Finite Group Databasebeta

← #31796747

Subgroups — #31796747

Order 17536.

252 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 66 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#22578483C211yesmaximal, characteristic
2#22578418C212yesmaximal, complemented
3#22578378C211yesmaximal, characteristic
4#22578278C212yesmaximal, complemented
5#22577386C211yesmaximal, characteristic
6#17984176—22no—
7#17984176—22no—
8C2×#14555520C2211yescharacteristic
9D137⋊M16—22no—
10D137⋊M16C2212yes—
11C2×#14555518C2211yescharacteristic
12#17984159—22no—
13#17984158—22no—
14#17984145—22no—
15C2×#14555511C2211yescharacteristic
16M16×D137C2212yes—
17#17984007C411yescharacteristic
18#17984004C412yes—
19#101×C137C411yescharacteristic
20#14555520—22no—
21#14555520—22no—
22#14555520—22no—
23#14555520D412yes—
24#14555520D412yes—
25#14555519—22no—
26#14555518—24no—
27#14555518—22no—
28#14555518—22no—
29C2×#11764047—21no—
30C2×#11764047—21no—
31#14555511—22no—
32#14555511—22no—
33C2×D548C2×C411yescharacteristic
34C2×D548Q812yes—
35C2×C4×D137D412yes—
36C2×C4×D137C2311yescharacteristic
37#14555494C2×C412yes—
38#14555493—22no—
39C8×D137—22no—
40C2×Q8×C137C2×C411yescharacteristic
41M16×C137C2×C412yes—
42#11764047—22no—
43#11764047—22no—
44#11764047—41no—
45#11764047—41no—
46C22×D137C4⋊C412yes—
47C22×D137C2×Q811yescharacteristic
48D548—24no—
49D548—22no—
50C2×Dic137C2×D411yescharacteristic
51C4×D137—22no—
52C4×D137—22no—
53C4×D137C2×D412yes—
54C4×D137#4112yes—
55#11764040—22no—
56Q8×C137—22no—
57C2×C4×C137C4⋊C412yes—
58C2×C4×C137C22×C411yescharacteristic
59C8×C137—22no—
60C22×C137C2×C4⋊C411yescharacteristic
61C2×D137—22no—
62C2×D137—41no—
63C2×D137—42no—
64C2×D137C22⋊Q812yes—
65C4×C137—22no—
66C4×C137C4×D412yes—
67Dic137#12011yescharacteristic
68Dic137#12311yescharacteristic
69C2×C137—21no—
70C2×C137#38511yescharacteristic
71D137—22no—
72D137—42no—
73C137#185811yescharacteristic, complemented, Sylow
74#1858—1371nomaximal, Sylow
75#568—1371no—
76#430—1372no—
77#413—1371no—
78#408—1371no—
79#341—1372no—
80C2×#41—1371no—
81#131—2742no—
82#131—2742no—
83C2×M16—2742no—
84C2×M16—1371no—
85C2×M16—1372no—
86C2×M16—1372no—
87C22×C8—1371no—
88#108—2742no—
89#108—2742no—
90#101—1371no—
91#101#1058773911yescharacteristic
92#100—1372no—
93C22⋊C8—2742no—
94#41—2742no—
95#41—2742no—
96C2×Q8C2×#1058773911yescharacteristic
97C2×D4—1372no—
98C2×D4—1371no—
99C22×C4—1372no—
100C22×C4—1371no—
101M16—2742no—
102M16—2742no—
103M16—2742no—
104M16—2742no—
105M16—2742no—
106M16—1372no—
107M16—1372no—
108M16—1372no—
109M16C2×#1058773912yes—
110C2×C8—2742no—
111C2×C8—2744no—
112C2×C8—2742no—
113C2×C8—2741no—
114C2×C8—2741no—
115C2×C8—2742no—
116C23—1372no—
117C23—1371no—
118Q8—22no—
119D4—2744no—
120D4—2742no—
121C2×C4—2742no—
122C2×C4—2742no—
123C2×C4—1371no—
124C2×C4—1372no—
125C2×C4—1372no—
126C2×C4C4⋊#1058773912yes—
127C2×C4C22×#1058773911yescharacteristic
128C8—22no—
129C8—2742no—
130C8—2742no—
131C8—2742no—
132C8—5481no—
133C8—5481no—
134C22—2742no—
135C22—5481no—
136C22—5482no—
137C22—1372no—
138C22C2×C4⋊#1058773911yescharacteristic
139C4—22no—
140C4—1371no—
141C4—1371no—
142C4D4×#1058773912yes—
143C2—21no—
144C2—2742no—
145C2—5482no—
146C2#2257835011yescharacteristic
subgroupG/H
centerC2#22578350
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#101×C137C4
socleC2×C137#385
prime radicalC2×#14555511C22
cosocleC2×C4×D137C23
hypercenter#101#10587739
hypocenterC137#1858
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
2. #22578418C2semidirect
4. #22578278C2semidirect
73. C137#1858semidirect

Maximal subgroups

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

subgroupindex
#225784832
#225784182
#225783782
#225782782
#225773862
#1858137