Subgroups — #25079715

Order 11008.

650 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145725	C₂	1	2	yes	maximal
2	C₂×#14583304	C₂	1	1	yes	maximal, characteristic, complemented
3	#19143308	C₂	1	1	yes	maximal, characteristic
4	#19134119	C₂	1	4	yes	maximal, complemented
5	#19133995	C₂	1	4	yes	maximal, complemented
6	#19133275	C₂	1	2	yes	maximal, complemented
7	C₄₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063331	—	2	1	no	—
9	#14583449	C₂²	1	1	yes	characteristic
10	C₂×#13063325	C₂²	1	2	yes	—
11	#14583304	—	2	1	no	—
12	#14583304	—	2	1	no	—
13	#14583304	C₂²	1	2	yes	—
14	#14583304	C₂²	1	2	yes	—
15	#14583297	—	2	2	no	—
16	#14583246	—	2	2	no	—
17	#14583218	—	2	4	no	—
18	#14583190	C₂²	1	8	yes	—
19	#14582978	—	2	1	no	—
20	#14582978	C₂²	1	2	yes	—
21	C₈×D₁₇₂	—	2	1	no	—
22	C₈×D₁₇₂	C₂²	1	2	yes	—
23	C₂×#13063259	—	2	1	no	—
24	C₂×#13063259	—	2	1	no	—
25	#14582923	C₂²	1	1	yes	characteristic
26	#14582865	—	2	1	no	—
27	#14582647	C₂²	1	8	yes	—
28	C₄₃×#431	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₄₃×#335	C₂²	1	2	yes	—
31	C₄₃×#334	C₂²	1	2	yes	—
32	C₂×#10621898	—	2	1	no	—
33	C₂²×Dic₈₆	—	2	2	no	—
34	#13063331	—	4	4	no	—
35	C₂×#10621877	—	2	1	no	—
36	C₂×#10621872	—	2	2	no	—
37	C₂×C₈×D₄₃	—	2	2	no	—
38	#13063325	—	2	2	no	—
39	#13063325	—	2	2	no	—
40	#13063325	—	2	2	no	—
41	#13063325	D₄	1	1	yes	characteristic
42	#13063325	C₂³	1	4	yes	—
43	#13063322	D₄	1	1	yes	characteristic
44	#13063313	—	2	1	no	—
45	#13063299	—	2	1	no	—
46	#13063299	D₄	1	2	yes	—
47	C₄×D₁₇₂	—	2	1	no	—
48	C₄×D₁₇₂	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₄₃	—	2	2	no	—
50	C₂×C₄⋊Dic₄₃	—	2	2	no	—
51	#13063259	—	2	2	no	—
52	#13063259	—	2	2	no	—
53	#13063259	—	2	2	no	—
54	#13063259	—	2	2	no	—
55	#13063257	—	2	8	no	—
56	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
57	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
58	#13063247	—	2	8	no	—
59	(C₂×D₄₃)⋊C₈	—	2	2	no	—
60	(C₂×D₄₃)⋊C₈	—	2	2	no	—
61	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₄₃	C₂³	1	4	yes	—
65	C₂²⋊C₈×C₄₃	—	2	2	no	—
66	C₄×C₈×C₄₃	—	2	2	no	—
67	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
68	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
74	C₂×C₂²⋊D₄₃	—	2	2	no	—
75	C₂²×Dic₄₃	—	4	2	no	—
76	#10621898	—	4	2	no	—
77	#10621898	—	4	2	no	—
78	C₂×D₁₇₂	—	2	1	no	—
79	C₂×D₁₇₂	—	2	1	no	—
80	C₂×D₁₇₂	C₂×D₄	1	2	yes	—
81	C₂×C₄×D₄₃	—	2	2	no	—
82	C₂×Dic₈₆	—	2	1	no	—
83	C₂×Dic₈₆	—	2	4	no	—
84	C₂×Dic₈₆	—	4	2	no	—
85	C₂×Dic₈₆	—	8	4	no	—
86	C₂×C₈×C₄₃	—	2	2	no	—
87	C₂×C₈×C₄₃	—	2	4	no	—
88	C₂×C₈×C₄₃	SD₁₆	1	4	yes	—
89	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₄₃	#41	1	1	yes	characteristic
96	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
97	(C₂×D₄₃)⋊C₄	—	2	2	no	—
98	(C₂×D₄₃)⋊C₄	—	2	2	no	—
99	C₄⋊Dic₄₃	—	2	1	no	—
100	C₄⋊Dic₄₃	—	2	4	no	—
101	C₄⋊Dic₄₃	—	2	4	no	—
102	C₄⋊Dic₄₃	—	4	2	no	—
103	C₄⋊Dic₄₃	—	4	2	no	—
104	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
105	#10621877	—	4	2	no	—
106	C₂×C₄₃⋊C₈	—	2	2	no	—
107	#10621872	—	4	4	no	—
108	C₈×D₄₃	—	4	4	no	—
109	C₂³×C₄₃	#121	1	1	yes	characteristic
110	C₂²×D₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₄₃	#137	1	2	yes	—
