Subgroups — #25005423

Order 11008.

606 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	C₂×#14582937	C₂	1	1	yes	maximal, characteristic
2	#19134327	C₂	1	2	yes	maximal
3	#19134312	C₂	1	1	yes	maximal, characteristic
4	#19134306	C₂	1	2	yes	maximal
5	C₄₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063284	C₂²	1	2	yes	—
7	C₂×#13063284	C₂²	1	2	yes	—
8	#14582937	—	2	2	no	—
9	#14582937	—	2	2	no	—
10	C₂×#13063259	C₄	1	2	yes	complemented
11	C₂×#13063257	C₄	1	2	yes	complemented
12	#14582665	—	2	4	no	—
13	#14582662	—	2	2	no	—
14	#14582660	—	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	#13063284	—	2	2	no	—
20	#13063284	—	2	2	no	—
21	#13063284	—	2	2	no	—
22	#13063284	—	2	2	no	—
23	#13063284	—	2	2	no	—
24	#13063284	—	2	2	no	—
25	#13063284	—	2	2	no	—
26	#13063284	Q₈	1	4	yes	—
27	C₂×C₄⋊Dic₄₃	—	2	2	no	—
28	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
30	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
31	C₂×C₄×Dic₄₃	—	2	2	no	—
32	C₂×C₄×Dic₄₃	—	2	2	no	—
33	#13063259	—	2	4	no	—
34	#13063257	—	2	4	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	2	no	—
41	C₂²×C₈×C₄₃	—	2	2	no	—
42	C₂²×C₈×C₄₃	—	2	1	no	—
43	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₃	D₄	1	2	yes	—
46	C₂²×C₈×C₄₃	D₄	1	2	yes	—
47	#117×C₄₃	—	2	2	no	—
48	#117×C₄₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
51	C₂×C₄²×C₄₃	—	2	1	no	—
52	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	—	2	2	no	—
54	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	#41	1	2	yes	—
62	C₂²×C₄×C₄₃	#41	1	2	yes	—
63	C₂²×C₄×C₄₃	#41	1	2	yes	—
64	C₂²×Dic₄₃	—	2	4	no	—
65	C₂²×Dic₄₃	—	4	2	no	—
66	M₁₆×C₄₃	—	2	4	no	—
67	M₁₆×C₄₃	—	4	2	no	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	—	2	2	no	—
70	C₂×C₈×C₄₃	—	2	2	no	—
71	C₂×C₈×C₄₃	—	2	2	no	—
72	C₂×C₈×C₄₃	—	2	4	no	—
73	C₂×C₈×C₄₃	—	2	2	no	—
74	C₂×C₈×C₄₃	—	4	1	no	—
75	C₂×C₈×C₄₃	—	4	1	no	—
76	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
78	C₄⋊C₄×C₄₃	—	4	2	no	—
79	C₂²⋊C₄×C₄₃	—	2	4	no	—
80	C₄²×C₄₃	—	4	1	no	—
81	C₄²×C₄₃	—	4	1	no	—
82	C₂²⋊Dic₄₃	—	2	4	no	—
83	C₂²⋊Dic₄₃	—	4	2	no	—
84	C₂²⋊Dic₄₃	—	4	2	no	—
85	C₂²⋊Dic₄₃	—	4	2	no	—
86	C₄⋊Dic₄₃	—	2	4	no	—
87	C₄⋊Dic₄₃	—	2	4	no	—
88	C₄⋊Dic₄₃	—	2	4	no	—
89	C₄⋊Dic₄₃	—	4	2	no	—
90	C₄×Dic₄₃	—	2	4	no	—
91	C₄×Dic₄₃	—	4	2	no	—
92	C₄×Dic₄₃	—	4	2	no	—
93	C₄×Dic₄₃	—	4	2	no	—
94	C₂³×C₄₃	—	4	2	no	—
95	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
96	C₂³×C₄₃	#121	1	1	yes	characteristic
97	C₂³×C₄₃	C₂²⋊Q₈	1	2	yes	—
98	C₂³×C₄₃	#123	1	2	yes	—
99	C₂³×C₄₃	#127	1	1	yes	characteristic
100	C₂×C₄×C₄₃	—	4	4	no	—
101	C₂×C₄×C₄₃	—	4	2	no	—
102	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₃	#117	1	2	yes	—
104	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₄×Q₈	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	#123	1	1	yes	characteristic
