Finite Group Databasebeta

← #31796303

Subgroups — #31796303

Order 17536.

240 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 70 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#22578421C211yesmaximal, characteristic
2#22578408C211yesmaximal, characteristic
3#22578263C214yesmaximal, complemented
4C4⋊Q8×D137C211yesmaximal, characteristic
5C2×#14555520—21no—
6#17984155—21no—
7#17984155C2212yes—
8#17984155C2212yes—
9#17984145—21no—
10C4×#11764047—21no—
11C2×Q8×D137C412yes—
12#17984088C411yescharacteristic
13#17984087—21no—
14#17984087—21no—
15#17984086C2212yes—
16C4⋊C4×D137—21no—
17C4⋊C4×D137—21no—
18C42×D137C2211yescharacteristic
19C4⋊Q8×C137C411yescharacteristic
20#14555520—42no—
21C2×#11764047—22no—
22C2×#11764047—22no—
23C2×Dic274—22no—
24C2×Dic274C2×C412yes—
25Q8×D137—44no—
26C2×C4×D137—22no—
27C2×C4×D137C2×C411yescharacteristic
28C2×C4×D137D411yescharacteristic
29C2×C4×D137D411yescharacteristic
30C4⋊Dic137—21no—
31C4⋊Dic137—21no—
32#14555489—22no—
33#14555489—22no—
34C4×Dic137D411yescharacteristic
35C4×Dic137D411yescharacteristic
36C4×Dic137C2311yescharacteristic
37C2×Q8×C137C2×C412yes—
38C4⋊C4×C137—21no—
39C4⋊C4×C137—21no—
40C42×C137C2×C411yescharacteristic
41#11764047—42no—
42#11764047—42no—
43C22×D137C22⋊C411yescharacteristic
44Dic274—44no—
45Dic274—42no—
46C2×Dic137—22no—
47C2×Dic137C22⋊C411yescharacteristic
48C2×Dic137C2×D411yescharacteristic
49C2×Dic137C2×D411yescharacteristic
50C4×D137—22no—
51C4×D137—22no—
52C4×D137—44no—
53C4×D137SD1612yes—
54C4×D137Q1612yes—
55Q8×C137—42no—
56C2×C4×C137—22no—
57C2×C4×C137C22⋊C411yescharacteristic
58C2×C4×C137C22⋊C411yescharacteristic
59C2×C4×C137C22×C411yescharacteristic
60C22×C137C2×C22⋊C411yescharacteristic
61C2×D137—21no—
62C2×D137—21no—
63C2×D137#10312yes—
64C4×C137—21no—
65C4×C137—21no—
66C4×C137—42no—
67C4×C137#10312yes—
68Dic137—42no—
69Dic137#10012yes—
70Dic137C2×SD1611yescharacteristic
71Dic137C2×Q1611yescharacteristic
72Dic137#13712yes—
73C2×C137C2×#10011yescharacteristic
74C2×C137C2×#10311yescharacteristic
75C2×C137#41911yescharacteristic
76D137—22no—
77C137#154811yescharacteristic, complemented, Sylow
78#1548—1371nomaximal, Sylow
79C2×C4⋊Q8—1371no—
80#431—1371no—
81#423—1371no—
82#333—1374no—
83C22×Q8—1372no—
84C2×M16—2741no—
85C4⋊Q8—2741no—
86C4⋊Q8—2741no—
87C4⋊Q8—1372no—
88C4⋊Q8—1371no—
89C4⋊Q8#1058773911yescharacteristic
90C2×C4⋊C4—2741no—
91C2×C4⋊C4—2741no—
92C2×C42—1371no—
93C4⋊C8—2741no—
94C4⋊C8—1372no—
95C4⋊C8—1372no—
96C22⋊C8—2741no—
97C4×C8—2741no—
98C2×Q8—2742no—
99C2×Q8—5484no—
100C2×Q8—1372no—
101C2×Q8C2×#1058773912yes—
102C22×C4—2742no—
103C22×C4—1371no—
104C22×C4—1371no—
105C22×C4—1371no—
106M16—5482no—
107C2×C8—2742no—
108C2×C8—2742no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—2742no—
112C4⋊C4—2742no—
113C4⋊C4—2741no—
114C4⋊C4—2741no—
115C42—1371no—
116C42—1371no—
117C42—1371no—
118C42C2×#1058773911yescharacteristic
119C23—1371no—
120Q8—42no—
121Q8—5482no—
122Q8—5484no—
123C2×C4—22no—
124C2×C4—2742no—
125C2×C4—2742no—
126C2×C4—2742no—
127C2×C4—5484no—
128C2×C4—1371no—
129C2×C4—1371no—
130C2×C4—1371no—
131C2×C4—1372no—
132C2×C4—1372no—
133C2×C4C22⋊#1058773911yescharacteristic
134C2×C4C22⋊#1058773911yescharacteristic
135C2×C4C22×#1058773911yescharacteristic
136C8—5482no—
137C8—5482no—
138C22—2741no—
139C22—2741no—
140C22—1372no—
141C22C2×C22⋊#1058773911yescharacteristic
142C4—21no—
143C4—21no—
144C4—42no—
145C4—5482no—
146C4—1372no—
147C4—1371no—
148C4—1371no—
149C4—1372no—
150C4#1798415112yes—
151C2—2742no—
152C2D137⋊#10011yescharacteristic
153C2C2×#1798415111yescharacteristic
154C2#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#1548
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. #22578263C2semidirect
77. C137#1548semidirect

Maximal subgroups

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

subgroupindex
#225784212
#225784082
#225782632
C4⋊Q8×D1372
#1548137