Finite Group Databasebeta

← #16934779

Subgroups — #16934779

Order 4352.

610 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 109 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#14539537C211yesmaximal, characteristic
2C2×#11763127C211yesmaximal, characteristic
3#14529620C211yesmaximal, characteristic
4#14528874C211yesmaximal, characteristic, complemented
5#14527356C211yesmaximal, characteristic, complemented
6#14526632C211yesmaximal, characteristic, complemented
7C17×#2502C211yesmaximal, characteristic, complemented
8C2×#10587616C2211yescharacteristic
9#11763137—21no—
10#11763127—21no—
11#11763127—22no—
12#11763127—22no—
13#11763040—22no—
14#11762959—21no—
15#11762948—21no—
16C2×#10587479—21no—
17C2×#10587480—21no—
18C2×#10587477—22no—
19C2×C22⋊#3622C2211yescharacteristic
20C2×C22⋊#3622C2211yescharacteristic
21#11762669C2211yescharacteristic
22#11762571—22no—
23#11762546—22no—
24C2×C17×C22⋊Q8C2211yescharacteristic
25C22×C4⋊C4×C17C2211yescharacteristic
26C17×#398—22no—
27C17×#391—21no—
28C17×#385—22no—
29C17×#384—21no—
30C4×C22⋊C4×C17C2211yescharacteristic
31C22×Q8×C17D411yescharacteristic
32C23×C4×C17—21no—
33C23×C4×C17C2311yescharacteristic
34C22×C22⋊D17—21no—
35C22×D68D411yescharacteristic
36C17×C22⋊Q8—21no—
37C17×C22⋊Q8—21no—
38C17×C22⋊Q8—21no—
39C17×C22⋊Q8—21no—
40C2×C4⋊C4×C17—21no—
41C2×C4⋊C4×C17—21no—
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17—21no—
44C2×C4⋊C4×C17—21no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17—24no—
47C2×C4⋊C4×C17D412yes—
48C2×C4⋊C4×C17D411yescharacteristic
49C2×C22⋊C4×C17—21no—
50C2×C22⋊C4×C17D412yes—
51C2×C42×C17—21no—
52C2×C22⋊Dic17—21no—
53C2×#63230—22no—
54#10587616—21no—
55#10587616—21no—
56#10587616—21no—
57#10587616—21no—
58C2×#63227—21no—
59C2×C4⋊Dic17—21no—
60C2×C4⋊Dic17D411yescharacteristic
61C22×#3622—21no—
62C22×#3622D412yes—
63C17×#95—24no—
64C17×#95—21no—
65#10587506—22no—
66#10587502—21no—
67C22⋊#3622—21no—
68C22⋊#3622—21no—
69C22⋊#3622—21no—
70C22⋊#3622D412yes—
71#10587480—22no—
72#10587480—24no—
73#10587479—22no—
74#10587479—24no—
75#10587477—44no—
76C24×C17C2×D411yescharacteristic
77C23×D17—21no—
78C2×Q8×C17—21no—
79C2×Q8×C17—21no—
80C2×Q8×C17—42no—
81C22×C4×C17—22no—
82C22×C4×C17—22no—
83C22×C4×C17—21no—
84C22×C4×C17—21no—
85C22×C4×C17—42no—
86C22×C4×C17—41no—
87C22×C4×C17—41no—
88C22×C4×C17—82no—
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17C2×D411yescharacteristic
94C22×C4×C17#4111yescharacteristic
95C2×C22⋊D17—41no—
96C2×C22⋊D17—41no—
97C2×C22⋊D17—42no—
98C2×C22⋊D17—42no—
99C22×Dic17—21no—
100C22×Dic17—41no—
101C2×D68—21no—
102C2×D68—21no—
103C2×D68—42no—
104C4⋊C4×C17—21no—
105C4⋊C4×C17—22no—
106C4⋊C4×C17—21no—
107C4⋊C4×C17—22no—
108C4⋊C4×C17—41no—
109C4⋊C4×C17—41no—
110C4⋊C4×C17—41no—
111C4⋊C4×C17—41no—
112C4⋊C4×C17—84no—
113C22⋊C4×C17—22no—
114C22⋊C4×C17—41no—
115C22⋊C4×C17—41no—
116C22⋊C4×C17SD1614yescomplemented
117C42×C17—42no—
118C22⋊Dic17—42no—
