Finite Group Databasebeta

← #17038216

Subgroups — #17038216

Order 4352.

1790 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762847C211yesmaximal, characteristic, complemented
2C2×#11762814C212yesmaximal, complemented
3C2×#11762764C211yesmaximal, characteristic, complemented
4#14531366C212yesmaximal, complemented
5#14529661C214yesmaximal, complemented
6#14529651C214yesmaximal, complemented
7C23×D8×C17C211yesmaximal, characteristic
8C22×#63223C2211yescharacteristic
9C2×#10587612C2212yes—
10C2×#10587612C2211yescharacteristic
11#11762847—24no—
12C2×#10587489C2212yescomplemented
13C2×#10587488C2212yescomplemented
14#11762814—28no—
15C2×#10587503C2218yescomplemented
16#11762764—24no—
17C2×#10587486C2212yescomplemented
18C2×C8×Dic17C2212yescomplemented
19#11762712—24no—
20#11762712C2218yescomplemented
21#11762665—24no—
22C23×D4×C17C2212yes—
23C22×D8×C17—24no—
24C22×D8×C17—24no—
25C22×D8×C17C418yescomplemented
26C22×D8×C17C2212yes—
27C22×D8×C17C2212yes—
28C23×C8×C17C2211yescharacteristic
29C25×C17—22no—
30C22×D4×C17—24no—
31C22×D4×C17—24no—
32C22×D4×C17—24no—
33C22×D4×C17—22no—
34C22×D4×C17—24no—
35C22×D4×C17—24no—
36C22×D4×C17C2×C418yes—
37C22×D4×C17C2314yes—
38C23×C4×C17C2311yescharacteristic
39C2×D8×C17—232no—
40C2×D8×C17—24no—
41C2×D8×C17—24no—
42C2×D8×C17—24no—
43C2×D8×C17—24no—
44C2×D8×C17C2×C418yes—
45C2×D8×C17C2×C418yes—
46C22×C8×C17C2×C414yes—
47C22×C8×C17D414yes—
48C22×C8×C17D414yes—
49C22×C8×C17C2311yescharacteristic
50C22×C8×C17C2311yescharacteristic
51C2×C22⋊Dic17—22no—
52C2×C22⋊Dic17—216no—
53C2×C22⋊Dic17D412yes—
54#10587612—22no—
55#10587612—22no—
56#10587612—24no—
57#10587612—24no—
58C2×C4⋊Dic17D412yes—
59C2×C4⋊Dic17C2314yes—
60C2×C4×Dic17—22no—
61C2×C4×Dic17C2312yes—
62C2×#63223—22no—
63C2×#63223—22no—
64C2×#63223D414yes—
65C22×#3622C2312yes—
66#10587503—216no—
67#10587489—24no—
68#10587488—24no—
69#10587486—24no—
70C8×Dic17—24no—
71C24×C17—24no—
72C24×C17—28no—
73C24×C17—44no—
74C24×C17—44no—
75C24×C17—88no—
76C24×C17C2×D411yescharacteristic
77C2×D4×C17—216no—
78C2×D4×C17—216no—
79C2×D4×C17—28no—
80C2×D4×C17—24no—
81C2×D4×C17—24no—
82C2×D4×C17—24no—
83C2×D4×C17—24no—
84C2×D4×C17—24no—
85C2×D4×C17—48no—
86C2×D4×C17—44no—
87C2×D4×C17—48no—
88C2×D4×C17—44no—
89C2×D4×C17C22×C418yes—
90C22×C4×C17C22×C412yes—
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17C2×D412yes—
93C22×C4×C17C2×D412yes—
94C22×C4×C17C2×D412yes—
95C22×C4×C17#4112yes—
96C22×C4×C17#4112yes—
97C22×C4×C17C2411yescharacteristic
98C22×Dic17—24no—
99C22×Dic17C2×D414yes—
100D8×C17—216no—
101D8×C17—216no—
102C2×C8×C17C22⋊C4116yescomplemented
103C2×C8×C17C22×C412yes—
104C2×C8×C17C22×C412yes—
105C2×C8×C17C2×D412yes—
106C2×C8×C17C2×D412yes—
107C2×C8×C17C2×D412yes—
108C2×C8×C17C2×D412yes—
109C22⋊Dic17—24no—
110C22⋊Dic17—44no—
111C22⋊Dic17—816no—
112C4⋊Dic17—24no—
113C4⋊Dic17—24no—
114C4⋊Dic17—24no—
115C4×Dic17—22no—
116C4×Dic17—22no—
117C4×Dic17—44no—
118#63223—28no—
119C2×#3622—22no—
120C2×#3622—22no—
121C2×#3622C2×D414yes—
122#21233—171nomaximal, Sylow
123C23×C17—24no—
124C23×C17—28no—
125C23×C17—44no—
126C23×C17—44no—
