Finite Group Databasebeta

← #31815775

Subgroups — #31815775

Order 17792.

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

normal subgroups only

This group has 736 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
2C2×#17984868C212yesmaximal, complemented
3#22581125C214yesmaximal, complemented
4#22581120C214yesmaximal, complemented
5#22580971C212yesmaximal, complemented
6#22580937C211yesmaximal, characteristic, complemented
7#535×C139C211yesmaximal, characteristic, complemented
8#17984890C2212yescomplemented
9#17984889—22no—
10#17984888—22no—
11#17984887—22no—
12#17984887C2212yescomplemented
13C2×#14555785—22no—
14C2×#14555785C2212yes—
15#41×D139—24no—
16#17984868—24no—
17#17984868C2218yescomplemented
18C2×#14555773—22no—
19C2×#14555770—22no—
20C2×#14555769C2212yescomplemented
21#17984770C2218yescomplemented
22#17984761C2212yescomplemented
23#17984759C2212yescomplemented
24C2×#41×C139C2211yescharacteristic
25C22×D4×C139C2211yescharacteristic, complemented
26#127×C139C2212yescomplemented
27#124×C139C2212yescomplemented
28#121×C139—22no—
29#120×C139—22no—
30#117×C139C2211yescharacteristic, complemented
31#14555786—24no—
32#14555785—24no—
33#14555785—24no—
34#14555785—22no—
35#14555785—24no—
36#14555785—22no—
37#14555785D414yescomplemented
38#14555785C2314yes—
39#14555784—28no—
40#14555784—24no—
41C2×Dic278—22no—
42C2×Dic278D412yescomplemented
43Q8×D139—24no—
44C2×D556—22no—
45C2×D556D412yescomplemented
46C2×C22⋊D139—24no—
47C2×C22⋊D139—22no—
48D4×D139—28no—
49D4×D139—24no—
50C2×C4×D139—24no—
51C2×C4×D139—22no—
52#14555773—28no—
53#14555770—28no—
54#14555769—22no—
55#14555769C2314yes—
56C2×C139⋊C8—22no—
57C24×C139—21no—
58#41×C139—22no—
59#41×C139—22no—
60C2×Q8×C139C2311yescharacteristic
61C2×D4×C139—24no—
62C2×D4×C139—22no—
63C2×D4×C139—22no—
64C2×D4×C139—21no—
65C2×D4×C139D412yes—
66C2×D4×C139C2312yes—
67C2×D4×C139C2311yescharacteristic
68C22×C4×C139D412yes—
69C22×C4×C139C2311yescharacteristic
70C4⋊C4×C139—21no—
71C22⋊C4×C139—21no—
72C22⋊C4×C139—24no—
73C42×C139C2312yes—
74C22×D139—24no—
75C22×D139—22no—
76Dic278—24no—
77Dic278—22no—
78Dic278C2×D414yes—
79D556—24no—
80D556—22no—
81D556C2×D414yes—
82C22⋊D139—28no—
83C22⋊D139—24no—
84C22⋊D139—24no—
85C2×Dic139—24no—
86C2×Dic139—22no—
87C4×D139—28no—
88C4×D139—24no—
89C4×D139—24no—
90C139⋊C8—24no—
91C23×C139—22no—
92C23×C139—42no—
93C23×C139—82no—
94C23×C139C2×D412yes—
95C23×C139C2×D411yescharacteristic
96Q8×C139—22no—
97D4×C139—24no—
98D4×C139—22no—
99D4×C139—24no—
100D4×C139—41no—
101D4×C139—41no—
102D4×C139—84no—
103C2×C4×C139—22no—
104C2×C4×C139—22no—
105C2×C4×C139—22no—
106C2×C4×C139C2×D412yes—
107C2×C4×C139C2×D412yes—
108C2×C4×C139C2×D412yes—
109C2×C4×C139C2×D411yescharacteristic
110C2×C4×C139C2411yescharacteristic
111C22×C139—22no—
112C22×C139—41no—
113C22×C139—42no—
114C22×C139—44no—
115C22×C139—42no—
116C22×C139—84no—
117C22×C139#12012yes—
118C22×C139C22×D411yescharacteristic
119C2×D139—24no—
120C2×D139—82no—
121C2×D139—84no—
122C4×C139—22no—
123C4×C139—42no—
124C4×C139#12011yescharacteristic
125C4×C139#12011yescharacteristic
126C4×C139C22×D412yes—
127Dic139—24no—
128C2×C139—21no—
129C2×C139—22no—
130C2×C139—42no—
131C2×C139—82no—
132C2×C139C2×#12011yescharacteristic
133D139—44no—
134C139#302111yescharacteristic, complemented, Sylow
135#3021—1391nomaximal, Sylow
136#585—1391no—
137C2×AGL(1,ℤ/8)—1392no—
138#535D13911yescharacteristic, complemented
139#454—1394no—
140#453—1394no—
