Finite Group Databasebeta

← #16935490

Subgroups — #16935490

Order 4352.

806 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763157C211yesmaximal, characteristic, complemented
2#14533730C211yesmaximal, characteristic
3#14529737C211yesmaximal, characteristic, complemented
4C2×#11762552C211yesmaximal, characteristic, complemented
5C2×D4⋊#3622C211yesmaximal, characteristic, complemented
6#14526623C211yesmaximal, characteristic, complemented
7C2×C17×#461C211yesmaximal, characteristic, complemented
8C2×#10587560C2211yescharacteristic
9C2×#10587626—21no—
10C2×#10587616—22no—
11#11763157—22no—
12#11763157—22no—
13C2×C4⋊#3622C2211yescharacteristic
14C2×#10587480—22no—
15C2×C4×#3622C2211yescharacteristic
16#11762718—22no—
17#11762710—21no—
18C4×C4⋊Dic17C2211yescharacteristic
19#11762571—22no—
20#11762552—22no—
21D4⋊#3622—22no—
22C22×SD16×C17—21no—
23C2×C17×C4⋊Q8C2211yescharacteristic
24C2×C4×D4×C17C2211yescharacteristic
25C17×#461—22no—
26C17×#461—22no—
27C2×C17×C4⋊C8C2211yescharacteristic
28C2×C17×#103—21no—
29C22×Q8×C17—21no—
30C22×D4×C17D411yescharacteristic
31C23×C4×C17—21no—
32C22×C22⋊D17—24no—
33C22×D68—21no—
34C22×D68—21no—
35C22×D68D412yes—
36C2×SD16×C17—24no—
37C2×SD16×C17—42no—
38C2×SD16×C17—42no—
39C22×C8×C17—21no—
40C17×C4⋊Q8—21no—
41C17×C4⋊Q8—21no—
42C17×C4⋊Q8—21no—
43C17×C4⋊Q8D412yescomplemented
44C4×D4×C17—21no—
45C4×D4×C17—21no—
46C4×D4×C17—21no—
47C4×D4×C17—21no—
48C2×C4⋊C4×C17—21no—
49C2×C4⋊C4×C17—21no—
50C2×C4⋊C4×C17D411yescharacteristic
51C2×C22⋊C4×C17—21no—
52C2×C42×C17C2311yescharacteristic
53#10587626—42no—
54#10587626—42no—
55#10587616—44no—
56#10587616—44no—
57C2×#63227—24no—
58C2×C4⋊Dic17D412yes—
59C2×C4×Dic17—21no—
60C22×#3622—21no—
61C22×#3622—21no—
62#10587560—21no—
63#10587560—21no—
64#10587560—21no—
65#10587560—21no—
66C17×C4⋊C8—21no—
67C17×C4⋊C8D412yescomplemented
68C17×#103—24no—
69#10587502—21no—
70#10587480—28no—
71C4⋊#3622—21no—
72C4⋊#3622—21no—
73C4×#3622—21no—
74C4×#3622D412yescomplemented
75C24×C17—21no—
76C23×D17—22no—
77C2×Q8×C17—22no—
78C2×Q8×C17—41no—
79C2×Q8×C17—41no—
80C2×Q8×C17—41no—
81C2×Q8×C17—41no—
82C2×Q8×C17—41no—
83C2×D4×C17—21no—
84C2×D4×C17—21no—
85C2×D4×C17—21no—
86C2×D4×C17—21no—
87C2×D4×C17—21no—
88C2×D4×C17D812yes—
89C22×C4×C17—21no—
90C22×C4×C17—41no—
91C22×C4×C17—41no—
92C22×C4×C17—41no—
93C22×C4×C17—42no—
94C22×C4×C17—42no—
95C22×C4×C17C2×D411yescharacteristic
96C22×C4×C17C2×D411yescharacteristic
97C22×C4×C17C2×D411yescharacteristic
98C2×C22⋊D17—48no—
99C2×C22⋊D17—48no—
100C2×C22⋊D17—84no—
101C22×Dic17—22no—
102C2×D68—22no—
103C2×D68—22no—
104C2×D68—44no—
105C2×D68—41no—
106C2×D68—41no—
107C2×D68—41no—
108C2×D68—41no—
109C2×D68—41no—
110C2×D68—42no—
111C2×D68—41no—
112C2×D68—41no—
113C2×D68—42no—
114C2×D68—41no—
115SD16×C17—48no—
116C2×C8×C17—22no—
117C2×C8×C17—41no—
118C2×C8×C17—41no—
119C4⋊C4×C17—22no—
120C4⋊C4×C17—21no—
121C4⋊C4×C17—22no—
122C4⋊C4×C17—21no—
123C4⋊C4×C17—41no—
124C4⋊C4×C17—41no—
125C22⋊C4×C17—41no—
126C22⋊C4×C17—41no—
127C42×C17C2×D411yescharacteristic
128C42×C17C2×D411yescharacteristic
129C42×C17C2×D411yescharacteristic
130C42×C17#4111yescharacteristic
131#63227—48no—
132C4⋊Dic17—22no—
133C4⋊Dic17SD1614yescomplemented
134C4×Dic17—41no—
135C4×Dic17—41no—
136C2×#3622—22no—
137C2×#3622—42no—
138C2×#3622—41no—
139C2×#3622—42no—
140#10062—171nomaximal, Sylow
141C23×C17—22no—
142C23×C17—41no—
143C23×C17—41no—
144C23×C17—42no—
145C23×C17—42no—
