Finite Group Databasebeta

← #16934791

Subgroups — #16934791

Order 4352.

594 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 101 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×#11763120C211yesmaximal, characteristic
2#14533270C211yesmaximal, characteristic, complemented
3#14529621C211yesmaximal, characteristic
4#14528871C211yesmaximal, characteristic, complemented
5#14527356C211yesmaximal, characteristic, complemented
6#14526631C211yesmaximal, characteristic, complemented
7C17×#3187C211yesmaximal, characteristic
8C22×C4⋊Dic17C2211yescharacteristic
9#11763120—22no—
10#11763120—22no—
11C2×#10587478—21no—
12C2×#10587503—21no—
13C2×#10587505—22no—
14C2×C22⋊#3622C2211yescharacteristic
15#11762748—22no—
16#11762716—21no—
17#11762674—21no—
18#11762670—22no—
19#11762669C2211yescharacteristic
20C4×C22⋊Dic17C2211yescharacteristic
21#11762572—22no—
22#11762546—22no—
23C2×C17×C22⋊Q8C2211yescharacteristic
24C2×C17×#121C2211yescharacteristic
25C17×#483—21no—
26C17×#481—21no—
27C17×#465—22no—
28C17×#450—21no—
29C17×#448—21no—
30C2×C17×C22⋊C8C2211yescharacteristic
31C22×Q8×C17D411yescharacteristic
32C22×D4×C17—21no—
33C22×D4×C17D411yescharacteristic
34C23×C4×C17C2311yescharacteristic
35C23×Dic17—21no—
36C2×SD16×C17—22no—
37C22×C8×C17D412yes—
38C17×C22⋊Q8—21no—
39C17×C22⋊Q8—21no—
40C17×C22⋊Q8—21no—
41C17×C22⋊Q8—21no—
42C17×#121—21no—
43C17×#121—21no—
44C17×#121—21no—
45C17×#121—21no—
46C2×C4⋊C4×C17—21no—
47C2×C4⋊C4×C17D411yescharacteristic
48C2×C4⋊C4×C17D411yescharacteristic
49C2×C22⋊C4×C17—21no—
50C2×C22⋊C4×C17—21no—
51C2×C22⋊Dic17—21no—
52C2×C22⋊Dic17D412yes—
53C2×C4⋊Dic17—21no—
54C2×C4⋊Dic17—21no—
55C2×C4⋊Dic17—21no—
56C2×C4⋊Dic17—21no—
57C2×C4⋊Dic17—21no—
58C2×C4⋊Dic17—21no—
59C2×C4⋊Dic17D412yes—
60C2×C4×Dic17—21no—
61C22×#3622—21no—
62C17×#106—22no—
63C17×#103—22no—
64C17×#102—22no—
65C17×C22⋊C8—21no—
66C17×C22⋊C8—21no—
67#10587505—44no—
68#10587503—24no—
69#10587502—24no—
70#10587502—24no—
71#10587502—21no—
72#10587502—21no—
73C22⋊#3622—21no—
74C22⋊#3622D412yes—
75#10587478—24no—
76C24×C17—21no—
77C24×C17C2×D411yescharacteristic
78C2×Q8×C17—21no—
79C2×Q8×C17—21no—
80C2×Q8×C17—42no—
81C2×D4×C17—21no—
82C2×D4×C17—21no—
83C2×D4×C17—42no—
84C2×D4×C17—41no—
85C2×D4×C17—41no—
86C2×D4×C17—42no—
87C2×D4×C17—42no—
88C22×C4×C17—21no—
89C22×C4×C17—21no—
90C22×C4×C17—21no—
91C22×C4×C17—42no—
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17C2×D411yescharacteristic
94C22×C4×C17C2×D411yescharacteristic
95C22×C4×C17C2×D411yescharacteristic
96C22×C4×C17C2×D411yescharacteristic
97C22×C4×C17#4111yescharacteristic
98C22×Dic17—22no—
99C22×Dic17—22no—
100C22×Dic17—41no—
101C22×Dic17—41no—
102C22×Dic17—42no—
103C22×Dic17—42no—
104C22×Dic17—41no—
105SD16×C17—82no—
106C2×C8×C17—22no—
107C2×C8×C17—42no—
108C4⋊C4×C17—21no—
109C4⋊C4×C17—21no—
110C4⋊C4×C17—21no—
111C4⋊C4×C17—21no—
112C4⋊C4×C17—41no—
113C4⋊C4×C17—41no—
114C22⋊C4×C17—41no—
115C22⋊C4×C17—41no—
116C22⋊C4×C17—41no—
117C22⋊C4×C17—41no—
