Finite Group Databasebeta

← #16944994

Subgroups — #16944994

Order 4352.

574 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 87 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#14528858C211yesmaximal, characteristic, complemented
2#14527111C211yesmaximal, characteristic, complemented
3#14526784C212yesmaximal, complemented
4#14526781C212yesmaximal, complemented
5C17×#2617C211yesmaximal, characteristic
6#11762773—21no—
7#11762665—21no—
8#11762576—21no—
9#11762576C2212yes—
10#11762533—21no—
11#11762533C2212yes—
12C2×C17×#121C411yescharacteristic, complemented
13C2×C17×#120—21no—
14C22×C22⋊C4×C17C2211yescharacteristic
15C17×#397C411yescharacteristic, complemented
16C17×#397C2211yescharacteristic
17C17×#394—21no—
18C17×#394—21no—
19C17×#392C411yescharacteristic, complemented
20C17×#392C2211yescharacteristic
21C17×#386—21no—
22C17×#380C411yescharacteristic, complemented
23C25×C17D411yescharacteristic
24C22×D4×C17—21no—
25C22×D4×C17—21no—
26C22×D4×C17—21no—
27C22×D4×C17C2×C411yescharacteristic
28C23×C4×C17C2×C411yescharacteristic
29C23×C4×C17D411yescharacteristic
30C17×#121—41no—
31C17×#121—41no—
32C17×#120—44no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17—41no—
42C2×C22⋊C4×C17—41no—
43C2×C22⋊C4×C17C2×C411yescharacteristic
44C2×C22⋊C4×C17D411yescharacteristic
45C2×C22⋊C4×C17D411yescharacteristic
46C2×C22⋊C4×C17C2311yescharacteristic
47C2×C22⋊Dic17—22no—
48C2×C22⋊Dic17—41no—
49C2×C22⋊Dic17—41no—
50C2×#63223—41no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95C2×C411yescharacteristic
54C17×#95C2×C411yescharacteristic
55C22⋊#3622—22no—
56C22⋊#3622—41no—
57C24×C17—21no—
58C24×C17—21no—
59C24×C17—22no—
60C24×C17—21no—
61C24×C17—41no—
62C24×C17—81no—
63C24×C17—81no—
64C24×C17C22⋊C411yescharacteristic
65C24×C17C22⋊C411yescharacteristic
66C24×C17C2×D411yescharacteristic
67C2×D4×C17—41no—
68C2×D4×C17—42no—
69C2×D4×C17—42no—
70C2×D4×C17—42no—
71C2×D4×C17—41no—
72C2×D4×C17—41no—
73C2×D4×C17—42no—
74C2×D4×C17—81no—
75C2×D4×C17—81no—
76C2×D4×C17—81no—
77C2×D4×C17—81no—
78C2×D4×C17—81no—
79C22×C4×C17—21no—
80C22×C4×C17—21no—
81C22×C4×C17—21no—
82C22×C4×C17—21no—
83C22×C4×C17—21no—
84C22×C4×C17—21no—
85C22×C4×C17—21no—
86C22×C4×C17—81no—
87C22×C4×C17—81no—
88C22×C4×C17C22⋊C411yescharacteristic
89C22×C4×C17C22⋊C411yescharacteristic
90C22×C4×C17C22×C411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C22×Dic17—42no—
93C4⋊C4×C17—41no—
94C4⋊C4×C17—81no—
95C22⋊C4×C17—41no—
96C22⋊C4×C17—41no—
97C22⋊C4×C17—41no—
98C22⋊C4×C17—41no—
99C22⋊C4×C17—42no—
100C22⋊C4×C17—42no—
101C22⋊C4×C17—42no—
102C22⋊C4×C17—42no—
103C22⋊C4×C17—81no—
104C22⋊C4×C17—81no—
105C22⋊Dic17—42no—
106C22⋊Dic17—42no—
107C22⋊Dic17—82no—
108C22⋊Dic17—82no—
109#63223—82no—
110C2×#3622—42no—
111#11799—171nomaximal, Sylow
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17—21no—
118C23×C17—21no—
119C23×C17—41no—
120C23×C17—41no—
121C23×C17—42no—
122C23×C17—42no—
123C23×C17—41no—
124C23×C17—41no—
125C23×C17—41no—
126C23×C17—41no—
127C23×C17—42no—
128C23×C17—42no—
129C23×C17—41no—
130C23×C17—41no—
131C23×C17—41no—
132C23×C17—41no—
133C23×C17—41no—
134C23×C17—41no—
135C23×C17—41no—
136C23×C17—81no—
137C23×C17—81no—
138C23×C17—84no—
139C23×C17—84no—
140C23×C17#9911yescharacteristic
141C23×C17#9911yescharacteristic
142C23×C17C2×C22⋊C411yescharacteristic
143C23×C17AGL(1,ℤ/8)12yes—
144D4×C17—82no—
145D4×C17—82no—
146D4×C17—82no—
147D4×C17—84no—
148D4×C17—84no—
149C2×C4×C17—21no—
150C2×C4×C17—21no—
151C2×C4×C17—21no—
152C2×C4×C17—21no—
153C2×C4×C17—42no—
154C2×C4×C17—41no—
155C2×C4×C17—41no—
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—41no—
163C2×C4×C17—41no—
164C2×C4×C17—41no—
165C2×C4×C17—41no—
166C2×C4×C17—41no—
167C2×C4×C17—81no—
168C2×C4×C17—81no—
169C2×C4×C17—81no—
170C2×Dic17—82no—
171C2×Dic17—82no—
172C2×Dic17—82no—
173#3622—82no—
174#2617D1711yescharacteristic
175#1819—171no—
176#1462—171no—
177#1344—172no—
178#1343—172no—
179C22×C17—21no—
180C22×C17—21no—
181C22×C17—21no—
182C22×C17—21no—
183C22×C17—21no—
184C22×C17—21no—
185C22×C17—22no—
186C22×C17—21no—
187C22×C17—21no—
188C22×C17—42no—
189C22×C17—42no—
190C22×C17—42no—
191C22×C17—42no—
192C22×C17—41no—
193C22×C17—41no—
194C22×C17—41no—
195C22×C17—41no—
196C22×C17—41no—
197C22×C17—81no—
198C22×C17—81no—
199C22×C17—81no—
200C22×C17—81no—
201C22×C17—81no—
202C22×C17—81no—
