Finite Group Databasebeta

← #31805716

Subgroups — #31805716

Order 17792.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#22581703C211yesmaximal, characteristic
2#22581362C212yesmaximal, complemented
3#22581348C211yesmaximal, characteristic
4#22580694C212yesmaximal, complemented
5#329×C139C211yesmaximal, characteristic
6C2×#14555775—21no—
7C2×#14555773—21no—
8#17984842C2211yescharacteristic
9#17984840—21no—
10#17984835—22no—
11#17984834C2212yes—
12#17984823—21no—
13#17984819—22no—
14#17984776—22no—
15#17984774—22no—
16#17984753C2211yescharacteristic
17#124×C139C2211yescharacteristic
18#97×C139C2212yes—
19C2×Dic278—21no—
20C2×Dic278—21no—
21C22×Dic139—21no—
22#14555775—42no—
23#14555773—42no—
24C2×C139⋊C8—21no—
25C4⋊Dic139—22no—
26C4⋊Dic139D412yes—
27C4⋊Dic139D412yes—
28Dic139⋊C4—22no—
29Dic139⋊C4—22no—
30Dic139⋊C4—22no—
31C22⋊Dic139—22no—
32C22⋊Dic139—21no—
33C4×Dic139—21no—
34C2×Q8×C139D411yescharacteristic
35C2×D4×C139D411yescharacteristic
36C2×C8×C139—22no—
37C22⋊C4×C139—22no—
38C42×C139C2311yescharacteristic
39Dic278—41no—
40Dic278—41no—
41Dic278—81no—
42C2×Dic139—22no—
43C2×Dic139—22no—
44C2×Dic139—41no—
45C2×Dic139—81no—
46C139⋊C8—22no—
47C23×C139—21no—
48Q8×C139—41no—
49D4×C139—41no—
50C2×C4×C139—21no—
51C2×C4×C139C2×D412yes—
52C2×C4×C139C2×D411yescharacteristic
53C8×C139—24no—
54C22×C139—41no—
55C22×C139—42no—
56C22×C139#12011yescharacteristic
57C4×C139—21no—
58C4×C139—22no—
59C4×C139—41no—
60Dic139—42no—
61Dic139—42no—
62Dic139—81no—
63C2×C139—81no—
64C2×C139#45311yescharacteristic
65C2×C139#45612yes—
66C139#166011yescharacteristic, complemented, Sylow
67#1660—1391nomaximal, Sylow
68#563—1391no—
69#501—1392no—
70#497—1391no—
71#330—1392no—
72#329D13911yescharacteristic
73C2×Q16—2781no—
74C2×SD16—2781no—
75C4⋊Q8—2781no—
76C4⋊Q8—1391no—
77#126—2782no—
78#125—1392no—
79#124C2×D13911yescharacteristic
80#123—2781no—
81C22⋊Q8—2782no—
82#107—2782no—
83#106—2782no—
84#97—1391no—
85#97C2×D13912yes—
86C2×Q8—2781no—
87C2×Q8—2781no—
88C2×Q8D55611yescharacteristic
89C2×D4D55611yescharacteristic
90C22×C4—2781no—
91Q16—5562no—
92SD16—5562no—
93C2×C8—22no—
94C2×C8—2781no—
95C4⋊C4—2782no—
96C4⋊C4—2782no—
97C4⋊C4—2782no—
98C4⋊C4—2782no—
99C4⋊C4—1392no—
100C4⋊C4—1392no—
101C22⋊C4—22no—
102C22⋊C4—2782no—
103C22⋊C4—2781no—
104C42—2781no—
105C42C22×D13911yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—5561no—
109Q8—5561no—
110Q8—11121no—
111D4—41no—
112C2×C4—21no—
113C2×C4—2782no—
114C2×C4—2782no—
115C2×C4—5561no—
116C2×C4—11121no—
117C2×C4D4×D13912yes—
118C2×C4C2×D55611yescharacteristic
119C8—24no—
120C8—2782no—
121C22—41no—
122C22—42no—
123C22#1798481211yescharacteristic
124C4—21no—
125C4—22no—
126C4—41no—
127C4—5562no—
128C4—5562no—
129C4—11121no—
130C2—81no—
131C2#2258112311yescharacteristic
132C2#2258113712yes—
subgroupG/H
centerC22#17984812
commutator, cosocleC42×C139C23
FrattiniC42C22×D139
Fitting#329×C139C2
socleC22×C139#120
prime radicalC2×D4×C139D4
hypercenter#329D139
hypocenterC139#1660
LCSG ▷ C42×C139 ▷ C22×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #22581362C2semidirect
4. #22580694C2semidirect
66. C139#1660semidirect

Maximal subgroups

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

subgroupindex
#225817032
#225813622
#225813482
#225806942
#329×C1392
#1660139