Finite Group Databasebeta

← #31815649

Subgroups — #31815649

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×#14555785C211yesmaximal, characteristic, complemented
2C2×#17984875C211yesmaximal, characteristic, complemented
3C2×#17984870C211yesmaximal, characteristic, complemented
4#22581464C211yesmaximal, characteristic, complemented
5#22581117C214yesmaximal, complemented
6#22581100C214yesmaximal, complemented
7#22580963C211yesmaximal, characteristic, complemented
8#22580959C211yesmaximal, characteristic, complemented
9#407×C139C211yesmaximal, characteristic, complemented
10C2×#14555785—24no—
11C2×#14555785—24no—
12C2×#14555785—22no—
13C2×#14555785C2214yes—
14C2×#14555785C2211yescharacteristic
15C22×Dic278C2211yescharacteristic
16C22×D556C2211yescharacteristic
17C22×C22⋊D139—21no—
18C22×C4×D139—21no—
19#17984875—24no—
20#17984870—24no—
21C2×Dic556C2212yescomplemented
22C2×#14555774C2212yescomplemented
23C2×#14555774C2212yescomplemented
24C2×D1112C2212yescomplemented
25#17984825—21no—
26#17984821C2212yescomplemented
27#17984816C2212yescomplemented
28#17984811—21no—
29C4×C22⋊D139—22no—
30#17984801C2211yescharacteristic
31#17984767C2214yescomplemented
32#17984764C2214yescomplemented
33C23×C4×C139C2211yescharacteristic
34C2×M16×C139C2211yescharacteristic
35C2×M16×C139C2211yescharacteristic
36C22⋊C8×C139C2214yescomplemented
37C23×D139—21no—
38#14555785—216no—
39#14555785—28no—
40#14555785—28no—
41#14555785—22no—
42#14555785—22no—
43C2×Dic278—22no—
44C2×Dic278—21no—
45C2×Dic278D414yescomplemented
46C2×Dic278C2311yescharacteristic
47C2×Dic278C2312yes—
48C2×D556—22no—
49C2×D556—21no—
50C2×D556D414yescomplemented
51C2×D556C2311yescharacteristic
52C2×D556C2312yes—
53C2×C22⋊D139—28no—
54C2×C22⋊D139—24no—
55C2×C22⋊D139—22no—
56C2×C22⋊D139—21no—
57C22×Dic139—21no—
58C2×C4×D139—28no—
59C2×C4×D139—24no—
60C2×C4×D139—21no—
61Dic556—24no—
62#14555774—24no—
63#14555774—24no—
64D1112—24no—
65C4⋊Dic139C2312yes—
66Dic139⋊C4—22no—
67C22⋊Dic139—21no—
68C22⋊Dic139—22no—
69(C2×D139)⋊C4—22no—
70C4×Dic139—21no—
71C24×C139D411yescharacteristic
72C22×C4×C139—22no—
73C22×C4×C139—22no—
74C22×C4×C139D412yes—
75C22×C4×C139D411yescharacteristic
76C22×C4×C139C2312yes—
77C22×C4×C139C2311yescharacteristic
78M16×C139—22no—
79M16×C139—22no—
80C2×C8×C139C2312yes—
81C2×C8×C139C2312yes—
82C22×D139—24no—
83C22×D139—22no—
84C22×D139—21no—
85C22×D139—44no—
86Dic278—24no—
87Dic278—22no—
88Dic278—21no—
89Dic278—21no—
90Dic278C2×D414yes—
91D556—24no—
92D556—22no—
93D556—21no—
94D556—21no—
95D556C2×D414yes—
96C22⋊D139—216no—
97C22⋊D139—44no—
98C22⋊D139—42no—
99C2×Dic139—24no—
100C2×Dic139—22no—
101C2×Dic139—22no—
102C2×Dic139—21no—
103C4×D139—216no—
104C4×D139—42no—
105C23×C139—22no—
106C23×C139—42no—
107C23×C139C2×D412yes—
108C23×C139C2×D411yescharacteristic
109C2×C4×C139—22no—
110C2×C4×C139—24no—
111C2×C4×C139—22no—
112C2×C4×C139—22no—
113C2×C4×C139C2×D412yes—
114C2×C4×C139C2×D412yes—
115C2×C4×C139C2×D412yes—
116C2×C4×C139C2×D411yescharacteristic
117C2×C4×C139C2411yescharacteristic
118C8×C139—22no—
119C8×C139—22no—
120C22×C139—22no—
121C22×C139—22no—
122C22×C139—22no—
123C22×C139—22no—
124C22×C139—44no—
125C22×C139#12012yes—
126C22×C139C22×D411yescharacteristic
127C2×D139—24no—
128C2×D139—48no—
129C2×D139—44no—
130C2×D139—41no—
131C2×D139—42no—
132C4×C139—22no—
133C4×C139#12012yes—
134C4×C139C22×D411yescharacteristic
135C4×C139C22×D411yescharacteristic
136Dic139—24no—
137Dic139—41no—
138Dic139—42no—
139C2×C139—22no—
140C2×C139—42no—
141C2×C139C2×#12011yescharacteristic
142C2×C139#57512yes—