127C23×C17—416no—
128C23×C17—416no—
129C23×C17—816no—
130C23×C17—816no—
131C23×C17—816no—
132C23×C17C4×D412yes—
133C23×C17#12112yes—
134C23×C17#12712yes—
135C23×C17C22×D411yescharacteristic
136D4×C17—28no—
137D4×C17—28no—
138D4×C17—44no—
139D4×C17—44no—
140D4×C17—416no—
141C2×C4×C17C2×C22⋊C414yes—
142C2×C4×C17C4×D414yes—
143C2×C4×C17C4×D412yes—
144C2×C4×C17#12114yes—
145C2×C4×C17#12112yes—
146C2×C4×C17#12414yes—
147C2×C4×C17#12712yes—
148C2×C4×C17C23×C411yescharacteristic
149C2×C4×C17C22×D411yescharacteristic
150C2×C4×C17C22×D411yescharacteristic
151C2×C4×C17C22×D411yescharacteristic
152C2×C4×C17C2×#4111yescharacteristic
153C2×C4×C17C2×#4111yescharacteristic
154C2×Dic17—24no—
155C2×Dic17—28no—
156C2×Dic17—44no—
157C2×Dic17—48no—
158C8×C17C2×C22⋊C414yes—
159C8×C17C2×C22⋊C414yes—
160#3622—24no—
161C23×D8D1711yescharacteristic
162C2×#427—171no—
163C2×#418—172no—
164C2×#405—171no—
165#2309—172no—
166#1972—174no—
167#1970—174no—
168C22×C17—24no—
169C22×C17—24no—
170C22×C17—24no—
171C22×C17—416no—
172C22×C17—48no—
173C22×C17—832no—
174C22×C17—88no—
175C22×C17—88no—
176C22×C17#39014yes—
177C22×C17C2×C4×D411yescharacteristic
178C22×C17C2×#12111yescharacteristic
179C22×C17C2×#12711yescharacteristic
180C22×C17#57614yes—
181C4×C17#39014yes—
182C4×C17C22×C22⋊C411yescharacteristic
183C4×C17C2×C4×D411yescharacteristic
184C4×C17C2×#12111yescharacteristic
185C4×C17C2×#12411yescharacteristic
186Dic17—44no—
187Dic17—84no—
188C23×D4C2×D1712yes—
189C22×D8—24no—
190C22×D8—24no—
191C22×D8Dic1718yescomplemented
192C22×D8C2×D1712yes—
193C22×D8C2×D1712yes—
194C22×M16—171no—
195C23×C8C2×D1711yescharacteristic
196C2×#117—171no—
197C2×#117—172no—
198#427—344no—
199C2×#107—172no—
200C2×#106—172no—
201#418—348no—
202C2×#102—178no—
203#405—344no—
204C2×#97—172no—
205C2×C4×C8—172no—
206#390—178no—
207#390—344no—
208#379—344no—
209C2×C17—24no—
210C2×C17—88no—
211C2×C17C2×#39011yescharacteristic
212C2×C17#294812yes—
213C2×C17#305812yes—
214C2×C17#315512yes—
215C25—22no—
216C22×D4—24no—
217C22×D4—22no—
218C22×D4—24no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—24no—
222C22×D4C2×Dic1718yes—
223C22×D4C22×D1714yes—
224C23×C4C22×D1711yescharacteristic
225C2×D8—232no—
226C2×D8—24no—
227C2×D8—24no—
228C2×D8—24no—
229C2×D8—24no—
230C2×D8C2×Dic1718yes—
231C2×D8C2×Dic1718yes—
232C2×M16—174no—
233C2×M16—342no—
234C2×M16—342no—
235C22×C8—172no—
236C22×C8C2×Dic1714yes—
237C22×C8C22⋊D1714yes—
238C22×C8C22⋊D1714yes—
239C22×C8C22×D1711yescharacteristic
240C22×C8C22×D1711yescharacteristic
241#117—342no—
242#117—342no—
243#117—344no—
244#117—344no—
245C2×C4⋊C4—172no—
246C2×C4⋊C4—174no—
247C2×C22⋊C4—172no—
248C2×C22⋊C4—342no—
249C2×C22⋊C4—3416no—
250C2×C42—172no—
251C2×C42—342no—
252#107—344no—
253#106—344no—
254#102—3416no—
255#97—344no—
256C4×C8—344no—
257C17#2123311yescharacteristic, complemented, Sylow
258C24—24no—
259C24—28no—
260C24—44no—
261C24—44no—
262C24—88no—
263C24D4×D1711yescharacteristic
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—216no—
270C2×D4—28no—
271C2×D4—216no—
272C2×D4—48no—
273C2×D4—44no—
274C2×D4—48no—