110	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
111	C₂×C₄×C₄₃	#123	1	2	yes	—
112	C₂×C₄×C₄₃	#123	1	2	yes	—
113	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
114	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
115	C₂×C₄×C₄₃	#125	1	2	yes	—
116	C₂×C₄×C₄₃	#125	1	2	yes	—
117	C₂×C₄×C₄₃	#126	1	2	yes	—
118	C₂×C₄×C₄₃	#126	1	2	yes	—
119	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×Dic₄₃	—	4	4	no	—
121	C₂×Dic₄₃	—	4	4	no	—
122	C₂×Dic₄₃	—	8	4	no	—
123	C₂×Dic₄₃	—	8	4	no	—
124	C₈×C₄₃	—	2	2	no	—
125	C₈×C₄₃	—	2	2	no	—
126	C₈×C₄₃	—	4	2	no	—
127	#11364	—	43	1	no	maximal, Sylow
128	C₂²×C₄₃	—	2	2	no	—
129	C₂²×C₄₃	—	2	2	no	—
130	C₂²×C₄₃	—	2	2	no	—
131	C₂²×C₄₃	—	4	2	no	—
132	C₂²×C₄₃	—	4	2	no	—
133	C₂²×C₄₃	#387	1	2	yes	—
134	C₂²×C₄₃	#390	1	1	yes	characteristic
135	C₂²×C₄₃	#397	1	1	yes	characteristic
136	C₂²×C₄₃	#398	1	1	yes	characteristic
137	C₂²×C₄₃	#399	1	2	yes	—
138	C₄×C₄₃	—	8	2	no	—
139	C₄×C₄₃	#381	1	1	yes	characteristic
140	C₄×C₄₃	#387	1	2	yes	—
141	C₄×C₄₃	#391	1	1	yes	characteristic
142	C₄×C₄₃	#397	1	1	yes	characteristic
143	C₄×C₄₃	#398	1	1	yes	characteristic
144	C₄×C₄₃	#400	1	2	yes	—
145	Dic₄₃	—	8	4	no	—
146	Dic₄₃	—	8	4	no	—
147	C₂×#427	—	43	1	no	—
148	#1740	—	43	2	no	—
149	#1735	—	43	1	no	—
150	#1733	—	43	2	no	—
151	#1729	D₄₃	1	1	yes	characteristic
152	C₂×C₄₃	—	2	2	no	—
153	C₂×C₄₃	—	2	1	no	—
154	C₂×C₄₃	—	2	2	no	—
155	C₂×C₄₃	—	2	1	no	—
156	C₂×C₄₃	#1445	1	1	yes	characteristic
157	C₂×C₄₃	#1976	1	2	yes	—
158	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
159	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
160	C₂×#117	—	43	2	no	—
161	C₂×#117	—	43	2	no	—
162	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
163	#427	—	86	2	no	—
164	#427	—	86	2	no	—
165	C₂×#107	—	43	2	no	—
166	C₂×#106	—	43	2	no	—
167	#342	—	86	4	no	—
168	#340	—	86	2	no	—
169	#339	—	86	4	no	—
170	#336	—	2	2	no	—
171	C₄₃	#11364	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	D₁₇₂	1	2	yes	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
181	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
182	C₂²×C₈	D₁₇₂	1	2	yes	—
183	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	43	4	no	—
187	#117	—	86	2	no	—
188	#117	—	86	2	no	—
189	#117	—	86	2	no	—
190	#117	—	86	2	no	—
191	#117	—	86	2	no	—
192	#117	—	86	2	no	—
193	#117	—	86	2	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	43	4	no	—
196	C₂×C₄⋊C₄	—	86	2	no	—
197	C₂×C₂²⋊C₄	—	86	2	no	—
198	C₂×C₂²⋊C₄	—	86	2	no	—
199	C₂×C₂²⋊C₄	D₁₇₂	1	2	yes	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	86	2	no	—
202	C₂×C₄²	—	86	2	no	—
203	#107	—	86	4	no	—
204	#106	—	86	4	no	—
205	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	86	4	no	—
208	C₂²×C₄	—	172	2	no	—
209	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
212	C₂²×C₄	#10621898	1	1	yes	characteristic
