Finite Group Databasebeta

← #31805521

Subgroups — #31805521

Order 17792.

314 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 57 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#22581588C211yesmaximal, characteristic
2#22581325C211yesmaximal, characteristic, complemented
3#22581244C212yesmaximal, complemented
4#22580680C212yesmaximal, complemented
5#331×C139C211yesmaximal, characteristic, complemented
6C2×D4×D139—22no—
7C2×#14555773—22no—
8#17984842C2211yescharacteristic, complemented
9#17984813—22no—
10#17984810—22no—
11C4×D556C2212yes—
12#17984774—22no—
13#17984764—22no—
14C4×C139⋊C8C2211yescharacteristic, complemented
15#127×C139C2211yescharacteristic
16C4⋊C8×C139C2212yescomplemented
17C23×D139—22no—
18C2×Dic278—21no—
19C2×D556D412yes—
20C2×C22⋊D139—24no—
21D4×D139—44no—
22D4×D139—44no—
23C2×C4×D139—22no—
24#14555773—44no—
25C2×C139⋊C8—21no—
26C4⋊Dic139—22no—
27C4⋊Dic139D412yes—
28C22⋊Dic139—22no—
29(C2×D139)⋊C4—22no—
30C2×D4×C139—22no—
31C2×D4×C139D412yes—
32C2×C8×C139—22no—
33C42×C139C2311yescharacteristic
34C22×D139—22no—
35C22×D139—48no—
36C22×D139—82no—
37Dic278—42no—
38D556—22no—
39D556SD1614yescomplemented
40C22⋊D139—48no—
41C2×Dic139—22no—
42C2×Dic139—41no—
43C4×D139—42no—
44C4×D139—42no—
45C139⋊C8—22no—
46C23×C139—22no—
47D4×C139—42no—
48D4×C139—42no—
49D4×C139—42no—
50C2×C4×C139C2×D412yes—
51C2×C4×C139C2×D411yescharacteristic
52C8×C139—42no—
53C22×C139—44no—
54C22×C139—42no—
55C22×C139#12011yescharacteristic
56C2×D139—24no—
57C2×D139—42no—
58C2×D139—42no—
59C2×D139—88no—
60C4×C139—21no—
61C4×C139C2×SD1612yes—
62C4×C139AGL(1,ℤ/8)12yes—
63Dic139—42no—
64Dic139—81no—
65C2×C139—82no—
66C2×C139#45012yes—
67C2×C139#45311yescharacteristic
68D139—44no—
69C139#162911yescharacteristic, complemented, Sylow
70#1629—1391nomaximal, Sylow
71D42—1391no—
72#492—1391no—
73#476—1392no—
74#331D13911yescharacteristic, complemented
75D4⋊C8—1392no—
76C22×D4—2782no—
77C2×SD16—2782no—
78C4⋊Q8—1391no—
79#127C2×D13911yescharacteristic
80#121—2782no—
81#120—2782no—
82C4×D4—1392no—
83#106—2782no—
84C4⋊C8C2×D13912yescomplemented
85#102—2782no—
86C4×C8—1391no—
87C24—2782no—
88C2×Q8—2781no—
89C2×D4—22no—
90C2×D4—2784no—
91C2×D4—5564no—
92C2×D4—5564no—
93C2×D4—1392no—
94C2×D4D55612yes—
95C22×C4—2782no—
96SD16—5564no—
97C2×C8—22no—
98C2×C8—2781no—
99C4⋊C4—2782no—
100C4⋊C4—1392no—
101C22⋊C4—2782no—
102C22⋊C4—2782no—
103C42C22×D13911yescharacteristic
104C23—22no—
105C23—2782no—
106C23—5568no—
107C23—11122no—
108Q8—5562no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—2782no—
113D4—5568no—
114D4—1394no—
115C2×C4—2782no—
116C2×C4—5562no—
117C2×C4—5562no—
118C2×C4—5561no—
119C2×C4D4×D13912yes—
120C2×C4C2×D55611yescharacteristic
121C8—42no—
122C8—2782no—
123C22—44no—
124C22—42no—
125C22—2784no—
126C22—5562no—
127C22—5562no—
128C22—11128no—
129C22#1798481211yescharacteristic
130C4—21no—
131C4—5562no—
132C4—11121no—
133C4SD16×D13912yes—
134C4#1798486612yes—
135C2—82no—
136C2—5564no—
137C2#2258110512yes—
138C2#2258112311yescharacteristic
subgroupG/H
centerC22#17984812
commutator, cosocleC42×C139C23
FrattiniC42C22×D139
Fitting#331×C139C2
socleC22×C139#120
prime radical#22581588C2
hypercenter#331D139
hypocenterC139#1629
LCSG ▷ C42×C139 ▷ C22×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #22581325C2semidirect
3. #22581244C2semidirect
4. #22580680C2semidirect
5. #331×C139C2semidirect
8. #17984842C22semidirect
14. C4×C139⋊C8C22semidirect
16. C4⋊C8×C139C22semidirect
39. D556SD16semidirect
69. C139#1629semidirect
74. #331D139semidirect
84. C4⋊C8C2×D139semidirect

Maximal subgroups

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

subgroupindex
#225815882
#225813252
#225812442
#225806802
#331×C1392
#1629139