Finite Group Databasebeta

← #31796063

Subgroups — #31796063

Order 17536.

204 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 62 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#22578424C211yesmaximal, characteristic
2#22578420C211yesmaximal, characteristic
3#22578249C214yesmaximal
4C2×#17984079C211yesmaximal, characteristic
5#17984155—21no—
6C4⋊#11764047—21no—
7C22⋊#11764047C2212yes—
8C22⋊#11764047C2212yes—
9#17984143—21no—
10C4×#11764047—21no—
11#17984079—21no—
12#17984079—21no—
13#17984079—21no—
14#17984079—21no—
15C2×C4⋊Dic137C412yes—
16C2×C4⋊Dic137C411yescharacteristic
17C2×#14555489C2212yes—
18C2×C4×Dic137C2211yescharacteristic
19C2×C4⋊C4×C137C411yescharacteristic
20C2×#11764047—22no—
21C2×#11764047—22no—
22C22×Dic137C2×C412yes—
23C22×Dic137C2×C411yescharacteristic
24C22×Dic137C2311yescharacteristic
25C4⋊Dic137—22no—
26C4⋊Dic137—22no—
27C4⋊Dic137—21no—
28C4⋊Dic137—21no—
29#14555489—22no—
30#14555489—22no—
31C4×Dic137D411yescharacteristic
32C4×Dic137D411yescharacteristic
33C4×Dic137D411yescharacteristic
34C4×Dic137D411yescharacteristic
35C22×C4×C137C2×C412yes—
36C22×C4×C137C2×C411yescharacteristic
37C4⋊C4×C137—21no—
38C4⋊C4×C137—21no—
39#11764047—42no—
40#11764047—42no—
41C2×Dic137—22no—
42C2×Dic137—22no—
43C2×Dic137—24no—
44C2×Dic137—41no—
45C2×Dic137C22⋊C411yescharacteristic
46C2×Dic137C22⋊C411yescharacteristic
47C2×Dic137C2×D411yescharacteristic
48C2×Dic137C2×D411yescharacteristic
49C23×C137C22×C411yescharacteristic
50C2×C4×C137—22no—
51C2×C4×C137—22no—
52C2×C4×C137—42no—
53C2×C4×C137C22⋊C411yescharacteristic
54C2×C4×C137C22⋊C411yescharacteristic
55C22×C137—21no—
56C22×C137—21no—
57C22×C137C2×C22⋊C411yescharacteristic
58C22×C137#13112yes—
59C4×C137—21no—
60C4×C137—42no—
61C4×C137#10112yes—
62Dic137—21no—
63Dic137—21no—
64Dic137—21no—
65Dic137—21no—
66Dic137—42no—
67C2×C137—22no—
68C2×C137#40811yescharacteristic
69C2×C137#41011yescharacteristic
70C2×C137C2×#10111yescharacteristic
71C137#147011yescharacteristic, complemented, Sylow
72#1470—1371nomaximal, Sylow
73C2×#125—1371no—
74#432—1371no—
75#431—1371no—
76#324—1374no—
77#125—2741no—
78#125—2741no—
79#125—2741no—
80#125—2741no—
81C2×C4⋊C4—1372no—
82C2×C4⋊C4—1372no—
83C2×C4⋊C4—1371no—
84C2×C4⋊C4#1058773911yescharacteristic
85C2×C42—1371no—
86C4⋊C8—2741no—
87C4⋊C8—2741no—
88C22⋊C8—1372no—
89C22⋊C8—1372no—
90#97—2741no—
91C4×C8—2741no—
92C22×C4—1372no—
93C22×C4—1371no—
94C22×C4—1371no—
95C22×C4C2×#1058773912yes—
96C22×C4C2×#1058773911yescharacteristic
97C2×C8—2742no—
98C2×C8—2742no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—2742no—
102C4⋊C4—2742no—
103C4⋊C4—2742no—
104C4⋊C4—2742no—
105C4⋊C4—2741no—
106C4⋊C4—2741no—
107C42—1371no—
108C42—1371no—
109C42—1371no—
110C42—1371no—
111C23C22×#1058773911yescharacteristic
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—42no—
115C2×C4—2742no—
116C2×C4—2742no—
117C2×C4—2744no—
118C2×C4—5481no—
119C2×C4—1371no—
120C2×C4—1371no—
121C2×C4—1371no—
122C2×C4—1371no—
123C2×C4C22⋊#1058773911yescharacteristic
124C2×C4C22⋊#1058773911yescharacteristic
125C8—5482no—
126C8—5482no—
127C22—21no—
128C22—21no—
129C22C2×C22⋊#1058773911yescharacteristic
130C22#1798417612yes—
131C4—21no—
132C4—42no—
133C4—2741no—
134C4—2741no—
135C4—2741no—
136C4—2741no—
137C4—5482no—
138C4#1798414912yes—
139C2—22no—
140C2#2257837811yescharacteristic
141C2#2257838311yescharacteristic
142C2C2×#1798414911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutator, prime radicalC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C4⋊C4×C137C4
socleC22×C137C2×C22⋊C4
cosocleC22×Dic137C23
hypercenterC2×C4⋊C4#10587739
hypocenterC137#1470
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
71. C137#1470semidirect

Maximal subgroups

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

subgroupindex
#225784242
#225784202
#225782492
C2×#179840792
#1470137