Subgroups — #25084578

Order 11008.

878 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145536	C₂	1	2	yes	maximal, complemented
2	#19145181	C₂	1	1	yes	maximal, characteristic, complemented
3	#19144969	C₂	1	2	yes	maximal, complemented
4	#19134233	C₂	1	4	yes	maximal, complemented
5	#19134200	C₂	1	2	yes	maximal, complemented
6	#19134184	C₂	1	2	yes	maximal, complemented
7	C₈×#10621877	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#493	C₂	1	1	yes	maximal, characteristic
9	#14583456	C₂²	1	2	yes	—
10	#14583304	C₂²	1	1	yes	characteristic, complemented
11	#14583279	C₂²	1	1	yes	characteristic, complemented
12	#14583271	C₂²	1	4	yes	complemented
13	#14583263	—	2	4	no	—
14	#14583250	C₂²	1	2	yes	complemented
15	#14583245	C₂²	1	4	yes	complemented
16	#14583245	C₂²	1	4	yes	complemented
17	#14583245	C₂²	1	4	yes	complemented
18	#14583140	—	2	2	no	—
19	#14583096	—	2	4	no	—
20	#14583050	—	2	4	no	—
21	C₄×Dic₁₇₂	—	2	2	no	—
22	#14582952	C₂²	1	1	yes	characteristic
23	#14582904	—	2	4	no	—
24	C₄×#10621877	C₂²	1	1	yes	characteristic
25	#14582621	C₂²	1	2	yes	complemented
26	C₈×C₄₃⋊C₈	C₂²	1	2	yes	complemented
27	C₂²×D₈×C₄₃	—	2	2	no	—
28	C₂×#127×C₄₃	C₂²	1	2	yes	—
29	C₄₃×#493	—	2	2	no	—
30	C₄₃×#493	C₂²	1	2	yes	—
31	C₄₃×#493	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×Dic₁₇₂	—	2	2	no	—
34	C₂×#10621883	—	2	8	no	—
35	C₂×#10621877	D₄	1	2	yes	—
36	#13063325	C₂³	1	2	yes	—
37	#13063325	C₂³	1	2	yes	—
38	#13063317	—	2	2	no	—
39	#13063304	—	2	2	no	—
40	#13063304	—	2	2	no	—
41	C₄×Dic₈₆	—	2	2	no	—
42	C₄×Dic₈₆	—	2	2	no	—
43	#13063284	—	2	2	no	—
44	#13063259	—	2	2	no	—
45	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
46	#13063250	—	2	4	no	—
47	#13063246	—	2	8	no	—
48	#13063246	—	2	8	no	—
49	#13063246	—	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	C₂²×D₄×C₄₃	—	2	2	no	—
55	C₂²×D₄×C₄₃	—	2	2	no	—
56	C₂²×D₄×C₄₃	—	2	2	no	—
57	C₂×D₈×C₄₃	—	2	4	no	—
58	C₂×D₈×C₄₃	—	2	4	no	—
59	C₂×D₈×C₄₃	—	4	8	no	—
60	C₂²×C₈×C₄₃	D₄	1	2	yes	—
61	#127×C₄₃	—	2	2	no	—
62	#127×C₄₃	—	2	2	no	—
63	#127×C₄₃	—	2	2	no	—
64	#127×C₄₃	C₂³	1	4	yes	—
65	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₄₃	D₄	1	2	yes	—
67	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₂⁴×C₄₃	—	4	2	no	—
70	C₂×D₄×C₄₃	—	2	4	no	—
71	C₂×D₄×C₄₃	—	2	4	no	—
72	C₂×D₄×C₄₃	—	2	4	no	—
73	C₂×D₄×C₄₃	—	4	2	no	—
74	C₂×D₄×C₄₃	—	4	2	no	—
75	C₂×D₄×C₄₃	—	4	2	no	—
76	C₂×D₄×C₄₃	—	4	8	no	—
77	C₂×D₄×C₄₃	—	4	8	no	—
78	C₂×D₄×C₄₃	—	8	4	no	—
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₈₆	—	2	2	no	—
83	C₂×Dic₈₆	—	2	2	no	—
84	D₈×C₄₃	—	4	8	no	—
85	D₈×C₄₃	—	4	8	no	—
86	C₂×C₈×C₄₃	—	2	4	no	—
87	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
89	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
93	C₂²⋊Dic₄₃	—	4	2	no	—
94	#10621883	—	8	8	no	—
95	C₄⋊Dic₄₃	—	2	2	no	—
96	C₄⋊Dic₄₃	—	2	2	no	—
97	C₄⋊Dic₄₃	—	2	2	no	—
98	C₄⋊Dic₄₃	—	2	2	no	—
99	Dic₄₃⋊C₄	—	4	2	no	—
100	Dic₄₃⋊C₄	—	4	2	no	—
101	C₄×Dic₄₃	—	4	2	no	—
102	#10621877	—	2	4	no	—
103	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
104	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
105	Dic₁₇₂	—	4	4	no	—
106	C₂³×C₄₃	—	4	4	no	—
107	C₂³×C₄₃	—	8	4	no	—
108	C₂³×C₄₃	—	8	4	no	—
