Finite Group Databasebeta

← #16937682

Subgroups — #16937682

Order 4352.

708 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533730C211yesmaximal, characteristic
2#14529589C211yesmaximal, characteristic, complemented
3#14529540C211yesmaximal, characteristic, complemented
4C2×#11762552C211yesmaximal, characteristic, complemented
5C2×D4⋊#3622C211yesmaximal, characteristic, complemented
6#14526623C211yesmaximal, characteristic, complemented
7C17×#1802C211yesmaximal, characteristic, complemented
8C2×#10587560C2211yescharacteristic
9C2×#10587626—21no—
10C2×#10587616—21no—
11C2×#10587616—21no—
12C2×#10587478—21no—
13C2×C4⋊#3622C2211yescharacteristic
14C2×#10587480—22no—
15C2×C4×#3622C2211yescharacteristic
16#11762718—22no—
17#11762710—21no—
18C4×C4⋊Dic17C2211yescharacteristic
19#11762571—21no—
20#11762552—22no—
21D4⋊#3622—22no—
22C2×C4×D4×C17C2211yescharacteristic
23C2×C17×C4⋊C8C2211yescharacteristic
24C2×C17×#102—21no—
25C17×#384C2211yescharacteristic
26C17×#340—21no—
27C22×D4×C17D411yescharacteristic
28C23×C4×C17—21no—
29C22×C22⋊D17—22no—
30C22×C22⋊D17—22no—
31C22×D68—21no—
32C22×D68—21no—
33C22×D68D412yes—
34C22×C8×C17—21no—
35C4×D4×C17—21no—
36C4×D4×C17—21no—
37C4×D4×C17—21no—
38C4×D4×C17—21no—
39C2×C4⋊C4×C17—21no—
40C2×C4⋊C4×C17—21no—
41C2×C4⋊C4×C17—21no—
42C2×C4⋊C4×C17D411yescharacteristic
43C2×C22⋊C4×C17—21no—
44C2×C42×C17C2311yescharacteristic
45#10587626—42no—
46#10587626—42no—
47#10587616—42no—
48#10587616—42no—
49#10587616—42no—
50#10587616—42no—
51C2×#63227—22no—
52C2×#63227—22no—
53C2×C4⋊Dic17D411yescharacteristic
54C2×C4⋊Dic17D411yescharacteristic
55C2×C4×Dic17—21no—
56C22×#3622—21no—
57C22×#3622—21no—
58#10587560—21no—
59#10587560—21no—
60#10587560—21no—
61#10587560—21no—
62C17×C4⋊C8—21no—
63C17×C4⋊C8—21no—
64C17×#102—42no—
65#10587502—21no—
66#10587480—44no—
67#10587478—42no—
68C4⋊#3622—21no—
69C4⋊#3622D412yescomplemented
70C4×#3622—21no—
71C4×#3622—21no—
72C24×C17—21no—
73C23×D17—22no—
74C2×D4×C17—21no—
75C2×D4×C17—21no—
76C2×D4×C17—21no—
77C2×D4×C17—21no—
78C2×D4×C17—21no—
79C2×D4×C17D812yes—
80C22×C4×C17—21no—
81C22×C4×C17—41no—
82C22×C4×C17—41no—
83C22×C4×C17—41no—
84C22×C4×C17—41no—
85C22×C4×C17—41no—
86C22×C4×C17—41no—
87C22×C4×C17—41no—
88C22×C4×C17C2×D411yescharacteristic
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C2×C22⋊D17—44no—
92C2×C22⋊D17—44no—
93C2×C22⋊D17—44no—
94C2×C22⋊D17—44no—
95C2×C22⋊D17—82no—
96C2×C22⋊D17—82no—
97C22×Dic17—21no—
98C22×Dic17—21no—
99C2×D68—22no—
100C2×D68—22no—
101C2×D68—42no—
102C2×D68—42no—
103C2×D68—41no—
104C2×D68—41no—
105C2×D68—41no—
106C2×D68—41no—
107C2×D68—41no—
108C2×D68—41no—
109C2×D68—41no—
110C2×D68—41no—
111C2×D68—41no—
112C2×D68—41no—
113C2×D68—41no—
114C2×D68—41no—
115C2×C8×C17—41no—
116C2×C8×C17—41no—
117C2×C8×C17—41no—
118C4⋊C4×C17—21no—
119C4⋊C4×C17—21no—
