Finite Group Databasebeta

← #17953618

Subgroups — #17953618

Order 4352.

622 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 124 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
1C2×#11763652C212yesmaximal
2#14543627C214yesmaximal, complemented
3#14531137C211yesmaximal, characteristic
4C2×C22⋊#3624C2214yes—
5#11763652—22no—
6#11763652—22no—
7C22×C22⋊Dic17C411yescharacteristic
8C22×C22⋊Dic17C2211yescharacteristic
9C22⋊C4×Dic17—22no—
10C22⋊C4×Dic17C412yes—
11C22⋊C4×Dic17C2212yes—
12C4×C22⋊Dic17—22no—
13C22×C22⋊C4×C17C411yescharacteristic
14C25×C17C2×C411yescharacteristic
15C23×C4×C17—21no—
16C23×Dic17—21no—
17C23×Dic17C2×C411yescharacteristic
18C23×Dic17C2311yescharacteristic
19C22×#3624—24no—
20C2×C22⋊C4×C17—21no—
21C2×C22⋊C4×C17—22no—
22C2×C22⋊C4×C17C814yescomplemented
23C2×C22⋊C4×C17C2×C412yes—
24C2×C22⋊Dic17—21no—
25C2×C22⋊Dic17—24no—
26C2×C22⋊Dic17—24no—
27C2×C22⋊Dic17C2×C412yes—
28C2×C22⋊Dic17D412yes—
29C2×C22⋊Dic17D412yes—
30C2×C4×Dic17—22no—
31C2×C4×Dic17—22no—
32C22⋊#3624—24no—
33C22⋊#3624—24no—
34#10587502—22no—
35#10587502—22no—
36C24×C17—22no—
37C24×C17—22no—
38C24×C17—81no—
39C24×C17—81no—
40C24×C17C22⋊C411yescharacteristic
41C24×C17C22⋊C411yescharacteristic
42C24×C17C2×C812yes—
43C24×C17M1612yes—
44C24×C17C22×C411yescharacteristic
45C22×C4×C17—22no—
46C22×C4×C17—24no—
47C22×C4×C17—81no—
48C22×C4×C17—81no—
49C22×Dic17—22no—
50C22×Dic17—24no—
51C22×Dic17—22no—
52C22×Dic17—81no—
53C22×Dic17—81no—
54C22×Dic17—81no—
55C22×Dic17—81no—
56C22×Dic17C22⋊C412yes—
57C22×Dic17C2×D412yes—
58C22×Dic17#4114yes—
59C2×#3624—44no—
60C2×#3624—84no—
61C22⋊C4×C17—24no—
62C22⋊C4×C17—22no—
63C22⋊C4×C17C2×C814yes—
64C22⋊Dic17—24no—
65C22⋊Dic17—22no—
66C22⋊Dic17—24no—
67C22⋊Dic17—24no—
68C22⋊Dic17M1614yes—
69C4×Dic17—82no—
70C4×Dic17—82no—
71#8734—171nomaximal, Sylow
72C23×C17—21no—
73C23×C17—21no—
74C23×C17—21no—
75C23×C17—21no—
76C23×C17—24no—
77C23×C17—24no—
78C23×C17—41no—
79C23×C17—41no—
80C23×C17—41no—
81C23×C17—41no—
82C23×C17—41no—
83C23×C17—41no—
84C23×C17—82no—
85C23×C17—82no—
86C23×C17—82no—
87C23×C17—82no—
88C23×C17—84no—
89C23×C17C22⋊C812yes—
90C23×C17C22⋊C812yes—
91C23×C17C2×C22⋊C411yescharacteristic
92C23×C17#11712yes—
93C23×C17C22×C811yescharacteristic
94C23×C17C2×M1612yes—
95C23×C17C2×M1611yescharacteristic
96C2×C4×C17—24no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17—84no—
100C2×Dic17—22no—
101C2×Dic17—24no—
102C2×Dic17—44no—
103C2×Dic17—42no—
104C2×Dic17—42no—
105C2×Dic17—42no—
106C2×Dic17—42no—
107C2×Dic17—82no—
108C2×Dic17—84no—
109C2×Dic17#12314yes—
110#3624—84no—
111#2273—171no—
112C2×#323—172no—
113#1282—174no—
114C22×C17—21no—
115C22×C17—22no—
116C22×C17—21no—
117C22×C17—22no—
118C22×C17—22no—
119C22×C17—22no—
120C22×C17—41no—
121C22×C17—42no—
122C22×C17—41no—
123C22×C17—42no—
124C22×C17—42no—
125C22×C17—42no—
126C22×C17—42no—
127C22×C17—41no—
128C22×C17—41no—
129C22×C17—41no—
130C22×C17—41no—
131C22×C17—84no—
132C22×C17—84no—
133C22×C17#38111yescharacteristic
134C22×C17C2×C22⋊C811yescharacteristic
135C22×C17#40711yescharacteristic
136C22×C17#43112yes—
137C22×C17#43212yes—
138C4×C17—44no—
139Dic17—24no—
140Dic17—44no—
141Dic17—42no—
