Finite Group Databasebeta

← #31796212

Subgroups — #31796212

Order 17536.

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

normal subgroups only

This group has 208 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
2#22578348C211yesmaximal, characteristic
3#22578256C214yesmaximal
4#22577958C211yesmaximal, characteristic
5#17984155—21no—
6#17984145C2212yes—
7#17984143—21no—
8(C2×#10587739)⋊C4—21no—
9(C2×#10587739)⋊C4—21no—
10(C2×#10587739)⋊C4C2212yes—
11#17984088C411yescharacteristic, complemented
12#17984079—21no—
13#17984068C2212yes—
14#17984064—21no—
15Q8×Dic137—21no—
16#17984053C412yescomplemented
17#17984046—21no—
18C42×D137C2211yescharacteristic
19C4⋊Q8×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21C2×#11764047—22no—
22C2×Dic274C2×C412yes—
23C2×C4×D137D411yescharacteristic
24C2×C4×D137D411yescharacteristic
25C2×C4×D137C2311yescharacteristic
26C4⋊Dic137—22no—
27C4⋊Dic137—21no—
28C4⋊Dic137—21no—
29#14555489—22no—
30#14555487—22no—
31C4×Dic137—22no—
32C4×Dic137C2×C411yescharacteristic
33C4×Dic137D411yescharacteristic
34C4×Dic137D411yescharacteristic
35C2×Q8×C137—21no—
36C4⋊C4×C137—21no—
37C4⋊C4×C137C2×C412yes—
38C42×C137C2×C411yescharacteristic
39C2×#10587739—42no—
40C2×#10587739—42no—
41C2×#10587739—42no—
42#11764047—42no—
43C22×D137C2×D411yescharacteristic
44Dic274—42no—
45C2×Dic137—22no—
46C2×Dic137C22⋊C411yescharacteristic
47C2×Dic137C22⋊C411yescharacteristic
48C2×Dic137C2×D411yescharacteristic
49C4×D137—22no—
50C4×D137—41no—
51C4×D137—41no—
52Q8×C137—42no—
53C2×C4×C137—22no—
54C2×C4×C137C22⋊C411yescharacteristic
55C2×C4×C137C22⋊C411yescharacteristic
56C2×C4×C137C22×C411yescharacteristic
57C22×C137C2×C22⋊C411yescharacteristic
58C2×D137—21no—
59C2×D137—21no—
60C2×D137#13712yes—
61#10587739—82no—
62C4×C137—21no—
63C4×C137—21no—
64C4×C137—21no—
65C4×C137—42no—
66Dic137—21no—
67Dic137—21no—
68Dic137—21no—
69Dic137—42no—
70C2×C137#41011yescharacteristic
71C2×C137#41911yescharacteristic
72C2×C137#42111yescharacteristic
73D137—22no—
74C137#151511yescharacteristic, complemented, Sylow
75#1515—1371nomaximal, Sylow
76#533—1371no—
77#432—1371no—
78#383—1371no—
79#328—1374no—
80C4⋊Q8—1371no—
81C4⋊Q8#1058773911yescharacteristic, complemented
82#125—2741no—
83C22⋊Q8—2741no—
84C22⋊Q8—1372no—
85C4×Q8—2741no—
86C4×Q8—1372no—
87#117—2741no—
88C2×C42—1371no—
89C4⋊C8—2741no—
90C22⋊C8—1372no—
91#97—2741no—
92#95—2741no—
93#95—2741no—
94#95—1372no—
95C2×Q8—21no—
96C2×Q8—1372no—
97C22×C4—2742no—
98C22×C4—1371no—
99C22×C4—1371no—
100C22×C4—1371no—
101C2×C8—2742no—
102C4⋊C4—21no—
103C4⋊C4—2742no—
104C4⋊C4—2742no—
105C4⋊C4—2741no—
106C4⋊C4—2741no—
107C4⋊C4C2×#1058773912yes—
108C22⋊C4—2742no—
109C42—2742no—
110C42—1371no—
111C42—1371no—
112C42—1371no—
113C42C2×#1058773911yescharacteristic
114C23—1371no—
115Q8—42no—
116Q8—5482no—
117C2×C4—22no—
118C2×C4—2742no—
119C2×C4—2742no—
120C2×C4—5481no—
121C2×C4—5481no—
122C2×C4—5482no—
123C2×C4—5482no—
124C2×C4—5482no—
125C2×C4—1371no—
126C2×C4—1371no—
127C2×C4—1371no—
128C2×C4C22⋊#1058773911yescharacteristic
129C2×C4C22⋊#1058773911yescharacteristic
130C2×C4C22×#1058773911yescharacteristic
131C8—5482no—
132C22—2741no—
133C22—2741no—
134C22—1372no—
135C22C2×C22⋊#1058773911yescharacteristic
136C4—21no—
137C4—21no—
138C4—21no—
139C4—42no—
140C4—2741no—
141C4—2741no—
142C4—2741no—
143C4—5482no—
144C4—10962no—
145C2—2742no—
146C2#2257838411yescharacteristic
147C2#2257839611yescharacteristic
148C2#2257840111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D137C2×D4
cosocleC2×C4×D137C23
hypercenterC4⋊Q8#10587739
hypocenterC137#1515
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
11. #17984088C4semidirect
16. #17984053C4semidirect
19. C4⋊Q8×C137C4semidirect
74. C137#1515semidirect
81. C4⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784262
#225783482
#225782562
#225779582
#1515137