119#63230—82no—
120#63227—41no—
121#63227—41no—
122C4⋊Dic17—21no—
123C4⋊Dic17—21no—
124C4⋊Dic17—42no—
125C2×#3622—22no—
126C2×#3622—22no—
127C2×#3622—42no—
128C2×#3622—42no—
129#9951—171nomaximal, Sylow
130C23×C17—41no—
131C23×C17—41no—
132C23×C17#12011yescharacteristic
133C23×C17#12111yescharacteristic
134C23×C17#12711yescharacteristic
135C23×C17C2×SD1612yes—
136C23×C17#13712yes—
137C22×D17—41no—
138C22×D17—42no—
139C22×D17—81no—
140C22×D17—81no—
141Q8×C17—42no—
142Q8×C17—81no—
143C2×C4×C17—22no—
144C2×C4×C17—22no—
145C2×C4×C17—22no—
146C2×C4×C17—24no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—42no—
151C2×C4×C17—42no—
152C2×C4×C17—42no—
153C2×C4×C17—44no—
154C2×C4×C17—42no—
155C2×C4×C17—41no—
156C2×C4×C17—41no—
157C2×C4×C17—42no—
158C2×C4×C17—82no—
159C2×C4×C17—82no—
160C2×C4×C17#12011yescharacteristic
161C2×C4×C17#12011yescharacteristic
162C2×C4×C17#12111yescharacteristic
163C2×C4×C17#12111yescharacteristic
164C22⋊D17—82no—
165C22⋊D17—81no—
166C22⋊D17—81no—
167C2×Dic17—41no—
168C2×Dic17—42no—
169C2×Dic17—81no—
170C2×Dic17—81no—
171C2×Dic17—81no—
172D68—42no—
173D68—42no—
174#3622—42no—
175#3622—42no—
176#3187—171no—
177C2×#481—171no—
178#2502D1711yescharacteristic, complemented
179#1964—171no—
180#1824—171no—
181#1520—171no—
182#1282—171no—
183C22×C17—21no—
184C22×C17—21no—
185C22×C17—21no—
186C22×C17—21no—
187C22×C17—21no—
188C22×C17—21no—
189C22×C17—42no—
190C22×C17—42no—
191C22×C17#39211yescharacteristic
192C22×C17#44812yes—
193C22×C17#46112yes—
194C22×C17#49211yescharacteristic
195C22×C17#49711yescharacteristic
196C2×D17—81no—
197C2×D17—81no—
198C2×D17—81no—
199C2×D17—162no—
200C4×C17—21no—
201C4×C17—21no—
202C4×C17—21no—
203C4×C17—44no—
204C4×C17—81no—
205C4×C17—82no—
206C4×C17—81no—
207C4×C17#45812yes—
208Dic17—81no—
209Dic17—161no—
210C2×C22⋊Q8C2×D1711yescharacteristic
211C2×#121—171no—
212C22×C4⋊C4C2×D1711yescharacteristic
213#483—341no—
214#481—342no—
215#481—342no—
216#481—341no—
217#465—342no—
218#450—341no—
219#448—341no—
220C2×#106—341no—
221C2×#102—341no—
222C2×#99—342no—
223C2×C22⋊C8—171no—
224C2×C22⋊C8—171no—
225#398—22no—
226#391—21no—
227#385—22no—
228#384—21no—
229#380—171no—
230C4×C22⋊C4C2×D1711yescharacteristic
231#340—342no—
232#328—342no—
233C2×C17—21no—
234C2×C17—22no—
235C2×C17—22no—
236C2×C17—21no—
237C2×C17#200811yescharacteristic
238C2×C17#201311yescharacteristic
239C2×C17#202211yescharacteristic
240D17—161no—
241C22×Q8C22⋊D1711yescharacteristic
242C22×D4—171no—
243C22×D4—341no—
244C23×C4—21no—
245C23×C4C22×D1711yescharacteristic
246C2×SD16—342no—
247C22×C8—172no—
248C22×C8—341no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253#121—341no—
254#121—341no—
255#121—341no—
256#121—341no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—171no—
265C2×C4⋊C4—341no—
266C2×C4⋊C4C22⋊D1712yes—
267C2×C4⋊C4C22⋊D1711yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—341no—
270C2×C22⋊C4—341no—
271C2×C22⋊C4C22⋊D1712yes—
