Finite Group Databasebeta

← #31796319

Subgroups — #31796319

Order 17536.

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

normal subgroups only

This group has 228 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#22578426C211yesmaximal, characteristic
2#22578426C211yesmaximal, characteristic
3#22578262C214yesmaximal, complemented
4C4⋊Q8×D137C211yesmaximal, characteristic
5#17984155C2212yes—
6#17984155C2212yes—
7#17984145—21no—
8#17984145—21no—
9#17984143—21no—
10#17984143—21no—
11C2×Q8×D137—21no—
12#17984088C411yescharacteristic
13#17984087—21no—
14#17984087C2212yes—
15#17984086—21no—
16C4⋊C4×D137—21no—
17C4⋊C4×D137C412yes—
18C42×D137C2211yescharacteristic
19C4⋊Q8×C137C411yescharacteristic
20C2×#11764047—22no—
21C2×#11764047—22no—
22C2×Dic274—22no—
23C2×Dic274—21no—
24Q8×D137—44no—
25C2×C4×D137—22no—
26C2×C4×D137C2×C411yescharacteristic
27C2×C4×D137D411yescharacteristic
28C2×C4×D137D411yescharacteristic
29C4⋊Dic137—21no—
30C4⋊Dic137C2×C412yes—
31#14555489—22no—
32#14555489—22no—
33C4×Dic137D411yescharacteristic
34C4×Dic137D411yescharacteristic
35C4×Dic137C2311yescharacteristic
36C2×Q8×C137—21no—
37C4⋊C4×C137—21no—
38C4⋊C4×C137C2×C412yes—
39C42×C137C2×C411yescharacteristic
40#11764047—42no—
41#11764047—42no—
42C22×D137C22⋊C411yescharacteristic
43Dic274—42no—
44Dic274—44no—
45C2×Dic137—22no—
46C2×Dic137C22⋊C411yescharacteristic
47C2×Dic137C2×D411yescharacteristic
48C2×Dic137C2×D411yescharacteristic
49C4×D137—22no—
50C4×D137—22no—
51C4×D137—22no—
52C4×D137—44no—
53Q8×C137—42no—
54C2×C4×C137—22no—
55C2×C4×C137C22⋊C411yescharacteristic
56C2×C4×C137C22⋊C411yescharacteristic
57C2×C4×C137C22×C411yescharacteristic
58C22×C137C2×C22⋊C411yescharacteristic
59C2×D137—21no—
60C2×D137—21no—
61C2×D137#10112yes—
62C4×C137—21no—
63C4×C137—21no—
64C4×C137—21no—
65C4×C137—42no—
66Dic137—21no—
67Dic137—42no—
68Dic137AGL(1,ℤ/8)12yes—
69Dic137#13712yes—
70C2×C137C2×#10111yescharacteristic
71C2×C137#41811yescharacteristic
72C2×C137#41911yescharacteristic
73D137—22no—
74C137#155611yescharacteristic, complemented, Sylow
75#1556—1371nomaximal, Sylow
76C2×C4⋊Q8—1371no—
77#432—1371no—
78#432—1371no—
79#332—1374no—
80C22×Q8—2741no—
81C4⋊Q8—2741no—
82C4⋊Q8—2741no—
83C4⋊Q8—1372no—
84C4⋊Q8—1371no—
85C4⋊Q8#1058773911yescharacteristic
86C2×C4⋊C4—2741no—
87C2×C4⋊C4—1372no—
88C2×C42—1371no—
89C4⋊C8—1372no—
90C4⋊C8—1372no—
91C22⋊C8—2741no—
92C22⋊C8—2741no—
93#97—2741no—
94#97—2741no—
95C2×Q8—21no—
96C2×Q8—2742no—
97C2×Q8—2741no—
98C2×Q8—5484no—
99C22×C4—2742no—
100C22×C4—1371no—
101C22×C4—1371no—
102C22×C4—1371no—
103C2×C8—2742no—
104C2×C8—2742no—
105C4⋊C4—21no—
106C4⋊C4—2742no—
107C4⋊C4—2742no—
108C4⋊C4—2741no—
109C4⋊C4—1372no—
110C4⋊C4C2×#1058773912yes—
111C42—1371no—
112C42—1371no—
113C42—1371no—
114C42C2×#1058773911yescharacteristic
115C23—1371no—
116Q8—42no—
117Q8—5482no—
118Q8—5484no—
119C2×C4—22no—
120C2×C4—2742no—
121C2×C4—2742no—
122C2×C4—2742no—
123C2×C4—2742no—
124C2×C4—5484no—
125C2×C4—1371no—
126C2×C4—1371no—
127C2×C4—1371no—
128C2×C4C22⋊#1058773911yescharacteristic
129C2×C4C22⋊#1058773911yescharacteristic
130C2×C4C22×#1058773911yescharacteristic
131C8—5482no—
132C8—5482no—
133C22—2741no—
134C22—2741no—
135C22—1372no—
136C22C2×C22⋊#1058773911yescharacteristic
137C4—21no—
138C4—21no—
139C4—21no—
140C4—42no—
141C4—2741no—
142C4—5482no—
143C4—1372no—
144C4—1372no—
145C2—2742no—
146C2D137⋊#10111yescharacteristic
147C2#2257839511yescharacteristic
148C2#2257839611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D137C22⋊C4
cosocleC4×Dic137C23
hypercenterC4⋊Q8#10587739
hypocenterC137#1556
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22578262C2semidirect
74. C137#1556semidirect

Maximal subgroups

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

subgroupindex
#225784262
#225782622
C4⋊Q8×D1372
#1556137