Subgroups — #25084528

Order 11008.

1050 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	#19145510	C₂	1	2	yes	maximal, complemented
2	#19145048	C₂	1	2	yes	maximal, complemented
3	C₂×#14583276	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134263	C₂	1	4	yes	maximal, complemented
5	#19134209	C₂	1	2	yes	maximal, complemented
6	#19134188	C₂	1	2	yes	maximal, complemented
7	C₈×#10621877	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063350	—	2	4	no	—
10	C₂²×D₃₄₄	—	2	2	no	—
11	C₂×#13063322	C₂²	1	2	yes	—
12	#14583289	—	2	4	no	—
13	#14583276	—	2	2	no	—
14	#14583276	C₂²	1	2	yes	—
15	#14583276	C₂²	1	2	yes	—
16	#14583275	C₂²	1	4	yes	complemented
17	#14583275	C₂²	1	4	yes	complemented
18	#14583273	C₂²	1	4	yes	complemented
19	#14583247	C₂²	1	4	yes	complemented
20	#14583094	—	2	4	no	—
21	#14583032	—	2	4	no	—
22	#14582952	C₂²	1	1	yes	characteristic
23	C₄×#10621877	C₂²	1	1	yes	characteristic
24	#14582621	C₂²	1	2	yes	complemented
25	C₈×C₄₃⋊C₈	C₂²	1	2	yes	complemented
26	C₄₃×#532	C₂²	1	2	yes	—
27	C₄₃×#500	C₂²	1	1	yes	characteristic, complemented
28	C₄₃×#493	C₂²	1	1	yes	characteristic, complemented
29	C₄₃×#486	C₂²	1	2	yes	complemented
30	C₄₃×#466	—	2	2	no	—
31	C₄×D₈×C₄₃	—	2	2	no	—
32	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₂²×D₁₇₂	—	2	2	no	—
34	C₂²×D₁₇₂	—	2	2	no	—
35	C₂²×D₁₇₂	—	2	2	no	—
36	#13063350	—	4	16	no	—
37	C₂×#10621884	—	2	8	no	—
38	C₂×D₃₄₄	—	2	4	no	—
39	C₂×D₃₄₄	—	2	4	no	—
40	C₂×D₃₄₄	—	4	8	no	—
41	C₂×#10621882	—	2	8	no	—
42	C₂×#10621882	—	2	8	no	—
43	C₂×#10621877	D₄	1	2	yes	—
44	#13063322	—	2	2	no	—
45	#13063322	—	2	2	no	—
46	#13063322	—	2	2	no	—
47	#13063322	C₂³	1	4	yes	—
48	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
49	#13063245	—	2	8	no	—
50	C₂²⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
51	#13063236	D₄	1	2	yes	—
52	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
53	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
54	C₂×#41×C₄₃	—	2	2	no	—
55	C₂×D₈×C₄₃	—	2	2	no	—
56	C₂²×C₈×C₄₃	D₄	1	2	yes	—
57	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
58	#127×C₄₃	C₂³	1	2	yes	—
59	#124×C₄₃	—	2	2	no	—
60	#121×C₄₃	—	2	2	no	—
61	#121×C₄₃	—	2	2	no	—
62	C₄×D₄×C₄₃	—	2	2	no	—
63	C₄×D₄×C₄₃	—	2	2	no	—
64	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
65	#107×C₄₃	—	2	2	no	—
66	#102×C₄₃	—	2	4	no	—
67	C₄×C₈×C₄₃	D₄	1	2	yes	—
68	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
70	C₂³×D₄₃	—	4	2	no	—
71	#41×C₄₃	—	4	8	no	—
72	C₂×Q₈×C₄₃	—	2	2	no	—
73	C₂×D₄×C₄₃	—	2	2	no	—
74	C₂×D₄×C₄₃	—	2	2	no	—
75	C₂×D₄×C₄₃	—	2	2	no	—
