Finite Group Databasebeta

← #17091359

Subgroups — #17091359

Order 4352.

1082 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533270C212yesmaximal, complemented
2#14533092C211yesmaximal, characteristic
3#14529950C214yesmaximal, complemented
4#14529770C212yesmaximal, complemented
5#14529197C214yesmaximal
6C4×#10587505C211yesmaximal, characteristic
7C17×#3501C211yesmaximal, characteristic
8C2×#10587621C2212yes—
9C22×C4⋊Dic17C2211yescharacteristic
10C2×#10587505C2212yes—
11#11762748—24no—
12#11762739—21no—
13#11762734—21no—
14#11762716—22no—
15#11762707—21no—
16#11762674—22no—
17#11762670C2218yes—
18#11762670C2212yes—
19#11762669—21no—
20#11762669C2211yescharacteristic
21C4×C22⋊Dic17C2212yes—
22#11762576C2212yes—
23#11762576C2218yes—
24C17×#554—22no—
25C17×#554—24no—
26D4×Q8×C17—22no—
27C17×#544—21no—
28C17×#543—21no—
29C17×#543C2212yes—
30C17×#533—21no—
31C2×C17×C22⋊Q8—21no—
32C2×C17×C22⋊Q8C2212yes—
33C17×#518C2212yes—
34C2×C4×D4×C17C2211yescharacteristic
35C22×Q8×C17—21no—
36C22×Q8×C17D412yes—
37C22×D4×C17C2311yescharacteristic
38C23×C4×C17D411yescharacteristic
39C23×C4×C17C2311yescharacteristic
40C23×Dic17C2312yes—
41C17×C4⋊Q8—21no—
42C17×C4⋊Q8—21no—
43C17×C4⋊Q8—24no—
44C17×#125—22no—
45C17×#125—24no—
46C17×C22⋊Q8—22no—
47C17×C22⋊Q8—22no—
48C17×C22⋊Q8—24no—
49C17×C22⋊Q8—24no—
50C17×C22⋊Q8—22no—
51C17×C22⋊Q8—22no—
52C17×C22⋊Q8—22no—
53C17×C22⋊Q8—22no—
54C17×C22⋊Q8—22no—
55C17×C22⋊Q8—22no—
56C17×C22⋊Q8—24no—
57C17×C22⋊Q8—24no—
58C4×Q8×C17—22no—
59C4×Q8×C17—22no—
60C4×D4×C17—22no—
61C4×D4×C17—22no—
62C4×D4×C17—24no—
63C4×D4×C17—24no—
64C17×#117—22no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—22no—
67C2×C4⋊C4×C17D411yescharacteristic
68C2×C4⋊C4×C17D414yes—
69C2×C4⋊C4×C17D412yes—
70C2×C22⋊C4×C17—22no—
71C2×C22⋊C4×C17D411yescharacteristic
72C2×C22⋊C4×C17C2311yescharacteristic
73C2×C22⋊C4×C17C2314yes—
74C2×C42×C17D411yescharacteristic
75C2×C22⋊Dic17—22no—
76C2×C22⋊Dic17C2314yes—
77C2×C22⋊Dic17C2312yes—
78#10587621—28no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17—22no—
82C2×C4⋊Dic17—21no—
83C2×C4⋊Dic17—22no—
84C2×#63225—22no—
85C2×C4×Dic17—22no—
86#10587505—24no—
87#10587502—22no—
88#10587502—28no—
89#10587502—21no—
90#10587502—22no—
91#10587502—22no—
92#10587502—28no—
93C24×C17C2×D411yescharacteristic
94C24×C17C2411yescharacteristic
95C2×Q8×C17—22no—
96C2×Q8×C17—22no—
97C2×Q8×C17—22no—
98C2×Q8×C17—22no—
99C2×Q8×C17—44no—
100C2×Q8×C17—82no—
101C2×D4×C17—22no—
102C2×D4×C17—22no—
103C2×D4×C17#4114yes—
104C22×C4×C17—22no—
105C22×C4×C17—22no—
106C22×C4×C17—21no—
107C22×C4×C17—21no—
108C22×C4×C17—42no—
109C22×C4×C17—42no—
110C22×C4×C17C2×D411yescharacteristic
111C22×C4×C17C2×D412yes—
112C22×C4×C17C2×D411yescharacteristic
113C22×C4×C17C2×D411yescharacteristic
114C22×C4×C17C2×D411yescharacteristic
115C22×C4×C17C2×D414yes—
116C22×C4×C17C2×D414yes—
117C22×C4×C17C2×D411yescharacteristic
118C22×C4×C17C2×D412yes—
119C22×Dic17—24no—
120C22×Dic17—22no—
121C22×Dic17—24no—
122C22×Dic17—22no—
123C22×Dic17—24no—
124C22×Dic17—24no—
125C4⋊C4×C17—24no—
126C4⋊C4×C17—22no—
127C4⋊C4×C17—22no—
128C4⋊C4×C17—24no—
129C4⋊C4×C17—22no—
130C4⋊C4×C17—22no—
131C4⋊C4×C17—22no—
132C4⋊C4×C17—24no—
133C4⋊C4×C17—24no—
134C4⋊C4×C17—22no—
135C4⋊C4×C17—22no—
136C22⋊C4×C17—24no—
137C22⋊C4×C17—24no—
138C22⋊C4×C17—22no—