109	C₂³×C₄₃	—	8	4	no	—
110	C₂³×C₄₃	#127	1	1	yes	characteristic
111	D₄×C₄₃	—	4	8	no	—
112	D₄×C₄₃	—	4	8	no	—
113	D₄×C₄₃	—	4	8	no	—
114	D₄×C₄₃	—	8	4	no	—
115	D₄×C₄₃	—	8	4	no	—
116	D₄×C₄₃	—	8	4	no	—
117	C₂×C₄×C₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	—	2	2	no	—
119	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
121	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#135	1	4	yes	—
123	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×Dic₄₃	—	4	2	no	—
127	C₂×Dic₄₃	—	4	2	no	—
128	Dic₈₆	—	4	4	no	—
129	Dic₈₆	—	4	4	no	—
130	C₈×C₄₃	—	2	2	no	—
131	C₈×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
132	C₄₃⋊C₈	—	2	4	no	—
133	C₄₃⋊C₈	—	2	4	no	—
134	#19635	—	43	1	no	maximal, Sylow
135	C₂²×C₄₃	—	2	1	no	—
136	C₂²×C₄₃	—	2	1	no	—
137	C₂²×C₄₃	—	8	4	no	—
138	C₂²×C₄₃	—	8	4	no	—
139	C₂²×C₄₃	—	8	4	no	—
140	C₂²×C₄₃	—	16	8	no	—
141	C₂²×C₄₃	#495	1	2	yes	—
142	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
143	C₄×C₄₃	—	2	1	no	—
144	C₄×C₄₃	—	2	1	no	—
145	C₄×C₄₃	#495	1	2	yes	—
146	C₄×C₄₃	C₂×#135	1	2	yes	—
147	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
148	C₄×C₄₃	#576	1	2	yes	—
149	Dic₄₃	—	8	2	no	—
150	Dic₄₃	—	8	2	no	—
151	#3208	—	43	2	no	—
152	#3170	—	43	1	no	—
153	C₂×#493	D₄₃	1	1	yes	characteristic
154	#3138	—	43	2	no	—
155	#1717	—	43	4	no	—
156	#1709	—	43	2	no	—
157	#1706	—	43	2	no	—
158	C₈×M₁₆	—	43	1	no	—
159	C₂×C₄₃	—	2	2	no	—
160	C₂×C₄₃	—	16	4	no	—
161	C₂×C₄₃	C₂×#495	1	1	yes	characteristic
162	C₂×C₄₃	#3151	1	1	yes	characteristic
163	C₂×C₄₃	#3155	1	1	yes	characteristic
164	C₂²×D₈	—	2	2	no	—
165	#533	—	43	2	no	—
166	C₂×#127	C₂×D₄₃	1	2	yes	—
167	#500	—	43	1	no	—
168	#494	—	43	1	no	—
169	#493	—	2	2	no	—
170	#493	C₂×D₄₃	1	2	yes	—
171	#493	C₂×D₄₃	1	2	yes	—
172	#492	—	43	4	no	—
173	#490	—	86	4	no	—
174	#487	—	43	2	no	—
175	#486	—	43	4	no	—
176	#486	—	43	4	no	—
177	#486	—	43	4	no	—
178	#467	—	86	2	no	—
179	#460	—	86	4	no	—
180	#450	—	86	4	no	—
181	C₄×Q₁₆	—	86	2	no	—
182	#431	—	43	1	no	—
183	#418	—	86	4	no	—
184	C₄×M₁₆	—	43	1	no	—
185	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
186	#322	—	43	2	no	—
187	C₈²	—	43	2	no	—
188	C₄₃	#19635	1	1	yes	characteristic, complemented, Sylow
189	C₂²×D₄	—	2	2	no	—
190	C₂²×D₄	—	2	2	no	—
191	C₂²×D₄	—	2	2	no	—
192	C₂×Q₁₆	—	86	2	no	—
193	C₂×SD₁₆	—	86	8	no	—
194	C₂×D₈	—	2	4	no	—
195	C₂×D₈	—	2	4	no	—
196	C₂×D₈	—	4	8	no	—
197	C₂×M₁₆	—	43	2	no	—
198	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
199	C₄⋊Q₈	—	43	2	no	—
200	C₄⋊Q₈	—	43	2	no	—
201	#127	—	2	2	no	—
202	#127	—	2	2	no	—
203	#127	—	2	2	no	—
204	#127	C₂²×D₄₃	1	4	yes	—
205	#125	—	86	2	no	—
206	C₂²⋊Q₈	—	86	2	no	—
207	C₂²⋊Q₈	—	86	2	no	—
208	C₄×Q₈	—	86	2	no	—
209	C₄×Q₈	—	86	2	no	—
210	#117	—	86	2	no	—
211	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
212	#107	—	86	2	no	—
213	C₄⋊C₈	—	43	2	no	—
214	#103	—	86	4	no	—
215	#102	—	86	8	no	—
216	#102	—	86	8	no	—
217	#102	—	86	8	no	—
218	C₂²⋊C₈	—	43	2	no	—
219	#97	—	43	2	no	—
220	C₄×C₈	—	43	2	no	—
221	C₄×C₈	—	43	2	no	—
222	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
223	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
224	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
