Finite Group Databasebeta

← #31782862

Subgroups — #31782862

Order 17536.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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#22577839C212yesmaximal, complemented
2#22577521C211yesmaximal, characteristic, complemented
3#22577224C211yesmaximal, characteristic, complemented
4#22577216C212yesmaximal, complemented
5#493×C137C211yesmaximal, characteristic, complemented
6#17984110C2211yescharacteristic
7#17984094—21no—
8#17984032—22no—
9#17984029—22no—
10#17984028C2212yes—
11#17984024C2211yescharacteristic
12#17984016—21no—
13C2×D8×C137—21no—
14C2×D8×C137C2212yes—
15#127×C137—21no—
16C4×C8×C137C2211yescharacteristic
17#14555510—21no—
18Dic548D411yescharacteristic, complemented
19#14555499—21no—
20D1096D411yescharacteristic, complemented
21#14555494—21no—
22#14555493—41no—
23C8×D137—41no—
24#14555485—22no—
25C2×D4×C137—22no—
26C2×D4×C137—42no—
27D8×C137—22no—
28D8×C137—22no—
29D8×C137—41no—
30D8×C137—41no—
31C2×C8×C137D411yescharacteristic
32C2×C8×C137C2311yescharacteristic
33C42×C137D411yescharacteristic
34Dic274—21no—
35D548—21no—
36C22⋊D137—41no—
37C4×D137—41no—
38#11764040—41no—
39C23×C137—42no—
40D4×C137—42no—
41D4×C137—42no—
42D4×C137—84no—
43C2×C4×C137—21no—
44C2×C4×C137C2×D411yescharacteristic
45C8×C137D811yescharacteristic
46C8×C137D811yescharacteristic
47C8×C137C2×D411yescharacteristic
48C8×C137#4111yescharacteristic
49C22×C137—84no—
50C22×C137—82no—
51C22×C137AGL(1,ℤ/8)11yescharacteristic
52C2×D137—41no—
53C4×C137—22no—
54C4×C137#12111yescharacteristic
55C4×C137C2×D811yescharacteristic
56Dic137—41no—
57C2×C137—21no—
58C2×C137—162no—
59C2×C137#45911yescharacteristic
60D137—81no—
61C137#221311yescharacteristic, complemented, Sylow
62#2213—1371nomaximal, Sylow
63#509—1372no—
64#493D13711yescharacteristic, complemented
65#443—1371no—
66#364—1371no—
67#361—1372no—
68#135—1371no—
69C2×D8—21no—
70C2×D8C2×D13712yes—
71#131—2741no—
72#127—21no—
73SD32—2742no—
74D16—2742no—
75M32—1372no—
76#108—1371no—
77#104—2741no—
78C4×C8C2×D13711yescharacteristic
79#41—2741no—
80C2×D4—22no—
81C2×D4—42no—
82Q16—1371no—
83SD16—2741no—
84D8—22no—
85D8—22no—
86D8—41no—
87D8—41no—
88D8—1371no—
89M16—2741no—
90M16—5481no—
91C2×C8—5481no—
92C2×C8C22⋊D13711yescharacteristic
93C2×C8C22×D13711yescharacteristic
94C42C22⋊D13711yescharacteristic
95C16—2742no—
96C23—42no—
97Q8—2741no—
98D4—42no—
99D4—42no—
100D4—84no—
101D4—2741no—
102D4—5481no—
103C2×C4—21no—
104C2×C4—5481no—
105C2×C4C2×C22⋊D13711yescharacteristic
106C8—5481no—
107C8#1455549511yescharacteristic
108C8#1455549511yescharacteristic
109C8D4×D13711yescharacteristic
110C8#1455550911yescharacteristic
111C22—84no—
112C22—82no—
113C22—5481no—
114C22#1798411411yescharacteristic
115C4—22no—
116C4—5481no—
117C4#1798405911yescharacteristic
118C4C2×#1455549511yescharacteristic
119C2—21no—
120C2—162no—
121C2—10961no—
122C2#2257759511yescharacteristic
subgroupG/H
centerC2#22577595
commutator, cosocleC2×C8×C137C23
FrattiniC2×C8C22×D137
Fitting#493×C137C2
socleC2×C137#459
hypercenter#493D137
hypocenterC137#2213
LCSG ▷ C2×C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #22577839C2semidirect
2. #22577521C2semidirect
3. #22577224C2semidirect
4. #22577216C2semidirect
5. #493×C137C2semidirect
18. Dic548D4semidirect
20. D1096D4semidirect
61. C137#2213semidirect
64. #493D137semidirect

Maximal subgroups

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

subgroupindex
#225778392
#225775212
#225772242
#225772162
#493×C1372
#2213137