Finite Group Databasebeta

← #16944610

Subgroups — #16944610

Order 4352.

592 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 107 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#14534589C211yesmaximal, characteristic
2C2×#11762576C211yesmaximal, characteristic
3#14526809C212yesmaximal, complemented
4#14526806C212yesmaximal, complemented
5C2×C17×#323C211yesmaximal, characteristic
6C2×#10587610C411yescharacteristic, complemented
7C22×C22⋊Dic17—21no—
8#11762745C411yescharacteristic, complemented
9#11762745C2211yescharacteristic
10#11762732—21no—
11#11762732—21no—
12#11762721C411yescharacteristic, complemented
13#11762721C2211yescharacteristic
14#11762682—21no—
15#11762669C411yescharacteristic, complemented
16#11762665—21no—
17#11762576—21no—
18#11762576C2212yes—
19C22×C22⋊C4×C17C2211yescharacteristic
20C2×C17×C22⋊C8—21no—
21C17×#323—21no—
22C17×#323C2212yes—
23C25×C17D411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C23×Dic17—41no—
27C22×Dic34C2×C411yescharacteristic, complemented
28C22×C8×C17—41no—
29C2×C22⋊C4×C17—41no—
30C2×C22⋊C4×C17—41no—
31C2×C22⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C22⋊C4×C17C2311yescharacteristic
35C2×C22⋊Dic17—24no—
36C2×C22⋊Dic17—22no—
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—21no—
39C2×C22⋊Dic17—21no—
40C2×C22⋊Dic17—21no—
41C2×C22⋊Dic17—21no—
42C2×C22⋊Dic17—21no—
43C2×C22⋊Dic17—41no—
44C2×C22⋊Dic17—42no—
45C2×C4⋊Dic17—21no—
46C2×C4⋊Dic17C2×C411yescharacteristic, complemented
47#10587610—41no—
48#10587610—41no—
49C2×#63225—21no—
50C2×#63225—21no—
51C2×#63225—21no—
52C17×C22⋊C8—22no—
53C17×C22⋊C8—41no—
54#10587502—21no—
55#10587502—21no—
56#10587502C2×C411yescharacteristic, complemented
57#10587502C2×C411yescharacteristic, complemented
58C24×C17—21no—
59C24×C17—21no—
60C24×C17—22no—
61C24×C17—81no—
62C24×C17—81no—
63C24×C17C22⋊C411yescharacteristic
64C24×C17C22⋊C411yescharacteristic
65C24×C17SD1612yes—
66C24×C17Q1612yes—
67C24×C17C2×D411yescharacteristic
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—81no—
73C22×C4×C17—81no—
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22×C411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×Dic17—21no—
79C22×Dic17—21no—
80C22×Dic17—21no—
81C22×Dic17—21no—
82C22×Dic17—44no—
83C22×Dic17—42no—
84C22×Dic17—82no—
85C22×Dic17—82no—
86C2×Dic34—41no—
87C2×Dic34—81no—
88C2×C8×C17—42no—
89C2×C8×C17—82no—
90C22⋊C4×C17—41no—
91C22⋊C4×C17—41no—
92C22⋊C4×C17—41no—
93C22⋊C4×C17—41no—
94C22⋊Dic17—24no—
95C22⋊Dic17—44no—
96C22⋊Dic17—41no—
97C22⋊Dic17—41no—
98C22⋊Dic17—42no—
99C22⋊Dic17—42no—
100C22⋊Dic17—81no—
101C22⋊Dic17—81no—
102C22⋊Dic17—81no—
103C22⋊Dic17—81no—
104C4⋊Dic17—41no—
105C4⋊Dic17—81no—
106#63225—42no—
107#63225—81no—
108#63225—81no—
109#11716—171nomaximal, Sylow
110C23×C17—22no—
111C23×C17—22no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—22no—
115C23×C17—22no—
116C23×C17—21no—
117C23×C17—21no—
118C23×C17—41no—
119C23×C17—41no—
120C23×C17—42no—
121C23×C17—42no—
