Finite Group Databasebeta

← #16937708

Subgroups — #16937708

Order 4352.

516 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533958C211yesmaximal, characteristic
2#14529545C211yesmaximal, characteristic, complemented
3#14528781C211yesmaximal, characteristic
4C2×#11762558C211yesmaximal, characteristic, complemented
5C2×#11762538C211yesmaximal, characteristic
6#14526624C211yesmaximal, characteristic, complemented
7C17×#1956C211yesmaximal, characteristic, complemented
8C2×C4×D68C2211yescharacteristic
9C2×#10587479—21no—
10C2×C4⋊#3622C2211yescharacteristic
11C2×C4⋊#3622C2211yescharacteristic
12C2×#10587480—21no—
13#11762720—22no—
14#11762710—21no—
15#11762674—21no—
16#11762666—21no—
17#11762666—21no—
18C4×C4⋊Dic17C2211yescharacteristic
19#11762571—21no—
20#11762571—21no—
21#11762558—22no—
22#11762538—22no—
23C2×C17×C4⋊Q8C2211yescharacteristic
24C2×C17×#103—22no—
25C2×C4×C8×C17C2211yescharacteristic
26C17×#384C2211yescharacteristic
27C22×Q8×C17D412yes—
28C22×D68D411yescharacteristic
29C22×C4×D17—21no—
30C22×C8×C17—21no—
31C17×C4⋊Q8—21no—
32C17×C4⋊Q8—21no—
33C17×C4⋊Q8—21no—
34C17×C4⋊Q8—21no—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17—21no—
39C2×C4⋊C4×C17—21no—
40C2×C42×C17C2311yescharacteristic
41C2×#63227—22no—
42C2×#63227—22no—
43C2×#63227—21no—
44C2×C4⋊Dic17D411yescharacteristic
45C2×C4⋊Dic17D411yescharacteristic
46C2×C4⋊Dic17D411yescharacteristic
47C2×C4×Dic17—21no—
48C22×#3622—21no—
49C22×#3622—21no—
50C4×D68—21no—
51C4×D68—21no—
52C4×D68—21no—
53C4×D68—21no—
54C17×#103—44no—
55C4×C8×C17—21no—
56C4×C8×C17—21no—
57#10587502—22no—
58#10587502—22no—
59#10587502—21no—
60#10587480—42no—
61#10587479—42no—
62C4⋊#3622—21no—
63C4⋊#3622—21no—
64C4⋊#3622—21no—
65C4⋊#3622D412yes—
66C23×D17—21no—
67C2×Q8×C17—22no—
68C2×Q8×C17—22no—
69C2×Q8×C17—42no—
70C2×Q8×C17—42no—
71C22×C4×C17—22no—
72C22×C4×C17—41no—
73C22×C4×C17C2×D411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C22×C4×C17C2×D411yescharacteristic
76C22×Dic17—21no—
77C22×Dic17—21no—
78C22×Dic17—21no—
79C2×D68—21no—
80C2×D68—21no—
81C2×D68—21no—
82C2×D68—21no—
83C2×D68—21no—
84C2×D68SD1612yes—
85C2×C4×D17—41no—
86C2×C4×D17—41no—
87C2×C4×D17—41no—
88C2×C4×D17—41no—
89C2×C4×D17—41no—
90C2×C4×D17—41no—
91C2×C8×C17—22no—
92C2×C8×C17—41no—
93C2×C8×C17—41no—
94C4⋊C4×C17—42no—
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—
102C4⋊C4×C17—41no—
103C42×C17C2×D411yescharacteristic
104C42×C17#4111yescharacteristic
105C42×C17#4111yescharacteristic
106C42×C17#4111yescharacteristic
107#63227—41no—
108#63227—41no—
109#63227—82no—
110#63227—82no—
111C4⋊Dic17—21no—
112C4⋊Dic17—21no—
113C4⋊Dic17—21no—
114C4⋊Dic17—21no—
115C4⋊Dic17—21no—
116C4⋊Dic17—21no—
117C4×Dic17—41no—
118C4×Dic17—41no—
119C2×#3622—22no—
120C2×#3622—41no—
121C2×#3622—41no—
122C2×#3622—41no—
123C2×#3622—41no—
124C2×#3622—41no—
125#10412—171nomaximal, Sylow
126C23×C17#12011yescharacteristic
127C22×D17—22no—
128C22×D17—41no—
129C22×D17—41no—
130C22×D17—41no—
131C22×D17—41no—
132C22×D17—41no—
133C22×D17—41no—
134Q8×C17—44no—
135Q8×C17—82no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—42no—
139C2×C4×C17—42no—
140C2×C4×C17—42no—
141C2×C4×C17—81no—
142C2×C4×C17—81no—
143C2×C4×C17—81no—
144C2×C4×C17#12111yescharacteristic
145C2×C4×C17#12111yescharacteristic
