Finite Group Databasebeta

← #16957036

Subgroups — #16957036

Order 4352.

2230 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762677C211yesmaximal, characteristic, complemented
2C2×#11762655C211yesmaximal, characteristic, complemented
3#14527052C212yesmaximal, complemented
4#14527025C212yesmaximal
5#14527021C214yesmaximal, complemented
6#14527007C214yesmaximal, complemented
7C22×C4×D4×C17C211yesmaximal, characteristic
8C22×C22⋊Dic17C2218yes—
9C22×C4×Dic17C2212yes—
10#11762677—24no—
11#11762655—24no—
12C2×#10587502C2212yes—
13C2×#10587502C2212yes—
14C2×#10587502C2212yes—
15C2×#10587502C2212yes—
16C2×#10587502C2212yes—
17C2×#10587502C2218yes—
18C23×D4×C17C2211yescharacteristic
19C24×C4×C17C2212yes—
20C2×C4×D4×C17—24no—
21C2×C4×D4×C17—24no—
22C2×C4×D4×C17C418yescomplemented
23C22×C4⋊C4×C17C2211yescharacteristic
24C22×C22⋊C4×C17C2212yes—
25C22×C42×C17C2211yescharacteristic
26C25×C17C2312yes—
27C22×D4×C17—28no—
28C22×D4×C17—22no—
29C22×D4×C17—22no—
30C22×D4×C17C2×C414yes—
31C23×C4×C17—24no—
32C23×C4×C17—24no—
33C23×C4×C17—48no—
34C23×C4×C17C2×C418yes—
35C23×C4×C17C2312yes—
36C23×C4×C17C2311yescharacteristic
37C23×C4×C17C2311yescharacteristic
38C23×C4×C17C2311yescharacteristic
39C23×Dic17C2314yes—
40C23×Dic17C2314yes—
41C4×D4×C17—232no—
42C2×C4⋊C4×C17—22no—
43C2×C4⋊C4×C17—22no—
44C2×C4⋊C4×C17C2×C414yes—
45C2×C22⋊C4×C17—24no—
46C2×C22⋊C4×C17—24no—
47C2×C22⋊C4×C17C2×C418yes—
48C2×C42×C17C2×C414yes—
49C2×C42×C17D414yes—
50C2×C42×C17D414yes—
51C2×C22⋊Dic17—216no—
52C2×C22⋊Dic17—216no—
53C2×C22⋊Dic17—48no—
54C2×C4×Dic17—24no—
55C2×C4×Dic17—28no—
56#10587502—24no—
57#10587502—24no—
58#10587502—24no—
59#10587502—24no—
60#10587502—24no—
61#10587502—48no—
62C24×C17—22no—
63C24×C17—22no—
64C24×C17—24no—
65C24×C17—28no—
66C24×C17—28no—
67C24×C17C22×C414yes—
68C24×C17#4118yes—
69C24×C17C2411yescharacteristic
70C2×D4×C17—216no—
71C2×D4×C17—216no—
72C2×D4×C17—216no—
73C2×D4×C17—28no—
74C22×C4×C17—24no—
75C22×C4×C17—24no—
76C22×C4×C17—24no—
77C22×C4×C17—22no—
78C22×C4×C17—22no—
79C22×C4×C17—28no—
80C22×C4×C17—216no—
81C22×C4×C17—416no—
82C22×C4×C17—416no—
83C22×C4×C17—416no—
84C22×C4×C17C22×C414yes—
85C22×C4×C17C22×C412yes—
86C22×C4×C17C22×C412yes—
87C22×C4×C17C22×C412yes—
88C22×C4×C17C2×D412yes—
89C22×C4×C17C2×D412yes—
90C22×C4×C17C2×D412yes—
91C22×C4×C17C2×D412yes—
92C22×C4×C17C2×D412yes—
93C22×C4×C17C2×D412yes—
94C22×Dic17—24no—
95C22×Dic17—24no—
96C22×Dic17—28no—
97C22×Dic17—28no—
98C22×Dic17—24no—
99C22×Dic17—28no—
100C22×Dic17—28no—
101C22×Dic17—24no—
102C22×Dic17—24no—
103C22×Dic17#4118yes—
104C4⋊C4×C17—28no—
105C22⋊C4×C17—216no—
106C42×C17C22⋊C4116yescomplemented
107C22⋊Dic17—432no—
108C4×Dic17—216no—
109#13959—171nomaximal, Sylow
110C23×C17—24no—
111C23×C17—24no—
112C23×C17—28no—
113C23×C17—28no—
114C23×C17—28no—
115C23×C17—28no—
116C23×C17—216no—
117C23×C17—216no—
118C23×C17—44no—
119C23×C17—44no—
120C23×C17—44no—
