Finite Group Databasebeta

← #31796327

Subgroups — #31796327

Order 17536.

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

normal subgroups only

This group has 254 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#22578408C212yesmaximal, complemented
2#22578263C212yesmaximal, complemented
3#22578261C212yesmaximal, complemented
4#22577952C211yesmaximal, characteristic
5C2×#14555520—22no—
6#17984155—22no—
7#17984155C2214yes—
8C2×#14555511C411yescharacteristic
9C2×#14555509C411yescharacteristic
10#17984086C2211yescharacteristic
11#17984084C2211yescharacteristic
12#17984076C411yescharacteristic
13#17984061—22no—
14#17984048—22no—
15C42×D137C2211yescharacteristic
16#124×C137C411yescharacteristic
17#14555520—44no—
18C2×#11764047—24no—
19#14555511—42no—
20#14555509—42no—
21C2×Dic274C2×C411yescharacteristic
22C2×D548—21no—
23C2×D548C2×C411yescharacteristic
24C2×C22⋊D137—21no—
25C2×C22⋊D137—22no—
26C22×Dic137—21no—
27C2×C4×D137—21no—
28C2×C4×D137C2×C411yescharacteristic
29C2×C4×D137D412yes—
30#14555489—22no—
31#14555487—22no—
32C4×Dic137D412yes—
33C4×Dic137C2311yescharacteristic
34C2×Q8×C137C2×C411yescharacteristic
35C2×D4×C137C2×C411yescharacteristic
36C22⋊C4×C137—22no—
37C42×C137C2×C411yescharacteristic
38#11764047—44no—
39C22×D137—21no—
40C22×D137C22⋊C411yescharacteristic
41Dic274—41no—
42D548—41no—
43D548—42no—
44C22⋊D137—42no—
45C22⋊D137—42no—
46C22⋊D137—42no—
47C2×Dic137—21no—
48C2×Dic137—42no—
49C2×Dic137C22⋊C411yescharacteristic
50C2×Dic137C2×D412yes—
51C4×D137—22no—
52C4×D137—22no—
53C4×D137—22no—
54C4×D137—42no—
55C23×C137—21no—
56Q8×C137—41no—
57D4×C137—41no—
58C2×C4×C137—21no—
59C2×C4×C137C22⋊C412yes—
60C2×C4×C137C22×C411yescharacteristic
61C22×C137—41no—
62C22×C137—42no—
63C22×C137C2×C22⋊C411yescharacteristic
64C2×D137—21no—
65C2×D137—22no—
66C2×D137—41no—
67C2×D137—42no—
68C4×C137—21no—
69C4×C137—22no—
70C4×C137—41no—
71Dic137—41no—
72Dic137#10012yes—
73Dic137AGL(1,ℤ/8)12yes—
74Dic137#13712yes—
75C2×C137—81no—
76C2×C137C2×#10011yescharacteristic
77C2×C137#41712yes—
78D137—41no—
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—1371no—
88C2×M16—2742no—
89C4⋊Q8—1371no—
90#127—1371no—
91#124—1371no—
92#124#1058773911yescharacteristic
93#121—2742no—
94C4×D4—2742no—
95C2×C42—1371no—
96C4⋊C8—2742no—
97C4⋊C8—1374no—
98#41—5482no—
99#41—5482no—
100C2×Q8—1371no—
101C2×Q8C2×#1058773911yescharacteristic
102C2×D4—2741no—
103C2×D4—2741no—
104C2×D4—2742no—
105C2×D4—1371no—
106C2×D4C2×#1058773911yescharacteristic
107C22×C4—2741no—
108C22×C4—2741no—
109C22×C4—1372no—
110C22×C4—1371no—
111M16—5484no—
112C2×C8—2744no—
113C4⋊C4—2742no—
114C22⋊C4—22no—
115C22⋊C4—2742no—
116C42—1372no—
117C42—1371no—
118C42C2×#1058773911yescharacteristic
119C23—21no—
120C23—2741no—
121C23—1371no—
122Q8—41no—
123Q8—5481no—
124D4—41no—
125D4—5481no—
126D4—5482no—
127D4—5482no—
128D4—5482no—
129D4—5482no—
130C2×C4—21no—
131C2×C4—2742no—
132C2×C4—2742no—
133C2×C4—2742no—
134C2×C4—2741no—
135C2×C4—5482no—
136C2×C4—5482no—
137C2×C4—1372no—
138C2×C4—1371no—
139C2×C4C22⋊#1058773912yes—
140C2×C4C22×#1058773911yescharacteristic
141C8—5484no—
142C22—41no—
143C22—42no—
144C22—2741no—
145C22—2742no—
146C22—5481no—
147C22—5482no—
148C22C2×C22⋊#1058773911yescharacteristic
149C4—21no—
150C4—22no—
151C4—41no—
152C4—5481no—
153C4—1372no—
154C4—1372no—
155C4—1372no—
156C2—81no—
157C2—5481no—
158C2—10961no—
159C2D137⋊#10011yescharacteristic
160C2#2257839412yes—
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#1560
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. #22578408C2semidirect
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
#225784082
#225782632
#225782612
#225779522
#1560137