Finite Group Databasebeta

← #17969025

Subgroups — #17969025

Order 4352.

890 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D17⋊#433C211yesmaximal, characteristic
2#14545246C211yesmaximal, characteristic
3#14544585C212yesmaximal, complemented
4#14544577C212yesmaximal, complemented
5#14544573C212yesmaximal, complemented
6C4⋊C8×#595C212yesmaximal, complemented
7#14544524C212yesmaximal
8C8×C22⋊#595C212yesmaximal
9D17×#424C211yesmaximal, characteristic
10#11763834C414yescomplemented
11#11763833C414yescomplemented
12#11763810C2212yes—
13#11763777C2212yes—
14#11763768C2211yescharacteristic
15C2×#10587532C2212yes—
16C4×#63242C2211yescharacteristic
17C2×C8×#595C2212yes—
18C2×C8×#595C2212yes—
19C2×C8×#595C2212yes—
20#11763759C414yes—
21#11763758C414yes—
22C4×C4⋊#595C2212yes—
23C4×C22⋊#595C2212yes—
24#11763745C2211yescharacteristic
25C42×#595C2211yescharacteristic
26#11763674C2212yes—
27#11763674C2212yes—
28#11763674C2212yes—
29#11763674C2212yes—
30C2×M16×D17C2211yescharacteristic
31C22×C8×D17C2211yescharacteristic
32#117×D17C2211yescharacteristic
33#11762837C411yescharacteristic, complemented
34#11762836C411yescharacteristic, complemented
35#11762835C411yescharacteristic, complemented
36#11762834C412yescomplemented
37#11762834C412yescomplemented
38C4⋊C8×D17C2212yes—
39C4⋊C8×D17C2212yes—
40C17×#424C411yescharacteristic, complemented
41C22×C4×D17C2311yescharacteristic
42C2×C22⋊#595C2×C414yes—
43C2×C4⋊#595C2×C414yes—
44C2×C4×#595C2×C414yes—
45C2×C4×#595C2312yes—
46C2×C4×#595C2312yes—
47C2×#63242C2312yes—
48C2×#63242C2312yes—
49C2×M16×C17C2×C411yescharacteristic
50C22×C8×C17C2×C411yescharacteristic
51C17×#117C2×C411yescharacteristic
52#10587612C2×C411yescharacteristic
53C2×#63223C2×C411yescharacteristic
54C22×#3622C2×C411yescharacteristic
55M16×D17—42no—
56C2×#63218C2×C412yes—
57C2×C8×D17—42no—
58C2×C8×D17C2×C412yes—
59C2×C8×D17C2312yes—
60C2×C8×D17C2312yes—
61#10587578C2×C412yes—
62C4⋊C4×D17C2×C412yes—
63C22⋊C4×D17C2×C412yes—
64#10587563C2×C412yes—
65#10587558C2×C411yescharacteristic
66#10587558C2×C411yescharacteristic
67C42×D17C2311yescharacteristic
68C42×D17C2311yescharacteristic
69C22⋊#3624C2×C414yes—
70(C2×#595)⋊C4C2×C414yes—
71(C2×#595)⋊C4C2×C414yes—
72(C2×#595)⋊C4C2×C414yes—
73#10587544C2×C414yes—
74#10587543C2×C414yes—
75#10587542C2×C414yes—
76C4⋊#3624C2×C414yes—
77C4×#3624C2×C414yes—
78#10587532—42no—
79C8×#595—42no—
80C8×#595—42no—
81C8×#595—42no—
82C17×C4⋊C8C2×C412yes—
83C17×C4⋊C8C2×C412yes—
84#10587485C2×C412yes—
85#10587485C2×C412yes—
86#10587485C2×C412yes—
87#10587485C2×C412yes—
88C4⋊#3622C2×C412yes—
89C4⋊#3622C2×C412yes—
90C23×D17C22×C411yescharacteristic
91C22×#595C22×C414yes—
92C22×#595C22×C414yes—
93C22×C4×C17C22×C411yescharacteristic
94C22×Dic17C22×C411yescharacteristic
95C2×C4×D17—22no—
96C2×C4×D17—22no—
97C2×C4×D17C22×C412yes—
98C2×C4×D17C22×C412yes—
99C2×C4×D17C22×C411yescharacteristic
100C2×C4×D17C22×C411yescharacteristic
101C2×C4×D17C22×C411yescharacteristic
102C2×C4×D17C2411yescharacteristic
103C22⋊#595—28no—
104C2×#3624C22×C414yes—
105C2×#3624C22×C414yes—
106C4⋊#595—28no—
107C4×#595—28no—
108C4×#595—22no—
109C4×#595—22no—
110C4×#595—22no—
111C4×#595—22no—
112#63242—22no—
113#63242—22no—
114#63242—22no—
115#63242—22no—
