Subgroups — #25249964

Order 11008.

682 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063269	C₂	1	1	yes	maximal, characteristic
2	#19143464	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136459	C₂	1	4	yes	maximal, complemented
4	C₄×#13063269	C₂	1	4	yes	maximal, complemented
5	#19136377	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136314	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136311	C₂	1	1	yes	maximal, characteristic
8	#19136192	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#2948	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
11	#14583376	—	2	1	no	—
12	C₂×#13063269	C₄	1	8	yes	complemented
13	C₂×#13063269	C₂²	1	2	yes	—
14	C₂×#13063269	C₂²	1	2	yes	—
15	C₂×#13063265	C₂²	1	1	yes	characteristic
16	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
17	#14583007	—	2	2	no	—
18	C₄×Dic₁₇₂	C₂²	1	4	yes	complemented
19	#14582937	C₂²	1	1	yes	characteristic
20	#14582909	—	2	1	no	—
21	#14582726	C₂²	1	2	yes	complemented
22	#14582724	C₂²	1	2	yes	complemented
23	#14582702	C₂²	1	4	yes	complemented
24	#14582700	C₂²	1	4	yes	—
25	#14582675	C₂²	1	2	yes	complemented
26	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	complemented
27	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#518	—	2	1	no	—
29	C₄₃×#442	—	2	2	no	—
30	C₄×D₈×C₄₃	C₂²	1	4	yes	—
31	C₄₃×#427	C₂²	1	1	yes	characteristic
32	C₄₃×#418	—	2	1	no	—
33	C₄₃×#405	C₂²	1	1	yes	characteristic
34	C₂²×Dic₈₆	—	2	1	no	—
35	C₂×Dic₁₇₂	—	2	1	no	—
36	C₂×Dic₁₇₂	—	2	2	no	—
37	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	complemented
38	C₂×Dic₁₇₂	C₂³	1	2	yes	—
39	C₄×Dic₈₆	—	2	4	no	—
40	#13063290	—	4	2	no	—
41	#13063284	—	2	1	no	—
42	#13063281	—	4	2	no	—
43	#13063269	C₂×C₄	1	8	yes	complemented
44	#13063269	C₂×C₄	1	8	yes	complemented
45	#13063265	—	2	2	no	—
46	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	complemented
47	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
48	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
49	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
50	#13063259	C₂³	1	2	yes	—
51	#13063257	—	2	1	no	—
52	#13063250	—	2	4	no	—
53	C₂²×D₄×C₄₃	—	2	1	no	—
54	C₂×D₈×C₄₃	—	2	1	no	—
55	C₂×D₈×C₄₃	—	2	2	no	—
56	C₂×D₈×C₄₃	C₂×C₄	1	4	yes	—
57	C₂×D₈×C₄₃	C₂³	1	2	yes	—
58	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₄×D₄×C₄₃	—	2	4	no	—
61	C₄×D₄×C₄₃	—	4	2	no	—
62	#117×C₄₃	—	2	1	no	—
63	#117×C₄₃	D₄	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₄₃	—	4	2	no	—
65	#107×C₄₃	C₂³	1	2	yes	—
66	#106×C₄₃	—	2	1	no	—
67	#102×C₄₃	—	2	4	no	—
68	#97×C₄₃	D₄	1	2	yes	—
69	C₄×C₈×C₄₃	C₂³	1	2	yes	—
70	C₂⁴×C₄₃	—	4	1	no	—
71	C₂×D₄×C₄₃	—	2	4	no	—
72	C₂×D₄×C₄₃	—	2	2	no	—
73	C₂×D₄×C₄₃	—	4	1	no	—
74	C₂×D₄×C₄₃	—	4	2	no	—
75	C₂²×C₄×C₄₃	—	4	4	no	—
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₄₃	—	4	1	no	—
78	C₂×Dic₈₆	—	2	4	no	—
79	C₂×Dic₈₆	—	2	2	no	—
80	C₂×Dic₈₆	—	4	1	no	—
81	C₂×Dic₈₆	—	4	2	no	—
82	D₈×C₄₃	—	2	4	no	—
83	D₈×C₄₃	—	2	2	no	—
84	D₈×C₄₃	C₂²×C₄	1	4	yes	—
85	M₁₆×C₄₃	—	2	2	no	—
86	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₃	#41	1	2	yes	—
90	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₄₃	—	2	2	no	—
92	C₄⋊C₄×C₄₃	D₈	1	2	yes	—
93	C₂²⋊C₄×C₄₃	—	4	4	no	—
94	C₂²⋊C₄×C₄₃	—	4	1	no	—
95	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
96	C₄²×C₄₃	—	4	1	no	—
