Subgroups — #25005314

Order 11008.

666 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582947	C₂	1	2	yes	maximal
2	#19143159	C₂	1	1	yes	maximal, characteristic
3	#19134334	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582664	C₂	1	2	yes	maximal
5	C₄₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂²×#10621877	C₂²	1	2	yes	—
8	#14582947	—	2	4	no	—
9	#14582947	—	2	4	no	—
10	C₂×#13063260	—	2	4	no	—
11	#14582926	—	2	4	no	—
12	#14582923	—	2	2	no	—
13	#14582668	C₄	1	4	yes	—
14	#14582664	—	2	4	no	—
15	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#336	—	2	2	no	—
19	C₂×#10621877	—	2	2	no	—
20	C₂×#10621877	—	2	2	no	—
21	C₂×#10621877	—	2	2	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	—	2	2	no	—
25	C₂×#10621877	—	2	2	no	—
26	C₂×#10621877	—	2	2	no	—
27	C₂×#10621877	C₂×C₄	1	4	yes	—
28	C₂×#10621877	D₄	1	4	yes	—
29	C₂×#10621877	D₄	1	4	yes	—
30	C₂×#10621877	Q₈	1	4	yes	—
31	C₂²×C₄₃⋊C₈	—	2	2	no	—
32	C₂²×C₄₃⋊C₈	—	2	2	no	—
33	#13063260	—	2	16	no	—
34	C₄⋊(C₄₃⋊C₈)	—	2	8	no	—
35	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₄₃	—	2	1	no	—
37	C₂×M₁₆×C₄₃	—	2	2	no	—
38	C₂×M₁₆×C₄₃	—	2	2	no	—
39	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
40	C₂²×C₈×C₄₃	—	2	1	no	—
41	C₂²×C₈×C₄₃	—	2	2	no	—
42	C₂²×C₈×C₄₃	—	2	1	no	—
43	C₂²×C₈×C₄₃	—	2	1	no	—
44	C₂²×C₈×C₄₃	—	2	2	no	—
45	C₂²×C₈×C₄₃	Q₈	1	2	yes	—
46	#117×C₄₃	—	2	2	no	—
47	#117×C₄₃	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₄₃	Q₈	1	2	yes	—
50	C₂×C₄²×C₄₃	D₄	1	2	yes	—
51	C₂⁴×C₄₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₄₃	#41	1	2	yes	—
61	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
63	M₁₆×C₄₃	—	2	4	no	—
64	M₁₆×C₄₃	—	4	2	no	—
65	C₂×C₈×C₄₃	—	2	2	no	—
66	C₂×C₈×C₄₃	—	2	4	no	—
67	C₂×C₈×C₄₃	—	2	2	no	—
68	C₂×C₈×C₄₃	—	4	1	no	—
69	C₂×C₈×C₄₃	—	4	1	no	—
70	C₂×C₈×C₄₃	—	4	2	no	—
71	C₂×C₈×C₄₃	—	4	1	no	—
72	C₂×C₈×C₄₃	—	4	1	no	—
73	C₄⋊C₄×C₄₃	—	2	4	no	—
74	C₂²⋊C₄×C₄₃	—	2	4	no	—
75	C₄²×C₄₃	—	2	2	no	—
76	C₄²×C₄₃	—	2	2	no	—
77	#10621877	—	2	4	no	—
78	#10621877	—	2	4	no	—
79	#10621877	—	2	4	no	—
80	#10621877	—	2	4	no	—
81	#10621877	—	2	4	no	—
82	#10621877	—	4	2	no	—
83	#10621877	—	4	2	no	—
84	#10621877	C₄⋊C₄	1	8	yes	—
85	C₂×C₄₃⋊C₈	—	2	4	no	—
86	C₂×C₄₃⋊C₈	—	2	4	no	—
87	C₂×C₄₃⋊C₈	—	2	4	no	—
88	C₂×C₄₃⋊C₈	—	2	4	no	—
89	C₂×C₄₃⋊C₈	—	4	2	no	—
90	C₂×C₄₃⋊C₈	—	4	2	no	—
91	C₂³×C₄₃	—	4	2	no	—
92	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₄₃	C₂²⋊Q₈	1	2	yes	—
94	C₂³×C₄₃	#125	1	1	yes	characteristic
95	C₂³×C₄₃	C₄⋊Q₈	1	2	yes	—
96	C₂³×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₄₃	—	2	4	no	—
98	C₂×C₄×C₄₃	—	4	4	no	—