118	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
121	C₂²⋊D₄₃	—	4	4	no	—
122	C₂×Dic₄₃	—	2	2	no	—
123	C₂×Dic₄₃	—	4	4	no	—
124	C₂×Dic₄₃	—	8	4	no	—
125	D₁₇₂	—	2	4	no	—
126	D₁₇₂	—	4	2	no	—
127	D₁₇₂	—	4	2	no	—
128	C₄×D₄₃	—	4	2	no	—
129	C₄×D₄₃	—	4	2	no	—
130	Dic₈₆	—	4	2	no	—
131	Dic₈₆	—	8	4	no	—
132	Dic₈₆	—	8	4	no	—
133	C₈×C₄₃	—	2	1	no	—
134	C₈×C₄₃	—	2	1	no	—
135	C₈×C₄₃	—	4	4	no	—
136	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
137	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
138	C₄₃⋊C₈	—	4	2	no	—
139	#19193	—	43	1	no	maximal, Sylow
140	C₂²×C₄₃	—	2	1	no	—
141	C₂²×C₄₃	—	2	1	no	—
142	C₂²×C₄₃	#461	1	2	yes	—
143	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
144	C₂×D₄₃	—	4	2	no	—
145	C₂×D₄₃	—	4	2	no	—
146	C₂×D₄₃	—	4	2	no	—
147	C₄×C₄₃	—	2	1	no	—
148	C₄×C₄₃	—	2	1	no	—
149	C₄×C₄₃	#461	1	2	yes	—
150	C₄×C₄₃	#472	1	2	yes	—
151	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
152	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
153	C₄×C₄₃	#578	1	2	yes	—
154	Dic₄₃	—	4	2	no	—
155	Dic₄₃	—	8	4	no	—
156	#3227	—	43	2	no	—
157	C₂×#500	—	43	1	no	—
158	#2931	—	43	1	no	—
159	#1695	—	43	4	no	—
160	#1675	—	43	4	no	—
161	#1564	D₄₃	1	1	yes	characteristic, complemented
162	#1548	—	43	2	no	—
163	C₂×C₄₃	—	2	2	no	—
164	C₂×C₄₃	C₂×#461	1	1	yes	characteristic
165	C₂×C₄₃	#3059	1	1	yes	characteristic
166	C₂×C₄₃	C₂×#472	1	1	yes	characteristic
167	D₄₃	—	8	2	no	—
168	C₂×#131	—	86	1	no	—
169	#532	—	43	1	no	—
170	C₂×C₄⋊Q₈	—	43	2	no	—
171	#500	—	43	2	no	—
172	#500	—	43	2	no	—
173	#500	—	86	1	no	—
174	#500	—	86	1	no	—
175	#498	—	86	2	no	—
176	#486	—	86	2	no	—
177	#481	—	86	4	no	—
178	#476	—	43	8	no	—
179	#436	—	43	2	no	—
180	#436	—	86	1	no	—
181	C₈×D₄	—	43	2	no	—
182	C₈×D₄	—	86	1	no	—
183	#431	C₂×D₄₃	1	2	yes	—
184	C₂×#107	—	86	1	no	—
185	C₂×#107	—	86	1	no	—
186	#423	—	43	1	no	—
187	#408	—	86	1	no	—
188	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
189	#335	C₂×D₄₃	1	2	yes	—
190	#334	C₂×D₄₃	1	2	yes	—
191	#333	—	43	8	no	—
192	C₄₃	#19193	1	1	yes	characteristic, complemented, Sylow
193	C₂×#41	—	86	1	no	—
194	C₂²×Q₈	—	86	2	no	—
195	#131	—	172	4	no	—
196	C₂×M₁₆	—	86	1	no	—
197	C₂×M₁₆	—	86	2	no	—
198	C₂²×C₈	—	86	2	no	—
199	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
200	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
201	C₄⋊Q₈	—	43	1	no	—
202	C₄⋊Q₈	—	43	4	no	—
203	C₄⋊Q₈	—	86	2	no	—
204	C₄⋊Q₈	—	86	2	no	—
205	C₄⋊Q₈	—	86	2	no	—
206	#127	—	43	1	no	—
207	#124	—	86	1	no	—
208	#121	—	43	2	no	—
209	#121	—	86	1	no	—
210	C₄×D₄	—	43	2	no	—
211	C₄×D₄	—	86	1	no	—
212	C₂×C₄⋊C₄	—	86	2	no	—
213	C₂×C₄⋊C₄	—	86	2	no	—
214	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
215	#107	—	86	2	no	—
216	#107	—	86	2	no	—
217	#107	—	86	2	no	—
218	#107	—	86	2	no	—
219	#106	—	86	8	no	—
220	C₄⋊C₈	—	43	2	no	—
221	C₄⋊C₈	—	86	1	no	—
222	C₄⋊C₈	C₂²×D₄₃	1	4	yes	—
223	#102	—	86	8	no	—
224	C₂²⋊C₈	—	2	2	no	—
225	C₂²⋊C₈	—	86	2	no	—
226	C₂²⋊C₈	—	86	2	no	—
227	C₄×C₈	—	2	2	no	—
228	C₄×C₈	D₁₇₂	1	1	yes	characteristic
229	C₄×C₈	D₁₇₂	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
232	#41	—	172	2	no	—
233	#41	—	172	2	no	—
