Subgroups — #25079676

Order 11008.

714 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145718	C₂	1	2	yes	maximal
2	C₂×#14583279	C₂	1	1	yes	maximal, characteristic, complemented
3	#19143308	C₂	1	1	yes	maximal, characteristic
4	#19134085	C₂	1	4	yes	maximal, complemented
5	#19133961	C₂	1	4	yes	maximal, complemented
6	#19133266	C₂	1	2	yes	maximal, complemented
7	C₄₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₁₇₂	—	2	1	no	—
9	C₂²×Dic₁₇₂	—	2	1	no	—
10	C₂×#13063331	—	2	1	no	—
11	#14583449	C₂²	1	1	yes	characteristic
12	C₂×#13063325	C₂²	1	2	yes	—
13	#14583304	—	2	2	no	—
14	#14583279	—	2	1	no	—
15	#14583279	—	2	1	no	—
16	#14583279	C₂²	1	2	yes	—
17	#14583279	C₂²	1	2	yes	—
18	#14583246	—	2	2	no	—
19	#14583213	—	2	4	no	—
20	#14583180	C₂²	1	8	yes	—
21	#14582978	—	2	1	no	—
22	#14582978	C₂²	1	2	yes	—
23	C₈×D₁₇₂	—	2	1	no	—
24	C₈×D₁₇₂	C₂²	1	2	yes	—
25	#14582923	C₂²	1	1	yes	characteristic
26	#14582865	—	2	1	no	—
27	#14582644	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	C₂×Dic₁₇₂	—	2	2	no	—
35	C₂×Dic₁₇₂	—	2	2	no	—
36	C₂×Dic₁₇₂	—	2	2	no	—
37	C₂×Dic₁₇₂	—	2	2	no	—
38	C₂×Dic₁₇₂	—	4	4	no	—
39	C₂×Dic₁₇₂	—	4	4	no	—
40	#13063331	—	4	4	no	—
41	C₂×#10621877	—	2	1	no	—
42	C₂×#10621872	—	2	2	no	—
43	C₂×C₈×D₄₃	—	2	2	no	—
44	#13063325	—	2	2	no	—
45	#13063325	—	2	2	no	—
46	#13063325	—	2	2	no	—
47	#13063325	D₄	1	1	yes	characteristic
48	#13063325	C₂³	1	4	yes	—
49	#13063322	D₄	1	1	yes	characteristic
50	#13063313	—	2	1	no	—
51	#13063299	—	2	1	no	—
52	#13063299	D₄	1	2	yes	—
53	C₄×D₁₇₂	—	2	1	no	—
54	C₄×D₁₇₂	C₂³	1	2	yes	—
55	C₂×C₄⋊Dic₄₃	—	2	2	no	—
56	C₂×C₄⋊Dic₄₃	—	2	2	no	—
57	#13063259	—	2	8	no	—
58	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
59	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
60	#13063247	—	2	8	no	—
61	(C₂×D₄₃)⋊C₈	—	2	2	no	—
62	(C₂×D₄₃)⋊C₈	—	2	2	no	—
63	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₄₃	C₂³	1	4	yes	—
67	C₂²⋊C₈×C₄₃	—	2	2	no	—
68	C₄×C₈×C₄₃	—	2	2	no	—
69	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
71	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
76	C₂×C₂²⋊D₄₃	—	2	2	no	—
77	C₂²×Dic₄₃	—	4	2	no	—
78	#10621898	—	4	2	no	—
79	#10621898	—	4	2	no	—
80	C₂×D₁₇₂	—	2	1	no	—
81	C₂×D₁₇₂	—	2	1	no	—
82	C₂×D₁₇₂	C₂×D₄	1	2	yes	—
83	C₂×C₄×D₄₃	—	2	2	no	—
84	C₂×Dic₈₆	—	2	1	no	—
85	C₂×Dic₈₆	—	2	4	no	—
86	C₂×Dic₈₆	—	4	4	no	—
87	C₂×Dic₈₆	—	4	4	no	—
88	C₂×Dic₈₆	—	4	2	no	—
89	C₂×C₈×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	—	2	4	no	—
91	C₂×C₈×C₄₃	D₈	1	4	yes	—
92	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₄₃	#41	1	1	yes	characteristic
99	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
100	(C₂×D₄₃)⋊C₄	—	2	2	no	—
101	(C₂×D₄₃)⋊C₄	—	2	2	no	—
102	C₄⋊Dic₄₃	—	2	1	no	—
103	C₄⋊Dic₄₃	—	2	4	no	—
104	C₄⋊Dic₄₃	—	2	4	no	—
105	C₄⋊Dic₄₃	—	4	2	no	—
106	C₄⋊Dic₄₃	—	4	2	no	—
107	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
108	#10621877	—	4	2	no	—
109	C₂×C₄₃⋊C₈	—	2	2	no	—
110	Dic₁₇₂	—	4	4	no	—
111	Dic₁₇₂	—	4	4	no	—
112	Dic₁₇₂	—	4	4	no	—
113	Dic₁₇₂	—	4	4	no	—
114	#10621872	—	4	4	no	—
115	C₈×D₄₃	—	4	4	no	—
116	C₂³×C₄₃	#121	1	1	yes	characteristic
117	C₂²×D₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	—	2	2	no	—
