Finite Group Databasebeta

← #16938151

Subgroups — #16938151

Order 4352.

506 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763015C211yesmaximal, characteristic
2#14529745C211yesmaximal, characteristic, complemented
3#14529578C211yesmaximal, characteristic
4#14528785C211yesmaximal, characteristic
5C2×#11762556C211yesmaximal, characteristic, complemented
6#14526668C211yesmaximal, characteristic, complemented
7C2×C17×#334C211yesmaximal, characteristic, complemented
8C22×#63219—21no—
9C2×#10587555C2211yescharacteristic
10C2×C4×D68C2211yescharacteristic
11#11763015—22no—
12#11763015—22no—
13C2×C4⋊#3622C2211yescharacteristic
14C2×#10587487—21no—
15C2×#10587492—21no—
16C2×#10587491—21no—
17C2×#10587491—21no—
18#11762679C2211yescharacteristic
19#11762572—21no—
20#11762572—21no—
21#11762572—21no—
22#11762556—22no—
23C2×C17×C4⋊C8C2211yescharacteristic
24C2×C17×C4⋊C8C2211yescharacteristic
25C2×C4×C8×C17C2211yescharacteristic
26C17×#334—22no—
27C22×D68D411yescharacteristic
28C22×C4×D17—21no—
29C22×Dic34—21no—
30C22×C8×C17—21no—
31C22×C8×C17—21no—
32C22×C8×C17D411yescharacteristic
33C22×C8×C17D411yescharacteristic
34C2×C42×C17C2311yescharacteristic
35C2×#63227—21no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17D411yescharacteristic
42C22×#3622—21no—
43C2×#63219—24no—
44C2×#63219—42no—
45C2×#63219—42no—
46C4×D68—21no—
47C4×D68—21no—
48C4×D68—21no—
49C4×D68—21no—
50#10587555—21no—
51#10587555—21no—
52#10587555—21no—
53#10587555D412yes—
54C17×C4⋊C8—21no—
55C17×C4⋊C8—21no—
56C17×C4⋊C8—21no—
57C17×C4⋊C8D412yes—
58C4×C8×C17—21no—
59C4×C8×C17—21no—
60#10587492—42no—
61#10587491—42no—
62#10587491—42no—
63#10587487—24no—
64C4⋊#3622—21no—
65C4⋊#3622—21no—
66C23×D17—21no—
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C22×C4×C17#4111yescharacteristic
70C22×Dic17—21no—
71C22×Dic17—41no—
72C22×Dic17—41no—
73C2×D68—21no—
74C2×D68—21no—
75C2×D68—21no—
76C2×D68—21no—
77C2×D68—21no—
78C2×D68SD1612yes—
79C2×C4×D17—41no—
80C2×C4×D17—41no—
81C2×C4×D17—41no—
82C2×C4×D17—41no—
83C2×C4×D17—41no—
84C2×C4×D17—41no—
85C2×Dic34—22no—
86C2×Dic34—41no—
87C2×Dic34—41no—
88C2×Dic34—41no—
89C2×Dic34—41no—
90C2×Dic34—41no—
91C2×C8×C17—21no—
92C2×C8×C17—21no—
93C2×C8×C17—22no—
94C2×C8×C17—21no—
95C2×C8×C17—21no—
96C2×C8×C17—21no—
97C2×C8×C17—41no—
98C2×C8×C17—41no—
99C2×C8×C17—41no—
100C2×C8×C17—41no—
101C2×C8×C17—41no—
102C2×C8×C17SD1612yescomplemented
103C42×C17C2×D411yescharacteristic
104C42×C17C2×D411yescharacteristic
105C42×C17#4111yescharacteristic
106C42×C17#4111yescharacteristic
107#63227—41no—
108#63227—41no—
109C4⋊Dic17—21no—
110C4⋊Dic17—22no—
111C4⋊Dic17—22no—
112C4⋊Dic17—21no—
113C4⋊Dic17—41no—
114C4⋊Dic17—41no—
115C4⋊Dic17—41no—
116C4⋊Dic17—41no—
117C4⋊Dic17—41no—
118C4⋊Dic17—41no—
119C4⋊Dic17—81no—
120C2×#3622—41no—
121C2×#3622—41no—
122C2×#3622—41no—
123#63219—48no—
124#10477—171nomaximal, Sylow
125C23×C17#12111yescharacteristic
126C22×D17—22no—
127C22×D17—41no—
128C22×D17—41no—
129C22×D17—41no—
130C22×D17—41no—
131C22×D17—41no—
132C22×D17—41no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17#12011yescharacteristic
136C2×C4×C17#12111yescharacteristic
137C2×C4×C17#12111yescharacteristic
138C2×C4×C17#12311yescharacteristic
139C2×C4×C17#12311yescharacteristic
140C2×C4×C17#12411yescharacteristic
141C2×C4×C17C2×SD1611yescharacteristic
