Finite Group Databasebeta

← #31818198

Subgroups — #31818198

Order 17792.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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#22581765C212yesmaximal, complemented
2#22581760C212yesmaximal, complemented
3#22581326C211yesmaximal, characteristic, complemented
4#22581217C212yesmaximal, complemented
5#22581216C212yesmaximal, complemented
6#22581137C212yesmaximal, complemented
7#22581122C212yesmaximal, complemented
8#22581071C211yesmaximal, characteristic, complemented
9#443×C139C211yesmaximal, characteristic, complemented
10#17984889C2212yes—
11#17984888C2212yescomplemented
12#17984875—22no—
13#17984875C2212yescomplemented
14#17984874—22no—
15#17984872C2212yescomplemented
16#17984870—22no—
17#17984870C2212yes—
18#17984867C2212yes—
19#17984866—22no—
20#17984864—22no—
21#17984864C2211yescharacteristic
22#17984863—22no—
23C2×#14555774—22no—
24C2×#14555774C2212yescomplemented
25C2×#14555774C2212yescomplemented
26SD16×D139—22no—
27#17984848C2212yescomplemented
28#17984842C2211yescharacteristic, complemented
29#17984839C2211yescharacteristic
30#17984778C2211yescharacteristic, complemented
31#17984770C2212yescomplemented
32#17984760C2212yescomplemented
33#17984758C2212yescomplemented
34#135×C139C2211yescharacteristic
35#131×C139C2212yescomplemented
36#108×C139C2211yescharacteristic, complemented
37#104×C139C2212yescomplemented
38C4×C8×C139C2211yescharacteristic, complemented
39#14555786—22no—
40#14555785—22no—
41#14555785—22no—
42#14555785C2312yes—
43#14555784—22no—
44#14555784—42no—
45C2×Dic278—22no—
46C2×Dic278C2312yes—
47Q8×D139—22no—
48C2×D556—22no—
49C2×D556—41no—
50C2×D556C2312yes—
51D4×D139—22no—
52D4×D139—42no—
53Dic556—22no—
54Dic556D411yescharacteristic, complemented
55#14555775—22no—
56#14555774—22no—
57#14555774—22no—
58#14555774—22no—
59#14555774—22no—
60#14555774—22no—
61#14555774—22no—
62#14555774D412yescomplemented
63#14555773—22no—
64#14555772—22no—
65D1112—22no—
66D1112D411yescharacteristic, complemented
67#14555770—22no—
68#14555769C2312yes—
69#14555768—22no—
70C8×D139—22no—
71C4⋊Dic139—41no—
72#41×C139C2312yes—
73Q16×C139D411yescharacteristic, complemented
74SD16×C139D412yescomplemented
75D8×C139D411yescharacteristic, complemented
76M16×C139—22no—
77M16×C139C2312yes—
78C2×C8×C139—22no—
79C2×C8×C139C2311yescharacteristic
80C2×C8×C139C2311yescharacteristic
81C42×C139C2311yescharacteristic
82C22×D139—22no—
83C22×D139—42no—
84Dic278—22no—
85Dic278—22no—
86Dic278—42no—
87Dic278C2×D412yes—
88D556—22no—
89D556—22no—
90D556—42no—
91D556—82no—
92D556C2×D412yes—
93C22⋊D139—22no—
94C22⋊D139—42no—
95C22⋊D139—42no—
96C2×Dic139—22no—
97C2×Dic139—42no—
98C4×D139—22no—
99C4×D139—42no—
100C4×D139—42no—
101C139⋊C8—22no—
102Q8×C139C2×D412yes—
103D4×C139—22no—
104D4×C139C2×D412yes—
105C2×C4×C139—21no—
106C2×C4×C139—22no—
107C2×C4×C139C2411yescharacteristic
108C8×C139—22no—
109C8×C139C2×D411yescharacteristic
110C8×C139C2×D411yescharacteristic
111C8×C139C2×D411yescharacteristic
112C8×C139C2×D411yescharacteristic
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×C139C22×D411yescharacteristic
122C4×C139C22×D411yescharacteristic
123Dic139—22no—
124Dic139—42no—
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—1392no—
138#453—1392no—
139#443—1391no—
140#443D13911yescharacteristic, complemented
1412−1+4—1392no—
142H(5,𝔽2)—1392no—
143#137—2782no—
144#137—2782no—
145#137—1392no—
146#137—1392no—
147AGL(1,ℤ/8)—2782no—
148AGL(1,ℤ/8)—2782no—
149AGL(1,ℤ/8)—1392no—
150AGL(1,ℤ/8)—1392no—
151#135—2782no—
152#135—2782no—
