Finite Group Databasebeta

← #31816884

Subgroups — #31816884

Order 17792.

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

normal subgroups only

This group has 596 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×#17984866C212yesmaximal, complemented
2C2×#17984861C211yesmaximal, characteristic, complemented
3C2×D8×D139C211yesmaximal, characteristic, complemented
4#22581516C212yesmaximal, complemented
5#22581341C212yesmaximal, complemented
6#22581340C212yesmaximal, complemented
7#22581337C211yesmaximal, characteristic, complemented
8#22581336C211yesmaximal, characteristic, complemented
9#22581328C211yesmaximal, characteristic, complemented
10#22580907C211yesmaximal, characteristic, complemented
11#493×C139C211yesmaximal, characteristic, complemented
12C2×#14555784C2212yes—
13C2×D4×D139C2212yes—
14#17984866—28no—
15#17984861—24no—
16C2×Dic556C2211yescharacteristic, complemented
17C2×#14555774C2212yescomplemented
18C2×#14555773C2212yescomplemented
19C2×#14555773C2212yescomplemented
20C2×D1112C2211yescharacteristic
21C2×#14555770C2212yes—
22C2×#14555770C2212yescomplemented
23D8×D139—24no—
24C2×#14555768C2211yescharacteristic
25C2×C8×D139C2211yescharacteristic
26#17984838C2212yescomplemented
27#17984830C2212yescomplemented
28#17984829C2212yescomplemented
29#17984814—24no—
30#17984810—24no—
31#17984799C2211yescharacteristic
32#17984753C2211yescharacteristic, complemented
33#17984752C2211yescharacteristic, complemented
34C8×Dic139C2211yescharacteristic, complemented
35C2×D8×C139C2211yescharacteristic
36C2×D8×C139C2211yescharacteristic, complemented
37C2×D8×C139C2212yescomplemented
38#127×C139C2212yescomplemented
39C4×C8×C139C2211yescharacteristic, complemented
40C23×D139—22no—
41#14555784—24no—
42#14555784—24no—
43C2×Dic278C2312yes—
44C2×D556C2312yes—
45C2×C22⋊D139—22no—
46C2×C22⋊D139—22no—
47C2×C22⋊D139—24no—
48D4×D139—24no—
49D4×D139—24no—
50C22×Dic139—22no—
51C2×C4×D139C2311yescharacteristic
52Dic556—22no—
53#14555774—24no—
54#14555773—24no—
55#14555773—24no—
56D1112—22no—
57#14555770—24no—
58#14555770—24no—
59#14555768D414yescomplemented
60C2×C139⋊C8C2311yescharacteristic
61C2×C139⋊C8C2311yescharacteristic
62C8×D139D414yescomplemented
63Dic139⋊C4D412yes—
64C22⋊Dic139—24no—
65(C2×D139)⋊C4—24no—
66(C2×D139)⋊C4D412yes—
67C4×Dic139C2311yescharacteristic
68C2×D4×C139—24no—
69C2×D4×C139C2312yes—
70C2×D4×C139C2312yes—
71D8×C139—22no—
72D8×C139—22no—
73D8×C139—24no—
74C2×C8×C139C2311yescharacteristic
75C2×C8×C139C2311yescharacteristic
76C42×C139C2311yescharacteristic
77C22×D139—22no—
78C22×D139—44no—
79C22×D139—84no—
80C22×D139C2×D411yescharacteristic
81Dic278—22no—
82Dic278—22no—
83D556—22no—
84D556—22no—
85C22⋊D139—44no—
86C22⋊D139—44no—
87C22⋊D139—84no—
88C2×Dic139—22no—
89C2×Dic139—44no—
90C2×Dic139C2×D412yes—
91C2×Dic139C2×D411yescharacteristic
92C4×D139C2×D412yes—
93C4×D139C2×D412yes—
94C139⋊C8C2×D412yes—
95C139⋊C8C2×D412yes—
96C23×C139—22no—
97C23×C139—22no—
98D4×C139—22no—
99D4×C139—22no—
100D4×C139—22no—
101D4×C139—22no—
102D4×C139—84no—
103C2×C4×C139C2×D412yes—
104C2×C4×C139C2411yescharacteristic
105C8×C139C2×D412yes—
106C8×C139C2×D412yes—
107C22×C139—42no—
108C22×C139—42no—
109C22×C139—44no—
110C22×C139—44no—
111C22×C139C22×D411yescharacteristic
112C2×D139—22no—
113C2×D139—42no—
114C2×D139—44no—
115C2×D139—88no—
116C2×D139#12712yes—
117C4×C139—22no—
118C4×C139C22×D411yescharacteristic
119C4×C139C22×D411yescharacteristic
120Dic139—22no—
121Dic139—42no—
122Dic139#12712yes—
123C2×C139—82no—
124C2×C139—82no—
125C2×C139C2×#12711yescharacteristic
126C2×C139#57612yes—
127D139—22no—
128D139—82no—
129C139#315511yescharacteristic, complemented, Sylow