234	C₂×Q₈	—	86	1	no	—
235	C₂×Q₈	—	86	4	no	—
236	C₂×Q₈	—	344	4	no	—
237	C₂×Q₈	—	172	2	no	—
238	C₂×D₄	—	43	2	no	—
239	C₂×D₄	—	86	2	no	—
240	C₂×D₄	—	86	1	no	—
241	C₂×D₄	—	86	1	no	—
242	C₂²×C₄	—	86	2	no	—
243	C₂²×C₄	—	172	2	no	—
244	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
245	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
246	C₂²×C₄	#10621902	1	1	yes	characteristic
247	M₁₆	—	172	2	no	—
248	M₁₆	—	172	4	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	86	2	no	—
252	C₂×C₈	—	172	4	no	—
253	C₂×C₈	#10621873	1	4	yes	—
254	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
255	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
256	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
257	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
258	C₄⋊C₄	—	43	2	no	—
259	C₄⋊C₄	—	86	1	no	—
260	C₄⋊C₄	—	86	4	no	—
261	C₄⋊C₄	—	86	4	no	—
262	C₄⋊C₄	—	172	2	no	—
263	C₄⋊C₄	—	172	2	no	—
264	C₂²⋊C₄	—	86	2	no	—
265	C₂²⋊C₄	—	86	2	no	—
266	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
267	C₄²	D₄×D₄₃	1	1	yes	characteristic
268	C₄²	#10621902	1	1	yes	characteristic
269	C₄²	C₂³×D₄₃	1	1	yes	characteristic
270	C₂³	—	86	2	no	—
271	C₂³	#13063300	1	1	yes	characteristic
272	Q₈	—	344	4	no	—
273	Q₈	—	344	4	no	—
274	Q₈	—	172	2	no	—
275	D₄	—	86	4	no	—
276	D₄	—	172	4	no	—
277	D₄	—	172	2	no	—
278	D₄	—	172	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	86	2	no	—
282	C₂×C₄	—	344	4	no	—
283	C₂×C₄	—	172	4	no	—
284	C₂×C₄	—	172	2	no	—
285	C₂×C₄	—	172	2	no	—
286	C₂×C₄	#13063300	1	1	yes	characteristic
287	C₂×C₄	#13063300	1	1	yes	characteristic
288	C₂×C₄	#13063300	1	1	yes	characteristic
289	C₂×C₄	C₂×#10621873	1	2	yes	—
290	C₂×C₄	#13063358	1	2	yes	—
291	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
293	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	4	4	no	—
297	C₈	—	172	2	no	—
298	C₈	C₂×#10621873	1	2	yes	—
299	C₈	C₂×#10621873	1	2	yes	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	172	2	no	—
303	C₂²	—	172	2	no	—
304	C₂²	—	172	2	no	—
305	C₂²	#14583105	1	2	yes	—
306	C₂²	C₂×#13063300	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	344	4	no	—
310	C₄	—	172	2	no	—
311	C₄	#14583105	1	2	yes	—
312	C₄	#14583170	1	2	yes	—
313	C₄	C₂²×#10621873	1	1	yes	characteristic
314	C₄	C₂×#13063358	1	1	yes	characteristic
315	C₄	#14583713	1	2	yes	—
316	C₂	—	2	2	no	—
317	C₂	—	344	2	no	—
318	C₂	C₂×#14583105	1	1	yes	characteristic
319	C₂	#19144251	1	1	yes	characteristic
320	C₂	C₂×#14583170	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063300
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1564	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	#14583449	C₂²
hypercenter	#1564	D₄₃
hypocenter	C₄₃	#19193
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
2. C₂×#14583304	C₂	semidirect
4. #19134119	C₂	semidirect
5. #19133995	C₂	semidirect
6. #19133275	C₂	semidirect
7. C₄₃×#1564	C₂	semidirect
161. #1564	D₄₃	semidirect
192. C₄₃	#19193	semidirect

Maximal subgroups

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

subgroup	index
#19145725	2
C₂×#14583304	2
#19143308	2
#19134119	2
#19133995	2
#19133275	2
C₄₃×#1564	2
#19193	43