Finite Group Databasebeta

← #31796326

Subgroups — #31796326

Order 17536.

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

normal subgroups only

This group has 254 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#22578406C212yesmaximal, complemented
2#22578263C212yesmaximal, complemented
3#22578261C212yesmaximal, complemented
4#22577949C211yesmaximal, characteristic
5C2×#14555519—22no—
6#17984155C2214yes—
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
17#14555519—44no—
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—44no—
39C22×D137—21no—
40Dic274—41no—
41D548—41no—
42C22⋊D137—42no—
43C22⋊D137—42no—
44C22⋊D137—42no—
45C2×Dic137—22no—
46C2×Dic137—22no—
47C2×Dic137—21no—
48C2×Dic137C22⋊C412yes—
49C2×Dic137C2×D412yes—
50C4×D137—42no—
51C23×C137—21no—
52C23×C137C22⋊C411yescharacteristic
53Q8×C137—41no—
54D4×C137—41no—
55D4×C137—42no—
56C2×C4×C137—22no—
57C2×C4×C137—21no—
58C2×C4×C137—42no—
59C2×C4×C137C22⋊C411yescharacteristic
60C2×C4×C137C22×C411yescharacteristic
61C22×C137—21no—
62C22×C137—22no—
63C22×C137—41no—
64C22×C137—42no—
65C22×C137C2×C22⋊C411yescharacteristic
66C2×D137—41no—
67C2×D137—42no—
68C4×C137—21no—
69C4×C137—41no—
70C4×C137#10012yes—
71Dic137—22no—
72Dic137—41no—
73Dic137AGL(1,ℤ/8)12yes—
74Dic137#13712yes—
75C2×C137—41no—
76C2×C137—81no—
77C2×C137C2×#10011yescharacteristic
78C2×C137#41712yes—
79D137—81no—
80C137#156011yescharacteristic, complemented, Sylow
81#1560—1371nomaximal, Sylow
82#532—1371no—
83#423—1372no—
84#333—1372no—
85#331—1372no—
86C2×#41—1371no—
87C2×#41#1058773911yescharacteristic
88C2×M16—2742no—
89C4⋊Q8—1371no—
90#127—1371no—
91#124—1371no—
92#124—1371no—
93#121—2742no—
94C4×D4—2742no—
95C2×C42—1371no—
96C4⋊C8—2742no—
97C4⋊C8—1374no—
98#41—42no—
99#41—5482no—
100C2×Q8—1371no—
101C2×Q8C2×#1058773911yescharacteristic
102C2×D4—21no—
103C2×D4—2741no—
104C2×D4—2742no—
105C2×D4—1371no—
106C2×D4C2×#1058773911yescharacteristic
107C22×C4—21no—
108C22×C4—2741no—
109C22×C4—1372no—
110C22×C4C2×#1058773911yescharacteristic
111M16—5484no—
112C2×C8—2744no—
113C4⋊C4—2742no—
114C22⋊C4—2742no—
115C22⋊C4—2742no—
116C42—1372no—
117C42—1371no—
118C42—1371no—
119C23—21no—
120C23—2741no—
121C23C22⋊#1058773911yescharacteristic
122Q8—41no—
123Q8—5481no—
124D4—41no—
125D4—42no—
126D4—5481no—
127D4—5482no—
128D4—5482no—
129D4—5482no—
130C2×C4—22no—
131C2×C4—21no—
132C2×C4—42no—
133C2×C4—2742no—
134C2×C4—2742no—
135C2×C4—2741no—
136C2×C4—5482no—
137C2×C4—1372no—
138C2×C4—1372no—
139C2×C4C22⋊#1058773911yescharacteristic
140C2×C4C22×#1058773911yescharacteristic
141C8—5484no—
142C22—21no—
143C22—22no—
144C22—41no—
145C22—42no—
146C22—5481no—
147C22—5482no—
148C22C2×C22⋊#1058773911yescharacteristic
149C4—21no—
150C4—41no—
151C4—2742no—
152C4—5481no—
153C4—1372no—
154C4—1372no—
155C4#1798414812yes—
156C2—41no—
157C2—81no—
158C2—10961no—
159C2C2×#1798414811yescharacteristic
160C2#2257839412yes—
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC2×#41×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577949C2
cosocleC4×Dic137C23
hypercenterC2×#41#10587739
hypocenterC137#1560
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #22578406C2semidirect
2. #22578263C2semidirect
3. #22578261C2semidirect
80. C137#1560semidirect

Maximal subgroups

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

subgroupindex
#225784062
#225782632
#225782612
#225779492
#1560137