120C4⋊C4×C17—42no—
121C4⋊C4×C17—41no—
122C4⋊C4×C17—41no—
123C4⋊C4×C17—41no—
124C4⋊C4×C17—41no—
125C22⋊C4×C17—41no—
126C22⋊C4×C17—41no—
127C42×C17C2×D411yescharacteristic
128C42×C17#4111yescharacteristic
129C42×C17#4111yescharacteristic
130C42×C17#4111yescharacteristic
131#63227—44no—
132#63227—44no—
133C4⋊Dic17—21no—
134C4⋊Dic17—21no—
135C4⋊Dic17D812yes—
136C4⋊Dic17SD1612yes—
137C4×Dic17—41no—
138C4×Dic17—41no—
139C2×#3622—22no—
140C2×#3622—22no—
141C2×#3622—41no—
142C2×#3622—41no—
143C2×#3622—41no—
144C2×#3622—41no—
145#10408—171nomaximal, Sylow
146C23×C17—22no—
147C23×C17—41no—
148C23×C17—41no—
149C23×C17—41no—
150C23×C17—41no—
151C23×C17—41no—
152C23×C17—41no—
153C23×C17#12011yescharacteristic
154C22×D17—42no—
155C22×D17—42no—
156C22×D17—42no—
157C22×D17—42no—
158C22×D17—42no—
159C22×D17—42no—
160C22×D17—42no—
161D4×C17—22no—
162D4×C17—22no—
163D4×C17—41no—
164D4×C17—41no—
165C2×C4×C17—21no—
166C2×C4×C17—21no—
167C2×C4×C17—41no—
168C2×C4×C17—41no—
169C2×C4×C17—41no—
170C2×C4×C17—81no—
171C2×C4×C17—81no—
172C2×C4×C17—81no—
173C2×C4×C17—81no—
174C2×C4×C17—81no—
175C2×C4×C17—81no—
176C2×C4×C17—81no—
177C2×C4×C17#12111yescharacteristic
178C2×C4×C17#12111yescharacteristic
179C2×C4×C17#12111yescharacteristic
180C2×C4×C17#12311yescharacteristic
181C2×C4×C17#12311yescharacteristic
182C2×C4×C17#12411yescharacteristic
183C2×C4×C17C2×D811yescharacteristic
184C2×C4×C17C2×D811yescharacteristic
185C2×C4×C17C2×SD1611yescharacteristic
186C2×C4×C17#13511yescharacteristic
187C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
188C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
189C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
190C2×C4×C17#13711yescharacteristic
191C22⋊D17—88no—
192C22⋊D17—88no—
193C2×Dic17—22no—
194C2×Dic17—22no—
195C2×Dic17—41no—
196C2×Dic17—41no—
197C2×Dic17—41no—
198C2×Dic17—41no—
199D68—44no—
200D68—82no—
201D68—81no—
202D68—81no—
203D68—81no—
204D68—81no—
205D68—81no—
206D68—81no—
207D68—81no—
208D68—81no—
209C8×C17—81no—
210#3622—42no—
211#3622—42no—
212#2541—171no—
213#1957—171no—
214#1947—171no—
215#1802D1711yescharacteristic, complemented
216C2×#331—171no—
217C2×D4⋊C8—171no—
218#1279—171no—
219C22×C17—42no—
220C22×C17—42no—
221C22×C17—42no—
222C22×C17—81no—
223C22×C17—81no—
224C22×C17—81no—
225C22×C17—81no—
226C22×C17#39411yescharacteristic
227C22×C17#44711yescharacteristic
228C22×C17#44711yescharacteristic
229C22×C17#45011yescharacteristic
230C22×C17#45211yescharacteristic
231C22×C17#45311yescharacteristic
232C22×C17#45511yescharacteristic
233C2×D17—82no—
234C2×D17—82no—
235C2×D17—82no—
236C2×D17—82no—
237C2×D17—82no—
238C2×D17—82no—
239C2×D17—82no—
240C4×C17—21no—
241C4×C17—21no—
242C4×C17—81no—
243C4×C17—161no—
244C4×C17#45911yescharacteristic
245C4×C17#46011yescharacteristic
246C4×C17#46611yescharacteristic
