Finite Group Databasebeta

← #31815670

Subgroups — #31815670

Order 17792.

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

normal subgroups only

This group has 784 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
1C2×#17984888C211yesmaximal, characteristic, complemented
2C2×#17984868C211yesmaximal, characteristic, complemented
3C22×#14555770C211yesmaximal, characteristic, complemented
4#22581096C214yesmaximal, complemented
5#22581086C214yesmaximal, complemented
6#22580957C211yesmaximal, characteristic, complemented
7#22580947C211yesmaximal, characteristic, complemented
8#22580920C211yesmaximal, characteristic, complemented
9#535×C139C211yesmaximal, characteristic, complemented
10#17984888—24no—
11#17984888—24no—
12C2×#14555786—21no—
13C2×#14555785C2211yescharacteristic
14C2×#14555785C2212yescomplemented
15C22×D556C2211yescharacteristic
16C22×D556C2212yescomplemented
17C2×D4×D139—22no—
18C2×D4×D139—21no—
19#17984868—24no—
20C2×#14555773C2212yescomplemented
21C2×#14555770—24no—
22C2×#14555770C2212yescomplemented
23C2×#14555770C2214yescomplemented
24C2×#14555769C2211yescharacteristic, complemented
25C22×C139⋊C8C2211yescharacteristic, complemented
26#17984765C2212yescomplemented
27#17984762C2212yescomplemented
28#17984762C2214yescomplemented
29C22⋊(C139⋊C8)C2214yescomplemented
30C2×#41×C139C2211yescharacteristic
31C22×D4×C139C2211yescharacteristic, complemented
32#127×C139—21no—
33#124×C139—21no—
34#121×C139C2214yescomplemented
35#120×C139—22no—
36#117×C139C2211yescharacteristic, complemented
37C23×D139—21no—
38C23×D139—22no—
39#14555786—28no—
40#14555785—24no—
41#14555785—24no—
42#14555785—22no—
43#14555785D414yescomplemented
44C2×Dic278C2311yescharacteristic
45C2×D556—24no—
46C2×D556—24no—
47C2×D556—21no—
48C2×D556—22no—
49C2×D556—22no—
50C2×D556D414yescomplemented
51C2×D556C2312yes—
52C2×D556C2311yescharacteristic
53C2×C22⋊D139—21no—
54C2×C22⋊D139—22no—
55D4×D139—216no—
56D4×D139—28no—
57C2×C4×D139—21no—
58C2×C4×D139—22no—
59#14555773—24no—
60#14555770—28no—
61#14555770—24no—
62#14555769—22no—
63C2×C139⋊C8—22no—
64C2×C139⋊C8C2312yes—
65C2×C139⋊C8C2312yes—
66C24×C139—21no—
67#41×C139—22no—
68#41×C139—22no—
69C2×Q8×C139D411yescharacteristic
70C2×D4×C139—24no—
71C2×D4×C139—22no—
72C2×D4×C139—22no—
73C2×D4×C139—21no—
74C2×D4×C139D412yes—
75C2×D4×C139D411yescharacteristic
76C2×D4×C139C2312yes—
77C22×C4×C139C2312yes—
78C22×C4×C139C2311yescharacteristic
79C4⋊C4×C139C2312yes—
80C22⋊C4×C139—21no—
81C22⋊C4×C139—24no—
82C42×C139—21no—
83C22×D139—21no—
84C22×D139—22no—
85C22×D139—28no—
86C22×D139—24no—
87C22×D139—82no—
88C22×D139—84no—
89Dic278—21no—
90Dic278C2×D412yes—
91D556—28no—
92D556—24no—
93D556—22no—
94D556—21no—
95D556C2×D412yes—
96D556C2×D414yes—
97C22⋊D139—28no—
98C22⋊D139—24no—
99C2×Dic139—21no—
100C4×D139—28no—
101C4×D139—24no—
102C139⋊C8—22no—
103C139⋊C8—22no—
104C23×C139—22no—
105C23×C139—42no—
106C23×C139—82no—
107C23×C139C2×D412yes—
108C23×C139C2×D411yescharacteristic
109Q8×C139—22no—
110D4×C139—24no—
111D4×C139—22no—
112D4×C139—22no—
113D4×C139—22no—
114D4×C139—42no—
115D4×C139—84no—
116C2×C4×C139—22no—
117C2×C4×C139—22no—
118C2×C4×C139—22no—
119C2×C4×C139C2×D412yes—
120C2×C4×C139C2×D412yes—
121C2×C4×C139C2×D412yes—
122C2×C4×C139C2×D411yescharacteristic
123C2×C4×C139C2411yescharacteristic
124C22×C139—22no—
125C22×C139—22no—
126C22×C139—24no—
127C22×C139—42no—
128C22×C139—44no—
129C22×C139—84no—
130C22×C139#12012yes—
131C22×C139C22×D411yescharacteristic
132C2×D139—24no—
133C2×D139—22no—
134C2×D139—44no—
135C2×D139—42no—
136C2×D139—84no—
137C2×D139—88no—
138C4×C139—22no—
139C4×C139—42no—
140C4×C139#12012yes—
141C4×C139C22×D411yescharacteristic
142C4×C139C22×D411yescharacteristic
143Dic139—22no—
144C2×C139—22no—
145C2×C139—42no—
146C2×C139—82no—
147C2×C139C2×#12011yescharacteristic
148C2×C139#57612yes—
149D139—44no—
150D139—42no—
