Finite Group Databasebeta

← #31816011

Subgroups — #31816011

Order 17792.

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 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×#17984861C211yesmaximal, characteristic, complemented
2C2×M16×D139C211yesmaximal, characteristic, complemented
3#22581463C212yesmaximal, complemented
4#22581212C211yesmaximal, characteristic, complemented
5#22581211C211yesmaximal, characteristic, complemented
6#22581203C211yesmaximal, characteristic, complemented
7#22581198C212yesmaximal, complemented
8#22581197C212yesmaximal, complemented
9#22580988C211yesmaximal, characteristic, complemented
10#22580946C212yesmaximal, complemented
11#469×C139C211yesmaximal, characteristic, complemented
12C2×#14555784C2212yes—
13C22×C4×D139C2211yescharacteristic
14#17984861—24no—
15C2×Dic556C2211yescharacteristic, complemented
16C2×#14555773C2212yescomplemented
17C2×#14555769C2211yescharacteristic
18C2×#14555768C2211yescharacteristic
19C2×#14555768C2211yescharacteristic
20M16×D139—24no—
21C2×C8×D139C2211yescharacteristic
22C2×C8×D139C2211yescharacteristic
23#17984824—22no—
24#17984822C2212yes—
25#17984821C2212yes—
26#17984813C2212yes—
27#17984810—22no—
28D4×Dic139—22no—
29(C22⋊D139)⋊C4—22no—
30#17984800C2212yes—
31#17984775C2212yescomplemented
32#17984774C2211yescharacteristic, complemented
33#17984767C2212yescomplemented
34#17984765C2212yescomplemented
35#17984763C2212yescomplemented
36C2×D8×C139C2211yescharacteristic, complemented
37C2×M16×C139C2211yescharacteristic
38#121×C139C2212yes—
39#106×C139C2211yescharacteristic, complemented
40#102×C139C2212yescomplemented
41C23×D139D411yescharacteristic
42#14555784—24no—
43#14555784—24no—
44C2×Dic278C2312yes—
45C2×C22⋊D139—22no—
46C2×C22⋊D139—22no—
47C22×Dic139—22no—
48C22×Dic139D411yescharacteristic
49C2×C4×D139—22no—
50C2×C4×D139—22no—
51C2×C4×D139D411yescharacteristic
52C2×C4×D139D411yescharacteristic
53C2×C4×D139C2311yescharacteristic
54C2×C4×D139C2311yescharacteristic
55Dic556—22no—
56#14555773—24no—
57#14555769—22no—
58#14555768—22no—
59#14555768—22no—
60C2×C139⋊C8C2311yescharacteristic
61C2×C139⋊C8C2311yescharacteristic
62C8×D139—22no—
63C8×D139D414yescomplemented
64C4⋊Dic139—22no—
65C4⋊Dic139C2312yes—
66Dic139⋊C4—22no—
67C22⋊Dic139—22no—
68C22⋊Dic139—22no—
69(C2×D139)⋊C4—22no—
70(C2×D139)⋊C4—22no—
71C4×Dic139—22no—
72C2×D4×C139—22no—
73C2×D4×C139C2312yes—
74C22×C4×C139C2311yescharacteristic
75D8×C139—22no—
76M16×C139—22no—
77C2×C8×C139C2311yescharacteristic
78C2×C8×C139C2311yescharacteristic
79C4⋊C4×C139C2312yes—
80C22⋊C4×C139—22no—
81C22×D139—22no—
82C22×D139—42no—
83C22×D139C2×D411yescharacteristic
84C22×D139C2×D411yescharacteristic
85Dic278—22no—
86Dic278—22no—
87C22⋊D139—44no—
88C22⋊D139—44no—
89C2×Dic139—22no—
90C2×Dic139—22no—
91C2×Dic139—22no—
92C2×Dic139—44no—
93C2×Dic139C2×D411yescharacteristic
94C2×Dic139C2×D411yescharacteristic
95C4×D139—22no—
96C4×D139—21no—
97C4×D139—21no—
98C4×D139—22no—
99C4×D139C2×D412yes—
100C4×D139#4112yes—
101C139⋊C8—21no—
102C139⋊C8C2×D412yes—
103C23×C139—22no—
104C23×C139C2×D411yescharacteristic
105D4×C139—22no—
106D4×C139—22no—
107D4×C139—44no—
108C2×C4×C139—21no—
109C2×C4×C139—21no—
110C2×C4×C139—22no—
111C2×C4×C139C2×D411yescharacteristic
112C2×C4×C139C2411yescharacteristic
113C8×C139—21no—
114C8×C139C2×D412yes—
115C22×C139—21no—
116C22×C139—22no—
117C22×C139—42no—
118C22×C139—42no—
119C22×C139—42no—
120C22×C139C22×D411yescharacteristic
121C2×D139—22no—
122C2×D139—21no—
123C2×D139—22no—
124C2×D139—44no—
125C2×D139#12112yes—
126C4×C139—21no—
127C4×C139—42no—
128C4×C139C22×D411yescharacteristic
129C4×C139C2×#4111yescharacteristic
130Dic139—21no—
131Dic139—42no—
132Dic139—42no—
133Dic139#12112yes—
134C2×C139—41no—
135C2×C139—82no—
136C2×C139C2×#12111yescharacteristic
