Subgroups — #25104802

Order 11008.

730 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144943	C₂	1	1	yes	maximal, characteristic
2	#19143530	C₂	1	1	yes	maximal, characteristic, complemented
3	#19143306	C₂	1	1	yes	maximal, characteristic
4	#19135427	C₂	1	1	yes	maximal, characteristic
5	#19135300	C₂	1	4	yes	maximal, complemented
6	#19134735	C₂	1	1	yes	maximal, characteristic, complemented
7	#19134436	C₂	1	4	yes	maximal, complemented
8	#19134282	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#1995	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13063333	C₂²	1	2	yes	—
11	#14583449	C₂²	1	1	yes	characteristic
12	#14583269	C₄	1	2	yes	complemented
13	#14583265	C₄	1	2	yes	complemented
14	#14583261	C₄	1	4	yes	complemented
15	#14583030	—	2	2	no	—
16	Q₈×C₄₃⋊C₈	—	2	2	no	—
17	#14582976	—	2	2	no	—
18	#14582964	—	2	2	no	—
19	#14582926	C₂²	1	2	yes	—
20	#14582923	C₂²	1	1	yes	characteristic
21	C₂×#13063253	C₂²	1	4	yes	—
22	#14582893	C₂²	1	4	yes	—
23	#14582867	C₂²	1	2	yes	—
24	#14582855	C₂²	1	2	yes	—
25	C₂×#13063236	C₂²	1	2	yes	—
26	C₂×#13063236	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
28	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
29	#14582663	C₂²	1	4	yes	—
30	C₄₃×#533	C₂²	1	1	yes	characteristic
31	C₄₃×#423	C₂²	1	1	yes	characteristic
32	C₄₃×#376	C₂²	1	1	yes	characteristic
33	C₄₃×#339	C₂²	1	4	yes	—
34	C₂×#10621898	C₂³	1	2	yes	—
35	#13063333	—	2	8	no	—
36	C₂×#10621877	C₂³	1	2	yes	—
37	C₂×#10621872	—	2	2	no	—
38	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
39	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
41	C₂×C₈×D₄₃	—	2	2	no	—
42	#13063325	C₂×C₄	1	1	yes	characteristic
43	#13063322	C₂×C₄	1	1	yes	characteristic
44	#13063313	C₂×C₄	1	2	yes	—
45	#13063299	—	2	2	no	—
46	C₄×D₁₇₂	—	2	2	no	—
47	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
48	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
49	#13063253	—	2	8	no	—
50	#13063248	C₂×C₄	1	8	yes	—
51	#13063245	C₂×C₄	1	8	yes	—
52	(C₂×D₄₃)⋊C₈	—	2	4	no	—
53	#13063236	—	2	2	no	—
54	#13063236	—	2	4	no	—
55	#13063236	C₂×C₄	1	2	yes	—
56	#13063236	C₂×C₄	1	2	yes	—
57	C₄×C₄₃⋊C₈	—	2	4	no	—
58	C₄×C₄₃⋊C₈	—	2	2	no	—
59	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
60	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
61	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₄₃	C₂×C₄	1	2	yes	—
63	#125×C₄₃	C₂×C₄	1	2	yes	—
64	C₂²⋊Q₈×C₄₃	—	2	2	no	—
65	C₄×Q₈×C₄₃	—	2	2	no	—
66	#117×C₄₃	C₂³	1	2	yes	—
67	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
68	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₄₃	—	2	2	no	—
70	#95×C₄₃	—	2	2	no	—
71	C₂×Q₈×C₄₃	—	2	2	no	—
72	C₂²×C₄×C₄₃	—	2	2	no	—
73	C₂²×C₄×C₄₃	#41	1	2	yes	—
74	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
75	C₂×C₂²⋊D₄₃	—	2	2	no	—
76	#10621898	—	2	4	no	—
77	#10621898	—	2	4	no	—
78	C₂×D₁₇₂	—	2	2	no	—
79	C₂×D₁₇₂	C₂²×C₄	1	2	yes	—
80	C₂×C₄×D₄₃	—	2	2	no	—
81	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
82	M₁₆×C₄₃	—	2	4	no	—
83	C₂×C₈×C₄₃	—	2	2	no	—
84	C₄⋊C₄×C₄₃	—	2	4	no	—
85	C₄⋊C₄×C₄₃	—	2	2	no	—
86	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
87	C₂²⋊C₄×C₄₃	—	2	2	no	—
88	C₄²×C₄₃	—	2	2	no	—
89	C₄²×C₄₃	—	2	2	no	—
