Finite Group Databasebeta

← #16937741

Subgroups — #16937741

Order 4352.

520 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533779C211yesmaximal, characteristic
2#14529745C211yesmaximal, characteristic, complemented
3#14529542C211yesmaximal, characteristic, complemented
4#14528785C211yesmaximal, characteristic
5C2×#11762556C211yesmaximal, characteristic, complemented
6C2×#11762538C211yesmaximal, characteristic
7C17×#1279C211yesmaximal, characteristic, complemented
8C2×#10587555C2211yescharacteristic
9C2×#10587561—21no—
10C2×C4×D68C2211yescharacteristic
11C2×#10587478—21no—
12C2×C4⋊#3622C2211yescharacteristic
13C2×#10587492—21no—
14#11762736—22no—
15#11762715—21no—
16#11762679C2211yescharacteristic
17#11762666—21no—
18#11762666—21no—
19#11762572—21no—
20#11762572—22no—
21#11762571—21no—
22#11762556—22no—
23#11762538—22no—
24C2×C17×C4⋊C8C2211yescharacteristic
25C2×C4×C8×C17C2211yescharacteristic
26C4×C4⋊C4×C17C2211yescharacteristic
27C22×D68D411yescharacteristic
28C22×C4×D17—21no—
29C22×Dic34—21no—
30C22×C8×C17—21no—
31C22×C8×C17—21no—
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C42×C17—21no—
35C2×C42×C17C2311yescharacteristic
36C2×#63227—22no—
37C2×#63227—22no—
38C2×#63227—21no—
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17—21no—
42C2×C4⋊Dic17—21no—
43C2×C4⋊Dic17D411yescharacteristic
44C2×C4⋊Dic17D412yes—
45C22×#3622—21no—
46#10587561—42no—
47#10587561—42no—
48C4×D68—21no—
49C4×D68—21no—
50C4×D68—21no—
51C4×D68—21no—
52#10587555—21no—
53#10587555—21no—
54#10587555—21no—
55#10587555—21no—
56C17×C4⋊C8—21no—
57C17×C4⋊C8—21no—
58C4×C8×C17—21no—
59C4×C8×C17—21no—
60C17×#95—21no—
61#10587502—22no—
62#10587502—22no—
63#10587492—42no—
64#10587478—42no—
65C4⋊#3622—21no—
66C4⋊#3622D412yes—
67C23×D17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×Dic17—22no—
74C22×Dic17—21no—
75C22×Dic17—41no—
76C2×D68—21no—
77C2×D68—21no—
78C2×D68—21no—
79C2×D68—21no—
80C2×D68—21no—
81C2×D68D812yes—
82C2×C4×D17—41no—
83C2×C4×D17—41no—
84C2×C4×D17—41no—
85C2×C4×D17—41no—
86C2×C4×D17—41no—
87C2×C4×D17—41no—
88C2×Dic34—41no—
89C2×Dic34—41no—
90C2×Dic34—41no—
91C2×Dic34—41no—
92C2×Dic34—41no—
93C2×Dic34—41no—
94C2×C8×C17—22no—
95C2×C8×C17—41no—
96C2×C8×C17—41no—
97C2×C8×C17—41no—
98C2×C8×C17—41no—
99C2×C8×C17—41no—
100C4⋊C4×C17—21no—
101C4⋊C4×C17—21no—
102C4⋊C4×C17—21no—
103C4⋊C4×C17—21no—
104C42×C17—41no—
105C42×C17—41no—
106C42×C17C2×D411yescharacteristic
107C42×C17#4111yescharacteristic
108C42×C17#4111yescharacteristic
109C42×C17#4111yescharacteristic
110#63227—44no—
111#63227—44no—
112#63227—41no—
113#63227—41no—
114C4⋊Dic17—22no—
115C4⋊Dic17—21no—
116C4⋊Dic17—22no—
117C4⋊Dic17—21no—
118C4⋊Dic17—41no—
119C4⋊Dic17—41no—
120C4⋊Dic17—42no—
121C4⋊Dic17—41no—
122C4⋊Dic17—41no—
123C4⋊Dic17—41no—
124C4⋊Dic17—41no—
125C2×#3622—22no—
126C2×#3622—41no—
127C2×#3622—41no—
128#10416—171nomaximal, Sylow
129C23×C17#12011yescharacteristic
130C22×D17—22no—
131C22×D17—41no—
132C22×D17—41no—
133C22×D17—41no—
134C22×D17—41no—
135C22×D17—41no—
136C22×D17—41no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17—41no—
144C2×C4×C17—41no—
145C2×C4×C17#12111yescharacteristic
146C2×C4×C17#12111yescharacteristic
147C2×C4×C17#12111yescharacteristic
148C2×C4×C17#12311yescharacteristic
149C2×C4×C17#12311yescharacteristic
