Finite Group Databasebeta

← #16953024

Subgroups — #16953024

Order 4352.

618 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587499C211yesmaximal, characteristic, complemented
2C2×#11763045C211yesmaximal, characteristic
3C2×#11762613C211yesmaximal, characteristic, complemented
4C2×#11762609C211yesmaximal, characteristic, complemented
5C2×C22⋊#63214C211yesmaximal, characteristic, complemented
6#14526875C211yesmaximal, characteristic, complemented
7C2×C17×#467C211yesmaximal, characteristic, complemented
8C22×D136C2211yescharacteristic
9C2×#10587616—21no—
10C2×#10587499—22no—
11C2×#10587499—22no—
12C2×#10587499—22no—
13C22×#63214C2211yescharacteristic
14C22×#63214C2211yescharacteristic
15#11763045—22no—
16#11763045—22no—
17C2×#10587489C2211yescharacteristic
18C2×#10587492—21no—
19#11762613—22no—
20#11762609—22no—
21C22⋊#63214—22no—
22C22×Q16×C17C2211yescharacteristic
23C23×C8×C17C2211yescharacteristic
24C2×C17×C22⋊Q8—21no—
25C17×#467—22no—
26C17×#467—22no—
27C2×C17×#107C2211yescharacteristic
28C2×C17×#103—21no—
29C22×Q8×C17—21no—
30C23×C4×C17D411yescharacteristic
31C22×C22⋊D17—41no—
32C22×D68—21no—
33C2×Q16×C17—21no—
34C2×Q16×C17—21no—
35C2×Q16×C17—21no—
36C2×Q16×C17—21no—
37C2×Q16×C17—21no—
38C2×Q16×C17D412yescomplemented
39C22×C8×C17—21no—
40C22×C8×C17—21no—
41C22×C8×C17—21no—
42C22×C8×C17—21no—
43C22×C8×C17—21no—
44C22×C8×C17—21no—
45C22×C8×C17D411yescharacteristic
46C22×C8×C17C2311yescharacteristic
47C17×C22⋊Q8—24no—
48C17×C22⋊Q8—42no—
49C2×C4⋊C4×C17—21no—
50C2×C4⋊C4×C17—41no—
51C2×C22⋊C4×C17—41no—
52#10587616—24no—
53#10587616—42no—
54C2×#63227—41no—
55C2×C4⋊Dic17—21no—
56C2×D136—21no—
57C2×D136—21no—
58C2×D136—21no—
59C2×D136—21no—
60C2×D136—21no—
61C2×D136D412yes—
62C17×#107—21no—
63C17×#107D412yescomplemented
64C17×#103—24no—
65#10587499—28no—
66#10587492—24no—
67#10587489—21no—
68#10587489D412yes—
69C2×#63214—21no—
70C2×#63214—21no—
71C2×#63214—21no—
72C2×#63214—21no—
73C2×#63214—21no—
74C2×#63214D412yescomplemented
75C24×C17D811yescharacteristic
76C23×D17—41no—
77C2×Q8×C17—22no—
78C2×Q8×C17—41no—
79C2×Q8×C17—41no—
80C2×Q8×C17—41no—
81C2×Q8×C17—41no—
82C2×Q8×C17—41no—
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—41no—
90C22×C4×C17—41no—
91C22×C4×C17D811yescharacteristic
92C22×C4×C17C2×D411yescharacteristic
93C2×C22⋊D17—44no—
94C2×C22⋊D17—82no—
95C2×C22⋊D17—82no—
96C22×Dic17—41no—
97C2×D68—22no—
98C2×D68—41no—
99C2×D68—41no—
100C2×D68—41no—
101C2×D68—41no—
102C2×D68—41no—
103Q16×C17—22no—
104Q16×C17—22no—
105Q16×C17—41no—
106Q16×C17—41no—
107C2×C8×C17—21no—
108C2×C8×C17—21no—
109C2×C8×C17—21no—
110C2×C8×C17—41no—
111C2×C8×C17—41no—
112C2×C8×C17—41no—
113C2×C8×C17—41no—
114C2×C8×C17C2×D411yescharacteristic
115C2×C8×C17C2×D411yescharacteristic
116C2×C8×C17C2×D411yescharacteristic
117C2×C8×C17C2×D411yescharacteristic
118C2×C8×C17C2×D411yescharacteristic
119C2×C8×C17#4111yescharacteristic
120C4⋊C4×C17—22no—
121C4⋊C4×C17—44no—
122C4⋊C4×C17—41no—
123C22⋊C4×C17—44no—
124#63227—44no—
125C4⋊Dic17—22no—
126C4⋊Dic17—41no—
127D136—22no—
128D136—22no—
129D136—41no—
130D136—41no—
131#63214—22no—
132#63214—41no—
133#13305—171nomaximal, Sylow
134C23×C17—21no—
135C23×C17—21no—
136C23×C17—21no—
137C23×C17—21no—
138C23×C17—21no—
139C23×C17—21no—
140C23×C17SD3212yescomplemented
141C23×C17C2×D811yescharacteristic
142C22×D17—42no—
143C22×D17—81no—
144C22×D17—81no—
