Finite Group Databasebeta

← #16944800

Subgroups — #16944800

Order 4352.

700 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 95 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×#11762576C211yesmaximal, characteristic, complemented
2#14527111C211yesmaximal, characteristic, complemented
3#14526808C212yesmaximal, complemented
4#14526781C212yesmaximal, complemented
5C17×#2539C211yesmaximal, characteristic
6C22×C22⋊Dic17—21no—
7#11762773—21no—
8#11762576—21no—
9#11762576C2212yes—
10#11762533—21no—
11#11762533C2212yes—
12C2×C17×#123—21no—
13C2×C17×#121C411yescharacteristic, complemented
14C2×C17×#120—21no—
15C22×C22⋊C4×C17C2211yescharacteristic
16C17×#399C411yescharacteristic, complemented
17C17×#399C2211yescharacteristic
18C17×#392C411yescharacteristic, complemented
19C17×#392C2211yescharacteristic
20C17×#390—21no—
21C17×#390—21no—
22C4×C22⋊C4×C17C411yescharacteristic, complemented
23C25×C17D411yescharacteristic
24C22×D4×C17—21no—
25C22×D4×C17—21no—
26C22×D4×C17—21no—
27C22×D4×C17C2×C411yescharacteristic, complemented
28C23×C4×C17C2×C411yescharacteristic
29C23×C4×C17D411yescharacteristic
30C23×Dic17—41no—
31C17×#123—44no—
32C17×#121—41no—
33C17×#121—41no—
34C17×#120—44no—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17C2×C411yescharacteristic, complemented
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17—21no—
42C2×C22⋊C4×C17—21no—
43C2×C22⋊C4×C17—21no—
44C2×C22⋊C4×C17—41no—
45C2×C22⋊C4×C17—41no—
46C2×C22⋊C4×C17C2×C411yescharacteristic
47C2×C22⋊C4×C17D411yescharacteristic
48C2×C22⋊C4×C17D411yescharacteristic
49C2×C22⋊C4×C17C2311yescharacteristic
50C2×C42×C17—21no—
51C2×C22⋊Dic17—24no—
52C2×C22⋊Dic17—22no—
53C2×C22⋊Dic17—41no—
54C2×C22⋊Dic17—42no—
55C2×#63223—41no—
56C17×#95C2×C411yescharacteristic, complemented
57C17×#95C2×C411yescharacteristic, complemented
58C22⋊#3622—22no—
59C22⋊#3622—41no—
60C24×C17—21no—
61C24×C17—21no—
62C24×C17—22no—
63C24×C17—21no—
64C24×C17—21no—
65C24×C17—21no—
66C24×C17—42no—
67C24×C17—81no—
68C24×C17C22⋊C411yescharacteristic
69C24×C17C22⋊C411yescharacteristic
70C24×C17C2×D411yescharacteristic
71C2×D4×C17—41no—
72C2×D4×C17—42no—
73C2×D4×C17—42no—
74C2×D4×C17—42no—
75C2×D4×C17—42no—
76C2×D4×C17—41no—
77C2×D4×C17—41no—
78C2×D4×C17—42no—
79C2×D4×C17—41no—
80C2×D4×C17—41no—
81C2×D4×C17—81no—
82C2×D4×C17—81no—
83C2×D4×C17—81no—
84C22×C4×C17—21no—
85C22×C4×C17—21no—
86C22×C4×C17—21no—
87C22×C4×C17—21no—
88C22×C4×C17—21no—
89C22×C4×C17—21no—
90C22×C4×C17—21no—
91C22×C4×C17—42no—
92C22×C4×C17—81no—
93C22×C4×C17C22⋊C411yescharacteristic
94C22×C4×C17C22⋊C411yescharacteristic
95C22×C4×C17C22×C411yescharacteristic
96C22×C4×C17C2×D411yescharacteristic
97C22×Dic17—44no—
98C22×Dic17—42no—
99C22×Dic17—82no—
100C22×Dic17—82no—
101C4⋊C4×C17—41no—
102C4⋊C4×C17—42no—
103C4⋊C4×C17—42no—
104C22⋊C4×C17—24no—
105C22⋊C4×C17—41no—
106C22⋊C4×C17—41no—
107C22⋊C4×C17—41no—
108C22⋊C4×C17—41no—
109C22⋊C4×C17—42no—
110C22⋊C4×C17—42no—
111C22⋊C4×C17—42no—
112C22⋊C4×C17—42no—
113C22⋊C4×C17—42no—
114C42×C17—81no—
115C22⋊Dic17—24no—
116C22⋊Dic17—44no—
117C22⋊Dic17—42no—
118#63223—82no—
119C2×#3622—42no—
120#11757—171nomaximal, Sylow
121C23×C17—22no—
122C23×C17—22no—
123C23×C17—22no—
124C23×C17—22no—
125C23×C17—22no—
126C23×C17—21no—
127C23×C17—21no—
128C23×C17—21no—
129C23×C17—21no—
130C23×C17—21no—
131C23×C17—21no—
132C23×C17—41no—
133C23×C17—42no—
134C23×C17—41no—
