Finite Group Databasebeta

← #16951809

Subgroups — #16951809

Order 4352.

722 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#11762793C211yesmaximal, characteristic
2C2×#11762599C212yesmaximal, complemented
3#14528702C211yesmaximal, characteristic
4#14526902C212yesmaximal, complemented
5C17×#2406C211yesmaximal, characteristic
6C22×#63223—21no—
7#11762793—24no—
8C2×#10587507—21no—
9C2×#10587504C2212yes—
10C2×#10587505—21no—
11C2×#10587505C2212yes—
12#11762672—21no—
13#11762599—24no—
14C22×#41×C17C2211yescharacteristic
15C2×C17×#123C411yescharacteristic, complemented
16C2×C17×#123C2211yescharacteristic
17C22×C4⋊C4×C17C411yescharacteristic, complemented
18C17×#395—21no—
19C17×#394—21no—
20C17×#386C411yescharacteristic, complemented
21C17×#386C2211yescharacteristic
22C17×#385—21no—
23C17×#385—21no—
24C17×#381C411yescharacteristic, complemented
25C2×#41×C17—22no—
26C2×#41×C17—22no—
27C2×#41×C17—41no—
28C2×#41×C17—41no—
29C2×#41×C17—42no—
30C22×Q8×C17D411yescharacteristic
31C22×D4×C17C2×C411yescharacteristic
32C22×D4×C17D411yescharacteristic
33C22×D4×C17C2311yescharacteristic
34C23×C4×C17C2×C411yescharacteristic
35C23×C4×C17C2×C411yescharacteristic
36C23×C4×C17C2×C411yescharacteristic
37C23×C4×C17D411yescharacteristic
38C23×C4×C17D411yescharacteristic
39C17×#123—22no—
40C17×#123—22no—
41C17×#123—42no—
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17—21no—
44C2×C4⋊C4×C17—21no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17—41no—
47C2×C4⋊C4×C17—41no—
48C2×C22⋊C4×C17—21no—
49C2×C22⋊C4×C17—21no—
50C2×C22⋊C4×C17—21no—
51C2×C22⋊C4×C17—21no—
52C2×C22⋊C4×C17C2×C411yescharacteristic
53C2×C22⋊C4×C17C2×C411yescharacteristic
54C2×C22⋊Dic17—22no—
55C2×#63223—24no—
56C2×#63223—22no—
57C2×#63223—42no—
58C2×#63223—41no—
59C22×#3622—41no—
60C17×#95—21no—
61C17×#95—21no—
62C17×#95—21no—
63C17×#95—21no—
64#10587507—24no—
65#10587505—22no—
66#10587505—22no—
67#10587505—42no—
68#10587504—22no—
69#10587504D414yes—
70#10587502—41no—
71#10587502—41no—
72C24×C17C22⋊C411yescharacteristic
73C24×C17C22×C411yescharacteristic
74C24×C17C2×D411yescharacteristic
75#41×C17—44no—
76#41×C17—44no—
77#41×C17—44no—
78#41×C17—44no—
79C2×Q8×C17—21no—
80C2×Q8×C17—21no—
81C2×Q8×C17—41no—
82C2×Q8×C17—41no—
83C2×D4×C17—22no—
84C2×D4×C17—22no—
85C2×D4×C17—22no—
86C2×D4×C17—22no—
87C2×D4×C17—41no—
88C2×D4×C17—41no—
89C2×D4×C17—41no—
90C2×D4×C17—41no—
91C2×D4×C17C2×D412yes—
92C2×D4×C17#4112yes—
93C22×C4×C17—22no—
94C22×C4×C17—21no—
95C22×C4×C17—21no—
96C22×C4×C17—21no—
97C22×C4×C17—21no—
98C22×C4×C17—21no—
99C22×C4×C17—22no—
100C22×C4×C17—22no—
101C22×C4×C17—21no—
102C22×C4×C17—21no—
103C22×C4×C17—21no—
104C22×C4×C17—21no—
105C22×C4×C17—21no—
106C22×C4×C17—21no—
107C22×C4×C17—21no—
108C22×C4×C17—22no—
109C22×C4×C17—41no—
110C22×C4×C17—41no—
111C22×C4×C17—42no—
112C22×C4×C17—41no—
113C22×C4×C17—41no—
114C22×C4×C17—41no—
115C22×C4×C17C22⋊C411yescharacteristic
116C22×C4×C17C22⋊C411yescharacteristic
117C22×C4×C17C22⋊C411yescharacteristic
118C22×C4×C17C2×D411yescharacteristic
119C22×Dic17—42no—
120C4⋊C4×C17—42no—
121C4⋊C4×C17—42no—
122C4⋊C4×C17—41no—
123C4⋊C4×C17—41no—
124C4⋊C4×C17—41no—
125C4⋊C4×C17—41no—