142C2×C4×C17C2×SD1611yescharacteristic
143C2×C4×C17#13511yescharacteristic
144C2×C4×C17#13511yescharacteristic
145C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C17#13711yescharacteristic
149C2×Dic17—42no—
150C2×Dic17—41no—
151C2×Dic17—41no—
152C2×Dic17—41no—
153C2×Dic17—81no—
154C2×Dic17—81no—
155C2×Dic17—81no—
156C2×Dic17—81no—
157C2×Dic17—81no—
158D68—22no—
159D68—22no—
160D68—41no—
161D68—41no—
162C4×D17—81no—
163C4×D17—81no—
164C4×D17—81no—
165C4×D17—81no—
166Dic34—42no—
167Dic34—42no—
168Dic34—81no—
169Dic34—81no—
170C8×C17—22no—
171C8×C17—42no—
172C8×C17—41no—
173C8×C17—81no—
174#3622—81no—
175C2×#461—171no—
176#1990—171no—
177#1954—171no—
178#1802—171no—
179C2×#334D1711yescharacteristic, complemented
180C2×#332—171no—
181#1296—171no—
182C22×C17#39411yescharacteristic
183C22×C17#46111yescharacteristic
184C22×C17#46311yescharacteristic
185C22×C17#46511yescharacteristic
186C22×C17#46911yescharacteristic
187C22×C17#47111yescharacteristic
188C22×C17#47311yescharacteristic
189C2×D17—42no—
190C2×D17—42no—
191C2×D17—42no—
192C2×D17—81no—
193C2×D17—81no—
194C2×D17—81no—
195C2×D17—81no—
196C4×C17—21no—
197C4×C17—21no—
198C4×C17#44911yescharacteristic
199C4×C17#45011yescharacteristic
200C4×C17#46011yescharacteristic
201C4×C17#46411yescharacteristic
202C4×C17#46511yescharacteristic
203C4×C17#46811yescharacteristic
204C4×C17#48111yescharacteristic
205C4×C17#48211yescharacteristic
206Dic17—81no—
207Dic17—82no—
208Dic17—161no—
209C22×SD16—341no—
210C2×C4⋊Q8—171no—
211C2×C4×D4—171no—
212#461—342no—
213#461—342no—
214C2×C4⋊C8—171no—
215C2×C4⋊C8C2×D1711yescharacteristic
216C2×C4⋊C8C2×D1711yescharacteristic
217C2×#103—341no—
218C2×#102—341no—
219C2×C22⋊C8—341no—
220C2×C22⋊C8—341no—
221C2×C4×C8C2×D1711yescharacteristic
222#384—171no—
223#340—341no—
224#340—341no—
225#340—341no—
226#334—22no—
227#332—342no—
228C2×C17#167111yescharacteristic
229C2×C17#167911yescharacteristic
230C2×C17#169311yescharacteristic
231C2×C17#170111yescharacteristic
232C2×C17#203411yescharacteristic
233C2×C17#205211yescharacteristic
234C2×C17#205511yescharacteristic
235D17—82no—
236C22×Q8—341no—
237C22×D4—171no—
238C23×C4—341no—
239C2×SD16—344no—
240C2×SD16—682no—
241C2×SD16—682no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—341no—
245C22×C8D6811yescharacteristic
246C22×C8D6811yescharacteristic
247C4⋊Q8—172no—
248C4⋊Q8—341no—
249C4⋊Q8—341no—
250C4⋊Q8—341no—
251C4×D4—341no—
252C4×D4—341no—
253C4×D4—341no—
254C4×D4—341no—
255C2×C4⋊C4—171no—
256C2×C4⋊C4—341no—
257C2×C4⋊C4—341no—
258C2×C4⋊C4—341no—
259C2×C4⋊C4—341no—
260C2×C4⋊C4—341no—
261C2×C22⋊C4—341no—
262C2×C42C22×D1711yescharacteristic
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—341no—
267C4⋊C8—341no—
268C4⋊C8C22⋊D1712yes—
269#103—344no—
270#102—682no—
271C22⋊C8—682no—
272C22⋊C8—682no—
273C4×C8—21no—
274C4×C8—21no—
275C17#1047711yescharacteristic, complemented, Sylow
276C24—341no—
277C2×Q8—342no—
278C2×Q8—681no—
279C2×Q8—681no—
280C2×Q8—681no—
281C2×Q8—681no—
282C2×Q8—681no—
283C2×D4—172no—
284C2×D4—341no—
285C2×D4—341no—
286C2×D4—341no—
287C2×D4—341no—
288C2×D4—341no—
289C22×C4—341no—
290C22×C4—681no—
291C22×C4—681no—
292C22×C4—681no—
293C22×C4—681no—
294C22×C4—681no—
295C22×C4—681no—
296C22×C4—681no—
297C22×C4—681no—
298C22×C4C2×D6811yescharacteristic