135C23×C17—41no—
136C23×C17—41no—
137C23×C17—41no—
138C23×C17—41no—
139C23×C17—41no—
140C23×C17—41no—
141C23×C17—41no—
142C23×C17—42no—
143C23×C17—41no—
144C23×C17—41no—
145C23×C17—41no—
146C23×C17—81no—
147C23×C17—81no—
148C23×C17—84no—
149C23×C17—82no—
150C23×C17—82no—
151C23×C17#9911yescharacteristic
152C23×C17#9911yescharacteristic
153C23×C17#10412yes—
154C23×C17C2×C22⋊C411yescharacteristic
155C23×C17AGL(1,ℤ/8)11yescharacteristic
156C23×C17#13711yescharacteristic
157D4×C17—84no—
158D4×C17—82no—
159D4×C17—82no—
160D4×C17—82no—
161D4×C17—82no—
162D4×C17—82no—
163C2×C4×C17—22no—
164C2×C4×C17—22no—
165C2×C4×C17—21no—
166C2×C4×C17—21no—
167C2×C4×C17—21no—
168C2×C4×C17—21no—
169C2×C4×C17—41no—
170C2×C4×C17—41no—
171C2×C4×C17—42no—
172C2×C4×C17—41no—
173C2×C4×C17—41no—
174C2×C4×C17—41no—
175C2×C4×C17—41no—
176C2×C4×C17—41no—
177C2×C4×C17—41no—
178C2×C4×C17—41no—
179C2×C4×C17—41no—
180C2×C4×C17—41no—
181C2×C4×C17—81no—
182C2×C4×C17—81no—
183C2×C4×C17—81no—
184C2×Dic17—44no—
185C2×Dic17—88no—
186C2×Dic17—82no—
187C2×Dic17—82no—
188#3622—82no—
189#2539D1711yescharacteristic
190C2×#342—171no—
191#1462—171no—
192#1352—172no—
193#1343—172no—
194C22×C17—21no—
195C22×C17—21no—
196C22×C17—21no—
197C22×C17—21no—
198C22×C17—21no—
199C22×C17—21no—
200C22×C17—21no—
201C22×C17—21no—
202C22×C17—22no—
203C22×C17—21no—
204C22×C17—42no—
205C22×C17—42no—
206C22×C17—42no—
207C22×C17—41no—
208C22×C17—41no—
209C22×C17—44no—
210C22×C17—44no—
211C22×C17—41no—
212C22×C17—41no—
213C22×C17—41no—
214C22×C17—41no—
215C22×C17—81no—
216C22×C17—81no—
217C22×C17—81no—
218C22×C17—81no—
219C22×C17—81no—
220C22×C17—81no—
221C22×C17—82no—
222C22×C17—82no—
223C22×C17—81no—
224C22×C17—81no—
225C22×C17—81no—
226C22×C17#35312yes—
227C22×C17C2×#9911yescharacteristic
228C22×C17#41711yescharacteristic
229C22×C17C2×#10411yescharacteristic
230C4×C17—42no—
231C4×C17—42no—
232C4×C17—41no—
233C4×C17—41no—
234C4×C17—41no—
235C4×C17—41no—
236C4×C17—81no—
237Dic17—84no—
238C2×#123—21no—
239C2×#121Dic1711yescharacteristic, complemented
240C2×#120—21no—
241C22×C22⋊C4—341no—
242C22×C22⋊C4C2×D1711yescharacteristic
243#407—341no—
244#399Dic1711yescharacteristic, complemented
245#399C2×D1711yescharacteristic
246#392Dic1711yescharacteristic, complemented
247#392C2×D1711yescharacteristic
248#390—21no—
249#390—21no—
250C4×C22⋊C4Dic1711yescharacteristic, complemented
251#342—172no—
252#342—341no—
253#323—172no—
254#323—341no—
255C2×C17—21no—
256C2×C17—21no—
257C2×C17—22no—
258C2×C17—41no—
259C2×C17—41no—
260C2×C17—42no—
261C2×C17—41no—
262C2×C17—161no—
263C2×C17#150411yescharacteristic
264C2×C17#212111yescharacteristic
265C2×C17C2×#35311yescharacteristic
266C25C22⋊D1711yescharacteristic
267C22×D4—21no—
268C22×D4—21no—
269C22×D4—21no—
270C22×D4C2×Dic1711yescharacteristic, complemented
271C23×C4—681no—
272C23×C4C2×Dic1711yescharacteristic
273C23×C4C22⋊D1711yescharacteristic
274C2×M16—681no—
275#123—44no—
276#121—41no—
277#121—41no—
278#120—44no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4C2×Dic1711yescharacteristic, complemented
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—344no—
291C2×C22⋊C4—342no—
292C2×C22⋊C4—681no—
293C2×C22⋊C4—682no—
294C2×C22⋊C4C2×Dic1711yescharacteristic
295C2×C22⋊C4C22⋊D1711yescharacteristic
296C2×C22⋊C4C22⋊D1711yescharacteristic
297C2×C22⋊C4C22×D1711yescharacteristic
298C2×C42—21no—
