Subgroups — #24989720

Order 11008.

482 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582964	C₂	1	1	yes	maximal, characteristic
2	#19134766	C₂	1	1	yes	maximal, characteristic, complemented
3	#19133341	C₂	1	1	yes	maximal, characteristic, complemented
4	#19133021	C₂	1	1	yes	maximal, characteristic
5	#19133019	C₂	1	1	yes	maximal, characteristic
6	#19132492	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₃	—	2	1	no	—
9	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
10	#14582964	—	2	1	no	—
11	#14582964	—	2	1	no	—
12	#14582964	C₄	1	2	yes	—
13	#14582964	C₄	1	2	yes	—
14	#14582952	C₂²	1	1	yes	characteristic
15	C₂×#13063260	—	2	1	no	—
16	C₂×#13063260	—	2	1	no	—
17	#14582923	C₂²	1	1	yes	characteristic
18	C₂×(C₂×D₄₃)⋊C₈	—	2	1	no	—
19	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	#14582651	—	2	1	no	—
21	#14582649	—	2	1	no	—
22	#14582628	—	2	2	no	—
23	#14582626	—	2	2	no	—
24	C₄₃×#431	C₂²	1	1	yes	characteristic
25	C₄₃×#423	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×D₁₇₂	Q₈	1	1	yes	characteristic
28	C₂²×C₄×D₄₃	—	2	1	no	—
29	C₂×#10621877	—	2	1	no	—
30	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
31	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
32	C₂×C₈×D₄₃	—	2	2	no	—
33	C₂×C₈×D₄₃	—	2	2	no	—
34	C₂×C₈×D₄₃	—	4	2	no	—
35	C₂×C₈×D₄₃	—	4	2	no	—
36	C₄×D₁₇₂	—	2	1	no	—
37	C₄×D₁₇₂	—	2	1	no	—
38	C₄×D₁₇₂	C₂×C₄	1	2	yes	—
39	C₄×D₁₇₂	D₄	1	2	yes	—
40	C₂×C₄⋊Dic₄₃	Q₈	1	1	yes	characteristic
41	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
42	#13063260	—	4	2	no	—
43	#13063260	—	4	2	no	—
44	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
45	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
46	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
47	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
48	(C₂×D₄₃)⋊C₈	—	2	2	no	—
49	(C₂×D₄₃)⋊C₈	—	2	2	no	—
50	C₄×C₄₃⋊C₈	—	2	1	no	—
51	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
54	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₄₃	—	2	1	no	—
56	C₂²×C₈×C₄₃	—	2	1	no	—
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₄₃	—	2	1	no	—
59	C₄⋊C₈×C₄₃	—	2	1	no	—
60	C₄⋊C₈×C₄₃	—	2	1	no	—
61	C₄⋊C₈×C₄₃	—	2	1	no	—
62	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
63	C₂²⋊C₈×C₄₃	—	2	1	no	—
64	C₄×C₈×C₄₃	—	2	1	no	—
65	C₂³×D₄₃	—	2	1	no	—
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
69	C₂²×Dic₄₃	—	2	1	no	—
70	C₂×D₁₇₂	—	2	1	no	—
71	C₂×D₁₇₂	—	2	4	no	—
72	C₂×D₁₇₂	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×D₄₃	—	2	2	no	—