150C2×C4×C17#12411yescharacteristic
151C2×C4×C17C2×D811yescharacteristic
152C2×C4×C17#13511yescharacteristic
153C2×C4×C17#13511yescharacteristic
154C2×C4×C17#13511yescharacteristic
155C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C17#13711yescharacteristic
158C2×C4×C17#13711yescharacteristic
159C2×Dic17—42no—
160C2×Dic17—41no—
161C2×Dic17—42no—
162C2×Dic17—41no—
163C2×Dic17—42no—
164C2×Dic17—41no—
165C2×Dic17—81no—
166C2×Dic17—81no—
167C2×Dic17—81no—
168D68—22no—
169D68—22no—
170D68—41no—
171D68—41no—
172C4×D17—81no—
173C4×D17—81no—
174C4×D17—81no—
175C4×D17—81no—
176Dic34—81no—
177Dic34—81no—
178Dic34—81no—
179Dic34—81no—
180C8×C17—42no—
181C8×C17—81no—
182#3622—42no—
183#2547—171no—
184#1990—171no—
185#1947—171no—
186#1802—171no—
187C2×#332—171no—
188C2×D4⋊C8—171no—
189#1279D1711yescharacteristic, complemented
190C22×C17#39411yescharacteristic
191C22×C17#44711yescharacteristic
192C22×C17#44911yescharacteristic
193C22×C17#45211yescharacteristic
194C22×C17#45211yescharacteristic
195C22×C17#45411yescharacteristic
196C22×C17#45611yescharacteristic
197C2×D17—42no—
198C2×D17—42no—
199C2×D17—42no—
200C2×D17—81no—
201C2×D17—81no—
202C2×D17—81no—
203C2×D17—81no—
204C4×C17—21no—
205C4×C17—21no—
206C4×C17—81no—
207C4×C17—81no—
208C4×C17#46311yescharacteristic
209C4×C17#46411yescharacteristic
210C4×C17#46611yescharacteristic
211C4×C17#46811yescharacteristic
212C4×C17#48011yescharacteristic
213C4×C17#48211yescharacteristic
214C4×C17#48811yescharacteristic
215C4×C17#48911yescharacteristic
216Dic17—82no—
217Dic17—81no—
218Dic17—161no—
219C2×C4⋊Q8—171no—
220C2×#124—341no—
221C2×C4×D4—171no—
222C2×#107—341no—
223C2×C4⋊C8—171no—
224C2×C4⋊C8C2×D1711yescharacteristic
225C2×#102—341no—
226C2×C4×C8C2×D1711yescharacteristic
227#397—342no—
228#391—341no—
229#384—171no—
230#380—341no—
231#380—341no—
232C4×C4⋊C4C2×D1711yescharacteristic
233#340—341no—
234#340—342no—
235#340—341no—
236#332—342no—
237D4⋊C8—342no—
238C2×C17#162511yescharacteristic
239C2×C17#163811yescharacteristic
240C2×C17#164711yescharacteristic
241C2×C17#165211yescharacteristic
242C2×C17#203311yescharacteristic
243C2×C17#203811yescharacteristic
244C2×C17#204211yescharacteristic
245D17—82no—
246C22×Q8—341no—
247C22×D4—171no—
248C23×C4—341no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—341no—
252C4⋊Q8—341no—
253C4⋊Q8—341no—
254C4⋊Q8—341no—
255C4⋊Q8—341no—
256#124—682no—
257#124—682no—
258C4×D4—341no—
259C4×D4—341no—
260C4×D4—341no—
261C4×D4—341no—
262C2×C4⋊C4—171no—
263C2×C4⋊C4—172no—
264C2×C4⋊C4—341no—
265C2×C4⋊C4—341no—
266C2×C4⋊C4—341no—
267C2×C4⋊C4—341no—
268C2×C4⋊C4D6811yescharacteristic
269C2×C4⋊C4D6811yescharacteristic
270C2×C22⋊C4—342no—
271C2×C22⋊C4—342no—
272C2×C22⋊C4—341no—
273C2×C42—21no—
274C2×C42C22×D1711yescharacteristic
275#107—682no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—172no—
279C4⋊C8—341no—
280#102—682no—
281C4×C8—21no—
282C4×C8—21no—
283#95—21no—
284#95—342no—
285#95—342no—
286C17#1041611yescharacteristic, complemented, Sylow
287C24—341no—
288C2×Q8—681no—
289C2×Q8—681no—
290C2×Q8—681no—
291C2×Q8—681no—
292C2×Q8—681no—
293C2×Q8—681no—
294C2×D4—172no—
295C2×D4—341no—
296C2×D4—341no—
297C2×D4—341no—
298C2×D4—341no—
299C2×D4—341no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—341no—
303C22×C4—342no—
304C22×C4—681no—
