Finite Group Databasebeta

← #31816867

Subgroups — #31816867

Order 17792.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#17984874C212yesmaximal, complemented
2C2×#17984869C212yesmaximal, complemented
3#22581504C211yesmaximal, characteristic, complemented
4#124×D139C211yesmaximal, characteristic, complemented
5#22581345C211yesmaximal, characteristic, complemented
6#22581343C211yesmaximal, characteristic, complemented
7#22581342C211yesmaximal, characteristic, complemented
8#22581338C212yesmaximal, complemented
9#22581334C212yesmaximal, complemented
10#22580905C211yesmaximal, characteristic, complemented
11#497×C139C211yesmaximal, characteristic, complemented
12C2×#14555786C2212yes—
13C2×Q8×D139C2211yescharacteristic
14C2×D4×D139C2211yescharacteristic
15#17984874—28no—
16#17984869—28no—
17C2×#14555775C2212yescomplemented
18C2×#14555774C2212yescomplemented
19C2×#14555772C2212yes—
20C2×#14555772C2212yescomplemented
21C2×D1112C2212yes—
22C2×#14555770C2212yes—
23C2×#14555768C2212yes—
24#17984839C2211yescharacteristic, complemented
25#17984831C2211yescharacteristic
26#17984831C2212yescomplemented
27#17984830C2211yescharacteristic
28#17984829C2211yescharacteristic
29#17984828—22no—
30#17984817—22no—
31#17984804—22no—
32C22⋊C4×D139—22no—
33C42×D139C2211yescharacteristic
34#17984753C2211yescharacteristic, complemented
35C8×Dic139C2212yescomplemented
36C2×Q16×C139C2212yescomplemented
37C2×SD16×C139C2212yes—
38C4⋊Q8×C139C2211yescharacteristic, complemented
39#124×C139C2211yescharacteristic
40#97×C139C2211yescharacteristic, complemented
41C23×D139—21no—
42#14555786—24no—
43#14555786—24no—
44C2×Dic278—21no—
45C2×Dic278C2311yescharacteristic
46Q8×D139—24no—
47C2×D556—22no—
48C2×D556—22no—
49C2×D556C2311yescharacteristic
50C2×D556C2312yes—
51C2×C22⋊D139—21no—
52D4×D139—24no—
53C2×C4×D139—21no—
54C2×C4×D139—22no—
55C2×C4×D139D412yes—
56C2×C4×D139C2311yescharacteristic
57#14555775—24no—
58#14555774—24no—
59#14555772—24no—
60#14555772—24no—
61D1112—24no—
62#14555770—24no—
63#14555768D418yescomplemented
64C2×C139⋊C8C2312yes—
65C22⋊Dic139—22no—
66(C2×D139)⋊C4—22no—
67(C2×D139)⋊C4—24no—
68(C2×D139)⋊C4—22no—
69C4×Dic139D412yes—
70C4×Dic139C2311yescharacteristic
71C2×Q8×C139C2311yescharacteristic
72C2×Q8×C139C2312yes—
73C2×D4×C139C2311yescharacteristic
74Q16×C139—24no—
75SD16×C139—24no—
76C2×C8×C139C2312yes—
77C4⋊C4×C139—22no—
78C22⋊C4×C139—22no—
79C42×C139C2311yescharacteristic
80C22×D139—21no—
81C22×D139—22no—
82C22×D139—44no—
83C22×D139—42no—
84C22×D139C2×D411yescharacteristic
85Dic278—22no—
86Dic278—42no—
87D556—22no—
88D556—22no—
89D556—22no—
90D556—44no—
91D556—44no—
92C22⋊D139—42no—
93C2×Dic139—21no—
94C2×Dic139C2×D412yes—
95C2×Dic139C2×D411yescharacteristic
96C4×D139—42no—
97C4×D139—42no—
98C4×D139—44no—
99C4×D139C2×D414yes—
100C139⋊C8C2×D414yes—
101C23×C139—21no—
102Q8×C139—22no—
103Q8×C139—22no—
104Q8×C139—22no—
105D4×C139—22no—
106C2×C4×C139—21no—
107C2×C4×C139—22no—
108C2×C4×C139C2×D412yes—
109C2×C4×C139C2411yescharacteristic
110C8×C139C2×D414yes—
111C22×C139—41no—
112C22×C139—42no—
113C22×C139C22×D411yescharacteristic
114C2×D139—22no—
115C2×D139—42no—
116C2×D139—41no—
117C2×D139—44no—
118C2×D139—42no—
119C2×D139—84no—
120C2×D139#12712yes—
121C4×C139—22no—
122C4×C139—42no—
123C4×C139—41no—
124C4×C139C22×D412yes—
125Dic139—22no—
126Dic139—41no—
127Dic139#12712yes—
128C2×C139—81no—
129C2×C139C2×#12711yescharacteristic
130C2×C139#57512yes—
131D139—22no—
132D139—82no—
133D139—81no—
134C139#315311yescharacteristic, complemented, Sylow