119	C₂×C₄×C₄₃	—	2	2	no	—
120	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
128	C₂²⋊D₄₃	—	4	4	no	—
129	C₂×Dic₄₃	—	2	2	no	—
130	C₂×Dic₄₃	—	4	4	no	—
131	C₂×Dic₄₃	—	8	4	no	—
132	D₁₇₂	—	2	4	no	—
133	D₁₇₂	—	4	2	no	—
134	D₁₇₂	—	4	2	no	—
135	C₄×D₄₃	—	4	2	no	—
136	C₄×D₄₃	—	4	2	no	—
137	Dic₈₆	—	4	2	no	—
138	Dic₈₆	—	4	4	no	—
139	Dic₈₆	—	4	4	no	—
140	Dic₈₆	—	8	4	no	—
141	C₈×C₄₃	—	2	1	no	—
142	C₈×C₄₃	—	2	1	no	—
143	C₈×C₄₃	—	4	4	no	—
144	C₈×C₄₃	C₂×D₈	1	2	yes	—
145	C₈×C₄₃	C₂×D₈	1	2	yes	—
146	C₄₃⋊C₈	—	4	2	no	—
147	#19188	—	43	1	no	maximal, Sylow
148	C₂²×C₄₃	—	2	1	no	—
149	C₂²×C₄₃	—	2	1	no	—
150	C₂²×C₄₃	#459	1	2	yes	—
151	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
152	C₂×D₄₃	—	4	2	no	—
153	C₂×D₄₃	—	4	2	no	—
154	C₂×D₄₃	—	4	2	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	—	2	1	no	—
157	C₄×C₄₃	#459	1	2	yes	—
158	C₄×C₄₃	#473	1	2	yes	—
159	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
160	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₄×C₄₃	#578	1	2	yes	—
162	Dic₄₃	—	4	2	no	—
163	Dic₄₃	—	8	4	no	—
164	#3226	—	43	2	no	—
165	C₂×#494	—	43	1	no	—
166	#2931	—	43	1	no	—
167	#1689	—	43	4	no	—
168	#1669	—	43	4	no	—
169	#1564	D₄₃	1	1	yes	characteristic, complemented
170	#1546	—	43	2	no	—
171	C₂×C₄₃	—	2	2	no	—
172	C₂×C₄₃	C₂×#459	1	1	yes	characteristic
173	C₂×C₄₃	#3059	1	1	yes	characteristic
174	C₂×C₄₃	C₂×#473	1	1	yes	characteristic
175	D₄₃	—	8	2	no	—
176	C₂²×Q₁₆	—	86	1	no	—
177	C₂²×Q₁₆	—	86	1	no	—
178	C₂×#131	—	86	1	no	—
179	#532	—	43	1	no	—
180	C₂×C₄⋊Q₈	—	43	2	no	—
181	#500	—	86	2	no	—
182	#494	—	43	2	no	—
183	#494	—	43	2	no	—
184	#494	—	86	1	no	—
185	#494	—	86	1	no	—
186	#486	—	86	2	no	—
187	#480	—	86	4	no	—
188	#474	—	43	8	no	—
189	#436	—	43	2	no	—
190	#436	—	86	1	no	—
191	C₈×D₄	—	43	2	no	—
192	C₈×D₄	—	86	1	no	—
193	#431	C₂×D₄₃	1	2	yes	—
194	#423	—	43	1	no	—
195	#408	—	86	1	no	—
196	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
197	#335	C₂×D₄₃	1	2	yes	—
198	#334	C₂×D₄₃	1	2	yes	—
199	#332	—	43	8	no	—
200	C₄₃	#19188	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	86	1	no	—
202	C₂²×Q₈	—	86	2	no	—
203	C₂×Q₁₆	—	86	2	no	—
204	C₂×Q₁₆	—	86	2	no	—
205	C₂×Q₁₆	—	86	2	no	—
206	C₂×Q₁₆	—	86	2	no	—
207	C₂×Q₁₆	—	172	4	no	—
208	C₂×Q₁₆	—	172	4	no	—
209	#131	—	172	4	no	—
210	C₂×M₁₆	—	86	1	no	—
211	C₂×M₁₆	—	86	2	no	—
212	C₂²×C₈	—	86	2	no	—
213	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
214	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
215	C₄⋊Q₈	—	43	1	no	—
216	C₄⋊Q₈	—	43	4	no	—
217	C₄⋊Q₈	—	86	2	no	—
218	C₄⋊Q₈	—	86	2	no	—
219	C₄⋊Q₈	—	86	2	no	—
220	#127	—	43	1	no	—
221	#124	—	86	1	no	—
222	#121	—	43	2	no	—
223	#121	—	86	1	no	—
224	C₄×D₄	—	43	2	no	—
225	C₄×D₄	—	86	1	no	—
226	C₂×C₄⋊C₄	—	86	2	no	—
227	C₂×C₄⋊C₄	—	86	2	no	—
228	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
229	#107	—	86	8	no	—
230	C₄⋊C₈	—	43	2	no	—
231	C₄⋊C₈	—	86	1	no	—
232	C₄⋊C₈	C₂²×D₄₃	1	4	yes	—
233	#102	—	86	8	no	—
234	C₂²⋊C₈	—	2	2	no	—
235	C₂²⋊C₈	—	86	2	no	—
236	C₂²⋊C₈	—	86	2	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	D₁₇₂	1	1	yes	characteristic
239	C₄×C₈	D₁₇₂	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