99	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
104	C₂×C₄×C₄₃	#120	1	2	yes	—
105	C₂×C₄×C₄₃	#121	1	2	yes	—
106	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#123	1	2	yes	—
113	C₂×C₄×C₄₃	#125	1	2	yes	—
114	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
117	C₈×C₄₃	—	4	2	no	—
118	C₈×C₄₃	—	4	2	no	—
119	C₄₃⋊C₈	—	2	8	no	—
120	C₄₃⋊C₈	—	4	4	no	—
121	#11343	—	43	1	no	maximal, Sylow
122	C₂²×C₄₃	—	2	2	no	—
123	C₂²×C₄₃	—	2	4	no	—
124	C₂²×C₄₃	—	4	1	no	—
125	C₂²×C₄₃	—	4	2	no	—
126	C₂²×C₄₃	—	4	1	no	—
127	C₂²×C₄₃	#384	1	1	yes	characteristic
128	C₂²×C₄₃	#385	1	1	yes	characteristic
129	C₂²×C₄₃	#389	1	1	yes	characteristic
130	C₂²×C₄₃	#395	1	2	yes	—
131	C₂²×C₄₃	#398	1	2	yes	—
132	C₄×C₄₃	—	4	4	no	—
133	C₄×C₄₃	#385	1	2	yes	—
134	C₄×C₄₃	#389	1	2	yes	—
135	C₄×C₄₃	#392	1	1	yes	characteristic
136	C₄×C₄₃	#395	1	1	yes	characteristic
137	C₄×C₄₃	#398	1	1	yes	characteristic
138	C₄×C₄₃	#399	1	1	yes	characteristic
139	C₂×#430	—	43	2	no	—
140	#2905	—	43	1	no	—
141	#1742	—	43	1	no	—
142	C₂×#341	—	43	2	no	—
143	#1729	D₄₃	1	1	yes	characteristic
144	C₂×C₄₃	—	2	1	no	—
145	C₂×C₄₃	—	2	1	no	—
146	C₂×C₄₃	—	2	2	no	—
147	C₂×C₄₃	—	2	2	no	—
148	C₂×C₄₃	#1444	1	1	yes	characteristic
149	C₂×C₄₃	#1858	1	2	yes	—
150	C₂²×M₁₆	—	43	2	no	—
151	C₂²×M₁₆	—	43	2	no	—
152	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
153	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
154	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
155	#430	—	86	4	no	—
156	#430	—	86	4	no	—
157	C₂×#108	—	86	4	no	—
158	#424	—	86	4	no	—
159	#423	—	86	2	no	—
160	#343	—	43	4	no	—
161	#341	—	86	4	no	—
162	#336	—	2	2	no	—
163	C₄₃	#11343	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
165	C₂×M₁₆	—	2	1	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	43	4	no	—
169	C₂×M₁₆	—	43	4	no	—
170	C₂×M₁₆	—	43	4	no	—
171	C₂×M₁₆	—	43	4	no	—
172	C₂×M₁₆	—	86	2	no	—
173	C₂×M₁₆	—	86	2	no	—
174	C₂×M₁₆	—	86	2	no	—
175	C₂×M₁₆	—	86	2	no	—
176	C₂×M₁₆	—	86	2	no	—
177	C₂×M₁₆	—	86	2	no	—
178	C₂×M₁₆	—	86	2	no	—
179	C₂×M₁₆	—	86	2	no	—
180	C₂×M₁₆	D₁₇₂	1	2	yes	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	1	no	—
185	C₂²×C₈	—	2	1	no	—
186	C₂²×C₈	—	86	2	no	—
187	C₂²×C₈	—	86	2	no	—
188	C₂²×C₈	Dic₈₆	1	2	yes	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	C₂×C₄⋊C₄	C₄×D₄₃	1	2	yes	—
192	C₂×C₂²⋊C₄	Dic₈₆	1	2	yes	—
193	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
194	#108	—	86	16	no	—
195	C₄⋊C₈	—	86	8	no	—
196	C₂⁴	C₂×Dic₈₆	1	1	yes	characteristic
197	C₂²×C₄	Dic₄₃⋊C₄	1	4	yes	—
198	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
201	C₂²×C₄	#10621898	1	1	yes	characteristic