247C4×C17#46811yescharacteristic
248C4×C17#48011yescharacteristic
249C4×C17#48211yescharacteristic
250C4×C17#48611yescharacteristic
251C4×C17#49011yescharacteristic
252Dic17—42no—
253Dic17—42no—
254C2×#127—171no—
255C2×#127—341no—
256C2×#121—341no—
257C2×#121—341no—
258C2×C4×D4C2×D1711yescharacteristic
259C2×#107—341no—
260C2×C4⋊C8—171no—
261C2×C4⋊C8C2×D1711yescharacteristic
262C2×#102—21no—
263C2×#102—342no—
264C2×C4×C8—171no—
265#392—342no—
266#390—341no—
267#384C2×D1711yescharacteristic
268C4×C4⋊C4—171no—
269#340—21no—
270#340—341no—
271#331—342no—
272D4⋊C8—342no—
273C2×C17—82no—
274C2×C17#161911yescharacteristic
275C2×C17#163111yescharacteristic
276C2×C17#163911yescharacteristic
277C2×C17#164111yescharacteristic
278C2×C17#203311yescharacteristic
279C2×C17#203711yescharacteristic
280C2×C17#204111yescharacteristic
281D17—162no—
282C22×D4—172no—
283C22×D4—342no—
284C22×D4—342no—
285C22×D4—341no—
286C22×D4—341no—
287C22×D4C22⋊D1711yescharacteristic
288C23×C4—21no—
289C22×C8—21no—
290C22×C8—341no—
291C22×C8—341no—
292#127—341no—
293#127—341no—
294#127—341no—
295#127—341no—
296#127—682no—
297#127—682no—
298#121—682no—
299#121—682no—
300#121—682no—
301#121—682no—
302C4×D4—21no—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—171no—
310C2×C4⋊C4—171no—
311C2×C4⋊C4C22⋊D1711yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—342no—
314C2×C22⋊C4—342no—
315C2×C42—341no—
316C2×C42C22×D1711yescharacteristic
317#107—682no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—172no—
321C4⋊C8—341no—
322#102—42no—
323#102—684no—
324C4×C8—341no—
325C4×C8—341no—
326#95—341no—
327C17#1040811yescharacteristic, complemented, Sylow
328C24—21no—
329C24—342no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—342no—
336C2×D4—342no—
337C2×D4—682no—
338C2×D4—682no—
339C2×D4—684no—
340C2×D4—684no—
341C2×D4—684no—
342C2×D4—684no—
343C2×D4—681no—
344C2×D4—681no—
345C2×D4—681no—
346C2×D4—681no—
347C2×D4—681no—
348C2×D4—681no—
349C2×D4—681no—
350C2×D4—681no—
351C2×D4—681no—
352C2×D4—681no—
353C2×D4—681no—
354C2×D4—681no—
355C2×D4—1362no—
356C2×D4—1362no—
357C2×D4#6322812yes—
358C22×C4—21no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—341no—
367C22×C4—341no—
368C22×C4D4×D1711yescharacteristic
369C22×C4D4×D1711yescharacteristic
370C22×C4C2×C22⋊D1711yescharacteristic
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—342no—
375C2×C8—342no—
376C2×C8—681no—
377C2×C8—681no—
378C2×C8—681no—
379C2×C8—681no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—42no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—172no—
388C4⋊C4—172no—
389C4⋊C4—341no—
390C4⋊C4—341no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—684no—
394C22⋊C4—684no—
395C42—681no—
396C42—681no—
397C42#6325211yescharacteristic
398C42#6325211yescharacteristic
