Finite Group Databasebeta

← #31818194

Subgroups — #31818194

Order 17792.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#22581762C212yesmaximal, complemented
2#22581761C212yesmaximal, complemented
3#22581327C211yesmaximal, characteristic, complemented
4#22581219C212yesmaximal, complemented
5#22581218C212yesmaximal, complemented
6#22581135C211yesmaximal, characteristic, complemented
7#22581122C212yesmaximal, complemented
8#22581067C212yesmaximal, complemented
9#456×C139C211yesmaximal, characteristic, complemented
10#17984890C2211yescharacteristic
11#17984888C2212yescomplemented
12#17984874—22no—
13#17984873—22no—
14#17984873C2212yescomplemented
15#17984870C2212yes—
16#17984869—22no—
17#17984867—22no—
18#17984867C2212yes—
19#17984865—22no—
20#17984862—22no—
21#17984862C2212yes—
22C2×#14555772—22no—
23C2×#14555772C2212yescomplemented
24C2×#14555772C2212yescomplemented
25SD16×D139—22no—
26#17984850C2212yescomplemented
27#17984849C2212yescomplemented
28#17984839C2211yescharacteristic
29#17984777C2212yescomplemented
30#17984770C2212yescomplemented
31#17984761C2211yescharacteristic, complemented
32#17984758C2212yescomplemented
33C4×C139⋊C8C2211yescharacteristic, complemented
342−1+4×C139C2211yescharacteristic
35#137×C139C2212yescomplemented
36C4⋊Q8×C139C2211yescharacteristic, complemented
37#104×C139C2212yescomplemented
38#101×C139C2211yescharacteristic, complemented
39#14555786—22no—
40#14555786—22no—
41#14555785—22no—
42#14555785—41no—
43#14555785C2312yes—
44Q8×D139—22no—
45C2×D556—22no—
46C2×D556—41no—
47C2×D556C2312yes—
48D4×D139—22no—
49D4×D139—42no—
50#14555775—22no—
51#14555775D412yescomplemented
52#14555774—22no—
53#14555773D412yescomplemented
54#14555772—22no—
55#14555772—22no—
56#14555772—22no—
57#14555772—22no—
58#14555772—22no—
59#14555772—22no—
60#14555772D412yescomplemented
61D1112—22no—
62#14555770—22no—
63#14555770D412yescomplemented
64#14555769—22no—
65#14555769C2312yes—
66#14555768—22no—
67C2×C139⋊C8—22no—
68C2×C139⋊C8C2312yes—
69C8×D139—22no—
70#41×C139—22no—
71#41×C139—41no—
72#41×C139C2312yes—
73C2×Q8×C139—22no—
74C2×Q8×C139C2311yescharacteristic
75C2×Q8×C139C2311yescharacteristic
76Q16×C139—22no—
77SD16×C139—22no—
78M16×C139C2312yes—
79C4⋊C4×C139—41no—
80C42×C139C2311yescharacteristic
81C22×D139—22no—
82C22×D139—42no—
83Dic278—41no—
84Dic278C2×D412yes—
85D556—22no—
86D556—22no—
87D556—42no—
88D556—41no—
89D556—82no—
90D556C2×D412yes—
91C22⋊D139—22no—
92C22⋊D139—42no—
93C4×D139—22no—
94C4×D139—42no—
95C4×D139—42no—
96C139⋊C8—22no—
97C139⋊C8C2×D412yes—
98C139⋊C8C2×D412yes—
99Q8×C139—22no—
100Q8×C139—22no—
101Q8×C139—41no—
102Q8×C139C2×D412yes—
103D4×C139—22no—
104D4×C139—41no—
105D4×C139C2×D412yes—
106C2×C4×C139—21no—
107C2×C4×C139—22no—
108C2×C4×C139—21no—
109C2×C4×C139—21no—
110C2×C4×C139—42no—
111C2×C4×C139C2411yescharacteristic
112C8×C139—22no—
113C22×C139—22no—
114C22×C139H(5,𝔽2)11yescharacteristic
115C2×D139—22no—
116C2×D139—42no—
117C2×D139—82no—
118C2×D139—162no—
119C4×C139—22no—
120C4×C139—22no—
121C4×C139—41no—
122C4×C139—41no—
123C4×C139C22×D412yes—
124Dic139—22no—
125C2×C139—21no—
126C2×C139—42no—
127C2×C139D4211yescharacteristic
128D139—42no—
129D139—82no—
130C139#329111yescharacteristic, complemented, Sylow
131#3291—1391nomaximal, Sylow
132#577—1392no—
133#577—1392no—
134#492—1391no—
135#471—1392no—
136#471—1392no—
137#456—1391no—
138#456D13911yescharacteristic, complemented