118C22⋊Dic17—22no—
119C22⋊Dic17—22no—
120C22⋊Dic17—42no—
121C4⋊Dic17—22no—
122C4⋊Dic17—22no—
123C4⋊Dic17—41no—
124C4⋊Dic17—41no—
125C4⋊Dic17—42no—
126C4×Dic17—42no—
127C2×#3622—22no—
128C2×#3622—42no—
129#9953—171nomaximal, Sylow
130C23×C17—41no—
131C23×C17—41no—
132C23×C17—41no—
133C23×C17—42no—
134C23×C17—81no—
135C23×C17—81no—
136C23×C17#12011yescharacteristic
137C23×C17#12111yescharacteristic
138C23×C17#12711yescharacteristic
139C23×C17#13512yes—
140C23×C17AGL(1,ℤ/8)12yes—
141Q8×C17—42no—
142Q8×C17—81no—
143D4×C17—42no—
144D4×C17—42no—
145D4×C17—82no—
146D4×C17—81no—
147D4×C17—81no—
148C2×C4×C17—22no—
149C2×C4×C17—22no—
150C2×C4×C17—22no—
151C2×C4×C17—41no—
152C2×C4×C17—41no—
153C2×C4×C17—41no—
154C2×C4×C17—42no—
155C2×C4×C17—42no—
156C2×C4×C17—41no—
157C2×C4×C17—41no—
158C2×C4×C17—41no—
159C2×C4×C17—42no—
160C2×C4×C17#12011yescharacteristic
161C2×C4×C17#12011yescharacteristic
162C2×C4×C17#12111yescharacteristic
163C2×C4×C17#12111yescharacteristic
164C2×Dic17—42no—
165C2×Dic17—42no—
166C2×Dic17—42no—
167C2×Dic17—42no—
168C2×Dic17—44no—
169C2×Dic17—82no—
170C2×Dic17—81no—
171C2×Dic17—81no—
172C2×Dic17—81no—
173C2×Dic17—81no—
174C2×Dic17—81no—
175C8×C17—42no—
176#3622—42no—
177#3187D1711yescharacteristic
178C2×#480—171no—
179#2503—171no—
180#1964—171no—
181#1823—171no—
182#1520—171no—
183#1282—171no—
184C22×C17—21no—
185C22×C17—21no—
186C22×C17—21no—
187C22×C17—21no—
188C22×C17—21no—
189C22×C17—21no—
190C22×C17—42no—
191C22×C17—42no—
192C22×C17—81no—
193C22×C17—81no—
194C22×C17—81no—
195C22×C17—162no—
196C22×C17#39211yescharacteristic
197C22×C17#44912yes—
198C22×C17#46312yes—
199C22×C17#49511yescharacteristic
200C22×C17#49611yescharacteristic
201C4×C17—21no—
202C4×C17—21no—
203C4×C17—21no—
204C4×C17—81no—
205C4×C17—81no—
206C4×C17—81no—
207C4×C17#45812yes—
208Dic17—82no—
209Dic17—82no—
210Dic17—161no—
211C2×C22⋊Q8C2×D1711yescharacteristic
212C2×#121C2×D1711yescharacteristic
213C22×C4⋊C4—171no—
214#483—21no—
215#481—21no—
216#480—342no—
217#480—342no—
218#465—22no—
219#450—21no—
220#448—21no—
221C2×#107—341no—
222C2×#102—341no—
223C2×#99—342no—
224C2×C22⋊C8—171no—
225C2×C22⋊C8C2×D1711yescharacteristic
226#400—342no—
227#391—341no—
228#382—341no—
229#381—342no—
230#380—171no—
231C4×C22⋊C4—171no—
232#340—342no—
233#328—342no—
234C2×C17—22no—
235C2×C17—21no—
236C2×C17—22no—
237C2×C17—21no—
238C2×C17—161no—
239C2×C17#200911yescharacteristic
240C2×C17#201211yescharacteristic
241C2×C17#202211yescharacteristic
242C22×Q8C22⋊D1711yescharacteristic
243C22×D4—21no—
244C22×D4C22⋊D1711yescharacteristic
245C23×C4—341no—
246C23×C4C22×D1711yescharacteristic
247C2×SD16—22no—
248C22×C8—341no—
249C22×C8C22⋊D1712yes—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254#121—21no—
255#121—21no—
256#121—21no—
257#121—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—172no—
260C2×C4⋊C4—341no—
261C2×C4⋊C4—341no—
262C2×C4⋊C4—341no—
263C2×C4⋊C4—341no—
264C2×C4⋊C4—341no—
265C2×C4⋊C4—341no—
