Finite Group Databasebeta

← #31780421

Subgroups — #31780421

Order 17536.

176 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 61 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#22577799C211yesmaximal, characteristic
2#22577799C211yesmaximal, characteristic
3#22577782C211yesmaximal, characteristic
4#22577754C211yesmaximal, characteristic
5#22577753C211yesmaximal, characteristic
6#22577125C211yesmaximal, characteristic
7#498×C137C211yesmaximal, characteristic
8C2×#14555500—22no—
9#17984088C2211yescharacteristic
10#17984088C2211yescharacteristic
11#17984019—22no—
12#17984019—22no—
13#17984013—22no—
14#17984012—22no—
15C4×#11764040C2211yescharacteristic
16C4×#11764040C2211yescharacteristic
17C4⋊Q8×C137C2211yescharacteristic
18C4⋊Q8×C137C2211yescharacteristic
19#106×C137—22no—
20C4×C8×C137C2211yescharacteristic
21C2×Dic274—21no—
22C2×Dic274—21no—
23#14555500—44no—
24C2×#11764040D412yes—
25C2×#11764040D412yes—
26C4⋊Dic137—21no—
27C4⋊Dic137—21no—
28C4⋊Dic137—21no—
29C4⋊Dic137—21no—
30C2×Q8×C137—21no—
31C2×Q8×C137—21no—
32C2×C8×C137D412yes—
33C4⋊C4×C137—21no—
34C4⋊C4×C137—21no—
35C4⋊C4×C137—21no—
36C4⋊C4×C137—21no—
37C42×C137C2311yescharacteristic
38Dic274—42no—
39Dic274—42no—
40C2×Dic137—41no—
41C2×Dic137—41no—
42#11764040—22no—
43#11764040SD1614yes—
44Q8×C137—42no—
45Q8×C137—42no—
46C2×C4×C137—41no—
47C2×C4×C137—41no—
48C2×C4×C137C2×D411yescharacteristic
49C2×C4×C137C2×D411yescharacteristic
50C2×C4×C137C2×D411yescharacteristic
51C8×C137—22no—
52C22×C137#12711yescharacteristic
53C4×C137—81no—
54C4×C137—81no—
55C4×C137C2×SD1612yes—
56C4×C137#13712yes—
57C4×C137#13712yes—
58Dic137—81no—
59Dic137—81no—
60C2×C137#49211yescharacteristic
61C2×C137#49711yescharacteristic
62C2×C137#49711yescharacteristic
63C137#171611yescharacteristic, complemented, Sylow
64#1716—1371nomaximal, Sylow
65#498—1371no—
66#498—1371no—
67#498D13711yescharacteristic
68#494—1371no—
69#487—1371no—
70#487—1371no—
71#322—1371no—
72C2×Q16—2742no—
73C4⋊Q8—1371no—
74C4⋊Q8—1371no—
75C4⋊Q8C2×D13711yescharacteristic
76C4⋊Q8C2×D13711yescharacteristic
77#106—22no—
78#106—2742no—
79#106—2742no—
80#103—2742no—
81#103—2742no—
82C4×C8—1371no—
83C4×C8—1371no—
84C4×C8C2×D13711yescharacteristic
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—2741no—
88C2×Q8—2741no—
89Q16—5484no—
90C2×C8—1372no—
91C2×C8—1372no—
92C2×C8C22⋊D13712yes—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—2741no—
98C4⋊C4—2741no—
99C4⋊C4—2741no—
100C4⋊C4—2741no—
101C42C22×D13711yescharacteristic
102Q8—42no—
103Q8—42no—
104Q8—5482no—
105Q8—5482no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—5481no—
109C2×C4—5481no—
110C2×C4D4×D13711yescharacteristic
111C2×C4D4×D13711yescharacteristic
112C2×C4C2×C22⋊D13711yescharacteristic
113C8—22no—
114C8—2742no—
115C8—1374no—
116C22#1798408411yescharacteristic
117C4—81no—
118C4—81no—
119C4—10961no—
120C4—10961no—
121C4SD16×D13712yes—
122C4#1798411712yes—
123C4#1798411912yes—
124C2#2257777411yescharacteristic
125C2#2257779411yescharacteristic
126C2#2257779511yescharacteristic
subgroupG/H
centerC22#17984084
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#498×C137C2
socle, prime radicalC22×C137#127
hypercenter#498D137
hypocenterC137#1716
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
63. C137#1716semidirect

Maximal subgroups

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

subgroupindex
#225777992
#225777822
#225777542
#225777532
#225771252
#498×C1372
#1716137