97	C₄²×C₄₃	C₂×D₄	1	2	yes	—
98	C₂²⋊Dic₄₃	—	4	1	no	—
99	C₄⋊Dic₄₃	—	2	2	no	—
100	Dic₄₃⋊C₄	—	4	4	no	—
101	C₄×Dic₄₃	—	4	4	no	—
102	C₄×Dic₄₃	—	4	1	no	—
103	Dic₁₇₂	—	2	4	no	—
104	Dic₁₇₂	—	2	2	no	—
105	Dic₁₇₂	C₂²×C₄	1	4	yes	—
106	C₄₃⋊C₁₆	—	2	2	no	—
107	C₄₃⋊C₁₆	C₂²×C₄	1	2	yes	—
108	C₄₃⋊C₁₆	C₂²×C₄	1	2	yes	—
109	C₂³×C₄₃	—	4	2	no	—
110	C₂³×C₄₃	—	4	4	no	—
111	C₂³×C₄₃	—	16	2	no	—
112	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
113	D₄×C₄₃	—	2	4	no	—
114	D₄×C₄₃	—	4	2	no	—
115	D₄×C₄₃	—	4	4	no	—
116	C₂×C₄×C₄₃	—	4	2	no	—
117	C₂×C₄×C₄₃	—	16	4	no	—
118	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₄₃	C₂×D₈	1	4	yes	—
120	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₄₃	#135	1	2	yes	—
122	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₄₃	—	4	2	no	—
124	C₂×Dic₄₃	—	4	4	no	—
125	C₂×Dic₄₃	—	4	2	no	—
126	Dic₈₆	—	2	4	no	—
127	Dic₈₆	—	4	2	no	—
128	Dic₈₆	—	4	4	no	—
129	C₈×C₄₃	—	2	2	no	—
130	C₈×C₄₃	C₄×D₄	1	2	yes	—
131	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
132	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
133	#33457	—	43	1	no	maximal, Sylow
134	C₂²×C₄₃	—	2	2	no	—
135	C₂²×C₄₃	—	4	4	no	—
136	C₂²×C₄₃	—	8	4	no	—
137	C₂²×C₄₃	—	16	4	no	—
138	C₂²×C₄₃	C₄×D₈	1	2	yes	—
139	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
140	C₄×C₄₃	—	2	4	no	—
141	C₄×C₄₃	—	8	2	no	—
142	C₄×C₄₃	C₄×D₈	1	2	yes	—
143	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
144	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
145	Dic₄₃	—	4	4	no	—
146	Dic₄₃	—	8	2	no	—
147	C₂²×SD₃₂	—	43	1	no	—
148	#2950	—	43	1	no	—
149	#2948	D₄₃	1	1	yes	characteristic
150	#2175	—	43	4	no	—
151	C₄×SD₃₂	—	43	4	no	—
152	#2158	—	43	1	no	—
153	#2141	—	43	1	no	—
154	#2140	—	43	1	no	—
155	#2108	—	43	1	no	—
156	C₂×C₄₃	—	2	2	no	—
157	C₂×C₄₃	—	8	4	no	—
158	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
159	C₂×C₄₃	#3416	1	2	yes	—
160	C₂²×Q₁₆	—	43	1	no	—
161	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
162	#519	—	86	1	no	—
163	#518	—	2	1	no	—
164	C₂×SD₃₂	—	43	2	no	—
165	C₂×SD₃₂	—	43	2	no	—
166	C₂×SD₃₂	—	43	8	no	—
167	C₂×M₃₂	—	43	1	no	—
168	C₂²×C₁₆	—	43	1	no	—
169	#442	—	2	2	no	—
170	#441	—	86	2	no	—
171	C₄×Q₁₆	—	43	4	no	—
172	C₄×D₈	C₂×D₄₃	1	4	yes	—
173	#427	—	43	1	no	—
174	#427	C₂×D₄₃	1	1	yes	characteristic
175	#419	—	86	1	no	—
176	#418	—	2	1	no	—
177	#405	C₂×D₄₃	1	1	yes	characteristic
178	#367	—	43	2	no	—
179	#366	—	43	2	no	—
180	#358	—	43	4	no	—
181	#357	—	43	4	no	—
182	#346	—	43	2	no	—
183	C₄×C₁₆	—	43	2	no	—
184	C₄₃	#33457	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₈	—	86	1	no	—
186	C₂²×D₄	—	2	1	no	—
187	C₂×Q₁₆	—	43	4	no	—
188	C₂×Q₁₆	—	43	2	no	—
189	C₂×Q₁₆	—	86	2	no	—
190	C₂×Q₁₆	—	86	1	no	—
191	C₂×D₈	—	2	2	no	—
192	C₂×D₈	—	2	1	no	—
193	C₂×D₈	C₄×D₄₃	1	4	yes	—
194	C₂×D₈	C₂²×D₄₃	1	2	yes	—
195	C₂×M₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
197	C₄×Q₈	—	86	4	no	—
198	C₄×Q₈	—	172	2	no	—
199	C₄×D₄	—	2	4	no	—
200	C₄×D₄	—	4	2	no	—
201	#117	—	2	1	no	—
202	#117	—	86	1	no	—
203	#117	—	172	2	no	—
204	#117	C₂²⋊D₄₃	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	—	4	2	no	—
206	SD₃₂	—	43	8	no	—
207	SD₃₂	—	43	8	no	—
208	M₃₂	—	86	2	no	—
209	C₂×C₁₆	—	43	4	no	—
210	C₂×C₁₆	—	43	1	no	—
211	C₂×C₁₆	—	43	1	no	—
212	C₂×C₁₆	—	43	2	no	—
213	#107	—	43	2	no	—
214	#107	C₂²×D₄₃	1	2	yes	—
215	#106	—	2	1	no	—
