Finite Group Databasebeta

← #31796209

Subgroups — #31796209

Order 17536.

252 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 56 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#22578426C211yesmaximal, characteristic, complemented
2#22578348C211yesmaximal, characteristic, complemented
3#22578254C214yesmaximal, complemented
4#22577951C211yesmaximal, characteristic
5#17984155—21no—
6#17984145C2212yes—
7#17984143—21no—
8(C2×#10587739)⋊C4—21no—
9(C2×#10587739)⋊C4—21no—
10(C2×#10587739)⋊C4C2212yes—
11C2×#14555509—21no—
12#17984088C411yescharacteristic, complemented
13#17984075—21no—
14#17984059—21no—
15#17984059C2212yes—
16D4×Dic137—21no—
17D4×Dic137C412yescomplemented
18C42×D137C2211yescharacteristic
19#127×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21C2×#11764047—22no—
22#14555509—44no—
23C2×Dic274—21no—
24C2×C22⋊D137—22no—
25C22×Dic137—22no—
26C2×C4×D137D411yescharacteristic
27C2×C4×D137D411yescharacteristic
28C2×C4×D137C2311yescharacteristic
29C4⋊Dic137—21no—
30C4⋊Dic137C2×C412yes—
31C22⋊Dic137—22no—
32C22⋊Dic137—22no—
33C4×Dic137C2×C411yescharacteristic
34C4×Dic137D411yescharacteristic
35C4×Dic137D411yescharacteristic
36C2×D4×C137—21no—
37C2×D4×C137—21no—
38C2×D4×C137C2×C412yes—
39C42×C137C2×C411yescharacteristic
40C2×#10587739—42no—
41C2×#10587739—42no—
42C2×#10587739—42no—
43#11764047—42no—
44C22×D137C2×D411yescharacteristic
45Dic274—42no—
46C22⋊D137—44no—
47C2×Dic137—22no—
48C2×Dic137—44no—
49C2×Dic137C22⋊C411yescharacteristic
50C2×Dic137C22⋊C411yescharacteristic
51C2×Dic137C2×D411yescharacteristic
52C4×D137—22no—
53C4×D137—41no—
54C4×D137—41no—
55C23×C137—22no—
56D4×C137—42no—
57D4×C137—42no—
58D4×C137—42no—
59C2×C4×C137C22⋊C411yescharacteristic
60C2×C4×C137C22⋊C411yescharacteristic
61C2×C4×C137C22×C411yescharacteristic
62C22×C137—42no—
63C22×C137—42no—
64C22×C137—42no—
65C22×C137C2×C22⋊C411yescharacteristic
66C2×D137—21no—
67C2×D137—21no—
68C2×D137AGL(1,ℤ/8)12yes—
69#10587739—82no—
70C4×C137—21no—
71C4×C137—21no—
72C4×C137—21no—
73Dic137—21no—
74Dic137—21no—
75Dic137—21no—
76Dic137—42no—
77C2×C137—82no—
78C2×C137#41011yescharacteristic
79C2×C137#41811yescharacteristic
80C2×C137#42111yescharacteristic
81D137—22no—
82C137#151411yescharacteristic, complemented, Sylow
83#1514—1371nomaximal, Sylow
84#532—1371no—
85#432—1371no—
86#383—1371no—
87#327—1374no—
88C2×#41—2741no—
89C4⋊Q8—1371no—
90#127#1058773911yescharacteristic, complemented
91#124—2741no—
92#121—2741no—
93#121—1372no—
94C4×D4—2741no—
95C4×D4—1372no—
96C2×C42—1371no—
97C4⋊C8—2741no—
98C22⋊C8—1372no—
99#97—2741no—
100#95—2741no—
101#95—2741no—
102#95—1372no—
103#41—5484no—
104C2×Q8—2741no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—2742no—
108C2×D4C2×#1058773912yes—
109C22×C4—2742no—
110C22×C4—2742no—
111C22×C4—1371no—
112C22×C4—1371no—
113C22×C4—1371no—
114C2×C8—2742no—
115C4⋊C4—2741no—
116C4⋊C4—1372no—
117C22⋊C4—2742no—
118C22⋊C4—2742no—
119C42—1371no—
120C42—1371no—
121C42—1371no—
122C42C2×#1058773911yescharacteristic
123C23—22no—
124C23—1371no—
125Q8—5482no—
126D4—42no—
127D4—42no—
128D4—42no—
129D4—5484no—
130C2×C4—2742no—
131C2×C4—2742no—
132C2×C4—5481no—
133C2×C4—5481no—
134C2×C4—5484no—
135C2×C4—5482no—
136C2×C4—5482no—
137C2×C4—5482no—
138C2×C4—1371no—
139C2×C4—1371no—
140C2×C4—1371no—
141C2×C4C22⋊#1058773911yescharacteristic
142C2×C4C22⋊#1058773911yescharacteristic
143C2×C4C22×#1058773911yescharacteristic
144C8—5482no—
145C22—42no—
146C22—42no—
147C22—42no—
148C22—2741no—
149C22—2741no—
150C22—1372no—
151C22C2×C22⋊#1058773911yescharacteristic
152C4—21no—
153C4—21no—
154C4—21no—
155C4—2741no—
156C4—2741no—
157C4—2741no—
158C4—5482no—
159C4—10962no—
160C2—82no—
161C2—2742no—
162C2#2257838311yescharacteristic
163C2#2257839511yescharacteristic
164C2#2257840111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#127×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577951C2
cosocleC2×C4×D137C23
hypercenter#127#10587739
hypocenterC137#1514
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #22578426C2semidirect
2. #22578348C2semidirect
3. #22578254C2semidirect
12. #17984088C4semidirect
17. D4×Dic137C4semidirect
19. #127×C137C4semidirect
82. C137#1514semidirect
90. #127#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784262
#225783482
#225782542
#225779512
#1514137