Finite Group Databasebeta

← #16952617

Subgroups — #16952617

Order 4352.

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

normal subgroups only

This group has 624 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
1C2×#11762599C212yesmaximal, complemented
2C2×#11762596C212yesmaximal
3#14528772C211yesmaximal, characteristic, complemented
4C4×#10587504C211yesmaximal, characteristic, complemented
5C17×#2534C211yesmaximal, characteristic
6C2×#10587504C2212yes—
7C2×#10587505C2212yes—
8C4×#63223—21no—
9#11762669—21no—
10#11762599—24no—
11#11762596—24no—
12#11762578—21no—
13#11762576—21no—
14C2×C17×#123C2211yescharacteristic
15C2×C17×#120C2211yescharacteristic
16C2×C4×D4×C17C2211yescharacteristic
17C22×C22⋊C4×C17C411yescharacteristic, complemented
18C17×#396—21no—
19C17×#394—21no—
20C17×#388—21no—
21C17×#386—21no—
22C17×#381C411yescharacteristic, complemented
23C17×#380C411yescharacteristic, complemented
24C17×#379C411yescharacteristic, complemented
25C25×C17C2×C411yescharacteristic
26C22×D4×C17—21no—
27C22×D4×C17C2311yescharacteristic
28C23×C4×C17C2×C411yescharacteristic
29C23×C4×C17C2×C411yescharacteristic
30C23×C4×C17D411yescharacteristic
31C17×#123—22no—
32C17×#123—22no—
33C17×#120—22no—
34C17×#120—22no—
35C4×D4×C17—41no—
36C4×D4×C17—41no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17—21no—
39C2×C4⋊C4×C17D411yescharacteristic
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17—21no—
42C2×C22⋊C4×C17—21no—
43C2×C22⋊C4×C17—21no—
44C2×C22⋊C4×C17—21no—
45C2×C22⋊C4×C17—21no—
46C2×C22⋊C4×C17—41no—
47C2×C22⋊C4×C17—41no—
48C2×C22⋊C4×C17C2×C411yescharacteristic
49C2×C22⋊C4×C17C2×C411yescharacteristic
50C2×C22⋊C4×C17C2×C411yescharacteristic
51C2×C22⋊C4×C17D411yescharacteristic
52C2×C42×C17D411yescharacteristic
53C2×C22⋊Dic17—22no—
54C2×C4⋊Dic17—41no—
55C2×#63223—22no—
56C17×#95—21no—
57C17×#95—21no—
58C17×#95—21no—
59#10587505—22no—
60#10587505—22no—
61#10587504—22no—
62#10587504D414yes—
63#10587502—41no—
64#10587474—41no—
65C4×#3622—41no—
66C24×C17—21no—
67C24×C17—22no—
68C24×C17—42no—
69C24×C17—41no—
70C24×C17—41no—
71C24×C17—81no—
72C24×C17C22×C411yescharacteristic
73C24×C17C2×D411yescharacteristic
74C2×D4×C17—41no—
75C2×D4×C17—41no—
76C2×D4×C17—42no—
77C2×D4×C17—41no—
78C2×D4×C17—41no—
79C2×D4×C17—81no—
80C2×D4×C17C2×D412yes—
81C2×D4×C17#4112yes—
82C22×C4×C17—21no—
83C22×C4×C17—21no—
84C22×C4×C17—21no—
85C22×C4×C17—21no—
86C22×C4×C17—21no—
87C22×C4×C17—21no—
88C22×C4×C17—21no—
89C22×C4×C17—22no—
90C22×C4×C17—41no—
91C22×C4×C17—41no—
92C22×C4×C17—81no—
93C22×C4×C17—81no—
94C22×C4×C17C22⋊C411yescharacteristic
95C22×C4×C17C22⋊C411yescharacteristic
96C22×C4×C17C22⋊C411yescharacteristic
97C22×C4×C17C22⋊C411yescharacteristic
98C22×C4×C17C2×D411yescharacteristic
99C22×Dic17—42no—
100C4⋊C4×C17—41no—
101C4⋊C4×C17—42no—
102C4⋊C4×C17—81no—
103C22⋊C4×C17—22no—
104C22⋊C4×C17—22no—
105C22⋊C4×C17—41no—
106C22⋊C4×C17—41no—
107C22⋊C4×C17—42no—
108C22⋊C4×C17—42no—
109C22⋊C4×C17—41no—
110C22⋊C4×C17—41no—
111C22⋊C4×C17—41no—
112C22⋊C4×C17—41no—
113C22⋊C4×C17—42no—
114C22⋊C4×C17—81no—
115C42×C17—21no—
116C42×C17—21no—