74	C₂×C₄×D₄₃	—	2	2	no	—
75	C₂×C₄×D₄₃	—	4	1	no	—
76	C₂×C₄×D₄₃	—	4	1	no	—
77	C₂×C₄×D₄₃	—	4	1	no	—
78	C₂×C₄×D₄₃	—	4	1	no	—
79	M₁₆×C₄₃	—	4	1	no	—
80	M₁₆×C₄₃	—	4	1	no	—
81	C₂×C₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	—	4	1	no	—
83	C₂×C₈×C₄₃	—	4	1	no	—
84	C₂×C₈×C₄₃	—	4	1	no	—
85	C₂×C₈×C₄₃	—	4	1	no	—
86	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₄₃	#41	1	1	yes	characteristic
90	(C₂×D₄₃)⋊C₄	—	2	2	no	—
91	(C₂×D₄₃)⋊C₄	—	2	2	no	—
92	C₄⋊Dic₄₃	—	2	1	no	—
93	C₄⋊Dic₄₃	C₄⋊C₄	1	2	yes	—
94	#10621877	—	4	1	no	—
95	#10621877	—	4	1	no	—
96	C₂×C₄₃⋊C₈	—	2	1	no	—
97	C₂×C₄₃⋊C₈	—	2	1	no	—
98	C₂×C₄₃⋊C₈	—	4	1	no	—
99	C₂×C₄₃⋊C₈	—	4	1	no	—
100	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄₃⋊C₈	D₈	1	2	yes	—
105	C₂×C₄₃⋊C₈	SD₁₆	1	2	yes	—
106	C₈×D₄₃	—	4	4	no	—
107	C₈×D₄₃	—	4	4	no	—
108	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂²×D₄₃	—	2	4	no	—
110	C₂²×D₄₃	—	2	2	no	—
111	C₂²×D₄₃	—	4	2	no	—
112	C₂²×D₄₃	—	4	1	no	—
113	C₂²×D₄₃	—	4	1	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	4	1	no	—
116	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
119	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
126	C₂×Dic₄₃	—	2	2	no	—
127	C₂×Dic₄₃	—	4	1	no	—
128	C₂×Dic₄₃	—	4	1	no	—
129	D₁₇₂	—	2	4	no	—
130	D₁₇₂	—	2	2	no	—
131	D₁₇₂	#108	1	4	yes	complemented
132	C₄×D₄₃	—	4	2	no	—
133	C₄×D₄₃	—	4	2	no	—
134	C₄×D₄₃	—	4	2	no	—
135	C₄×D₄₃	—	4	2	no	—
136	C₈×C₄₃	—	4	2	no	—
137	C₈×C₄₃	—	8	1	no	—
138	C₈×C₄₃	—	8	1	no	—
139	C₄₃⋊C₈	—	2	1	no	—
140	C₄₃⋊C₈	—	2	1	no	—
141	C₄₃⋊C₈	—	8	1	no	—
142	C₄₃⋊C₈	—	8	1	no	—
143	C₄₃⋊C₈	#102	1	2	yes	—
144	C₄₃⋊C₈	#102	1	2	yes	—
145	#8658	—	43	1	no	maximal, Sylow
146	C₂²×C₄₃	—	2	1	no	—
147	C₂²×C₄₃	—	2	1	no	—
148	C₂²×C₄₃	#380	1	1	yes	characteristic
149	C₂²×C₄₃	#474	1	1	yes	characteristic
150	C₂²×C₄₃	#476	1	1	yes	characteristic
151	C₂×D₄₃	—	2	4	no	—
152	C₂×D₄₃	—	4	4	no	—
153	C₂×D₄₃	—	4	4	no	—
154	C₂×D₄₃	—	4	2	no	—
155	C₂×D₄₃	—	4	2	no	—
156	C₄×C₄₃	—	2	1	no	—
157	C₄×C₄₃	—	2	1	no	—
158	C₄×C₄₃	—	2	1	no	—
159	C₄×C₄₃	C₂×#102	1	1	yes	characteristic
160	C₄×C₄₃	#417	1	1	yes	characteristic
161	C₄×C₄₃	C₂×#108	1	2	yes	—
162	C₄×C₄₃	#459	1	1	yes	characteristic
163	C₄×C₄₃	#461	1	1	yes	characteristic
164	Dic₄₃	—	4	2	no	—
165	C₂×C₈×D₄	—	43	1	no	—
166	#1848	—	43	1	no	—
167	#1564	—	43	1	no	—
168	#1486	—	43	1	no	—
169	#1486	—	43	1	no	—
170	#1301	—	43	1	no	—
171	#1301	D₄₃	1	1	yes	characteristic, complemented
172	C₂×C₄₃	—	2	1	no	—
173	C₂×C₄₃	—	2	1	no	—
174	C₂×C₄₃	#1802	1	1	yes	characteristic
