Finite Group Databasebeta

← #31797073

Subgroups — #31797073

Order 17536.

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

normal subgroups only

This group has 238 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
2#22578310C214yesmaximal, complemented
3#22577979C211yesmaximal, characteristic
4#17984167—22no—
5#17984147C2214yes—
6#17984146—22no—
7C2×#14555510C2211yescharacteristic
8C22×Dic274C411yescharacteristic, complemented
9#17984088C412yescomplemented
10#17984076C2212yes—
11#17984072—22no—
12#17984068—22no—
13#117×C137C411yescharacteristic, complemented
14C4⋊#10587739—42no—
15C22⋊#10587739—24no—
16C4×#10587739—42no—
17#14555510—42no—
18C2×Dic274—21no—
19C2×Dic274—42no—
20C2×Dic274C2×C412yes—
21C2×Dic274C2×C411yescharacteristic
22C2×D548C2311yescharacteristic
23C2×C22⋊D137D412yes—
24C22×Dic137—21no—
25C2×C4×D137D412yes—
26C4⋊Dic137—24no—
27#14555487—24no—
28C22×C4×C137C2×C411yescharacteristic
29C4⋊C4×C137—21no—
30C22⋊C4×C137—21no—
31C42×C137C2×C412yes—
32C2×#10587739—44no—
33C22×D137C2×D412yes—
34Dic274—41no—
35Dic274—42no—
36Dic274—42no—
37D548—41no—
38C22⋊D137—42no—
39C2×Dic137—22no—
40C2×Dic137—42no—
41C2×Dic137C22⋊C412yes—
42C4×D137—22no—
43C4×D137—22no—
44C23×C137C22⋊C411yescharacteristic
45C2×C4×C137—22no—
46C2×C4×C137—22no—
47C2×C4×C137C22⋊C411yescharacteristic
48C2×C4×C137C22×C411yescharacteristic
49C22×C137—21no—
50C22×C137—41no—
51C22×C137C2×C22⋊C411yescharacteristic
52C2×D137—22no—
53C2×D137—42no—
54#10587739—84no—
55C4×C137—21no—
56C4×C137—42no—
57C4×C137#9912yes—
58Dic137—22no—
59Dic137—42no—
60C2×C137—21no—
61C2×C137—41no—
62C2×C137C2×#9911yescharacteristic
63D137—42no—
64C137#212911yescharacteristic, complemented, Sylow
65#2129—1371nomaximal, Sylow
66#536—1371no—
67#410—1372no—
68#354—1374no—
69C2×#41—1371no—
70C22×Q8—1371no—
71C4⋊Q8—1372no—
72#124—1372no—
73#123—2742no—
74C22⋊Q8—2742no—
75#117—2742no—
76#117#1058773911yescharacteristic, complemented
77#99—2742no—
78#99—1374no—
79#41—5482no—
80C2×Q8—2741no—
81C2×Q8—5482no—
82C2×Q8—1372no—
83C2×Q8—1371no—
84C2×D4—1372no—
85C2×D4—1371no—
86C22×C4—2741no—
87C22×C4—1372no—
88C22×C4C2×#1058773911yescharacteristic
89C4⋊C4—21no—
90C4⋊C4—2744no—
91C4⋊C4—5482no—
92C22⋊C4—21no—
93C22⋊C4—2744no—
94C22⋊C4—2744no—
95C42—5482no—
96C42C2×#1058773912yes—
97C23—1372no—
98C23C22⋊#1058773911yescharacteristic
99Q8—5481no—
100Q8—5482no—
101Q8—5482no—
102D4—5481no—
103D4—5482no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—2742no—
107C2×C4—2742no—
108C2×C4—2742no—
109C2×C4—5482no—
110C2×C4—5484no—
111C2×C4—1372no—
112C2×C4C22⋊#1058773911yescharacteristic
113C2×C4C22×#1058773911yescharacteristic
114C22—21no—
115C22—41no—
116C22—2742no—
117C22—5482no—
118C22C2×C22⋊#1058773911yescharacteristic
119C4—21no—
120C4—42no—
121C4—2742no—
122C4—5482no—
123C4—10964no—
124C4#1798414712yes—
125C2—21no—
126C2—41no—
127C2—5482no—
128C2C2×#1798414711yescharacteristic
subgroupG/H
centerC2C2×#17984147
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#117×C137C4
socleC2×C137C2×#99
prime radicalC2×#14555510C22
cosocleC2×D548C23
hypercenter#117#10587739
hypocenterC137#2129
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #22578310C2semidirect
8. C22×Dic274C4semidirect
9. #17984088C4semidirect
13. #117×C137C4semidirect
64. C137#2129semidirect
76. #117#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783822
#225783102
#225779792
#2129137