272C2×C42—21no—
273#106—344no—
274#106—342no—
275#103—342no—
276#102—344no—
277#102—342no—
278#99—684no—
279C22⋊C8—172no—
280C22⋊C8—341no—
281C22⋊C8—341no—
282C22⋊C8—341no—
283#95—24no—
284#95—21no—
285#95—341no—
286C17#995111yescharacteristic, complemented, Sylow
287C24—341no—
288C24C2×C22⋊D1711yescharacteristic
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—42no—
292C2×D4—341no—
293C2×D4—341no—
294C2×D4—682no—
295C2×D4—681no—
296C2×D4—682no—
297C2×D4—682no—
298C2×D4—681no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—82no—
307C22×C4—341no—
308C22×C4—681no—
309C22×C4D4×D1711yescharacteristic
310C22×C4D4×D1711yescharacteristic
311C22×C4D4×D1711yescharacteristic
312C22×C4#6325411yescharacteristic
313C22×C4C2×C22⋊D1711yescharacteristic
314C22×C4C2×C22⋊D1711yescharacteristic
315SD16—1362no—
316C2×C8—342no—
317C2×C8—342no—
318C2×C8—682no—
319C2×C8—682no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—84no—
329C4⋊C4—341no—
330C4⋊C4—341no—
331C4⋊C4—682no—
332C22⋊C4—22no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—681no—
336C22⋊C4—681no—
337C22⋊C4—682no—
338C22⋊C4#6323014yescomplemented
339C42—42no—
340C23—41no—
341C23—41no—
342C23—682no—
343C23—681no—
344C23—1361no—
345C23—1361no—
346C23#1058762411yescharacteristic
347C23#1058762611yescharacteristic
348C23C2×#6323012yes—
349C23#1058762812yes—
350C23#1058763911yescharacteristic
351Q8—42no—
352Q8—81no—
353D4—682no—
354D4—682no—
355D4—1361no—
356D4—1362no—
357D4—1361no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—681no—
376C2×C4—682no—
377C2×C4—1361no—
378C2×C4—1361no—
379C2×C4—1361no—
380C2×C4#1058762311yescharacteristic
381C2×C4#1058762311yescharacteristic
382C2×C4#1058762511yescharacteristic
383C2×C4#1058762511yescharacteristic
384C8—682no—
385C8—682no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—42no—
393C22—42no—
394C22—2722no—
395C22—1361no—
396C22—1361no—
397C22—1361no—
398C22#1176271711yescharacteristic
399C22#1176294812yes—
400C22#1176301112yes—
401C22#1176318311yescharacteristic
402C22#1176320111yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—44no—
407C4—82no—
408C4—81no—
409C4—81no—
410C4—2721no—
411C4—1361no—
412C4#1176299812yes—
413C2—22no—
414C2—21no—
415C2—22no—
416C2—21no—
417C2—2721no—
418C2#1452983711yescharacteristic
419C2#1452986711yescharacteristic
420C2#1452991211yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#2502C2
socleC22×C17#392
prime radicalC2×#10587616C22
hypercenter#2502D17
hypocenterC17#9951
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
4. #14528874C2semidirect
5. #14527356C2semidirect
6. #14526632C2semidirect
7. C17×#2502C2semidirect
116. C22⋊C4×C17SD16semidirect
178. #2502D17semidirect
286. C17#9951semidirect
338. C22⋊C4#63230semidirect

Maximal subgroups

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

subgroupindex
#145395372
C2×#117631272
#145296202
#145288742
#145273562
#145266322
C17×#25022
#995117