203C22×C17—84no—
204C22×C17—84no—
205C22×C17—81no—
206C22×C17—81no—
207C22×C17—81no—
208C22×C17C2×#9911yescharacteristic
209C22×C17#41811yescharacteristic
210C22×C17#42111yescharacteristic
211C4×C17—41no—
212C4×C17—41no—
213C4×C17—41no—
214C4×C17—41no—
215C4×C17—81no—
216C4×C17—81no—
217C4×C17—81no—
218Dic17—162no—
219C2×#121Dic1711yescharacteristic, complemented
220C2×#120—21no—
221C22×C22⋊C4C2×D1711yescharacteristic
222#407—341no—
223#397Dic1711yescharacteristic, complemented
224#397C2×D1711yescharacteristic
225#394—21no—
226#394—21no—
227#392Dic1711yescharacteristic, complemented
228#392C2×D1711yescharacteristic
229#386—21no—
230#380Dic1711yescharacteristic, complemented
231#379—341no—
232#342—172no—
233#342—341no—
234#323—172no—
235#323—341no—
236C2×C17—21no—
237C2×C17—21no—
238C2×C17—42no—
239C2×C17—41no—
240C2×C17—41no—
241C2×C17—81no—
242C2×C17—161no—
243C2×C17#151011yescharacteristic
244C2×C17#212111yescharacteristic
245C2×C17#212711yescharacteristic
246C25C22⋊D1711yescharacteristic
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4C2×Dic1711yescharacteristic
251C23×C4C2×Dic1711yescharacteristic
252C23×C4C22⋊D1711yescharacteristic
253C2×M16—681no—
254#121—41no—
255#121—41no—
256#120—44no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C2×Dic1711yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—342no—
268C2×C22⋊C4—681no—
269C2×C22⋊C4—681no—
270C2×C22⋊C4C2×Dic1711yescharacteristic
271C2×C22⋊C4C22⋊D1711yescharacteristic
272C2×C22⋊C4C22⋊D1711yescharacteristic
273C2×C22⋊C4C22×D1711yescharacteristic
274C22⋊C8—342no—
275C22⋊C8—681no—
276#95—21no—
277#95—21no—
278#95C2×Dic1711yescharacteristic
279#95C2×Dic1711yescharacteristic
280C17#1179911yescharacteristic, complemented, Sylow
281C24—21no—
282C24—21no—
283C24—22no—
284C24—21no—
285C24—41no—
286C24—81no—
287C24—81no—
288C24C22⋊Dic1711yescharacteristic
289C24C22⋊Dic1711yescharacteristic
290C24C2×C22⋊D1711yescharacteristic
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—81no—
299C2×D4—81no—
300C2×D4—81no—
301C2×D4—81no—
302C2×D4—81no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4—682no—
313C22×C4C22⋊Dic1711yescharacteristic
314C22×C4C22⋊Dic1711yescharacteristic
315C22×C4C22×Dic1711yescharacteristic
316C22×C4C2×C22⋊D1711yescharacteristic
317M16—1362no—
318C2×C8—682no—
319C4⋊C4—41no—
320C4⋊C4—81no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—81no—
330C22⋊C4—81no—
331C22⋊C4—682no—
332C22⋊C4—682no—
333C22⋊C4—1362no—
334C22⋊C4—1362no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—41no—
343C23—41no—
344C23—42no—
345C23—42no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—81no—
360C23—81no—
361C23—84no—
362C23—84no—
363C23#1058750511yescharacteristic
364C23#1058750511yescharacteristic
365C23#1058761812yes—
366C23C2×C22⋊Dic1711yescharacteristic
367D4—82no—
368D4—82no—
369D4—82no—
370D4—84no—
371D4—84no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—1362no—
394C2×C4—1362no—
395C2×C4—1362no—
396C8—1362no—
397C22—21no—
398C22—21no—
399C22—22no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—84no—
422C22—84no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22C2×#1058750511yescharacteristic
427C22#1176281411yescharacteristic
428C22#1176282611yescharacteristic
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—81no—
434C4—81no—
435C4—81no—
436C4—2722no—
437C2—21no—
438C2—21no—
439C2—42no—
440C2—41no—
441C2—41no—
442C2—81no—
443C2—161no—
444C2#1452731311yescharacteristic
445C2#1453043011yescharacteristic
446C2#1453045311yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
Fitting, prime radicalC17×#2617C2
socleC22×C17C2×#99
cosocleC2×C22⋊C4×C17C23
hypercenter#2617D17
hypocenterC17#11799
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2617

Complements

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

NH ≅ G/Ntype
1. #14528858C2semidirect
2. #14527111C2semidirect
3. #14526784C2semidirect
4. #14526781C2semidirect
12. C2×C17×#121C4semidirect
15. C17×#397C4semidirect
19. C17×#392C4semidirect
22. C17×#380C4semidirect
219. C2×#121Dic17semidirect
223. #397Dic17semidirect
227. #392Dic17semidirect
230. #380Dic17semidirect
280. C17#11799semidirect

Maximal subgroups

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

subgroupindex
#145288582
#145271112
#145267842
#145267812
C17×#26172
#1179917