76	C₂×D₄×C₄₃	—	4	2	no	—
77	C₂²×C₄×C₄₃	—	4	2	no	—
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂×D₁₇₂	—	2	4	no	—
82	C₂×D₁₇₂	—	2	4	no	—
83	C₂×D₁₇₂	—	2	4	no	—
84	C₂×D₁₇₂	—	4	2	no	—
85	C₂×D₁₇₂	—	4	2	no	—
86	C₂×D₁₇₂	—	4	2	no	—
87	C₂×D₁₇₂	—	4	8	no	—
88	C₂×D₁₇₂	—	4	8	no	—
89	C₂×D₁₇₂	—	8	4	no	—
90	D₈×C₄₃	—	4	4	no	—
91	C₂×C₈×C₄₃	—	2	4	no	—
92	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₄₃	—	2	2	no	—
95	C₄⋊C₄×C₄₃	—	2	2	no	—
96	C₂²⋊C₄×C₄₃	—	4	2	no	—
97	C₂²⋊C₄×C₄₃	—	4	2	no	—
98	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
102	#10621884	—	4	16	no	—
103	#10621882	—	4	16	no	—
104	#10621882	—	8	8	no	—
105	#10621877	D₈	1	8	yes	complemented
106	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
107	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
108	D₃₄₄	—	4	8	no	—
109	D₃₄₄	—	4	8	no	—
110	C₂³×C₄₃	—	4	2	no	—
111	C₂³×C₄₃	#127	1	1	yes	characteristic
112	C₂²×D₄₃	—	4	4	no	—
113	C₂²×D₄₃	—	8	4	no	—
114	C₂²×D₄₃	—	8	4	no	—
115	C₂²×D₄₃	—	8	4	no	—
116	Q₈×C₄₃	—	4	4	no	—
117	D₄×C₄₃	—	4	4	no	—
118	D₄×C₄₃	—	4	4	no	—
119	D₄×C₄₃	—	4	4	no	—
120	D₄×C₄₃	—	8	4	no	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	—	4	2	no	—
124	C₂×C₄×C₄₃	—	8	4	no	—
125	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
126	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
127	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×D₈	1	4	yes	—
129	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
132	D₁₇₂	—	4	8	no	—
133	D₁₇₂	—	4	8	no	—
134	D₁₇₂	—	4	8	no	—
135	D₁₇₂	—	8	4	no	—
136	D₁₇₂	—	8	4	no	—
137	D₁₇₂	—	8	4	no	—
138	C₈×C₄₃	—	2	2	no	—
139	C₈×C₄₃	AGL(1,ℤ/8)	1	4	yes	complemented
140	C₄₃⋊C₈	—	2	4	no	—
141	C₄₃⋊C₈	C₂×D₈	1	8	yes	complemented
142	#19629	—	43	1	no	maximal, Sylow
143	C₂²×C₄₃	—	2	1	no	—
144	C₂²×C₄₃	—	2	1	no	—
145	C₂²×C₄₃	—	8	2	no	—
146	C₂²×C₄₃	—	8	2	no	—
147	C₂²×C₄₃	—	8	2	no	—
148	C₂²×C₄₃	#493	1	2	yes	—
149	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
150	C₂×D₄₃	—	8	4	no	—
151	C₂×D₄₃	—	8	4	no	—
152	C₂×D₄₃	—	8	4	no	—
153	C₂×D₄₃	—	16	8	no	—
154	C₄×C₄₃	—	2	1	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	—	8	2	no	—
157	C₄×C₄₃	#493	1	2	yes	—
158	C₄×C₄₃	C₂²×D₈	1	2	yes	—
159	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
160	C₄×C₄₃	#576	1	2	yes	—
161	#3205	—	43	2	no	—
162	#3169	D₄₃	1	1	yes	characteristic, complemented
163	#3151	—	43	2	no	—
164	C₂×#493	—	43	1	no	—
165	#1721	—	43	4	no	—
166	#1712	—	43	2	no	—
167	#1706	—	43	2	no	—
168	C₈×M₁₆	—	43	1	no	—
169	C₂×C₄₃	—	2	2	no	—
170	C₂×C₄₃	—	16	2	no	—
171	C₂×C₄₃	C₂×#493	1	1	yes	characteristic