116M16×C17—41no—
117C2×C8×C17—41no—
118C2×C8×C17C22×C412yes—
119C2×C8×C17C22×C412yes—
120C4⋊C4×C17C4212yes—
121C22⋊C4×C17C4212yes—
122C42×C17C22×C411yescharacteristic
123C42×C17C22×C411yescharacteristic
124C22⋊Dic17C4212yes—
125#63227C4212yes—
126#63227C4212yes—
127C4⋊Dic17C4212yes—
128#63225C4212yes—
129#63225C4212yes—
130C4×Dic17C22×C411yescharacteristic
131C4×Dic17C22×C411yescharacteristic
132C4×Dic17C22×C412yes—
133C4×Dic17C22×C412yes—
134#63223—41no—
135C2×#3622—41no—
136C2×#3622C22×C412yes—
137C2×#3622C22×C412yes—
138#63218—42no—
139C8×D17—22no—
140C8×D17—22no—
141C8×D17—22no—
142C8×D17—22no—
143C8×D17—42no—
144#20552—171nomaximal, Sylow
145C23×C17C2×C4211yescharacteristic
146C22×D17—22no—
147C22×D17—24no—
148C22×D17C2×C4211yescharacteristic
149C22×D17C23×C411yescharacteristic
150C2×#595—24no—
151C2×#595—24no—
152C2×#595—24no—
153C2×#595—24no—
154C2×#595—24no—
155C2×#595#13118yes—
156C2×C4×C17—21no—
157C2×C4×C17—21no—
158C2×C4×C17C2×C4212yes—
159C2×C4×C17C2×C4212yes—
160C2×C4×C17C2×C4211yescharacteristic
161C2×C4×C17C23×C411yescharacteristic
162C2×Dic17—22no—
163C2×Dic17C2×C4212yes—
164C2×Dic17C2×C4212yes—
165C2×Dic17C2×C4211yescharacteristic
166C2×Dic17C23×C411yescharacteristic
167C4×D17—22no—
168C4×D17—24no—
169C4×D17—24no—
170C4×D17—21no—
171C4×D17—21no—
172C4×D17—22no—
173C4×D17H(5,𝔽2)12yes—
174C4×D172−1+412yes—
175#3624—24no—
176#3624—24no—
177C8×C17—22no—
178C8×C17—22no—
179#3622—22no—
180#3622—22no—
181C2×#433—171no—
182C2×#424—171no—
183#2298—171no—
184#1771—172no—
185C8×C4⋊C4—172no—
186#1768—172no—
187C4×C4⋊C8—172no—
188#1753—172no—
189C8×C22⋊C4—172no—
190C22×C17—22no—
191C22×C17—21no—
192C22×C17C22×C4211yescharacteristic
193C2×D17—22no—
194C2×D17—21no—
195C2×D17—44no—
196C2×D17#52012yes—
197C2×D17C2×#13114yes—
198#595—28no—
199#595—44no—
200C4×C17—22no—
201C4×C17—22no—
202C4×C17—21no—
203C4×C17#51811yescharacteristic
204C4×C17#51911yescharacteristic
205Dic17—22no—
206Dic17—22no—
207Dic17—21no—
208Dic17#52012yes—
209C22×M16—171no—
210C23×C8—171no—
211C2×#117—171no—
212#433—174no—
213#433—174no—
214#424—171no—
215#424—171no—
216#424—171no—
217#424—172no—
218#424—172no—
219#424#59511yescharacteristic, complemented
220C2×C4⋊C8—172no—
221C2×C4⋊C8—172no—
222C2×C4⋊C8—172no—
223C2×C22⋊C8—172no—
224C2×#97—171no—
225C2×#97—172no—
226C2×C4×C8—171no—
227C2×C4×C8—172no—
228C2×C4×C8—172no—
229C2×C4×C8—172no—
230#388—174no—
231#387—174no—
232C4×C4⋊C4—172no—
233C4×C22⋊C4—172no—
234#376—171no—
235C43—171no—
236#336—172no—
237#336—172no—
238#336—172no—
239#336—172no—
240C2×C17—41no—
241C2×C17#227411yescharacteristic
242C2×C17C4×#13112yes—
243D17—41no—
244D17#299414yes—
245C23×C4—171no—
246C2×M16—171no—
247C2×M16—172no—
248C2×M16—682no—
249C2×M16C2×#59511yescharacteristic
250C22×C8—172no—
251C22×C8—172no—
252C22×C8—171no—
253C22×C8—172no—
254C22×C8—172no—
255C22×C8—172no—
256C22×C8—682no—
257C22×C8C2×#59511yescharacteristic
258#117—171no—
259#117—172no—
260#117—172no—
261#117—171no—
262#117—171no—
263#117C2×#59511yescharacteristic
264C2×C4⋊C4—172no—
265C2×C4⋊C4—174no—
266C2×C22⋊C4—172no—
267C2×C22⋊C4—174no—
268C2×C42—171no—
269C2×C42—171no—
270C2×C42—174no—
271C2×C42—172no—
272C2×C42—172no—
273C4⋊C8—174no—
274C4⋊C8—174no—
275C4⋊C8—172no—
