Finite Group Databasebeta

← #31796250

Subgroups — #31796250

Order 17536.

264 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 62 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#22578424C211yesmaximal, characteristic, complemented
2C4×#14555519C211yesmaximal, characteristic, complemented
3#22578259C214yesmaximal, complemented
4C2×#17984077C211yesmaximal, characteristic
5C2×#14555519—21no—
6C4⋊#11764047—21no—
7C22⋊#11764047—21no—
8#17984143C2212yescomplemented
9#17984143C2212yescomplemented
10C4×#11764047—21no—
11C22×D548C411yescharacteristic
12#17984077—21no—
13#17984077—21no—
14#17984077C412yes—
15#17984077C2212yes—
16C2×#14555487—21no—
17C2×#14555487—21no—
18C2×C4×Dic137C2211yescharacteristic
19C22×Q8×C137C411yescharacteristic
20#14555519—24no—
21C2×#11764047—22no—
22C2×#11764047—22no—
23C23×D137—21no—
24C2×D548—21no—
25C2×D548—42no—
26C2×D548C2×C412yes—
27C22×Dic137D411yescharacteristic
28C22×Dic137D411yescharacteristic
29#14555487—22no—
30#14555487—22no—
31#14555487—22no—
32#14555487—22no—
33C4×Dic137C2×C411yescharacteristic
34C4×Dic137D411yescharacteristic
35C4×Dic137D411yescharacteristic
36C4×Dic137C2311yescharacteristic
37C2×Q8×C137—21no—
38C2×Q8×C137—42no—
39C2×Q8×C137C2×C412yes—
40C22×C4×C137—21no—
41C22×C4×C137C2×C411yescharacteristic
42#11764047—24no—
43#11764047—42no—
44C22×D137—22no—
45C22×D137—42no—
46C22×D137—42no—
47C22×D137—42no—
48D548—42no—
49D548—42no—
50C2×Dic137—22no—
51C2×Dic137—41no—
52C2×Dic137C22⋊C411yescharacteristic
53C2×Dic137C22⋊C411yescharacteristic
54C2×Dic137C2×D411yescharacteristic
55C2×Dic137C2×D411yescharacteristic
56C23×C137C22⋊C411yescharacteristic
57Q8×C137—42no—
58Q8×C137—42no—
59C2×C4×C137—22no—
60C2×C4×C137—22no—
61C2×C4×C137—42no—
62C2×C4×C137C22⋊C411yescharacteristic
63C2×C4×C137C22×C411yescharacteristic
64C22×C137—21no—
65C22×C137—21no—
66C22×C137#10412yes—
67C22×C137C2×C22⋊C411yescharacteristic
68C2×D137—42no—
69C2×D137—42no—
70C2×D137—42no—
71C2×D137—84no—
72C4×C137—21no—
73C4×C137—42no—
74C4×C137#10012yes—
75Dic137—21no—
76Dic137—21no—
77Dic137—21no—
78Dic137—21no—
79C2×C137—22no—
80C2×C137C2×#10011yescharacteristic
81C2×C137C2×#10411yescharacteristic
82C2×C137#42111yescharacteristic
83D137—82no—
84C137#152711yescharacteristic, complemented, Sylow
85#1527—1371nomaximal, Sylow
86C2×#124—1371no—
87#432—1371no—
88C4×M16—1371no—
89#329—1374no—
90C22×Q8#1058773911yescharacteristic
91C22×D4—1371no—
92C2×M16—2741no—
93#124—2741no—
94#124—2741no—
95#124—1372no—
96#124—1372no—
97C2×C22⋊C4—2741no—
98C2×C22⋊C4—2741no—
99C2×C42—1371no—
100C4⋊C8—2741no—
101C22⋊C8—2741no—
102#97—1372no—
103#97—1372no—
104C4×C8—2741no—
105C24—2741no—
106C2×Q8—21no—
107C2×Q8—42no—
108C2×Q8C2×#1058773912yes—
109C2×D4—2741no—
110C2×D4—5482no—
111C2×D4—1372no—
112C22×C4—21no—
113C22×C4—1371no—
114C22×C4—1371no—
115C22×C4C2×#1058773911yescharacteristic
116M16—2744no—
117C2×C8—2742no—
118C2×C8—2742no—
119C22⋊C4—2742no—
120C22⋊C4—2742no—
121C22⋊C4—2742no—
122C22⋊C4—2742no—
123C42—1371no—
124C42—1371no—
125C42—1371no—
126C42—1371no—
127C23—2742no—
128C23—5482no—
129C23—5482no—
130C23—5482no—
131C23C22⋊#1058773911yescharacteristic
132Q8—42no—
133Q8—42no—
134D4—5482no—
135D4—5482no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—2742no—
140C2×C4—5481no—
141C2×C4—1371no—
142C2×C4—1371no—
143C2×C4—1371no—
144C2×C4—1371no—
145C2×C4C22⋊#1058773911yescharacteristic
146C2×C4C22×#1058773911yescharacteristic
147C8—2744no—
148C8—5482no—
149C22—21no—
150C22—21no—
151C22—5482no—
152C22—5482no—
153C22—5482no—
154C22—10964no—
155C22#1798415312yes—
156C22C2×C22⋊#1058773911yescharacteristic
157C4—21no—
158C4—42no—
159C4—2741no—
160C4—2741no—
161C4—2741no—
162C4—2741no—
163C4#1798414812yes—
164C2—22no—
165C2—10962no—
166C2C2×#1798414811yescharacteristic
167C2C2×#1798415311yescharacteristic
168C2#2257840211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22×Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D548C4
cosocleC4×Dic137C23
hypercenterC22×Q8#10587739
hypocenterC137#1527
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. #22578424C2semidirect
2. C4×#14555519C2semidirect
3. #22578259C2semidirect
8. #17984143C22semidirect
9. #17984143C22semidirect
84. C137#1527semidirect

Maximal subgroups

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

subgroupindex
#225784242
C4×#145555192
#225782592
C2×#179840772
#1527137