Finite Group Databasebeta

← #31796298

Subgroups — #31796298

Order 17536.

240 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 72 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#22578420C211yesmaximal, characteristic
2#22578406C211yesmaximal, characteristic
3#22578262C214yesmaximal, complemented
4C2×#17984087C211yesmaximal, characteristic
5C2×#14555519—21no—
6#17984155C2212yes—
7#17984155C2212yes—
8C4⋊#11764047—21no—
9C22⋊#11764047—21no—
10C4×#11764047—21no—
11C22×Dic274—21no—
12#17984087—21no—
13#17984087—21no—
14#17984087C412yes—
15#17984087C2212yes—
16C2×C4⋊Dic137C411yescharacteristic
17C2×#14555489—21no—
18C2×C4×Dic137C2211yescharacteristic
19C2×C4⋊C4×C137C411yescharacteristic
20#14555519—42no—
21C2×#11764047—22no—
22C2×#11764047—22no—
23C2×Dic274—22no—
24C2×Dic274—22no—
25C2×Dic274—42no—
26C2×Dic274—42no—
27C22×Dic137—21no—
28C22×Dic137D411yescharacteristic
29C22×Dic137D411yescharacteristic
30C4⋊Dic137—21no—
31C4⋊Dic137C2×C412yes—
32#14555489—22no—
33#14555489—22no—
34C4×Dic137C2×C411yescharacteristic
35C4×Dic137D411yescharacteristic
36C4×Dic137D411yescharacteristic
37C4×Dic137C2311yescharacteristic
38C22×C4×C137—21no—
39C22×C4×C137C2×C411yescharacteristic
40C4⋊C4×C137—21no—
41C4⋊C4×C137C2×C412yes—
42#11764047—42no—
43#11764047—42no—
44Dic274—44no—
45Dic274—42no—
46Dic274—42no—
47C2×Dic137—22no—
48C2×Dic137—22no—
49C2×Dic137—42no—
50C2×Dic137C22⋊C411yescharacteristic
51C2×Dic137C22⋊C411yescharacteristic
52C2×Dic137D812yes—
53C2×Dic137SD1612yes—
54C2×Dic137C2×D411yescharacteristic
55C2×Dic137C2×D411yescharacteristic
56C23×C137C22⋊C411yescharacteristic
57C2×C4×C137—22no—
58C2×C4×C137—22no—
59C2×C4×C137—42no—
60C2×C4×C137C22⋊C411yescharacteristic
61C2×C4×C137C22×C411yescharacteristic
62C22×C137—21no—
63C22×C137—21no—
64C22×C137#10212yes—
65C22×C137C2×C22⋊C411yescharacteristic
66C4×C137—21no—
67C4×C137—42no—
68C4×C137#10112yes—
69Dic137—21no—
70Dic137—42no—
71Dic137#10212yes—
72Dic137C2×D811yescharacteristic
73Dic137C2×SD1611yescharacteristic
74Dic137#13712yes—
75C2×C137—22no—
76C2×C137C2×#10111yescharacteristic
77C2×C137C2×#10211yescharacteristic
78C2×C137#41911yescharacteristic
79C137#154611yescharacteristic, complemented, Sylow
80#1546—1371nomaximal, Sylow
81C2×C4⋊Q8—1371no—
82#431—1371no—
83#423—1371no—
84#332—1374no—
85C22×Q8—2741no—
86C2×M16—2741no—
87C4⋊Q8—2741no—
88C4⋊Q8—2741no—
89C4⋊Q8—1372no—
90C4⋊Q8—1372no—
91C2×C4⋊C4—2741no—
92C2×C4⋊C4—1371no—
93C2×C4⋊C4#1058773911yescharacteristic
94C2×C42—1371no—
95C4⋊C8—2741no—
96C4⋊C8—1372no—
97C4⋊C8—1372no—
98C22⋊C8—2741no—
99C4×C8—2741no—
100C2×Q8—2742no—
101C2×Q8—2742no—
102C2×Q8—5482no—
103C2×Q8—5482no—
104C22×C4—21no—
105C22×C4—2741no—
106C22×C4—1371no—
107C22×C4—1371no—
108C22×C4C2×#1058773911yescharacteristic
109M16—5482no—
110C2×C8—2742no—
111C2×C8—2742no—
112C4⋊C4—21no—
113C4⋊C4—2742no—
114C4⋊C4—2742no—
115C4⋊C4—2741no—
116C4⋊C4—1372no—
117C4⋊C4C2×#1058773912yes—
118C42—1371no—
119C42—1371no—
120C42—1371no—
121C42—1371no—
122C23C22⋊#1058773911yescharacteristic
123Q8—5482no—
124Q8—5484no—
125Q8—5482no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—42no—
129C2×C4—2742no—
130C2×C4—2742no—
131C2×C4—5482no—
132C2×C4—1371no—
133C2×C4—1372no—
134C2×C4—1372no—
135C2×C4—1371no—
136C2×C4—1371no—
137C2×C4—1371no—
138C2×C4C22⋊#1058773911yescharacteristic
139C2×C4C22×#1058773911yescharacteristic
140C8—5482no—
141C8—5482no—
142C22—21no—
143C22—21no—
144C22#1798415012yes—
145C22C2×C22⋊#1058773911yescharacteristic
146C4—21no—
147C4—42no—
148C4—2741no—
149C4—5482no—
150C4—1371no—
151C4—1372no—
152C4—1371no—
153C4—1372no—
154C4#1798414912yes—
155C2—22no—
156C2C2×#1798414911yescharacteristic
157C2C2×#1798415011yescharacteristic
158C2#2257839611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC2×C4⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC23×C137C22⋊C4
cosocleC4×Dic137C23
hypercenterC2×C4⋊C4#10587739
hypocenterC137#1546
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #22578262C2semidirect
79. C137#1546semidirect

Maximal subgroups

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

subgroupindex
#225784202
#225784062
#225782622
C2×#179840872
#1546137