122C23×C17—41no—
123C23×C17—41no—
124C23×C17—41no—
125C23×C17—41no—
126C23×C17—81no—
127C23×C17—81no—
128C23×C17—82no—
129C23×C17—82no—
130C23×C17—84no—
131C23×C17#9911yescharacteristic
132C23×C17#9911yescharacteristic
133C23×C17#10312yes—
134C23×C17#10312yes—
135C23×C17#10412yes—
136C23×C17C2×C22⋊C411yescharacteristic
137C23×C17C2×SD1611yescharacteristic
138C23×C17C2×Q1611yescharacteristic
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—42no—
144C2×C4×C17—41no—
145C2×C4×C17—41no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—81no—
149C2×C4×C17—81no—
150C2×C4×C17—81no—
151C2×Dic17—44no—
152C2×Dic17—41no—
153C2×Dic17—41no—
154C2×Dic17—41no—
155C2×Dic17—41no—
156C2×Dic17—41no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159C2×Dic17—41no—
160C2×Dic17—41no—
161C2×Dic17—41no—
162C2×Dic17—41no—
163C2×Dic17—41no—
164C2×Dic17—88no—
165C2×Dic17—82no—
166C2×Dic17—82no—
167Dic34—82no—
168C8×C17—82no—
169#2626—171no—
170C2×#342—171no—
171C2×#323D1711yescharacteristic
172#1352—172no—
173#1351—172no—
174C22×C17—21no—
175C22×C17—21no—
176C22×C17—21no—
177C22×C17—21no—
178C22×C17—21no—
179C22×C17—21no—
180C22×C17—21no—
181C22×C17—22no—
182C22×C17—21no—
183C22×C17—42no—
184C22×C17—42no—
185C22×C17—42no—
186C22×C17—41no—
187C22×C17—41no—
188C22×C17—44no—
189C22×C17—44no—
190C22×C17—41no—
191C22×C17—41no—
192C22×C17—41no—
193C22×C17—41no—
194C22×C17—81no—
195C22×C17—81no—
196C22×C17—82no—
197C22×C17—82no—
198C22×C17#32812yes—
199C22×C17#32812yes—
200C22×C17C2×#9911yescharacteristic
201C22×C17C2×#10311yescharacteristic
202C22×C17C2×#10411yescharacteristic
203C4×C17—41no—
204C4×C17—41no—
205C4×C17—41no—
206C4×C17—41no—
207Dic17—81no—
208Dic17—81no—
209Dic17—84no—
210Dic17—81no—
211Dic17—81no—
212C2×C22⋊Q8—171no—
213C22×C22⋊C4—341no—
214C22×C22⋊C4C2×D1711yescharacteristic
215C2×C22⋊C8—21no—
216#399—171no—
217#399—171no—
218#396—341no—
219#396—341no—
220#393—171no—
221#393—171no—
222#385—341no—
223#380—171no—
224#379—341no—
225#342—172no—
226#342—341no—
227#323—21no—
228#323C2×D1712yes—
229C2×C17—21no—
230C2×C17—22no—
231C2×C17—21no—
232C2×C17—22no—
233C2×C17—41no—
234C2×C17—42no—
235C2×C17—41no—
236C2×C17C2×#32811yescharacteristic
237C2×C17#212111yescharacteristic
238C2×C17#212711yescharacteristic
239C25D6811yescharacteristic
240C22×Q8—171no—
241C23×C4—681no—
242C23×C4C4×D1711yescharacteristic
243C23×C4C22⋊D1711yescharacteristic
244C22×C8—41no—
245C22⋊Q8—681no—
246C22⋊Q8—681no—
247C2×C4⋊C4—171no—
248C2×C4⋊C4—341no—
249C2×C4⋊C4—341no—
250C2×C4⋊C4—341no—
251C2×C4⋊C4—341no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—344no—
255C2×C22⋊C4—342no—
256C2×C22⋊C4—341no—
257C2×C22⋊C4—341no—
258C2×C22⋊C4—341no—
259C2×C22⋊C4—341no—
260C2×C22⋊C4—341no—
261C2×C22⋊C4—341no—
262C2×C22⋊C4—681no—
263C2×C22⋊C4—682no—
264C2×C22⋊C4C4×D1711yescharacteristic
265C2×C22⋊C4D6811yescharacteristic
266C2×C22⋊C4C22⋊D1711yescharacteristic
267C2×C22⋊C4C22×D1711yescharacteristic
268C22⋊C8—22no—
269C22⋊C8—41no—
270#95—171no—