137C2×C139#57512yes—
138D139—22no—
139D139—41no—
140C139#305411yescharacteristic, complemented, Sylow
141#3054—1391nomaximal, Sylow
142C2×#135—1391no—
143C22×M16—1391no—
144#525—1392no—
145#470—1391no—
146#470—1391no—
147#469—1391no—
148#469D13911yescharacteristic, complemented
149#468—1392no—
150#468—1392no—
151#427—1391no—
152#416—1392no—
153C2×#41—1392no—
154C23×C4—1391no—
155#135—2784no—
156C2×Q16—1391no—
157C2×SD16—1392no—
158C2×D8C2×D13911yescharacteristic, complemented
159C2×M16—2784no—
160C2×M16—1391no—
161C2×M16—1391no—
162C2×M16—1391no—
163C2×M16C2×D13911yescharacteristic
164C22×C8—1391no—
165C22×C8—1391no—
166#123—2782no—
167C22⋊Q8—1392no—
168C22⋊Q8—1392no—
169#121—1392no—
170#121C2×D13912yes—
171#120—2782no—
172C4×D4—2782no—
173C4×D4—2782no—
174#117—1392no—
175#107—1392no—
176#106—1391no—
177#106C2×D13911yescharacteristic, complemented
178#103—1392no—
179#103—1392no—
180#102—1392no—
181#102C2×D13912yescomplemented
182C24—1391no—
183#41—2784no—
184#41—2784no—
185C2×Q8—1392no—
186C2×D4—22no—
187C2×D4—2782no—
188C2×D4—2782no—
189C2×D4C22×D13912yes—
190C22×C4—2782no—
191C22×C4—2782no—
192C22×C4—2782no—
193C22×C4—1391no—
194C22×C4—1391no—
195C22×C4—1391no—
196C22×C4—1391no—
197C22×C4—1391no—
198C22×C4C22×D13911yescharacteristic
199Q16—2782no—
200SD16—2784no—
201D8—22no—
202M16—22no—
203M16—2782no—
204M16—2782no—
205M16—2782no—
206C2×C8—2782no—
207C2×C8—1391no—
208C2×C8—1391no—
209C2×C8—1394no—
210C2×C8C22×D13911yescharacteristic
211C2×C8C22×D13911yescharacteristic
212C4⋊C4—2782no—
213C4⋊C4—2782no—
214C4⋊C4—1392no—
215C4⋊C4C22×D13912yes—
216C22⋊C4—22no—
217C22⋊C4—2782no—
218C22⋊C4—2782no—
219C22⋊C4—2782no—
220C22⋊C4—2782no—
221C42—2782no—
222C23—22no—
223C23—2782no—
224C23—5562no—
225C23—1391no—
226C23—1391no—
227C23D4×D13911yescharacteristic
228Q8—2782no—
229Q8—2782no—
230D4—22no—
231D4—22no—
232D4—44no—
233D4—5564no—
234D4—5564no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—2782no—
239C2×C4—2782no—
240C2×C4—2782no—
241C2×C4—2781no—
242C2×C4—2781no—
243C2×C4—2782no—
244C2×C4—2782no—
245C2×C4—5564no—
246C2×C4—1391no—
247C2×C4—1391no—
248C2×C4—1392no—
249C2×C4—1392no—
250C2×C4D4×D13911yescharacteristic
251C2×C4C23×D13911yescharacteristic
252C8—21no—
253C8—2781no—
254C8—1392no—
255C8D4×D13912yes—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—2782no—
262C22—2781no—
263C22—2782no—
264C22—5564no—
265C22—1392no—
266C22C2×D4×D13911yescharacteristic
267C4—21no—
268C4—42no—
269C4—2781no—
270C4—5562no—
271C4—5562no—
272C4—1392no—
273C4C2×D4×D13911yescharacteristic
274C4#41×D13911yescharacteristic
275C2—41no—
276C2—82no—
277C2—2782no—
278C2—5561no—
279C2#121×D13911yescharacteristic
280C2#2258174912yes—
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#469×C139C2
socleC22×C139C22×D4
hypercenter#469D139
hypocenterC139#3054
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#17984861C2semidirect
2. C2×M16×D139C2semidirect
3. #22581463C2semidirect
4. #22581212C2semidirect
5. #22581211C2semidirect
6. #22581203C2semidirect
7. #22581198C2semidirect
8. #22581197C2semidirect
9. #22580988C2semidirect
10. #22580946C2semidirect
11. #469×C139C2semidirect
15. C2×Dic556C22semidirect
16. C2×#14555773C22semidirect
31. #17984775C22semidirect
32. #17984774C22semidirect
33. #17984767C22semidirect
34. #17984765C22semidirect
35. #17984763C22semidirect
36. C2×D8×C139C22semidirect
39. #106×C139C22semidirect
40. #102×C139C22semidirect
63. C8×D139D4semidirect
140. C139#3054semidirect
148. #469D139semidirect
158. C2×D8C2×D139semidirect
177. #106C2×D139semidirect
181. #102C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848612
C2×M16×D1392
#225814632
#225812122
#225812112
#225812032
#225811982
#225811972
#225809882
#225809462
#469×C1392
#3054139