202	C₂²×C₄	#10621898	1	1	yes	characteristic
203	C₂²×C₄	D₄×D₄₃	1	2	yes	—
204	C₂²×C₄	D₄×D₄₃	1	2	yes	—
205	C₂²×C₄	#10621900	1	2	yes	—
206	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
207	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	43	8	no	—
211	M₁₆	—	86	4	no	—
212	M₁₆	—	86	4	no	—
213	M₁₆	—	86	4	no	—
214	M₁₆	—	86	4	no	—
215	M₁₆	—	86	4	no	—
216	M₁₆	—	172	2	no	—
217	M₁₆	—	172	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	86	4	no	—
227	C₂×C₈	—	86	4	no	—
228	C₂×C₈	—	86	4	no	—
229	C₂×C₈	—	86	4	no	—
230	C₂×C₈	—	172	2	no	—
231	C₂×C₈	—	172	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
238	C₂³	#13063302	1	2	yes	—
239	C₂³	#13063317	1	1	yes	characteristic
240	C₂³	#13063324	1	2	yes	—
241	C₂³	#13063325	1	1	yes	characteristic
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
245	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
246	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
247	C₂×C₄	C₄×Dic₈₆	1	1	yes	characteristic
248	C₂×C₄	#13063293	1	1	yes	characteristic
249	C₂×C₄	#13063295	1	2	yes	—
250	C₂×C₄	#13063298	1	2	yes	—
251	C₂×C₄	#13063299	1	1	yes	characteristic
252	C₂×C₄	#13063302	1	2	yes	—
253	C₂×C₄	#13063303	1	2	yes	—
254	C₂×C₄	#13063304	1	1	yes	characteristic
255	C₂×C₄	#13063305	1	2	yes	—
256	C₂×C₄	#13063306	1	1	yes	characteristic
257	C₂×C₄	#13063307	1	2	yes	—
258	C₂×C₄	#13063315	1	2	yes	—
259	C₂×C₄	#13063321	1	1	yes	characteristic
260	C₂×C₄	#13063323	1	1	yes	characteristic
261	C₂×C₄	#13063324	1	2	yes	—
262	C₈	—	4	2	no	—
263	C₈	—	4	2	no	—
264	C₈	—	86	8	no	—
265	C₈	—	172	4	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	#14582768	1	1	yes	characteristic
272	C₂²	#14582771	1	1	yes	characteristic
273	C₂²	#14582797	1	1	yes	characteristic
274	C₂²	#14582820	1	2	yes	—
275	C₂²	#14582833	1	2	yes	—
276	C₄	—	4	4	no	—
277	C₄	#14582773	1	2	yes	—
278	C₄	#14582796	1	2	yes	—
279	C₄	#14582808	1	1	yes	characteristic
280	C₄	#14582819	1	1	yes	characteristic
281	C₄	#14582829	1	1	yes	characteristic
282	C₄	#14582835	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	2	no	—
285	C₂	—	2	2	no	—
286	C₂	—	2	1	no	—
287	C₂	#19132859	1	1	yes	characteristic
288	C₂	#19134814	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13063293
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄₃×#1729	C₂
socle	C₂²×C₄₃	#385
prime radical	C₂⁴×C₄₃	C₂×Q₈
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	#1729	D₄₃
hypocenter	C₄₃	#11343
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
163. C₄₃	#11343	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582947	2
#19143159	2
#19134334	2
C₂×#14582664	2
C₄₃×#1729	2
#11343	43