213	C₂²×C₄	#10621898	1	1	yes	characteristic
214	C₂²×C₄	#10621900	1	2	yes	—
215	C₂²×C₄	#10621900	1	2	yes	—
216	C₂²×C₄	#10621900	1	2	yes	—
217	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
218	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
230	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	86	4	no	—
233	C₄⋊C₄	—	86	4	no	—
234	C₄⋊C₄	—	86	4	no	—
235	C₄⋊C₄	—	172	2	no	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₂²⋊C₄	—	86	4	no	—
238	C₂²⋊C₄	—	172	2	no	—
239	C₂²⋊C₄	—	172	2	no	—
240	C₂²⋊C₄	—	172	2	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	86	4	no	—
244	C₄²	—	172	2	no	—
245	C₄²	—	172	2	no	—
246	C₄²	—	172	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
249	C₂³	#13063299	1	1	yes	characteristic
250	C₂³	#13063305	1	2	yes	—
251	C₂³	#13063309	1	2	yes	—
252	C₂³	#13063322	1	1	yes	characteristic
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	344	4	no	—
256	C₂×C₄	—	344	4	no	—
257	C₂×C₄	—	172	4	no	—
258	C₂×C₄	—	172	4	no	—
259	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	#13063281	1	2	yes	—
261	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
262	C₂×C₄	#13063290	1	2	yes	—
263	C₂×C₄	#13063299	1	1	yes	characteristic
264	C₂×C₄	#13063305	1	2	yes	—
265	C₂×C₄	#13063305	1	2	yes	—
266	C₂×C₄	#13063306	1	1	yes	characteristic
267	C₂×C₄	#13063306	1	1	yes	characteristic
268	C₂×C₄	#13063309	1	2	yes	—
269	C₂×C₄	#13063309	1	2	yes	—
270	C₂×C₄	#13063312	1	1	yes	characteristic
271	C₂×C₄	#13063313	1	1	yes	characteristic
272	C₂×C₄	#13063315	1	2	yes	—
273	C₂×C₄	#13063315	1	2	yes	—
274	C₂×C₄	#13063318	1	2	yes	—
275	C₂×C₄	#13063318	1	2	yes	—
276	C₂×C₄	#13063323	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	#14582784	1	2	yes	—
286	C₂²	#14582801	1	1	yes	characteristic
287	C₂²	#14582824	1	1	yes	characteristic
288	C₂²	#14582829	1	1	yes	characteristic
289	C₂²	#14582836	1	2	yes	—
290	C₄	—	8	2	no	—
291	C₄	—	344	4	no	—
292	C₄	—	344	4	no	—
293	C₄	#14582760	1	1	yes	characteristic
294	C₄	#14582784	1	2	yes	—
295	C₄	#14582804	1	1	yes	characteristic
296	C₄	#14582824	1	1	yes	characteristic
297	C₄	#14582829	1	1	yes	characteristic
298	C₄	#14582839	1	2	yes	—
299	C₂	—	2	1	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	2	no	—
302	C₂	—	2	1	no	—
303	C₂	#19132862	1	1	yes	characteristic
304	C₂	#19135493	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13063299
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄₃×#1729	C₂
socle	C₂²×C₄₃	#390
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	#1729	D₄₃
hypocenter	C₄₃	#11364
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
10. C₂×#13063259	C₄	semidirect
11. C₂×#13063257	C₄	semidirect
171. C₄₃	#11364	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582937	2
#19134327	2
#19134312	2
#19134306	2
C₄₃×#1729	2
#11364	43