Finite Group Databasebeta

← #31797071

Subgroups — #31797071

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#22578309C214yesmaximal, complemented
3#22577977C211yesmaximal, characteristic
4#17984167—22no—
5#17984147C2214yes—
6#17984146—22no—
7C2×#14555510C2211yescharacteristic
8#17984086C412yescomplemented
9#17984077C2212yes—
10#17984071—22no—
11#17984064—22no—
12#17984047C411yescharacteristic, complemented
13C22×Q8×C137C411yescharacteristic, complemented
14C4⋊#10587739—42no—
15C22⋊#10587739—24no—
16C4×#10587739—42no—
17#14555510—42no—
18C2×Dic274C2×C411yescharacteristic
19C2×D548C2311yescharacteristic
20C2×C22⋊D137D412yes—
21C2×C4×D137D412yes—
22C4⋊Dic137—21no—
23#14555489—24no—
24C22⋊Dic137—21no—
25#14555487—24no—
26C4×Dic137C2×C412yes—
27C2×Q8×C137—21no—
28C2×Q8×C137—42no—
29C2×Q8×C137C2×C412yes—
30C22×C4×C137—21no—
31C22×C4×C137C2×C411yescharacteristic
32C2×#10587739—44no—
33C22×D137C2×D412yes—
34Dic274—41no—
35D548—41no—
36C22⋊D137—42no—
37C2×Dic137—22no—
38C2×Dic137C22⋊C412yes—
39C4×D137—22no—
40C4×D137—22no—
41C23×C137C22⋊C411yescharacteristic
42Q8×C137—42no—
43Q8×C137—42no—
44C2×C4×C137—22no—
45C2×C4×C137—22no—
46C2×C4×C137—42no—
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#1058773911yescharacteristic, complemented
71C4⋊Q8—1372no—
72#124—1372no—
73#123—2742no—
74C22⋊Q8—2742no—
75#117—2742no—
76#117—1371no—
77#99—2742no—
78#99—1374no—
79#41—5482no—
80C2×Q8—21no—
81C2×Q8—42no—
82C2×Q8—1371no—
83C2×Q8C2×#1058773912yes—
84C2×D4—1372no—
85C2×D4—1371no—
86C22×C4—21no—
87C22×C4—1372no—
88C22×C4C2×#1058773911yescharacteristic
89C4⋊C4—2744no—
90C4⋊C4—2741no—
91C4⋊C4—5482no—
92C22⋊C4—2744no—
93C22⋊C4—2741no—
94C22⋊C4—2744no—
95C42—5482no—
96C42—1372no—
97C23—1372no—
98C23C22⋊#1058773911yescharacteristic
99Q8—42no—
100Q8—42no—
101Q8—5481no—
102D4—5481no—
103D4—5482no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—42no—
107C2×C4—2742no—
108C2×C4—2742no—
109C2×C4—2742no—
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#1798414612yes—
125C2—21no—
126C2—41no—
127C2—5482no—
128C2C2×#1798414611yescharacteristic
subgroupG/H
centerC2C2×#17984146
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22×Q8×C137C4
socleC2×C137C2×#99
prime radicalC2×#14555510C22
cosocleC2×D548C23
hypercenterC22×Q8#10587739
hypocenterC137#2129
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. #22578309C2semidirect
8. #17984086C4semidirect
12. #17984047C4semidirect
13. C22×Q8×C137C4semidirect
64. C137#2129semidirect
70. C22×Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783822
#225783092
#225779772
#2129137