117C22⋊Dic17—42no—
118C22⋊Dic17—42no—
119C4⋊Dic17—82no—
120#63223—24no—
121#63223—42no—
122C2×#3622—42no—
123#13219—171nomaximal, Sylow
124C23×C17—21no—
125C23×C17—21no—
126C23×C17—21no—
127C23×C17—21no—
128C23×C17—22no—
129C23×C17—41no—
130C23×C17—41no—
131C23×C17—41no—
132C23×C17—41no—
133C23×C17—41no—
134C23×C17—41no—
135C23×C17—41no—
136C23×C17—42no—
137C23×C17—42no—
138C23×C17—41no—
139C23×C17—41no—
140C23×C17—41no—
141C23×C17—82no—
142C23×C17—84no—
143C23×C17—84no—
144C23×C17—82no—
145C23×C17C2×C22⋊C411yescharacteristic
146C23×C17C4×D412yes—
147C23×C17#12411yescharacteristic
148C23×C17#12711yescharacteristic
149D4×C17—42no—
150D4×C17—42no—
151D4×C17—82no—
152D4×C17—82no—
153C2×C4×C17—21no—
154C2×C4×C17—21no—
155C2×C4×C17—21no—
156C2×C4×C17—21no—
157C2×C4×C17—21no—
158C2×C4×C17—41no—
159C2×C4×C17—42no—
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—41no—
168C2×C4×C17—41no—
169C2×C4×C17—41no—
170C2×C4×C17—41no—
171C2×C4×C17—41no—
172C2×C4×C17—81no—
173C2×C4×C17—81no—
174C2×C4×C17—81no—
175C2×C4×C17—82no—
176C2×C4×C17#9911yescharacteristic
177C2×C4×C17#9911yescharacteristic
178C2×C4×C17#12112yes—
179C2×Dic17—82no—
180C2×Dic17—82no—
181C2×Dic17—82no—
182#3622—44no—
183#2534D1711yescharacteristic
184C2×#352—172no—
185C2×AΣL(1,𝔽16)—172no—
186#1799—171no—
187C4×#100—171no—
188C22×C17—21no—
189C22×C17—21no—
190C22×C17—21no—
191C22×C17—22no—
192C22×C17—21no—
193C22×C17—41no—
194C22×C17—41no—
195C22×C17—42no—
196C22×C17—42no—
197C22×C17—42no—
198C22×C17—42no—
199C22×C17—41no—
200C22×C17—41no—
201C22×C17—41no—
202C22×C17—41no—
203C22×C17—42no—
204C22×C17—81no—
205C22×C17—81no—
206C22×C17—81no—
207C22×C17—84no—
208C22×C17—84no—
209C22×C17—81no—
210C22×C17—81no—
211C22×C17#39011yescharacteristic
212C22×C17C2×#9911yescharacteristic
213C22×C17#41011yescharacteristic
214C4×C17—21no—
215C4×C17—21no—
216C4×C17—22no—
217C4×C17—41no—
218C4×C17—41no—
219C4×C17—41no—
220C4×C17—81no—
221C4×C17—81no—
222C4×C17—81no—
223Dic17—162no—
224C2×#123C2×D1711yescharacteristic
225C2×#120C2×D1711yescharacteristic
226C2×C4×D4C2×D1711yescharacteristic
227C22×C22⋊C4Dic1711yescharacteristic, complemented
228C2×#100—172no—
229C2×#99—172no—
230C4×M16—341no—
231#396—21no—
232#394—21no—
233#388—21no—
234#386—21no—
235#381Dic1711yescharacteristic, complemented
236#380—341no—
237#380Dic1711yescharacteristic, complemented
238#379Dic1711yescharacteristic, complemented
239#352—344no—
240AΣL(1,𝔽16)—344no—
241#343—341no—
242#342—341no—
243C2×C17—21no—
244C2×C17—21no—
245C2×C17—41no—
246C2×C17—42no—
247C2×C17—41no—
248C2×C17—82no—
249C2×C17#196411yescharacteristic
250C2×C17#212011yescharacteristic
251C2×C17#212111yescharacteristic
252C25C2×Dic1711yescharacteristic
253C22×D4—21no—
254C22×D4C22×D1711yescharacteristic
255C23×C4C2×Dic1711yescharacteristic
256C23×C4C2×Dic1711yescharacteristic
257C23×C4C22⋊D1711yescharacteristic
258C2×M16—342no—
259#123—22no—
260#123—22no—
261#120—22no—
262#120—22no—
263C4×D4—41no—
264C4×D4—41no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—681no—
268C2×C4⋊C4C22⋊D1711yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—342no—
