Finite Group Databasebeta

← #31796180

Subgroups — #31796180

Order 17536.

272 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 60 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#22578339C211yesmaximal, characteristic, complemented
3#22578255C214yesmaximal, complemented
4#22577954C211yesmaximal, characteristic
5C2×C4×#10587739—21no—
6C4⋊#11764047—21no—
7#17984145C2212yes—
8#17984143—21no—
9(C2×#10587739)⋊C4—21no—
10(C2×#10587739)⋊C4C2212yes—
11C2×#14555511—21no—
12#17984085C411yescharacteristic, complemented
13#17984077—21no—
14#17984063—21no—
15#17984063C2212yes—
16#17984050—21no—
17#17984050C412yescomplemented
18C42×D137C2211yescharacteristic
19C4⋊Q8×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21C2×#11764047—22no—
22C4×#10587739—24no—
23#14555511—44no—
24C2×D548—22no—
25C2×D548—21no—
26C2×D548—21no—
27C2×D548C2×C412yes—
28C2×C4×D137—22no—
29C2×C4×D137D411yescharacteristic
30C2×C4×D137D411yescharacteristic
31C2×C4×D137C2311yescharacteristic
32#14555487—22no—
33#14555487—22no—
34C4×Dic137C2×C411yescharacteristic
35C4×Dic137D411yescharacteristic
36C4×Dic137D411yescharacteristic
37C2×Q8×C137—21no—
38C4⋊C4×C137—21no—
39C4⋊C4×C137C2×C412yes—
40C42×C137C2×C411yescharacteristic
41C2×#10587739—24no—
42C2×#10587739—42no—
43C2×#10587739—42no—
44#11764047—42no—
45C22×D137—22no—
46C22×D137C2×D411yescharacteristic
47D548—42no—
48D548—42no—
49D548—44no—
50D548—42no—
51C2×Dic137C22⋊C411yescharacteristic
52C2×Dic137C22⋊C411yescharacteristic
53C2×Dic137C2×D411yescharacteristic
54C4×D137—22no—
55C4×D137—22no—
56C4×D137—41no—
57C4×D137—44no—
58Q8×C137—42no—
59C2×C4×C137—22no—
60C2×C4×C137C22⋊C411yescharacteristic
61C2×C4×C137C22⋊C411yescharacteristic
62C2×C4×C137C22×C411yescharacteristic
63C22×C137C2×C22⋊C411yescharacteristic
64C2×D137—21no—
65C2×D137—21no—
66C2×D137—42no—
67C2×D137—42no—
68C2×D137—42no—
69C2×D137AGL(1,ℤ/8)12yes—
70#10587739—44no—
71C4×C137—21no—
72C4×C137—21no—
73C4×C137—42no—
74C4×C137#10412yes—
75Dic137—21no—
76Dic137—21no—
77Dic137—21no—
78C2×C137#41011yescharacteristic
79C2×C137#41811yescharacteristic
80C2×C137C2×#10411yescharacteristic
81D137—22no—
82D137—82no—
83C137#150511yescharacteristic, complemented, Sylow
84#1505—1371nomaximal, Sylow
85#532—1371no—
86#432—1371no—
87#376—1371no—
88#327—1374no—
89C2×#41—2741no—
90C4⋊Q8#1058773911yescharacteristic, complemented
91#127—1371no—
92#124—2741no—
93#121—2741no—
94#121—1372no—
95C4×D4—2741no—
96C4×D4—1372no—
97C2×C42—2741no—
98C2×C42—1371no—
99C4⋊C8—2741no—
100C22⋊C8—1372no—
101#97—2741no—
102#95—2741no—
103#95—1372no—
104#41—5484no—
105C2×Q8—21no—
106C2×D4—2742no—
107C2×D4—2741no—
108C2×D4—2741no—
109C2×D4—1372no—
110C22×C4—2742no—
111C22×C4—2742no—
112C22×C4—1371no—
113C22×C4—1371no—
114C22×C4—1371no—
115C2×C8—2742no—
116C4⋊C4—21no—
117C4⋊C4C2×#1058773912yes—
118C22⋊C4—2742no—
119C22⋊C4—2742no—
120C42—2744no—
121C42—1371no—
122C42—1371no—
123C42—1371no—
124C42C2×#1058773911yescharacteristic
125C23—2742no—
126C23—1371no—
127Q8—42no—
128D4—5482no—
129D4—5482no—
130D4—5484no—
131D4—5482no—
132C2×C4—22no—
133C2×C4—2742no—
134C2×C4—2742no—
135C2×C4—2744no—
136C2×C4—5481no—
137C2×C4—5484no—
138C2×C4—5482no—
139C2×C4—5482no—
140C2×C4—1371no—
141C2×C4—1371no—
142C2×C4—1371no—
143C2×C4C22⋊#1058773911yescharacteristic
144C2×C4C22⋊#1058773911yescharacteristic
145C2×C4C22×#1058773911yescharacteristic
146C8—5482no—
147C22—2741no—
148C22—2741no—
149C22—5482no—
150C22—5482no—
151C22—5482no—
152C22—1372no—
153C22C2×C22⋊#1058773911yescharacteristic
154C4—21no—
155C4—21no—
156C4—42no—
157C4—2741no—
158C4—2741no—
159C4—2741no—
160C4—5484no—
161C4#1798415312yes—
162C2—2742no—
163C2—10962no—
164C2#2257838411yescharacteristic
165C2#2257839511yescharacteristic
166C2C2×#1798415311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577954C2
cosocleC2×C4×D137C23
hypercenterC4⋊Q8#10587739
hypocenterC137#1505
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
1. #22578425C2semidirect
2. #22578339C2semidirect
3. #22578255C2semidirect
12. #17984085C4semidirect
17. #17984050C4semidirect
19. C4⋊Q8×C137C4semidirect
83. C137#1505semidirect
90. C4⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784252
#225783392
#225782552
#225779542
#1505137