Subgroups — #25104753

Order 11008.

770 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	#19144920	C₂	1	1	yes	maximal, characteristic
2	#19143530	C₂	1	1	yes	maximal, characteristic, complemented
3	#19143306	C₂	1	1	yes	maximal, characteristic
4	#19135423	C₂	1	1	yes	maximal, characteristic
5	#19135300	C₂	1	4	yes	maximal, complemented
6	#19134732	C₂	1	1	yes	maximal, characteristic, complemented
7	#19134429	C₂	1	4	yes	maximal, complemented
8	C₄²×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#1994	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13063333	C₂²	1	2	yes	—
11	#14583449	C₂²	1	1	yes	characteristic
12	#14583256	C₄	1	4	yes	complemented
13	#14583251	C₄	1	2	yes	complemented
14	#14583247	C₄	1	2	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×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
26	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
27	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
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×C₄₃	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	C₄×C₄₃⋊C₈	—	2	4	no	—
55	C₄×C₄₃⋊C₈	—	2	2	no	—
56	C₄×C₄₃⋊C₈	—	2	4	no	—
57	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
58	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
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	—	2	2	no	—
68	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
69	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×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	8	no	—
89	C₄²×C₄₃	—	2	2	no	—
90	C₄²×C₄₃	—	2	2	no	—
91	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₄₃	C₂×D₄	1	2	yes	—
94	C₄²×C₄₃	#41	1	4	yes	—
95	(C₂×D₄₃)⋊C₄	—	2	4	no	—
96	C₄⋊Dic₄₃	—	2	2	no	—
97	#10621877	—	2	4	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	—	2	2	no	—
102	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
103	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
104	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄₃⋊C₈	#41	1	4	yes	—
107	#10621872	—	4	4	no	—
108	C₈×D₄₃	—	4	4	no	—
109	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
110	C₂²×D₄₃	—	2	2	no	—
111	Q₈×C₄₃	—	4	4	no	—
112	C₂×C₄×C₄₃	—	2	8	no	—
113	C₂×C₄×C₄₃	—	2	4	no	—
114	C₂×C₄×C₄₃	—	2	4	no	—
115	C₂×C₄×C₄₃	—	2	4	no	—
116	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
121	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
122	C₂²⋊D₄₃	—	4	4	no	—
123	C₂×Dic₄₃	—	2	2	no	—
124	D₁₇₂	—	2	4	no	—
125	D₁₇₂	—	4	4	no	—
126	C₄×D₄₃	—	4	2	no	—
127	C₄×D₄₃	—	4	2	no	—
128	Dic₈₆	—	2	4	no	—
129	C₈×C₄₃	—	4	2	no	—
130	C₄₃⋊C₈	—	2	4	no	—
131	C₄₃⋊C₈	—	2	2	no	—
132	C₄₃⋊C₈	—	4	2	no	—
133	C₄₃⋊C₈	#124	1	2	yes	—
134	C₄₃⋊C₈	#124	1	2	yes	—
135	#21658	—	43	1	no	maximal, Sylow
136	C₂²×C₄₃	—	2	2	no	—
137	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₄₃	#529	1	2	yes	—
139	C₂×D₄₃	—	4	4	no	—
140	C₂×D₄₃	—	4	2	no	—
141	C₄×C₄₃	—	2	8	no	—
142	C₄×C₄₃	—	2	2	no	—
143	C₄×C₄₃	—	4	4	no	—
144	C₄×C₄₃	#443	1	4	yes	—
145	C₄×C₄₃	C₄×#41	1	2	yes	—
146	C₄×C₄₃	C₂×#124	1	1	yes	characteristic
147	C₄×C₄₃	#532	1	1	yes	characteristic
148	Dic₄₃	—	4	2	no	—
149	#3131	—	43	1	no	—
150	#2960	—	43	1	no	—
151	#2931	—	43	1	no	—
152	#1994	D₄₃	1	1	yes	characteristic, complemented
153	#1962	—	43	1	no	—
154	#1937	—	43	4	no	—
155	#1837	—	43	1	no	—
156	#1764	—	43	4	no	—
157	C₄²×C₈	—	43	1	no	—
158	C₂×C₄₃	—	2	2	no	—
159	C₂×C₄₃	C₄×#124	1	1	yes	characteristic
160	C₂×C₄₃	C₂×#443	1	2	yes	—
161	D₄₃	—	8	2	no	—
162	C₂×#131	—	43	2	no	—
163	#533	C₂×D₄₃	1	1	yes	characteristic
164	#532	—	43	1	no	—
165	#488	—	43	4	no	—
166	#487	—	43	2	no	—
167	#486	—	43	2	no	—
168	#446	—	86	2	no	—
169	C₈×Q₈	—	86	2	no	—
170	#436	—	86	2	no	—
171	C₈×D₄	—	86	2	no	—
172	#424	—	43	2	no	—
173	#423	—	43	1	no	—
174	#423	C₂×D₄₃	1	1	yes	characteristic
175	C₂×#104	—	43	4	no	—
176	#416	—	43	4	no	—
177	#408	—	43	2	no	—
178	#405	—	43	2	no	—
179	C₂×C₄×C₈	—	43	1	no	—
180	C₂×C₄×C₈	—	43	1	no	—
181	C₂×C₄×C₈	—	43	2	no	—
182	C₂×C₄×C₈	—	43	2	no	—
