Finite Group Databasebeta

← #31815776

Subgroups — #31815776

Order 17792.

736 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 161 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×#17984867C212yesmaximal, complemented
2#22581517C211yesmaximal, characteristic, complemented
3#22581126C214yesmaximal, complemented
4#22581121C214yesmaximal, complemented
5#22580970C212yesmaximal, complemented
6#22580937C211yesmaximal, characteristic, complemented
7#585×C139C211yesmaximal, characteristic, complemented
8C22×D556C2211yescharacteristic, complemented
9#17984867—24no—
10#17984867C2218yescomplemented
11C2×#14555772—22no—
12C2×#14555770—22no—
13C2×#14555769C2212yescomplemented
14#17984838C2212yescomplemented
15#17984831C2212yescomplemented
16#17984816—22no—
17#17984810—22no—
18#17984801C2211yescharacteristic, complemented
19#17984769C2218yescomplemented
20#17984761C2212yescomplemented
21#17984759C2212yescomplemented
222−1+4×C139—22no—
232−1+4×C139C2212yescomplemented
24H(5,𝔽2)×C139—22no—
25H(5,𝔽2)×C139—22no—
26H(5,𝔽2)×C139C2212yescomplemented
27C2×#41×C139—24no—
28C2×#41×C139—22no—
29C2×#41×C139C2212yes—
30C2×#41×C139C2211yescharacteristic
31C23×D139—21no—
32C2×D556—22no—
33C2×D556—22no—
34C2×D556—21no—
35C2×D556C2312yes—
36C2×C22⋊D139—24no—
37#14555772—28no—
38#14555770—28no—
39#14555769—22no—
40#14555769C2314yes—
41C2×C139⋊C8—22no—
42C4⋊Dic139—21no—
43C22⋊Dic139—21no—
44(C2×D139)⋊C4—24no—
45C4×Dic139C2312yes—
46#41×C139—24no—
47#41×C139—28no—
48#41×C139—24no—
49#41×C139—24no—
50#41×C139—24no—
51#41×C139—22no—
52#41×C139—22no—
53#41×C139—24no—
54#41×C139—22no—
55#41×C139—22no—
56#41×C139D414yescomplemented
57#41×C139C2314yes—
58C2×Q8×C139—22no—
59C2×Q8×C139—24no—
60C2×Q8×C139D412yescomplemented
61C2×Q8×C139C2311yescharacteristic
62C2×D4×C139—28no—
63C2×D4×C139—22no—
64C2×D4×C139—24no—
65C2×D4×C139—22no—
66C2×D4×C139—24no—
67C2×D4×C139D412yes—
68C2×D4×C139D412yescomplemented
69C2×D4×C139C2311yescharacteristic
70C22×C4×C139—24no—
71C22×C4×C139—22no—
72C22×C4×C139D412yes—
73C22×C4×C139C2311yescharacteristic
74C22×D139—22no—
75C22×D139—42no—
76C22×D139—82no—
77D556—24no—
78D556—41no—
79D556—41no—
80C22⋊D139—84no—
81C2×Dic139—22no—
82C139⋊C8—24no—
83C23×C139—24no—
84C23×C139—22no—
85C23×C139C2×D412yes—
86C23×C139C2×D411yescharacteristic
87Q8×C139—24no—
88Q8×C139—22no—
89Q8×C139—22no—
90Q8×C139C2×D414yes—
91D4×C139—28no—
92D4×C139—24no—
93D4×C139—24no—
94D4×C139—24no—
95D4×C139—24no—
96D4×C139—22no—
97D4×C139—22no—
98D4×C139C2×D414yes—
99C2×C4×C139—24no—
100C2×C4×C139—28no—
101C2×C4×C139—22no—
102C2×C4×C139—24no—
103C2×C4×C139—22no—
104C2×C4×C139—22no—
105C2×C4×C139—24no—
106C2×C4×C139C2×D412yes—
107C2×C4×C139C2×D411yescharacteristic
108C2×C4×C139C2×D412yes—
109C2×C4×C139C2×D412yes—
110C2×C4×C139C2411yescharacteristic
111C22×C139—22no—
112C22×C139—24no—
113C22×C139—41no—
114C22×C139—42no—
115C22×C139—82no—
116C22×C139—84no—
117C22×C139#12012yes—
118C22×C139C22×D411yescharacteristic
119C2×D139—44no—
120C2×D139—42no—
121C2×D139—84no—
122C4×C139—24no—
123C4×C139—22no—
124C4×C139#12011yescharacteristic
125C4×C139#12011yescharacteristic
126C4×C139C22×D412yes—
127Dic139—42no—
128C2×C139—21no—
129C2×C139—22no—
130C2×C139—42no—
131C2×C139—44no—
132C2×C139C2×#12011yescharacteristic
133D139—82no—
134C139#302111yescharacteristic, complemented, Sylow
135#3021—1391nomaximal, Sylow
136#585D13911yescharacteristic, complemented
137C2×AGL(1,ℤ/8)—1392no—
138#535—1391no—
139#454—1394no—
