Finite Group Databasebeta

← #31796183

Subgroups — #31796183

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#22578425C211yesmaximal, characteristic
2#22578339C211yesmaximal, characteristic
3#22578257C214yesmaximal
4#22577958C211yesmaximal, characteristic
5C2×C4×#10587739—21no—
6C4⋊#11764047—21no—
7#17984145C2212yes—
8#17984143—21no—
9(C2×#10587739)⋊C4—21no—
10(C2×#10587739)⋊C4C2212yes—
11#17984088C411yescharacteristic, complemented
12#17984079—21no—
13#17984068—21no—
14#17984064C2212yes—
15Q8×Dic137C412yescomplemented
16#17984053—21no—
17#17984046—21no—
18C42×D137C2211yescharacteristic
19C4⋊Q8×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21C2×#11764047—22no—
22C4×#10587739—24no—
23C2×Dic274—21no—
24C2×C4×D137D411yescharacteristic
25C2×C4×D137D411yescharacteristic
26C2×C4×D137C2311yescharacteristic
27C4⋊Dic137—22no—
28C4⋊Dic137—21no—
29C4⋊Dic137C2×C412yes—
30#14555489—22no—
31#14555487—22no—
32C4×Dic137—22no—
33C4×Dic137C2×C411yescharacteristic
34C4×Dic137D411yescharacteristic
35C4×Dic137D411yescharacteristic
36C2×Q8×C137C2×C412yes—
37C4⋊C4×C137—21no—
38C4⋊C4×C137—21no—
39C42×C137C2×C411yescharacteristic
40C2×#10587739—24no—
41C2×#10587739—42no—
42C2×#10587739—42no—
43#11764047—42no—
44C22×D137C2×D411yescharacteristic
45Dic274—42no—
46C2×Dic137—22no—
47C2×Dic137C22⋊C411yescharacteristic
48C2×Dic137C22⋊C411yescharacteristic
49C2×Dic137C2×D411yescharacteristic
50C4×D137—22no—
51C4×D137—22no—
52C4×D137—41no—
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#13712yes—
62#10587739—44no—
63C4×C137—21no—
64C4×C137—21no—
65C4×C137—42no—
66C4×C137#10412yes—
67Dic137—21no—
68Dic137—21no—
69Dic137—21no—
70Dic137—42no—
71C2×C137#41011yescharacteristic
72C2×C137#41911yescharacteristic
73C2×C137C2×#10411yescharacteristic
74D137—22no—
75C137#150611yescharacteristic, complemented, Sylow
76#1506—1371nomaximal, Sylow
77#533—1371no—
78#432—1371no—
79#376—1371no—
80#328—1374no—
81C4⋊Q8—1371no—
82C4⋊Q8#1058773911yescharacteristic, complemented
83#125—2741no—
84C22⋊Q8—2741no—
85C22⋊Q8—1372no—
86C4×Q8—2741no—
87C4×Q8—1372no—
88#117—2741no—
89C2×C42—2741no—
90C2×C42—1371no—
91C4⋊C8—2741no—
92C22⋊C8—1372no—
93#97—2741no—
94#95—2741no—
95#95—1372no—
96C2×Q8—2741no—
97C2×Q8C2×#1058773912yes—
98C22×C4—2742no—
99C22×C4—1371no—
100C22×C4—1371no—
101C22×C4—1371no—
102C2×C8—2742no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—2742no—
106C4⋊C4—2742no—
107C4⋊C4—2741no—
108C4⋊C4—1372no—
109C22⋊C4—2742no—
110C42—2742no—
111C42—2744no—
112C42—1371no—
113C42—1371no—
114C42—1371no—
115C42C2×#1058773911yescharacteristic
116C23—1371no—
117Q8—42no—
118Q8—5482no—
119C2×C4—22no—
120C2×C4—2742no—
121C2×C4—2742no—
122C2×C4—2744no—
123C2×C4—2742no—
124C2×C4—5481no—
125C2×C4—5482no—
126C2×C4—5482no—
127C2×C4—1371no—
128C2×C4—1371no—
129C2×C4—1371no—
130C2×C4C22⋊#1058773911yescharacteristic
131C2×C4C22⋊#1058773911yescharacteristic
132C2×C4C22×#1058773911yescharacteristic
133C8—5482no—
134C22—2741no—
135C22—2741no—
136C22—1372no—
137C22C2×C22⋊#1058773911yescharacteristic
138C4—21no—
139C4—21no—
140C4—42no—
141C4—2741no—
142C4—2741no—
143C4—2741no—
144C4—5484no—
145C4—5482no—
146C4#1798415312yes—
147C2—2742no—
148C2#2257838311yescharacteristic
149C2#2257839611yescharacteristic
150C2C2×#1798415311yescharacteristic
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#1506
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
15. Q8×Dic137C4semidirect
19. C4⋊Q8×C137C4semidirect
75. C137#1506semidirect
82. C4⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784252
#225783392
#225782572
#225779582
#1506137