Finite Group Databasebeta

← #31805715

Subgroups — #31805715

Order 17792.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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#22581363C212yesmaximal, complemented
2#22581348C211yesmaximal, characteristic
3#22580695C212yesmaximal, complemented
4#22580693C211yesmaximal, characteristic
5#563×C139C211yesmaximal, characteristic
6C2×#14555775—21no—
7C2×#14555773—21no—
8#17984842C2211yescharacteristic
9#17984775—22no—
10#17984773—22no—
11#17984753C2211yescharacteristic
12#17984753C2212yes—
13C4⋊Q8×C139—21no—
14#126×C139—22no—
15#125×C139C2212yes—
16#124×C139C2211yescharacteristic
17#123×C139—21no—
18C22⋊Q8×C139—22no—
19C2×Dic278—21no—
20#14555775—42no—
21#14555773—42no—
22C2×C139⋊C8—22no—
23C2×C139⋊C8—21no—
24C4⋊Dic139—22no—
25C2×Q8×C139—21no—
26C2×Q8×C139D411yescharacteristic
27C2×D4×C139D411yescharacteristic
28C22×C4×C139—21no—
29C4⋊C4×C139—22no—
30C4⋊C4×C139—22no—
31C4⋊C4×C139—22no—
32C4⋊C4×C139D412yes—
33C4⋊C4×C139D412yes—
34C22⋊C4×C139—22no—
35C22⋊C4×C139—21no—
36C22⋊C4×C139—22no—
37C42×C139—21no—
38C42×C139C2311yescharacteristic
39Dic278—41no—
40Dic278—41no—
41C2×Dic139—41no—
42C139⋊C8—24no—
43C139⋊C8—22no—
44C23×C139—21no—
45Q8×C139—41no—
46Q8×C139—81no—
47D4×C139—41no—
48C2×C4×C139—22no—
49C2×C4×C139—21no—
50C2×C4×C139—22no—
51C2×C4×C139—81no—
52C2×C4×C139C2×D412yes—
53C2×C4×C139C2×D411yescharacteristic
54C22×C139—41no—
55C22×C139—42no—
56C22×C139#12011yescharacteristic
57C4×C139—21no—
58C4×C139—22no—
59C4×C139—42no—
60C4×C139—41no—
61C4×C139—42no—
62Dic139—81no—
63C2×C139—81no—
64C2×C139#45311yescharacteristic
65C2×C139#45612yes—
66C139#166011yescharacteristic, complemented, Sylow
67#1660—1391nomaximal, Sylow
68#563D13911yescharacteristic
69#501—1392no—
70#497—1391no—
71#330—1392no—
72#329—1391no—
73C2×Q16—2781no—
74C2×SD16—2781no—
75C4⋊Q8—21no—
76C4⋊Q8—1391no—
77#126—22no—
78#125C2×D13912yes—
79#124C2×D13911yescharacteristic
80#123—21no—
81C22⋊Q8—22no—
82#107—2782no—
83#106—2782no—
84#97—1392no—
85#97—1391no—
86C2×Q8—21no—
87C2×Q8—2781no—
88C2×Q8C22⋊D13911yescharacteristic
89C2×D4C22⋊D13911yescharacteristic
90C22×C4—21no—
91Q16—5562no—
92SD16—5562no—
93C2×C8—2782no—
94C2×C8—2781no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—2782no—
99C4⋊C4C22⋊D13912yes—
100C4⋊C4C22⋊D13912yes—
101C22⋊C4—22no—
102C22⋊C4—21no—
103C22⋊C4—22no—
104C42—21no—
105C42C22×D13911yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—81no—
109Q8—5561no—
110Q8—5561no—
111D4—41no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—81no—
116C2×C4—5561no—
117C2×C4C2×C22⋊D13912yes—
118C2×C4C2×C22⋊D13911yescharacteristic
119C8—2782no—
120C8—2784no—
121C22—41no—
122C22—42no—
123C22#1798481111yescharacteristic
124C4—21no—
125C4—22no—
126C4—42no—
127C4—41no—
128C4—42no—
129C4—11121no—
130C2—81no—
131C2#2258112111yescharacteristic
132C2#2258113612yes—
subgroupG/H
centerC22#17984811
commutator, cosocleC42×C139C23
FrattiniC42C22×D139
Fitting#563×C139C2
socleC22×C139#120
prime radicalC2×D4×C139D4
hypercenter#563D139
hypocenterC139#1660
LCSG ▷ C42×C139 ▷ C22×C139 ▷ C139
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #22581363C2semidirect
3. #22580695C2semidirect
66. C139#1660semidirect

Maximal subgroups

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

subgroupindex
#225813632
#225813482
#225806952
#225806932
#563×C1392
#1660139