Finite Group Databasebeta

← #16934610

Subgroups — #16934610

Order 4352.

654 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 113 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×#11763137C211yesmaximal, characteristic
2#14534599C211yesmaximal, characteristic
3C2×#11762741C211yesmaximal, characteristic
4#14528871C211yesmaximal, characteristic
5C2×#11762534C211yesmaximal, characteristic
6#14526663C211yesmaximal, characteristic
7C2×C17×#328C211yesmaximal, characteristic
8C22×C4⋊Dic17—21no—
9C22×C4⋊Dic17C2211yescharacteristic
10C22×#63225C2211yescharacteristic
11#11763137—22no—
12#11763137—22no—
13C2×#10587479—21no—
14C2×#10587506—21no—
15C2×C22⋊#3622C2211yescharacteristic
16#11762747—21no—
17#11762742—22no—
18#11762741—22no—
19#11762741—22no—
20#11762728—21no—
21#11762681—21no—
22#11762670—21no—
23#11762669C2211yescharacteristic
24C2×#10587502—21no—
25#11762572—22no—
26#11762534—22no—
27C2×C17×C22⋊Q8C2211yescharacteristic
28C2×C17×C22⋊C8C2211yescharacteristic
29C2×C17×#95C2211yescharacteristic
30C17×#328—22no—
31C22×Q8×C17D411yescharacteristic
32C23×C4×C17—21no—
33C23×C4×C17C2311yescharacteristic
34C23×Dic17—21no—
35C23×Dic17D411yescharacteristic
36C23×Dic17D411yescharacteristic
37C22×C8×C17—21no—
38C17×C22⋊Q8—21no—
39C17×C22⋊Q8—21no—
40C17×C22⋊Q8—21no—
41C17×C22⋊Q8—21no—
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17D411yescharacteristic
44C2×C22⋊C4×C17—21no—
45C2×C22⋊Dic17—21no—
46C2×C4⋊Dic17—24no—
47C2×C4⋊Dic17—21no—
48C2×C4⋊Dic17—21no—
49C2×C4⋊Dic17—21no—
50C2×C4⋊Dic17—21no—
51C2×C4⋊Dic17—21no—
52C2×C4⋊Dic17—42no—
53C2×C4⋊Dic17—42no—
54C2×C4⋊Dic17D412yes—
55C2×C4⋊Dic17D412yes—
56C2×#63225—22no—
57C2×#63225—21no—
58C2×#63225—21no—
59C2×#63225—21no—
60C2×#63225—21no—
61C2×#63225—21no—
62C2×#63225—21no—
63C22×#3622—21no—
64C17×C22⋊C8—21no—
65C17×C22⋊C8—21no—
66C17×#95—21no—
67C17×#95D412yes—
68#10587506—24no—
69#10587502—22no—
70#10587502—24no—
71#10587502—22no—
72#10587502—22no—
73#10587502—21no—
74C22⋊#3622—21no—
75C22⋊#3622D412yes—
76#10587479—24no—
77C24×C17C2×D411yescharacteristic
78C2×Q8×C17—21no—
79C2×Q8×C17—21no—
80C2×Q8×C17—42no—
81C22×C4×C17—24no—
82C22×C4×C17—22no—
83C22×C4×C17—21no—
84C22×C4×C17—21no—
85C22×C4×C17—42no—
86C22×C4×C17—42no—
87C22×C4×C17—41no—
88C22×C4×C17—41no—
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17C2×D411yescharacteristic
94C22×C4×C17#4111yescharacteristic
95C22×Dic17—21no—
96C22×Dic17—21no—
97C22×Dic17—21no—
98C22×Dic17—21no—
99C22×Dic17—21no—
100C22×Dic17—21no—
101C22×Dic17—22no—
102C22×Dic17—21no—
103C22×Dic17—21no—
104C22×Dic17—22no—
105C22×Dic17—21no—
106C22×Dic17—42no—
107C22×Dic17—41no—
108C22×Dic17—41no—
109C22×Dic17—41no—
110C22×Dic17—41no—
111C22×Dic17—41no—
112C22×Dic17—41no—
113C22×Dic17SD1612yes—
114C2×C8×C17—42no—
115C2×C8×C17—41no—
116C4⋊C4×C17—21no—
117C4⋊C4×C17—21no—
118C4⋊C4×C17—41no—
119C4⋊C4×C17—41no—
120C22⋊C4×C17—41no—
121C22⋊C4×C17—41no—
122C22⋊Dic17—41no—
123C22⋊Dic17—41no—
124C4⋊Dic17—22no—
125C4⋊Dic17—22no—
126C4⋊Dic17—48no—
127C4⋊Dic17—41no—
128C4⋊Dic17—41no—
129C4⋊Dic17—42no—
130#63225—41no—
131#63225—41no—
132#63225—41no—
133#63225—41no—
134#63225—82no—
135C2×#3622—22no—
136C2×#3622—42no—
137#9927—171nomaximal, Sylow
