Finite Group Databasebeta

← #31796304

Subgroups — #31796304

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—44no—
26C22×Dic137—21no—
27C22×Dic137D411yescharacteristic
28C22×Dic137D411yescharacteristic
29C4⋊Dic137—21no—
30C4⋊Dic137C2×C412yes—
31#14555489—22no—
32#14555489—22no—
33C4×Dic137C2×C411yescharacteristic
34C4×Dic137D411yescharacteristic
35C4×Dic137D411yescharacteristic
36C4×Dic137C2311yescharacteristic
37C22×C4×C137—21no—
38C22×C4×C137C2×C411yescharacteristic
39C4⋊C4×C137—21no—
40C4⋊C4×C137C2×C412yes—
41#11764047—42no—
42#11764047—42no—
43Dic274—44no—
44Dic274—44no—
45C2×Dic137—22no—
46C2×Dic137—22no—
47C2×Dic137—42no—
48C2×Dic137C22⋊C411yescharacteristic
49C2×Dic137C22⋊C411yescharacteristic
50C2×Dic137SD1612yes—
51C2×Dic137Q1612yes—
52C2×Dic137C2×D411yescharacteristic
53C2×Dic137C2×D411yescharacteristic
54C23×C137C22⋊C411yescharacteristic
55C2×C4×C137—22no—
56C2×C4×C137—22no—
57C2×C4×C137—42no—
58C2×C4×C137C22⋊C411yescharacteristic
59C2×C4×C137C22×C411yescharacteristic
60C22×C137—21no—
61C22×C137—21no—
62C22×C137#10312yes—
63C22×C137C2×C22⋊C411yescharacteristic
64C4×C137—21no—
65C4×C137—42no—
66C4×C137#10112yes—
67Dic137—21no—
68Dic137—42no—
69Dic137#10312yes—
70Dic137C2×SD1611yescharacteristic
71Dic137C2×Q1611yescharacteristic
72Dic137AGL(1,ℤ/8)12yes—
73C2×C137—22no—
74C2×C137C2×#10111yescharacteristic
75C2×C137C2×#10311yescharacteristic
76C2×C137#41811yescharacteristic
77C137#154911yescharacteristic, complemented, Sylow
78#1549—1371nomaximal, Sylow
79C2×C4⋊Q8—1371no—
80#431—1371no—
81#423—1371no—
82#332—1374no—
83C22×Q8—2741no—
84C2×M16—2741no—
85C4⋊Q8—2741no—
86C4⋊Q8—2741no—
87C4⋊Q8—1372no—
88C4⋊Q8—1372no—
89C2×C4⋊C4—2741no—
90C2×C4⋊C4—1371no—
91C2×C4⋊C4#1058773911yescharacteristic
92C2×C42—1371no—
93C4⋊C8—2741no—
94C4⋊C8—1372no—
95C4⋊C8—1372no—
96C22⋊C8—2741no—
97C4×C8—2741no—
98C2×Q8—2742no—
99C2×Q8—2742no—
100C2×Q8—5484no—
101C22×C4—21no—
102C22×C4—2741no—
103C22×C4—1371no—
104C22×C4—1371no—
105C22×C4C2×#1058773911yescharacteristic
106M16—5482no—
107C2×C8—2742no—
108C2×C8—2742no—
109C4⋊C4—21no—
110C4⋊C4—2742no—
111C4⋊C4—2742no—
112C4⋊C4—2741no—
113C4⋊C4—1372no—
114C4⋊C4C2×#1058773912yes—
115C42—1371no—
116C42—1371no—
117C42—1371no—
118C42—1371no—
119C23C22⋊#1058773911yescharacteristic
120Q8—5484no—
121Q8—5484no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—42no—
125C2×C4—2742no—
126C2×C4—2742no—
127C2×C4—5482no—
128C2×C4—1371no—
129C2×C4—1372no—
130C2×C4—1372no—
131C2×C4—1371no—
132C2×C4—1371no—
133C2×C4—1371no—
134C2×C4C22⋊#1058773911yescharacteristic
135C2×C4C22×#1058773911yescharacteristic
136C8—5482no—
137C8—5482no—
138C22—21no—
139C22—21no—
140C22#1798415112yes—
141C22C2×C22⋊#1058773911yescharacteristic
142C4—21no—
143C4—42no—
144C4—2741no—
145C4—5482no—
146C4—1371no—
147C4—1372no—
148C4—1371no—
149C4—1372no—
150C4#1798414912yes—
151C2—22no—
152C2C2×#1798414911yescharacteristic
153C2C2×#1798415111yescharacteristic
154C2#2257839511yescharacteristic
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#1549
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
77. C137#1549semidirect

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
#1549137