Subgroups — #25084527

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	#19145515	C₂	1	2	yes	maximal, complemented
2	#19145174	C₂	1	1	yes	maximal, characteristic, complemented
3	#19145047	C₂	1	2	yes	maximal, complemented
4	C₂×#14583276	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134261	C₂	1	4	yes	maximal, complemented
6	#19134210	C₂	1	2	yes	maximal, complemented
7	#19134187	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063353	—	2	4	no	—
10	C₂²×D₃₄₄	—	2	2	no	—
11	#14583449	C₂²	1	2	yes	—
12	C₂×#13063322	C₂²	1	2	yes	—
13	#14583304	C₂²	1	1	yes	characteristic, complemented
14	#14583288	—	2	4	no	—
15	#14583276	—	2	2	no	—
16	#14583276	C₂²	1	2	yes	—
17	#14583276	C₂²	1	2	yes	—
18	#14583276	C₂²	1	1	yes	characteristic, complemented
19	#14583276	C₂²	1	4	yes	complemented
20	#14583276	C₂²	1	4	yes	complemented
21	#14583272	C₂²	1	4	yes	complemented
22	#14583246	C₂²	1	4	yes	complemented
23	#14583246	C₂²	1	2	yes	complemented
24	#14583135	—	2	2	no	—
25	#14583095	—	2	4	no	—
26	#14583035	—	2	4	no	—
27	C₄×D₃₄₄	—	2	2	no	—
28	C₄₃×#431	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₄₃×#322	C₂²	1	2	yes	complemented
32	C₈²×C₄₃	C₂²	1	2	yes	complemented
33	C₂×#10621898	—	2	2	no	—
34	C₂²×D₁₇₂	—	2	2	no	—
35	C₂²×D₁₇₂	—	2	2	no	—
36	C₂²×D₁₇₂	—	2	2	no	—
37	#13063353	—	4	16	no	—
38	C₂×#10621873	—	2	8	no	—
39	C₂×D₃₄₄	—	2	2	no	—
40	C₂×D₃₄₄	—	2	4	no	—
41	C₂×D₃₄₄	—	2	4	no	—
42	C₂×D₃₄₄	—	2	8	no	—
43	C₂×D₃₄₄	—	2	8	no	—
44	C₂×D₃₄₄	—	4	8	no	—
45	#13063325	C₂³	1	2	yes	—
46	#13063322	—	2	2	no	—
47	#13063322	—	2	2	no	—
48	#13063322	—	2	2	no	—
49	#13063322	C₂³	1	4	yes	—
50	#13063322	C₂³	1	2	yes	—
51	#13063313	—	2	2	no	—
52	#13063299	—	2	2	no	—
53	#13063299	—	2	2	no	—
54	C₄×D₁₇₂	—	2	2	no	—
55	C₄×D₁₇₂	—	2	2	no	—
56	#13063259	—	2	2	no	—
57	#13063247	—	2	8	no	—
58	#13063247	—	2	4	no	—
59	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
60	C₂²×C₈×C₄₃	D₄	1	2	yes	—
61	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
63	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
64	#97×C₄₃	D₄	1	2	yes	—
65	C₄×C₈×C₄₃	D₄	1	2	yes	—
66	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₃	C₂³	1	2	yes	—
68	C₄×C₈×C₄₃	C₂³	1	2	yes	—
69	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
70	C₂³×D₄₃	—	4	2	no	—
71	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₄₃	—	4	2	no	—
75	#10621898	—	4	8	no	—
76	C₂×D₁₇₂	—	2	2	no	—
77	C₂×D₁₇₂	—	2	4	no	—
78	C₂×D₁₇₂	—	2	2	no	—
79	C₂×D₁₇₂	—	2	2	no	—
80	C₂×D₁₇₂	—	2	4	no	—
81	C₂×D₁₇₂	—	2	4	no	—
82	C₂×D₁₇₂	—	4	2	no	—
83	C₂×D₁₇₂	—	4	2	no	—
84	C₂×D₁₇₂	—	4	2	no	—
85	C₂×D₁₇₂	—	4	8	no	—
86	C₂×D₁₇₂	—	4	8	no	—
87	C₂×D₁₇₂	—	8	4	no	—
88	C₂×C₄×D₄₃	—	4	2	no	—
89	C₂×Dic₈₆	—	2	2	no	—
90	M₁₆×C₄₃	D₈	1	8	yes	complemented
91	C₂×C₈×C₄₃	—	2	4	no	—
92	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
94	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
95	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
96	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
100	(C₂×D₄₃)⋊C₄	—	4	2	no	—
101	(C₂×D₄₃)⋊C₄	—	4	2	no	—
102	C₄⋊Dic₄₃	—	2	2	no	—
103	C₄⋊Dic₄₃	—	2	2	no	—
104	D₃₄₄	—	4	8	no	—
105	D₃₄₄	—	4	16	no	—
106	D₃₄₄	—	4	4	no	—
107	D₃₄₄	—	4	8	no	—
108	D₃₄₄	—	8	8	no	—
109	#10621873	—	4	16	no	—
110	C₂³×C₄₃	#127	1	1	yes	characteristic
111	C₂²×D₄₃	—	4	2	no	—
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	C₂×C₄×C₄₃	—	2	2	no	—