399C42D4×D1711yescharacteristic
400C42#6325411yescharacteristic
401C23—22no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—682no—
409C23—682no—
410C23—682no—
411C23—682no—
412C23—682no—
413C23—682no—
414C23—682no—
415C23#1058762311yescharacteristic
416D4—22no—
417D4—22no—
418D4—41no—
419D4—41no—
420D4—684no—
421D4—1362no—
422D4—1368no—
423D4—1368no—
424D4—1361no—
425D4—1361no—
426D4—1361no—
427D4—1361no—
428D4—1361no—
429D4—1361no—
430D4—1361no—
431D4—1361no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—342no—
445C2×C4—342no—
446C2×C4—681no—
447C2×C4—681no—
448C2×C4—681no—
449C2×C4—681no—
450C2×C4#1058756911yescharacteristic
451C2×C4#1058757011yescharacteristic
452C2×C4#1058757011yescharacteristic
453C2×C4#1058757111yescharacteristic
454C2×C4D8×D1711yescharacteristic
455C2×C4#1058759711yescharacteristic
456C2×C4SD16×D1711yescharacteristic
457C2×C4#1058760011yescharacteristic
458C2×C4#1058761511yescharacteristic
459C2×C4C2×#6322811yescharacteristic
460C2×C4#1058762511yescharacteristic
461C2×C4#1058762811yescharacteristic
462C2×C4#1058763511yescharacteristic
463C2×C4#1058763611yescharacteristic
464C8—81no—
465C8—682no—
466C8—682no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—1362no—
475C22—1362no—
476C22—1362no—
477C22—1362no—
478C22—1362no—
479C22—1362no—
480C22—1362no—
481C22#1176272411yescharacteristic
482C22#1176294111yescharacteristic
483C22#1176294211yescharacteristic
484C22#1176295811yescharacteristic
485C22#1176296911yescharacteristic
486C22#1176297511yescharacteristic
487C22#1176298511yescharacteristic
488C4—21no—
489C4—21no—
490C4—81no—
491C4—161no—
492C4—682no—
493C4—682no—
494C4#1176300311yescharacteristic
495C4#1176300811yescharacteristic
496C4#1176304311yescharacteristic
497C4#1176305711yescharacteristic
498C4#1176312211yescharacteristic
499C4#1176313311yescharacteristic
500C4#1176315411yescharacteristic
501C4#1176317211yescharacteristic
502C2—82no—
503C2—2722no—
504C2#1452783711yescharacteristic
505C2#1452791011yescharacteristic
506C2#1452796111yescharacteristic
507C2#1452797611yescharacteristic
508C2#1452996611yescharacteristic
509C2#1452999211yescharacteristic
510C2#1453001611yescharacteristic
subgroupG/H
centerC23#10587623
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1802C2
socleC23×C17#120
prime radical#14533730C2
hypercenter#1802D17
hypocenterC17#10408
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. #14529589C2semidirect
3. #14529540C2semidirect
4. C2×#11762552C2semidirect
5. C2×D4⋊#3622C2semidirect
6. #14526623C2semidirect
7. C17×#1802C2semidirect
69. C4⋊#3622D4semidirect
215. #1802D17semidirect
327. C17#10408semidirect

Maximal subgroups

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

subgroupindex
#145337302
#145295892
#145295402
C2×#117625522
C2×D4⋊#36222
#145266232
C17×#18022
#1040817