Finite Group Databasebeta

← #31795891

Subgroups — #31795891

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#22578384C211yesmaximal, characteristic
2#22578378C211yesmaximal, characteristic
3#22577386C211yesmaximal, characteristic
4C2×#14555520C411yescharacteristic, complemented
5C2×#14555518C411yescharacteristic, complemented
6#17984167—21no—
7#17984147—21no—
8#17984147C412yes—
9#17984145—21no—
10#17984145—21no—
11C2×#14555511C2211yescharacteristic
12M16×D137—21no—
13#17984007C411yescharacteristic, complemented
14#17984004—21no—
15#101×C137C411yescharacteristic, complemented
16#14555520—41no—
17#14555518—41no—
18C2×#11764047—21no—
19C2×#11764047—21no—
20C4⋊#10587739—41no—
21C22⋊#10587739—22no—
22C4×#10587739—41no—
23#14555511—41no—
24#14555511—41no—
25C2×D548C2×C411yescharacteristic
26C2×D548D411yescharacteristic
27C2×D548Q811yescharacteristic
28C2×C4×D137C2×C411yescharacteristic
29C2×C4×D137D411yescharacteristic
30C2×C4×D137D411yescharacteristic
31#14555494—21no—
32#14555493—41no—
33C8×D137—41no—
34C2×Q8×C137C2×C411yescharacteristic
35M16×C137—21no—
36C2×#10587739—42no—
37#11764047—41no—
38#11764047—41no—
39C22×D137C22⋊C411yescharacteristic
40C22×D137C4⋊C411yescharacteristic
41C22×D137C4⋊C411yescharacteristic
42D548—41no—
43D548—41no—
44D548—41no—
45C2×Dic137C22⋊C411yescharacteristic
46C4×D137—21no—
47C4×D137—21no—
48C4×D137—21no—
49C4×D137—21no—
50C4×D137—21no—
51C4×D137—21no—
52#11764040—41no—
53Q8×C137—41no—
54C2×C4×C137C4211yescharacteristic
55C2×C4×C137C22⋊C411yescharacteristic
56C2×C4×C137C4⋊C411yescharacteristic
57C8×C137—41no—
58C22×C137#9511yescharacteristic
59C2×D137—21no—
60C2×D137—21no—
61C2×D137—21no—
62C2×D137—41no—
63C2×D137—41no—
64C2×D137—41no—
65#10587739—82no—
66C4×C137—21no—
67C4×C137—21no—
68C4×C137—21no—
69Dic137#9911yescharacteristic
70Dic137#9911yescharacteristic
71C2×C137—21no—
72C2×C137#34211yescharacteristic
73D137—41no—
74D137—41no—
75D137—41no—
76C137#139511yescharacteristic, complemented, Sylow
77#1395—1371nomaximal, Sylow
78#413—1371no—
79#410—1371no—
80#408—1371no—
81C2×#41—1371no—
82C2×M16—2741no—
83C2×M16—1371no—
84C22×C8—1371no—
85#117—2741no—
86#101—1371no—
87#101#1058773911yescharacteristic, complemented
88#100—2741no—
89#99—2741no—
90#99—1372no—
91C22⋊C8—2741no—
92C22⋊C8—2741no—
93#41—5481no—
94#41—5481no—
95C2×Q8C2×#1058773911yescharacteristic
96C2×D4—1371no—
97C2×D4—1371no—
98C2×D4—1371no—
99C22×C4—1371no—
100C22×C4—1371no—
101C22×C4—1371no—
102M16—21no—
103M16—2741no—
104M16—5481no—
105M16—5481no—
106C2×C8—2741no—
107C2×C8—2741no—
108C2×C8—5481no—
109C2×C8—5481no—
110C4⋊C4—5481no—
111C22⋊C4—2742no—
112C42—5481no—
113C23—1371no—
114C23—1371no—
115C23—1371no—
116Q8—41no—
117D4—5481no—
118D4—5481no—
119D4—5481no—
120C2×C4—2741no—
121C2×C4—2741no—
122C2×C4—2741no—
123C2×C4—2741no—
124C2×C4—2741no—
125C2×C4—2741no—
126C2×C4—5482no—
127C2×C4—1371no—
128C2×C4C4×#1058773911yescharacteristic
129C2×C4C22⋊#1058773911yescharacteristic
130C2×C4C4⋊#1058773911yescharacteristic
131C8—41no—
132C8—5481no—
133C8—5481no—
134C8—5481no—
135C22—2741no—
136C22—2741no—
137C22—2741no—
138C22—5481no—
139C22—5481no—
140C22—5481no—
141C22(C2×#10587739)⋊C411yescharacteristic
142C4—21no—
143C4—21no—
144C4—21no—
145C4—10962no—
146C4—1371no—
147C4—1371no—
148C2—21no—
149C2—5481no—
150C2—5481no—
151C2—5481no—
152C2#2257828011yescharacteristic
subgroupG/H
centerC2#22578280
commutatorC2×C4×C137C42
FrattiniC2×Q8C2×#10587739
Fitting#101×C137C4
socleC2×C137#342
prime radical, cosocleC2×#14555511C22
hypercenter#101#10587739
hypocenterC137#1395
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #101

Complements

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

NH ≅ G/Ntype
4. C2×#14555520C4semidirect
5. C2×#14555518C4semidirect
13. #17984007C4semidirect
15. #101×C137C4semidirect
76. C137#1395semidirect
87. #101#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783842
#225783782
#225773862
#1395137