145C22×D17—81no—
146C22×D17—81no—
147C22×D17—81no—
148C22×D17—81no—
149Q8×C17—42no—
150Q8×C17—42no—
151Q8×C17—81no—
152Q8×C17—81no—
153C2×C4×C17—21no—
154C2×C4×C17—21no—
155C2×C4×C17—21no—
156C2×C4×C17—41no—
157C2×C4×C17—41no—
158C2×C4×C17—41no—
159C2×C4×C17—41no—
160C2×C4×C17—42no—
161C2×C4×C17—42no—
162C2×C4×C17—81no—
163C2×C4×C17—81no—
164C2×C4×C17—81no—
165C2×C4×C17—81no—
166C2×C4×C17#12011yescharacteristic
167C2×C4×C17#12011yescharacteristic
168C2×C4×C17#12111yescharacteristic
169C2×C4×C17#13511yescharacteristic
170C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
172C22⋊D17—88no—
173C2×Dic17—42no—
174C2×Dic17—81no—
175C2×Dic17—81no—
176D68—42no—
177D68—42no—
178D68—81no—
179D68—81no—
180C8×C17—41no—
181C8×C17#12011yescharacteristic
182C8×C17#12111yescharacteristic
183C8×C17#12111yescharacteristic
184C8×C17#12711yescharacteristic
185C22×SD32—171no—
186C2×#467D1711yescharacteristic, complemented
187C2×#466—171no—
188C2×#358—171no—
189C2×#357—171no—
190C2×C22⋊C16—171no—
191#1380—171no—
192C22×C17—22no—
193C22×C17—22no—
194C22×C17—22no—
195C22×C17—41no—
196C22×C17—41no—
197C22×C17—41no—
198C22×C17—41no—
199C22×C17#44711yescharacteristic
200C22×C17#44711yescharacteristic
201C22×C17#46611yescharacteristic
202C22×C17C2×SD3211yescharacteristic
203C22×C17#50811yescharacteristic
204C22×C17#50911yescharacteristic
205C22×C17#51011yescharacteristic
206C2×D17—82no—
207C2×D17—82no—
208C2×D17—82no—
209C2×D17—161no—
210C2×D17—161no—
211C2×D17—161no—
212C2×D17—161no—
213C4×C17—41no—
214C4×C17—82no—
215C4×C17—82no—
216C4×C17#39211yescharacteristic
217C4×C17#44911yescharacteristic
218C4×C17#44911yescharacteristic
219C4×C17#46911yescharacteristic
220Dic17—82no—
221C22×Q16C2×D1711yescharacteristic
222C22×D8—171no—
223C23×C8C2×D1711yescharacteristic
224C2×C22⋊Q8—21no—
225C2×#121—341no—
226C2×SD32—342no—
227C2×SD32—342no—
228C2×SD32—342no—
229C22×C16—171no—
230C22×C16—171no—
231#467—22no—
232#467—22no—
233#466—342no—
234#466—342no—
235C2×#107—171no—
236C2×#107C2×D1711yescharacteristic
237C2×#103—21no—
238C2×#102—341no—
239#358—342no—
240#357—342no—
241C22⋊C16—342no—
242C2×C17—42no—
243C2×C17#199911yescharacteristic
244C2×C17#218511yescharacteristic
245C2×C17#218611yescharacteristic
246C2×C17#218711yescharacteristic
247C2×C17#218911yescharacteristic
248C2×C17#221711yescharacteristic
249C2×C17#222011yescharacteristic
250D17—162no—
251C22×Q8—21no—
252C22×D4—341no—
253C22×D4—681no—
254C23×C4C22⋊D1711yescharacteristic
255C2×Q16—21no—
256C2×Q16—21no—
257C2×Q16—21no—
258C2×Q16—21no—
259C2×Q16—21no—
260C2×Q16C22⋊D1712yescomplemented
261C2×D8—172no—
262C2×D8—341no—
263C2×D8—341no—
264C2×D8—341no—
265C2×D8—341no—
266C2×D8—341no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8C22⋊D1711yescharacteristic
274C22×C8C22×D1711yescharacteristic
275C22⋊Q8—24no—
276C22⋊Q8—42no—
277#121—344no—
278#121—682no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4—341no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—681no—
284SD32—348no—
285C2×C16—172no—
286C2×C16—341no—
287C2×C16—341no—
288C2×C16—341no—
289C2×C16—341no—
290C2×C16—341no—
291#107—21no—
292#107—172no—
293#107—341no—
294#107C22⋊D1712yescomplemented
295#103—24no—
296#102—344no—
297C17#1330511yescharacteristic, complemented, Sylow
298C24—681no—
299C24#6322811yescharacteristic
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×D4—342no—
