Finite Group Databasebeta

← #16928481

Subgroups — #16928481

Order 4352.

618 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531684C211yesmaximal, characteristic
2#14529540C211yesmaximal, characteristic, complemented
3#14528781C211yesmaximal, characteristic
4C2×#11762538C211yesmaximal, characteristic
5#14526667C211yesmaximal, characteristic, complemented
6#14526624C211yesmaximal, characteristic, complemented
7C17×#1845C211yesmaximal, characteristic, complemented
8C2×C4×D68C2211yescharacteristic
9C2×#10587579—21no—
10C2×C4⋊C4×D17—21no—
11C2×#10587478—21no—
12C2×C4⋊#3622C2211yescharacteristic
13C2×C4⋊#3622C2211yescharacteristic
14C2×#10587480—21no—
15#11762705—21no—
16#11762680—22no—
17#11762666—21no—
18C4×C4⋊Dic17C2211yescharacteristic
19#11762571—21no—
20#11762571—21no—
21#11762538C414yescomplemented
22C2×C17×#107—22no—
23C2×C4×C8×C17C2211yescharacteristic
24C17×#384C2211yescharacteristic
25C17×#384C2211yescharacteristic
26C22×D68D411yescharacteristic
27C22×C4×D17—21no—
28C22×C4×D17—22no—
29C22×C8×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17D412yes—
36C2×C42×C17C2311yescharacteristic
37C2×#63227—22no—
38C2×#63227—21no—
39C2×C4⋊Dic17D411yescharacteristic
40C2×C4⋊Dic17D411yescharacteristic
41C2×C4⋊Dic17D411yescharacteristic
42C2×#63225—22no—
43C2×C4×Dic17—21no—
44C22×#3622—21no—
45C22×#3622—21no—
46#10587579—42no—
47#10587579—42no—
48C4⋊C4×D17—42no—
49C4⋊C4×D17—42no—
50C4×D68—21no—
51C4×D68—21no—
52C4×D68C2×C412yes—
53C4×D68Q812yes—
54C17×#107—44no—
55C4×C8×C17—21no—
56C4×C8×C17C2×C412yescomplemented
57#10587502—22no—
58#10587502—21no—
59#10587480—42no—
60#10587478—42no—
61C4⋊#3622—21no—
62C4⋊#3622—21no—
63C4⋊#3622—21no—
64C4⋊#3622C2×C412yescomplemented
65C23×D17—21no—
66C22×C4×C17—22no—
67C22×C4×C17—41no—
68C22×C4×C17C2×D411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C2×D68—21no—
75C2×D68—21no—
76C2×D68—21no—
77C2×D68C4⋊C412yes—
78C2×D68D812yes—
79C2×D68Q1612yes—
80C2×C4×D17—22no—
81C2×C4×D17—22no—
82C2×C4×D17—41no—
83C2×C4×D17—41no—
84C2×C4×D17—41no—
85C2×C4×D17—41no—
86C2×C4×D17—44no—
87C2×C4×D17—44no—
88C2×C4×D17—82no—
89C2×C8×C17—22no—
90C2×C8×C17—22no—
91C2×C8×C17—22no—
92C4⋊C4×C17—42no—
93C4⋊C4×C17—42no—
94C4⋊C4×C17—41no—
95C4⋊C4×C17—41no—
96C4⋊C4×C17—41no—
97C4⋊C4×C17—41no—
98C4⋊C4×C17—41no—
99C4⋊C4×C17—41no—
100C4⋊C4×C17—41no—
101C4⋊C4×C17—41no—
102C42×C17C22×C411yescharacteristic
103C42×C17C2×Q811yescharacteristic
104C42×C17#4111yescharacteristic
105C42×C17#4111yescharacteristic
106#63227—22no—
107#63227—22no—
108#63227—44no—
109C4⋊Dic17—21no—
110C4⋊Dic17—21no—
111C4⋊Dic17—21no—
112C4⋊Dic17—21no—
113C4⋊Dic17C4⋊C412yes—
114C4⋊Dic17D812yes—
115#63225—44no—
116C4×Dic17—41no—
117C4×Dic17—41no—
118C2×#3622—22no—
119C2×#3622—41no—
120C2×#3622—41no—
121C2×#3622—41no—
122C2×#3622—41no—
123C2×#3622—41no—
124#8882—171nomaximal, Sylow
125C23×C17#12011yescharacteristic
126C22×D17—22no—
127C22×D17—22no—
128C22×D17—22no—
129C22×D17—41no—
130C22×D17—41no—
131C22×D17—41no—
132C22×D17—41no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—42no—
136C2×C4×C17—42no—
137C2×C4×C17—42no—
138C2×C4×C17—81no—
139C2×C4×C17—81no—
140C2×C4×C17—81no—
141C2×C4×C17C2×C4⋊C411yescharacteristic
142C2×C4×C17C4×D411yescharacteristic
143C2×C4×C17C4×D411yescharacteristic
144C2×C4×C17C22⋊Q811yescharacteristic
145C2×C4×C17C22⋊Q811yescharacteristic
