Finite Group Databasebeta

← #31796090

Subgroups — #31796090

Order 17536.

312 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 84 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#22578419C211yesmaximal, characteristic, complemented
2#22578419C211yesmaximal, characteristic, complemented
3#22578251C214yesmaximal, complemented
4C2×#17984050C211yesmaximal, characteristic
5C4⋊#11764047—21no—
6C4⋊#11764047C2212yescomplemented
7C22⋊#11764047—21no—
8C22⋊#11764047—21no—
9C4×#11764047—21no—
10C4×#11764047C2212yescomplemented
11C22×D548C411yescharacteristic
12C22×C4×D137—21no—
13#17984050—21no—
14#17984050—21no—
15#17984050C412yes—
16#17984050C2212yes—
17C2×#14555487—21no—
18C2×C4×Dic137C2211yescharacteristic
19C2×C4⋊C4×C137C411yescharacteristic
20C2×#11764047—22no—
21C2×#11764047—22no—
22C23×D137—21no—
23C2×D548—21no—
24C2×D548—22no—
25C2×D548C814yescomplemented
26C2×D548C2×C412yes—
27C22×Dic137D411yescharacteristic
28C22×Dic137D411yescharacteristic
29C2×C4×D137—22no—
30C2×C4×D137—22no—
31C2×C4×D137—44no—
32#14555487—22no—
33#14555487—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—
44C22×D137—24no—
45C22×D137—22no—
46C22×D137—42no—
47C22×D137—42no—
48D548—24no—
49D548—22no—
50D548C2×C814yes—
51C2×Dic137—22no—
52C2×Dic137—41no—
53C2×Dic137C22⋊C411yescharacteristic
54C2×Dic137C22⋊C411yescharacteristic
55C2×Dic137C2×D411yescharacteristic
56C2×Dic137C2×D411yescharacteristic
57C4×D137—44no—
58C4×D137—44no—
59C23×C137C22⋊C411yescharacteristic
60C2×C4×C137—22no—
61C2×C4×C137—42no—
62C2×C4×C137C22⋊C411yescharacteristic
63C2×C4×C137C2×C812yes—
64C2×C4×C137M1612yes—
65C2×C4×C137C22×C411yescharacteristic
66C22×C137—21no—
67C22×C137—21no—
68C22×C137C22⋊C812yes—
69C22×C137C2×C22⋊C411yescharacteristic
70C2×D137—24no—
71C2×D137—48no—
72C2×D137—42no—
73C2×D137—42no—
74C4×C137—42no—
75C4×C137C22⋊C812yes—
76C4×C137C22×C811yescharacteristic
77C4×C137C2×M1611yescharacteristic
78Dic137—21no—
79Dic137—21no—
80Dic137—21no—
81Dic137AGL(1,ℤ/8)12yes—
82C2×C137—22no—
83C2×C137C2×C22⋊C811yescharacteristic
84C2×C137#41811yescharacteristic
85C2×C137#42111yescharacteristic
86D137—44no—
87C137#147811yescharacteristic, complemented, Sylow
88#1478—1371nomaximal, Sylow
89C2×C4×D4—1371no—
90#431—1371no—
91#431—1371no—
92D4⋊C8—1374no—
93C22×D4—1371no—
94C23×C4—2741no—
95C4×D4—2741no—
96C4×D4—2741no—
97C4×D4—1372no—
98C4×D4—1372no—
99C2×C4⋊C4#1058773911yescharacteristic
100C2×C22⋊C4—2741no—
101C2×C42—1371no—
102C4⋊C8—2741no—
103C4⋊C8—1372no—
104C22⋊C8—2741no—
105C22⋊C8—2741no—
106C4×C8—2741no—
107C4×C8—1372no—
108C24—2741no—
109C2×D4—2741no—
110C2×D4—2742no—
111C2×D4—1374no—
112C2×D4—1372no—
113C22×C4—21no—
114C22×C4—2742no—
115C22×C4—2742no—
116C22×C4—5484no—
117C22×C4—1371no—
118C22×C4—1371no—
119C22×C4C2×#1058773911yescharacteristic
120C2×C8—2742no—
121C2×C8—2742no—
122C4⋊C4—21no—
123C4⋊C4C2×#1058773912yes—
124C22⋊C4—2742no—
125C22⋊C4—2742no—
126C42—1371no—
127C42—1371no—
128C42—1371no—
129C42—1371no—
130C23—2744no—
131C23—2742no—
132C23—5482no—
133C23—5482no—
134C23C22⋊#1058773911yescharacteristic
135D4—2744no—
136D4—2742no—
137D4—1374no—
138C2×C4—22no—
139C2×C4—42no—
140C2×C4—2742no—
141C2×C4—5484no—
142C2×C4—5484no—
143C2×C4—5481no—
144C2×C4—1371no—
145C2×C4—1371no—
146C2×C4—1371no—
147C2×C4—1371no—
148C2×C4C22⋊#1058773911yescharacteristic
149C2×C4#1455551812yes—
150C2×C4#1455552012yes—
151C2×C4C22×#1058773911yescharacteristic
152C8—5482no—
153C8—5482no—
154C22—21no—
155C22—21no—
156C22—2744no—
157C22—5488no—
158C22—5482no—
159C22—5482no—
160C22#1798414512yes—
161C22C2×C22⋊#1058773911yescharacteristic
162C4—42no—
163C4—2741no—
164C4—2741no—
165C4—2741no—
166C4—1372no—
167C4#1798414512yes—
168C4C2×#1455551811yescharacteristic
169C4C2×#1455552011yescharacteristic
170C2—22no—
171C2—5484no—
172C2C2×#1798414511yescharacteristic
173C2#2257839511yescharacteristic
174C2#2257840211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC4×C137C22×C8
FrattiniC2×C4C22×#10587739
FittingC2×C4⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D548C4
cosocleC4×Dic137C23
hypercenterC2×C4⋊C4#10587739
hypocenterC137#1478
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #22578419C2semidirect
2. #22578419C2semidirect
3. #22578251C2semidirect
6. C4⋊#11764047C22semidirect
10. C4×#11764047C22semidirect
25. C2×D548C8semidirect
87. C137#1478semidirect

Maximal subgroups

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

subgroupindex
#225784192
#225782512
C2×#179840502
#1478137