307C2×D4—684no—
308C2×D4—681no—
309C2×D4—681no—
310C2×D4—681no—
311C2×D4—681no—
312C2×D4—681no—
313C2×D4—1362no—
314C2×D4—1362no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—681no—
324C22×C4#6322811yescharacteristic
325C22×C4C2×C22⋊D1711yescharacteristic
326Q16—22no—
327Q16—22no—
328Q16—41no—
329Q16—41no—
330D8—342no—
331D8—342no—
332D8—681no—
333D8—681no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8D4×D1711yescharacteristic
342C2×C8D4×D1711yescharacteristic
343C2×C8D4×D1711yescharacteristic
344C2×C8#6325411yescharacteristic
345C2×C8C2×C22⋊D1711yescharacteristic
346C2×C8C2×C22⋊D1711yescharacteristic
347C4⋊C4—22no—
348C4⋊C4—44no—
349C4⋊C4—41no—
350C4⋊C4—342no—
351C4⋊C4—681no—
352C22⋊C4—44no—
353C22⋊C4—684no—
354C16—342no—
355C16—681no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—682no—
363C23—1361no—
364C23—1361no—
365C23—1361no—
366C23—1361no—
367C23—1361no—
368C23—1361no—
369C23#1058749912yescomplemented
370C23C2×#6322811yescharacteristic
371Q8—42no—
372Q8—42no—
373Q8—81no—
374Q8—81no—
375D4—682no—
376D4—682no—
377D4—1368no—
378D4—1361no—
379D4—1361no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—682no—
394C2×C4—1361no—
395C2×C4—1361no—
396C2×C4#1058761811yescharacteristic
397C2×C4#1058762311yescharacteristic
398C2×C4#1058762511yescharacteristic
399C2×C4#1058763511yescharacteristic
400C2×C4#1058763611yescharacteristic
401C2×C4#1058763911yescharacteristic
402C8—41no—
403C8#1058762311yescharacteristic
404C8#1058762411yescharacteristic
405C8#1058762511yescharacteristic
406C8#1058762611yescharacteristic
407C22—22no—
408C22—22no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—2721no—
415C22—2721no—
416C22—2721no—
417C22—2721no—
418C22—1362no—
419C22—1362no—
420C22—1362no—
421C22#1176294211yescharacteristic
422C22#1176294311yescharacteristic
423C22#1176304311yescharacteristic
424C22C2×#1058749911yescharacteristic
425C22#1176323911yescharacteristic
426C22#1176324411yescharacteristic
427C22#1176325011yescharacteristic
428C4—41no—
429C4—82no—
430C4—82no—
431C4—1362no—
432C4#1176271711yescharacteristic
433C4#1176295211yescharacteristic
434C4#1176295411yescharacteristic
435C4#1176306011yescharacteristic
436C2—42no—
437C2—2722no—
438C2#1452978011yescharacteristic
439C2#1453070111yescharacteristic
440C2#1453070711yescharacteristic
441C2#1453071211yescharacteristic
442C2#1453072211yescharacteristic
443C2#1453088211yescharacteristic
444C2#1453089911yescharacteristic
subgroupG/H
centerC23C2×#63228
commutator, cosocleC22×C8×C17C23
FrattiniC22×C8C22×D17
FittingC2×C17×#467C2
socleC23×C17C2×D8
prime radicalC2×#11763045C2
hypercenterC2×#467D17
hypocenterC17#13305
LCSG ▷ C22×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
1. C22×#10587499C2semidirect
3. C2×#11762613C2semidirect
4. C2×#11762609C2semidirect
5. C2×C22⋊#63214C2semidirect
6. #14526875C2semidirect
7. C2×C17×#467C2semidirect
38. C2×Q16×C17D4semidirect
63. C17×#107D4semidirect
74. C2×#63214D4semidirect
140. C23×C17SD32semidirect
186. C2×#467D17semidirect
260. C2×Q16C22⋊D17semidirect
294. #107C22⋊D17semidirect
297. C17#13305semidirect
369. C23#10587499semidirect

Maximal subgroups

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

subgroupindex
C22×#105874992
C2×#117630452
C2×#117626132
C2×#117626092
C2×C22⋊#632142
#145268752
C2×C17×#4672
#1330517