299C22×C4D4×D1711yescharacteristic
300C22×C4#6325611yescharacteristic
301SD16—688no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—681no—
314C2×C8—681no—
315C2×C8—681no—
316C2×C8#6321912yescomplemented
317C4⋊C4—341no—
318C4⋊C4—342no—
319C4⋊C4—342no—
320C4⋊C4—341no—
321C4⋊C4—681no—
322C4⋊C4—681no—
323C4⋊C4—681no—
324C4⋊C4—681no—
325C4⋊C4—681no—
326C4⋊C4—681no—
327C4⋊C4—1361no—
328C22⋊C4—681no—
329C22⋊C4—681no—
330C42#6325211yescharacteristic
331C42D4×D1711yescharacteristic
332C42#6325611yescharacteristic
333C42C2×C22⋊D1711yescharacteristic
334C23—342no—
335C23—681no—
336C23—681no—
337C23—681no—
338C23—681no—
339C23—681no—
340C23—681no—
341C23#1058758111yescharacteristic
342Q8—682no—
343Q8—682no—
344Q8—1361no—
345Q8—1361no—
346D4—342no—
347D4—342no—
348D4—681no—
349D4—681no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—681no—
353C2×C4—682no—
354C2×C4—681no—
355C2×C4—681no—
356C2×C4—1361no—
357C2×C4—1361no—
358C2×C4—1361no—
359C2×C4—1361no—
360C2×C4—1361no—
361C2×C4—1361no—
362C2×C4—1361no—
363C2×C4—1361no—
364C2×C4—1361no—
365C2×C4#1058758011yescharacteristic
366C2×C4#1058758011yescharacteristic
367C2×C4#1058758111yescharacteristic
368C2×C4C2×#6321911yescharacteristic
369C2×C4#1058759011yescharacteristic
370C2×C4#1058759411yescharacteristic
371C2×C4#1058759511yescharacteristic
372C2×C4SD16×D1711yescharacteristic
373C2×C4#1058760011yescharacteristic
374C2×C4#1058760011yescharacteristic
375C2×C4#1058760211yescharacteristic
376C2×C4#1058761611yescharacteristic
377C2×C4#1058762311yescharacteristic
378C2×C4#1058763311yescharacteristic
379C8—22no—
380C8—42no—
381C8—41no—
382C8—81no—
383C8—1361no—
384C22—682no—
385C22—682no—
386C22—682no—
387C22—1361no—
388C22—1361no—
389C22—1361no—
390C22—1361no—
391C22#1176272611yescharacteristic
392C22#1176301511yescharacteristic
393C22#1176302511yescharacteristic
394C22#1176303911yescharacteristic
395C22#1176306311yescharacteristic
396C22#1176307311yescharacteristic
397C22#1176308511yescharacteristic
398C4—21no—
399C4—21no—
400C4—2721no—
401C4—1361no—
402C4—1362no—
403C4#1176295311yescharacteristic
404C4#1176295811yescharacteristic
405C4#1176301011yescharacteristic
406C4#1176303211yescharacteristic
407C4#1176303811yescharacteristic
408C4#1176305511yescharacteristic
409C4#1176312911yescharacteristic
410C4#1176313511yescharacteristic
411C2—1362no—
412C2#1452816011yescharacteristic
413C2#1452820811yescharacteristic
414C2#1452829811yescharacteristic
415C2#1452834611yescharacteristic
416C2#1452997511yescharacteristic
417C2#1453008511yescharacteristic
418C2#1453010211yescharacteristic
subgroupG/H
centerC23#10587581
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×#334C2
socleC23×C17#121
prime radicalC22×D68D4
hypercenterC2×#334D17
hypocenterC17#10477
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#334

Complements

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

NH ≅ G/Ntype
2. #14529745C2semidirect
5. C2×#11762556C2semidirect
6. #14526668C2semidirect
7. C2×C17×#334C2semidirect
102. C2×C8×C17SD16semidirect
179. C2×#334D17semidirect
275. C17#10477semidirect
316. C2×C8#63219semidirect

Maximal subgroups

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

subgroupindex
C2×#117630152
#145297452
#145295782
#145287852
C2×#117625562
#145266682
C2×C17×#3342
#1047717