂×C₈	#10621900	1	2	yes	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	86	2	no	—
256	C₄⋊C₄	—	86	1	no	—
257	C₂²⋊C₄	—	2	4	no	—
258	C₂²⋊C₄	—	86	2	no	—
259	C₂²⋊C₄	—	86	1	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	43	2	no	—
262	C₄²	—	43	1	no	—
263	C₄²	—	43	1	no	—
264	C₄²	—	86	2	no	—
265	C₄²	—	86	1	no	—
266	C₂³	#13063283	1	2	yes	—
267	C₂³	C₂×#10621900	1	1	yes	characteristic
268	Q₈	—	2	2	no	—
269	D₄	—	2	4	no	—
270	D₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	86	1	no	—
275	C₂×C₄	—	86	2	no	—
276	C₂×C₄	—	86	1	no	—
277	C₂×C₄	—	344	1	no	—
278	C₂×C₄	—	344	2	no	—
279	C₂×C₄	#13063283	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
281	C₂×C₄	#41×D₄₃	1	2	yes	—
282	C₂×C₄	C₂×#10621900	1	2	yes	—
283	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	43	2	no	—
286	C₈	—	43	2	no	—
287	C₈	—	86	2	no	—
288	C₈	—	172	4	no	—
289	C₈	#13063281	1	1	yes	characteristic
290	C₈	#13063281	1	1	yes	characteristic
291	C₈	#13063281	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₂×#13063283	1	1	yes	characteristic
296	C₂²	#14583430	1	2	yes	—
297	C₄	—	2	2	no	—
298	C₄	—	8	4	no	—
299	C₄	—	172	1	no	—
300	C₄	—	172	2	no	—
301	C₄	—	172	1	no	—
302	C₄	#117×D₄₃	1	1	yes	characteristic
303	C₄	C₂×#13063281	1	1	yes	characteristic
304	C₄	#14583435	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	1	no	—
307	C₂	—	4	2	no	—
308	C₂	#19136990	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1812	C₂
socle	C₂×C₄₃	#2283
prime radical	C₂×#41×C₄₃	C₂³
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1812	D₄₃
hypocenter	C₄₃	#20788
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #1812

Complements

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

N	H ≅ G/N	type
2. #19134742	C₂	semidirect
3. #19134718	C₂	semidirect
6. #19134481	C₂	semidirect
8. #19134349	C₂	semidirect
14. #14582857	C₄	semidirect
17. #14582854	C₄	semidirect
19. M₁₆×Dic₄₃	C₄	semidirect
21. C₂×#13063235	C₄	semidirect
185. C₄₃	#20788	semidirect

Maximal subgroups

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

subgroup	index
#19143021	2
#19134742	2
#19134718	2
#19134696	2
#19134645	2
#19134481	2
#19134397	2
#19134349	2
C₄₃×#1812	2
#20788	43