Finite Group Databasebeta

← #497×D139 #31816840

Subgroups — #497×D139 #31816840

Order 17792.

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

normal subgroups only

This group has 564 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
1C2×Q16×D139C212yesmaximal, complemented
2C2×SD16×D139C212yesmaximal
3C4⋊Q8×D139C211yesmaximal, characteristic, complemented
4#124×D139C211yesmaximal, characteristic
5#22581348C211yesmaximal, characteristic, complemented
6#22581347C211yesmaximal, characteristic, complemented
7#22581346C212yesmaximal, complemented
8#22581345C211yesmaximal, characteristic, complemented
9#22581344C212yesmaximal, complemented
10#97×D139C211yesmaximal, characteristic, complemented
11#497×C139C211yesmaximal, characteristic, complemented
12C2×Q8×D139C2211yescharacteristic
13C2×Q8×D139C2212yes—
14C2×D4×D139C2211yescharacteristic
15C2×Dic556C2212yescomplemented
16C2×#14555775C2212yescomplemented
17C2×#14555775C2212yescomplemented
18Q16×D139—28no—
19C2×#14555774C2212yes—
20C2×#14555773C2212yes—
21C2×#14555772C2212yes—
22SD16×D139—28no—
23C2×C8×D139C2212yes—
24#17984842C2211yescharacteristic, complemented
25#17984841—22no—
26#17984840C2212yescomplemented
27#17984831C2211yescharacteristic
28#17984830C2211yescharacteristic
29#17984829C2211yescharacteristic
30#17984828—22no—
31C4⋊C4×D139—22no—
32C22⋊C4×D139—22no—
33C42×D139C2211yescharacteristic
34#17984753C2211yescharacteristic, complemented
35#17984752C2212yescomplemented
36C2×Q16×C139C2212yescomplemented
37C2×SD16×C139C2212yes—
38C4⋊Q8×C139C2211yescharacteristic, complemented
39#124×C139C2211yescharacteristic
40#97×C139C2211yescharacteristic, complemented
41C23×D139—21no—
42C2×Dic278—21no—
43C2×Dic278—22no—
44C2×Dic278C2311yescharacteristic
45C2×Dic278C2312yes—
46Q8×D139—24no—
47Q8×D139—24no—
48Q8×D139—24no—
49C2×D556C2311yescharacteristic
50C2×C22⋊D139—21no—
51D4×D139—24no—
52C2×C4×D139—21no—
53C2×C4×D139—22no—
54C2×C4×D139D412yes—
55C2×C4×D139C2311yescharacteristic
56Dic556—24no—
57#14555775—24no—
58#14555775—24no—
59#14555774—24no—
60#14555773—24no—
61#14555772—24no—
62C2×C139⋊C8C2312yes—
63C8×D139D418yes—
64C4⋊Dic139—22no—
65Dic139⋊C4—24no—
66C22⋊Dic139—22no—
67(C2×D139)⋊C4—22no—
68(C2×D139)⋊C4—22no—
69C4×Dic139D412yes—
70C4×Dic139C2311yescharacteristic
71C2×Q8×C139C2311yescharacteristic
72C2×Q8×C139C2312yes—
73C2×D4×C139C2311yescharacteristic
74Q16×C139—24no—
75SD16×C139—24no—
76C2×C8×C139C2312yes—
77C4⋊C4×C139—22no—
78C22⋊C4×C139—22no—
79C42×C139C2311yescharacteristic
80C22×D139—21no—
81C22×D139—42no—
82C22×D139—44no—
83C22×D139C2×D411yescharacteristic
84Dic278—22no—
85Dic278—22no—
86Dic278—22no—
87Dic278—44no—
88Dic278—42no—
89D556—22no—
90C22⋊D139—42no—
91C2×Dic139—21no—
92C2×Dic139—22no—
93C2×Dic139C2×D412yes—
94C2×Dic139C2×D411yescharacteristic
95C4×D139—24no—
96C4×D139—42no—
97C4×D139—44no—
98C4×D139C2×D414yes—
99C139⋊C8C2×D414yes—
100C23×C139—21no—
101Q8×C139—22no—
102Q8×C139—22no—
103Q8×C139—22no—
104D4×C139—22no—
105C2×C4×C139—21no—
106C2×C4×C139—22no—
107C2×C4×C139C2×D412yes—
108C2×C4×C139C2411yescharacteristic
109C8×C139C2×D414yes—
110C22×C139—41no—
111C22×C139—42no—
112C22×C139C22×D411yescharacteristic
113C2×D139—41no—
114C2×D139—42no—
115C2×D139—84no—
116C2×D139#12712yes—
117C2×D139#13714yes—
118C4×C139—22no—
119C4×C139—41no—
120C4×C139—42no—
121C4×C139C22×D412yes—
122Dic139—22no—
123Dic139—41no—
124Dic139—42no—
125Dic139#12712yes—
126C2×C139—81no—
127C2×C139C2×#12711yescharacteristic
128C2×C139C2×#13712yes—
129D139—81no—
130D139#49714yesdirect product factor