126C22⋊C4×C17—22no—
127C22⋊C4×C17—42no—
128C22⋊C4×C17—42no—
129C22⋊C4×C17—42no—
130C22⋊C4×C17—42no—
131C22⋊C4×C17—41no—
132C22⋊Dic17—42no—
133C22⋊Dic17—42no—
134#63223—24no—
135#63223—44no—
136#63223—42no—
137C2×#3622—44no—
138C2×#3622—42no—
139#13081—171nomaximal, Sylow
140C23×C17—21no—
141C23×C17—21no—
142C23×C17—21no—
143C23×C17—21no—
144C23×C17—21no—
145C23×C17—21no—
146C23×C17—41no—
147C23×C17—41no—
148C23×C17—41no—
149C23×C17—41no—
150C23×C17—41no—
151C23×C17—41no—
152C23×C17#9912yes—
153C23×C17C2×C22⋊C411yescharacteristic
154C23×C17C4×D412yes—
155C23×C17#12112yes—
156Q8×C17—42no—
157Q8×C17—42no—
158D4×C17—42no—
159D4×C17—42no—
160D4×C17—44no—
161D4×C17—44no—
162C2×C4×C17—22no—
163C2×C4×C17—22no—
164C2×C4×C17—22no—
165C2×C4×C17—22no—
166C2×C4×C17—21no—
167C2×C4×C17—21no—
168C2×C4×C17—21no—
169C2×C4×C17—21no—
170C2×C4×C17—22no—
171C2×C4×C17—21no—
172C2×C4×C17—21no—
173C2×C4×C17—21no—
174C2×C4×C17—21no—
175C2×C4×C17—21no—
176C2×C4×C17—21no—
177C2×C4×C17—42no—
178C2×C4×C17—42no—
179C2×C4×C17—42no—
180C2×C4×C17—42no—
181C2×C4×C17—41no—
182C2×C4×C17—42no—
183C2×C4×C17—41no—
184C2×C4×C17—42no—
185C2×C4×C17—41no—
186C2×C4×C17—41no—
187C2×C4×C17—41no—
188C2×C4×C17—41no—
189C2×C4×C17—41no—
190C2×C4×C17—41no—
191C2×C4×C17—41no—
192C2×C4×C17—41no—
193C2×C4×C17—41no—
194C2×C4×C17—82no—
195C2×C4×C17#9911yescharacteristic
196C2×C4×C17#9911yescharacteristic
197C2×C4×C17#12011yescharacteristic
198C2×C4×C17#12311yescharacteristic
199C2×Dic17—82no—
200C2×Dic17—82no—
201C2×Dic17—82no—
202#3622—44no—
203C2×#413—171no—
204#2406D1711yescharacteristic
205C2×#352—172no—
206#1782—171no—
207#1390—172no—
208C22×C17—21no—
209C22×C17—21no—
210C22×C17—22no—
211C22×C17—22no—
212C22×C17—22no—
213C22×C17—42no—
214C22×C17—42no—
215C22×C17—42no—
216C22×C17—42no—
217C22×C17—41no—
218C22×C17—41no—
219C22×C17—41no—
220C22×C17—41no—
221C22×C17—41no—
222C22×C17—41no—
223C22×C17—81no—
224C22×C17—81no—
225C22×C17#38611yescharacteristic
226C22×C17C2×#9911yescharacteristic
227C22×C17C2×#9911yescharacteristic
228C4×C17—21no—
229C4×C17—21no—
230C4×C17—21no—
231C4×C17—21no—
232C4×C17—42no—
233C4×C17—41no—
234C4×C17—41no—
235C4×C17—81no—
236C4×C17—81no—
237C4×C17—81no—
238Dic17—162no—
239C22×#41C2×D1711yescharacteristic
240C22×M16—341no—
241C2×#123Dic1711yescharacteristic, complemented
242C2×#123C2×D1711yescharacteristic
243C22×C4⋊C4Dic1711yescharacteristic, complemented
244#413—344no—
245C2×#101—341no—
246C2×#100—172no—
247C2×#99—172no—
248C2×#99—341no—
249#395—21no—
250#394—21no—
251#386Dic1711yescharacteristic, complemented
252#386C2×D1711yescharacteristic
253#385—21no—
254#385—21no—
255#381—341no—
256#381Dic1711yescharacteristic, complemented
257#352—344no—
258C2×C17—21no—
259C2×C17—21no—
260C2×C17—42no—
261C2×C17—42no—
262C2×C17—42no—
263C2×C17#189811yescharacteristic
264C2×C17#212011yescharacteristic
265C2×C17#212311yescharacteristic
266C2×#41—22no—
267C2×#41—22no—
268C2×#41—41no—
269C2×#41—41no—
270C2×#41—42no—
271C22×Q8C22⋊D1711yescharacteristic
272C22×D4C2×Dic1711yescharacteristic
273C22×D4C22⋊D1711yescharacteristic
274C22×D4C22×D1711yescharacteristic
275C23×C4C2×Dic1711yescharacteristic