175	C₂×C₄₃	#1810	1	1	yes	characteristic
176	C₂×C₄₃	#1814	1	1	yes	characteristic
177	D₄₃	—	4	4	no	—
178	C₂³×C₈	—	86	1	no	—
179	C₂×C₄×D₄	—	43	1	no	—
180	C₈×D₄	—	43	2	no	—
181	C₈×D₄	—	43	2	no	—
182	C₈×D₄	—	86	1	no	—
183	C₈×D₄	—	86	1	no	—
184	#431	—	43	1	no	—
185	#431	C₂×D₄₃	1	1	yes	characteristic
186	C₂×#108	—	86	1	no	—
187	C₂×#108	—	86	1	no	—
188	#423	—	43	1	no	—
189	#423	C₂×D₄₃	1	1	yes	characteristic
190	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
191	C₂×C₂²⋊C₈	—	86	1	no	—
192	C₂×C₄×C₈	—	43	1	no	—
193	#335	—	86	1	no	—
194	#334	—	86	1	no	—
195	D₄⋊C₈	—	86	2	no	—
196	D₄⋊C₈	—	86	2	no	—
197	C₄₃	#8658	1	1	yes	characteristic, complemented, Sylow
198	C₂²×D₄	—	43	1	no	—
199	C₂³×C₄	—	86	1	no	—
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	—	86	1	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	43	1	no	—
204	C₂²×C₈	—	43	1	no	—
205	C₂²×C₈	—	86	2	no	—
206	C₂²×C₈	—	86	2	no	—
207	C₂²×C₈	—	172	2	no	—
208	C₂²×C₈	—	172	2	no	—
209	C₄×D₄	—	43	2	no	—
210	C₄×D₄	—	43	2	no	—
211	C₄×D₄	—	86	1	no	—
212	C₄×D₄	—	86	1	no	—
213	C₂×C₄⋊C₄	—	43	1	no	—
214	C₂×C₂²⋊C₄	—	86	1	no	—
215	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
216	#108	—	172	2	no	—
217	#108	—	172	2	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	86	1	no	—
223	C₄⋊C₈	—	86	1	no	—
224	C₄⋊C₈	—	86	1	no	—
225	C₄⋊C₈	C₄×D₄₃	1	2	yes	—
226	C₂²⋊C₈	—	2	1	no	—
227	C₂²⋊C₈	—	86	2	no	—
228	C₂²⋊C₈	—	86	2	no	—
229	C₂²⋊C₈	—	86	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	43	1	no	—
232	C₄×C₈	—	43	1	no	—
233	C₄×C₈	—	43	1	no	—
234	C₄×C₈	—	43	1	no	—
235	C₄×C₈	—	86	1	no	—
236	C₂⁴	—	86	1	no	—
237	C₂×D₄	—	43	2	no	—
238	C₂×D₄	—	86	4	no	—
239	C₂×D₄	—	86	1	no	—
240	C₂²×C₄	—	86	1	no	—
241	C₂²×C₄	—	86	2	no	—
242	C₂²×C₄	—	86	2	no	—
243	C₂²×C₄	—	172	1	no	—
244	C₂²×C₄	—	172	1	no	—
245	C₂²×C₄	—	172	1	no	—
246	C₂²×C₄	—	172	1	no	—
247	C₂²×C₄	#10621898	1	1	yes	characteristic
248	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
249	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	172	1	no	—
253	M₁₆	—	172	1	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	43	1	no	—
260	C₂×C₈	—	43	1	no	—
261	C₂×C₈	—	43	2	no	—
262	C₂×C₈	—	43	2	no	—
263	C₂×C₈	—	43	1	no	—
264	C₂×C₈	—	43	1	no	—
265	C₂×C₈	—	86	1	no	—
266	C₂×C₈	—	86	1	no	—
267	C₂×C₈	—	172	4	no	—
268	C₂×C₈	—	172	4	no	—
269	C₂×C₈	—	172	1	no	—
270	C₂×C₈	—	172	1	no	—
271	C₄⋊C₄	—	43	2	no	—
272	C₄⋊C₄	—	86	1	no	—
273	C₂²⋊C₄	—	86	2	no	—
274	C₂²⋊C₄	—	86	2	no	—
275	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
