Subgroups — #25014006

Order 11008.

470 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147698	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14582712	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136311	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134649	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132709	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132698	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#472	C₂	1	1	yes	maximal, characteristic
8	#14583316	—	2	2	no	—
9	#14583316	—	2	2	no	—
10	C₂×#13063265	—	2	2	no	—
11	C₂×#13063265	C₂²	1	1	yes	characteristic
12	C₂×#13063265	C₂²	1	1	yes	characteristic
13	C₂×#13063261	C₂²	1	1	yes	characteristic
14	#14582937	C₂²	1	1	yes	characteristic
15	#14582916	—	2	1	no	—
16	#14582712	—	2	2	no	—
17	#14582700	—	2	2	no	—
18	#14582671	—	2	1	no	—
19	C₂×AGL(1,ℤ/8)×C₄₃	—	2	1	no	—
20	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#472	—	2	2	no	—
23	C₄₃×#472	C₄	1	4	yes	complemented
24	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#100×C₄₃	—	2	1	no	—
26	C₂×#10621877	—	2	1	no	—
27	#13063284	—	2	1	no	—
28	#13063265	—	2	4	no	—
29	#13063265	—	2	1	no	—
30	#13063265	—	2	1	no	—
31	#13063265	—	2	1	no	—
32	#13063265	D₄	1	2	yes	—
33	C₂×C₄₃⋊C₁₆	—	2	4	no	—
34	C₂×C₄₃⋊C₁₆	—	2	1	no	—
35	C₂×C₄₃⋊C₁₆	D₄	1	2	yes	—
36	#13063261	—	2	1	no	—
37	#13063261	—	2	1	no	—
38	#13063259	—	2	1	no	—
39	#13063257	—	2	1	no	—
40	#13063252	—	4	2	no	—
41	C₂×#41×C₄₃	D₄	1	1	yes	characteristic
42	C₂²×D₄×C₄₃	—	2	1	no	—
43	AGL(1,ℤ/8)×C₄₃	—	2	4	no	—
44	AGL(1,ℤ/8)×C₄₃	—	4	2	no	—
45	C₂×SD₁₆×C₄₃	—	4	1	no	—
46	C₂×D₈×C₄₃	—	2	1	no	—
47	C₂×D₈×C₄₃	—	2	1	no	—
48	C₂×D₈×C₄₃	—	2	1	no	—
49	C₂×D₈×C₄₃	—	2	1	no	—
50	C₂×D₈×C₄₃	—	2	1	no	—
51	C₂×D₈×C₄₃	—	4	1	no	—
52	C₂×D₈×C₄₃	C₂×C₄	1	2	yes	—
53	#131×C₄₃	—	2	1	no	—
54	#131×C₄₃	—	2	1	no	—
55	#131×C₄₃	C₂×C₄	1	2	yes	—
56	#131×C₄₃	D₄	1	2	yes	—
57	C₂×M₁₆×C₄₃	—	2	1	no	—
58	C₂×M₁₆×C₄₃	—	2	1	no	—
59	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₄₃	—	2	1	no	—
61	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	#108×C₄₃	—	2	1	no	—
63	#108×C₄₃	C₂×C₄	1	2	yes	—
64	#100×C₄₃	—	2	4	no	—
65	C₂⁴×C₄₃	—	4	1	no	—
66	#41×C₄₃	—	2	1	no	—
67	#41×C₄₃	—	2	1	no	—
68	#41×C₄₃	C₂²⋊C₄	1	2	yes	—
69	#41×C₄₃	D₈	1	2	yes	—
70	C₂×Q₈×C₄₃	SD₁₆	1	1	yes	characteristic
71	C₂×D₄×C₄₃	—	2	1	no	—
72	C₂×D₄×C₄₃	—	2	2	no	—
73	C₂×D₄×C₄₃	—	4	1	no	—
74	C₂×D₄×C₄₃	—	4	1	no	—
75	C₂×D₄×C₄₃	—	4	1	no	—
76	C₂×D₄×C₄₃	—	4	1	no	—
77	C₂×D₄×C₄₃	—	4	1	no	—
78	C₂×D₄×C₄₃	SD₁₆	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	—	2	1	no	—
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	SD₁₆×C₄₃	—	4	4	no	—
82	D₈×C₄₃	—	2	2	no	—
83	D₈×C₄₃	—	2	2	no	—
84	D₈×C₄₃	—	4	4	no	—
85	D₈×C₄₃	—	4	1	no	—
86	D₈×C₄₃	—	4	1	no	—
87	M₁₆×C₄₃	—	2	1	no	—
88	M₁₆×C₄₃	—	2	2	no	—
89	M₁₆×C₄₃	—	2	2	no	—
90	M₁₆×C₄₃	—	2	2	no	—
91	M₁₆×C₄₃	—	4	1	no	—
92	M₁₆×C₄₃	C₂²⋊C₄	1	2	yes	—
93	C₂×C₈×C₄₃	—	2	1	no	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₂×C₈×C₄₃	—	2	2	no	—
96	C₂×C₈×C₄₃	—	4	1	no	—
97	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
103	C₂²⋊Dic₄₃	—	4	1	no	—
104	C₄⋊Dic₄₃	—	4	1	no	—
105	C₄×Dic₄₃	—	4	1	no	—
106	#10621877	—	4	1	no	—
107	#10621877	—	4	1	no	—
108	C₂×C₄₃⋊C₈	—	4	1	no	—
109	C₄₃⋊C₁₆	—	2	2	no	—
110	C₄₃⋊C₁₆	—	2	2	no	—
111	C₂³×C₄₃	—	2	1	no	—