146C23×C17#12011yescharacteristic
147C22×D17—42no—
148C22×D17—42no—
149C22×D17—42no—
150C22×D17—44no—
151C22×D17—44no—
152Q8×C17—42no—
153Q8×C17—42no—
154Q8×C17—81no—
155Q8×C17—81no—
156D4×C17—22no—
157D4×C17—22no—
158D4×C17—41no—
159D4×C17—41no—
160C2×C4×C17—22no—
161C2×C4×C17—41no—
162C2×C4×C17—42no—
163C2×C4×C17—42no—
164C2×C4×C17—81no—
165C2×C4×C17—81no—
166C2×C4×C17—82no—
167C2×C4×C17—81no—
168C2×C4×C17—81no—
169C2×C4×C17#12011yescharacteristic
170C2×C4×C17#12011yescharacteristic
171C2×C4×C17#12111yescharacteristic
172C2×C4×C17#12111yescharacteristic
173C2×C4×C17#12111yescharacteristic
174C2×C4×C17#12711yescharacteristic
175C2×C4×C17C2×D811yescharacteristic
176C2×C4×C17C2×SD1612yes—
177C2×C4×C17#13511yescharacteristic
178C2×C4×C17AGL(1,ℤ/8)12yes—
179C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
181C22⋊D17—816no—
182C2×Dic17—24no—
183C2×Dic17—42no—
184C2×Dic17—42no—
185D68—44no—
186D68—82no—
187D68—81no—
188D68—81no—
189D68—82no—
190D68—81no—
191D68—81no—
192D68—81no—
193D68—81no—
194C8×C17—42no—
195#3622—42no—
196#3622—81no—
197C2×#486—171no—
198C2×#461D1711yescharacteristic, complemented
199#2541—171no—
200#1988—171no—
201C2×#331—171no—
202C2×D4⋊C8—171no—
203#1279—171no—
204C22×C17—44no—
205C22×C17—42no—
206C22×C17—82no—
207C22×C17—81no—
208C22×C17—81no—
209C22×C17#39211yescharacteristic
210C22×C17#44711yescharacteristic
211C22×C17#44911yescharacteristic
212C22×C17#45012yes—
213C22×C17#45512yes—
214C2×D17—84no—
215C2×D17—84no—
216C2×D17—82no—
217C2×D17—82no—
218C2×D17—82no—
219C4×C17—21no—
220C4×C17—21no—
221C4×C17—82no—
222C4×C17—81no—
223C4×C17#44711yescharacteristic
224C4×C17#44911yescharacteristic
225C4×C17#46012yes—
226C4×C17#46611yescharacteristic
227C4×C17#46911yescharacteristic
228C4×C17#49211yescharacteristic
229C4×C17#49611yescharacteristic
230Dic17—44no—
231C22×SD16—21no—
232C2×C4⋊Q8C2×D1711yescharacteristic
233C2×#127—171no—
234C2×#127—341no—
235C2×#121—342no—
236C2×C4×D4C2×D1711yescharacteristic
237#486—342no—
238#486—342no—
239#461—22no—
240#461—22no—
241C2×C4⋊C8—171no—
242C2×C4⋊C8C2×D1711yescharacteristic
243C2×#103—21no—
244C2×#102—342no—
245C2×C4×C8—171no—
246#392—342no—
247#390—341no—
248C4×C4⋊C4—171no—
249#340—342no—
250#331—342no—
251D4⋊C8—342no—
252C2×C17—82no—
253C2×C17#163912yes—
254C2×C17#164212yes—
255C2×C17#199911yescharacteristic
256C2×C17#200411yescharacteristic
257C2×C17#201411yescharacteristic
258D17—162no—
259C22×Q8—21no—
260C22×D4—172no—
261C22×D4—344no—
262C22×D4—341no—
263C22×D4—341no—
264C22×D4C22⋊D1711yescharacteristic
265C23×C4—21no—
266C2×SD16—24no—
267C2×SD16—42no—
268C2×SD16—42no—
269C22×C8—21no—
270C22×C8—341no—
271C22×C8—341no—
272C4⋊Q8—21no—
273C4⋊Q8—21no—
274C4⋊Q8—21no—
275C4⋊Q8C22⋊D1712yescomplemented
276#127—341no—
277#127—341no—
278#127—341no—
279#127—341no—
280#127—682no—
281#127—682no—
282#121—684no—
283#121—684no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—172no—
291C2×C4⋊C4C22⋊D1711yescharacteristic
292C2×C22⋊C4—21no—
293C2×C22⋊C4—344no—
294C2×C42—341no—
295C2×C42C22×D1711yescharacteristic
296C4⋊C8—21no—
297C4⋊C8—341no—
298C4⋊C8—341no—
299C4⋊C8C22⋊D1712yescomplemented
300#103—24no—
301#102—348no—
302C4×C8—172no—
303C4×C8—341no—
304#95—341no—
305C17#1006211yescharacteristic, complemented, Sylow
306C24—21no—
307C24—342no—
308C2×Q8—22no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—342no—
320C2×D4—342no—
321C2×D4—684no—
322C2×D4—688no—