90	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₄₃	C₂×D₄	1	2	yes	—
93	C₄²×C₄₃	#41	1	4	yes	—
94	(C₂×D₄₃)⋊C₄	—	2	4	no	—
95	C₄⋊Dic₄₃	—	2	2	no	—
96	#10621877	—	2	4	no	—
97	C₂×C₄₃⋊C₈	—	2	2	no	—
98	C₂×C₄₃⋊C₈	—	2	2	no	—
99	C₂×C₄₃⋊C₈	—	2	2	no	—
100	C₂×C₄₃⋊C₈	—	2	2	no	—
101	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
102	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
103	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄₃⋊C₈	#41	1	4	yes	—
106	#10621872	—	4	4	no	—
107	C₈×D₄₃	—	4	4	no	—
108	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
109	C₂²×D₄₃	—	2	2	no	—
110	Q₈×C₄₃	—	4	4	no	—
111	C₂×C₄×C₄₃	—	2	4	no	—
112	C₂×C₄×C₄₃	—	2	4	no	—
113	C₂×C₄×C₄₃	—	4	2	no	—
114	C₂×C₄×C₄₃	—	4	4	no	—
115	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
120	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
121	C₂²⋊D₄₃	—	4	4	no	—
122	C₂×Dic₄₃	—	2	2	no	—
123	D₁₇₂	—	2	4	no	—
124	D₁₇₂	—	4	4	no	—
125	C₄×D₄₃	—	4	2	no	—
126	C₄×D₄₃	—	4	2	no	—
127	Dic₈₆	—	2	4	no	—
128	C₈×C₄₃	—	4	2	no	—
129	C₄₃⋊C₈	—	2	4	no	—
130	C₄₃⋊C₈	—	2	2	no	—
131	C₄₃⋊C₈	—	4	2	no	—
132	C₄₃⋊C₈	#124	1	2	yes	—
133	C₄₃⋊C₈	#124	1	2	yes	—
134	#21663	—	43	1	no	maximal, Sylow
135	C₂²×C₄₃	—	2	2	no	—
136	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₄₃	#529	1	2	yes	—
138	C₂×D₄₃	—	4	4	no	—
139	C₂×D₄₃	—	4	2	no	—
140	C₄×C₄₃	—	2	2	no	—
141	C₄×C₄₃	—	2	2	no	—
142	C₄×C₄₃	—	4	4	no	—
143	C₄×C₄₃	—	4	4	no	—
144	C₄×C₄₃	C₄×#41	1	2	yes	—
145	C₄×C₄₃	C₂×#124	1	1	yes	characteristic
146	C₄×C₄₃	#532	1	1	yes	characteristic
147	Dic₄₃	—	4	2	no	—
148	#3135	—	43	1	no	—
149	#2960	—	43	1	no	—
150	#2931	—	43	1	no	—
151	#1995	D₄₃	1	1	yes	characteristic, complemented
152	#1963	—	43	1	no	—
153	#1937	—	43	4	no	—
154	#1838	—	43	1	no	—
155	#1765	—	43	4	no	—
156	#1726	—	43	1	no	—
157	C₂×C₄₃	—	2	2	no	—
158	C₂×C₄₃	C₄×#124	1	1	yes	characteristic
159	C₂×C₄₃	#2955	1	2	yes	—
160	D₄₃	—	8	2	no	—
161	C₂×#131	—	43	2	no	—
162	#533	C₂×D₄₃	1	1	yes	characteristic
163	#532	—	43	1	no	—
164	#491	—	43	2	no	—
165	#490	—	43	2	no	—
166	#489	—	43	4	no	—
167	#446	—	86	2	no	—
168	C₈×Q₈	—	86	2	no	—
169	#436	—	86	2	no	—
170	C₈×D₄	—	86	2	no	—
171	#424	—	43	2	no	—
172	#423	—	43	1	no	—
173	#423	C₂×D₄₃	1	1	yes	characteristic
174	C₂×#104	—	43	4	no	—
175	#416	—	43	4	no	—
176	#408	—	43	2	no	—
177	#405	—	43	2	no	—
178	C₂×#97	—	43	1	no	—
179	C₂×#97	—	43	2	no	—
180	C₂×C₄×C₈	—	43	1	no	—
181	C₂×C₄×C₈	—	43	2	no	—
182	#376	C₂×D₄₃	1	1	yes	characteristic
183	#341	—	43	4	no	—
184	#339	C₂×D₄₃	1	4	yes	—
185	C₄₃	#21663	1	1	yes	characteristic, complemented, Sylow
186	C₂×#41	—	43	2	no	—
187	#131	—	86	8	no	—
188	C₂×M₁₆	—	43	2	no	—
189	C₂×M₁₆	—	86	2	no	—
190	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
191	C₂²×C₈	—	43	1	no	—
192	C₂²×C₈	—	43	2	no	—
193	C₂²×C₈	—	43	1	no	—
194	C₂²×C₈	—	86	2	no	—
195	C₄⋊Q₈	—	43	1	no	—
196	C₄⋊Q₈	C₄×D₄₃	1	2	yes	—
197	#127	—	43	1	no	—
198	#125	C₄×D₄₃	1	2	yes	—
199	#124	—	43	2	no	—
200	C₂²⋊Q₈	—	2	2	no	—
201	#121	—	86	2	no	—
202	C₄×Q₈	—	2	2	no	—
203	C₄×D₄	—	86	2	no	—
204	#117	C₂²×D₄₃	1	2	yes	—
205	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
206	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	—	86	4	no	—
209	C₄⋊C₈	—	86	2	no	—
210	#104	—	86	8	no	—
