Finite Group Databasebeta

← #34056489

Subgroups — #34056489

Order 19072.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750430C211yesmaximal, characteristic, complemented
2#23750429C212yesmaximal, complemented
3#23750426C211yesmaximal, characteristic, complemented
4#23750000C211yesmaximal, characteristic, complemented
5#23749896C211yesmaximal, characteristic, complemented
6#23749894C212yesmaximal, complemented
7#23749795C212yesmaximal, complemented
8#23749791C212yesmaximal, complemented
9#23749741C211yesmaximal, characteristic, complemented
10#23749740C211yesmaximal, characteristic, complemented
11#472×C149C211yesmaximal, characteristic, complemented
12#18011957C2212yescomplemented
13#18011956C2212yescomplemented
14#18011939—22no—
15#18011939C2212yescomplemented
16#18011939C2212yescomplemented
17#18011938—22no—
18#18011937C2212yescomplemented
19#18011935—22no—
20#18011935—22no—
21#18011933—21no—
22#18011933C2211yescharacteristic
23#18011932—22no—
24#18011931C2211yescharacteristic
25#18011930—21no—
26C2×D1192—21no—
27C2×D1192C2211yescharacteristic, complemented
28C2×#14558721C2212yescomplemented
29D8×D149—21no—
30D8×D149—22no—
31#18011919C2212yescomplemented
32#18011917C2211yescharacteristic, complemented
33#18011907C2212yescomplemented
34#18011846C2211yescharacteristic, complemented
35#18011838C2211yescharacteristic, complemented
36#18011837C2211yescharacteristic, complemented
37#18011837C2212yescomplemented
38#18011827C2212yescomplemented
39C8×Dic149C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C149C2212yescomplemented
41C2×D8×C149C2211yescharacteristic, complemented
42#131×C149C2211yescharacteristic, complemented
43#108×C149C2211yescharacteristic, complemented
44#100×C149C2212yescomplemented
45#14558737—22no—
46#14558736—22no—
47#14558736C2311yescharacteristic
48#14558736C2312yes—
49#14558735—22no—
50#14558735—22no—
51C2×D596—22no—
52C2×D596C2312yes—
53C2×C22⋊D149—42no—
54C2×C22⋊D149—42no—
55D4×D149—22no—
56D4×D149—22no—
57D4×D149—22no—
58D4×D149—42no—
59Dic596D411yescharacteristic, complemented
60#14558726D411yescharacteristic, complemented
61#14558725—22no—
62#14558725—22no—
63#14558725D412yescomplemented
64#14558724—22no—
65#14558724D411yescharacteristic, complemented
66#14558723D411yescharacteristic, complemented
67D1192—22no—
68D1192—22no—
69D1192—21no—
70D1192—21no—
71D1192D411yescharacteristic, complemented
72#14558721—22no—
73#14558721—22no—
74#14558721—22no—
75#14558721D411yescharacteristic, complemented
76#14558720C2311yescharacteristic
77#14558719—22no—
78#14558719—21no—
79C2×#11924746C2311yescharacteristic
80C8×D149—22no—
81C8×D149—21no—
82C4×Dic149C2311yescharacteristic
83#41×C149C2311yescharacteristic
84C2×D4×C149C2312yes—
85SD16×C149—22no—
86D8×C149—22no—
87D8×C149—21no—
88D8×C149—21no—
89M16×C149—21no—
90M16×C149C2312yes—
91M16×C149C2311yescharacteristic
92C2×C8×C149—21no—
93C2×C8×C149C2311yescharacteristic
94C22×D149—22no—
95C22×D149—42no—
96C22×D149—42no—
97Dic298C2×D412yes—
98Dic298C2×D411yescharacteristic
99D596—22no—
100D596—22no—
101D596—42no—
102D596C2×D412yes—
103D596C2×D411yescharacteristic
104C22⋊D149—22no—
105C22⋊D149—21no—
106C22⋊D149—42no—
107C22⋊D149—42no—
108C22⋊D149—42no—
109C22⋊D149—84no—
110C2×Dic149—21no—
111C2×Dic149—42no—
112C4×D149—21no—
113C4×D149—22no—
114C4×D149—42no—
115#11924746—21no—
116#11924746C2×D411yescharacteristic
117#11924746C2×D411yescharacteristic
118C23×C149—22no—
119Q8×C149C2×D411yescharacteristic
120D4×C149—21no—
121D4×C149—22no—
122D4×C149—22no—
123D4×C149C2×D411yescharacteristic
124C2×C4×C149—21no—
125C2×C4×C149C2411yescharacteristic
126C8×C149—21no—
127C8×C149—22no—
128C8×C149C2×D411yescharacteristic
129C8×C149C2×D411yescharacteristic
130C22×C149—21no—
131C22×C149—42no—
132C22×C149—82no—
133C22×C149H(5,𝔽2)11yescharacteristic
134C2×D149—21no—
135C2×D149—22no—
136C2×D149—42no—
137C2×D149—82no—
138C2×D149—162no—
139C2×D149—162no—
140C4×C149—21no—
141C4×C149C22×D411yescharacteristic
142C4×C149C22×D411yescharacteristic
143Dic149—22no—
144Dic149—22no—
145Dic149—21no—
146C2×C149—21no—
147C2×C149—41no—
148C2×C149—82no—
149C2×C149D4211yescharacteristic
150D149—41no—
151D149—42no—
152D149—82no—
153C149#329211yescharacteristic, complemented, Sylow
154#3292—1491nomaximal, Sylow
155#576—1492no—
156#576—1491no—
157#576—1491no—
