Finite Group Databasebeta

← #31780098

Subgroups — #31780098

Order 17536.

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#22578153C211yesmaximal, characteristic
2#22577809C212yesmaximal, complemented
3#22577144C212yesmaximal, complemented
4#22577143C211yesmaximal, characteristic
5#497×C137C211yesmaximal, characteristic
6#17984086—21no—
7#17984081—22no—
8#17984080C2212yes—
9#17984069—21no—
10#17984065—22no—
11#17984021—22no—
12#17984019—22no—
13#17983999C2212yes—
14C2×Q16×C137—21no—
15C2×SD16×C137—21no—
16C4⋊Q8×C137C2211yescharacteristic
17#124×C137C2211yescharacteristic
18#97×C137C2211yescharacteristic
19C2×Dic274—21no—
20C22×Dic137—21no—
21C2×#11764040—22no—
22C4⋊Dic137D412yes—
23C4⋊Dic137D412yes—
24#14555489—22no—
25#14555489—22no—
26#14555489—22no—
27C22⋊Dic137—22no—
28C22⋊Dic137—21no—
29C4×Dic137—21no—
30C2×Q8×C137—21no—
31C2×Q8×C137D411yescharacteristic
32C2×D4×C137D411yescharacteristic
33Q16×C137—42no—
34SD16×C137—42no—
35C2×C8×C137—21no—
36C4⋊C4×C137—22no—
37C22⋊C4×C137—22no—
38C42×C137C2311yescharacteristic
39Dic274—81no—
40C2×Dic137—22no—
41C2×Dic137—22no—
42C2×Dic137—81no—
43#11764040—24no—
44C23×C137—21no—
45Q8×C137—41no—
46Q8×C137—41no—
47Q8×C137—41no—
48D4×C137—41no—
49C2×C4×C137—21no—
50C2×C4×C137—41no—
51C2×C4×C137C2×D412yes—
52C2×C4×C137C2×D411yescharacteristic
53C8×C137—22no—
54C22×C137—41no—
55C22×C137—42no—
56C22×C137#12011yescharacteristic
57C4×C137—21no—
58C4×C137—22no—
59C4×C137—41no—
60C4×C137—81no—
61Dic137—42no—
62Dic137—42no—
63C2×C137—81no—
64C2×C137#45311yescharacteristic
65C2×C137#45612yes—
66C137#166011yescharacteristic, complemented, Sylow
67#1660—1371nomaximal, Sylow
68#563—1371no—
69#501—1372no—
70#497D13711yescharacteristic
71#330—1372no—
72#329—1371no—
73C2×Q16—21no—
74C2×SD16—21no—
75C4⋊Q8—2741no—
76C4⋊Q8C2×D13711yescharacteristic
77#126—2742no—
78#125—1372no—
79#124C2×D13711yescharacteristic
80#123—2741no—
81C22⋊Q8—2742no—
82#107—2742no—
83#106—2742no—
84#97—1372no—
85#97C2×D13711yescharacteristic
86C2×Q8—21no—
87C2×Q8—2741no—
88C2×Q8C22⋊D13711yescharacteristic
89C2×D4C22⋊D13711yescharacteristic
90C22×C4—2741no—
91Q16—42no—
92SD16—42no—
93C2×C8—21no—
94C2×C8—2742no—
95C4⋊C4—22no—
96C4⋊C4—2742no—
97C4⋊C4—2742no—
98C4⋊C4—2742no—
99C4⋊C4—1372no—
100C4⋊C4—1372no—
101C22⋊C4—22no—
102C22⋊C4—2742no—
103C22⋊C4—2741no—
104C42—2741no—
105C42C22×D13711yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—41no—
109Q8—41no—
110Q8—10961no—
111D4—41no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—2742no—
115C2×C4—2742no—
116C2×C4—10961no—
117C2×C4D4×D13712yes—
118C2×C4C2×C22⋊D13711yescharacteristic
119C8—22no—
120C8—2744no—
121C22—41no—
122C22—42no—
123C22#1798405611yescharacteristic
124C4—21no—
125C4—22no—
126C4—41no—
127C4—81no—
128C4—5482no—
129C4—5482no—
130C2—81no—
131C2#2257757011yescharacteristic
132C2#2257758412yes—
subgroupG/H
centerC22#17984056
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#497×C137C2
socleC22×C137#120
prime radicalC2×D4×C137D4
hypercenter#497D137
hypocenterC137#1660
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #22577809C2semidirect
3. #22577144C2semidirect
66. C137#1660semidirect

Maximal subgroups

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

subgroupindex
#225781532
#225778092
#225771442
#225771432
#497×C1372
#1660137