183	C₄³	C₂×D₄₃	1	1	yes	characteristic
184	#341	—	43	4	no	—
185	#339	C₂×D₄₃	1	4	yes	—
186	C₄₃	#21658	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	43	2	no	—
188	#131	—	86	8	no	—
189	C₂×M₁₆	—	43	2	no	—
190	C₂×M₁₆	—	86	2	no	—
191	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
192	C₂²×C₈	—	43	1	no	—
193	C₂²×C₈	—	43	2	no	—
194	C₂²×C₈	—	43	1	no	—
195	C₂²×C₈	—	86	2	no	—
196	C₄⋊Q₈	—	43	1	no	—
197	C₄⋊Q₈	C₄×D₄₃	1	2	yes	—
198	#127	—	43	1	no	—
199	#125	C₄×D₄₃	1	2	yes	—
200	#124	—	43	2	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	#121	—	86	2	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×D₄	—	86	2	no	—
205	#117	C₂²×D₄₃	1	2	yes	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
208	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	—	86	4	no	—
211	C₄⋊C₈	—	86	2	no	—
212	#104	—	86	8	no	—
213	#103	—	43	8	no	—
214	#102	—	43	8	no	—
215	C₂²⋊C₈	—	86	4	no	—
216	#97	—	86	2	no	—
217	C₄×C₈	—	43	2	no	—
218	C₄×C₈	—	43	2	no	—
219	C₄×C₈	—	43	2	no	—
220	C₄×C₈	—	43	2	no	—
221	C₄×C₈	—	86	4	no	—
222	C₄×C₈	—	86	2	no	—
223	C₄×C₈	—	86	4	no	—
224	#41	—	86	4	no	—
225	#41	—	86	4	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	43	2	no	—
228	C₂×D₄	—	43	2	no	—
229	C₂×D₄	—	86	2	no	—
230	C₂×D₄	—	86	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	86	2	no	—
233	C₂²×C₄	#10621900	1	2	yes	—
234	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
235	M₁₆	—	2	4	no	—
236	M₁₆	—	86	4	no	—
237	M₁₆	—	172	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	43	1	no	—
240	C₂×C₈	—	43	1	no	—
241	C₂×C₈	—	43	4	no	—
242	C₂×C₈	—	43	4	no	—
243	C₂×C₈	—	43	2	no	—
244	C₂×C₈	—	86	2	no	—
245	C₂×C₈	—	86	2	no	—
246	C₂×C₈	—	86	2	no	—
247	C₂×C₈	—	86	2	no	—
248	C₂×C₈	—	172	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	86	2	no	—
252	C₄⋊C₄	C₂×C₄×D₄₃	1	4	yes	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	86	4	no	—
255	C₄²	—	2	8	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
260	C₄²	D₄×D₄₃	1	2	yes	—
261	C₄²	#10621902	1	4	yes	—
262	C₂³	—	86	2	no	—
263	C₂³	C₂×#10621900	1	1	yes	characteristic
264	Q₈	—	4	4	no	—
265	Q₈	—	86	4	no	—
266	D₄	—	86	4	no	—
267	D₄	—	172	4	no	—
268	D₄	—	172	4	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	86	2	no	—
274	C₂×C₄	—	172	2	no	—
275	C₂×C₄	—	172	2	no	—
276	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
277	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
278	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
279	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
280	C₂×C₄	#41×D₄₃	1	2	yes	—
281	C₂×C₄	C₂×#10621902	1	2	yes	—
282	C₈	—	4	2	no	—
283	C₈	—	43	2	no	—
284	C₈	—	43	2	no	—
285	C₈	—	86	4	no	—
286	C₈	—	86	2	no	—
287	C₈	—	172	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	172	4	no	—
290	C₂²	—	172	2	no	—
291	C₂²	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
292	C₂²	#14583435	1	2	yes	—
293	C₄	—	2	8	no	—
294	C₄	—	2	2	no	—
295	C₄	—	4	4	no	—
296	C₄	—	172	2	no	—
297	C₄	#14583013	1	4	yes	—
298	C₄	C₄×#10621902	1	2	yes	—
299	C₄	#124×D₄₃	1	1	yes	characteristic
300	C₄	#14583450	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	344	2	no	—
303	C₂	#19137138	1	1	yes	characteristic
304	C₂	C₂×#14583013	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×#1994	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	#14583449	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1994	D₄₃
hypocenter	C₄₃	#21658
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1994

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. #19134732	C₂	semidirect
7. #19134429	C₂	semidirect
8. C₄²×C₄₃⋊C₈	C₂	semidirect
9. C₄₃×#1994	C₂	semidirect
12. #14583256	C₄	semidirect
13. #14583251	C₄	semidirect
14. #14583247	C₄	semidirect
152. #1994	D₄₃	semidirect
186. C₄₃	#21658	semidirect

Maximal subgroups

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

subgroup	index
#19144920	2
#19143530	2
#19143306	2
#19135423	2
#19135300	2
#19134732	2
#19134429	2
C₄²×C₄₃⋊C₈	2
C₄₃×#1994	2
#21658	43