139C22⋊C4×C17—24no—
140C22⋊C4×C17—22no—
141C42×C17—24no—
142C42×C17—21no—
143C42×C17—21no—
144C22⋊Dic17—24no—
145C22⋊Dic17—24no—
146C22⋊Dic17—24no—
147C22⋊Dic17—44no—
148C4⋊Dic17—42no—
149C4⋊Dic17—41no—
150C4⋊Dic17—42no—
151C4⋊Dic17—41no—
152#63225—44no—
153C4×Dic17—44no—
154#25948—171nomaximal, Sylow
155C23×C17—21no—
156C23×C17—21no—
157C23×C17—41no—
158C23×C17—41no—
159C23×C17—41no—
160C23×C17—41no—
161C23×C17#12412yes—
162C23×C17C22×D411yescharacteristic
163C23×C17C22×D412yes—
164C23×C17C2×#4112yes—
165C23×C172−1+412yes—
166Q8×C17—44no—
167Q8×C17—82no—
168Q8×C17—82no—
169Q8×C17—82no—
170D4×C17—28no—
171C2×C4×C17—21no—
172C2×C4×C17—21no—
173C2×C4×C17—22no—
174C2×C4×C17—24no—
175C2×C4×C17—21no—
176C2×C4×C17—24no—
177C2×C4×C17—24no—
178C2×C4×C17—21no—
179C2×C4×C17—42no—
180C2×C4×C17—41no—
181C2×C4×C17—42no—
182C2×C4×C17—41no—
183C2×C4×C17—44no—
184C2×C4×C17—41no—
185C2×C4×C17—44no—
186C2×C4×C17—81no—
187C2×C4×C17—82no—
188C2×C4×C17—82no—
189C2×C4×C17#12011yescharacteristic
190C2×C4×C17#12012yes—
191C2×C4×C17#12011yescharacteristic
192C2×C4×C17#12412yes—
193C2×Dic17—44no—
194C2×Dic17—42no—
195C2×Dic17—44no—
196C2×Dic17—42no—
197C2×Dic17—42no—
198C2×Dic17—42no—
199C2×Dic17—42no—
200C2×Dic17—42no—
201C2×Dic17—44no—
202C2×Dic17—48no—
203#3501D1711yescharacteristic
204#2503—172no—
205#2486—171no—
206#2030—174no—
207#1996—172no—
208#1890—174no—
209C4×#99—171no—
210C22×C17—22no—
211C22×C17—21no—
212C22×C17—22no—
213C22×C17—21no—
214C22×C17—22no—
215C22×C17—42no—
216C22×C17—42no—
217C22×C17—42no—
218C22×C17—41no—
219C22×C17—41no—
220C22×C17—81no—
221C22×C17C2×#12011yescharacteristic
222C22×C17C2×#12411yescharacteristic
223C22×C17#53611yescharacteristic
224C22×C17#54814yes—
225C4×C17—22no—
226C4×C17—21no—
227C4×C17—22no—
228C4×C17—41no—
229C4×C17—44no—
230C4×C17—44no—
231C4×C17—41no—
232C4×C17#45812yes—
233Dic17—82no—
234Dic17—84no—
235Dic17—82no—
236#554—22no—
237#554—24no—
238D4×Q8—22no—
239#544—21no—
240#543—21no—
241#543C2×D1712yes—
242#533—21no—
243C2×#123—172no—
244C2×C22⋊Q8—21no—
245C2×C22⋊Q8C2×D1712yes—
246#518C2×D1712yes—
247C2×C4×D4C2×D1711yescharacteristic
248C22×C4⋊C4—171no—
249C2×#99—172no—
250#400—344no—
251#398—341no—
252#397—341no—
253#391—342no—
254#389—341no—
255#382—342no—
256#381—178no—
257#381—172no—
258#380—171no—
259#380—341no—
260C4×C22⋊C4—172no—
261#342—178no—
262#342—172no—
263C2×C17—21no—
264C2×C17—24no—
265C2×C17—21no—
266C2×C17—42no—
267C2×C17#243611yescharacteristic
268C2×C17C2×#45811yescharacteristic
269C2×C17#302811yescharacteristic
270C22×Q8—21no—
271C22×Q8C22⋊D1712yes—
272C22×D4C22×D1711yescharacteristic
273C23×C4—172no—
274C23×C4C22⋊D1711yescharacteristic
275C23×C4C22×D1711yescharacteristic
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—24no—
279#125—22no—
280#125—24no—
281#123—348no—
282C22⋊Q8—22no—
283C22⋊Q8—22no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286C22⋊Q8—22no—
287C22⋊Q8—22no—
288C22⋊Q8—22no—
289C22⋊Q8—22no—
290C22⋊Q8—22no—
291C22⋊Q8—22no—
292C22⋊Q8—24no—
293C22⋊Q8—24no—
294C4×Q8—22no—
295C4×Q8—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—24no—
299C4×D4—24no—
300#117—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—341no—
304C2×C4⋊C4—342no—
305C2×C4⋊C4—341no—
306C2×C4⋊C4—342no—
307C2×C4⋊C4—342no—
308C2×C4⋊C4—341no—
309C2×C4⋊C4C22⋊D1711yescharacteristic
