Finite Group Databasebeta

← #31796314

Subgroups — #31796314

Order 17536.

198 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 58 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#22578426C211yesmaximal, characteristic
2#22578421C211yesmaximal, characteristic
3#22578263C212yesmaximal, complemented
4#22578262C212yesmaximal, complemented
5#22577957C211yesmaximal, characteristic
6#17984155C2212yes—
7#17984155C2212yes—
8#17984145—21no—
9#17984145—21no—
10#17984143—21no—
11C4×#11764047—21no—
12#17984087C2211yescharacteristic
13#17984086C2211yescharacteristic
14#17984078—21no—
15#17984068C411yescharacteristic
16#17984066—21no—
17#17984064C411yescharacteristic
18C4×Dic274—21no—
19C4×Dic274C411yescharacteristic
20#17984045—21no—
21C42×D137C2211yescharacteristic
22C4×Q8×C137C411yescharacteristic
23C2×#11764047—22no—
24C2×#11764047—22no—
25C2×Dic274—21no—
26C2×Dic274C2×C411yescharacteristic
27C2×C4×D137C2×C411yescharacteristic
28C2×C4×D137D411yescharacteristic
29C2×C4×D137D411yescharacteristic
30C4⋊Dic137—21no—
31C4⋊Dic137C2×C411yescharacteristic
32#14555489—21no—
33#14555489—21no—
34#14555489—21no—
35#14555489—21no—
36#14555489—21no—
37#14555487—21no—
38#14555487—21no—
39C4×Dic137—21no—
40C4×Dic137D411yescharacteristic
41C4×Dic137D411yescharacteristic
42C4×Dic137C2311yescharacteristic
43C2×Q8×C137C2×C411yescharacteristic
44C4⋊C4×C137—21no—
45C4⋊C4×C137C2×C411yescharacteristic
46C42×C137—21no—
47C42×C137C2×C411yescharacteristic
48#11764047—42no—
49#11764047—42no—
50C22×D137C22⋊C411yescharacteristic
51Dic274—42no—
52Dic274—41no—
53C2×Dic137—21no—
54C2×Dic137—21no—
55C2×Dic137C22⋊C411yescharacteristic
56C2×Dic137C2×D411yescharacteristic
57C2×Dic137C2×D411yescharacteristic
58C4×D137—22no—
59C4×D137—22no—
60C4×D137—41no—
61Q8×C137—41no—
62C2×C4×C137—21no—
63C2×C4×C137—21no—
64C2×C4×C137C22⋊C411yescharacteristic
65C2×C4×C137C22⋊C411yescharacteristic
66C2×C4×C137C22×C411yescharacteristic
67C22×C137C2×C22⋊C411yescharacteristic
68C2×D137—21no—
69C2×D137—21no—
70C2×D137—21no—
71C4×C137—21no—
72C4×C137—21no—
73C4×C137—21no—
74C4×C137—41no—
75C4×C137—41no—
76Dic137—21no—
77Dic137—41no—
78Dic137—41no—
79Dic137#13512yes—
80Dic137#13712yes—
81C2×C137#41311yescharacteristic
82C2×C137#41611yescharacteristic
83C2×C137#41911yescharacteristic
84D137—41no—
85C137#155411yescharacteristic, complemented, Sylow
86#1554—1371nomaximal, Sylow
87#533—1371no—
88#432—1371no—
89#431—1371no—
90#333—1372no—
91#332—1372no—
92C4⋊Q8—1371no—
93C4⋊Q8—1371no—
94#125—2741no—
95C22⋊Q8—2741no—
96C22⋊Q8—1371no—
97C22⋊Q8—1371no—
98C4×Q8—2741no—
99C4×Q8—1371no—
100C4×Q8#1058773911yescharacteristic
101#117—2741no—
102C2×C42—1371no—
103C4⋊C8—1372no—
104C4⋊C8—1372no—
105C22⋊C8—2741no—
106C22⋊C8—2741no—
107#97—2741no—
108C4×C8—2741no—
109C2×Q8—2741no—
110C2×Q8—1371no—
111C2×Q8C2×#1058773911yescharacteristic
112C22×C4—1371no—
113C22×C4—1371no—
114C22×C4—1371no—
115C2×C8—2742no—
116C2×C8—2742no—
117C4⋊C4—21no—
118C4⋊C4—2741no—
119C4⋊C4—2741no—
120C4⋊C4—2741no—
121C4⋊C4—2741no—
122C4⋊C4—2741no—
123C4⋊C4—2741no—
124C4⋊C4—1371no—
125C4⋊C4C2×#1058773911yescharacteristic
126C22⋊C4—2741no—
127C22⋊C4—2741no—
128C42—21no—
129C42—2741no—
130C42—1371no—
131C42—1371no—
132C42—1371no—
133C42C2×#1058773911yescharacteristic
134C23—1371no—
135Q8—41no—
136Q8—5481no—
137Q8—5482no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—2742no—
141C2×C4—2742no—
142C2×C4—2741no—
143C2×C4—2741no—
144C2×C4—5481no—
145C2×C4—1371no—
146C2×C4—1371no—
147C2×C4—1371no—
148C2×C4C22⋊#1058773911yescharacteristic
149C2×C4C22⋊#1058773911yescharacteristic
150C2×C4C22×#1058773911yescharacteristic
151C8—5482no—
152C8—5482no—
153C22—2741no—
154C22—2741no—
155C22—2741no—
156C22C2×C22⋊#1058773911yescharacteristic
157C4—21no—
158C4—21no—
159C4—21no—
160C4—41no—
161C4—41no—
162C4—2741no—
163C4—5481no—
164C4—5481no—
165C4—1372no—
166C4—1372no—
167C2—5481no—
168C2#2257838911yescharacteristic
169C2#2257839311yescharacteristic
170C2#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#1554
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
4. #22578262C2semidirect
85. C137#1554semidirect

Maximal subgroups

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

subgroupindex
#225784262
#225784212
#225782632
#225782622
#225779572
#1554137