Finite Group Databasebeta

← #31796293

Subgroups — #31796293

Order 17536.

246 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 60 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#22578421C212yesmaximal, complemented
2#22578263C212yesmaximal, complemented
3#22578261C212yesmaximal, complemented
4#22577952C211yesmaximal, characteristic
5#17984155C2214yes—
6#17984145—22no—
7C4×#11764047—22no—
8C2×#14555511C411yescharacteristic
9C2×#14555509C411yescharacteristic
10#17984086C2211yescharacteristic
11#17984084C2211yescharacteristic
12#17984076C411yescharacteristic
13#17984061—22no—
14#17984048—22no—
15C42×D137C2211yescharacteristic
16#124×C137C411yescharacteristic
17C2×#11764047—24no—
18#14555511—42no—
19#14555509—42no—
20C2×Dic274C2×C411yescharacteristic
21C2×D548—21no—
22C2×D548C2×C411yescharacteristic
23C2×C22⋊D137—21no—
24C2×C22⋊D137—22no—
25C22×Dic137—21no—
26C2×C4×D137—21no—
27C2×C4×D137C2×C411yescharacteristic
28C2×C4×D137D412yes—
29#14555489—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×D137C22⋊C411yescharacteristic
40Dic274—41no—
41D548—41no—
42D548—42no—
43C22⋊D137—42no—
44C22⋊D137—44no—
45C2×Dic137—21no—
46C2×Dic137—42no—
47C2×Dic137C22⋊C411yescharacteristic
48C2×Dic137C2×D412yes—
49C4×D137—22no—
50C4×D137—24no—
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—21no—
62C2×D137—22no—
63C2×D137—41no—
64C2×D137—42no—
65C4×C137—21no—
66C4×C137—22no—
67C4×C137—41no—
68Dic137—41no—
69Dic137#10012yes—
70Dic137#13514yes—
71C2×C137—81no—
72C2×C137C2×#10011yescharacteristic
73C2×C137#41612yes—
74D137—41no—
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—1371no—
84C4⋊Q8—1371no—
85#127—1371no—
86#124—1371no—
87#124#1058773911yescharacteristic
88#121—2742no—
89C4×D4—2742no—
90C2×C42—1371no—
91C4⋊C8—1374no—
92C22⋊C8—2742no—
93C4×C8—2742no—
94#41—5482no—
95#41—5482no—
96C2×Q8—1371no—
97C2×Q8C2×#1058773911yescharacteristic
98C2×D4—2741no—
99C2×D4—2741no—
100C2×D4—2742no—
101C2×D4—1371no—
102C2×D4C2×#1058773911yescharacteristic
103C22×C4—2741no—
104C22×C4—2741no—
105C22×C4—1372no—
106C22×C4—1371no—
107C2×C8—2744no—
108C4⋊C4—2742no—
109C22⋊C4—22no—
110C22⋊C4—2742no—
111C42—1372no—
112C42—1371no—
113C42C2×#1058773911yescharacteristic
114C23—21no—
115C23—2741no—
116C23—1371no—
117Q8—41no—
118Q8—5481no—
119D4—41no—
120D4—5481no—
121D4—5482no—
122D4—5482no—
123D4—5484no—
124C2×C4—21no—
125C2×C4—2742no—
126C2×C4—2744no—
127C2×C4—2741no—
128C2×C4—5482no—
129C2×C4—5482no—
130C2×C4—1372no—
131C2×C4—1371no—
132C2×C4C22⋊#1058773912yes—
133C2×C4C22×#1058773911yescharacteristic
134C8—5484no—
135C22—41no—
136C22—42no—
137C22—2741no—
138C22—2742no—
139C22—5481no—
140C22—5482no—
141C22C2×C22⋊#1058773911yescharacteristic
142C4—21no—
143C4—22no—
144C4—41no—
145C4—5481no—
146C4—1372no—
147C4—1374no—
148C2—81no—
149C2—5481no—
150C2—10961no—
151C2D137⋊#10011yescharacteristic
152C2#2257839312yes—
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#124×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577952C2
cosocleC4×Dic137C23
hypercenter#124#10587739
hypocenterC137#1543
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. #22578421C2semidirect
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
#225784212
#225782632
#225782612
#225779522
#1543137