Finite Group Databasebeta

← #17958157

Subgroups — #17958157

Order 4352.

686 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 104 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#14544015C212yesmaximal, complemented
2#14543713C214yesmaximal, complemented
3#14531364C211yesmaximal, characteristic
4#11763825—22no—
5#11763654C2214yescomplemented
6#11763652—22no—
7C23×D68C411yescharacteristic
8C2×#10587612C2211yescharacteristic
9#11762711C412yes—
10#11762711C2212yes—
11#11762710—22no—
12#11762664—22no—
13C2×C17×#117C411yescharacteristic
14C24×D17—21no—
15C23×C4×C17C2×C411yescharacteristic
16C22×D68—21no—
17C22×D68—22no—
18C22×D68—42no—
19C22×D68—42no—
20C22×D68C814yescomplemented
21C22×D68C2×C412yes—
22C17×#117—42no—
23C2×C4⋊C4×C17—21no—
24C2×C22⋊C4×C17—21no—
25C2×C42×C17C2×C412yes—
26C2×C22⋊Dic17D412yes—
27C2×#63227—24no—
28C2×#63227—24no—
29#10587612—42no—
30C2×C4⋊Dic17C2×C411yescharacteristic
31C2×C4⋊Dic17C2311yescharacteristic
32C2×C4×Dic17D412yes—
33C22⋊#3624—24no—
34C4⋊#3624—42no—
35C4×#3624—42no—
36C24×C17C22⋊C411yescharacteristic
37C23×D17—22no—
38C23×D17—24no—
39C23×D17—42no—
40C23×D17—42no—
41C23×D17—82no—
42C23×D17—82no—
43C22×C4×C17—22no—
44C22×C4×C17—22no—
45C22×C4×C17—22no—
46C22×C4×C17C22⋊C411yescharacteristic
47C22×C4×C17C2×C812yes—
48C22×C4×C17M1612yes—
49C22×C4×C17C22×C411yescharacteristic
50C22×Dic17C22⋊C412yes—
51C22×Dic17C2×D412yes—
52C2×D68—24no—
53C2×D68—22no—
54C2×D68—42no—
55C2×D68—42no—
56C2×D68—48no—
57C2×D68—42no—
58C2×D68—42no—
59C2×D68—48no—
60C2×D68C2×C814yes—
61C2×#3624—44no—
62C4⋊C4×C17—42no—
63C22⋊C4×C17—42no—
64C42×C17—42no—
65C22⋊Dic17—42no—
66#63227—84no—
67#63227—84no—
68C4⋊Dic17—41no—
69C4⋊Dic17—41no—
70C4×Dic17—22no—
71C4×Dic17—22no—
72#11433—171nomaximal, Sylow
73C23×C17—21no—
74C23×C17—21no—
75C23×C17—41no—
76C23×C17—41no—
77C23×C17C22⋊C812yes—
78C23×C17C2×C22⋊C411yescharacteristic
79C22×D17—24no—
80C22×D17—48no—
81C22×D17—44no—
82C22×D17—44no—
83C22×D17—42no—
84C22×D17—42no—
85C22×D17—88no—
86C22×D17—88no—
87C22×D17—84no—
88C22×D17—84no—
89C2×C4×C17—21no—
90C2×C4×C17—21no—
91C2×C4×C17—24no—
92C2×C4×C17—24no—
93C2×C4×C17—42no—
94C2×C4×C17—42no—
95C2×C4×C17—42no—
96C2×C4×C17C22⋊C812yes—
97C2×C4×C17#9912yes—
98C2×C4×C17#10012yes—
99C2×C4×C17C22×C811yescharacteristic
100C2×C4×C17C2×M1611yescharacteristic
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—42no—
104C2×Dic17—42no—
105D68—44no—
106D68—44no—
107D68—48no—
108#3624—84no—
109#2309—171no—
110#1458—172no—
111#1318—174no—
112C22×C17—21no—
113C22×C17—21no—
114C22×C17—22no—
115C22×C17—42no—
116C22×C17—42no—
117C22×C17—41no—
118C22×C17—41no—
119C22×C17C2×C22⋊C811yescharacteristic
120C22×C17C2×#9911yescharacteristic
121C22×C17C2×#10011yescharacteristic
122C2×D17—48no—
123C2×D17—44no—
124C2×D17—84no—
125C2×D17—816no—
126C2×D17—82no—
127C2×D17—82no—
128C4×C17—21no—
129C4×C17—21no—
130C4×C17—82no—
131C4×C17#32314yes—
132Dic17—42no—
133Dic17—42no—
134C23×D4—171no—
135C2×#117—171no—
136C2×#117#59511yescharacteristic
137#431—342no—
