Finite Group Databasebeta

← #31815805

Subgroups — #31815805

Order 17792.

772 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#22581789C211yesmaximal, characteristic, complemented
2#22581520C212yesmaximal, complemented
3#22581139C218yesmaximal, complemented
4#22580927C211yesmaximal, characteristic, complemented
5#22580926C212yesmaximal, complemented
6#535×C139C211yesmaximal, characteristic, complemented
7#17984890—21no—
8#17984889—22no—
9#17984888—22no—
10#17984887—21no—
11#17984887C2214yes—
12C2×#14555785—22no—
13C2×#14555785C2212yes—
14#41×D139—24no—
15#17984838—22no—
16#17984829—22no—
17#17984813—24no—
18#17984811C2218yescomplemented
19#17984801C2212yescomplemented
20#17984757C2214yescomplemented
21#17984756C2218yescomplemented
22C2×#41×C139C2211yescharacteristic
23C22×D4×C139C2211yescharacteristic, complemented
24#127×C139—21no—
25#124×C139—21no—
26#121×C139—22no—
27#120×C139C2214yescomplemented
28#117×C139C2211yescharacteristic, complemented
29#14555786—24no—
30#14555785—24no—
31#14555785—24no—
32#14555785—22no—
33#14555785—22no—
34#14555785—24no—
35#14555785—22no—
36#14555785—22no—
37#14555784—24no—
38#14555784—28no—
39C2×Dic278—22no—
40C2×Dic278D412yescomplemented
41Q8×D139—24no—
42C2×D556—22no—
43C2×D556D412yescomplemented
44C2×C22⋊D139—28no—
45C2×C22⋊D139—24no—
46C2×C22⋊D139C2314yes—
47D4×D139—24no—
48D4×D139—28no—
49C2×C4×D139—24no—
50C2×C4×D139D414yescomplemented
51C4⋊Dic139—22no—
52C22⋊Dic139—28no—
53C22⋊Dic139C2314yes—
54C4×Dic139—22no—
55C24×C139C2312yes—
56#41×C139—22no—
57#41×C139—22no—
58C2×Q8×C139D411yescharacteristic, complemented
59C2×D4×C139—24no—
60C2×D4×C139—22no—
61C2×D4×C139—22no—
62C2×D4×C139—21no—
63C2×D4×C139—21no—
64C2×D4×C139D411yescharacteristic, complemented
65C2×D4×C139C2312yes—
66C22×C4×C139D412yescomplemented
67C22×C4×C139C2311yescharacteristic
68C4⋊C4×C139—21no—
69C22⋊C4×C139—24no—
70C22⋊C4×C139C2312yes—
71C42×C139—21no—
72C22×D139—24no—
73C22×D139C2×D414yes—
74Dic278—24no—
75Dic278—22no—
76Dic278—22no—
77D556—24no—
78D556—22no—
79D556—22no—
80C22⋊D139—28no—
81C22⋊D139—24no—
82C22⋊D139—24no—
83C22⋊D139—88no—
84C2×Dic139—24no—
85C2×Dic139—24no—
86C2×Dic139C2×D414yes—
87C4×D139—28no—
88C4×D139—24no—
89C4×D139—24no—
90C23×C139—22no—
91C23×C139—24no—
92C23×C139—82no—
93C23×C139C2×D412yes—
94C23×C139C2411yescharacteristic
95Q8×C139—22no—
96D4×C139—24no—
97D4×C139—22no—
98D4×C139—41no—
99D4×C139—41no—
100D4×C139—42no—
101D4×C139—84no—
102C2×C4×C139—22no—
103C2×C4×C139—22no—
104C2×C4×C139—22no—
105C2×C4×C139C2×D412yes—
106C2×C4×C139C2×D412yes—
107C2×C4×C139C2×D412yes—
108C2×C4×C139C2×D411yescharacteristic
109C2×C4×C139C2×D411yescharacteristic
110C22×C139—22no—
111C22×C139—41no—
112C22×C139—42no—
113C22×C139—44no—
114C22×C139—48no—
115C22×C139—42no—
116C22×C139C22×D412yes—
117C22×C139C22×D411yescharacteristic
118C2×D139—24no—
119C2×D139—44no—
120C2×D139—84no—
121C4×C139—22no—
122C4×C139—42no—
123C4×C139#12011yescharacteristic
124C4×C139#12011yescharacteristic
125C4×C139#12012yes—
126Dic139—24no—
127Dic139—44no—
128C2×C139—22no—
129C2×C139—21no—
130C2×C139—42no—
131C2×C139—82no—
132C2×C139C2×#12011yescharacteristic
133D139—44no—
134C139#302711yescharacteristic, complemented, Sylow
135#3027—1391nomaximal, Sylow
136#585—1391no—
137#535—1392no—
138#535D13911yescharacteristic, complemented
139B(4,𝔽2)—1398no—
140#410—1391no—
141#410—1392no—
1422−1+4—2782no—
1432−1+4—2781no—