141#421—1392no—
142#413—1391no—
1432−1+4—2782no—
1442−1+4—1392no—
145H(5,𝔽2)—2782no—
146H(5,𝔽2)—2782no—
147H(5,𝔽2)—1392no—
148C2×#41—2784no—
149C2×#41—2782no—
150C2×#41—1392no—
151C2×#41C2×D13911yescharacteristic
152C22×D4C2×D13911yescharacteristic, complemented
153AGL(1,ℤ/8)—2784no—
154AGL(1,ℤ/8)—1398no—
155C2×SD16—2782no—
156C2×D8—2782no—
157C2×M16—1392no—
158#127C2×D13912yescomplemented
159#124C2×D13912yescomplemented
160#121—22no—
161#120—22no—
162#117C2×D13911yescharacteristic, complemented
163#104—1398no—
164#101—1392no—
165#100—1392no—
166C24—21no—
167#41—22no—
168#41—22no—
169#41—2784no—
170#41—2784no—
171#41—2788no—
172#41—2784no—
173#41—2784no—
174#41—2782no—
175#41—2782no—
176#41—2784no—
177#41—1394no—
178#41—1394no—
179C2×Q8—2782no—
180C2×Q8—2784no—
181C2×Q8—1392no—
182C2×Q8C22×D13911yescharacteristic
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—21no—
187C2×D4—2782no—
188C2×D4—2788no—
189C2×D4—2784no—
190C2×D4—2784no—
191C2×D4—2782no—
192C2×D4—1392no—
193C2×D4C22⋊D13912yes—
194C2×D4C22×D13912yes—
195C2×D4C22×D13911yescharacteristic
196C22×C4—2784no—
197C22×C4—2782no—
198C22×C4C22⋊D13912yes—
199C22×C4C22×D13911yescharacteristic
200SD16—2788no—
201D8—2788no—
202M16—2782no—
203M16—1394no—
204C2×C8—2782no—
205C4⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—24no—
208C42C22×D13912yes—
209C23—22no—
210C23—42no—
211C23—82no—
212C23—2784no—
213C23—2782no—
214C23C2×C22⋊D13912yes—
215C23C2×C22⋊D13911yescharacteristic
216Q8—22no—
217Q8—2784no—
218Q8—2782no—
219Q8—1394no—
220D4—24no—
221D4—22no—
222D4—24no—
223D4—41no—
224D4—41no—
225D4—84no—
226D4—2784no—
227D4—2788no—
228D4—2784no—
229D4—2784no—
230D4—2782no—
231D4—1394no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—2788no—
236C2×C4—2784no—
237C2×C4—2782no—
238C2×C4—2784no—
239C2×C4—2784no—
240C2×C4D4×D13912yes—
241C2×C4D4×D13912yes—
242C2×C4C2×C22⋊D13912yes—
243C2×C4C2×C22⋊D13911yescharacteristic
244C2×C4C23×D13911yescharacteristic
245C8—2784no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—44no—
250C22—42no—
251C22—84no—
252C22—2784no—
253C22—11122no—
254C22—11124no—
255C22#1798481012yes—
256C22C22×C22⋊D13911yescharacteristic
257C4—22no—
258C4—42no—
259C4—2784no—
260C4#1798481011yescharacteristic
261C4#1798481011yescharacteristic
262C4C2×D4×D13912yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—5564no—
268C2C2×#1798481011yescharacteristic
subgroupG/H
centerC2C2×#17984810
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#535×C139C2
socleC2×C139C2×#120
hypercenter#535D139
hypocenterC139#3021
LCSG ▷ C2×C4×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. C2×#17984868C2semidirect
3. #22581125C2semidirect
4. #22581120C2semidirect
5. #22580971C2semidirect
6. #22580937C2semidirect
7. #535×C139C2semidirect
8. #17984890C22semidirect
12. #17984887C22semidirect
17. #17984868C22semidirect
20. C2×#14555769C22semidirect
21. #17984770C22semidirect
22. #17984761C22semidirect
23. #17984759C22semidirect
25. C22×D4×C139C22semidirect
26. #127×C139C22semidirect
27. #124×C139C22semidirect
30. #117×C139C22semidirect
37. #14555785D4semidirect
42. C2×Dic278D4semidirect
45. C2×D556D4semidirect
134. C139#3021semidirect
138. #535D139semidirect
152. C22×D4C2×D139semidirect
158. #127C2×D139semidirect
159. #124C2×D139semidirect
162. #117C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225817892
C2×#179848682
#225811252
#225811202
#225809712
#225809372
#535×C1392
#3021139