278C2×C22⋊C4C2×Dic1711yescharacteristic
279C2×C22⋊C4C2×Dic1711yescharacteristic
280C2×C22⋊C4C2×Dic1711yescharacteristic
281C2×C22⋊C4C22⋊D1711yescharacteristic
282C2×C42C22⋊D1711yescharacteristic
283#100—174no—
284#100—342no—
285#99—342no—
286#99—342no—
287#97—681no—
288C4×C8—681no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—681no—
293C17#1321911yescharacteristic, complemented, Sylow
294C24—21no—
295C24—22no—
296C24—41no—
297C24—42no—
298C24—41no—
299C24—81no—
300C24C22×Dic1711yescharacteristic
301C24C2×C22⋊D1711yescharacteristic
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—81no—
308C2×D4D4×D1712yes—
309C2×D4#6325412yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4—682no—
323C22×C4C22⋊Dic1711yescharacteristic
324C22×C4C22⋊Dic1711yescharacteristic
325C22×C4C22⋊Dic1711yescharacteristic
326C22×C4C22⋊Dic1711yescharacteristic
327C22×C4C2×C22⋊D1711yescharacteristic
328M16—344no—
329M16—682no—
330C2×C8—682no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—81no—
334C4⋊C4—1362no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—42no—
346C22⋊C4—81no—
347C22⋊C4—682no—
348C22⋊C4—682no—
349C42—21no—
350C42—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—82no—
369C23—84no—
370C23—84no—
371C23—82no—
372C23D4×Dic1712yes—
373C23#1058762211yescharacteristic
374C23#1058762611yescharacteristic
375C23C2×C22⋊Dic1711yescharacteristic
376D4—42no—
377D4—42no—
378D4—82no—
379D4—82no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—1362no—
404C2×C4—1362no—
405C2×C4—1362no—
406C2×C4#1058750511yescharacteristic
407C2×C4#1058750511yescharacteristic
408C2×C4#1058762412yes—
409C8—684no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—21no—
414C22—21no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—42no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—84no—
430C22—84no—
431C22—81no—
432C22—81no—
433C22#1176271211yescharacteristic
434C22C2×#1058750511yescharacteristic
435C22#1176278311yescharacteristic
436C4—22no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—2722no—
446C2—21no—
447C2—21no—
448C2—41no—
449C2—42no—
450C2—41no—
451C2—82no—
452C2#1452962111yescharacteristic
453C2#1453042511yescharacteristic
454C2#1453043011yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC24×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#2534C2
socleC22×C17C2×#99
prime radicalC2×C17×#120C22
cosocleC22×D4×C17C23
hypercenter#2534D17
hypocenterC17#13219
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2534

Complements

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

NH ≅ G/Ntype
1. C2×#11762599C2semidirect
3. #14528772C2semidirect
4. C4×#10587504C2semidirect
17. C22×C22⋊C4×C17C4semidirect
22. C17×#381C4semidirect
23. C17×#380C4semidirect
24. C17×#379C4semidirect
227. C22×C22⋊C4Dic17semidirect
235. #381Dic17semidirect
237. #380Dic17semidirect
238. #379Dic17semidirect
293. C17#13219semidirect

Maximal subgroups

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

subgroupindex
C2×#117625992
C2×#117625962
#145287722
C4×#105875042
C17×#25342
#1321917