276	C₄²	#10621898	1	1	yes	characteristic
277	C₄²	D₄×D₄₃	1	1	yes	characteristic
278	C₄²	D₄×D₄₃	1	1	yes	characteristic
279	C₂³	—	86	4	no	—
280	C₂³	—	86	2	no	—
281	C₂³	—	172	2	no	—
282	C₂³	—	172	1	no	—
283	C₂³	—	172	1	no	—
284	C₂³	#13063303	1	1	yes	characteristic
285	D₄	—	43	4	no	—
286	D₄	—	86	4	no	—
287	D₄	—	86	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	86	2	no	—
291	C₂×C₄	—	172	1	no	—
292	C₂×C₄	—	172	1	no	—
293	C₂×C₄	—	172	2	no	—
294	C₂×C₄	—	172	2	no	—
295	C₂×C₄	—	172	2	no	—
296	C₂×C₄	—	172	2	no	—
297	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
298	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
299	C₂×C₄	#13063282	1	1	yes	characteristic
300	C₂×C₄	#13063298	1	1	yes	characteristic
301	C₂×C₄	#13063298	1	1	yes	characteristic
302	C₂×C₄	#13063303	1	1	yes	characteristic
303	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
304	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
305	C₂×C₄	#13063352	1	1	yes	characteristic
306	C₂×C₄	#13063357	1	1	yes	characteristic
307	C₈	—	4	2	no	—
308	C₈	—	8	1	no	—
309	C₈	—	8	1	no	—
310	C₈	—	43	2	no	—
311	C₈	—	43	2	no	—
312	C₈	—	86	1	no	—
313	C₈	—	86	1	no	—
314	C₈	—	344	1	no	—
315	C₈	—	344	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	86	4	no	—
319	C₂²	—	172	4	no	—
320	C₂²	—	172	4	no	—
321	C₂²	—	172	2	no	—
322	C₂²	—	172	2	no	—
323	C₂²	#14582758	1	1	yes	characteristic
324	C₂²	#14583178	1	1	yes	characteristic
325	C₂²	#14583188	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	172	2	no	—
330	C₄	#102×D₄₃	1	1	yes	characteristic
331	C₄	#14582899	1	1	yes	characteristic
332	C₄	#108×D₄₃	1	2	yes	—
333	C₄	#14583093	1	1	yes	characteristic
334	C₄	#14583103	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	172	4	no	—
338	C₂	#19134594	1	1	yes	characteristic
339	C₂	#19134632	1	1	yes	characteristic
340	C₂	#19134648	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063298
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1301	C₂
socle	C₂²×C₄₃	#380
prime radical	C₂²×D₁₇₂	Q₈
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1301	D₄₃
hypocenter	C₄₃	#8658
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
2. #19134766	C₂	semidirect
3. #19133341	C₂	semidirect
6. #19132492	C₂	semidirect
7. C₄₃×#1301	C₂	semidirect
131. D₁₇₂	#108	semidirect
171. #1301	D₄₃	semidirect
197. C₄₃	#8658	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582964	2
#19134766	2
#19133341	2
#19133021	2
#19133019	2
#19132492	2
C₄₃×#1301	2
#8658	43