140#453—1394no—
141#421—1392no—
142#413—1391no—
1432−1+4—22no—
1442−1+4C2×D13912yescomplemented
145H(5,𝔽2)—22no—
146H(5,𝔽2)—22no—
147H(5,𝔽2)C2×D13912yescomplemented
148C2×#41—24no—
149C2×#41—22no—
150C2×#41C2×D13912yes—
151C2×#41C2×D13911yescharacteristic
152C22×D4—1391no—
153AGL(1,ℤ/8)—2784no—
154AGL(1,ℤ/8)—1398no—
155C2×SD16—2782no—
156C2×D8—2782no—
157C2×M16—1392no—
158#127—1392no—
159#124—1392no—
160#121—2782no—
161#120—2782no—
162#117—1391no—
163#104—1398no—
164#101—1392no—
165#100—1392no—
166C24—2781no—
167#41—24no—
168#41—28no—
169#41—24no—
170#41—24no—
171#41—24no—
172#41—22no—
173#41—22no—
174#41—24no—
175#41—22no—
176#41—22no—
177#41C22⋊D13914yescomplemented
178#41C22×D13914yes—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8C22⋊D13912yescomplemented
182C2×Q8C22×D13911yescharacteristic
183C2×D4—28no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4—22no—
187C2×D4—24no—
188C2×D4—2784no—
189C2×D4—2782no—
190C2×D4—2782no—
191C2×D4—2781no—
192C2×D4—1392no—
193C2×D4C22⋊D13912yes—
194C2×D4C22⋊D13912yescomplemented
195C2×D4C22×D13911yescharacteristic
196C22×C4—24no—
197C22×C4—22no—
198C22×C4C22⋊D13912yes—
199C22×C4C22×D13911yescharacteristic
200SD16—2788no—
201D8—2788no—
202M16—2782no—
203M16—1394no—
204C2×C8—2782no—
205C4⋊C4—2781no—
206C22⋊C4—2781no—
207C22⋊C4—2784no—
208C42—1392no—
209C23—24no—
210C23—22no—
211C23—2782no—
212C23—5562no—
213C23—11122no—
214C23C2×C22⋊D13912yes—
215C23C2×C22⋊D13911yescharacteristic
216Q8—24no—
217Q8—22no—
218Q8—22no—
219Q8C2×C22⋊D13914yes—
220D4—28no—
221D4—24no—
222D4—24no—
223D4—24no—
224D4—24no—
225D4—22no—
226D4—22no—
227D4—2784no—
228D4—5561no—
229D4—5561no—
230D4—11124no—
231D4C2×C22⋊D13914yes—
232C2×C4—24no—
233C2×C4—28no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—2782no—
240C2×C4C2×C22⋊D13912yes—
241C2×C4C2×C22⋊D13911yescharacteristic
242C2×C4C2×C22⋊D13912yes—
243C2×C4C2×C22⋊D13912yes—
244C2×C4C23×D13911yescharacteristic
245C8—2784no—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—42no—
250C22—82no—
251C22—84no—
252C22—5564no—
253C22—5562no—
254C22—11124no—
255C22#1798481112yes—
256C22C22×C22⋊D13911yescharacteristic
257C4—24no—
258C4—22no—
259C4—5562no—
260C4#1798481111yescharacteristic
261C4#1798481111yescharacteristic
262C4C22×C22⋊D13912yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—44no—
267C2—11122no—
268C2C2×#1798481111yescharacteristic
subgroupG/H
centerC2C2×#17984811
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#585×C139C2
socleC2×C139C2×#120
hypercenter#585D139
hypocenterC139#3021
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#17984867C2semidirect
2. #22581517C2semidirect
3. #22581126C2semidirect
4. #22581121C2semidirect
5. #22580970C2semidirect
6. #22580937C2semidirect
7. #585×C139C2semidirect
8. C22×D556C22semidirect
10. #17984867C22semidirect
13. C2×#14555769C22semidirect
14. #17984838C22semidirect
15. #17984831C22semidirect
18. #17984801C22semidirect
19. #17984769C22semidirect
20. #17984761C22semidirect
21. #17984759C22semidirect
23. 2−1+4×C139C22semidirect
26. H(5,𝔽2)×C139C22semidirect
56. #41×C139D4semidirect
60. C2×Q8×C139D4semidirect
68. C2×D4×C139D4semidirect
134. C139#3021semidirect
136. #585D139semidirect
144. 2−1+4C2×D139semidirect
147. H(5,𝔽2)C2×D139semidirect
177. #41C22⋊D139semidirect
181. C2×Q8C22⋊D139semidirect
194. C2×D4C22⋊D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848672
#225815172
#225811262
#225811212
#225809702
#225809372
#585×C1392
#3021139