117	C₂×C₄×C₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
119	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×D₈	1	4	yes	—
122	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
125	C₂²⋊D₄₃	—	8	4	no	—
126	C₂×Dic₄₃	—	4	2	no	—
127	D₁₇₂	—	4	8	no	—
128	D₁₇₂	—	4	4	no	—
129	D₁₇₂	—	4	8	no	—
130	D₁₇₂	—	4	4	no	—
131	D₁₇₂	—	4	4	no	—
132	D₁₇₂	—	4	8	no	—
133	D₁₇₂	—	8	4	no	—
134	D₁₇₂	—	8	4	no	—
135	D₁₇₂	—	8	4	no	—
136	C₄×D₄₃	—	8	4	no	—
137	Dic₈₆	—	4	4	no	—
138	C₈×C₄₃	—	2	2	no	—
139	C₈×C₄₃	—	2	4	no	—
140	C₈×C₄₃	C₂×D₈	1	8	yes	complemented
141	C₈×C₄₃	AGL(1,ℤ/8)	1	4	yes	complemented
142	#19629	—	43	1	no	maximal, Sylow
143	C₂²×C₄₃	—	2	1	no	—
144	C₂²×C₄₃	—	2	1	no	—
145	C₂²×C₄₃	#493	1	2	yes	—
146	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
147	C₂×D₄₃	—	8	4	no	—
148	C₂×D₄₃	—	8	2	no	—
149	C₂×D₄₃	—	8	4	no	—
150	C₂×D₄₃	—	8	2	no	—
151	C₂×D₄₃	—	8	4	no	—
152	C₂×D₄₃	—	8	2	no	—
153	C₂×D₄₃	—	16	8	no	—
154	C₄×C₄₃	—	2	1	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	#493	1	2	yes	—
157	C₄×C₄₃	C₂²×D₈	1	2	yes	—
158	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
159	C₄×C₄₃	#576	1	2	yes	—
160	Dic₄₃	—	8	2	no	—
161	#3205	—	43	2	no	—
162	#3169	—	43	1	no	—
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	D₄₃	1	1	yes	characteristic, complemented
169	C₂×C₄₃	—	2	2	no	—
170	C₂×C₄₃	C₂×#493	1	1	yes	characteristic
171	C₂×C₄₃	#3151	1	1	yes	characteristic
172	C₂×C₄₃	#3155	1	1	yes	characteristic
173	D₄₃	—	16	2	no	—
174	D₄₃	—	16	4	no	—
175	C₂×AGL(1,ℤ/8)	—	86	4	no	—
176	C₂²×D₈	—	86	2	no	—
177	#532	—	43	2	no	—
178	C₂×#127	—	43	2	no	—
179	#500	—	43	1	no	—
180	#496	—	86	4	no	—
181	#493	—	43	2	no	—
182	#493	—	43	2	no	—
183	#493	—	43	4	no	—
184	#493	—	43	1	no	—
185	#493	—	43	4	no	—
186	#493	—	86	2	no	—
187	#492	—	43	4	no	—
188	#486	—	43	2	no	—
189	#486	—	43	4	no	—
190	#466	—	86	2	no	—
191	#459	—	86	4	no	—
192	#447	—	86	4	no	—
193	C₄×D₈	—	86	2	no	—
194	#431	C₂×D₄₃	1	1	yes	characteristic
195	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
197	#322	C₂×D₄₃	1	2	yes	complemented
198	C₈²	C₂×D₄₃	1	2	yes	complemented
199	C₄₃	#19629	1	1	yes	characteristic, complemented, Sylow
200	C₂×#41	—	86	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	—	86	8	no	—
207	C₂×D₈	—	86	4	no	—
208	C₂×D₈	—	86	4	no	—
209	C₂×D₈	—	86	8	no	—
210	C₂×D₈	—	86	2	no	—
211	C₂×D₈	—	172	8	no	—
212	C₂×M₁₆	D₁₇₂	1	2	yes	—
213	C₂²×C₈	D₁₇₂	1	2	yes	—
214	C₄⋊Q₈	—	43	2	no	—
215	#127	—	43	4	no	—
216	#127	—	43	2	no	—
217	#127	—	86	2	no	—
218	#127	—	86	2	no	—
219	#127	—	86	2	no	—
220	#124	—	86	2	no	—
221	#121	—	86	2	no	—
222	#121	—	86	2	no	—
223	C₄×D₄	—	86	2	no	—
224	C₄×D₄	—	86	2	no	—
225	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
226	#107	—	86	2	no	—
227	C₄⋊C₈	D₁₇₂	1	2	yes	—
228	#102	—	86	4	no	—
229	#102	—	86	8	no	—
230	C₂²⋊C₈	D₁₇₂	1	2	yes	—
231	#97	D₁₇₂	1	2	yes	—
232	C₄×C₈	D₁₇₂	1	2	yes	—
233	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₄₃	1	2	yes	—
235	C₄×C₈	C₂²×D₄₃	1	2	yes	—
236	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
237	C₂⁴	—	172	2	no	—
238	#41	—	172	8	no	—
239	C₂×Q₈	—	86	2	no	—
240	C₂×D₄	—	86	2	no	—
241	C₂×D₄	—	86	4	no	—
242	C₂×D₄	—	86	2	no	—
243	C₂×D₄	—	86	2	no	—
244	C₂×D₄	—	86	4	no	—
245	C₂×D₄	—	86	4	no	—
246	C₂×D₄	—	344	4	no	—
247	C₂×D₄	—	172	8	no	—
248	C₂×D₄	—	172	2	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	—	172	2	no	—