144H(5,𝔽2)—2782no—
145H(5,𝔽2)—2781no—
146H(5,𝔽2)—1394no—
147C2×#41—2784no—
148C2×#41—2782no—
149C2×#41—1392no—
150C2×#41C2×D13911yescharacteristic
151C22×D4C2×D13911yescharacteristic, complemented
152#127—21no—
153#127—2782no—
154#124—21no—
155#124—2782no—
156#121—22no—
157#121—2784no—
158#120—1398no—
159#120C2×D13914yescomplemented
160#117—1392no—
161#117C2×D13911yescharacteristic, complemented
162#99—1394no—
163#99—1398no—
164C24C22×D13912yes—
165#41—22no—
166#41—22no—
167#41—2784no—
168#41—2784no—
169#41—2784no—
170#41—2784no—
171#41—2788no—
172#41—2782no—
173#41—2782no—
174#41—2782no—
175#41—2784no—
176#41—2782no—
177C2×Q8—2782no—
178C2×Q8—2784no—
179C2×Q8—1392no—
180C2×Q8C22⋊D13911yescharacteristic, complemented
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—2782no—
187C2×D4—2784no—
188C2×D4—2788no—
189C2×D4—2784no—
190C2×D4—2788no—
191C2×D4—1394no—
192C2×D4—1392no—
193C2×D4C22⋊D13911yescharacteristic, complemented
194C2×D4C22×D13912yes—
195C22×C4—2784no—
196C22×C4—1394no—
197C22×C4C22⋊D13912yescomplemented
198C22×C4C22×D13911yescharacteristic
199C4⋊C4—21no—
200C4⋊C4—2782no—
201C22⋊C4—24no—
202C22⋊C4—2788no—
203C22⋊C4—1394no—
204C22⋊C4C22×D13912yes—
205C42—21no—
206C42—2782no—
207C23—24no—
208C23—22no—
209C23—82no—
210C23—2784no—
211C23—1394no—
212C23C2×C22⋊D13912yes—
213C23C23×D13911yescharacteristic
214Q8—22no—
215Q8—2784no—
216Q8—2782no—
217Q8—2782no—
218D4—24no—
219D4—22no—
220D4—41no—
221D4—41no—
222D4—42no—
223D4—84no—
224D4—2784no—
225D4—2788no—
226D4—2784no—
227D4—2784no—
228D4—2782no—
229D4—2782no—
230D4—11128no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—2788no—
235C2×C4—2784no—
236C2×C4—2784no—
237C2×C4—2784no—
238C2×C4—2784no—
239C2×C4—1394no—
240C2×C4D4×D13912yes—
241C2×C4D4×D13912yes—
242C2×C4C2×C22⋊D13912yes—
243C2×C4C2×C22⋊D13911yescharacteristic
244C2×C4C2×C22⋊D13911yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—44no—
249C22—48no—
250C22—42no—
251C22—2784no—
252C22—5564no—
253C22—11124no—
254C22C2×D4×D13912yes—
255C22C22×C22⋊D13911yescharacteristic
256C4—22no—
257C4—42no—
258C4—2784no—
259C4—5564no—
260C4#1798481011yescharacteristic
261C4#1798481011yescharacteristic
262C4#1798481012yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—5564no—
268C2C2×#1798481011yescharacteristic
subgroupG/H
centerC2C2×#17984810
commutator, cosocleC23×C139C24
FrattiniC23C23×D139
Fitting#535×C139C2
socleC2×C139C2×#120
hypercenter#535D139
hypocenterC139#3027
LCSG ▷ C23×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #22581789C2semidirect
2. #22581520C2semidirect
3. #22581139C2semidirect
4. #22580927C2semidirect
5. #22580926C2semidirect
6. #535×C139C2semidirect
18. #17984811C22semidirect
19. #17984801C22semidirect
20. #17984757C22semidirect
21. #17984756C22semidirect
23. C22×D4×C139C22semidirect
27. #120×C139C22semidirect
28. #117×C139C22semidirect
40. C2×Dic278D4semidirect
43. C2×D556D4semidirect
50. C2×C4×D139D4semidirect
58. C2×Q8×C139D4semidirect
64. C2×D4×C139D4semidirect
66. C22×C4×C139D4semidirect
134. C139#3027semidirect
138. #535D139semidirect
151. C22×D4C2×D139semidirect
159. #120C2×D139semidirect
161. #117C2×D139semidirect
180. C2×Q8C22⋊D139semidirect
193. C2×D4C22⋊D139semidirect
197. C22×C4C22⋊D139semidirect

Maximal subgroups

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

subgroupindex
#225817892
#225815202
#225811392
#225809272
#225809262
#535×C1392
#3027139