Finite Group Databasebeta

← #31797068

Subgroups — #31797068

Order 17536.

270 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 54 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#22578382C212yesmaximal, complemented
2#22578307C214yesmaximal, complemented
3#22577970C211yesmaximal, characteristic
4#17984167—22no—
5#17984147C2214yescomplemented
6#17984146—22no—
7C2×#14555510C2211yescharacteristic
8#17984084C2212yes—
9#17984075C412yescomplemented
10#17984059—22no—
11#17984057—22no—
12#17984047C411yescharacteristic, complemented
13C22×D4×C137C411yescharacteristic, complemented
14C4⋊#10587739—42no—
15C22⋊#10587739—24no—
16C4×#10587739—42no—
17#14555510—42no—
18C2×Dic274C2×C411yescharacteristic
19C2×D548C2311yescharacteristic
20C2×C22⋊D137—24no—
21C2×C22⋊D137D412yes—
22C2×C4×D137D412yes—
23C4⋊Dic137—21no—
24C22⋊Dic137—24no—
25C22⋊Dic137—21no—
26C4×Dic137C2×C412yes—
27C24×C137—21no—
28C2×D4×C137—21no—
29C2×D4×C137—42no—
30C2×D4×C137C2×C412yes—
31C22×C4×C137C2×C411yescharacteristic
32C2×#10587739—44no—
33C22×D137C2×D412yes—
34Dic274—41no—
35D548—41no—
36C22⋊D137—42no—
37C22⋊D137—84no—
38C2×Dic137—22no—
39C2×Dic137C22⋊C412yes—
40C4×D137—22no—
41C4×D137—22no—
42C23×C137—22no—
43C23×C137—42no—
44C23×C137—82no—
45C23×C137C22⋊C411yescharacteristic
46D4×C137—42no—
47D4×C137—42no—
48C2×C4×C137—22no—
49C2×C4×C137C22⋊C411yescharacteristic
50C2×C4×C137C22×C411yescharacteristic
51C22×C137—21no—
52C22×C137—41no—
53C22×C137—44no—
54C22×C137—42no—
55C22×C137—84no—
56C22×C137C2×C22⋊C411yescharacteristic
57C2×D137—22no—
58C2×D137—42no—
59#10587739—84no—
60C4×C137—21no—
61C4×C137#9912yes—
62Dic137—22no—
63Dic137—42no—
64C2×C137—21no—
65C2×C137—41no—
66C2×C137—82no—
67C2×C137C2×#9911yescharacteristic
68D137—42no—
69C137#212811yescharacteristic, complemented, Sylow
70#2128—1371nomaximal, Sylow
71#535—1371no—
72#410—1372no—
73#353—1374no—
74C2×#41—1371no—
75C22×D4#1058773911yescharacteristic, complemented
76#127—1372no—
77#124—1372no—
78#121—2742no—
79#120—2742no—
80#117—2742no—
81#117—1371no—
82#99—2742no—
83#99—1374no—
84C24—21no—
85#41—5482no—
86C2×Q8—1371no—
87C2×D4—21no—
88C2×D4—42no—
89C2×D4—2744no—
90C2×D4—1372no—
91C2×D4—1371no—
92C2×D4C2×#1058773912yes—
93C22×C4—1372no—
94C22×C4C2×#1058773911yescharacteristic
95C4⋊C4—2741no—
96C4⋊C4—5482no—
97C22⋊C4—2741no—
98C22⋊C4—2744no—
99C22⋊C4—2744no—
100C42—5482no—
101C42—1372no—
102C23—22no—
103C23—42no—
104C23—82no—
105C23—1372no—
106C23C22⋊#1058773911yescharacteristic
107Q8—5481no—
108D4—42no—
109D4—42no—
110D4—5481no—
111D4—5482no—
112D4—10964no—
113C2×C4—22no—
114C2×C4—2742no—
115C2×C4—2742no—
116C2×C4—2742no—
117C2×C4—5484no—
118C2×C4—1372no—
119C2×C4C22⋊#1058773911yescharacteristic
120C2×C4C22×#1058773911yescharacteristic
121C22—21no—
122C22—41no—
123C22—44no—
124C22—42no—
125C22—84no—
126C22—2742no—
127C22—5482no—
128C22C2×C22⋊#1058773911yescharacteristic
129C4—21no—
130C4—2742no—
131C4—5482no—
132C4—10964no—
133C4#1798414612yes—
134C2—21no—
135C2—41no—
136C2—82no—
137C2—5482no—
138C2C2×#1798414611yescharacteristic
subgroupG/H
centerC2C2×#17984146
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22×D4×C137C4
socleC2×C137C2×#99
prime radical#22577970C2
cosocleC2×D548C23
hypercenterC22×D4#10587739
hypocenterC137#2128
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #22578382C2semidirect
2. #22578307C2semidirect
5. #17984147C22semidirect
9. #17984075C4semidirect
12. #17984047C4semidirect
13. C22×D4×C137C4semidirect
69. C137#2128semidirect
75. C22×D4#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783822
#225783072
#225779702
#2128137