121C23×C17—44no—
122C23×C17—48no—
123C23×C17—48no—
124C23×C17#11718yes—
125C23×C17C23×C411yescharacteristic
126C23×C17C22×D411yescharacteristic
127C23×C17C22×D411yescharacteristic
128C23×C17C2×#4114yes—
129C23×C17C2×#4114yes—
130C23×C17H(5,𝔽2)14yes—
131D4×C17—232no—
132C2×C4×C17—24no—
133C2×C4×C17—24no—
134C2×C4×C17—24no—
135C2×C4×C17—24no—
136C2×C4×C17—24no—
137C2×C4×C17—24no—
138C2×C4×C17—24no—
139C2×C4×C17—24no—
140C2×C4×C17—24no—
141C2×C4×C17—24no—
142C2×C4×C17—28no—
143C2×C4×C17—432no—
144C2×C4×C17—44no—
145C2×C4×C17—44no—
146C2×C4×C17C2×C22⋊C414yes—
147C2×C4×C17C2×C22⋊C414yes—
148C2×C4×C17C2×C22⋊C414yes—
149C2×Dic17—28no—
150C2×Dic17—216no—
151C2×Dic17—44no—
152C2×Dic17—44no—
153C2×Dic17—48no—
154C2×Dic17—44no—
155C2×Dic17—44no—
156C2×Dic17—44no—
157C2×Dic17—48no—
158C2×Dic17—48no—
159C22×C4×D4D1711yescharacteristic
160C2×#383—171no—
161C2×#376—171no—
162#1445—172no—
163#1438—172no—
164#1437—174no—
165C22⋊#95—174no—
166C22×C17—216no—
167C22×C17—28no—
168C22×C17—44no—
169C22×C17—416no—
170C22×C17—416no—
171C22×C17—48no—
172C22×C17—44no—
173C22×C17C22×C22⋊C411yescharacteristic
174C22×C17C2×#11712yes—
175C22×C17C4×#4112yes—
176C22×C17#51812yes—
177C22×C17#52512yes—
178C22×C17#52512yes—
179C22×C17#52512yes—
180C22×C17#53212yes—
181C22×C17#53512yes—
182C22×C17#53512yes—
183C22×C17#55118yes—
184C22×C17#55318yes—
185C4×C17—24no—
186C4×C17—24no—
187C4×C17—24no—
188C4×C17—44no—
189Dic17—48no—
190Dic17—84no—
191C23×D4C2×D1711yescharacteristic
192C24×C4C2×D1712yes—
193C2×C4×D4—24no—
194C2×C4×D4—24no—
195C2×C4×D4Dic1718yescomplemented
196C22×C4⋊C4C2×D1711yescharacteristic
197C22×C22⋊C4—178no—
198C22×C22⋊C4C2×D1712yes—
199C22×C42—172no—
200C22×C42C2×D1711yescharacteristic
201#383—344no—
202#376—344no—
203C2×#95—172no—
204C2×#95—172no—
205C2×#95—172no—
206C2×#95—172no—
207C2×#95—172no—
208C2×#95—178no—
209C2×C17—48no—
210C2×C17#228011yescharacteristic
211C2×C17#229711yescharacteristic
212C2×C17#230911yescharacteristic
213C2×C17#235314yes—
214C2×C17#259814yes—
215C2×C17#260314yes—
216C25C22×D1712yes—
217C22×D4—28no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4C2×Dic1714yes—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—48no—
224C23×C4—174no—
225C23×C4—174no—
226C23×C4C2×Dic1718yes—
227C23×C4C22×D1712yes—
228C23×C4C22×D1711yescharacteristic
229C23×C4C22×D1711yescharacteristic
230C23×C4C22×D1711yescharacteristic
231C4×D4—232no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4C2×Dic1714yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—3416no—
238C2×C22⋊C4—3416no—
239C2×C22⋊C4—688no—
240C2×C22⋊C4C2×Dic1718yes—
241C2×C42—348no—
242C2×C42—344no—
243C2×C42C2×Dic1714yes—
244C2×C42C22⋊D1714yes—
245C2×C42C22⋊D1714yes—
246#95—344no—
247#95—344no—
248#95—344no—
249#95—344no—
250#95—344no—
251#95—688no—
252C17#1395911yescharacteristic, complemented, Sylow
253C24—24no—
254C24—28no—
255C24—28no—
256C24—22no—
257C24—22no—
