Finite Group Databasebeta

← #17949756

Subgroups — #17949756

Order 4352.

846 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 132 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#14544674C212yesmaximal, complemented
2#14543600C214yesmaximal, complemented
3C2×#11762712C211yesmaximal, characteristic
4C2×C22⋊#3624—22no—
5#11763675—22no—
6#11763675C2214yescomplemented
7C22×C4⋊Dic17C411yescharacteristic
8C22×C22⋊Dic17—22no—
9C22×C4×Dic17C2211yescharacteristic
10#11762712—22no—
11#11762712C412yes—
12#11762712C2212yes—
13C23×D4×C17C411yescharacteristic
14C25×C17—21no—
15C22×D4×C17—21no—
16C22×D4×C17—22no—
17C22×D4×C17—42no—
18C22×D4×C17—42no—
19C22×D4×C17C814yescomplemented
20C22×D4×C17C2×C412yes—
21C23×C4×C17C2×C411yescharacteristic
22C23×Dic17—21no—
23C23×Dic17D412yes—
24C22×#3624—24no—
25C2×C22⋊Dic17—24no—
26C2×C22⋊Dic17—24no—
27C2×C22⋊Dic17—48no—
28C2×C4⋊Dic17—21no—
29C2×C4⋊Dic17—42no—
30C2×C4⋊Dic17C2×C412yes—
31C2×C4×Dic17—42no—
32C2×C4×Dic17C2×C411yescharacteristic
33C2×C4×Dic17D412yes—
34C2×C4×Dic17C2311yescharacteristic
35C22⋊#3624—44no—
36C24×C17—24no—
37C24×C17—22no—
38C24×C17—44no—
39C24×C17—44no—
40C24×C17—42no—
41C24×C17—42no—
42C24×C17C22⋊C411yescharacteristic
43C2×D4×C17—24no—
44C2×D4×C17—22no—
45C2×D4×C17—48no—
46C2×D4×C17—42no—
47C2×D4×C17—42no—
48C2×D4×C17—48no—
49C2×D4×C17—42no—
50C2×D4×C17—42no—
51C2×D4×C17C2×C814yes—
52C22×C4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17C22⋊C411yescharacteristic
55C22×C4×C17C2×C812yes—
56C22×C4×C17M1612yes—
57C22×C4×C17C22×C411yescharacteristic
58C22×Dic17—24no—
59C22×Dic17—24no—
60C22×Dic17—22no—
61C22×Dic17—42no—
62C22×Dic17—44no—
63C22×Dic17—42no—
64C22×Dic17C22⋊C412yes—
65C22×Dic17C2×D412yes—
66C2×#3624—44no—
67C2×#3624—44no—
68C2×#3624—44no—
69C22⋊Dic17—48no—
70C22⋊Dic17—48no—
71C4⋊Dic17—42no—
72C4⋊Dic17—42no—
73C4×Dic17—41no—
74C4×Dic17—42no—
75C4×Dic17—41no—
76#7592—171nomaximal, Sylow
77C23×C17—22no—
78C23×C17—22no—
79C23×C17—24no—
80C23×C17—416no—
81C23×C17—48no—
82C23×C17—42no—
83C23×C17—42no—
84C23×C17—44no—
85C23×C17—44no—
86C23×C17—84no—
87C23×C17—88no—
88C23×C17—84no—
89C23×C17C22⋊C812yes—
90C23×C17#9912yes—
91C23×C17#10012yes—
92C23×C17C2×C22⋊C411yescharacteristic
93D4×C17—44no—
94D4×C17—48no—
95D4×C17—44no—
96C2×C4×C17—24no—
97C2×C4×C17—24no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17C22⋊C812yes—
103C2×C4×C17C22×C811yescharacteristic
104C2×C4×C17C2×M1611yescharacteristic
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—48no—
111C2×Dic17—42no—
112C2×Dic17—44no—
113C2×Dic17—84no—
114C2×Dic17AGL(1,ℤ/8)14yes—
115#3624—84no—
116C2×#390—171no—
117#1817—172no—
118#1270—174no—
119C22×C17—22no—
120C22×C17—24no—
121C22×C17—24no—
122C22×C17—22no—
123C22×C17—44no—
124C22×C17—48no—
125C22×C17—816no—
126C22×C17—82no—
127C22×C17—84no—
128C22×C17—82no—
129C22×C17#32314yes—
130C22×C17C2×C22⋊C811yescharacteristic
131C22×C17C2×#9911yescharacteristic
132C22×C17C2×#10011yescharacteristic
133C22×C17#41812yes—
134C22×C17#42112yes—
135C4×C17—22no—
136C4×C17—21no—
137C4×C17—21no—
138Dic17—24no—
139Dic17—42no—
140Dic17—82no—
141C23×D4#59511yescharacteristic
142C22×C4⋊C4—171no—
143C22×C22⋊C4—342no—