146C2×C4×C17#12111yescharacteristic
147C2×C4×C17#12311yescharacteristic
148C2×C4×C17#12311yescharacteristic
149C2×C4×C17#12411yescharacteristic
150C2×C4×C17C2×SD1611yescharacteristic
151C2×C4×C17#13511yescharacteristic
152C2×C4×C17#13511yescharacteristic
153C2×C4×C17#13511yescharacteristic
154C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C17#13711yescharacteristic
156C2×C4×C17#13711yescharacteristic
157C2×C4×C17#13711yescharacteristic
158C2×Dic17—41no—
159C2×Dic17—41no—
160C2×Dic17—41no—
161C2×Dic17—41no—
162C2×Dic17—41no—
163C2×Dic17—41no—
164C2×Dic17—41no—
165C2×Dic17—41no—
166C2×Dic17—41no—
167D68—22no—
168D68—22no—
169D68—41no—
170D68—41no—
171C4×D17—81no—
172C4×D17—81no—
173C4×D17—81no—
174C4×D17—81no—
175C8×C17—42no—
176#3622—42no—
177#3622—81no—
178#2563—171no—
179#1956D1711yescharacteristic, complemented
180#1948—171no—
181#1802—171no—
182C2×#333—171no—
183C2×D4⋊C8—171no—
184#1279—171no—
185C22×C17#39411yescharacteristic
186C22×C17#44811yescharacteristic
187C22×C17#44911yescharacteristic
188C22×C17#45211yescharacteristic
189C22×C17#45211yescharacteristic
190C22×C17#45311yescharacteristic
191C22×C17#45511yescharacteristic
192C2×D17—42no—
193C2×D17—42no—
194C2×D17—42no—
195C2×D17—81no—
196C2×D17—81no—
197C2×D17—81no—
198C2×D17—81no—
199C4×C17—21no—
200C4×C17—21no—
201C4×C17—82no—
202C4×C17—161no—
203C4×C17#46311yescharacteristic
204C4×C17#46411yescharacteristic
205C4×C17#46511yescharacteristic
206C4×C17#47011yescharacteristic
207C4×C17#48311yescharacteristic
208C4×C17#48511yescharacteristic
209C4×C17#48811yescharacteristic
210C4×C17#48911yescharacteristic
211Dic17—81no—
212Dic17—81no—
213Dic17—81no—
214C2×C4⋊Q8C2×D1711yescharacteristic
215C2×C4×D4—171no—
216C2×#106—341no—
217C2×C4⋊C8—171no—
218C2×C4⋊C8—171no—
219C2×#103—22no—
220C2×#102—341no—
221C2×C4×C8C2×D1711yescharacteristic
222#393—342no—
223#390—341no—
224#384C2×D1711yescharacteristic
225#382—341no—
226#380—341no—
227#380—341no—
228C4×C4⋊C4—171no—
229#340—341no—
230#340—341no—
231#333—342no—
232D4⋊C8—342no—
233C2×C17#162211yescharacteristic
234C2×C17#163411yescharacteristic
235C2×C17#164311yescharacteristic
236C2×C17#164611yescharacteristic
237C2×C17#203511yescharacteristic
238C2×C17#203711yescharacteristic
239C2×C17#204211yescharacteristic
240D17—82no—
241C22×Q8C22⋊D1712yes—
242C22×D4—171no—
243C23×C4—341no—
244C22×C8—21no—
245C22×C8—341no—
246C22×C8—341no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4×D4—341no—
252C4×D4—341no—
253C4×D4—341no—
254C4×D4—341no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—171no—
261C2×C4⋊C4—171no—
262C2×C4⋊C4—171no—
263C2×C22⋊C4—341no—
264C2×C22⋊C4—342no—
265C2×C22⋊C4—342no—
266C2×C42—341no—
267C2×C42C22×D1711yescharacteristic
268#106—682no—
269C4⋊C8—172no—
270C4⋊C8—341no—
271C4⋊C8—341no—
272C4⋊C8—341no—
273#103—44no—
274#102—682no—
275C4×C8—21no—
276C4×C8—21no—
277#95—342no—
278#95—342no—
279#95—341no—
280C17#1041211yescharacteristic, complemented, Sylow
281C24—341no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×D4—172no—
287C2×D4—341no—
288C2×D4—341no—
289C2×D4—341no—
290C2×D4—341no—
291C2×D4—341no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—341no—
295C22×C4—341no—
296C22×C4—341no—
297C22×C4—681no—
298C22×C4—681no—