254	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
257	SD₁₆	—	172	16	no	—
258	D₈	—	344	8	no	—
259	D₈	—	172	8	no	—
260	D₈	—	172	16	no	—
261	D₈	—	172	8	no	—
262	D₈	—	172	4	no	—
263	M₁₆	D₃₄₄	1	8	yes	complemented
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
266	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
267	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
268	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
269	C₄⋊C₄	—	86	2	no	—
270	C₄⋊C₄	—	86	2	no	—
271	C₂²⋊C₄	—	172	2	no	—
272	C₂²⋊C₄	—	172	2	no	—
273	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
274	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
275	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
276	C₄²	C₂³×D₄₃	1	1	yes	characteristic
277	C₂³	—	344	4	no	—
278	C₂³	—	344	4	no	—
279	C₂³	—	344	4	no	—
280	C₂³	—	172	2	no	—
281	C₂³	—	172	4	no	—
282	C₂³	#13063322	1	1	yes	characteristic
283	Q₈	—	172	4	no	—
284	D₄	—	344	4	no	—
285	D₄	—	344	4	no	—
286	D₄	—	344	4	no	—
287	D₄	—	344	4	no	—
288	D₄	—	172	4	no	—
289	D₄	—	172	8	no	—
290	D₄	—	172	8	no	—
291	D₄	—	172	4	no	—
292	D₄	—	172	4	no	—
293	D₄	—	172	8	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	344	4	no	—
297	C₂×C₄	—	172	2	no	—
298	C₂×C₄	#13063322	1	1	yes	characteristic
299	C₂×C₄	#13063322	1	1	yes	characteristic
300	C₂×C₄	#13063322	1	1	yes	characteristic
301	C₂×C₄	C₂×D₃₄₄	1	4	yes	—
302	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	2	4	no	—
307	C₈	C₂×D₃₄₄	1	8	yes	complemented
308	C₈	#13063353	1	4	yes	complemented
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	344	2	no	—
312	C₂²	—	344	4	no	—
313	C₂²	—	344	4	no	—
314	C₂²	—	344	2	no	—
315	C₂²	—	344	4	no	—
316	C₂²	—	344	2	no	—
317	C₂²	—	688	8	no	—
318	C₂²	#14583276	1	2	yes	—
319	C₂²	C₂×#13063322	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	344	2	no	—
323	C₄	#14583276	1	2	yes	—
324	C₄	C₂²×D₃₄₄	1	2	yes	—
325	C₄	C₂×#13063353	1	2	yes	—
326	C₄	#14583704	1	2	yes	—
327	C₂	—	2	2	no	—
328	C₂	—	688	4	no	—
329	C₂	—	688	2	no	—
330	C₂	C₂×#14583276	1	1	yes	characteristic
331	C₂	#19145047	1	1	yes	characteristic
332	C₂	#19145081	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063322
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₈×M₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂×#127
hypercenter	C₈×M₁₆	D₄₃
hypocenter	C₄₃	#19629
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #19145515	C₂	semidirect
2. #19145174	C₂	semidirect
3. #19145047	C₂	semidirect
4. C₂×#14583276	C₂	semidirect
5. #19134261	C₂	semidirect
6. #19134210	C₂	semidirect
7. #19134187	C₂	semidirect
8. C₈×M₁₆×C₄₃	C₂	semidirect
13. #14583304	C₂²	semidirect
18. #14583276	C₂²	semidirect
19. #14583276	C₂²	semidirect
20. #14583276	C₂²	semidirect
21. #14583272	C₂²	semidirect
22. #14583246	C₂²	semidirect
23. #14583246	C₂²	semidirect
31. C₄₃×#322	C₂²	semidirect
32. C₈²×C₄₃	C₂²	semidirect
90. M₁₆×C₄₃	D₈	semidirect
140. C₈×C₄₃	C₂×D₈	semidirect
141. C₈×C₄₃	AGL(1,ℤ/8)	semidirect
168. C₈×M₁₆	D₄₃	semidirect
197. #322	C₂×D₄₃	semidirect
198. C₈²	C₂×D₄₃	semidirect
199. C₄₃	#19629	semidirect
263. M₁₆	D₃₄₄	semidirect
307. C₈	C₂×D₃₄₄	semidirect
308. C₈	#13063353	semidirect

Maximal subgroups

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

subgroup	index
#19145515	2
#19145174	2
#19145047	2
C₂×#14583276	2
#19134261	2
#19134210	2
#19134187	2
C₈×M₁₆×C₄₃	2
#19629	43