Finite Group Databasebeta

← #31779521

Subgroups — #31779521

Order 17536.

254 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 67 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#22577951C211yesmaximal, characteristic
2#22577425C212yesmaximal, complemented
3#22577151C212yesmaximal, complemented
4#22577147C211yesmaximal, characteristic, complemented
5#331×C137C211yesmaximal, characteristic, complemented
6C2×#14555509C412yes—
7C2×#14555493—22no—
8#17984088C2211yescharacteristic
9#17984075C412yes—
10#17984059—22no—
11D4×Dic137—22no—
12C42×D137C2211yescharacteristic
13C4⋊#11764040C2212yes—
14#17984017—22no—
15#127×C137C2211yescharacteristic
16C4⋊C8×C137C2212yes—
17#14555509—44no—
18C2×Dic274C2×C412yes—
19C2×C22⋊D137—22no—
20C22×Dic137—22no—
21C2×C4×D137C2×C411yescharacteristic
22C2×C4×D137D412yes—
23#14555493—44no—
24C2×#11764040—22no—
25C4⋊Dic137—22no—
26C22⋊Dic137—24no—
27C4×Dic137C2×C411yescharacteristic
28C4×Dic137D412yes—
29C2×D4×C137—22no—
30C2×D4×C137C2×C412yes—
31C2×C8×C137—22no—
32C42×C137C2311yescharacteristic
33C22×D137C22⋊C411yescharacteristic
34Dic274—42no—
35C22⋊D137—44no—
36C2×Dic137—22no—
37C2×Dic137—44no—
38C2×Dic137C22⋊C412yes—
39C2×Dic137C22⋊C411yescharacteristic
40C4×D137—22no—
41C4×D137—22no—
42C4×D137—22no—
43#11764040—42no—
44C23×C137—22no—
45D4×C137—42no—
46D4×C137—42no—
47D4×C137—42no—
48C2×C4×C137C22×C411yescharacteristic
49C2×C4×C137C2×D412yes—
50C8×C137—42no—
51C22×C137—42no—
52C22×C137—44no—
53C22×C137C2×C22⋊C411yescharacteristic
54C2×D137—21no—
55C2×D137—22no—
56C4×C137—21no—
57C4×C137AGL(1,ℤ/8)12yes—
58C4×C137#13712yes—
59Dic137—22no—
60Dic137—42no—
61Dic137#10012yes—
62C2×C137—82no—
63C2×C137C2×#10011yescharacteristic
64C2×C137#41712yes—
65D137—41no—
66C137#156011yescharacteristic, complemented, Sylow
67#1560—1371nomaximal, Sylow
68#532—1371no—
69#423—1372no—
70#333—1372no—
71#331—1371no—
72#331D13711yescharacteristic, complemented
73C2×#41—1372no—
74C2×M16—2742no—
75C4⋊Q8—1371no—
76#127C2×D13711yescharacteristic
77#124—1372no—
78#121—2742no—
79C4×D4—2742no—
80C2×C42—1371no—
81C4⋊C8—2742no—
82C4⋊C8—1372no—
83C4⋊C8C2×D13712yes—
84#41—5484no—
85C2×Q8—1372no—
86C2×D4—22no—
87C2×D4—2742no—
88C2×D4C4×D13712yes—
89C22×C4—2742no—
90C22×C4—1371no—
91C22×C4—1372no—
92M16—5484no—
93C2×C8—22no—
94C2×C8—2742no—
95C4⋊C4—2742no—
96C22⋊C4—2744no—
97C42—1371no—
98C42—1372no—
99C42C22×D13711yescharacteristic
100C23—22no—
101C23—1371no—
102Q8—5482no—
103D4—42no—
104D4—42no—
105D4—42no—
106D4—5484no—
107C2×C4—2742no—
108C2×C4—2742no—
109C2×C4—2742no—
110C2×C4—2742no—
111C2×C4—5484no—
112C2×C4—1372no—
113C2×C4—1371no—
114C2×C4C2×C4×D13711yescharacteristic
115C2×C4D4×D13712yes—
116C8—42no—
117C8—5482no—
118C22—42no—
119C22—44no—
120C22—2741no—
121C22—2742no—
122C22C22⋊C4×D13711yescharacteristic
123C4—21no—
124C4—2742no—
125C4—5482no—
126C4—1372no—
127C4#1798411212yes—
128C4#1798411712yes—
129C2—82no—
130C2—5481no—
131C2#100×D13711yescharacteristic
132C2#2257740112yes—
subgroupG/H
centerC22C22⋊C4×D137
commutatorC2×C4×C137C22×C4
FrattiniC42C22×D137
Fitting#331×C137C2
socleC22×C137C2×C22⋊C4
prime radical#22577951C2
cosocleC42×C137C23
hypercenter#331D137
hypocenterC137#1560
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #22577425C2semidirect
3. #22577151C2semidirect
4. #22577147C2semidirect
5. #331×C137C2semidirect
66. C137#1560semidirect
72. #331D137semidirect

Maximal subgroups

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

subgroupindex
#225779512
#225774252
#225771512
#225771472
#331×C1372
#1560137