146C2×C4×C17#12311yescharacteristic
147C2×C4×C17C2×D811yescharacteristic
148C2×C4×C17C2×D811yescharacteristic
149C2×C4×C17C2×Q1611yescharacteristic
150C2×C4×C17#13511yescharacteristic
151C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C17#13711yescharacteristic
154C2×C4×C17#13711yescharacteristic
155C2×Dic17—22no—
156C2×Dic17—22no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159C2×Dic17—41no—
160C2×Dic17—41no—
161C2×Dic17—41no—
162C2×Dic17—41no—
163C2×Dic17—41no—
164D68—22no—
165D68—22no—
166D68—22no—
167D68#10714yescomplemented
168C4×D17—42no—
169C4×D17—42no—
170C4×D17—42no—
171C4×D17—42no—
172C4×D17—88no—
173C8×C17—24no—
174#3622—42no—
175#3622—81no—
176D4×C4⋊C4—171no—
177#1947—171no—
178#1845D1711yescharacteristic, complemented
179#1802—171no—
180C2×D4⋊C8—171no—
181#1296—171no—
182#1279—171no—
183C22×C17#38511yescharacteristic
184C22×C17#44711yescharacteristic
185C22×C17#44711yescharacteristic
186C22×C17#45111yescharacteristic
187C22×C17#45211yescharacteristic
188C22×C17#45311yescharacteristic
189C22×C17#45611yescharacteristic
190C2×D17—24no—
191C2×D17—42no—
192C2×D17—42no—
193C2×D17—42no—
194C2×D17—42no—
195C2×D17—42no—
196C2×D17—42no—
197C4×C17—21no—
198C4×C17—21no—
199C4×C17—82no—
200C4×C17—161no—
201C4×C17C2×#10711yescharacteristic
202C4×C17#42811yescharacteristic
203C4×C17C4×D811yescharacteristic
204C4×C17#44011yescharacteristic
205C4×C17#47411yescharacteristic
206C4×C17#47811yescharacteristic
207C4×C17#48011yescharacteristic
208C4×C17#48411yescharacteristic
209Dic17—42no—
210Dic17—42no—
211Dic17—81no—
212C2×C4×D4—171no—
213C2×C4×D4—341no—
214C22×C4⋊C4—341no—
215C2×#107—22no—
216C2×#107—341no—
217C2×C4⋊C8—171no—
218C2×C4⋊C8—171no—
219C2×#102—341no—
220C2×C4×C8C2×D1711yescharacteristic
221#389—341no—
222#385—342no—
223#384C2×D1711yescharacteristic
224#384C2×D1711yescharacteristic
225#380—341no—
226C4×C4⋊C4—171no—
227#340—341no—
228#340—341no—
229D4⋊C8—174no—
230C2×C17#161911yescharacteristic
231C2×C17#163211yescharacteristic
232C2×C17#164711yescharacteristic
233C2×C17#164911yescharacteristic
234C2×C17#185211yescharacteristic
235C2×C17#185911yescharacteristic
236C2×C17#186211yescharacteristic
237D17—44no—
238C22×D4—171no—
239C23×C4—342no—
240C23×C4—341no—
241C22×C8—21no—
242C22×C8—341no—
243C22×C8—341no—
244C4×D4—172no—
245C4×D4—172no—
246C4×D4—341no—
247C4×D4—341no—
248C4×D4—682no—
249C4×D4—682no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—171no—
256C2×C4⋊C4—171no—
257C2×C4⋊C4—171no—
258C2×C4⋊C4—342no—
259C2×C4⋊C4—682no—
260C2×C4⋊C4—682no—
261C2×C4⋊C4C22⋊D1712yes—
262C2×C22⋊C4—341no—
263C2×C22⋊C4—342no—
264C2×C42—341no—
265C2×C42C22×D1711yescharacteristic
266#107—44no—
267#107—682no—
268C4⋊C8—172no—
269C4⋊C8—341no—
270C4⋊C8—341no—
271C4⋊C8—341no—
272#102—682no—
273C4×C8—21no—
274C4×C8C4×D1712yescomplemented
275#95—342no—
276#95—341no—
277C17#888211yescharacteristic, complemented, Sylow
278C24—341no—
279C2×D4—172no—
280C2×D4—172no—
281C2×D4—172no—
282C2×D4—341no—
283C2×D4—341no—
284C2×D4—341no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—341no—
288C22×C4—341no—
289C22×C4—342no—
290C22×C4—342no—
291C22×C4—341no—
292C22×C4—684no—
293C22×C4—684no—
294C22×C4—681no—
295C22×C4—681no—
296C22×C4—681no—
297C22×C4—681no—
298C22×C4—1362no—
299C22×C4D4×D1711yescharacteristic
300C22×C4D4×D1711yescharacteristic