138C23×C17—22no—
139C23×C17—22no—
140C23×C17#12011yescharacteristic
141C23×C17#12011yescharacteristic
142C23×C17#12111yescharacteristic
143C23×C17C2×SD1611yescharacteristic
144C23×C17#13511yescharacteristic
145C23×C17AGL(1,ℤ/8)11yescharacteristic
146C23×C17#13711yescharacteristic
147Q8×C17—42no—
148Q8×C17—81no—
149C2×C4×C17—22no—
150C2×C4×C17—22no—
151C2×C4×C17—24no—
152C2×C4×C17—41no—
153C2×C4×C17—41no—
154C2×C4×C17—41no—
155C2×C4×C17—41no—
156C2×C4×C17—44no—
157C2×C4×C17—44no—
158C2×C4×C17—42no—
159C2×C4×C17—42no—
160C2×C4×C17—41no—
161C2×C4×C17—41no—
162C2×C4×C17—42no—
163C2×C4×C17—42no—
164C2×C4×C17#12011yescharacteristic
165C2×C4×C17#12111yescharacteristic
166C2×C4×C17#12111yescharacteristic
167C2×C4×C17#12711yescharacteristic
168C2×Dic17—22no—
169C2×Dic17—22no—
170C2×Dic17—22no—
171C2×Dic17—41no—
172C2×Dic17—41no—
173C2×Dic17—41no—
174C2×Dic17—41no—
175C2×Dic17—42no—
176C2×Dic17—41no—
177C2×Dic17—42no—
178C2×Dic17—42no—
179C2×Dic17—42no—
180C2×Dic17—42no—
181C2×Dic17—42no—
182C2×Dic17—42no—
183C2×Dic17—42no—
184C2×Dic17—81no—
185C2×Dic17—81no—
186C2×Dic17—81no—
187C2×Dic17—81no—
188C2×Dic17—81no—
189C8×C17—81no—
190#3622—42no—
191C2×#483—171no—
192#2627—171no—
193C2×#398—171no—
194#1823—171no—
195C2×#328D1711yescharacteristic
196C2×#324—171no—
197#1294—171no—
198C22×C17—22no—
199C22×C17—22no—
200C22×C17—22no—
201C22×C17—22no—
202C22×C17—21no—
203C22×C17—21no—
204C22×C17—21no—
205C22×C17—21no—
206C22×C17#39211yescharacteristic
207C22×C17#44811yescharacteristic
208C22×C17#44911yescharacteristic
209C22×C17#45011yescharacteristic
210C22×C17#45211yescharacteristic
211C22×C17#45612yes—
212C22×C17#45812yes—
213C22×C17#46511yescharacteristic
214C22×C17#46811yescharacteristic
215C4×C17—21no—
216C4×C17—21no—
217C4×C17—21no—
218C4×C17—21no—
219C4×C17—44no—
220C4×C17—81no—
221C4×C17—81no—
222Dic17—42no—
223Dic17—82no—
224Dic17—82no—
225Dic17—81no—
226C2×C22⋊Q8C2×D1711yescharacteristic
227C22×C4⋊C4—171no—
228C22×C4⋊C4—171no—
229C22×C4⋊C4—341no—
230#483—342no—
231#483—342no—
232C2×#106—341no—
233C2×#103—341no—
234C2×C22⋊C8—171no—
235C2×C22⋊C8C2×D1711yescharacteristic
236#400—341no—
237#398—342no—
238#398—342no—
239#398—342no—
240#395—341no—
241#385—341no—
242#381—341no—
243#380—171no—
244C2×#95—341no—
245C2×#95C2×D1711yescharacteristic
246#340—342no—
247#328—22no—
248#324—342no—
249C2×C17—21no—
250C2×C17—21no—
251C2×C17—21no—
252C2×C17—21no—
253C2×C17#160912yes—
254C2×C17#161112yes—
255C2×C17#200011yescharacteristic
256C2×C17#200511yescharacteristic
257C2×C17#202211yescharacteristic
258C22×Q8D6811yescharacteristic
259C23×C4—21no—
260C23×C4—171no—
261C23×C4—171no—
262C23×C4—341no—
263C23×C4C22×D1711yescharacteristic
264C22×C8—21no—
265C22×C8—341no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—172no—
272C2×C4⋊C4—172no—
273C2×C4⋊C4—341no—
274C2×C4⋊C4—341no—
275C2×C4⋊C4—341no—
276C2×C4⋊C4—341no—
277C2×C4⋊C4—344no—
278C2×C4⋊C4—341no—
279C2×C4⋊C4—341no—
280C2×C4⋊C4—341no—
281C2×C4⋊C4—341no—
282C2×C4⋊C4—341no—
283C2×C4⋊C4—341no—
284C2×C4⋊C4—341no—
285C2×C4⋊C4—342no—
286C2×C4⋊C4—682no—
287C2×C4⋊C4—682no—
288C2×C4⋊C4D6811yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—341no—
291#106—344no—
