Finite Group Databasebeta

← #16930366

Subgroups — #16930366

Order 4352.

684 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 113 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#14534592C211yesmaximal, characteristic
2#14531397C211yesmaximal, characteristic
3#14527357C211yesmaximal, characteristic, complemented
4#14527355C211yesmaximal, characteristic, complemented
5C2×#11762532C211yesmaximal, characteristic
6#14526651C211yesmaximal, characteristic, complemented
7C17×#1289C211yesmaximal, characteristic, complemented
8C2×#10587610C2211yescharacteristic
9C22×#63227C2211yescharacteristic
10C2×C22⋊C4×D17—21no—
11C2×C22⋊#3622C2211yescharacteristic
12#11762746C412yescomplemented
13#11762731—22no—
14#11762722C412yescomplemented
15#11762684—21no—
16#11762683—22no—
17#11762680—22no—
18#11762669C2211yescharacteristic
19#11762667—21no—
20#11762664—21no—
21#11762547—22no—
22#11762545—22no—
23#11762532—22no—
24C2×C17×C22⋊C8C2211yescharacteristic
25C17×#380C2211yescharacteristic
26C17×#380C2211yescharacteristic
27C24×D17D411yescharacteristic
28C23×C4×C17C2311yescharacteristic
29C23×Dic17D411yescharacteristic
30C22×C4×D17—22no—
31C22×Dic34C2×C411yescharacteristic, complemented
32C22×C8×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—21no—
39C2×#63227—22no—
40C2×#63227—22no—
41C2×#63227—22no—
42C2×#63227—21no—
43C2×#63227—21no—
44C2×#63227—21no—
45C2×#63227—21no—
46C2×#63227C2×C412yes—
47C2×#63227D412yes—
48C2×C4⋊Dic17—21no—
49C2×C4⋊Dic17—22no—
50C2×C4⋊Dic17C2×C411yescharacteristic, complemented
51#10587610—21no—
52#10587610—21no—
53#10587610—21no—
54#10587610—21no—
55C2×#63225—22no—
56C2×#63225—21no—
57C22×#3622—21no—
58C22⋊C4×D17—42no—
59C22⋊C4×D17—42no—
60C17×C22⋊C8—21no—
61C17×C22⋊C8—21no—
62C17×#95C2×C412yescomplemented
63C17×#95D412yes—
64#10587502—22no—
65#10587502—22no—
66#10587502D412yes—
67C22⋊#3622—21no—
68C22⋊#3622—21no—
69C24×C17C2×D411yescharacteristic
70C23×D17—21no—
71C23×D17—21no—
72C23×D17—21no—
73C23×D17—21no—
74C23×D17—81no—
75C23×D17—81no—
76C23×D17C22⋊C412yes—
77C23×D17SD1612yes—
78C23×D17Q1612yes—
79C22×C4×C17—22no—
80C22×C4×C17—22no—
81C22×C4×C17—81no—
82C22×C4×C17C22×C411yescharacteristic
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D411yescharacteristic
85C22×C4×C17C2×D411yescharacteristic
86C22×C4×C17#4111yescharacteristic
87C22×C4×C17#4111yescharacteristic
88C22×Dic17—21no—
89C22×Dic17—21no—
90C22×Dic17—21no—
91C22×Dic17—21no—
92C22×Dic17—22no—
93C22×Dic17—81no—
94C22×Dic17C22⋊C412yes—
95C2×C4×D17—44no—
96C2×C4×D17—82no—
97C2×C4×D17—82no—
98C2×Dic34—21no—
99C2×Dic34—21no—
100C2×Dic34—81no—
101C2×C8×C17—41no—
102C2×C8×C17—81no—
103C4⋊C4×C17—81no—
104C4⋊C4×C17—81no—
105C22⋊C4×C17—41no—
106C22⋊C4×C17—41no—
107C22⋊C4×C17—41no—
108C22⋊C4×C17—41no—
109C22⋊Dic17—41no—
110C22⋊Dic17—41no—
111C22⋊Dic17—41no—
112C22⋊Dic17—41no—
113#63227—22no—
114#63227—22no—
115#63227—22no—
116#63227—22no—
117#63227—44no—
118#63227—82no—
119#63227—82no—
120C4⋊Dic17—21no—
121C4⋊Dic17—21no—
122C4⋊Dic17—82no—
123C4⋊Dic17—81no—
124#63225—41no—
125#63225—41no—
126#63225—82no—
127C2×#3622—41no—
128C2×#3622—81no—
129#9199—171nomaximal, Sylow
130C23×C17—41no—
131C23×C17—41no—
132C23×C17C2×C22⋊C411yescharacteristic
133C23×C17#12011yescharacteristic
134C23×C17#12311yescharacteristic
135C23×C17C2×SD1611yescharacteristic
136C23×C17C2×Q1611yescharacteristic
137C23×C17AGL(1,ℤ/8)11yescharacteristic