301C22×C4C2×C22⋊D1711yescharacteristic
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—342no—
306C2×C8—681no—
307C2×C8—681no—
308C2×C8—681no—
309C2×C8—681no—
310C2×C8—681no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—172no—
322C4⋊C4—172no—
323C4⋊C4—341no—
324C4⋊C4—341no—
325C4⋊C4—341no—
326C4⋊C4—341no—
327C4⋊C4—684no—
328C22⋊C4—342no—
329C22⋊C4—342no—
330C22⋊C4—684no—
331C42—681no—
332C42—681no—
333C42C2×C4×D1711yescharacteristic
334C42#6325211yescharacteristic
335C42Q8×D1711yescharacteristic
336C42#6325611yescharacteristic
337C23—342no—
338C23—342no—
339C23—342no—
340C23—681no—
341C23—681no—
342C23—681no—
343C23—681no—
344C23#1058762311yescharacteristic
345D4—174no—
346D4—342no—
347D4—342no—
348D4—342no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—342no—
358C2×C4—342no—
359C2×C4—681no—
360C2×C4—681no—
361C2×C4—681no—
362C2×C4—681no—
363C2×C4—682no—
364C2×C4—682no—
365C2×C4—682no—
366C2×C4—682no—
367C2×C4—681no—
368C2×C4—681no—
369C2×C4—681no—
370C2×C4—1368no—
371C2×C4C4⋊C4×D1711yescharacteristic
372C2×C4#1058757911yescharacteristic
373C2×C4#1058758011yescharacteristic
374C2×C4#1058758211yescharacteristic
375C2×C4D8×D1711yescharacteristic
376C2×C4D8×D1711yescharacteristic
377C2×C4#1058759711yescharacteristic
378C2×C4#1058759711yescharacteristic
379C2×C4Q16×D1711yescharacteristic
380C2×C4#1058760411yescharacteristic
381C2×C4#1058760411yescharacteristic
382C2×C4#1058760511yescharacteristic
383C2×C4C4×C22⋊D1711yescharacteristic
384C2×C4#1058763211yescharacteristic
385C8—24no—
386C8—682no—
387C8—1361no—
388C22—344no—
389C22—682no—
390C22—682no—
391C22—682no—
392C22—682no—
393C22—682no—
394C22—682no—
395C22#1176268011yescharacteristic
396C22#1176294111yescharacteristic
397C22#1176294111yescharacteristic
398C22#1176296311yescharacteristic
399C22#1176297011yescharacteristic
400C22#1176297611yescharacteristic
401C22#1176299111yescharacteristic
402C4—21no—
403C4—21no—
404C4—82no—
405C4—161no—
406C4—682no—
407C4—682no—
408C4—1361no—
409C4#107×D1711yescharacteristic
410C4#1176285011yescharacteristic
411C4#1176289111yescharacteristic
412C4#1176290811yescharacteristic
413C4#1176308811yescharacteristic
414C4#1176311211yescharacteristic
415C4#1176312411yescharacteristic
416C4#1176314611yescharacteristic
417C2—684no—
418C2#1452783611yescharacteristic
419C2#1452791511yescharacteristic
420C2#1452801611yescharacteristic
421C2#1452803011yescharacteristic
422C2#1452897011yescharacteristic
423C2#1452900911yescharacteristic
424C2#1452902811yescharacteristic
subgroupG/H
centerC23#10587623
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1845C2
socleC23×C17#120
prime radicalC22×D68D4
cosocleC2×C42×C17C23
hypercenter#1845D17
hypocenterC17#8882
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1845

Complements

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

NH ≅ G/Ntype
2. #14529540C2semidirect
5. #14526667C2semidirect
6. #14526624C2semidirect
7. C17×#1845C2semidirect
21. #11762538C4semidirect
56. C4×C8×C17C2×C4semidirect
64. C4⋊#3622C2×C4semidirect
167. D68#107semidirect
178. #1845D17semidirect
274. C4×C8C4×D17semidirect
277. C17#8882semidirect

Maximal subgroups

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

subgroupindex
#145316842
#145295402
#145287812
C2×#117625382
#145266672
#145266242
C17×#18452
#888217