299C22×C4—681no—
300C22×C4—681no—
301C22×C4—681no—
302C22×C4—681no—
303C22×C4D4×D1711yescharacteristic
304C22×C4D4×D1711yescharacteristic
305C22×C4C2×C22⋊D1711yescharacteristic
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—342no—
310C2×C8—681no—
311C2×C8—681no—
312C2×C8—681no—
313C2×C8—681no—
314C2×C8—681no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—341no—
325C4⋊C4—341no—
326C4⋊C4—341no—
327C4⋊C4—341no—
328C4⋊C4—341no—
329C4⋊C4—341no—
330C22⋊C4—681no—
331C22⋊C4—681no—
332C22⋊C4—1362no—
333C22⋊C4—1362no—
334C42—681no—
335C42—681no—
336C42#6325211yescharacteristic
337C42#6325211yescharacteristic
338C42D4×D1711yescharacteristic
339C42#6325411yescharacteristic
340C23—342no—
341C23—681no—
342C23—681no—
343C23—681no—
344C23—681no—
345C23—681no—
346C23—681no—
347C23#1058762311yescharacteristic
348Q8—44no—
349Q8—82no—
350D4—342no—
351D4—342no—
352D4—681no—
353D4—681no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—681no—
363C2×C4—681no—
364C2×C4—681no—
365C2×C4—681no—
366C2×C4—681no—
367C2×C4—681no—
368C2×C4—681no—
369C2×C4—681no—
370C2×C4—681no—
371C2×C4—1361no—
372C2×C4—1361no—
373C2×C4—1361no—
374C2×C4—1361no—
375C2×C4#1058756911yescharacteristic
376C2×C4#1058757011yescharacteristic
377C2×C4#1058757011yescharacteristic
378C2×C4#1058757111yescharacteristic
379C2×C4SD16×D1711yescharacteristic
380C2×C4#1058760011yescharacteristic
381C2×C4#1058760111yescharacteristic
382C2×C4#1058760211yescharacteristic
383C2×C4#1058760411yescharacteristic
384C2×C4#1058760411yescharacteristic
385C2×C4#1058760511yescharacteristic
386C2×C4#1058760511yescharacteristic
387C2×C4#1058761511yescharacteristic
388C2×C4#1058762511yescharacteristic
389C8—42no—
390C8—682no—
391C8—1361no—
392C22—682no—
393C22—682no—
394C22—682no—
395C22—1361no—
396C22—1361no—
397C22—1361no—
398C22—1361no—
399C22#1176272411yescharacteristic
400C22#1176294611yescharacteristic
401C22#1176295311yescharacteristic
402C22#1176297011yescharacteristic
403C22#1176297011yescharacteristic
404C22#1176297611yescharacteristic
405C22#1176298611yescharacteristic
406C4—21no—
407C4—21no—
408C4—82no—
409C4—161no—
410C4—1361no—
411C4—1361no—
412C4—1361no—
413C4#1176302711yescharacteristic
414C4#1176303411yescharacteristic
415C4#1176304111yescharacteristic
416C4#1176306611yescharacteristic
417C4#1176313811yescharacteristic
418C4#1176314911yescharacteristic
419C4#1176316511yescharacteristic
420C4#1176317011yescharacteristic
421C2—1362no—
422C2#1452785311yescharacteristic
423C2#1452792711yescharacteristic
424C2#1452798811yescharacteristic
425C2#1452800911yescharacteristic
426C2#1452997811yescharacteristic
427C2#1452999411yescharacteristic
428C2#1453002111yescharacteristic
subgroupG/H
centerC23#10587623
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1956C2
socleC23×C17#120
prime radicalC22×D68D4
hypercenter#1956D17
hypocenterC17#10412
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
2. #14529545C2semidirect
4. C2×#11762558C2semidirect
6. #14526624C2semidirect
7. C17×#1956C2semidirect
179. #1956D17semidirect
280. C17#10412semidirect

Maximal subgroups

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

subgroupindex
#145339582
#145295452
#145287812
C2×#117625582
C2×#117625382
#145266242
C17×#19562
#1041217