151C139#300811yescharacteristic, complemented, Sylow
152#3008—1391nomaximal, Sylow
153C2×H(5,𝔽2)—1391no—
154C2×AGL(1,ℤ/8)—1391no—
155C22×D8—1391no—
156#535D13911yescharacteristic, complemented
157#449—1394no—
158#447—1394no—
159#418—1391no—
160#416—1391no—
161#408—1391no—
162H(5,𝔽2)—2784no—
163H(5,𝔽2)—2784no—
164C2×#41—2781no—
165C2×#41—1392no—
166C2×#41—1391no—
167C2×#41C2×D13911yescharacteristic
168C22×D4—2782no—
169C22×D4—2781no—
170C22×D4—1392no—
171C22×D4—1391no—
172C22×D4C2×D13911yescharacteristic, complemented
173AGL(1,ℤ/8)—2784no—
174C2×SD16—1392no—
175C2×D8—2784no—
176C2×D8—1394no—
177C2×D8—1392no—
178C2×M16—1391no—
179C22×C8—1391no—
180#127—21no—
181#124—21no—
182#121C2×D13914yescomplemented
183#120—22no—
184#117C2×D13911yescharacteristic, complemented
185#103—1392no—
186#102—1392no—
187#102—1394no—
188C22⋊C8—1394no—
189C24—21no—
190C24—2782no—
191C24—2781no—
192#41—22no—
193#41—22no—
194#41—2784no—
195#41—2784no—
196#41—2788no—
197#41—2782no—
198#41—1394no—
199C2×Q8—1391no—
200C2×Q8C22⋊D13911yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—2784no—
206C2×D4—2784no—
207C2×D4—27816no—
208C2×D4—2788no—
209C2×D4—2782no—
210C2×D4—2781no—
211C2×D4—2782no—
212C2×D4—2782no—
213C2×D4—2781no—
214C2×D4—1394no—
215C2×D4—1391no—
216C2×D4—1392no—
217C2×D4C22⋊D13912yes—
218C2×D4C22⋊D13911yescharacteristic
219C2×D4C22×D13912yes—
220C22×C4—2782no—
221C22×C4—2781no—
222C22×C4C22×D13912yes—
223C22×C4C22×D13911yescharacteristic
224SD16—2784no—
225D8—2788no—
226D8—2784no—
227M16—2782no—
228C2×C8—2782no—
229C2×C8—1392no—
230C2×C8—1392no—
231C4⋊C4C22×D13912yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—2788no—
239C23—2782no—
240C23—2781no—
241C23—2784no—
242C23—11122no—
243C23—11124no—
244C23C2×C22⋊D13912yes—
245C23C2×C22⋊D13911yescharacteristic
246Q8—22no—
247Q8—2781no—
248Q8—1392no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—2788no—
256D4—2788no—
257D4—2784no—
258D4—2784no—
259D4—2782no—
260D4—2781no—
261D4—1392no—
262D4—1394no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—2788no—
267C2×C4—2781no—
268C2×C4—2784no—
269C2×C4D4×D13912yes—
270C2×C4D4×D13912yes—
271C2×C4C2×C22⋊D13912yes—
272C2×C4C2×C22⋊D13911yescharacteristic
273C2×C4C23×D13911yescharacteristic
274C8—2782no—
275C8—2782no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—2784no—
283C22—2782no—
284C22—5564no—
285C22—5562no—
286C22—11124no—
287C22—11128no—
288C22#1798481012yes—
289C22C22×C22⋊D13911yescharacteristic
290C4—22no—
291C4—42no—
292C4—2782no—
293C4#1798481012yes—
294C4C2×D4×D13911yescharacteristic
295C4C2×D4×D13911yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—5564no—
300C2—5562no—
301C2C2×#1798481011yescharacteristic
302C2#2258175612yes—
subgroupG/H
centerC22C22×C22⋊D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#535×C139C2
socleC22×C139C22×D4
hypercenter#535D139
hypocenterC139#3008
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#17984888C2semidirect
2. C2×#17984868C2semidirect
3. C22×#14555770C2semidirect
4. #22581096C2semidirect
5. #22581086C2semidirect
6. #22580957C2semidirect
7. #22580947C2semidirect
8. #22580920C2semidirect
9. #535×C139C2semidirect
14. C2×#14555785C22semidirect
16. C22×D556C22semidirect
20. C2×#14555773C22semidirect
22. C2×#14555770C22semidirect
23. C2×#14555770C22semidirect
24. C2×#14555769C22semidirect
25. C22×C139⋊C8C22semidirect
26. #17984765C22semidirect
27. #17984762C22semidirect
28. #17984762C22semidirect
29. C22⋊(C139⋊C8)C22semidirect
31. C22×D4×C139C22semidirect
34. #121×C139C22semidirect
36. #117×C139C22semidirect
43. #14555785D4semidirect
50. C2×D556D4semidirect
151. C139#3008semidirect
156. #535D139semidirect
172. C22×D4C2×D139semidirect
182. #121C2×D139semidirect
184. #117C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848882
C2×#179848682
C22×#145557702
#225810962
#225810862
#225809572
#225809472
#225809202
#535×C1392
#3008139