Finite Group Databasebeta

← #31796055

Subgroups — #31796055

Order 17536.

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

normal subgroups only

This group has 280 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#22578374C211yesmaximal, characteristic, complemented
2#22578371C211yesmaximal, characteristic, complemented
3#22578248C212yesmaximal, complemented
4#22578246C212yesmaximal, complemented
5C2×#17984070C211yesmaximal, characteristic
6C2×#14555520—21no—
7C2×#14555518—21no—
8#17984145—21no—
9#17984145—21no—
10C22⋊#11764047C2212yes—
11C22⋊#11764047C2212yes—
12C22×C22⋊D137C411yescharacteristic
13C22×C4×D137C2211yescharacteristic
14#17984070—21no—
15#17984070—21no—
16#17984070—21no—
17#17984070—21no—
18C2×C4⋊Dic137C2211yescharacteristic
19C2×#14555489C411yescharacteristic
20C2×C22⋊Dic137C2211yescharacteristic
21C2×#14555487C411yescharacteristic
22C2×C22⋊C4×C137C411yescharacteristic
23#14555520—42no—
24#14555518—42no—
25C2×#11764047—22no—
26C2×#11764047—22no—
27C23×D137C2×C411yescharacteristic
28C2×C22⋊D137—21no—
29C2×C22⋊D137—21no—
30C2×C22⋊D137—42no—
31C22×Dic137C2×C411yescharacteristic
32C22×Dic137C2×C411yescharacteristic
33C22×Dic137C2311yescharacteristic
34C2×C4×D137—42no—
35C2×C4×D137D411yescharacteristic
36C2×C4×D137D411yescharacteristic
37C2×C4×D137D411yescharacteristic
38C2×C4×D137D411yescharacteristic
39C4⋊Dic137—21no—
40C4⋊Dic137—21no—
41#14555489—21no—
42#14555489—21no—
43C22⋊Dic137—21no—
44C22⋊Dic137—21no—
45#14555487—21no—
46#14555487—21no—
47C24×C137C2×C411yescharacteristic
48C22×C4×C137C2×C411yescharacteristic
49C22×C4×C137C2×C411yescharacteristic
50C22⋊C4×C137—21no—
51C22⋊C4×C137—21no—
52#11764047—42no—
53#11764047—42no—
54C22×D137—22no—
55C22×D137—22no—
56C22×D137—41no—
57C22×D137C22⋊C411yescharacteristic
58C22×D137C22⋊C411yescharacteristic
59C22⋊D137—42no—
60C22⋊D137—42no—
61C2×Dic137—21no—
62C2×Dic137—21no—
63C2×Dic137—41no—
64C2×Dic137—41no—
65C2×Dic137—41no—
66C2×Dic137C2×D411yescharacteristic
67C2×Dic137C2×D411yescharacteristic
68C4×D137—22no—
69C4×D137—22no—
70C4×D137—41no—
71C4×D137—41no—
72C23×C137—21no—
73C23×C137—22no—
74C23×C137—41no—
75C23×C137—41no—
76C23×C137C22×C411yescharacteristic
77C2×C4×C137—21no—
78C2×C4×C137—22no—
79C2×C4×C137—41no—
80C2×C4×C137C22⋊C411yescharacteristic
81C2×C4×C137C22⋊C411yescharacteristic
82C22×C137—21no—
83C22×C137—21no—
84C22×C137—41no—
85C22×C137—41no—
86C22×C137—41no—
87C22×C137—82no—
88C22×C137C2×C22⋊C411yescharacteristic
89C22×C137#13112yes—
90C2×D137—21no—
91C2×D137—21no—
92C2×D137—21no—
93C2×D137—21no—
94C2×D137—21no—
95C2×D137—44no—
96C2×D137#9912yes—
97C4×C137—21no—
98C4×C137—21no—
99C4×C137—41no—
100Dic137—21no—
101Dic137—21no—
102Dic137—41no—
103Dic137—41no—
104C2×C137—22no—
105C2×C137—81no—
106C2×C137#40811yescharacteristic
107C2×C137C2×#9911yescharacteristic
108C2×C137#41311yescharacteristic
109D137—22no—
110D137—41no—
111C137#146811yescharacteristic, complemented, Sylow
112#1468—1371nomaximal, Sylow
113C2×#123—1371no—
114#407—1371no—
115C2×C22⋊C8—1371no—
116#324—1372no—
117#323—1372no—
118C22×D4—1371no—
119C23×C4—1371no—
120C2×M16—2741no—
121C22×C8—2741no—
122#123—2741no—
123#123—2741no—
124#123—2741no—
125#123—2741no—
126C2×C4⋊C4—1371no—
127C2×C4⋊C4—1371no—
128C2×C22⋊C4—1371no—
129C2×C22⋊C4—1371no—
130C2×C22⋊C4#1058773911yescharacteristic
131C22⋊C8—2741no—
132C22⋊C8—2741no—
133C22⋊C8—1372no—
134C22⋊C8—1372no—
135C24—1371no—
136C24C2×#1058773911yescharacteristic
137C2×D4—2741no—
138C2×D4—2741no—
139C2×D4—5482no—
140C22×C4—5482no—
141C22×C4—1371no—
142C22×C4—1371no—
143C22×C4—1371no—
144C22×C4—1371no—
145C22×C4—1371no—
146C22×C4—1371no—
147C22×C4—1371no—
148C22×C4C2×#1058773911yescharacteristic
149C22×C4C2×#1058773911yescharacteristic
150M16—5482no—
151C2×C8—2742no—
152C2×C8—2742no—
153C2×C8—5482no—
154C4⋊C4—2741no—
155C4⋊C4—2741no—
156C4⋊C4—2741no—
157C4⋊C4—2741no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—2741no—
161C22⋊C4—2741no—
162C22⋊C4—2741no—
163C22⋊C4—2741no—
164C23—21no—
165C23—22no—
166C23—41no—
167C23—41no—
168C23—2742no—
169C23—2742no—
170C23—5481no—
171C23—1371no—
172C23—1371no—
173C23C22×#1058773911yescharacteristic
174D4—5482no—
175D4—5482no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—2742no—
180C2×C4—2742no—
181C2×C4—2741no—
182C2×C4—2741no—
183C2×C4—5481no—
184C2×C4—5481no—
185C2×C4—5481no—
186C2×C4—5481no—
187C2×C4—5481no—
188C2×C4—1371no—
189C2×C4—1371no—
190C2×C4C22⋊#1058773911yescharacteristic
191C2×C4C22⋊#1058773911yescharacteristic
192C8—5482no—
193C8—5482no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—82no—
200C22—2741no—
201C22—2741no—
202C22—2741no—
203C22—2741no—
204C22—2741no—
205C22—5484no—
206C22—1372no—
207C22C2×C22⋊#1058773911yescharacteristic
208C22#1798417512yes—
209C4—21no—
210C4—21no—
211C4—41no—
212C4—2741no—
213C4—2741no—
214C4—5481no—
215C4—5481no—
216C2—22no—
217C2—81no—
218C2—2742no—
219C2—5481no—
220C2#2257837711yescharacteristic
221C2D137⋊#9911yescharacteristic
222C2#2257838911yescharacteristic
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#1468
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. #22578374C2semidirect
2. #22578371C2semidirect
3. #22578248C2semidirect
4. #22578246C2semidirect
111. C137#1468semidirect

Maximal subgroups

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

subgroupindex
#225783742
#225783712
#225782482
#225782462
C2×#179840702
#1468137