112	C₂³×C₄₃	—	4	2	no	—
113	C₂³×C₄₃	—	8	1	no	—
114	C₂³×C₄₃	—	8	1	no	—
115	C₂³×C₄₃	—	16	2	no	—
116	C₂³×C₄₃	#135	1	1	yes	characteristic
117	Q₈×C₄₃	—	2	1	no	—
118	Q₈×C₄₃	#102	1	2	yes	—
119	D₄×C₄₃	—	2	2	no	—
120	D₄×C₄₃	—	2	2	no	—
121	D₄×C₄₃	—	2	1	no	—
122	D₄×C₄₃	—	4	2	no	—
123	D₄×C₄₃	—	4	2	no	—
124	D₄×C₄₃	—	8	1	no	—
125	D₄×C₄₃	—	8	1	no	—
126	D₄×C₄₃	#102	1	2	yes	—
127	C₂×C₄×C₄₃	—	2	1	no	—
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	2	2	no	—
130	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
136	C₂×Dic₄₃	—	8	1	no	—
137	C₈×C₄₃	—	2	2	no	—
138	C₈×C₄₃	—	4	2	no	—
139	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
140	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
141	C₈×C₄₃	#120	1	1	yes	characteristic
142	C₈×C₄₃	#123	1	1	yes	characteristic
143	C₄₃⋊C₈	—	8	1	no	—
144	#13068	—	43	1	no	maximal, Sylow
145	C₂²×C₄₃	—	2	2	no	—
146	C₂²×C₄₃	—	4	1	no	—
147	C₂²×C₄₃	—	8	1	no	—
148	C₂²×C₄₃	—	8	2	no	—
149	C₂²×C₄₃	—	16	2	no	—
150	C₂²×C₄₃	—	16	2	no	—
151	C₂²×C₄₃	—	16	2	no	—
152	C₂²×C₄₃	#416	1	1	yes	characteristic
153	C₂²×C₄₃	#465	1	1	yes	characteristic
154	C₂²×C₄₃	#466	1	1	yes	characteristic
155	C₄×C₄₃	—	2	2	no	—
156	C₄×C₄₃	#386	1	1	yes	characteristic
157	C₄×C₄₃	C₂×#102	1	1	yes	characteristic
158	C₄×C₄₃	#465	1	1	yes	characteristic
159	C₄×C₄₃	#466	1	1	yes	characteristic
160	Dic₄₃	—	16	1	no	—
161	#3407	—	43	1	no	—
162	C₂×#472	D₄₃	1	1	yes	characteristic
163	C₂×#361	—	43	1	no	—
164	#2140	—	43	1	no	—
165	#1814	—	43	1	no	—
166	#1389	—	43	1	no	—
167	#1385	—	43	1	no	—
168	C₂×C₄₃	—	2	1	no	—
169	C₂×C₄₃	—	2	1	no	—
170	C₂×C₄₃	—	2	1	no	—
171	C₂×C₄₃	—	4	2	no	—
172	C₂×C₄₃	—	16	2	no	—
173	C₂×C₄₃	#1893	1	1	yes	characteristic
174	C₂×AGL(1,ℤ/8)	—	2	1	no	—
175	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
176	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
177	#504	—	86	2	no	—
178	#504	—	86	2	no	—
179	C₂×M₃₂	—	43	1	no	—
180	C₂×M₃₂	—	43	1	no	—
181	C₂×M₃₂	—	86	2	no	—
182	#472	—	2	2	no	—
183	#472	Dic₄₃	1	4	yes	complemented
184	C₂×#108	—	43	1	no	—
185	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
186	#427	—	43	1	no	—
187	#421	—	86	1	no	—
188	C₂×#100	—	2	1	no	—
189	#361	—	86	2	no	—
190	#357	—	86	2	no	—
191	#344	—	86	1	no	—
192	C₄₃	#13068	1	1	yes	characteristic, complemented, Sylow
193	C₂×#41	C₂²⋊D₄₃	1	1	yes	characteristic
194	C₂²×D₄	—	2	1	no	—
195	AGL(1,ℤ/8)	—	2	4	no	—
196	AGL(1,ℤ/8)	—	4	2	no	—
197	C₂×SD₁₆	—	4	1	no	—
198	C₂×D₈	—	2	1	no	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	4	1	no	—
204	C₂×D₈	C₂×Dic₄₃	1	2	yes	—
205	#131	—	2	1	no	—
206	#131	—	2	1	no	—
207	#131	C₂×Dic₄₃	1	2	yes	—
208	#131	C₂²⋊D₄₃	1	2	yes	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	86	1	no	—
212	C₂×M₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
213	C₂²×C₈	—	2	1	no	—
214	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
215	#117	—	86	1	no	—
216	M₃₂	—	43	2	no	—
217	M₃₂	—	86	4	no	—
218	M₃₂	—	86	1	no	—
219	M₃₂	—	86	1	no	—
220	M₃₂	—	86	1	no	—
221	C₂×C₁₆	—	43	2	no	—
222	C₂×C₁₆	—	86	4	no	—
223	C₂×C₁₆	—	86	1	no	—
224	#108	—	2	1	no	—
225	#108	—	86	1	no	—
226	#108	—	86	1	no	—
227	#108	C₂×Dic₄₃	1	2	yes	—
228	#107	—	86	1	no	—
229	#106	—	86	1	no	—
230	#104	—	172	2	no	—
231	#100	—	2	4	no	—
232	C₂⁴	—	4	1	no	—
233	#41	—	2	1	no	—
234	#41	—	2	1	no	—
235	#41	#10621882	1	2	yes	—