138C23×C17#13711yescharacteristic
139C22×D17—22no—
140C22×D17—22no—
141C22×D17—22no—
142C22×D17—22no—
143C22×D17—22no—
144C22×D17—22no—
145C22×D17—42no—
146C22×D17—42no—
147C22×D17—42no—
148C22×D17—42no—
149C22×D17—82no—
150C22×D17—82no—
151C22×D17—81no—
152C22×D17—81no—
153C22×D17—82no—
154C22×D17—82no—
155C22×D17#10314yes—
156C2×C4×C17—41no—
157C2×C4×C17—41no—
158C2×C4×C17—41no—
159C2×C4×C17—41no—
160C2×C4×C17—41no—
161C2×C4×C17—41no—
162C2×C4×C17—42no—
163C2×C4×C17—42no—
164C2×C4×C17—42no—
165C2×C4×C17—42no—
166C2×C4×C17—42no—
167C2×C4×C17—42no—
168C2×C4×C17C4×D411yescharacteristic
169C2×C4×C17C4×D411yescharacteristic
170C2×C4×C17#12111yescharacteristic
171C2×C4×C17#12111yescharacteristic
172C2×Dic17—22no—
173C2×Dic17—42no—
174C2×Dic17—42no—
175C2×Dic17—42no—
176C2×Dic17—42no—
177C2×Dic17—42no—
178C2×Dic17—42no—
179C2×Dic17—41no—
180C2×Dic17—41no—
181C2×Dic17—42no—
182C2×Dic17—81no—
183C4×D17—88no—
184Dic34—81no—
185Dic34—81no—
186C8×C17—81no—
187#3622—81no—
188#2626—171no—
189#2315—171no—
190#1520—171no—
191#1520—171no—
192C2×#323—171no—
193#1289—171no—
194#1289D1711yescharacteristic, complemented
195C22×C17—21no—
196C22×C17—21no—
197C22×C17—21no—
198C22×C17—21no—
199C22×C17—21no—
200C22×C17—21no—
201C22×C17—41no—
202C22×C17—41no—
203C22×C17—41no—
204C22×C17—41no—
205C22×C17#38611yescharacteristic
206C22×C17C2×#10311yescharacteristic
207C22×C17#41711yescharacteristic
208C22×C17#45011yescharacteristic
209C22×C17#45111yescharacteristic
210C22×C17#48311yescharacteristic
211C22×C17#48411yescharacteristic
212C2×D17—24no—
213C2×D17—44no—
214C2×D17—44no—
215C2×D17—42no—
216C2×D17—42no—
217C2×D17—44no—
218C2×D17—82no—
219C2×D17—82no—
220C2×D17—82no—
221C4×C17—21no—
222C4×C17—21no—
223C4×C17—21no—
224C4×C17—21no—
225C4×C17—82no—
226C4×C17—82no—
227Dic17—42no—
228Dic17—82no—
229Dic17—81no—
230Dic17—81no—
231C2×C22⋊Q8—171no—
232C22×C22⋊C4—171no—
233C22×C22⋊C4—341no—
234C2×C22⋊C8—171no—
235C2×C22⋊C8C2×D1711yescharacteristic
236#399—172no—
237#396—342no—
238#393—172no—
239#385—342no—
240#385—341no—
241#385—342no—
242#380—171no—
243#380—341no—
244#380C2×D1711yescharacteristic
245#380C2×D1711yescharacteristic
246#379—341no—
247#328—342no—
248#328—342no—
249#323—342no—
250C2×C17—21no—
251C2×C17—21no—
252C2×C17—21no—
253C2×C17—21no—
254C2×C17—21no—
255C2×C17—21no—
256C2×C17#187211yescharacteristic
257C2×C17#191111yescharacteristic
258C2×C17#191311yescharacteristic
259D17—44no—
260C25—171no—
261C22×Q8—171no—
262C23×C4—171no—
263C23×C4—342no—
264C23×C4C22×D1711yescharacteristic
265C22×C8—21no—
266C22×C8—341no—
267C22⋊Q8—341no—
268C22⋊Q8—341no—
269C22⋊Q8—341no—
270C22⋊Q8—341no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—171no—
274C2×C4⋊C4—341no—
275C2×C4⋊C4—342no—
276C2×C4⋊C4—342no—
277C2×C4⋊C4—341no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—172no—
281C2×C22⋊C4—172no—
282C2×C22⋊C4—341no—
283C2×C22⋊C4—341no—
284C2×C22⋊C4—341no—
285C2×C22⋊C4—341no—
286C2×C22⋊C4—342no—
287C2×C22⋊C4—341no—
288C2×C22⋊C4—341no—
289C2×C22⋊C4—342no—
290C2×C22⋊C4—342no—
291C2×C22⋊C4—682no—
292C2×C22⋊C4—682no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—341no—
296C22⋊C8—341no—
297#95—172no—
298#95—342no—
299#95—342no—
300#95C4×D1712yescomplemented
301#95D6812yes—
302C17#919911yescharacteristic, complemented, Sylow
303C24—172no—
304C24—172no—
305C24—172no—
306C24—341no—
307C24—341no—
308C24—341no—
309C24—341no—
310C24—1361no—
