Finite Group Databasebeta

← #31796058

Subgroups — #31796058

Order 17536.

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

normal subgroups only

This group has 256 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#22578423C211yesmaximal, characteristic, complemented
2#22578419C211yesmaximal, characteristic, complemented
3#22578247C212yesmaximal, complemented
4#22578246C212yesmaximal, complemented
5C2×#17984074C211yesmaximal, characteristic
6#17984155—21no—
7C4⋊#11764047—21no—
8C22⋊#11764047C2212yes—
9C22⋊#11764047C2212yes—
10#17984143—21no—
11C4×#11764047—21no—
12C22×Dic274C411yescharacteristic
13C22×C22⋊D137C411yescharacteristic
14#17984074—21no—
15#17984074—21no—
16#17984074—21no—
17#17984074—21no—
18C2×C22⋊Dic137C2211yescharacteristic
19C2×#14555487C411yescharacteristic
20C2×#14555487C2211yescharacteristic
21C2×C4×Dic137C2211yescharacteristic
22C2×C22⋊C4×C137C411yescharacteristic
23C2×#11764047—22no—
24C2×#11764047—22no—
25C23×D137C2×C411yescharacteristic
26C2×Dic274—21no—
27C2×Dic274—21no—
28C2×Dic274—42no—
29C2×C22⋊D137—21no—
30C2×C22⋊D137—21no—
31C2×C22⋊D137—42no—
32C22×Dic137C2×C411yescharacteristic
33C22×Dic137C2×C411yescharacteristic
34C22×Dic137C2311yescharacteristic
35C22⋊Dic137—21no—
36C22⋊Dic137—21no—
37#14555487—21no—
38#14555487—21no—
39#14555487—21no—
40#14555487—21no—
41C4×Dic137D411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137D411yescharacteristic
44C4×Dic137D411yescharacteristic
45C24×C137C2×C411yescharacteristic
46C22×C4×C137C2×C411yescharacteristic
47C22×C4×C137C2×C411yescharacteristic
48C22⋊C4×C137—21no—
49C22⋊C4×C137—21no—
50#11764047—42no—
51#11764047—42no—
52C22×D137—21no—
53C22×D137—22no—
54C22×D137—41no—
55C22×D137—41no—
56Dic274—42no—
57Dic274—42no—
58C22⋊D137—42no—
59C22⋊D137—42no—
60C2×Dic137—22no—
61C2×Dic137—21no—
62C2×Dic137—41no—
63C2×Dic137—41no—
64C2×Dic137C22⋊C411yescharacteristic
65C2×Dic137C22⋊C411yescharacteristic
66C2×Dic137C2×D411yescharacteristic
67C2×Dic137C2×D411yescharacteristic
68C23×C137—21no—
69C23×C137—22no—
70C23×C137—41no—
71C23×C137—41no—
72C23×C137C22×C411yescharacteristic
73C2×C4×C137—22no—
74C2×C4×C137—21no—
75C2×C4×C137—41no—
76C2×C4×C137C22⋊C411yescharacteristic
77C2×C4×C137C22⋊C411yescharacteristic
78C22×C137—21no—
79C22×C137—21no—
80C22×C137—41no—
81C22×C137—41no—
82C22×C137—41no—
83C22×C137—82no—
84C22×C137C2×C22⋊C411yescharacteristic
85C22×C137#13112yes—
86C2×D137—41no—
87C2×D137—41no—
88C2×D137—41no—
89C2×D137—82no—
90C4×C137—21no—
91C4×C137—21no—
92C4×C137—41no—
93Dic137—21no—
94Dic137—21no—
95Dic137—21no—
96Dic137—41no—
97Dic137#10012yes—
98C2×C137—22no—
99C2×C137—81no—
100C2×C137#40811yescharacteristic
101C2×C137#41011yescharacteristic
102C2×C137C2×#10011yescharacteristic
103D137—81no—
104C137#146911yescharacteristic, complemented, Sylow
105#1469—1371nomaximal, Sylow
106C2×#124—1371no—
107#432—1371no—
108#431—1371no—
109#323—1372no—
110#323—1372no—
111C22×Q8—1371no—
112C22×D4—1371no—
113#124—2741no—
114#124—2741no—
115#124—2741no—
116#124—2741no—
117C2×C22⋊C4—1371no—
118C2×C22⋊C4—1371no—
119C2×C22⋊C4—1371no—
120C2×C22⋊C4#1058773911yescharacteristic
121C2×C42—1371no—
122C4⋊C8—2741no—
123C4⋊C8—2741no—
124C22⋊C8—1372no—
125C22⋊C8—1372no—
126#97—2741no—
127C4×C8—2741no—
128C24—1371no—
129C24C2×#1058773911yescharacteristic
130C2×Q8—2741no—
131C2×Q8—2741no—
132C2×Q8—5482no—
133C2×D4—2741no—
134C2×D4—2741no—
135C2×D4—5482no—
136C22×C4—1371no—
137C22×C4—1371no—
138C22×C4—1371no—
139C22×C4C2×#1058773911yescharacteristic
140C22×C4C2×#1058773911yescharacteristic
141C2×C8—2742no—
142C2×C8—2742no—
143C22⋊C4—21no—
144C22⋊C4—21no—
145C22⋊C4—2741no—
146C22⋊C4—2741no—
147C22⋊C4—2741no—
148C22⋊C4—2741no—
149C22⋊C4—2741no—
150C22⋊C4—2741no—
151C42—1371no—
152C42—1371no—
153C42—1371no—
154C42—1371no—
155C23—21no—
156C23—22no—
157C23—41no—
158C23—41no—
159C23—2741no—
160C23—2742no—
161C23—5481no—
162C23—5481no—
163C23C22×#1058773911yescharacteristic
164Q8—5482no—
165Q8—5482no—
166D4—5482no—
167D4—5482no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—2742no—
172C2×C4—2741no—
173C2×C4—5481no—
174C2×C4—5481no—
175C2×C4—1371no—
176C2×C4—1371no—
177C2×C4—1371no—
178C2×C4—1371no—
179C2×C4C22⋊#1058773911yescharacteristic
180C2×C4C22⋊#1058773911yescharacteristic
181C8—5482no—
182C8—5482no—
183C22—21no—
184C22—21no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22—82no—
189C22—5481no—
190C22—5481no—
191C22—5481no—
192C22—10962no—
193C22C2×C22⋊#1058773911yescharacteristic
194C22#1798417512yes—
195C4—21no—
196C4—21no—
197C4—41no—
198C4—2741no—
199C4—2741no—
200C4—2741no—
201C4—5481no—
202C4—1372no—
203C2—22no—
204C2—81no—
205C2—10961no—
206C2#2257837611yescharacteristic
207C2#2257838211yescharacteristic
208C2D137⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C22⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×C22⋊D137C4
cosocleC22×Dic137C23
hypercenterC2×C22⋊C4#10587739
hypocenterC137#1469
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #22578423C2semidirect
2. #22578419C2semidirect
3. #22578247C2semidirect
4. #22578246C2semidirect
104. C137#1469semidirect

Maximal subgroups

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

subgroupindex
#225784232
#225784192
#225782472
#225782462
C2×#179840742
#1469137