135#3153—1391nomaximal, Sylow
136C2×#137—1392no—
137C2×AGL(1,ℤ/8)—1392no—
138#532—1391no—
139C2×#124—1391no—
140#497—1391no—
141#497D13911yescharacteristic, complemented
142#496—1391no—
143#496—1391no—
144#495—1392no—
145#495—1392no—
146C4×M16—1391no—
147C2×#41—1392no—
148C22×Q8—1391no—
149C22×D4—1391no—
150#137—2788no—
151AGL(1,ℤ/8)—2788no—
152C2×Q16—1392no—
153C2×Q16C2×D13912yescomplemented
154C2×SD16—1392no—
155C2×SD16—1392no—
156C2×SD16—1392no—
157C2×SD16C2×D13912yes—
158C2×D8—1392no—
159C2×D8—1392no—
160C2×M16—1392no—
161C4⋊Q8C2×D13911yescharacteristic, complemented
162#127—1391no—
163#124—2782no—
164#124—1391no—
165#124—1392no—
166#124—1391no—
167#124—1391no—
168#124C2×D13911yescharacteristic
169#121—2782no—
170C4×D4—2782no—
171C2×C22⋊C4—2782no—
172C2×C42—1391no—
173#97—1391no—
174#97C2×D13911yescharacteristic, complemented
175C4×C8—1392no—
176C24—2781no—
177#41—2784no—
178#41—2784no—
179C2×Q8—2781no—
180C2×Q8—2784no—
181C2×Q8—1391no—
182C2×Q8C22×D13911yescharacteristic
183C2×Q8C22×D13912yes—
184C2×D4—2782no—
185C2×D4—2784no—
186C2×D4—2781no—
187C2×D4—2782no—
188C2×D4—1391no—
189C2×D4—1392no—
190C2×D4C22×D13911yescharacteristic
191C22×C4—2781no—
192C22×C4—2782no—
193C22×C4—1392no—
194C22×C4—1391no—
195Q16—24no—
196Q16—2784no—
197SD16—24no—
198SD16—2784no—
199SD16—2784no—
200SD16—2784no—
201D8—2784no—
202D8—2784no—
203M16—1398no—
204C2×C8—1392no—
205C2×C8C22×D13912yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—2782no—
209C22⋊C4—2784no—
210C22⋊C4—2782no—
211C22⋊C4—2782no—
212C42—1392no—
213C42—1391no—
214C42C22×D13911yescharacteristic
215C23—21no—
216C23—2781no—
217C23—2782no—
218C23—5564no—
219C23—5562no—
220C23—1391no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—2782no—
225Q8—5562no—
226D4—22no—
227D4—2782no—
228D4—2782no—
229D4—2782no—
230D4—5564no—
231D4—5564no—
232D4—5562no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—2781no—
236C2×C4—5562no—
237C2×C4—5562no—
238C2×C4—5564no—
239C2×C4—1392no—
240C2×C4—1391no—
241C2×C4—1394no—
242C2×C4D4×D13912yes—
243C2×C4C23×D13911yescharacteristic
244C8—1394no—
245C8D4×D13914yes—
246C22—41no—
247C22—42no—
248C22—2782no—
249C22—5562no—
250C22—5561no—
251C22—5564no—
252C22—5562no—
253C22—11124no—
254C22—1392no—
255C22C2×D4×D13911yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—2782no—
260C4—5561no—
261C4—1392no—
262C4C2×D4×D13912yes—
263C2—81no—
264C2—2782no—
265C2—11121no—
266C2—11122no—
267C2#127×D13911yescharacteristic
268C2#2258175212yes—
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#497×C139C2
socleC22×C139C22×D4
hypercenter#497D139
hypocenterC139#3153
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#17984874C2semidirect
2. C2×#17984869C2semidirect
3. #22581504C2semidirect
4. #124×D139C2semidirect
5. #22581345C2semidirect
6. #22581343C2semidirect
7. #22581342C2semidirect
8. #22581338C2semidirect
9. #22581334C2semidirect
10. #22580905C2semidirect
11. #497×C139C2semidirect
17. C2×#14555775C22semidirect
18. C2×#14555774C22semidirect
20. C2×#14555772C22semidirect
24. #17984839C22semidirect
26. #17984831C22semidirect
34. #17984753C22semidirect
35. C8×Dic139C22semidirect
36. C2×Q16×C139C22semidirect
38. C4⋊Q8×C139C22semidirect
40. #97×C139C22semidirect
63. #14555768D4semidirect
134. C139#3153semidirect
141. #497D139semidirect
153. C2×Q16C2×D139semidirect
161. C4⋊Q8C2×D139semidirect
174. #97C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848742
C2×#179848692
#225815042
#124×D1392
#225813452
#225813432
#225813422
#225813382
#225813342
#225809052
#497×C1392
#3153139