311C24—1361no—
312C24D4×D1711yescharacteristic
313C2×Q8—341no—
314C2×Q8—341no—
315C2×Q8—1361no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—81no—
319C22×C4—172no—
320C22×C4—341no—
321C22×C4—341no—
322C22×C4—341no—
323C22×C4—341no—
324C22×C4—342no—
325C22×C4—684no—
326C22×C4—1362no—
327C22×C4—1362no—
328C22×C4—1361no—
329C22×C4C2×C4×D1711yescharacteristic
330C22×C4C2×D6811yescharacteristic
331C22×C4#6325211yescharacteristic
332C22×C4D4×D1711yescharacteristic
333C22×C4D4×D1711yescharacteristic
334C22×C4#6325611yescharacteristic
335C2×C8—41no—
336C2×C8—81no—
337C2×C8—681no—
338C2×C8—1361no—
339C4⋊C4—81no—
340C4⋊C4—81no—
341C4⋊C4—341no—
342C4⋊C4—341no—
343C4⋊C4—681no—
344C4⋊C4—681no—
345C4⋊C4—1362no—
346C4⋊C4—1361no—
347C4⋊C4—1362no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—342no—
353C22⋊C4—342no—
354C22⋊C4—342no—
355C22⋊C4—342no—
356C22⋊C4—684no—
357C22⋊C4—681no—
358C22⋊C4—681no—
359C22⋊C4—681no—
360C22⋊C4—681no—
361C22⋊C4—1362no—
362C22⋊C4—1362no—
363C23—41no—
364C23—41no—
365C23—174no—
366C23—342no—
367C23—342no—
368C23—342no—
369C23—342no—
370C23—342no—
371C23—342no—
372C23—682no—
373C23—682no—
374C23—682no—
375C23—682no—
376C23—1361no—
377C23—1361no—
378C23—1362no—
379C23—1362no—
380C23—1362no—
381C23—1362no—
382C23C22⋊C4×D1711yescharacteristic
383C23#1058756811yescharacteristic
384C23#1058758011yescharacteristic
385C23#1058759711yescharacteristic
386C23SD16×D1711yescharacteristic
387C23#1058760111yescharacteristic
388C23Q16×D1711yescharacteristic
389Q8—1361no—
390Q8—1361no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—342no—
404C2×C4—682no—
405C2×C4—682no—
406C2×C4—682no—
407C2×C4—682no—
408C2×C4—682no—
409C2×C4—682no—
410C2×C4—682no—
411C2×C4—681no—
412C2×C4—681no—
413C2×C4—1368no—
414C2×C4—1361no—
415C2×C4C4×D6811yescharacteristic
416C2×C4#1058757011yescharacteristic
417C2×C4#1058757911yescharacteristic
418C2×C4#1058758111yescharacteristic
419C8—81no—
420C8—1361no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—344no—
432C22—684no—
433C22—684no—
434C22—684no—
435C22—682no—
436C22—682no—
437C22—1362no—
438C22—1362no—
439C22—1362no—
440C22#1176268711yescharacteristic
441C22#103×D1711yescharacteristic
442C22#1176280711yescharacteristic
443C22#1176296111yescharacteristic
444C22#1176296611yescharacteristic
445C22#1176314111yescharacteristic
446C22#1176314611yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—82no—
452C4—82no—
453C4—682no—
454C4—1362no—
455C4—1361no—
456C4—1361no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—21no—
463C2—684no—
464C2#1452909011yescharacteristic
465C2#1452931511yescharacteristic
466C2#1452932511yescharacteristic
subgroupG/H
centerC22#11762687
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1289C2
socleC22×C17#386
prime radicalC24×D17D4
cosocleC23×C4×C17C23
hypercenter#1289D17
hypocenterC17#9199
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
3. #14527357C2semidirect
4. #14527355C2semidirect
6. #14526651C2semidirect
7. C17×#1289C2semidirect
12. #11762746C4semidirect
14. #11762722C4semidirect
31. C22×Dic34C2×C4semidirect
50. C2×C4⋊Dic17C2×C4semidirect
62. C17×#95C2×C4semidirect
194. #1289D17semidirect
300. #95C4×D17semidirect
302. C17#9199semidirect

Maximal subgroups

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

subgroupindex
#145345922
#145313972
#145273572
#145273552
C2×#117625322
#145266512
C17×#12892
#919917