292#103—344no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—172no—
296C22⋊C8—341no—
297#95—21no—
298#95—344no—
299#95—342no—
300#95—342no—
301#95—342no—
302#95—341no—
303#95C22⋊D1712yes—
304C17#992711yescharacteristic, complemented, Sylow
305C24D4×D1711yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—42no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—172no—
318C22×C4—341no—
319C22×C4—341no—
320C22×C4—341no—
321C22×C4—341no—
322C22×C4—341no—
323C22×C4—341no—
324C22×C4—342no—
325C22×C4—341no—
326C22×C4—341no—
327C22×C4—342no—
328C22×C4—341no—
329C22×C4—682no—
330C22×C4—681no—
331C22×C4—681no—
332C22×C4—681no—
333C22×C4—681no—
334C22×C4—681no—
335C22×C4—681no—
336C22×C4C2×D6811yescharacteristic
337C22×C4#6325211yescharacteristic
338C22×C4D4×D1711yescharacteristic
339C22×C4D4×D1711yescharacteristic
340C22×C4D4×D1711yescharacteristic
341C22×C4C2×C22⋊D1711yescharacteristic
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—342no—
345C2×C8—682no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—342no—
351C4⋊C4—342no—
352C4⋊C4—682no—
353C4⋊C4—688no—
354C4⋊C4—681no—
355C4⋊C4—681no—
356C4⋊C4—681no—
357C4⋊C4—681no—
358C4⋊C4—681no—
359C4⋊C4—681no—
360C4⋊C4—1362no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—681no—
364C22⋊C4—681no—
365C23—22no—
366C23—22no—
367C23#1058756811yescharacteristic
368C23#1058757011yescharacteristic
369C23SD16×D1711yescharacteristic
370C23#1058760011yescharacteristic
371C23#1058760411yescharacteristic
372C23#1058760511yescharacteristic
373C23#1058762311yescharacteristic
374Q8—42no—
375Q8—81no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—342no—
392C2×C4—342no—
393C2×C4—342no—
394C2×C4—681no—
395C2×C4—681no—
396C2×C4—681no—
397C2×C4—681no—
398C2×C4—682no—
399C2×C4—682no—
400C2×C4—682no—
401C2×C4—682no—
402C2×C4—682no—
403C2×C4—681no—
404C2×C4—682no—
405C2×C4—682no—
406C2×C4—682no—
407C2×C4—1361no—
408C2×C4—1361no—
409C2×C4—1361no—
410C2×C4—1361no—
411C2×C4—1361no—
412C2×C4#1058756811yescharacteristic
413C2×C4#1058757011yescharacteristic
414C2×C4#1058761611yescharacteristic
415C2×C4#1058762611yescharacteristic
416C8—81no—
417C8—682no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22#1176271811yescharacteristic
427C22#1176294911yescharacteristic
428C22#1176295711yescharacteristic
429C22#1176295811yescharacteristic
430C22#1176297011yescharacteristic
431C22#1176299312yes—
432C22#1176300012yes—
433C22#1176304111yescharacteristic
434C22#1176305811yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—44no—
440C4—81no—
441C4—81no—
442C4—682no—
443C4—1362no—
444C4—1361no—
445C4—1362no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2#1452778412yes—
451C2#1452779312yes—
452C2#1452978611yescharacteristic
453C2#1452981911yescharacteristic
454C2#1452991411yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC2×C17×#328C2
socleC23×C17#120
prime radicalC24×C17C2×D4
hypercenterC2×#328D17
hypocenterC17#9927
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
304. C17#9927semidirect

Maximal subgroups

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

subgroupindex
C2×#117631372
#145345992
C2×#117627412
#145288712
C2×#117625342
#145266632
C2×C17×#3282
#992717