276C4⋊C8—172no—
277C4⋊C8—172no—
278C4⋊C8—172no—
279C4⋊C8—172no—
280C4⋊C8—172no—
281C4⋊C8C2×#59512yes—
282C4⋊C8C2×#59512yes—
283C22⋊C8—174no—
284C22⋊C8—174no—
285#97—174no—
286#97—682no—
287C4×C8—174no—
288C4×C8—682no—
289C4×C8—682no—
290C4×C8—682no—
291#95—174no—
292#95—174no—
293#95—174no—
294C17#2055211yescharacteristic, complemented, Sylow
295C24—171no—
296C22×C4—171no—
297C22×C4—174no—
298C22×C4—171no—
299C22×C4—171no—
300C22×C4—172no—
301C22×C4—172no—
302C22×C4—171no—
303C22×C4—171no—
304C22×C4—174no—
305C22×C4—342no—
306C22×C4—342no—
307C22×C4C22×#59511yescharacteristic
308M16—41no—
309M16—682no—
310M16—681no—
311C2×C8—41no—
312C2×C8—174no—
313C2×C8—174no—
314C2×C8—172no—
315C2×C8—172no—
316C2×C8—342no—
317C2×C8—342no—
318C2×C8—342no—
319C2×C8—342no—
320C2×C8—342no—
321C2×C8—342no—
322C2×C8—342no—
323C2×C8—342no—
324C2×C8—682no—
325C2×C8—681no—
326C2×C8C22×#59512yes—
327C2×C8C22×#59512yes—
328C4⋊C4—172no—
329C4⋊C4—172no—
330C4⋊C4—172no—
331C4⋊C4—348no—
332C4⋊C4C4×#59512yes—
333C22⋊C4—172no—
334C22⋊C4—172no—
335C22⋊C4—172no—
336C22⋊C4—348no—
337C22⋊C4C4×#59512yes—
338C42—172no—
339C42—172no—
340C42—171no—
341C42—171no—
342C42—342no—
343C42—342no—
344C42—342no—
345C42—342no—
346C42—348no—
347C42C22×#59511yescharacteristic
348C42C22×#59511yescharacteristic
349C23—171no—
350C23—171no—
351C23—344no—
352C23—342no—
353C23C2×C4×#59511yescharacteristic
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—171no—
357C2×C4—172no—
358C2×C4—172no—
359C2×C4—178no—
360C2×C4—172no—
361C2×C4—172no—
362C2×C4—171no—
363C2×C4—342no—
364C2×C4—344no—
365C2×C4—344no—
366C2×C4—342no—
367C2×C4—342no—
368C2×C4—344no—
369C2×C4—344no—
370C2×C4—341no—
371C2×C4—341no—
372C2×C4—344no—
373C2×C4—344no—
374C2×C4—344no—
375C2×C4C2×C4×#59512yes—
376C2×C4C2×C4×#59512yes—
377C2×C4C2×C4×#59511yescharacteristic
378C2×C4C23×#59511yescharacteristic
379C8—22no—
380C8—22no—
381C8—344no—
382C8—344no—
383C8—342no—
384C8—342no—
385C22—21no—
386C22—22no—
387C22—172no—
388C22—174no—
389C22—341no—
390C22—342no—
391C22—684no—
392C22C22×C4×#59511yescharacteristic
393C4—21no—
394C4—22no—
395C4—22no—
396C4—172no—
397C4—348no—
398C4—341no—
399C4—342no—
400C4—342no—
401C4—684no—
402C4#1176387611yescharacteristic
403C4#1176387711yescharacteristic
404C2—41no—
405C2—174no—
406C2—681no—
407C2#1454522211yescharacteristic
408C2#131×#59512yes—
subgroupG/H
center, FrattiniC2×C4C23×#595
commutator, socleC22×C17C22×C42
FittingC17×#424C4
prime radicalC23×D17C22×C4
cosocleC2×C4×D17C24
hypercenter#424#595
hypocenterC17#20552
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ #424

Complements

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

NH ≅ G/Ntype
3. #14544585C2semidirect
4. #14544577C2semidirect
5. #14544573C2semidirect
6. C4⋊C8×#595C2semidirect
10. #11763834C4semidirect
11. #11763833C4semidirect
33. #11762837C4semidirect
34. #11762836C4semidirect
35. #11762835C4semidirect
36. #11762834C4semidirect
37. #11762834C4semidirect
40. C17×#424C4semidirect
219. #424#595semidirect
294. C17#20552semidirect

Maximal subgroups

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

subgroupindex
D17⋊#4332
#145452462
#145445852
#145445772
#145445732
C4⋊C8×#5952
#145445242
C8×C22⋊#5952
D17×#4242
#2055217