266C2×C4⋊C4C22⋊D1711yescharacteristic
267C2×C4⋊C4C22⋊D1711yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—172no—
271C2×C22⋊C4—341no—
272C2×C42—341no—
273#107—344no—
274#106—22no—
275#103—22no—
276#102—22no—
277#102—344no—
278#99—684no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281C22⋊C8—172no—
282C22⋊C8—341no—
283#95—344no—
284#95—344no—
285#95—341no—
286#95—341no—
287C17#995311yescharacteristic, complemented, Sylow
288C24—21no—
289C24D4×D1711yescharacteristic
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—42no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—42no—
304C22×C4—342no—
305C22×C4—342no—
306C22×C4—682no—
307C22×C4—682no—
308C22×C4—681no—
309C22×C4—681no—
310C22×C4—681no—
311C22×C4D4×D1711yescharacteristic
312C22×C4D4×D1711yescharacteristic
313C22×C4#6325411yescharacteristic
314C22×C4C2×C22⋊D1711yescharacteristic
315C22×C4C2×C22⋊D1711yescharacteristic
316C22×C4C2×C22⋊D1711yescharacteristic
317SD16—82no—
318C2×C8—22no—
319C2×C8—42no—
320C2×C8—342no—
321C2×C8—682no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—342no—
329C4⋊C4—342no—
330C4⋊C4—682no—
331C4⋊C4—681no—
332C4⋊C4—681no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—342no—
338C22⋊C4—342no—
339C22⋊C4—682no—
340C42—682no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—42no—
345C23—81no—
346C23—81no—
347C23#1058760012yes—
348C23#1058760212yes—
349C23#1058762311yescharacteristic
350C23#1058762511yescharacteristic
351C23#1058762611yescharacteristic
352Q8—42no—
353Q8—81no—
354D4—42no—
355D4—42no—
356D4—82no—
357D4—81no—
358D4—81no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—682no—
372C2×C4—682no—
373C2×C4—682no—
374C2×C4—684no—
375C2×C4—682no—
376C2×C4—1361no—
377C2×C4—1361no—
378C2×C4—1361no—
379C2×C4—1362no—
380C2×C4—1361no—
381C2×C4—1361no—
382C2×C4#1058762311yescharacteristic
383C2×C4#1058762411yescharacteristic
384C2×C4#1058762511yescharacteristic
385C2×C4#1058763911yescharacteristic
386C8—42no—
387C8—682no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—162no—
400C22#1176271711yescharacteristic
401C22#1176295312yes—
402C22#1176302612yes—
403C22#1176319011yescharacteristic
404C22#1176319511yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—81no—
409C4—81no—
410C4—81no—
411C4—2721no—
412C4—1362no—
413C4—1362no—
414C4#1176299912yes—
415C2—21no—
416C2—22no—
417C2—22no—
418C2—21no—
419C2—161no—
420C2#1452984111yescharacteristic
421C2#1452985811yescharacteristic
422C2#1452991311yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#3187C2
socleC22×C17#392
prime radicalC2×C17×#121C22
hypercenter#3187D17
hypocenterC17#9953
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
2. #14533270C2semidirect
4. #14528871C2semidirect
5. #14527356C2semidirect
6. #14526631C2semidirect
287. C17#9953semidirect

Maximal subgroups

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

subgroupindex
C2×#117631202
#145332702
#145296212
#145288712
#145273562
#145266312
C17×#31872
#995317