153#135—1391no—
154#135C2×D13911yescharacteristic
155C2×SD16—2782no—
156C2×SD16—2782no—
157C2×SD16—1392no—
158C2×SD16—1392no—
159#131—1392no—
160#131C2×D13912yescomplemented
161C4⋊Q8—1391no—
162#127—1391no—
163#108—1391no—
164#108C2×D13911yescharacteristic, complemented
165#104—1392no—
166#104C2×D13912yescomplemented
167#101—1392no—
168#100—1392no—
169C4×C8C2×D13911yescharacteristic, complemented
170#41—2782no—
171#41—2782no—
172#41—2782no—
173#41—2782no—
174#41—5562no—
175#41—1392no—
176#41C22×D13912yes—
177C2×Q8—2782no—
178C2×Q8—2782no—
179C2×Q8—1392no—
180C2×D4—2782no—
181C2×D4—2782no—
182C2×D4—5561no—
183C2×D4—5562no—
184C2×D4—1392no—
185Q16—2782no—
186Q16—2782no—
187Q16—1391no—
188Q16D55611yescharacteristic, complemented
189SD16—2782no—
190SD16—2782no—
191SD16—2782no—
192SD16—2782no—
193SD16—2782no—
194SD16—2782no—
195SD16—2782no—
196SD16—2782no—
197SD16—1392no—
198SD16D55612yescomplemented
199D8—2782no—
200D8—2782no—
201D8—1391no—
202D8D55611yescharacteristic, complemented
203M16—22no—
204M16—2782no—
205M16—1392no—
206M16C22×D13912yes—
207C2×C8—22no—
208C2×C8—2782no—
209C2×C8C22×D13911yescharacteristic
210C2×C8C22×D13911yescharacteristic
211C4⋊C4—5561no—
212C42C22×D13911yescharacteristic
213C23—2782no—
214C23—5562no—
215Q8—2782no—
216Q8—2782no—
217Q8—5562no—
218Q8—1392no—
219Q8C2×D55612yes—
220D4—22no—
221D4—2782no—
222D4—2782no—
223D4—2782no—
224D4—5562no—
225D4—5562no—
226D4—5562no—
227D4—11122no—
228D4—1392no—
229D4C2×D55612yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—2782no—
233C2×C4—2782no—
234C2×C4—5562no—
235C2×C4—5562no—
236C2×C4—5562no—
237C2×C4C23×D13911yescharacteristic
238C8—22no—
239C8—2782no—
240C8D4×D13911yescharacteristic
241C8D4×D13911yescharacteristic
242C8C2×D55611yescharacteristic
243C8C2×D55611yescharacteristic
244C22—22no—
245C22—2782no—
246C22—5562no—
247C22—11122no—
248C22—22242no—
249C22#1798488811yescharacteristic
250C4—22no—
251C4—22no—
252C4—2782no—
253C4—5562no—
254C4C2×D4×D13911yescharacteristic
255C4C22×D55611yescharacteristic
256C2—21no—
257C2—42no—
258C2—5562no—
259C2—11122no—
260C2D4×D55611yescharacteristic
subgroupG/H
centerC2D4×D556
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
Fitting#443×C139C2
socleC2×C139D42
hypercenter#443D139
hypocenterC139#3291
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #22581765C2semidirect
2. #22581760C2semidirect
3. #22581326C2semidirect
4. #22581217C2semidirect
5. #22581216C2semidirect
6. #22581137C2semidirect
7. #22581122C2semidirect
8. #22581071C2semidirect
9. #443×C139C2semidirect
11. #17984888C22semidirect
13. #17984875C22semidirect
15. #17984872C22semidirect
24. C2×#14555774C22semidirect
25. C2×#14555774C22semidirect
27. #17984848C22semidirect
28. #17984842C22semidirect
30. #17984778C22semidirect
31. #17984770C22semidirect
32. #17984760C22semidirect
33. #17984758C22semidirect
35. #131×C139C22semidirect
36. #108×C139C22semidirect
37. #104×C139C22semidirect
38. C4×C8×C139C22semidirect
54. Dic556D4semidirect
62. #14555774D4semidirect
66. D1112D4semidirect
73. Q16×C139D4semidirect
74. SD16×C139D4semidirect
75. D8×C139D4semidirect
130. C139#3291semidirect
140. #443D139semidirect
160. #131C2×D139semidirect
164. #108C2×D139semidirect
166. #104C2×D139semidirect
169. C4×C8C2×D139semidirect
188. Q16D556semidirect
198. SD16D556semidirect
202. D8D556semidirect

Maximal subgroups

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

subgroupindex
#225817652
#225817602
#225813262
#225812172
#225812162
#225811372
#225811222
#225810712
#443×C1392
#3291139