Finite Group Databasebeta

← #31804556

Subgroups — #31804556

Order 17792.

112 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 23 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#22581125C211yesmaximal, characteristic
2#22580748C211yesmaximal, characteristic, complemented
3#329×C139C211yesmaximal, characteristic, complemented
4#17984890C411yescharacteristic, complemented
5#17984868—21no—
6#17984770—21no—
7#17984761C411yescharacteristic, complemented
8#17984757—21no—
9#124×C139C2211yescharacteristic
10#97×C139—21no—
11#14555786—41no—
12#14555785—21no—
13#14555785—41no—
14Q8×D139—41no—
15#14555773—41no—
16#14555770—41no—
17#14555769—21no—
18C22⋊Dic139—41no—
19C2×Q8×C139C2×C411yescharacteristic
20C2×D4×C139D411yescharacteristic
21C2×C8×C139—41no—
22C22⋊C4×C139—41no—
23C42×C139D411yescharacteristic
24Dic278—21no—
25Dic278—41no—
26D556—21no—
27D556—41no—
28C22⋊D139—41no—
29C2×Dic139—81no—
30C4×D139—41no—
31C4×D139—42no—
32C139⋊C8—41no—
33C23×C139—21no—
34Q8×C139—41no—
35D4×C139—41no—
36C2×C4×C139—21no—
37C2×C4×C139—21no—
38C2×C4×C139C22⋊C411yescharacteristic
39C8×C139—42no—
40C22×C139—41no—
41C22×C139—81no—
42C22×C139#9911yescharacteristic
43C2×D139—41no—
44C4×C139—21no—
45C4×C139—41no—
46C4×C139—41no—
47Dic139—41no—
48Dic139—161no—
49C2×C139—21no—
50C2×C139—81no—
51C2×C139AΣL(1,𝔽16)11yescharacteristic
52D139—81no—
53C139#140411yescharacteristic, complemented, Sylow
54#1404—1391nomaximal, Sylow
55#454—1391no—
56#354—1391no—
57#329D13911yescharacteristic, complemented
582−1+4—1391no—
59AGL(1,ℤ/8)—2781no—
60#124C2×D13911yescharacteristic
61#104—2781no—
62#101—1391no—
63#99—2781no—
64#97—21no—
65#41—2781no—
66#41—5561no—
67#41—5561no—
68C2×Q8—5561no—
69C2×Q8C4×D13911yescharacteristic
70C2×D4D55611yescharacteristic
71SD16—5561no—
72D8—5561no—
73M16—2781no—
74C2×C8—41no—
75C22⋊C4—41no—
76C22⋊C4—5561no—
77C42C22⋊D13911yescharacteristic
78C23—21no—
79Q8—41no—
80Q8—2781no—
81Q8—5561no—
82D4—41no—
83D4—2781no—
84D4—5561no—
85D4—5561no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—5561no—
89C2×C4—5562no—
90C2×C4—11121no—
91C2×C4(C2×D139)⋊C411yescharacteristic
92C8—42no—
93C8—5561no—
94C22—41no—
95C22—81no—
96C22—5561no—
97C22#1798475611yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—5561no—
102C4—22241no—
103C2—21no—
104C2—81no—
105C2—11121no—
106C2#2258074111yescharacteristic
subgroupG/H
centerC2#22580741
commutatorC2×Q8×C139C2×C4
Frattini#124C2×D139
Fitting#329×C139C2
socleC2×C139AΣL(1,𝔽16)
prime radical#22581125C2
cosocle#124×C139C22
hypercenter#329D139
hypocenterC139#1404
LCSG ▷ C2×Q8×C139 ▷ C2×C4×C139 ▷ C22×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #22580748C2semidirect
3. #329×C139C2semidirect
4. #17984890C4semidirect
7. #17984761C4semidirect
53. C139#1404semidirect
57. #329D139semidirect

Maximal subgroups

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

subgroupindex
#225811252
#225807482
#329×C1392
#1404139