242	#41	—	172	2	no	—
243	#41	—	172	2	no	—
244	C₂×Q₈	—	86	1	no	—
245	C₂×Q₈	—	86	4	no	—
246	C₂×Q₈	—	172	4	no	—
247	C₂×Q₈	—	172	4	no	—
248	C₂×Q₈	—	172	2	no	—
249	C₂×D₄	—	43	2	no	—
250	C₂×D₄	—	86	2	no	—
251	C₂×D₄	—	86	1	no	—
252	C₂×D₄	—	86	1	no	—
253	C₂²×C₄	—	86	2	no	—
254	C₂²×C₄	—	172	2	no	—
255	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
256	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
257	C₂²×C₄	#10621902	1	1	yes	characteristic
258	Q₁₆	—	172	4	no	—
259	Q₁₆	—	172	4	no	—
260	Q₁₆	—	172	4	no	—
261	Q₁₆	—	172	4	no	—
262	M₁₆	—	172	2	no	—
263	M₁₆	—	172	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	86	2	no	—
267	C₂×C₈	—	172	4	no	—
268	C₂×C₈	D₃₄₄	1	4	yes	—
269	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
270	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
271	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
272	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
273	C₄⋊C₄	—	43	2	no	—
274	C₄⋊C₄	—	86	1	no	—
275	C₄⋊C₄	—	86	4	no	—
276	C₄⋊C₄	—	86	4	no	—
277	C₄⋊C₄	—	172	2	no	—
278	C₄⋊C₄	—	172	2	no	—
279	C₂²⋊C₄	—	86	2	no	—
280	C₂²⋊C₄	—	86	2	no	—
281	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
282	C₄²	D₄×D₄₃	1	1	yes	characteristic
283	C₄²	#10621902	1	1	yes	characteristic
284	C₄²	C₂³×D₄₃	1	1	yes	characteristic
285	C₂³	—	86	2	no	—
286	C₂³	#13063300	1	1	yes	characteristic
287	Q₈	—	344	4	no	—
288	Q₈	—	172	4	no	—
289	Q₈	—	172	4	no	—
290	Q₈	—	172	2	no	—
291	D₄	—	86	4	no	—
292	D₄	—	172	4	no	—
293	D₄	—	172	2	no	—
294	D₄	—	172	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	86	2	no	—
298	C₂×C₄	—	344	4	no	—
299	C₂×C₄	—	172	4	no	—
300	C₂×C₄	—	172	2	no	—
301	C₂×C₄	—	172	2	no	—
302	C₂×C₄	#13063300	1	1	yes	characteristic
303	C₂×C₄	#13063300	1	1	yes	characteristic
304	C₂×C₄	#13063300	1	1	yes	characteristic
305	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
306	C₂×C₄	#13063353	1	2	yes	—
307	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
309	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	4	no	—
313	C₈	—	172	2	no	—
314	C₈	C₂×D₃₄₄	1	2	yes	—
315	C₈	C₂×D₃₄₄	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	172	2	no	—
319	C₂²	—	172	2	no	—
320	C₂²	—	172	2	no	—
321	C₂²	#14583095	1	2	yes	—
322	C₂²	C₂×#13063300	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	344	4	no	—
326	C₄	—	172	2	no	—
327	C₄	#14583095	1	2	yes	—
328	C₄	#14583175	1	2	yes	—
329	C₄	C₂²×D₃₄₄	1	1	yes	characteristic
330	C₄	C₂×#13063353	1	1	yes	characteristic
331	C₄	#14583713	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	344	2	no	—
334	C₂	C₂×#14583095	1	1	yes	characteristic
335	C₂	#19144251	1	1	yes	characteristic
336	C₂	C₂×#14583175	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	#19188
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×#14583279	C₂	semidirect
4. #19134085	C₂	semidirect
5. #19133961	C₂	semidirect
6. #19133266	C₂	semidirect
7. C₄₃×#1564	C₂	semidirect
169. #1564	D₄₃	semidirect
200. C₄₃	#19188	semidirect

Maximal subgroups

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

subgroup	index
#19145718	2
C₂×#14583279	2
#19143308	2
#19134085	2
#19133961	2
#19133266	2
C₄₃×#1564	2
#19188	43