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#419—1391no—
154#418—1391no—
155#407D13911yescharacteristic, complemented
156C2×#41—2784no—
157C2×#41—2784no—
158C2×#41—2782no—
159C2×#41—1391no—
160C2×#41—1394no—
161C22×Q8—1391no—
162C22×D4—2781no—
163C22×D4—1391no—
164C23×C4—2781no—
165C23×C4C2×D13911yescharacteristic
166#137—2784no—
167AGL(1,ℤ/8)—2784no—
168C2×Q16—1392no—
169C2×SD16—1392no—
170C2×SD16—1392no—
171C2×D8—1392no—
172C2×M16C2×D13911yescharacteristic
173C2×M16C2×D13911yescharacteristic
174#123—2781no—
175C22⋊Q8—1392no—
176#121—1392no—
177#120—2781no—
178C4×D4—2782no—
179#117—1391no—
180#103—1394no—
181#102—1394no—
182C22⋊C8C2×D13914yescomplemented
183C24—2781no—
184C24D55611yescharacteristic
185#41—27816no—
186#41—2788no—
187#41—2788no—
188#41—2782no—
189#41—2782no—
190C2×Q8—2782no—
191C2×Q8—2781no—
192C2×Q8—1394no—
193C2×Q8—1391no—
194C2×Q8—1392no—
195C2×D4—2788no—
196C2×D4—2784no—
197C2×D4—2782no—
198C2×D4—2781no—
199C2×D4—2782no—
200C2×D4—2781no—
201C2×D4—1394no—
202C2×D4—1391no—
203C2×D4—1392no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—2788no—
207C22×C4—2781no—
208C22×C4—2784no—
209C22×C4—2781no—
210C22×C4D55612yes—
211C22×C4D55611yescharacteristic
212C22×C4C22×D13912yes—
213C22×C4C22×D13911yescharacteristic
214Q16—2784no—
215SD16—2784no—
216SD16—2784no—
217D8—2784no—
218M16—22no—
219M16—22no—
220C2×C8C22×D13912yes—
221C2×C8C22×D13912yes—
222C4⋊C4—2782no—
223C4⋊C4—1392no—
224C22⋊C4—2781no—
225C22⋊C4—2782no—
226C22⋊C4—2782no—
227C42—2781no—
228C23—22no—
229C23—42no—
230C23—2784no—
231C23—2782no—
232C23—2781no—
233C23—5564no—
234C23C2×D55612yes—
235C23C2×D55611yescharacteristic
236Q8—2784no—
237Q8—2782no—
238Q8—2781no—
239Q8—2781no—
240Q8—1394no—
241D4—27816no—
242D4—2784no—
243D4—2782no—
244D4—2781no—
245D4—2781no—
246D4—5564no—
247D4—5562no—
248D4—1394no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—27816no—
254C2×C4—2784no—
255C2×C4—2782no—
256C2×C4—2782no—
257C2×C4—2781no—
258C2×C4—5562no—
259C2×C4D4×D13912yes—
260C2×C4D4×D13912yes—
261C2×C4C2×D55612yes—
262C2×C4C2×D55611yescharacteristic
263C2×C4C23×D13911yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—2784no—
272C22—5568no—
273C22—5564no—
274C22—5561no—
275C22—5562no—
276C22#1798481212yes—
277C22C22×D55611yescharacteristic
278C4—22no—
279C4—2784no—
280C4—5561no—
281C4—5562no—
282C4#1798481212yes—
283C4C2×D4×D13911yescharacteristic
284C4C2×D4×D13911yescharacteristic
285C2—22no—
286C2—42no—
287C2—5564no—
288C2—11121no—
289C2C2×#1798481211yescharacteristic
290C2#2258175312yes—
subgroupG/H
centerC22C22×D556
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#407×C139C2
socleC22×C139C22×D4
hypercenter#407D139
hypocenterC139#3005
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#14555785C2semidirect
2. C2×#17984875C2semidirect
3. C2×#17984870C2semidirect
4. #22581464C2semidirect
5. #22581117C2semidirect
6. #22581100C2semidirect
7. #22580963C2semidirect
8. #22580959C2semidirect
9. #407×C139C2semidirect
21. C2×Dic556C22semidirect
22. C2×#14555774C22semidirect
23. C2×#14555774C22semidirect
24. C2×D1112C22semidirect
26. #17984821C22semidirect
27. #17984816C22semidirect
31. #17984767C22semidirect
32. #17984764C22semidirect
36. C22⋊C8×C139C22semidirect
45. C2×Dic278D4semidirect
50. C2×D556D4semidirect
145. C139#3005semidirect
155. #407D139semidirect
182. C22⋊C8C2×D139semidirect

Maximal subgroups

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

subgroupindex
C22×#145557852
C2×#179848752
C2×#179848702
#225814642
#225811172
#225811002
#225809632
#225809592
#407×C1392
#3005139