138#390—172no—
139#390—172no—
140#390—342no—
141#379—342no—
142#324—174no—
143#323—342no—
144C2×C17—21no—
145C2×C17—21no—
146C2×C17—42no—
147C2×C17C2×#32311yescharacteristic
148C2×C17#212811yescharacteristic
149C2×C17#213411yescharacteristic
150D17—84no—
151C25—341no—
152C22×D4—174no—
153C22×D4—172no—
154C22×D4—342no—
155C22×D4—341no—
156C22×D4—682no—
157C22×D4—682no—
158C23×C4C2×#59511yescharacteristic
159#117—42no—
160#117—682no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—171no—
163C2×C4⋊C4—171no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—172no—
166C2×C22⋊C4—344no—
167C2×C22⋊C4—344no—
168C2×C42—172no—
169C2×C42C2×#59512yes—
170C4⋊C8—682no—
171C22⋊C8—344no—
172C4×C8—682no—
173C17#1143311yescharacteristic, complemented, Sylow
174C24—342no—
175C24—344no—
176C24—682no—
177C24—682no—
178C24—1362no—
179C24—1362no—
180C24C22⋊#59511yescharacteristic
181C2×D4—174no—
182C2×D4—342no—
183C2×D4—344no—
184C2×D4—682no—
185C2×D4—682no—
186C2×D4—688no—
187C2×D4—688no—
188C2×D4—682no—
189C2×D4—682no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—172no—
194C22×C4—172no—
195C22×C4#6324212yes—
196C22×C4#6324312yes—
197C22×C4C22⋊#59511yescharacteristic
198C22×C4C22×#59511yescharacteristic
199C2×C8—684no—
200C4⋊C4—42no—
201C4⋊C4—681no—
202C4⋊C4—681no—
203C22⋊C4—42no—
204C22⋊C4—682no—
205C22⋊C4—1364no—
206C22⋊C4—1364no—
207C42—42no—
208C42—342no—
209C42—342no—
210C23—21no—
211C23—21no—
212C23—41no—
213C23—41no—
214C23—344no—
215C23—684no—
216C23—684no—
217C23—688no—
218C23—682no—
219C23—682no—
220C23—1368no—
221C23—1368no—
222C23—1364no—
223C23—1364no—
224C23#1058754312yes—
225C23C2×C22⋊#59511yescharacteristic
226D4—684no—
227D4—684no—
228D4—688no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—342no—
237C2×C4—342no—
238C2×C4—682no—
239C2×C4—682no—
240C2×C4#1058753912yes—
241C2×C4#1058754312yes—
242C2×C4#1058755312yes—
243C2×C4C2×#6324211yescharacteristic
244C2×C4C2×#6324311yescharacteristic
245C8—1364no—
246C22—21no—
247C22—21no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—41no—
252C22—41no—
253C22—688no—
254C22—684no—
255C22—1364no—
256C22—13616no—
257C22—1362no—
258C22—1362no—
259C22C2×#1058754311yescharacteristic
260C22C2×#1058753911yescharacteristic
261C22C2×#1058755311yescharacteristic
262C4—21no—
263C4—21no—
264C4—82no—
265C4—682no—
266C4—682no—
267C4#1176365314yes—
268C2—21no—
269C2—21no—
270C2—42no—
271C2—1364no—
272C2C2×#1176365311yescharacteristic
273C2#1454506511yescharacteristic
274C2#1454508511yescharacteristic
subgroupG/H
centerC22C2×#10587539
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C17×#117C4
socleC22×C17C2×#99
prime radicalC23×D68C4
cosocleC2×C4⋊Dic17C23
hypercenterC2×#117#595
hypocenterC17#11433
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #14544015C2semidirect
2. #14543713C2semidirect
5. #11763654C22semidirect
20. C22×D68C8semidirect
173. C17#11433semidirect

Maximal subgroups

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

subgroupindex
#145440152
#145437132
#145313642
#1143317