Finite Group Databasebeta

← #31796208

Subgroups — #31796208

Order 17536.

252 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 56 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#22578425C211yesmaximal, characteristic, complemented
2#22578348C211yesmaximal, characteristic, complemented
3#22578255C212yesmaximal, complemented
4#22578254C212yesmaximal, complemented
5#22577950C211yesmaximal, characteristic
6C4⋊#11764047—21no—
7#17984145C2212yes—
8#17984143—21no—
9(C2×#10587739)⋊C4—21no—
10(C2×#10587739)⋊C4—21no—
11(C2×#10587739)⋊C4C2212yes—
12C2×#14555510—21no—
13#17984087C411yescharacteristic, complemented
14#17984084C411yescharacteristic, complemented
15#17984074—21no—
16#17984063C2211yescharacteristic
17#17984061—21no—
18#17984059C2211yescharacteristic
19C4×D548C411yescharacteristic, complemented
20C4×C22⋊D137—21no—
21C42×D137C2211yescharacteristic
22C4×D4×C137C411yescharacteristic, complemented
23C22×#10587739—22no—
24C2×#11764047—22no—
25#14555510—44no—
26C2×Dic274—21no—
27C2×D548—21no—
28C2×D548C2×C411yescharacteristic
29C2×C22⋊D137—21no—
30C2×C22⋊D137—21no—
31C2×C22⋊D137—21no—
32C2×C4×D137—21no—
33C2×C4×D137D411yescharacteristic
34C2×C4×D137D411yescharacteristic
35C2×C4×D137C2311yescharacteristic
36C4⋊Dic137C2×C411yescharacteristic
37#14555489—21no—
38C22⋊Dic137—21no—
39#14555487—21no—
40#14555487—21no—
41C4×Dic137C2×C411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137D411yescharacteristic
44C2×D4×C137C2×C411yescharacteristic
45C22×C4×C137—21no—
46C4⋊C4×C137C2×C411yescharacteristic
47C22⋊C4×C137—21no—
48C42×C137C2×C411yescharacteristic
49C2×#10587739—42no—
50C2×#10587739—42no—
51C2×#10587739—42no—
52#11764047—42no—
53C22×D137—21no—
54C22×D137C2×D411yescharacteristic
55Dic274—42no—
56D548—41no—
57D548—42no—
58C22⋊D137—42no—
59C22⋊D137—42no—
60C22⋊D137—42no—
61C2×Dic137—21no—
62C2×Dic137C22⋊C411yescharacteristic
63C2×Dic137C22⋊C411yescharacteristic
64C2×Dic137C2×D411yescharacteristic
65C4×D137—22no—
66C4×D137—41no—
67C4×D137—41no—
68C4×D137—42no—
69C23×C137—21no—
70D4×C137—41no—
71C2×C4×C137—21no—
72C2×C4×C137—42no—
73C2×C4×C137C22⋊C411yescharacteristic
74C2×C4×C137C22⋊C411yescharacteristic
75C2×C4×C137C22×C411yescharacteristic
76C22×C137—41no—
77C22×C137—41no—
78C22×C137—41no—
79C22×C137C2×C22⋊C411yescharacteristic
80C2×D137—21no—
81C2×D137—21no—
82C2×D137—41no—
83C2×D137—41no—
84C2×D137—41no—
85C2×D137AGL(1,ℤ/8)12yes—
86#10587739—82no—
87C4×C137—21no—
88C4×C137—21no—
89C4×C137—21no—
90C4×C137—41no—
91Dic137—21no—
92Dic137—21no—
93Dic137—21no—
94Dic137—41no—
95C2×C137—81no—
96C2×C137#41011yescharacteristic
97C2×C137#41811yescharacteristic
98C2×C137#42111yescharacteristic
99D137—22no—
100D137—81no—
101C137#151411yescharacteristic, complemented, Sylow
102#1514—1371nomaximal, Sylow
103#532—1371no—
104#432—1371no—
105#383—1371no—
106#327—1372no—
107#327—1372no—
108C2×#41—2741no—
109C4⋊Q8—1371no—
110#127—1371no—
111#124—2741no—
112#121—2741no—
113#121—1371no—
114#121—1371no—
115C4×D4—2741no—
116C4×D4—1371no—
117C4×D4#1058773911yescharacteristic, complemented
118C2×C42—1371no—
119C4⋊C8—2741no—
120C22⋊C8—1372no—
121#97—2741no—
122#95—2741no—
123#95—2741no—
124#95—1372no—
125#41—5484no—
126C2×Q8—2741no—
127C2×D4—2741no—
128C2×D4—2741no—
129C2×D4—2741no—
130C2×D4—2741no—
131C2×D4—1371no—
132C2×D4C2×#1058773911yescharacteristic
133C22×C4—21no—
134C22×C4—2741no—
135C22×C4—2742no—
136C22×C4—1371no—
137C22×C4—1371no—
138C22×C4—1371no—
139C2×C8—2742no—
140C4⋊C4—2741no—
141C4⋊C4—1371no—
142C4⋊C4C2×#1058773911yescharacteristic
143C22⋊C4—21no—
144C22⋊C4—2741no—
145C22⋊C4—2741no—
146C22⋊C4—2741no—
147C42—1371no—
148C42—1371no—
149C42—1371no—
150C42C2×#1058773911yescharacteristic
151C23—21no—
152C23—2741no—
153C23—1371no—
154Q8—5482no—
155D4—41no—
156D4—5481no—
157D4—5482no—
158D4—5482no—
159D4—5482no—
160D4—5482no—
161C2×C4—21no—
162C2×C4—42no—
163C2×C4—2742no—
164C2×C4—2741no—
165C2×C4—5481no—
166C2×C4—5481no—
167C2×C4—5482no—
168C2×C4—5482no—
169C2×C4—5482no—
170C2×C4—5482no—
171C2×C4—1371no—
172C2×C4—1371no—
173C2×C4—1371no—
174C2×C4C22⋊#1058773911yescharacteristic
175C2×C4C22⋊#1058773911yescharacteristic
176C2×C4C22×#1058773911yescharacteristic
177C8—5482no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—2741no—
182C22—2741no—
183C22—5481no—
184C22—5481no—
185C22—5481no—
186C22—1372no—
187C22C2×C22⋊#1058773911yescharacteristic
188C4—21no—
189C4—21no—
190C4—21no—
191C4—41no—
192C4—2741no—
193C4—2741no—
194C4—2741no—
195C4—5481no—
196C4—10962no—
197C2—81no—
198C2—2742no—
199C2—10961no—
200C2#2257838211yescharacteristic
201C2#2257839511yescharacteristic
202C2#2257840011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4×D4×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577950C2
cosocleC2×C4×D137C23
hypercenterC4×D4#10587739
hypocenterC137#1514
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #22578425C2semidirect
2. #22578348C2semidirect
3. #22578255C2semidirect
4. #22578254C2semidirect
13. #17984087C4semidirect
14. #17984084C4semidirect
19. C4×D548C4semidirect
22. C4×D4×C137C4semidirect
101. C137#1514semidirect
117. C4×D4#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784252
#225783482
#225782552
#225782542
#225779502
#1514137