Finite Group Databasebeta

← #31815650

Subgroups — #31815650

Order 17792.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#14555786C211yesmaximal, characteristic, complemented
2C2×#17984873C211yesmaximal, characteristic, complemented
3C2×#17984870C211yesmaximal, characteristic, complemented
4#22581465C211yesmaximal, characteristic, complemented
5#22581118C214yesmaximal, complemented
6#22581101C214yesmaximal, complemented
7#22580957C211yesmaximal, characteristic, complemented
8#22580916C211yesmaximal, characteristic, complemented
9#419×C139C211yesmaximal, characteristic, complemented
10C2×#14555786—24no—
11C2×#14555786—22no—
12C2×#14555786—24no—
13C2×#14555786C2214yes—
14C2×#14555785C2211yescharacteristic
15C22×D556—21no—
16C22×D556C2211yescharacteristic
17C22×C4×D139—21no—
18C22×C4×D139C2211yescharacteristic
19#17984873—24no—
20#17984870—24no—
21C2×#14555775C2212yescomplemented
22C2×#14555774C2212yescomplemented
23C2×#14555772C2212yescomplemented
24C2×D1112C2212yescomplemented
25C2×#14555769C2211yescharacteristic
26#17984826—21no—
27#17984822C2212yescomplemented
28#17984817C2212yescomplemented
29#17984812—21no—
30C4×D556—22no—
31#17984762C2214yescomplemented
32(C2×D139)⋊C8C2214yescomplemented
33C22×Q8×C139C2211yescharacteristic
34C2×M16×C139C2211yescharacteristic
35#117×C139C2211yescharacteristic
36#103×C139C2214yescomplemented
37C23×D139—21no—
38C23×D139D411yescharacteristic
39#14555786—216no—
40#14555786—28no—
41#14555786—28no—
42#14555785—22no—
43#14555785—22no—
44C2×Dic278C2311yescharacteristic
45C2×D556—28no—
46C2×D556—24no—
47C2×D556—22no—
48C2×D556—21no—
49C2×D556—22no—
50C2×D556D414yescomplemented
51C2×D556C2311yescharacteristic
52C2×D556C2312yes—
53C2×C22⋊D139—21no—
54C22×Dic139D411yescharacteristic
55C2×C4×D139—28no—
56C2×C4×D139—24no—
57C2×C4×D139—22no—
58C2×C4×D139—22no—
59C2×C4×D139—21no—
60C2×C4×D139D412yes—
61C2×C4×D139C2312yes—
62#14555775—24no—
63#14555774—24no—
64#14555772—24no—
65D1112—24no—
66#14555769—22no—
67C2×C139⋊C8C2312yes—
68C4⋊Dic139—22no—
69(C2×D139)⋊C4—22no—
70(C2×D139)⋊C4—22no—
71C2×Q8×C139—22no—
72C2×Q8×C139—21no—
73C2×Q8×C139D414yescomplemented
74C2×Q8×C139C2312yes—
75C22×C4×C139—21no—
76C22×C4×C139C2311yescharacteristic
77M16×C139—22no—
78C2×C8×C139C2312yes—
79C4⋊C4×C139C2312yes—
80C22⋊C4×C139—21no—
81C42×C139—21no—
82C22×D139—22no—
83C22×D139—24no—
84C22×D139—22no—
85C22×D139—21no—
86C22×D139—42no—
87C22×D139—44no—
88C22×D139C2×D412yes—
89Dic278—21no—
90Dic278—21no—
91D556—216no—
92D556—24no—
93D556—22no—
94D556—21no—
95D556—21no—
96D556—44no—
97D556C2×D414yes—
98C22⋊D139—42no—
99C2×Dic139—22no—
100C2×Dic139—21no—
101C2×Dic139C2×D412yes—
102C4×D139—216no—
103C4×D139—24no—
104C4×D139—22no—
105C4×D139—22no—
106C4×D139—42no—
107C139⋊C8—22no—
108C23×C139C2×D411yescharacteristic
109Q8×C139—24no—
110Q8×C139—22no—
111Q8×C139C2×D414yes—
112C2×C4×C139—24no—
113C2×C4×C139—22no—
114C2×C4×C139—22no—
115C2×C4×C139C2×D411yescharacteristic
116C2×C4×C139C2×D412yes—
117C2×C4×C139C2×D412yes—
118C2×C4×C139C2411yescharacteristic
119C8×C139—22no—
120C22×C139—22no—
121C22×C139#12012yes—
122C22×C139C22×D411yescharacteristic
123C2×D139—22no—
124C2×D139—24no—
125C2×D139—22no—
126C2×D139—22no—
127C2×D139—44no—
128C2×D139—48no—
129C2×D139—44no—
130C2×D139—41no—
131C2×D139—42no—
132C4×C139—24no—
133C4×C139—42no—
134C4×C139#12012yes—
135C4×C139C22×D411yescharacteristic
136C4×C139C22×D411yescharacteristic
137Dic139—22no—
138Dic139—41no—
139C2×C139—22no—
140C2×C139C2×#12011yescharacteristic
141C2×C139#57512yes—
142D139—42no—
143D139—44no—
144D139—81no—
145C139#300511yescharacteristic, complemented, Sylow
146#3005—1391nomaximal, Sylow
147C22×#41—1391no—