236	#41	C₂²⋊Dic₄₃	1	2	yes	—
237	C₂×Q₈	#10621883	1	1	yes	characteristic
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	#10621883	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
248	SD₁₆	—	4	4	no	—
249	D₈	—	2	2	no	—
250	D₈	—	2	2	no	—
251	D₈	—	4	4	no	—
252	D₈	—	4	1	no	—
253	D₈	—	4	1	no	—
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	2	2	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	172	1	no	—
260	M₁₆	—	172	1	no	—
261	M₁₆	C₂²⋊Dic₄₃	1	2	yes	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	172	1	no	—
267	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
268	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
269	C₂×C₈	#10621900	1	1	yes	characteristic
270	C₂×C₈	#10621900	1	1	yes	characteristic
271	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
273	C₄⋊C₄	—	172	1	no	—
274	C₂²⋊C₄	—	172	1	no	—
275	C₄²	—	172	1	no	—
276	C₁₆	—	86	2	no	—
277	C₁₆	—	86	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	8	1	no	—
281	C₂³	—	8	1	no	—
282	C₂³	—	16	2	no	—
283	C₂³	#13063345	1	1	yes	characteristic
284	Q₈	—	2	1	no	—
285	Q₈	#13063246	1	2	yes	—
286	D₄	—	2	2	no	—
287	D₄	—	2	2	no	—
288	D₄	—	2	1	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	8	1	no	—
292	D₄	—	8	1	no	—
293	D₄	#13063246	1	2	yes	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	344	1	no	—
298	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
299	C₂×C₄	#13063297	1	1	yes	characteristic
300	C₂×C₄	#13063297	1	1	yes	characteristic
301	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
302	C₂×C₄	C₂×#10621883	1	1	yes	characteristic
303	C₂×C₄	#13063345	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	4	2	no	—
306	C₈	—	344	1	no	—
307	C₈	D₄×Dic₄₃	1	1	yes	characteristic
308	C₈	D₄×Dic₄₃	1	1	yes	characteristic
309	C₈	#13063293	1	1	yes	characteristic
310	C₈	#13063306	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	8	1	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	16	2	no	—
316	C₂²	—	16	2	no	—
317	C₂²	—	16	2	no	—
318	C₂²	#14582895	1	1	yes	characteristic
319	C₂²	#14583126	1	1	yes	characteristic
320	C₂²	#14583133	1	1	yes	characteristic
321	C₄	—	2	2	no	—
322	C₄	—	688	1	no	—
323	C₄	#14582779	1	1	yes	characteristic
324	C₄	C₂×#13063246	1	1	yes	characteristic
325	C₄	#14583126	1	1	yes	characteristic
326	C₄	#14583133	1	1	yes	characteristic
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	4	2	no	—
331	C₂	—	16	2	no	—
332	C₂	#19135024	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19135024
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting, prime radical	C₂×C₄₃×#472	C₂
socle	C₂×C₄₃	#1893
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#472	D₄₃
hypocenter	C₄₃	#13068
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#472

Complements

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

N	H ≅ G/N	type
1. #19147698	C₂	semidirect
2. C₂×#14582712	C₂	semidirect
3. #19136311	C₂	semidirect
4. #19134649	C₂	semidirect
5. #19132709	C₂	semidirect
6. #19132698	C₂	semidirect
23. C₄₃×#472	C₄	semidirect
183. #472	Dic₄₃	semidirect
192. C₄₃	#13068	semidirect

Maximal subgroups

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

subgroup	index
#19147698	2
C₂×#14582712	2
#19136311	2
#19134649	2
#19132709	2
#19132698	2
C₂×C₄₃×#472	2
#13068	43