131C139C2×#49711yescharacteristic, complemented, Sylow
132C2×#497—1391nomaximal, Sylow
133C22×Q16—1392no—
134C22×SD16—1392no—
135C2×C4⋊Q8—1391no—
136C2×#124—1391no—
137#497—1392no—
138#497—1392no—
139#497—1391no—
140#497—1391no—
141#497—1391no—
142#497D13911yescharacteristic, direct product factor
143C2×#97—1391no—
144C22×Q8—1391no—
145C22×Q8—1392no—
146C22×D4—1391no—
147C2×Q16—2788no—
148C2×Q16—1392no—
149C2×Q16—1392no—
150C2×Q16—1392no—
151C2×Q16C2×D13912yescomplemented
152C2×SD16—2788no—
153C2×SD16—1392no—
154C2×SD16—1392no—
155C2×SD16—1392no—
156C2×SD16C2×D13912yes—
157C22×C8—1392no—
158C4⋊Q8—2782no—
159C4⋊Q8—1392no—
160C4⋊Q8—1391no—
161C4⋊Q8C2×D13911yescharacteristic, complemented
162#124—2782no—
163#124—1391no—
164#124—1391no—
165#124—1391no—
166#124C2×D13911yescharacteristic
167C2×C4⋊C4—2782no—
168C2×C22⋊C4—2782no—
169C2×C42—1391no—
170#97—1392no—
171#97—1391no—
172#97C2×D13911yescharacteristic, complemented
173C24—2781no—
174C2×Q8—2781no—
175C2×Q8—2782no—
176C2×Q8—2784no—
177C2×Q8—2784no—
178C2×Q8—2784no—
179C2×Q8—1391no—
180C2×Q8—1392no—
181C2×Q8C22×D13911yescharacteristic
182C2×Q8C22×D13912yes—
183C2×D4—2784no—
184C2×D4—2781no—
185C2×D4—1391no—
186C2×D4C22×D13911yescharacteristic
187C22×C4—2781no—
188C22×C4—2782no—
189C22×C4—1392no—
190C22×C4—1391no—
191Q16—24no—
192Q16—2784no—
193Q16—2784no—
194Q16—2784no—
195SD16—24no—
196SD16—2784no—
197SD16—2784no—
198SD16—2784no—
199C2×C8—1392no—
200C2×C8—1398no—
201C2×C8C22×D13912yes—
202C4⋊C4—22no—
203C4⋊C4—2784no—
204C4⋊C4—2782no—
205C22⋊C4—22no—
206C22⋊C4—2782no—
207C22⋊C4—2782no—
208C22⋊C4—2782no—
209C42—1392no—
210C42—1391no—
211C42C22×D13911yescharacteristic
212C23—21no—
213C23—2781no—
214C23—5562no—
215C23—5564no—
216C23—1391no—
217Q8—22no—
218Q8—22no—
219Q8—22no—
220Q8—2782no—
221Q8—2782no—
222Q8—2782no—
223Q8—5562no—
224Q8—5564no—
225D4—22no—
226D4—2782no—
227D4—5562no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—2784no—
231C2×C4—2781no—
232C2×C4—2782no—
233C2×C4—5562no—
234C2×C4—5564no—
235C2×C4—1392no—
236C2×C4—1391no—
237C2×C4—1394no—
238C2×C4D4×D13912yes—
239C2×C4C23×D13911yescharacteristic
240C8—1394no—
241C8D4×D13914yes—
242C22—42no—
243C22—41no—
244C22—5562no—
245C22—5561no—
246C22—11124no—
247C22—1394no—
248C22—1392no—
249C22C2×D4×D13911yescharacteristic
250C4—22no—
251C4—41no—
252C4—42no—
253C4—2782no—
254C4—5561no—
255C4—5562no—
256C4—1392no—
257C4C2×D4×D13912yes—
258C2—81no—
259C2—11121no—
260C2—1394no—
261C2#127×D13911yescharacteristic
262C2#137×D13912yes—
subgroupG/H
centerC22C2×D4×D139
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#497×C139C2
socleC22×C139C22×D4
prime radicalC2×D4×D139C22
hypercenter#497D139
hypocenterC139C2×#497
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D139C2semidirect
3. C4⋊Q8×D139C2semidirect
5. #22581348C2semidirect
6. #22581347C2semidirect
7. #22581346C2semidirect
8. #22581345C2semidirect
9. #22581344C2semidirect
10. #97×D139C2semidirect
11. #497×C139C2semidirect
15. C2×Dic556C22semidirect
16. C2×#14555775C22semidirect
17. C2×#14555775C22semidirect
24. #17984842C22semidirect
26. #17984840C22semidirect
34. #17984753C22semidirect
35. #17984752C22semidirect
36. C2×Q16×C139C22semidirect
38. C4⋊Q8×C139C22semidirect
40. #97×C139C22semidirect
130. D139#497direct (N × H)
131. C139C2×#497semidirect
142. #497D139direct (N × H)
151. C2×Q16C2×D139semidirect
161. C4⋊Q8C2×D139semidirect
172. #97C2×D139semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1392
C2×SD16×D1392
C4⋊Q8×D1392
#124×D1392
#225813482
#225813472
#225813462
#225813452
#225813442
#97×D1392
#497×C1392
C2×#497139