Finite Group Databasebeta

← #31796243

Subgroups — #31796243

Order 17536.

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

normal subgroups only

This group has 270 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
1C4×#14555519C212yesmaximal, complemented
2#22578259C212yesmaximal, complemented
3#22578258C212yesmaximal, complemented
4#22577949C211yesmaximal, characteristic
5C2×#14555519—22no—
6#17984143C2214yes—
7C4×#11764047—22no—
8C2×#14555510C411yescharacteristic
9#17984086C411yescharacteristic
10#17984084C411yescharacteristic
11#17984077C2211yescharacteristic
12#17984075C2211yescharacteristic
13#17984060—22no—
14C4×C22⋊D137—22no—
15C2×C4×Dic137C2211yescharacteristic
16C2×#41×C137C411yescharacteristic
17#14555519—28no—
18C2×#11764047—24no—
19#14555510—42no—
20C2×Dic274C2×C411yescharacteristic
21C2×D548C2×C411yescharacteristic
22C2×C22⋊D137—22no—
23C2×C22⋊D137—21no—
24C22×Dic137D412yes—
25C2×C4×D137—21no—
26#14555489—22no—
27C22⋊Dic137—22no—
28#14555487—22no—
29C4×Dic137C2×C411yescharacteristic
30C4×Dic137D412yes—
31C4×Dic137C2311yescharacteristic
32#41×C137—42no—
33C2×Q8×C137C2×C411yescharacteristic
34C2×D4×C137—21no—
35C2×D4×C137C2×C411yescharacteristic
36C22×C4×C137—21no—
37C22×C4×C137C2×C411yescharacteristic
38#11764047—28no—
39C22×D137—21no—
40Dic274—41no—
41D548—41no—
42C22⋊D137—42no—
43C22⋊D137—44no—
44C2×Dic137—24no—
45C2×Dic137—21no—
46C2×Dic137C22⋊C412yes—
47C2×Dic137C2×D412yes—
48C4×D137—42no—
49C23×C137—21no—
50C23×C137C22⋊C411yescharacteristic
51Q8×C137—41no—
52D4×C137—41no—
53D4×C137—42no—
54C2×C4×C137—22no—
55C2×C4×C137—21no—
56C2×C4×C137—42no—
57C2×C4×C137C22⋊C411yescharacteristic
58C2×C4×C137C22×C411yescharacteristic
59C22×C137—21no—
60C22×C137—22no—
61C22×C137—41no—
62C22×C137—42no—
63C22×C137C2×C22⋊C411yescharacteristic
64C2×D137—41no—
65C2×D137—42no—
66C4×C137—21no—
67C4×C137—41no—
68C4×C137#10012yes—
69Dic137—22no—
70Dic137—41no—
71Dic137#10414yes—
72C2×C137—41no—
73C2×C137—81no—
74C2×C137C2×#10011yescharacteristic
75C2×C137C2×#10412yes—
76D137—81no—
77C137#152511yescharacteristic, complemented, Sylow
78#1525—1371nomaximal, Sylow
79#532—1371no—
80C4×M16—1372no—
81#329—1372no—
82#329—1372no—
83C2×#41—1371no—
84C2×#41#1058773911yescharacteristic
85C2×M16—2742no—
86C4⋊Q8—1371no—
87#127—1371no—
88#124—1371no—
89#124—1371no—
90#121—2742no—
91C4×D4—2742no—
92C2×C42—1371no—
93#97—1374no—
94C4×C8—2742no—
95#41—42no—
96#41—5482no—
97C2×Q8—1371no—
98C2×Q8C2×#1058773911yescharacteristic
99C2×D4—21no—
100C2×D4—2741no—
101C2×D4—2742no—
102C2×D4—1371no—
103C2×D4C2×#1058773911yescharacteristic
104C22×C4—21no—
105C22×C4—2741no—
106C22×C4—1372no—
107C22×C4C2×#1058773911yescharacteristic
108M16—2748no—
109C2×C8—2744no—
110C4⋊C4—2742no—
111C22⋊C4—2742no—
112C22⋊C4—2742no—
113C42—1372no—
114C42—1371no—
115C42—1371no—
116C23—21no—
117C23—2741no—
118C23C22⋊#1058773911yescharacteristic
119Q8—41no—
120Q8—5481no—
121D4—41no—
122D4—42no—
123D4—5481no—
124D4—5482no—
125D4—5484no—
126C2×C4—22no—
127C2×C4—21no—
128C2×C4—42no—
129C2×C4—2744no—
130C2×C4—2741no—
131C2×C4—5482no—
132C2×C4—1372no—
133C2×C4—1372no—
134C2×C4C22⋊#1058773911yescharacteristic
135C2×C4C22×#1058773911yescharacteristic
136C8—2748no—
137C22—21no—
138C22—22no—
139C22—41no—
140C22—42no—
141C22—5481no—
142C22—5482no—
143C22C2×C22⋊#1058773911yescharacteristic
144C4—21no—
145C4—41no—
146C4—2742no—
147C4—5481no—
148C4—1374no—
149C4#1798414812yes—
150C2—41no—
151C2—81no—
152C2—10961no—
153C2C2×#1798414811yescharacteristic
154C2D137⋊#10412yes—
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#1525
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. C4×#14555519C2semidirect
2. #22578259C2semidirect
3. #22578258C2semidirect
77. C137#1525semidirect

Maximal subgroups

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

subgroupindex
C4×#145555192
#225782592
#225782582
#225779492
#1525137