258C24#6325218yes—
259C24C22×Dic1714yes—
260C24C23×D1711yescharacteristic
261C2×D4—216no—
262C2×D4—216no—
263C2×D4—216no—
264C2×D4—28no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—28no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—216no—
272C22×C4—416no—
273C22×C4—416no—
274C22×C4—416no—
275C22×C4—178no—
276C22×C4—348no—
277C22×C4—348no—
278C22×C4—344no—
279C22×C4—344no—
280C22×C4—348no—
281C22×C4—344no—
282C22×C4—344no—
283C22×C4—348no—
284C22×C4—344no—
285C22×C4C22×Dic1714yes—
286C22×C4C22×Dic1712yes—
287C22×C4C22×Dic1712yes—
288C22×C4C22×Dic1712yes—
289C22×C4C2×C22⋊D1712yes—
290C22×C4C2×C22⋊D1712yes—
291C22×C4C2×C22⋊D1712yes—
292C22×C4C2×C22⋊D1712yes—
293C22×C4C2×C22⋊D1712yes—
294C22×C4C2×C22⋊D1712yes—
295C4⋊C4—28no—
296C22⋊C4—216no—
297C22⋊C4—6832no—
298C42—3416no—
299C42C22⋊Dic17116yescomplemented
300C23—216no—
301C23—216no—
302C23—24no—
303C23—24no—
304C23—28no—
305C23—28no—
306C23—28no—
307C23—28no—
308C23—48no—
309C23—48no—
310C23—44no—
311C23—44no—
312C23—44no—
313C23—44no—
314C23#1058761218yes—
315C23C2×#6325214yes—
316C23#1058769114yes—
317C23#41×D1714yes—
318C23C23×Dic1711yescharacteristic
319C23C22×C22⋊D1711yescharacteristic
320C23C22×C22⋊D1711yescharacteristic
321D4—232no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—28no—
333C2×C4—432no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—3416no—
337C2×C4—348no—
338C2×C4—688no—
339C2×C4—684no—
340C2×C4—684no—
341C2×C4—688no—
342C2×C4—688no—
343C2×C4—684no—
344C2×C4—684no—
345C2×C4—684no—
346C2×C4C2×C22⋊Dic1714yes—
347C2×C4C2×C22⋊Dic1714yes—
348C2×C4C2×C22⋊Dic1714yes—
349C22—28no—
350C22—216no—
351C22—416no—
352C22—44no—
353C22—48no—
354C22—416no—
355C22—44no—
356C22C22×C22⋊Dic1711yescharacteristic
357C22C2×#1058761212yes—
358C22#41×Dic1712yes—
359C22#1176328412yes—
360C22#1176331612yes—
361C22#1176332012yes—
362C22#1176332012yes—
363C22#1176335512yes—
364C22#1176337612yes—
365C22#1176337612yes—
366C22#1176348818yes—
367C22#1176350818yes—
368C4—24no—
369C4—24no—
370C4—24no—
371C4—44no—
372C4—688no—
373C4—1364no—
374C2—48no—
375C2#1453116711yescharacteristic
376C2#1453127511yescharacteristic
377C2#1453136611yescharacteristic
378C2#1453172414yes—
379C2#1453434414yes—
380C2#1453440514yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC22×C4×D4×C17C2
socle, cosocleC24×C17C24
prime radicalC23×D4×C17C22
hypercenterC22×C4×D4D17
hypocenterC17#13959
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#11762677C2semidirect
2. C2×#11762655C2semidirect
3. #14527052C2semidirect
5. #14527021C2semidirect
6. #14527007C2semidirect
22. C2×C4×D4×C17C4semidirect
106. C42×C17C22⋊C4semidirect
195. C2×C4×D4Dic17semidirect
252. C17#13959semidirect
299. C42C22⋊Dic17semidirect

Maximal subgroups

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

subgroupindex
C2×#117626772
C2×#117626552
#145270522
#145270252
#145270212
#145270072
C22×C4×D4×C172
#1395917