276C23×C4C2×Dic1711yescharacteristic
277C23×C4C2×Dic1711yescharacteristic
278C23×C4C22⋊D1711yescharacteristic
279C23×C4C22⋊D1711yescharacteristic
280C2×M16—344no—
281C2×M16—342no—
282C2×M16—682no—
283C2×M16—681no—
284C22×C8—681no—
285#123—22no—
286#123—22no—
287#123—42no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—41no—
293C2×C4⋊C4—41no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—342no—
299C2×C22⋊C4C2×Dic1711yescharacteristic
300C2×C22⋊C4C2×Dic1711yescharacteristic
301#101—344no—
302#100—174no—
303#100—342no—
304#99—342no—
305#99—342no—
306#99—682no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—681no—
312#95—681no—
313C17#1308111yescharacteristic, complemented, Sylow
314C24C22⋊Dic1711yescharacteristic
315C24C22×Dic1711yescharacteristic
316C24C2×C22⋊D1711yescharacteristic
317#41—44no—
318#41—44no—
319#41—44no—
320#41—44no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4D4×D1712yes—
334C2×D4#6325412yes—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—682no—
358C22×C4C22⋊Dic1711yescharacteristic
359C22×C4C22⋊Dic1711yescharacteristic
360C22×C4C22⋊Dic1711yescharacteristic
361C22×C4C2×C22⋊D1711yescharacteristic
362M16—344no—
363M16—684no—
364M16—682no—
365C2×C8—684no—
366C2×C8—682no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C22⋊C4—22no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—41no—
379C22⋊C4—682no—
380C22⋊C4—682no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23#1058750512yes—
394C23D4×Dic1712yes—
395C23#1058762512yes—
396C23C2×C22⋊Dic1711yescharacteristic
397Q8—42no—
398Q8—42no—
399D4—42no—
400D4—42no—
401D4—44no—
402D4—44no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—82no—
436C2×C4—1362no—
437C2×C4—1362no—
438C2×C4—1362no—
439C2×C4#1058750511yescharacteristic
440C2×C4#1058750511yescharacteristic
441C2×C4#1058762111yescharacteristic
442C2×C4#1058762311yescharacteristic
443C8—684no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—81no—
460C22—81no—
461C22#1176268911yescharacteristic
462C22C2×#1058750511yescharacteristic
463C22C2×#1058750511yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—42no—
469C4—41no—
470C4—41no—
471C4—81no—
472C4—81no—
473C4—81no—
474C4—2722no—
475C2—21no—
476C2—21no—
477C2—42no—
478C2—42no—
479C2—42no—
480C2#1452923811yescharacteristic
481C2#1453042511yescharacteristic
482C2#1453043811yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC24×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#2406C2
socleC22×C17C2×#99
prime radicalC22×#41×C17C22
cosocleC22×D4×C17C23
hypercenter#2406D17
hypocenterC17#13081
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2406

Complements

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

NH ≅ G/Ntype
2. C2×#11762599C2semidirect
4. #14526902C2semidirect
15. C2×C17×#123C4semidirect
17. C22×C4⋊C4×C17C4semidirect
20. C17×#386C4semidirect
24. C17×#381C4semidirect
241. C2×#123Dic17semidirect
243. C22×C4⋊C4Dic17semidirect
251. #386Dic17semidirect
256. #381Dic17semidirect
313. C17#13081semidirect

Maximal subgroups

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

subgroupindex
C2×#117627932
C2×#117625992
#145287022
#145269022
C17×#24062
#1308117