142C22×C22⋊C4—171no—
143C22×C22⋊C4—171no—
144C22×C22⋊C4#59511yescharacteristic
145C2×C22⋊C8—174no—
146C4×C22⋊C4—172no—
147C4×C22⋊C4—172no—
148C4×C22⋊C4—342no—
149C4×C22⋊C4—342no—
150#323—342no—
151#323—342no—
152C2×C17—21no—
153C2×C17—21no—
154C2×C17—22no—
155C2×C17—22no—
156C2×C17—42no—
157C2×C17—42no—
158C2×C17#181711yescharacteristic
159C2×C17#182811yescharacteristic
160C2×C17#182911yescharacteristic
161C25C2×#59511yescharacteristic
162C23×C4—21no—
163C23×C4—171no—
164C23×C4—171no—
165C23×C4—341no—
166C22×C8—344no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—172no—
170C2×C22⋊C4—172no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4—344no—
173C2×C22⋊C4—344no—
174C2×C22⋊C4—341no—
175C2×C22⋊C4#362414yescomplemented
176C2×C22⋊C4C2×#59512yes—
177C2×C42—342no—
178C2×C42—342no—
179C22⋊C8—344no—
180C22⋊C8—344no—
181#95—342no—
182#95—342no—
183C17#873411yescharacteristic, complemented, Sylow
184C24—22no—
185C24—22no—
186C24—81no—
187C24—81no—
188C24C2×#362412yes—
189C24#6324712yes—
190C24C22⋊#59511yescharacteristic
191C24C22⋊#59511yescharacteristic
192C24C22×#59511yescharacteristic
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—174no—
198C22×C4—172no—
199C22×C4—172no—
200C22×C4—342no—
201C22×C4—344no—
202C22×C4—342no—
203C22×C4—1361no—
204C22×C4—1361no—
205C22×C4—1361no—
206C22×C4—1361no—
207C2×C8—684no—
208C2×C8—1364no—
209C22⋊C4—22no—
210C22⋊C4—24no—
211C22⋊C4—174no—
212C22⋊C4—344no—
213C22⋊C4—344no—
214C22⋊C4—344no—
215C22⋊C4—342no—
216C22⋊C4C2×#362414yes—
217C42—1362no—
218C42—1362no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—24no—
224C23—24no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—82no—
232C23—82no—
233C23—82no—
234C23—82no—
235C23—84no—
236C23C22⋊#362412yes—
237C23C22⋊#362412yes—
238C23D17⋊M1612yes—
239C23#1058767712yes—
240C23C22×#362411yescharacteristic
241C23C2×#6324711yescharacteristic
242C23C2×C22⋊#59511yescharacteristic
243C2×C4—24no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—84no—
247C2×C4—174no—
248C2×C4—342no—
249C2×C4—344no—
250C2×C4—682no—
251C2×C4—684no—
252C2×C4—682no—
253C2×C4—682no—
254C2×C4—682no—
255C2×C4—1362no—
256C2×C4—1364no—
257C8—1364no—
258C22—21no—
259C22—22no—
260C22—21no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—84no—
276C22—84no—
277C22#1176375211yescharacteristic
278C22C2×C22⋊#362411yescharacteristic
279C22#1176378111yescharacteristic
280C22#1176382512yes—
281C22#1176382912yes—
282C4—44no—
283C4—344no—
284C4—684no—
285C4—682no—
286C2—21no—
287C2—21no—
288C2—22no—
289C2—22no—
290C2—42no—
291C2—42no—
292C2#1454467411yescharacteristic
293C2#1454469511yescharacteristic
294C2#1454469711yescharacteristic
subgroupG/H
centerC22#11763752
commutatorC23×C17C22×C8
FrattiniC24C22×#595
FittingC22×C22⋊C4×C17C4
socleC22×C17#381
prime radicalC25×C17C2×C4
cosocleC23×Dic17C23
hypercenterC22×C22⋊C4#595
hypocenterC17#8734
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #14543627C2semidirect
22. C2×C22⋊C4×C17C8semidirect
175. C2×C22⋊C4#3624semidirect
183. C17#8734semidirect

Maximal subgroups

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

subgroupindex
C2×#117636522
#145436272
#145311372
#873417