172	C₂×C₄₃	#3151	1	1	yes	characteristic
173	C₂×C₄₃	#3155	1	1	yes	characteristic
174	D₄₃	—	16	4	no	—
175	C₂×AGL(1,ℤ/8)	—	86	4	no	—
176	C₂²×D₈	—	86	2	no	—
177	#532	C₂×D₄₃	1	2	yes	—
178	C₂×#127	—	43	2	no	—
179	#500	C₂×D₄₃	1	1	yes	characteristic, complemented
180	#496	—	86	4	no	—
181	#493	—	43	4	no	—
182	#493	—	43	4	no	—
183	#493	—	43	2	no	—
184	#493	—	43	2	no	—
185	#493	—	86	2	no	—
186	#493	C₂×D₄₃	1	1	yes	characteristic, complemented
187	#492	—	43	4	no	—
188	#486	—	43	4	no	—
189	#486	C₂×D₄₃	1	2	yes	complemented
190	#466	—	2	2	no	—
191	#459	—	86	4	no	—
192	#447	—	86	4	no	—
193	C₄×D₈	—	2	2	no	—
194	#431	—	43	1	no	—
195	C₄×M₁₆	—	43	1	no	—
196	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
197	#322	—	43	2	no	—
198	C₈²	—	43	2	no	—
199	C₄₃	#19629	1	1	yes	characteristic, complemented, Sylow
200	C₂×#41	—	2	2	no	—
201	C₂²×D₄	—	86	2	no	—
202	C₂²×D₄	—	86	2	no	—
203	C₂²×D₄	—	86	2	no	—
204	AGL(1,ℤ/8)	—	172	16	no	—
205	C₂×SD₁₆	—	86	8	no	—
206	C₂×D₈	—	2	2	no	—
207	C₂×D₈	—	86	8	no	—
208	C₂×D₈	—	86	8	no	—
209	C₂×D₈	—	86	4	no	—
210	C₂×D₈	—	86	4	no	—
211	C₂×D₈	—	172	8	no	—
212	C₂×M₁₆	—	43	2	no	—
213	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
214	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
215	#127	—	43	4	no	—
216	#127	—	86	2	no	—
217	#127	—	86	2	no	—
218	#127	—	86	2	no	—
219	#127	C₂²×D₄₃	1	2	yes	—
220	#124	—	2	2	no	—
221	#121	—	2	2	no	—
222	#121	—	2	2	no	—
223	C₄×D₄	—	2	2	no	—
224	C₄×D₄	—	2	2	no	—
225	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
226	#107	—	2	2	no	—
227	C₄⋊C₈	—	43	2	no	—
228	#102	—	2	4	no	—
229	#102	—	86	8	no	—
230	C₂²⋊C₈	—	43	2	no	—
231	#97	—	43	2	no	—
232	C₄×C₈	—	43	2	no	—
233	C₄×C₈	—	43	2	no	—
234	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
235	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
237	C₂⁴	—	172	2	no	—
238	#41	—	4	8	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	86	4	no	—
245	C₂×D₄	—	86	4	no	—
246	C₂×D₄	—	86	4	no	—
247	C₂×D₄	—	344	4	no	—
248	C₂×D₄	—	172	8	no	—
249	C₂×D₄	—	172	2	no	—
250	C₂×D₄	—	172	2	no	—
251	C₂×D₄	—	172	2	no	—
252	C₂×D₄	—	172	8	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
255	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
257	SD₁₆	—	172	16	no	—
258	D₈	—	4	4	no	—
259	D₈	—	344	8	no	—
260	D₈	—	172	8	no	—
261	D₈	—	172	16	no	—
262	D₈	—	172	8	no	—
263	M₁₆	—	43	8	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	43	4	no	—
266	C₂×C₈	—	43	4	no	—
267	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
268	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	D₄×D₄₃	1	1	yes	characteristic