310C2×C4⋊C4C22⋊D1712yes—
311C2×C4⋊C4C22⋊D1714yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—172no—
314C2×C22⋊C4—174no—
315C2×C22⋊C4—342no—
316C2×C22⋊C4C22⋊D1711yescharacteristic
317C2×C22⋊C4C22×D1711yescharacteristic
318C2×C22⋊C4C22×D1714yes—
319C2×C42—342no—
320C2×C42C22⋊D1711yescharacteristic
321#99—344no—
322#95—342no—
323#95—341no—
324#95—342no—
325#95—348no—
326#95—342no—
327#95—348no—
328C17#2594811yescharacteristic, complemented, Sylow
329C24C2×C22⋊D1711yescharacteristic
330C24C23×D1711yescharacteristic
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—44no—
336C2×Q8—82no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4#6325614yes—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—344no—
347C22×C4—344no—
348C22×C4—344no—
349C22×C4—344no—
350C22×C4—342no—
351C22×C4—342no—
352C22×C4C2×C22⋊D1711yescharacteristic
353C22×C4C2×C22⋊D1712yes—
354C22×C4C2×C22⋊D1711yescharacteristic
355C22×C4C2×C22⋊D1711yescharacteristic
356C22×C4C2×C22⋊D1714yes—
357C22×C4C2×C22⋊D1714yes—
358C22×C4C2×C22⋊D1711yescharacteristic
359C22×C4C2×C22⋊D1711yescharacteristic
360C22×C4C2×C22⋊D1712yes—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—682no—
373C4⋊C4—684no—
374C4⋊C4—681no—
375C4⋊C4—682no—
376C4⋊C4—681no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—22no—
380C22⋊C4—24no—
381C22⋊C4—22no—
382C22⋊C4—344no—
383C22⋊C4—344no—
384C22⋊C4—344no—
385C22⋊C4—684no—
386C42—24no—
387C42—21no—
388C42—21no—
389C42—684no—
390C23—21no—
391C23—21no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23#1058763312yes—
397C23C2×#6325612yes—
398C23#1058769412yes—
399C23C22×C22⋊D1712yes—
400C23C22×C22⋊D1711yescharacteristic
401Q8—44no—
402Q8—82no—
403Q8—82no—
404Q8—82no—
405D4—28no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—21no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—41no—
421C2×C4—81no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—684no—
425C2×C4—682no—
426C2×C4—684no—
427C2×C4—682no—
428C2×C4—684no—
429C2×C4—688no—
430C2×C4—682no—
431C2×C4—682no—
432C2×C4—682no—
433C2×C4—682no—
434C2×C4#1058763312yes—
435C2×C4#1058763911yescharacteristic
436C2×C4#1058763912yes—
437C2×C4#1058763911yescharacteristic
438C22—22no—
439C22—21no—
440C22—22no—
441C22—21no—
442C22—22no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—41no—
447C22—41no—
448C22—81no—
449C22C2×#1058763911yescharacteristic
450C22C2×#1058763311yescharacteristic
451C22#1176338311yescharacteristic
452C22#1176347014yes—
453C4—22no—
454C4—21no—
455C4—22no—
456C4—41no—
457C4—44no—
458C4—44no—
459C4—41no—
460C4—1362no—
461C4—1364no—
462C4—1362no—
463C4#1176299912yes—
464C2—21no—
465C2—24no—
466C2—21no—
467C2—42no—
468C2#1453243811yescharacteristic
469C2C2×#1176299911yescharacteristic
470C2#1453819011yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC24×C17C24
FrattiniC24C23×D17
FittingC17×#3501C2
socleC22×C17C2×#120
prime radicalC22×D4×C17C23
hypercenter#3501D17
hypocenterC17#25948
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #14533270C2semidirect
3. #14529950C2semidirect
4. #14529770C2semidirect
328. C17#25948semidirect

Maximal subgroups

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

subgroupindex
#145332702
#145330922
#145299502
#145297702
#145291972
C4×#105875052
C17×#35012
#2594817