Finite Group Databasebeta

← #31796269

Subgroups — #31796269

Order 17536.

248 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 56 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#22578425C212yesmaximal, complemented
2#22578259C212yesmaximal, complemented
3#22578258C212yesmaximal, complemented
4#124×D137C211yesmaximal, characteristic
5C4⋊#11764047—22no—
6#17984145—22no—
7#17984143C2214yescomplemented
8C2×Q8×D137C411yescharacteristic
9C2×D4×D137C411yescharacteristic
10#17984077C2211yescharacteristic
11#17984076C411yescharacteristic
12#17984075C2211yescharacteristic
13#17984074—22no—
14C22⋊C4×D137—22no—
15C42×D137C2211yescharacteristic
16#124×C137C411yescharacteristic
17C2×#11764047—24no—
18C23×D137—21no—
19C2×Dic274—21no—
20C2×Dic274C2×C411yescharacteristic
21Q8×D137—42no—
22C2×D548C2×C411yescharacteristic
23C2×C22⋊D137—21no—
24D4×D137—42no—
25C2×C4×D137—21no—
26C2×C4×D137C2×C411yescharacteristic
27C2×C4×D137D412yes—
28C22⋊Dic137—22no—
29#14555487—22no—
30#14555487—22no—
31C4×Dic137D412yes—
32C4×Dic137C2311yescharacteristic
33C2×Q8×C137C2×C411yescharacteristic
34C2×D4×C137C2×C411yescharacteristic
35C22⋊C4×C137—22no—
36C42×C137C2×C411yescharacteristic
37#11764047—44no—
38C22×D137—21no—
39C22×D137—44no—
40C22×D137—42no—
41C22×D137C22⋊C411yescharacteristic
42Dic274—42no—
43Dic274—41no—
44D548—41no—
45C22⋊D137—42no—
46C2×Dic137—21no—
47C2×Dic137C22⋊C411yescharacteristic
48C2×Dic137C2×D412yes—
49C4×D137—22no—
50C4×D137—42no—
51C4×D137—42no—
52C23×C137—21no—
53Q8×C137—41no—
54D4×C137—41no—
55C2×C4×C137—21no—
56C2×C4×C137C22⋊C412yes—
57C2×C4×C137C22×C411yescharacteristic
58C22×C137—41no—
59C22×C137—42no—
60C22×C137C2×C22⋊C411yescharacteristic
61C2×D137—22no—
62C2×D137—41no—
63C2×D137—42no—
64C2×D137—84no—
65C2×D137#10012yes—
66C4×C137—21no—
67C4×C137—22no—
68C4×C137—41no—
69Dic137—21no—
70Dic137—22no—
71Dic137—41no—
72C2×C137—81no—
73C2×C137C2×#10011yescharacteristic
74C2×C137#42112yes—
75D137—22no—
76D137—81no—
77C137#153311yescharacteristic, complemented, Sylow
78#1533—1371nomaximal, Sylow
79C2×#124—1371no—
80#432—1372no—
81#329—1372no—
82#329—1372no—
83C22×Q8—1371no—
84C22×D4—1371no—
85#124—2742no—
86#124—1371no—
87#124—1371no—
88#124—1371no—
89#124#1058773911yescharacteristic
90C2×C22⋊C4—2742no—
91C2×C42—1371no—
92C4⋊C8—2742no—
93C22⋊C8—2742no—
94#97—1374no—
95C24—2741no—
96C2×Q8—2741no—
97C2×Q8—5482no—
98C2×Q8—1371no—
99C2×Q8C2×#1058773911yescharacteristic
100C2×D4—2741no—
101C2×D4—5482no—
102C2×D4—1371no—
103C2×D4C2×#1058773911yescharacteristic
104C22×C4—2741no—
105C22×C4—1371no—
106C22×C4—1372no—
107C2×C8—2744no—
108C22⋊C4—22no—
109C22⋊C4—2742no—
110C22⋊C4—2742no—
111C22⋊C4—2742no—
112C42—1372no—
113C42—1371no—
114C42C2×#1058773911yescharacteristic
115C23—21no—
116C23—2741no—
117C23—5484no—
118C23—5482no—
119C23—1371no—
120Q8—41no—
121Q8—5481no—
122Q8—5482no—
123D4—41no—
124D4—5481no—
125D4—5482no—
126C2×C4—21no—
127C2×C4—2742no—
128C2×C4—2741no—
129C2×C4—5482no—
130C2×C4—5482no—
131C2×C4—1372no—
132C2×C4—1371no—
133C2×C4C22⋊#1058773912yes—
134C2×C4C22×#1058773911yescharacteristic
135C8—5484no—
136C22—41no—
137C22—42no—
138C22—2742no—
139C22—5481no—
140C22—5482no—
141C22—10964no—
142C22—1372no—
143C22C2×C22⋊#1058773911yescharacteristic
144C4—21no—
145C4—22no—
146C4—41no—
147C4—2741no—
148C4—2742no—
149C4—5481no—
150C2—81no—
151C2—2742no—
152C2—10961no—
153C2D137⋊#10011yescharacteristic
154C2#2257840012yes—
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#124×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC2×D4×D137C4
cosocleC4×Dic137C23
hypercenter#124#10587739
hypocenterC137#1533
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #22578425C2semidirect
2. #22578259C2semidirect
3. #22578258C2semidirect
7. #17984143C22semidirect
77. C137#1533semidirect

Maximal subgroups

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

subgroupindex
#225784252
#225782592
#225782582
#124×D1372
#1533137