299C22⋊C8—342no—
300C22⋊C8—681no—
301#95C2×Dic1711yescharacteristic, complemented
302#95C2×Dic1711yescharacteristic, complemented
303C17#1175711yescharacteristic, complemented, Sylow
304C24—21no—
305C24—21no—
306C24—22no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—42no—
311C24—81no—
312C24C22⋊Dic1711yescharacteristic
313C24C22⋊Dic1711yescharacteristic
314C24C2×C22⋊D1711yescharacteristic
315C2×D4—41no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—42no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—81no—
326C2×D4—81no—
327C2×D4—81no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—42no—
336C22×C4—81no—
337C22×C4—684no—
338C22×C4—682no—
339C22×C4—1362no—
340C22×C4—1362no—
341C22×C4C22⋊Dic1711yescharacteristic
342C22×C4C22⋊Dic1711yescharacteristic
343C22×C4C22×Dic1711yescharacteristic
344C22×C4C2×C22⋊D1711yescharacteristic
345M16—1362no—
346C2×C8—682no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C22⋊C4—24no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—344no—
361C22⋊C4—684no—
362C22⋊C4—682no—
363C42—81no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—22no—
373C23—21no—
374C23—21no—
375C23—41no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—81no—
390C23—81no—
391C23—84no—
392C23—82no—
393C23—82no—
394C23#1058750511yescharacteristic
395C23#1058750511yescharacteristic
396C23#1058750812yes—
397C23#1058763511yescharacteristic
398C23#1058763711yescharacteristic
399C23C2×C22⋊Dic1711yescharacteristic
400D4—84no—
401D4—82no—
402D4—82no—
403D4—82no—
404D4—82no—
405D4—82no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—684no—
428C2×C4—1368no—
429C2×C4—1362no—
430C2×C4—1362no—
431C8—1362no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—41no—
446C22—41no—
447C22—44no—
448C22—44no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—82no—
460C22—82no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22#1176260212yes—
465C22C2×#1058750511yescharacteristic
466C22#1176281011yescharacteristic
467C22C2×#1058750811yescharacteristic
468C4—42no—
469C4—42no—
470C4—41no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—81no—
475C4—1364no—
476C2—21no—
477C2—21no—
478C2—22no—
479C2—41no—
480C2—41no—
481C2—42no—
482C2—41no—
483C2—161no—
484C2#1452728811yescharacteristic
485C2#1453043011yescharacteristic
486C2C2×#1176260211yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
Fitting, prime radicalC17×#2539C2
socleC22×C17C2×#99
cosocleC2×C22⋊C4×C17C23
hypercenter#2539D17
hypocenterC17#11757
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2539

Complements

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

NH ≅ G/Ntype
1. C2×#11762576C2semidirect
2. #14527111C2semidirect
3. #14526808C2semidirect
4. #14526781C2semidirect
13. C2×C17×#121C4semidirect
16. C17×#399C4semidirect
18. C17×#392C4semidirect
22. C4×C22⋊C4×C17C4semidirect
27. C22×D4×C17C2×C4semidirect
37. C2×C4⋊C4×C17C2×C4semidirect
56. C17×#95C2×C4semidirect
57. C17×#95C2×C4semidirect
239. C2×#121Dic17semidirect
244. #399Dic17semidirect
246. #392Dic17semidirect
250. C4×C22⋊C4Dic17semidirect
270. C22×D4C2×Dic17semidirect
281. C2×C4⋊C4C2×Dic17semidirect
301. #95C2×Dic17semidirect
302. #95C2×Dic17semidirect
303. C17#11757semidirect

Maximal subgroups

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

subgroupindex
C2×#117625762
#145271112
#145268082
#145267812
C17×#25392
#1175717