271#95—171no—
272#95—341no—
273#95—341no—
274C17#1171611yescharacteristic, complemented, Sylow
275C24—21no—
276C24—21no—
277C24—22no—
278C24—81no—
279C24—81no—
280C24#6321912yes—
281C24Dic6812yes—
282C24#6322711yescharacteristic
283C24#6322711yescharacteristic
284C24C2×D6811yescharacteristic
285C2×Q8—681no—
286C2×Q8—1361no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—341no—
294C22×C4—341no—
295C22×C4—341no—
296C22×C4—341no—
297C22×C4—682no—
298C22×C4—684no—
299C22×C4—1362no—
300C22×C4—1362no—
301C22×C4#6322711yescharacteristic
302C22×C4#6322711yescharacteristic
303C22×C4C2×C4×D1711yescharacteristic
304C22×C4C2×C22⋊D1711yescharacteristic
305C2×C8—42no—
306C2×C8—82no—
307C4⋊C4—681no—
308C4⋊C4—682no—
309C4⋊C4—1361no—
310C4⋊C4—1361no—
311C4⋊C4—1361no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—344no—
317C22⋊C4—684no—
318C22⋊C4—681no—
319C22⋊C4—681no—
320C22⋊C4—682no—
321C22⋊C4—682no—
322C22⋊C4—1361no—
323C22⋊C4—1361no—
324C22⋊C4—1361no—
325C22⋊C4—1361no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—81no—
343C23—81no—
344C23—82no—
345C23—82no—
346C23—84no—
347C23#1058747711yescharacteristic
348C23#1058747711yescharacteristic
349C23#1058748712yes—
350C23#1058748712yes—
351C23#1058749612yes—
352C23C2×#6321911yescharacteristic
353C23C2×Dic6811yescharacteristic
354C23C2×#6322711yescharacteristic
355Q8—1362no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—684no—
369C2×C4—681no—
370C2×C4—681no—
371C2×C4—681no—
372C2×C4—681no—
373C2×C4—681no—
374C2×C4—681no—
375C2×C4—681no—
376C2×C4—681no—
377C2×C4—681no—
378C2×C4—681no—
379C2×C4—681no—
380C2×C4—681no—
381C2×C4—1368no—
382C2×C4—1362no—
383C2×C4—1362no—
384C8—82no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—44no—
400C22—44no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22—82no—
408C22—82no—
409C22#1176254712yes—
410C22#1176254712yes—
411C22C2×#1058747711yescharacteristic
412C22C2×#1058748711yescharacteristic
413C22C2×#1058749611yescharacteristic
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—1361no—
419C4—1361no—
420C4—1364no—
421C4—1361no—
422C4—1361no—
423C2—21no—
424C2—21no—
425C2—22no—
426C2—22no—
427C2—41no—
428C2—42no—
429C2—41no—
430C2C2×#1176254711yescharacteristic
431C2#1453043111yescharacteristic
432C2#1453045111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC2×C17×#323C2
socleC22×C17C2×#99
prime radicalC25×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenterC2×#323D17
hypocenterC17#11716
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #14526809C2semidirect
4. #14526806C2semidirect
6. C2×#10587610C4semidirect
8. #11762745C4semidirect
12. #11762721C4semidirect
15. #11762669C4semidirect
27. C22×Dic34C2×C4semidirect
46. C2×C4⋊Dic17C2×C4semidirect
56. #10587502C2×C4semidirect
57. #10587502C2×C4semidirect
274. C17#11716semidirect

Maximal subgroups

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

subgroupindex
#145345892
C2×#117625762
#145268092
#145268062
C2×C17×#3232
#1171617