211	#103	—	43	8	no	—
212	#102	—	43	8	no	—
213	C₂²⋊C₈	—	86	4	no	—
214	#97	—	43	2	no	—
215	#97	—	43	2	no	—
216	#97	—	86	4	no	—
217	#97	—	86	2	no	—
218	C₄×C₈	—	43	2	no	—
219	C₄×C₈	—	43	2	no	—
220	C₄×C₈	—	86	2	no	—
221	C₄×C₈	—	86	4	no	—
222	#95	—	2	2	no	—
223	#41	—	86	4	no	—
224	#41	—	86	4	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	43	2	no	—
227	C₂×D₄	—	43	2	no	—
228	C₂×D₄	—	86	2	no	—
229	C₂×D₄	—	86	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	86	2	no	—
232	C₂²×C₄	#10621900	1	2	yes	—
233	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
234	M₁₆	—	2	4	no	—
235	M₁₆	—	86	4	no	—
236	M₁₆	—	172	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	43	1	no	—
239	C₂×C₈	—	43	1	no	—
240	C₂×C₈	—	43	4	no	—
241	C₂×C₈	—	43	4	no	—
242	C₂×C₈	—	43	2	no	—
243	C₂×C₈	—	86	2	no	—
244	C₂×C₈	—	86	2	no	—
245	C₂×C₈	—	86	2	no	—
246	C₂×C₈	—	86	2	no	—
247	C₂×C₈	—	172	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	86	2	no	—
251	C₄⋊C₄	C₂×C₄×D₄₃	1	4	yes	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	—	86	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
257	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
258	C₄²	D₄×D₄₃	1	2	yes	—
259	C₄²	#10621902	1	4	yes	—
260	C₂³	—	86	2	no	—
261	C₂³	C₂×#10621900	1	1	yes	characteristic
262	Q₈	—	4	4	no	—
263	Q₈	—	86	4	no	—
264	D₄	—	86	4	no	—
265	D₄	—	172	4	no	—
266	D₄	—	172	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	86	2	no	—
272	C₂×C₄	—	172	2	no	—
273	C₂×C₄	—	172	2	no	—
274	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
275	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
276	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
278	C₂×C₄	#41×D₄₃	1	2	yes	—
279	C₂×C₄	C₂×#10621902	1	2	yes	—
280	C₈	—	4	2	no	—
281	C₈	—	43	2	no	—
282	C₈	—	43	2	no	—
283	C₈	—	86	4	no	—
284	C₈	—	86	2	no	—
285	C₈	—	172	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	172	4	no	—
288	C₂²	—	172	2	no	—
289	C₂²	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
290	C₂²	#14583435	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	4	no	—
294	C₄	—	4	4	no	—
295	C₄	—	172	2	no	—
296	C₄	C₄×#10621902	1	2	yes	—
297	C₄	#124×D₄₃	1	1	yes	characteristic
298	C₄	#14583450	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	344	2	no	—
301	C₂	#19137138	1	1	yes	characteristic
302	C₂	#19143494	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1995	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	#14583449	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1995	D₄₃
hypocenter	C₄₃	#21663
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

Complements

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

N	H ≅ G/N	type
2. #19143530	C₂	semidirect
5. #19135300	C₂	semidirect
6. #19134735	C₂	semidirect
7. #19134436	C₂	semidirect
8. #19134282	C₂	semidirect
9. C₄₃×#1995	C₂	semidirect
12. #14583269	C₄	semidirect
13. #14583265	C₄	semidirect
14. #14583261	C₄	semidirect
151. #1995	D₄₃	semidirect
185. C₄₃	#21663	semidirect

Maximal subgroups

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

subgroup	index
#19144943	2
#19143530	2
#19143306	2
#19135427	2
#19135300	2
#19134735	2
#19134436	2
#19134282	2
C₄₃×#1995	2
#21663	43