Finite Group Databasebeta

← #31816015

Subgroups — #31816015

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×#17984865C211yesmaximal, characteristic, complemented
2C2×M16×D139C211yesmaximal, characteristic, complemented
3#22581466C212yesmaximal, complemented
4#22581208C211yesmaximal, characteristic, complemented
5#22581207C211yesmaximal, characteristic, complemented
6#22581206C211yesmaximal, characteristic, complemented
7#22581202C212yesmaximal, complemented
8#22581201C212yesmaximal, complemented
9#22580988C211yesmaximal, characteristic, complemented
10#22580948C212yesmaximal, complemented
11#470×C139C211yesmaximal, characteristic, complemented
12C2×#14555786C2212yes—
13C22×C4×D139C2211yescharacteristic
14#17984865—24no—
15C2×#14555772C2212yescomplemented
16C2×D1112C2211yescharacteristic, complemented
17C2×#14555769C2211yescharacteristic
18C2×#14555768C2211yescharacteristic
19C2×#14555768C2211yescharacteristic
20M16×D139—24no—
21C2×C8×D139C2211yescharacteristic
22C2×C8×D139C2211yescharacteristic
23#17984827—22no—
24#17984818C2212yes—
25#17984817C2212yes—
26#17984816C2212yes—
27#17984812—22no—
28#17984804—22no—
29(C22⋊D139)⋊C4—22no—
30#17984800C2212yes—
31#17984775C2212yescomplemented
32#17984774C2211yescharacteristic, complemented
33#17984766C2212yescomplemented
34#17984764C2212yescomplemented
35#17984762C2212yescomplemented
36C2×Q16×C139C2211yescharacteristic, complemented
37C2×M16×C139C2211yescharacteristic
38C22⋊Q8×C139C2212yes—
39#106×C139C2211yescharacteristic, complemented
40#103×C139C2212yescomplemented
41C23×D139D411yescharacteristic
42#14555786—24no—
43#14555786—24no—
44C2×D556—22no—
45C2×D556—22no—
46C2×D556C2312yes—
47C2×C22⋊D139—22no—
48C22×Dic139D411yescharacteristic
49C2×C4×D139—22no—
50C2×C4×D139—22no—
51C2×C4×D139—22no—
52C2×C4×D139D411yescharacteristic
53C2×C4×D139D411yescharacteristic
54C2×C4×D139C2311yescharacteristic
55C2×C4×D139C2311yescharacteristic
56#14555772—24no—
57D1112—22no—
58#14555769—22no—
59#14555768—22no—
60#14555768—22no—
61C2×C139⋊C8C2311yescharacteristic
62C2×C139⋊C8C2311yescharacteristic
63C8×D139—22no—
64C8×D139D414yescomplemented
65C4⋊Dic139C2312yes—
66Dic139⋊C4—22no—
67(C2×D139)⋊C4—22no—
68(C2×D139)⋊C4—22no—
69(C2×D139)⋊C4—22no—
70(C2×D139)⋊C4—22no—
71C4×Dic139—22no—
72C2×Q8×C139C2312yes—
73C22×C4×C139C2311yescharacteristic
74Q16×C139—22no—
75M16×C139—22no—
76C2×C8×C139C2311yescharacteristic
77C2×C8×C139C2311yescharacteristic
78C4⋊C4×C139—22no—
79C4⋊C4×C139C2312yes—
80C22⋊C4×C139—22no—
81C22×D139—22no—
82C22×D139—22no—
83C22×D139—42no—
84C22×D139C2×D411yescharacteristic
85C22×D139C2×D411yescharacteristic
86D556—22no—
87D556—22no—
88D556—44no—
89D556—44no—
90C22⋊D139—44no—
91C2×Dic139—22no—
92C2×Dic139—22no—
93C2×Dic139C2×D411yescharacteristic
94C2×Dic139C2×D411yescharacteristic
95C4×D139—22no—
96C4×D139—21no—
97C4×D139—21no—
98C4×D139—22no—
99C4×D139—44no—
100C4×D139C2×D412yes—
101C4×D139#4112yes—
102C139⋊C8—21no—
103C139⋊C8C2×D412yes—
104C23×C139C2×D411yescharacteristic
105Q8×C139—22no—
106Q8×C139—22no—
107C2×C4×C139—21no—
108C2×C4×C139—21no—
109C2×C4×C139—22no—
110C2×C4×C139—22no—
111C2×C4×C139C2×D411yescharacteristic
112C2×C4×C139C2411yescharacteristic
113C8×C139—21no—
114C8×C139C2×D412yes—
115C22×C139—21no—
116C22×C139—22no—
117C22×C139C22×D411yescharacteristic