130#3155—1391nomaximal, Sylow
131C2×AGL(1,ℤ/8)—1392no—
132C2×#135—1391no—
133C22×D8—1391no—
134#535—1392no—
135#496—1392no—
136#496—1392no—
137#495—1391no—
138#495—1391no—
139#493—1391no—
140#493D13911yescharacteristic, complemented
141#405—1391no—
142C2×#41—1392no—
143C22×D4—1392no—
144AGL(1,ℤ/8)—2788no—
145#135—2784no—
146C2×Q16—1391no—
147C2×SD16—1392no—
148C2×SD16—1392no—
149C2×SD16—1392no—
150C2×D8—2784no—
151C2×D8—1392no—
152C2×D8—1392no—
153C2×D8—1391no—
154C2×D8C2×D13911yescharacteristic
155C2×D8C2×D13911yescharacteristic, complemented
156C2×D8C2×D13912yescomplemented
157C2×M16—1391no—
158C22×C8—1391no—
159#127—1392no—
160#127C2×D13912yescomplemented
161#124—1392no—
162#124—1392no—
163#121—2784no—
164#120—2784no—
165#117—1391no—
166#97—1391no—
167#97—1391no—
168C4×C8—1391no—
169C4×C8C2×D13911yescharacteristic, complemented
170C24—2782no—
171#41—2784no—
172#41—2784no—
173C2×Q8—1392no—
174C2×D4—24no—
175C2×D4—2782no—
176C2×D4—2784no—
177C2×D4—2784no—
178C2×D4—2782no—
179C2×D4—2784no—
180C2×D4—1392no—
181C2×D4C22×D13912yes—
182C2×D4C22×D13912yes—
183C22×C4—2782no—
184C22×C4—1391no—
185Q16—2782no—
186SD16—2784no—
187SD16—2784no—
188SD16—2784no—
189D8—22no—
190D8—22no—
191D8—24no—
192D8—2784no—
193D8—2784no—
194D8—2782no—
195M16—1394no—
196C2×C8—1391no—
197C2×C8—1391no—
198C2×C8—1394no—
199C2×C8C22×D13911yescharacteristic
200C2×C8C22×D13911yescharacteristic
201C4⋊C4—1392no—
202C22⋊C4—2784no—
203C22⋊C4—2784no—
204C22⋊C4—1392no—
205C42—1391no—
206C42C22×D13911yescharacteristic
207C23—22no—
208C23—22no—
209C23—2782no—
210C23—5564no—
211C23—11124no—
212C23—1391no—
213Q8—2782no—
214Q8—2782no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—84no—
220D4—2782no—
221D4—2782no—
222D4—5564no—
223D4—5564no—
224D4—11124no—
225C2×C4—2782no—
226C2×C4—5564no—
227C2×C4—1392no—
228C2×C4—1391no—
229C2×C4—1392no—
230C2×C4—1392no—
231C2×C4D4×D13912yes—
232C2×C4C23×D13911yescharacteristic
233C8—1392no—
234C8—1392no—
235C8D4×D13912yes—
236C8D4×D13912yes—
237C22—42no—
238C22—42no—
239C22—44no—
240C22—44no—
241C22—2782no—
242C22—5562no—
243C22—5564no—
244C22—11128no—
245C22—1392no—
246C22C2×D4×D13911yescharacteristic
247C4—22no—
248C4—2782no—
249C4—5562no—
250C4—1392no—
251C4C2×D4×D13911yescharacteristic
252C4C2×D4×D13911yescharacteristic
253C2—82no—
254C2—82no—
255C2—2782no—
256C2—11122no—
257C2#127×D13911yescharacteristic
258C2#2258175412yes—
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#493×C139C2
socleC22×C139C22×D4
hypercenter#493D139
hypocenterC139#3155
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#17984866C2semidirect
2. C2×#17984861C2semidirect
3. C2×D8×D139C2semidirect
4. #22581516C2semidirect
5. #22581341C2semidirect
6. #22581340C2semidirect
7. #22581337C2semidirect
8. #22581336C2semidirect
9. #22581328C2semidirect
10. #22580907C2semidirect
11. #493×C139C2semidirect
16. C2×Dic556C22semidirect
17. C2×#14555774C22semidirect
18. C2×#14555773C22semidirect
19. C2×#14555773C22semidirect
22. C2×#14555770C22semidirect
26. #17984838C22semidirect
27. #17984830C22semidirect
28. #17984829C22semidirect
32. #17984753C22semidirect
33. #17984752C22semidirect
34. C8×Dic139C22semidirect
36. C2×D8×C139C22semidirect
37. C2×D8×C139C22semidirect
38. #127×C139C22semidirect
39. C4×C8×C139C22semidirect
59. #14555768D4semidirect
62. C8×D139D4semidirect
129. C139#3155semidirect
140. #493D139semidirect
155. C2×D8C2×D139semidirect
156. C2×D8C2×D139semidirect
160. #127C2×D139semidirect
169. C4×C8C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848662
C2×#179848612
C2×D8×D1392
#225815162
#225813412
#225813402
#225813372
#225813362
#225813282
#225809072
#493×C1392
#3155139