158#493—1491no—
159#472—1491no—
160#472—1492no—
161#472D14911yescharacteristic, complemented
162#453—1492no—
163#453—1492no—
164#443—1491no—
165#443—1491no—
166H(5,𝔽2)—1492no—
167H(5,𝔽2)—1492no—
168AGL(1,ℤ/8)—2982no—
169AGL(1,ℤ/8)—2982no—
170AGL(1,ℤ/8)—2982no—
171AGL(1,ℤ/8)—2982no—
172AGL(1,ℤ/8)—1492no—
173AGL(1,ℤ/8)—1492no—
174AGL(1,ℤ/8)—1492no—
175AGL(1,ℤ/8)C2×D14912yescomplemented
176#135—2981no—
177#135—2981no—
178#135—2982no—
179#135—1491no—
180#135—1491no—
181C2×D8—2981no—
182C2×D8—2982no—
183C2×D8—2981no—
184C2×D8—1492no—
185C2×D8—1491no—
186C2×D8C2×D14911yescharacteristic, complemented
187#131—1492no—
188#131—1491no—
189#131C2×D14911yescharacteristic, complemented
190#127—1492no—
191#108—1491no—
192#108C2×D14911yescharacteristic, complemented
193#104—1491no—
194#104—1492no—
195#104—1491no—
196#100—1492no—
197#100C2×D14912yescomplemented
198C4×C8—1491no—
199#41—2982no—
200#41—2982no—
201#41—2982no—
202#41—2982no—
203#41—1491no—
204#41—1492no—
205#41C22×D14911yescharacteristic
206C2×D4—2982no—
207C2×D4—2982no—
208C2×D4—2982no—
209C2×D4—2982no—
210C2×D4—5962no—
211C2×D4—5962no—
212C2×D4—5962no—
213C2×D4—1492no—
214C2×D4C22×D14912yes—
215Q16—1491no—
216Q16—1491no—
217SD16—22no—
218SD16—2982no—
219SD16—2982no—
220SD16—2982no—
221SD16—1491no—
222SD16—1491no—
223SD16—1492no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—2982no—
228D8—2982no—
229D8—2982no—
230D8—2982no—
231D8—2982no—
232D8—2981no—
233D8—2981no—
234D8—1491no—
235D8—1491no—
236M16—21no—
237M16—2982no—
238M16—2981no—
239M16—1491no—
240M16C22×D14912yes—
241M16C22×D14911yescharacteristic
242C2×C8—21no—
243C2×C8—2982no—
244C2×C8—2981no—
245C2×C8—1491no—
246C2×C8C22×D14911yescharacteristic
247C42—1491no—
248C23—22no—
249C23—2982no—
250C23—5962no—
251C23—5962no—
252Q8—1491no—
253Q8—1492no—
254Q8D4×D14911yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—2982no—
259D4—2981no—
260D4—2982no—
261D4—2982no—
262D4—5962no—
263D4—5962no—
264D4—5962no—
265D4—5962no—
266D4—1491no—
267D4—1492no—
268D4—11924no—
269D4D4×D14911yescharacteristic
270C2×C4—21no—
271C2×C4—2981no—
272C2×C4—2982no—
273C2×C4—2981no—
274C2×C4—5962no—
275C2×C4—5962no—
276C2×C4C23×D14911yescharacteristic
277C8—21no—
278C8—22no—
279C8—2981no—
280C8—1491no—
281C8—1491no—
282C8D4×D14911yescharacteristic
283C8D4×D14911yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—2982no—
288C22—2981no—
289C22—23842no—
290C22—23842no—
291C22—5962no—
292C22—11922no—
293C22#1801195711yescharacteristic
294C4—21no—
295C4—2982no—
296C4—2982no—
297C4—2981no—
298C4C2×D4×D14911yescharacteristic
299C4C2×D4×D14911yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—5962no—
304C2—5961no—
305C2—11922no—
306C2#2375025911yescharacteristic
subgroupG/H
centerC2#23750259
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#472×C149C2
socleC2×C149D42
hypercenter#472D149
hypocenterC149#3292
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #23750430C2semidirect
2. #23750429C2semidirect
3. #23750426C2semidirect
4. #23750000C2semidirect
5. #23749896C2semidirect
6. #23749894C2semidirect
7. #23749795C2semidirect
8. #23749791C2semidirect
9. #23749741C2semidirect
10. #23749740C2semidirect
11. #472×C149C2semidirect
12. #18011957C22semidirect
13. #18011956C22semidirect
15. #18011939C22semidirect
16. #18011939C22semidirect
18. #18011937C22semidirect
27. C2×D1192C22semidirect
28. C2×#14558721C22semidirect
31. #18011919C22semidirect
32. #18011917C22semidirect
33. #18011907C22semidirect
34. #18011846C22semidirect
35. #18011838C22semidirect
36. #18011837C22semidirect
37. #18011837C22semidirect
38. #18011827C22semidirect
39. C8×Dic149C22semidirect
40. AGL(1,ℤ/8)×C149C22semidirect
41. C2×D8×C149C22semidirect
42. #131×C149C22semidirect
43. #108×C149C22semidirect
44. #100×C149C22semidirect
59. Dic596D4semidirect
60. #14558726D4semidirect
63. #14558725D4semidirect
65. #14558724D4semidirect
66. #14558723D4semidirect
71. D1192D4semidirect
75. #14558721D4semidirect
153. C149#3292semidirect
161. #472D149semidirect
175. AGL(1,ℤ/8)C2×D149semidirect
186. C2×D8C2×D149semidirect
189. #131C2×D149semidirect
192. #108C2×D149semidirect
197. #100C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237504302
#237504292
#237504262
#237500002
#237498962
#237498942
#237497952
#237497912
#237497412
#237497402
#472×C1492
#3292149