305C22×C4—681no—
306C22×C4—681no—
307C22×C4—681no—
308C22×C4—681no—
309C22×C4—681no—
310C22×C4—681no—
311C22×C4C2×D6811yescharacteristic
312C22×C4D4×D1711yescharacteristic
313C22×C4D4×D1711yescharacteristic
314C2×C8—22no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—342no—
321C2×C8—681no—
322C2×C8—681no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—341no—
328C4⋊C4—342no—
329C4⋊C4—341no—
330C4⋊C4—342no—
331C4⋊C4—682no—
332C4⋊C4—681no—
333C4⋊C4—681no—
334C4⋊C4—681no—
335C4⋊C4—681no—
336C4⋊C4—681no—
337C4⋊C4—681no—
338C22⋊C4—684no—
339C22⋊C4—684no—
340C22⋊C4—681no—
341C22⋊C4—681no—
342C42—41no—
343C42—41no—
344C42#6325211yescharacteristic
345C42#6325211yescharacteristic
346C42D4×D1711yescharacteristic
347C42#6325611yescharacteristic
348C23—342no—
349C23—681no—
350C23—681no—
351C23—681no—
352C23—681no—
353C23—681no—
354C23—681no—
355C23#1058756811yescharacteristic
356Q8—1361no—
357Q8—1361no—
358Q8—1361no—
359Q8—1361no—
360D4—342no—
361D4—342no—
362D4—681no—
363D4—681no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—682no—
373C2×C4—681no—
374C2×C4—681no—
375C2×C4—682no—
376C2×C4—681no—
377C2×C4—682no—
378C2×C4—1361no—
379C2×C4—1361no—
380C2×C4—1361no—
381C2×C4—1361no—
382C2×C4—1361no—
383C2×C4—1361no—
384C2×C4—1361no—
385C2×C4#1058756111yescharacteristic
386C2×C4#1058757011yescharacteristic
387C2×C4#1058757011yescharacteristic
388C2×C4#1058758011yescharacteristic
389C2×C4#1058758011yescharacteristic
390C2×C4#1058758111yescharacteristic
391C2×C4#1058759011yescharacteristic
392C2×C4#1058759011yescharacteristic
393C2×C4#1058759411yescharacteristic
394C2×C4#1058759511yescharacteristic
395C2×C4D8×D1711yescharacteristic
396C2×C4#1058760011yescharacteristic
397C2×C4#1058760211yescharacteristic
398C2×C4#1058760411yescharacteristic
399C8—42no—
400C8—81no—
401C8—682no—
402C22—682no—
403C22—682no—
404C22—682no—
405C22—1361no—
406C22—1361no—
407C22—1361no—
408C22—1361no—
409C22#1176272711yescharacteristic
410C22#1176294411yescharacteristic
411C22#1176295611yescharacteristic
412C22#1176297311yescharacteristic
413C22#1176297411yescharacteristic
414C22#1176298311yescharacteristic
415C22#1176299311yescharacteristic
416C4—21no—
417C4—21no—
418C4—81no—
419C4—81no—
420C4—2721no—
421C4—1362no—
422C4—1361no—
423C4#1176302511yescharacteristic
424C4#1176303211yescharacteristic
425C4#1176304411yescharacteristic
426C4#1176305811yescharacteristic
427C4#1176312411yescharacteristic
428C4#1176313511yescharacteristic
429C4#1176316411yescharacteristic
430C4#1176316911yescharacteristic
431C2—1362no—
432C2#1452787611yescharacteristic
433C2#1452795911yescharacteristic
434C2#1452802211yescharacteristic
435C2#1452805611yescharacteristic
436C2#1452996911yescharacteristic
437C2#1453000411yescharacteristic
438C2#1453002611yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1279C2
socleC23×C17#120
prime radicalC22×D68D4
hypercenter#1279D17
hypocenterC17#10416
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
2. #14529745C2semidirect
3. #14529542C2semidirect
5. C2×#11762556C2semidirect
7. C17×#1279C2semidirect
189. #1279D17semidirect
286. C17#10416semidirect

Maximal subgroups

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

subgroupindex
#145337792
#145297452
#145295422
#145287852
C2×#117625562
C2×#117625382
C17×#12792
#1041617