216	#106	—	86	1	no	—
217	#103	—	86	4	no	—
218	#102	—	2	4	no	—
219	#97	C₂²⋊D₄₃	1	2	yes	—
220	C₄×C₈	C₂²×D₄₃	1	2	yes	—
221	C₂⁴	—	4	1	no	—
222	C₂×Q₈	—	86	4	no	—
223	C₂×Q₈	—	86	2	no	—
224	C₂×Q₈	—	172	1	no	—
225	C₂×Q₈	—	172	2	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	—	172	1	no	—
232	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
233	Q₁₆	—	43	4	no	—
234	Q₁₆	—	86	4	no	—
235	Q₁₆	—	86	2	no	—
236	D₈	—	2	4	no	—
237	D₈	—	2	2	no	—
238	D₈	C₂×C₄×D₄₃	1	4	yes	—
239	M₁₆	—	2	2	no	—
240	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
241	C₂×C₈	#10621898	1	2	yes	—
242	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
243	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
244	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	86	2	no	—
247	C₄⋊C₄	—	172	4	no	—
248	C₄⋊C₄	#10621882	1	2	yes	—
249	C₂²⋊C₄	—	4	1	no	—
250	C₂²⋊C₄	—	4	4	no	—
251	C₂²⋊C₄	—	172	1	no	—
252	C₂²⋊C₄	#10621882	1	2	yes	—
253	C₄²	—	4	1	no	—
254	C₄²	—	172	1	no	—
255	C₄²	—	172	4	no	—
256	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
257	C₁₆	—	43	2	no	—
258	C₁₆	—	43	2	no	—
259	C₁₆	—	86	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	16	2	no	—
263	C₂³	C₂×#10621882	1	1	yes	characteristic
264	Q₈	—	86	4	no	—
265	Q₈	—	172	2	no	—
266	Q₈	—	172	4	no	—
267	D₄	—	2	4	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	16	4	no	—
272	C₂×C₄	—	172	2	no	—
273	C₂×C₄	—	172	2	no	—
274	C₂×C₄	—	172	4	no	—
275	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
276	C₂×C₄	C₂×#10621882	1	4	yes	—
277	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
278	C₂×C₄	#13063345	1	2	yes	—
279	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
280	C₈	—	2	2	no	—
281	C₈	C₄×C₂²⋊D₄₃	1	2	yes	—
282	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
283	C₈	C₂×#10621898	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	—	16	4	no	—
288	C₂²	C₄×#10621882	1	2	yes	—
289	C₂²	C₂²×#10621882	1	1	yes	characteristic
290	C₄	—	2	4	no	—
291	C₄	—	8	2	no	—
292	C₄	—	344	2	no	—
293	C₄	—	172	4	no	—
294	C₄	C₄×#10621882	1	2	yes	—
295	C₄	C₂×C₄×C₂²⋊D₄₃	1	1	yes	characteristic
296	C₄	C₂×#13063345	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	8	4	no	—
299	C₂	C₂×C₄×#10621882	1	1	yes	characteristic
300	C₂	#19147742	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2948	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂²×D₈×C₄₃	C₂²
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	#2948	D₄₃
hypocenter	C₄₃	#33457
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2948

Complements

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

N	H ≅ G/N	type
2. #19143464	C₂	semidirect
3. #19136459	C₂	semidirect
4. C₄×#13063269	C₂	semidirect
5. #19136377	C₂	semidirect
6. #19136314	C₂	semidirect
8. #19136192	C₂	semidirect
12. C₂×#13063269	C₄	semidirect
18. C₄×Dic₁₇₂	C₂²	semidirect
21. #14582726	C₂²	semidirect
22. #14582724	C₂²	semidirect
23. #14582702	C₂²	semidirect
25. #14582675	C₂²	semidirect
26. C₄×C₄₃⋊C₁₆	C₂²	semidirect
37. C₂×Dic₁₇₂	C₂×C₄	semidirect
43. #13063269	C₂×C₄	semidirect
44. #13063269	C₂×C₄	semidirect
46. C₂×C₄₃⋊C₁₆	C₂×C₄	semidirect
184. C₄₃	#33457	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063269	2
#19143464	2
#19136459	2
C₄×#13063269	2
#19136377	2
#19136314	2
#19136311	2
#19136192	2
C₄₃×#2948	2
#33457	43