Finite Group Databasebeta

← #31796056

Subgroups — #31796056

Order 17536.

280 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 62 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#22578373C211yesmaximal, characteristic, complemented
2C2×C22⋊#11764047C211yesmaximal, characteristic, complemented
3#22578248C212yesmaximal, complemented
4#22578247C212yesmaximal, complemented
5C2×#17984072C211yesmaximal, characteristic
6C2×#14555519—21no—
7C22×#11764047—21no—
8C22⋊#11764047—21no—
9C22⋊#11764047—21no—
10C22⋊#11764047C2212yes—
11C22⋊#11764047C2212yes—
12C22×C22⋊D137C411yescharacteristic
13C23×Dic137C2211yescharacteristic
14#17984072—21no—
15#17984072—21no—
16#17984072—21no—
17#17984072—21no—
18C2×C4⋊Dic137C411yescharacteristic
19C2×C4⋊Dic137C2211yescharacteristic
20C2×#14555487C411yescharacteristic
21C2×#14555487C2211yescharacteristic
22C2×C22⋊C4×C137C411yescharacteristic
23#14555519—42no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C2×#11764047—42no—
27C23×D137C2×C411yescharacteristic
28C2×C22⋊D137—21no—
29C2×C22⋊D137—21no—
30C2×C22⋊D137—42no—
31C22×Dic137—42no—
32C22×Dic137C2×C411yescharacteristic
33C22×Dic137C2×C411yescharacteristic
34C22×Dic137D411yescharacteristic
35C22×Dic137D411yescharacteristic
36C22×Dic137D411yescharacteristic
37C22×Dic137D411yescharacteristic
38C22×Dic137C2311yescharacteristic
39C4⋊Dic137—21no—
40C4⋊Dic137—21no—
41C4⋊Dic137—21no—
42C4⋊Dic137—21no—
43#14555487—21no—
44#14555487—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—21no—
55C22×D137—22no—
56C22×D137—41no—
57C22×D137—41no—
58C22⋊D137—42no—
59C22⋊D137—42no—
60C2×Dic137—22no—
61C2×Dic137—22no—
62C2×Dic137—21no—
63C2×Dic137—41no—
64C2×Dic137—41no—
65C2×Dic137—41no—
66C2×Dic137—41no—
67C2×Dic137—41no—
68C2×Dic137C22⋊C411yescharacteristic
69C2×Dic137C22⋊C411yescharacteristic
70C2×Dic137C2×D411yescharacteristic
71C2×Dic137C2×D411yescharacteristic
72C23×C137—22no—
73C23×C137—22no—
74C23×C137—41no—
75C23×C137C22⋊C411yescharacteristic
76C23×C137C22⋊C411yescharacteristic
77C23×C137C22×C411yescharacteristic
78C2×C4×C137—21no—
79C2×C4×C137—21no—
80C2×C4×C137—22no—
81C2×C4×C137—41no—
82C22×C137—21no—
83C22×C137—21no—
84C22×C137—21no—
85C22×C137—21no—
86C22×C137—21no—
87C22×C137—21no—
88C22×C137—21no—
89C22×C137—44no—
90C22×C137#9912yes—
91C22×C137C2×C22⋊C411yescharacteristic
92C22×C137#13112yes—
93C2×D137—41no—
94C2×D137—41no—
95C2×D137—41no—
96C2×D137—82no—
97C4×C137—41no—
98C4×C137—41no—
99Dic137—21no—
100Dic137—21no—
101Dic137—21no—
102Dic137—21no—
103Dic137—41no—
104C2×C137—22no—
105C2×C137—22no—
106C2×C137—41no—
107C2×C137#40811yescharacteristic
108C2×C137C2×#9911yescharacteristic
109C2×C137#41311yescharacteristic
110D137—81no—
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—22no—
165C23—22no—
166C23—41no—
167C23—2741no—
168C23—2742no—
169C23—5481no—
170C23—5481no—
171C23C22⋊#1058773911yescharacteristic
172C23C22⋊#1058773911yescharacteristic
173C23C22×#1058773911yescharacteristic
174D4—5482no—
175D4—5482no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—41no—
180C2×C4—2742no—
181C2×C4—2742no—
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×C4—1371no—
191C2×C4—1371no—
192C8—5482no—
193C8—5482no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—5481no—
203C22—5481no—
204C22—5481no—
205C22—10962no—
206C22#1798414712yes—
207C22C2×C22⋊#1058773911yescharacteristic
208C22#1798417512yes—
209C4—41no—
210C4—41no—
211C4—2741no—
212C4—2741no—
213C4—2741no—
214C4—2741no—
215C4—5481no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—10961no—
220C2#2257837611yescharacteristic
221C2C2×#1798414711yescharacteristic
222C2#2257839011yescharacteristic
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 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #22578373C2semidirect
2. C2×C22⋊#11764047C2semidirect
3. #22578248C2semidirect
4. #22578247C2semidirect
111. C137#1468semidirect

Maximal subgroups

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

subgroupindex
#225783732
C2×C22⋊#117640472
#225782482
#225782472
C2×#179840722
#1468137