118C2×D139—22no—
119C2×D139—21no—
120C2×D139—22no—
121C2×D139—44no—
122C2×D139—42no—
123C2×D139—42no—
124C2×D139—42no—
125C2×D139#12112yes—
126C4×C139—21no—
127C4×C139—42no—
128C4×C139—42no—
129C4×C139C22×D411yescharacteristic
130C4×C139C2×#4111yescharacteristic
131Dic139—21no—
132Dic139—42no—
133Dic139#12112yes—
134C2×C139—41no—
135C2×C139C2×#12111yescharacteristic
136C2×C139#57512yes—
137D139—22no—
138D139—41no—
139D139—82no—
140C139#305411yescharacteristic, complemented, Sylow
141#3054—1391nomaximal, Sylow
142C2×#135—1391no—
143C22×M16—1391no—
144#525—1392no—
145#470—1391no—
146#470D13911yescharacteristic, complemented
147#469—1391no—
148#469—1391no—
149#468—1392no—
150#468—1392no—
151#427—1391no—
152#416—1392no—
153C2×#41—1392no—
154C23×C4—1391no—
155#135—2784no—
156C2×Q16C2×D13911yescharacteristic, complemented
157C2×SD16—1392no—
158C2×D8—1391no—
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⋊Q8C2×D13912yes—
169#121—1392no—
170#121—1392no—
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#103C2×D13912yescomplemented
180#102—1392no—
181#102—1392no—
182C24—1391no—
183#41—2784no—
184#41—2784no—
185C2×Q8C22×D13912yes—
186C2×D4—2782no—
187C2×D4—2782no—
188C2×D4—2782no—
189C2×D4—1392no—
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—22no—
200SD16—2784no—
201D8—2782no—
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—22no—
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—2782no—
223C23—2782no—
224C23—5562no—
225C23—1391no—
226C23—1391no—
227C23D4×D13911yescharacteristic
228Q8—22no—
229Q8—22no—
230D4—2782no—
231D4—2782no—
232D4—5564no—
233D4—5564no—
234D4—5564no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—2782no—
240C2×C4—2782no—
241C2×C4—2782no—
242C2×C4—2781no—
243C2×C4—2781no—
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—2782no—
259C22—2781no—
260C22—2782no—
261C22—5564no—
262C22—5562no—
263C22—5562no—
264C22—5562no—
265C22—1392no—
266C22C2×D4×D13911yescharacteristic
267C4—21no—
268C4—42no—
269C4—42no—
270C4—2781no—
271C4—5562no—
272C4—1392no—
273C4C2×D4×D13911yescharacteristic
274C4#41×D13911yescharacteristic
275C2—41no—
276C2—2782no—
277C2—5561no—
278C2—11122no—
279C2#121×D13911yescharacteristic
280C2#2258175212yes—
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#470×C139C2
socleC22×C139C22×D4
hypercenter#470D139
hypocenterC139#3054
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#17984865C2semidirect
2. C2×M16×D139C2semidirect
3. #22581466C2semidirect
4. #22581208C2semidirect
5. #22581207C2semidirect
6. #22581206C2semidirect
7. #22581202C2semidirect
8. #22581201C2semidirect
9. #22580988C2semidirect
10. #22580948C2semidirect
11. #470×C139C2semidirect
15. C2×#14555772C22semidirect
16. C2×D1112C22semidirect
31. #17984775C22semidirect
32. #17984774C22semidirect
33. #17984766C22semidirect
34. #17984764C22semidirect
35. #17984762C22semidirect
36. C2×Q16×C139C22semidirect
39. #106×C139C22semidirect
40. #103×C139C22semidirect
64. C8×D139D4semidirect
140. C139#3054semidirect
146. #470D139semidirect
156. C2×Q16C2×D139semidirect
177. #106C2×D139semidirect
179. #103C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×#179848652
C2×M16×D1392
#225814662
#225812082
#225812072
#225812062
#225812022
#225812012
#225809882
#225809482
#470×C1392
#3054139