Finite Group Databasebeta

← #31796292

Subgroups — #31796292

Order 17536.

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

normal subgroups only

This group has 246 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#22578420C212yesmaximal, complemented
2#22578263C212yesmaximal, complemented
3#22578261C212yesmaximal, complemented
4#22577949C211yesmaximal, characteristic
5#17984155C2214yes—
6C22⋊#11764047—22no—
7C4×#11764047—22no—
8C2×#14555510C411yescharacteristic
9#17984086C2211yescharacteristic
10#17984084C2211yescharacteristic
11#17984077C411yescharacteristic
12#17984075C411yescharacteristic
13#17984060—22no—
14C4×C22⋊D137—22no—
15C2×C4×Dic137C2211yescharacteristic
16C2×#41×C137C411yescharacteristic
17C2×#11764047—24no—
18#14555510—42no—
19C2×Dic274C2×C411yescharacteristic
20C2×D548C2×C411yescharacteristic
21C2×C22⋊D137—22no—
22C2×C22⋊D137—21no—
23C22×Dic137D412yes—
24C2×C4×D137—21no—
25#14555489—22no—
26C22⋊Dic137—22no—
27#14555487—22no—
28C4×Dic137C2×C411yescharacteristic
29C4×Dic137D412yes—
30C4×Dic137C2311yescharacteristic
31#41×C137—42no—
32C2×Q8×C137C2×C411yescharacteristic
33C2×D4×C137—21no—
34C2×D4×C137C2×C411yescharacteristic
35C22×C4×C137—21no—
36C22×C4×C137C2×C411yescharacteristic
37#11764047—44no—
38C22×D137—21no—
39Dic274—41no—
40D548—41no—
41C22⋊D137—42no—
42C22⋊D137—44no—
43C2×Dic137—24no—
44C2×Dic137—21no—
45C2×Dic137C22⋊C412yes—
46C2×Dic137C2×D412yes—
47C4×D137—42no—
48C23×C137—21no—
49C23×C137C22⋊C411yescharacteristic
50Q8×C137—41no—
51D4×C137—41no—
52D4×C137—42no—
53C2×C4×C137—22no—
54C2×C4×C137—21no—
55C2×C4×C137—42no—
56C2×C4×C137C22⋊C411yescharacteristic
57C2×C4×C137C22×C411yescharacteristic
58C22×C137—21no—
59C22×C137—22no—
60C22×C137—41no—
61C22×C137—42no—
62C22×C137C2×C22⋊C411yescharacteristic
63C2×D137—41no—
64C2×D137—42no—
65C4×C137—21no—
66C4×C137—41no—
67C4×C137#10012yes—
68Dic137—22no—
69Dic137—41no—
70Dic137#13514yes—
71C2×C137—41no—
72C2×C137—81no—
73C2×C137C2×#10011yescharacteristic
74C2×C137#41612yes—
75D137—81no—
76C137#154311yescharacteristic, complemented, Sylow
77#1543—1371nomaximal, Sylow
78#532—1371no—
79#431—1372no—
80#333—1372no—
81#331—1372no—
82C2×#41—1371no—
83C2×#41#1058773911yescharacteristic
84C4⋊Q8—1371no—
85#127—1371no—
86#124—1371no—
87#124—1371no—
88#121—2742no—
89C4×D4—2742no—
90C2×C42—1371no—
91C4⋊C8—1374no—
92C22⋊C8—2742no—
93C4×C8—2742no—
94#41—42no—
95#41—5482no—
96C2×Q8—1371no—
97C2×Q8C2×#1058773911yescharacteristic
98C2×D4—21no—
99C2×D4—2741no—
100C2×D4—2742no—
101C2×D4—1371no—
102C2×D4C2×#1058773911yescharacteristic
103C22×C4—21no—
104C22×C4—2741no—
105C22×C4—1372no—
106C22×C4C2×#1058773911yescharacteristic
107C2×C8—2744no—
108C4⋊C4—2742no—
109C22⋊C4—2742no—
110C22⋊C4—2742no—
111C42—1372no—
112C42—1371no—
113C42—1371no—
114C23—21no—
115C23—2741no—
116C23C22⋊#1058773911yescharacteristic
117Q8—41no—
118Q8—5481no—
119D4—41no—
120D4—42no—
121D4—5481no—
122D4—5482no—
123D4—5484no—
124C2×C4—22no—
125C2×C4—21no—
126C2×C4—42no—
127C2×C4—2744no—
128C2×C4—2741no—
129C2×C4—5482no—
130C2×C4—1372no—
131C2×C4—1372no—
132C2×C4C22⋊#1058773911yescharacteristic
133C2×C4C22×#1058773911yescharacteristic
134C8—5484no—
135C22—21no—
136C22—22no—
137C22—41no—
138C22—42no—
139C22—5481no—
140C22—5482no—
141C22C2×C22⋊#1058773911yescharacteristic
142C4—21no—
143C4—41no—
144C4—2742no—
145C4—5481no—
146C4—1374no—
147C4#1798414812yes—
148C2—41no—
149C2—81no—
150C2—10961no—
151C2C2×#1798414811yescharacteristic
152C2#2257839312yes—
subgroupG/H
centerC2×C4C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC2×#41×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577949C2
cosocleC4×Dic137C23
hypercenterC2×#41#10587739
hypocenterC137#1543
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #22578420C2semidirect
2. #22578263C2semidirect
3. #22578261C2semidirect
76. C137#1543semidirect

Maximal subgroups

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

subgroupindex
#225784202
#225782632
#225782612
#225779492
#1543137