323C2×D4—688no—
324C2×D4—682no—
325C2×D4—682no—
326C2×D4—681no—
327C2×D4—681no—
328C2×D4—681no—
329C2×D4—681no—
330C2×D4—681no—
331C2×D4—681no—
332C2×D4—681no—
333C2×D4—681no—
334C2×D4—1364no—
335C2×D4#6322812yes—
336C22×C4—21no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—342no—
343C22×C4D4×D1711yescharacteristic
344C22×C4D4×D1711yescharacteristic
345C22×C4C2×C22⋊D1711yescharacteristic
346SD16—48no—
347C2×C8—22no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—342no—
351C2×C8—682no—
352C2×C8—682no—
353C2×C8—681no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—174no—
361C4⋊C4—342no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—688no—
365C42—681no—
366C42—681no—
367C42D4×D1711yescharacteristic
368C42#6325411yescharacteristic
369C42C2×C22⋊D1711yescharacteristic
370C42C2×C22⋊D1711yescharacteristic
371C23—22no—
372C23—41no—
373C23—42no—
374C23—41no—
375C23—42no—
376C23—682no—
377C23—684no—
378C23—682no—
379C23—684no—
380C23—682no—
381C23#1058762311yescharacteristic
382Q8—42no—
383Q8—42no—
384Q8—81no—
385Q8—81no—
386D4—22no—
387D4—22no—
388D4—41no—
389D4—41no—
390D4—684no—
391D4—1362no—
392D4—13616no—
393D4—1361no—
394D4—1362no—
395D4—1361no—
396D4—1361no—
397D4—1361no—
398D4—1361no—
399D4—1361no—
400C2×C4—22no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—82no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—344no—
410C2×C4—682no—
411C2×C4—682no—
412C2×C4SD16×D1712yes—
413C2×C4#1058760012yes—
414C2×C4C2×#6322811yescharacteristic
415C2×C4#1058761811yescharacteristic
416C2×C4#1058762311yescharacteristic
417C2×C4#1058762411yescharacteristic
418C2×C4#1058762511yescharacteristic
419C2×C4#1058762511yescharacteristic
420C2×C4#1058762611yescharacteristic
421C2×C4#1058763511yescharacteristic
422C2×C4#1058763611yescharacteristic
423C2×C4#1058763911yescharacteristic
424C8—42no—
425C8—682no—
426C8—1361no—
427C22—44no—
428C22—42no—
429C22—81no—
430C22—81no—
431C22—82no—
432C22—1362no—
433C22—1362no—
434C22—1364no—
435C22—1364no—
436C22—1362no—
437C22#1176271711yescharacteristic
438C22#1176294211yescharacteristic
439C22#1176295211yescharacteristic
440C22#1176295812yes—
441C22#1176298512yes—
442C4—21no—
443C4—21no—
444C4—81no—
445C4—82no—
446C4—684no—
447C4#1176294311yescharacteristic
448C4#1176295411yescharacteristic
449C4#1176300712yes—
450C4#1176304311yescharacteristic
451C4#1176306011yescharacteristic
452C4#1176318011yescharacteristic
453C4#1176319511yescharacteristic
454C2—82no—
455C2—2722no—
456C2#1452796112yes—
457C2#1452798212yes—
458C2#1452978011yescharacteristic
459C2#1452981011yescharacteristic
460C2#1452987011yescharacteristic
subgroupG/H
centerC23#10587623
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×#461C2
socleC23×C17#120
prime radical#14533730C2
hypercenterC2×#461D17
hypocenterC17#10062
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×#11763157C2semidirect
3. #14529737C2semidirect
4. C2×#11762552C2semidirect
5. C2×D4⋊#3622C2semidirect
6. #14526623C2semidirect
7. C2×C17×#461C2semidirect
43. C17×C4⋊Q8D4semidirect
67. C17×C4⋊C8D4semidirect
74. C4×#3622D4semidirect
133. C4⋊Dic17SD16semidirect
198. C2×#461D17semidirect
275. C4⋊Q8C22⋊D17semidirect
299. C4⋊C8C22⋊D17semidirect
305. C17#10062semidirect

Maximal subgroups

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

subgroupindex
C2×#117631572
#145337302
#145297372
C2×#117625522
C2×D4⋊#36222
#145266232
C2×C17×#4612
#1006217