225	C₂⁴	—	4	2	no	—
226	C₂×Q₈	—	86	2	no	—
227	C₂×Q₈	—	86	2	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	4	8	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	8	no	—
236	C₂×D₄	—	8	4	no	—
237	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
238	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
240	Q₁₆	—	172	4	no	—
241	SD₁₆	—	344	8	no	—
242	D₈	—	4	8	no	—
243	D₈	—	4	8	no	—
244	M₁₆	—	86	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	43	4	no	—
247	C₂×C₈	—	43	4	no	—
248	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
249	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
250	C₄⋊C₄	—	86	2	no	—
251	C₄⋊C₄	—	86	2	no	—
252	C₄⋊C₄	—	86	2	no	—
253	C₄⋊C₄	—	86	2	no	—
254	C₄⋊C₄	—	172	2	no	—
255	C₄⋊C₄	—	172	2	no	—
256	C₂²⋊C₄	—	172	2	no	—
257	C₄²	—	172	2	no	—
258	C₄²	D₄×D₄₃	1	1	yes	characteristic
259	C₄²	D₄×D₄₃	1	1	yes	characteristic
260	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
261	C₄²	C₂³×D₄₃	1	1	yes	characteristic
262	C₂³	—	4	4	no	—
263	C₂³	—	8	4	no	—
264	C₂³	—	8	4	no	—
265	C₂³	—	8	4	no	—
266	C₂³	#13063321	1	1	yes	characteristic
267	Q₈	—	172	4	no	—
268	Q₈	—	172	4	no	—
269	D₄	—	4	8	no	—
270	D₄	—	4	8	no	—
271	D₄	—	4	8	no	—
272	D₄	—	8	4	no	—
273	D₄	—	8	4	no	—
274	D₄	—	8	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	172	2	no	—
278	C₂×C₄	—	172	2	no	—
279	C₂×C₄	#13063321	1	1	yes	characteristic
280	C₂×C₄	#13063321	1	1	yes	characteristic
281	C₂×C₄	#13063321	1	1	yes	characteristic
282	C₂×C₄	#13063344	1	4	yes	—
283	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
284	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	86	4	no	—
288	C₈	—	86	4	no	—
289	C₈	#13063351	1	4	yes	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	8	4	no	—
293	C₂²	—	8	4	no	—
294	C₂²	—	8	4	no	—
295	C₂²	—	16	8	no	—
296	C₂²	#14583282	1	2	yes	—
297	C₂²	C₂×#13063321	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	344	2	no	—
301	C₄	—	344	2	no	—
302	C₄	#14583282	1	2	yes	—
303	C₄	C₂×#13063344	1	2	yes	—
304	C₄	C₂×#13063351	1	2	yes	—
305	C₄	#14583700	1	2	yes	—
306	C₂	—	2	2	no	—
307	C₂	—	16	4	no	—
308	C₂	C₂×#14583282	1	1	yes	characteristic
309	C₂	#19145045	1	1	yes	characteristic
310	C₂	#19145076	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063321
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting, prime radical	C₂×C₄₃×#493	C₂
socle	C₂²×C₄₃	C₂×#127
hypercenter	C₂×#493	D₄₃
hypocenter	C₄₃	#19635
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #19145536	C₂	semidirect
2. #19145181	C₂	semidirect
3. #19144969	C₂	semidirect
4. #19134233	C₂	semidirect
5. #19134200	C₂	semidirect
6. #19134184	C₂	semidirect
7. C₈×#10621877	C₂	semidirect
10. #14583304	C₂²	semidirect
11. #14583279	C₂²	semidirect
12. #14583271	C₂²	semidirect
14. #14583250	C₂²	semidirect
15. #14583245	C₂²	semidirect
16. #14583245	C₂²	semidirect
17. #14583245	C₂²	semidirect
25. #14582621	C₂²	semidirect
26. C₈×C₄₃⋊C₈	C₂²	semidirect
188. C₄₃	#19635	semidirect

Maximal subgroups

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

subgroup	index
#19145536	2
#19145181	2
#19144969	2
#19134233	2
#19134200	2
#19134184	2
C₈×#10621877	2
C₂×C₄₃×#493	2
#19635	43