Finite Group Databasebeta

← #31795888

Subgroups — #31795888

Order 17536.

162 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 38 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#22578390C211yesmaximal, characteristic
2#22578377C211yesmaximal, characteristic
3#22577375C211yesmaximal, characteristic
4C2×#14555520C411yescharacteristic, complemented
5C2×#14555519—21no—
6C2×#14555518C411yescharacteristic, complemented
7#17984149C412yescomplemented
8#17984148—21no—
9C22⋊#11764047—21no—
10C22⋊#11764047—21no—
11C2×#14555509C2211yescharacteristic
12#17984044—21no—
13#17984003C411yescharacteristic
14#17984002—21no—
15#99×C137C411yescharacteristic
16#14555520—41no—
17#14555519—22no—
18#14555519—41no—
19#14555518—41no—
20C2×#11764047—21no—
21C2×#11764047—21no—
22C2×#11764047—41no—
23#14555509—41no—
24#14555509—41no—
25C2×Dic274C2×C411yescharacteristic
26C2×C22⋊D137D411yescharacteristic
27C2×C22⋊D137Q811yescharacteristic
28C22×Dic137D411yescharacteristic
29C22×Dic137D411yescharacteristic
30C2×C4×D137C2×C411yescharacteristic
31#14555489—41no—
32C22⋊Dic137—21no—
33C4×Dic137—41no—
34C2×D4×C137C2×C411yescharacteristic
35C22⋊C4×C137—21no—
36#11764047—42no—
37#11764047—41no—
38#11764047—41no—
39C22×D137C4⋊C411yescharacteristic
40Dic274—41no—
41C22⋊D137—41no—
42C22⋊D137—41no—
43C2×Dic137—21no—
44C2×Dic137—21no—
45C2×Dic137—21no—
46C2×Dic137—21no—
47C2×Dic137—41no—
48C2×Dic137C22⋊C411yescharacteristic
49C2×Dic137C22⋊C411yescharacteristic
50C2×Dic137C4⋊C411yescharacteristic
51C4×D137—21no—
52C4×D137—21no—
53C23×C137C22⋊C411yescharacteristic
54C23×C137C4⋊C411yescharacteristic
55D4×C137—41no—
56C2×C4×C137—41no—
57C2×C4×C137C4211yescharacteristic
58C22×C137—21no—
59C22×C137—21no—
60C22×C137—41no—
61C22×C137—41no—
62C22×C137#9511yescharacteristic
63C2×D137—21no—
64C2×D137—41no—
65C4×C137—21no—
66C4×C137—81no—
67Dic137—21no—
68Dic137—21no—
69Dic137—81no—
70Dic137#9911yescharacteristic
71Dic137#9911yescharacteristic
72C2×C137—21no—
73C2×C137—41no—
74C2×C137—41no—
75C2×C137#34211yescharacteristic
76D137—41no—
77C137#139511yescharacteristic, complemented, Sylow
78#1395—1371nomaximal, Sylow
79#413—1371no—
80#410—1371no—
81#408—1371no—
82C2×#41—1371no—
83C2×M16—2741no—
84C2×M16—1371no—
85C22×C8—1371no—
86#117—2741no—
87#101—1372no—
88#100—2741no—
89#99—2741no—
90#99—1371no—
91#99#1058773911yescharacteristic
92C22⋊C8—2741no—
93C22⋊C8—2741no—
94#41—5481no—
95#41—5481no—
96C2×Q8—1371no—
97C2×D4—1371no—
98C2×D4—1371no—
99C2×D4C2×#1058773911yescharacteristic
100C22×C4—1371no—
101C22×C4—1371no—
102C22×C4—1371no—
103M16—2742no—
104M16—5481no—
105M16—5481no—
106C2×C8—2741no—
107C2×C8—2741no—
108C2×C8—5481no—
109C2×C8—5481no—
110C4⋊C4—5481no—
111C22⋊C4—21no—
112C22⋊C4—2741no—
113C42—5481no—
114C23—1371no—
115C23C22⋊#1058773911yescharacteristic
116C23C4⋊#1058773911yescharacteristic
117Q8—5481no—
118D4—41no—
119D4—5481no—
120D4—5481no—
121C2×C4—41no—
122C2×C4—2741no—
123C2×C4—2741no—
124C2×C4—2741no—
125C2×C4—2741no—
126C2×C4—2741no—
127C2×C4—2741no—
128C2×C4—5481no—
129C2×C4—1371no—
130C2×C4—1371no—
131C2×C4—1371no—
132C2×C4C4×#1058773911yescharacteristic
133C8—5482no—
134C8—5481no—
135C8—5481no—
136C22—21no—
137C22—21no—
138C22—41no—
139C22—41no—
140C22—2741no—
141C22—5481no—
142C22(C2×#10587739)⋊C411yescharacteristic
143C4—21no—
144C4—81no—
145C4—2741no—
146C4—2741no—
147C4—10961no—
148C4—1371no—
149C4—1371no—
150C2—21no—
151C2—41no—
152C2—41no—
153C2—5481no—
154C2#2257828011yescharacteristic
subgroupG/H
centerC2#22578280
commutatorC2×C4×C137C42
FrattiniC2×D4C2×#10587739
Fitting#99×C137C4
socleC2×C137#342
prime radical, cosocleC2×#14555509C22
hypercenter#99#10587739
hypocenterC137#1395
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #99

Complements

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

NH ≅ G/Ntype
4. C2×#14555520C4semidirect
6. C2×#14555518C4semidirect
7. #17984149C4semidirect
77. C137#1395semidirect

Maximal subgroups

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

subgroupindex
#225783902
#225783772
#225773752
#1395137