139#453—1392no—
140#443—1392no—
1412−1+4—1391no—
1422−1+4C2×D13911yescharacteristic
143H(5,𝔽2)—1392no—
144#137—2782no—
145#137—2782no—
146#137—1392no—
147#137C2×D13912yescomplemented
148AGL(1,ℤ/8)—2782no—
149AGL(1,ℤ/8)—2782no—
150AGL(1,ℤ/8)—1392no—
151AGL(1,ℤ/8)—1392no—
152#135—2782no—
153#135—2782no—
154#135—1392no—
155C2×SD16—2782no—
156C2×SD16—2782no—
157C2×SD16—1392no—
158C2×SD16—1392no—
159#131—1392no—
160#131—1392no—
161C4⋊Q8C2×D13911yescharacteristic, complemented
162#127—1391no—
163#108—1392no—
164#104—1392no—
165#104C2×D13912yescomplemented
166#101—1391no—
167#101C2×D13911yescharacteristic, complemented
168#100—1392no—
169C4×C8—1391no—
170#41—22no—
171#41—41no—
172#41—2782no—
173#41—2782no—
174#41—2782no—
175#41—5561no—
176#41—1392no—
177#41C22×D13912yes—
178C2×Q8—22no—
179C2×Q8—2782no—
180C2×Q8C22×D13911yescharacteristic
181C2×Q8C22×D13911yescharacteristic
182C2×D4—2782no—
183C2×D4—2782no—
184C2×D4—5561no—
185C2×D4—5562no—
186C2×D4—1392no—
187Q16—22no—
188Q16—2782no—
189Q16—1392no—
190SD16—22no—
191SD16—2782no—
192SD16—2782no—
193SD16—2782no—
194SD16—2782no—
195SD16—2782no—
196SD16—2782no—
197SD16—2782no—
198SD16—1392no—
199SD16—1392no—
200D8—2782no—
201D8—2782no—
202D8—1392no—
203M16—2782no—
204M16—2782no—
205M16—1392no—
206M16C22×D13912yes—
207C2×C8—2782no—
208C2×C8—2782no—
209C2×C8—1392no—
210C4⋊C4—41no—
211C42C22×D13911yescharacteristic
212C23—2782no—
213C23—5562no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—5561no—
218Q8—1392no—
219Q8D4×D13912yes—
220D4—22no—
221D4—41no—
222D4—2782no—
223D4—2782no—
224D4—2782no—
225D4—5562no—
226D4—5561no—
227D4—5562no—
228D4—11122no—
229D4—1392no—
230D4D4×D13912yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—2782no—
237C2×C4—5562no—
238C2×C4—5562no—
239C2×C4C23×D13911yescharacteristic
240C8—22no—
241C8—2782no—
242C8—1392no—
243C8—1392no—
244C22—22no—
245C22—2782no—
246C22—5562no—
247C22—11122no—
248C22—22242no—
249C22#1798488711yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—2782no—
255C4C2×D4×D13912yes—
256C2—21no—
257C2—42no—
258C2—5562no—
259C2—11122no—
260C2#2258158511yescharacteristic
subgroupG/H
centerC2#22581585
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#456×C139C2
socleC2×C139D42
hypercenter#456D139
hypocenterC139#3291
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #22581762C2semidirect
2. #22581761C2semidirect
3. #22581327C2semidirect
4. #22581219C2semidirect
5. #22581218C2semidirect
6. #22581135C2semidirect
7. #22581122C2semidirect
8. #22581067C2semidirect
9. #456×C139C2semidirect
11. #17984888C22semidirect
14. #17984873C22semidirect
23. C2×#14555772C22semidirect
24. C2×#14555772C22semidirect
26. #17984850C22semidirect
27. #17984849C22semidirect
29. #17984777C22semidirect
30. #17984770C22semidirect
31. #17984761C22semidirect
32. #17984758C22semidirect
33. C4×C139⋊C8C22semidirect
35. #137×C139C22semidirect
36. C4⋊Q8×C139C22semidirect
37. #104×C139C22semidirect
38. #101×C139C22semidirect
51. #14555775D4semidirect
53. #14555773D4semidirect
60. #14555772D4semidirect
63. #14555770D4semidirect
130. C139#3291semidirect
138. #456D139semidirect
147. #137C2×D139semidirect
161. C4⋊Q8C2×D139semidirect
165. #104C2×D139semidirect
167. #101C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225817622
#225817612
#225813272
#225812192
#225812182
#225811352
#225811222
#225810672
#456×C1392
#3291139