144C22×C42—171no—
145C2×C22⋊C8—342no—
146#390—172no—
147#390—172no—
148#390—342no—
149#336—174no—
150#336—342no—
151C2×C17—24no—
152C2×C17—84no—
153C2×C17C2×#32311yescharacteristic
154C2×C17#147812yes—
155C2×C17#151012yes—
156C2×C17#153311yescharacteristic
157C2×C17#155511yescharacteristic
158C25—21no—
159C22×D4—22no—
160C22×D4—21no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4#362414yescomplemented
164C22×D4C2×#59512yes—
165C23×C4—172no—
166C23×C4—341no—
167C23×C4C2×#59511yescharacteristic
168C22×C8—344no—
169C2×C4⋊C4—172no—
170C2×C4⋊C4—341no—
171C2×C4⋊C4—682no—
172C2×C22⋊C4—344no—
173C2×C22⋊C4—344no—
174C2×C22⋊C4—688no—
175C2×C42—172no—
176C2×C42—171no—
177C2×C42—171no—
178C2×C42—682no—
179C22⋊C8—684no—
180C17#759211yescharacteristic, complemented, Sylow
181C24—22no—
182C24—24no—
183C24—44no—
184C24—44no—
185C24—42no—
186C24—42no—
187C24C22⋊#59511yescharacteristic
188C2×D4—22no—
189C2×D4—24no—
190C2×D4—48no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—48no—
196C2×D4C2×#362414yes—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—172no—
200C22×C4—172no—
201C22×C4—344no—
202C22×C4—344no—
203C22×C4—342no—
204C22×C4—682no—
205C22×C4—684no—
206C22×C4—682no—
207C22×C4C2×#362412yes—
208C22×C4#6324712yes—
209C22×C4C22⋊#59511yescharacteristic
210C22×C4C22×#59511yescharacteristic
211C2×C8—684no—
212C2×C8—684no—
213C2×C8—684no—
214C4⋊C4—682no—
215C4⋊C4—682no—
216C22⋊C4—688no—
217C22⋊C4—688no—
218C42—682no—
219C42—681no—
220C42—681no—
221C23—22no—
222C23—22no—
223C23—24no—
224C23—416no—
225C23—42no—
226C23—42no—
227C23—44no—
228C23—44no—
229C23—48no—
230C23—88no—
231C23—84no—
232C23—84no—
233C23#1058755112yes—
234C23C22⋊#362412yes—
235C23#1058755312yes—
236C23C2×C22⋊#59511yescharacteristic
237D4—44no—
238D4—44no—
239D4—48no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—174no—
247C2×C4—342no—
248C2×C4—342no—
249C2×C4—342no—
250C2×C4—342no—
251C2×C4—342no—
252C2×C4—688no—
253C2×C4—682no—
254C2×C4—684no—
255C2×C4—1364no—
256C2×C4C22⋊#362412yes—
257C2×C4C22×#362411yescharacteristic
258C2×C4C2×#6324711yescharacteristic
259C8—1364no—
260C22—24no—
261C22—22no—
262C22—22no—
263C22—24no—
264C22—44no—
265C22—48no—
266C22—816no—
267C22—84no—
268C22—82no—
269C22—82no—
270C22#1176365214yes—
271C22C2×C22⋊#362411yescharacteristic
272C22C2×#1058755111yescharacteristic
273C22C2×#1058755311yescharacteristic
274C22#1176380112yes—
275C22#1176380712yes—
276C4—21no—
277C4—21no—
278C4—22no—
279C4—344no—
280C4—682no—
281C4—1362no—
282C2—24no—
283C2—84no—
284C2C2×#1176365211yescharacteristic
285C2#1454408312yes—
286C2#1454418912yes—
287C2#1454426311yescharacteristic
288C2#1454431111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
Fitting, prime radicalC23×D4×C17C4
socleC23×C17C2×C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC23×D4#595
hypocenterC17#7592
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #14544674C2semidirect
2. #14543600C2semidirect
6. #11763675C22semidirect
19. C22×D4×C17C8semidirect
163. C22×D4#3624semidirect
180. C17#7592semidirect

Maximal subgroups

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

subgroupindex
#145446742
#145436002
C2×#117627122
#759217