148C2×#137—1391no—
149C2×AGL(1,ℤ/8)—1391no—
150#525—1391no—
151#452—1394no—
152#449—1394no—
153#419D13911yescharacteristic, complemented
154#418—1391no—
155#407—1391no—
156C2×#41—2784no—
157C2×#41—2784no—
158C2×#41—2782no—
159C2×#41—1394no—
160C2×#41—1391no—
161C22×Q8C2×D13911yescharacteristic
162C22×D4—2781no—
163C22×D4—1391no—
164C23×C4—2781no—
165C23×C4—1391no—
166#137—2784no—
167AGL(1,ℤ/8)—2784no—
168C2×Q16—1392no—
169C2×SD16—1392no—
170C2×SD16—1392no—
171C2×D8—1392no—
172C2×M16—1391no—
173C2×M16C2×D13911yescharacteristic
174#123—2781no—
175C22⋊Q8—1392no—
176#121—1392no—
177#120—2781no—
178C4×D4—2782no—
179#117C2×D13911yescharacteristic
180#103C2×D13914yescomplemented
181#102—1394no—
182C22⋊C8—1394no—
183C24—2781no—
184C24—1391no—
185#41—27816no—
186#41—2788no—
187#41—2788no—
188#41—2782no—
189#41—2782no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—1391no—
193C2×Q8D55614yescomplemented
194C2×Q8C22×D13912yes—
195C2×D4—2788no—
196C2×D4—2784no—
197C2×D4—2782no—
198C2×D4—2781no—
199C2×D4—2781no—
200C2×D4—2782no—
201C2×D4—1394no—
202C2×D4—1391no—
203C2×D4—1392no—
204C22×C4—21no—
205C22×C4—2788no—
206C22×C4—2784no—
207C22×C4—2782no—
208C22×C4—2782no—
209C22×C4—2781no—
210C22×C4—1391no—
211C22×C4—1392no—
212C22×C4—1392no—
213C22×C4C22×D13911yescharacteristic
214Q16—2784no—
215SD16—2784no—
216SD16—2784no—
217D8—2784no—
218M16—22no—
219M16—2782no—
220C2×C8—1392no—
221C2×C8C22×D13912yes—
222C4⋊C4—2782no—
223C4⋊C4C22×D13912yes—
224C22⋊C4—21no—
225C22⋊C4—2782no—
226C22⋊C4—2782no—
227C42—21no—
228C23—2782no—
229C23—2784no—
230C23—2782no—
231C23—2781no—
232C23—5562no—
233C23—5564no—
234C23—1392no—
235C23D4×D13911yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—2781no—
239Q8—2781no—
240Q8C2×D55614yes—
241D4—27816no—
242D4—2784no—
243D4—2781no—
244D4—2782no—
245D4—2781no—
246D4—5564no—
247D4—5562no—
248D4—1394no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—27816no—
253C2×C4—2782no—
254C2×C4—2784no—
255C2×C4—2782no—
256C2×C4—2782no—
257C2×C4—2781no—
258C2×C4—5562no—
259C2×C4—1392no—
260C2×C4D4×D13911yescharacteristic
261C2×C4D4×D13912yes—
262C2×C4C2×D55612yes—
263C2×C4C23×D13911yescharacteristic
264C8—22no—
265C8—2782no—
266C22—22no—
267C22—2782no—
268C22—2784no—
269C22—2782no—
270C22—2782no—
271C22—5564no—
272C22—5568no—
273C22—5564no—
274C22—5561no—
275C22—5562no—
276C22#1798481212yes—
277C22C2×D4×D13911yescharacteristic
278C4—24no—
279C4—42no—
280C4—2782no—
281C4—5561no—
282C4#1798481212yes—
283C4C2×D4×D13911yescharacteristic
284C4C22×D55611yescharacteristic
285C2—22no—
286C2—5562no—
287C2—5564no—
288C2—11121no—
289C2C2×#1798481211yescharacteristic
290C2#2258175212yes—
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#419×C139C2
socleC22×C139C22×D4
hypercenter#419D139
hypocenterC139#3005
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#14555786C2semidirect
2. C2×#17984873C2semidirect
3. C2×#17984870C2semidirect
4. #22581465C2semidirect
5. #22581118C2semidirect
6. #22581101C2semidirect
7. #22580957C2semidirect
8. #22580916C2semidirect
9. #419×C139C2semidirect
21. C2×#14555775C22semidirect
22. C2×#14555774C22semidirect
23. C2×#14555772C22semidirect
24. C2×D1112C22semidirect
27. #17984822C22semidirect
28. #17984817C22semidirect
31. #17984762C22semidirect
32. (C2×D139)⋊C8C22semidirect
36. #103×C139C22semidirect
50. C2×D556D4semidirect
73. C2×Q8×C139D4semidirect
145. C139#3005semidirect
153. #419D139semidirect
180. #103C2×D139semidirect
193. C2×Q8D556semidirect

Maximal subgroups

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

subgroupindex
C22×#145557862
C2×#179848732
C2×#179848702
#225814652
#225811182
#225811012
#225809572
#225809162
#419×C1392
#3005139