275C2×D4—44no—
276C2×D4C22×Dic1718yes—
277C22×C4—174no—
278C22×C4—344no—
279C22×C4D4×D1712yes—
280C22×C4D4×D1711yescharacteristic
281C22×C4#6325412yes—
282C22×C4#6325412yes—
283C22×C4C22×Dic1712yes—
284C22×C4C2×C22⋊D1712yes—
285C22×C4C2×C22⋊D1712yes—
286C22×C4C23×D1711yescharacteristic
287D8—216no—
288D8—216no—
289M16—348no—
290C2×C8—174no—
291C2×C8—342no—
292C2×C8—342no—
293C2×C8C22⋊Dic17116yescomplemented
294C2×C8C22×Dic1712yes—
295C2×C8C22×Dic1712yes—
296C2×C8C2×C22⋊D1712yes—
297C2×C8C2×C22⋊D1712yes—
298C2×C8C2×C22⋊D1712yes—
299C2×C8C2×C22⋊D1712yes—
300C4⋊C4—344no—
301C4⋊C4—344no—
302C4⋊C4—344no—
303C22⋊C4—344no—
304C22⋊C4—684no—
305C22⋊C4—13616no—
306C42—342no—
307C42—342no—
308C42—684no—
309C23—24no—
310C23—28no—
311C23—416no—
312C23—44no—
313C23—44no—
314C23—416no—
315C23—816no—
316C23—816no—
317C23—816no—
318C23D4×Dic1712yes—
319C23#1058762412yes—
320C23#1058762612yes—
321C23C2×D4×D1711yescharacteristic
322D4—28no—
323D4—28no—
324D4—416no—
325D4—44no—
326D4—44no—
327C2×C4—344no—
328C2×C4—348no—
329C2×C4—684no—
330C2×C4—688no—
331C2×C4D4×Dic1712yes—
332C2×C4D4×Dic1714yes—
333C2×C4#1058762214yes—
334C2×C4#1058762412yes—
335C2×C4#1058762414yes—
336C2×C4#1058762612yes—
337C2×C4C2×C22⋊Dic1714yes—
338C2×C4C2×D4×D1711yescharacteristic
339C2×C4C2×#6325411yescharacteristic
340C2×C4C2×#6325411yescharacteristic
341C2×C4C23×Dic1711yescharacteristic
342C2×C4C22×C22⋊D1711yescharacteristic
343C2×C4C22×C22⋊D1711yescharacteristic
344C8—344no—
345C8C2×C22⋊Dic1714yes—
346C8C2×C22⋊Dic1714yes—
347C22—24no—
348C22—24no—
349C22—24no—
350C22—416no—
351C22—48no—
352C22—832no—
353C22—88no—
354C22—88no—
355C22#1176271214yes—
356C22C2×D4×Dic1711yescharacteristic
357C22C2×#1058762411yescharacteristic
358C22C2×#1058762611yescharacteristic
359C22#1176361014yes—
360C4—684no—
361C4—1364no—
362C4#1176271214yes—
363C4C22×C22⋊Dic1711yescharacteristic
364C4C2×D4×Dic1711yescharacteristic
365C4C2×#1058762411yescharacteristic
366C4C2×#1058762211yescharacteristic
367C2—24no—
368C2—88no—
369C2C2×#1176271211yescharacteristic
370C2#1453763612yes—
371C2#1453842812yes—
372C2#1453926512yes—
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
Fitting, prime radicalC23×D8×C17C2
socleC23×C17C22×D4
cosocleC22×C4×C17C24
hypercenterC23×D8D17
hypocenterC17#21233
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D8

Complements

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

NH ≅ G/Ntype
1. C2×#11762847C2semidirect
2. C2×#11762814C2semidirect
3. C2×#11762764C2semidirect
4. #14531366C2semidirect
5. #14529661C2semidirect
6. #14529651C2semidirect
12. C2×#10587489C22semidirect
13. C2×#10587488C22semidirect
15. C2×#10587503C22semidirect
17. C2×#10587486C22semidirect
18. C2×C8×Dic17C22semidirect
20. #11762712C22semidirect
25. C22×D8×C17C4semidirect
102. C2×C8×C17C22⋊C4semidirect
191. C22×D8Dic17semidirect
257. C17#21233semidirect
293. C2×C8C22⋊Dic17semidirect

Maximal subgroups

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

subgroupindex
C2×#117628472
C2×#117628142
C2×#117627642
#145313662
#145296612
#145296512
C23×D8×C172
#2123317