274	C₄²	D₄×D₄₃	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
276	C₄²	C₂³×D₄₃	1	1	yes	characteristic
277	C₂³	—	4	2	no	—
278	C₂³	—	344	4	no	—
279	C₂³	—	344	4	no	—
280	C₂³	—	344	4	no	—
281	C₂³	—	172	4	no	—
282	C₂³	#13063321	1	1	yes	characteristic
283	Q₈	—	4	4	no	—
284	D₄	—	4	4	no	—
285	D₄	—	4	4	no	—
286	D₄	—	4	4	no	—
287	D₄	—	8	4	no	—
288	D₄	—	344	4	no	—
289	D₄	—	344	4	no	—
290	D₄	—	344	4	no	—
291	D₄	—	172	8	no	—
292	D₄	—	172	8	no	—
293	D₄	—	172	8	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	#13063321	1	1	yes	characteristic
299	C₂×C₄	#13063321	1	1	yes	characteristic
300	C₂×C₄	#13063321	1	1	yes	characteristic
301	C₂×C₄	D₈×D₄₃	1	4	yes	—
302	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	43	8	no	—
307	C₈	—	86	4	no	—
308	C₈	#13063350	1	4	yes	complemented
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	344	4	no	—
315	C₂²	—	344	4	no	—
316	C₂²	—	344	4	no	—
317	C₂²	—	688	8	no	—
318	C₂²	#14583274	1	2	yes	—
319	C₂²	C₂×#13063321	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	2	no	—
323	C₄	#14583274	1	2	yes	—
324	C₄	C₂×D₈×D₄₃	1	2	yes	—
325	C₄	C₂×#13063350	1	2	yes	—
326	C₄	#14583701	1	2	yes	—
327	C₂	—	2	2	no	—
328	C₂	—	16	2	no	—
329	C₂	—	688	4	no	—
330	C₂	C₂×#14583274	1	1	yes	characteristic
331	C₂	#19145048	1	1	yes	characteristic
332	C₂	#19145082	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063321
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#3169	C₂
socle	C₂²×C₄₃	C₂×#127
hypercenter	#3169	D₄₃
hypocenter	C₄₃	#19629
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #19145510	C₂	semidirect
2. #19145048	C₂	semidirect
3. C₂×#14583276	C₂	semidirect
4. #19134263	C₂	semidirect
5. #19134209	C₂	semidirect
6. #19134188	C₂	semidirect
7. C₈×#10621877	C₂	semidirect
8. C₄₃×#3169	C₂	semidirect
16. #14583275	C₂²	semidirect
17. #14583275	C₂²	semidirect
18. #14583273	C₂²	semidirect
19. #14583247	C₂²	semidirect
24. #14582621	C₂²	semidirect
25. C₈×C₄₃⋊C₈	C₂²	semidirect
27. C₄₃×#500	C₂²	semidirect
28. C₄₃×#493	C₂²	semidirect
29. C₄₃×#486	C₂²	semidirect
105. #10621877	D₈	semidirect
139. C₈×C₄₃	AGL(1,ℤ/8)	semidirect
141. C₄₃⋊C₈	C₂×D₈	semidirect
162. #3169	D₄₃	semidirect
179. #500	C₂×D₄₃	semidirect
186. #493	C₂×D₄₃	semidirect
189. #486	C₂×D₄₃	semidirect
199. C₄₃	#19629	semidirect
308. C₈	#13063350	semidirect

Maximal subgroups

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

subgroup	index
#19145510	2
#19145048	2
C₂×#14583276	2
#19134263	2
#19134209	2
#19134188	2
C₈×#10621877	2
C₄₃×#3169	2
#19629	43