Finite Group Databasebeta

← #16935501

Subgroups — #16935501

Order 4352.

670 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763156C211yesmaximal, characteristic
2C2×#11763019C211yesmaximal, characteristic, complemented
3#14533783C211yesmaximal, characteristic
4#14529735C211yesmaximal, characteristic, complemented
5C2×#11762552C211yesmaximal, characteristic
6#14526624C211yesmaximal, characteristic, complemented
7C2×C17×Q8⋊C8C211yesmaximal, characteristic, complemented
8C22×#63231—21no—
9C2×#10587555C2211yescharacteristic
10C2×#10587560C2211yescharacteristic
11C2×#10587633—21no—
12#11763156—22no—
13#11763156—22no—
14#11763019—22no—
15#11763019—22no—
16C2×C4⋊#3622C2211yescharacteristic
17C2×#10587481—21no—
18C2×#10587492—22no—
19#11762726—22no—
20#11762711—22no—
21#11762710—21no—
22C4×C4⋊Dic17C2211yescharacteristic
23#11762572—22no—
24#11762552—22no—
25C2×C4×Q8×C17C2211yescharacteristic
26C2×C17×C4⋊C8C2211yescharacteristic
27C2×C4×C8×C17C2211yescharacteristic
28C17×Q8⋊C8—22no—
29C22×Q8×C17D411yescharacteristic
30C22×D68—21no—
31C22×D68—21no—
32C22×D68D412yes—
33C22×Dic34—21no—
34C22×C8×C17—21no—
35C22×C8×C17—21no—
36C4×Q8×C17—21no—
37C4×Q8×C17—21no—
38C4×Q8×C17—21no—
39C4×Q8×C17—21no—
40C2×C4⋊C4×C17—21no—
41C2×C4⋊C4×C17D411yescharacteristic
42C2×C42×C17—21no—
43C2×C42×C17C2311yescharacteristic
44#10587633—42no—
45#10587633—42no—
46C2×#63231—24no—
47C2×#63231—42no—
48C2×#63231—42no—
49C2×#63227—24no—
50C2×#63227—24no—
51C2×C4⋊Dic17—21no—
52C2×C4⋊Dic17—21no—
53C2×C4⋊Dic17D412yes—
54C2×C4×Dic17—21no—
55C22×#3622—21no—
56#10587560—21no—
57#10587560—21no—
58#10587560—21no—
59#10587560—21no—
60#10587555—21no—
61#10587555—21no—
62#10587555—21no—
63#10587555D412yes—
64C17×C4⋊C8—21no—
65C17×C4⋊C8—21no—
66C4×C8×C17—21no—
67C4×C8×C17D412yes—
68#10587502—21no—
69#10587492—28no—
70#10587481—24no—
71C4⋊#3622—21no—
72C4⋊#3622D412yes—
73C23×D17—22no—
74C2×Q8×C17—21no—
75C2×Q8×C17—21no—
76C2×Q8×C17—21no—
77C2×Q8×C17—21no—
78C2×Q8×C17—21no—
79C2×Q8×C17D812yes—
80C22×C4×C17—21no—
81C22×C4×C17—21no—
82C22×C4×C17C2×D411yescharacteristic
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D411yescharacteristic
85C22×Dic17—22no—
86C22×Dic17—41no—
87C2×D68—22no—
88C2×D68—22no—
89C2×D68—44no—
90C2×D68—41no—
91C2×D68—41no—
92C2×D68—41no—
93C2×D68—41no—
94C2×D68—41no—
95C2×D68—41no—
96C2×D68—42no—
97C2×D68—41no—
98C2×D68—41no—
99C2×D68—42no—
100C2×Dic34—22no—
101C2×Dic34—41no—
102C2×Dic34—41no—
103C2×Dic34—41no—
104C2×Dic34—41no—
105C2×Dic34—41no—
106C2×C8×C17—22no—
107C2×C8×C17—42no—
108C2×C8×C17—41no—
109C2×C8×C17—42no—
110C4⋊C4×C17—21no—
111C4⋊C4×C17—21no—
112C4⋊C4×C17—41no—
113C4⋊C4×C17—41no—
114C42×C17—41no—
115C42×C17—41no—
116C42×C17C2×D411yescharacteristic
117C42×C17C2×D411yescharacteristic
118C42×C17C2×D411yescharacteristic
119C42×C17#4111yescharacteristic
120#63231—48no—
121#63227—48no—
122#63227—84no—
123C4⋊Dic17—22no—
124C4⋊Dic17—22no—
125C4⋊Dic17—22no—
126C4⋊Dic17—22no—
127C4⋊Dic17—41no—
128C4⋊Dic17—41no—
129C4×Dic17—41no—
130C4×Dic17—41no—
131C2×#3622—22no—
132C2×#3622—41no—
133C2×#3622—41no—
134#10063—171nomaximal, Sylow
135C23×C17#12011yescharacteristic
136C22×D17—42no—
137C22×D17—42no—
138C22×D17—42no—
139C22×D17—44no—
140C22×D17—44no—
141Q8×C17—22no—
142Q8×C17—22no—
143Q8×C17—41no—
144Q8×C17—41no—
145C2×C4×C17—22no—
146C2×C4×C17—22no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—42no—
151C2×C4×C17#12011yescharacteristic
152C2×C4×C17#12011yescharacteristic
153C2×C4×C17#12111yescharacteristic
154C2×C4×C17#12111yescharacteristic
155C2×C4×C17#12111yescharacteristic
156C2×C4×C17#12711yescharacteristic
157C2×C4×C17C2×D811yescharacteristic
158C2×C4×C17#13511yescharacteristic
159C2×C4×C17#13512yes—
160C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C17#13712yes—
163C2×Dic17—42no—
164C2×Dic17—42no—
165C2×Dic17—42no—
166C2×Dic17—42no—
167C2×Dic17—81no—
168C2×Dic17—81no—
169D68—44no—
170D68—82no—
171D68—81no—
172D68—81no—
173D68—82no—
174D68—81no—
175D68—81no—
176D68—81no—
177D68—81no—
178Dic34—42no—
179Dic34—42no—
180Dic34—81no—
181Dic34—81no—
182C8×C17—42no—
183C8×C17—81no—
184#3622—42no—
185C2×#486—171no—
186C2×#462—171no—
187#2548—171no—
188#1988—171no—
189C2×#331—171no—
190C2×Q8⋊C8D1711yescharacteristic, complemented
191#1279—171no—
192C22×C17#39211yescharacteristic
193C22×C17#44711yescharacteristic
194C22×C17#44911yescharacteristic
195C22×C17#45212yes—
196C22×C17#45512yes—
197C2×D17—84no—
198C2×D17—84no—
199C2×D17—82no—
200C2×D17—82no—
201C2×D17—82no—
202C4×C17—21no—
203C4×C17—21no—
204C4×C17—42no—
205C4×C17—81no—
206C4×C17#44711yescharacteristic
207C4×C17#44911yescharacteristic
208C4×C17#46412yes—
209C4×C17#46611yescharacteristic
210C4×C17#46911yescharacteristic
211C4×C17#49511yescharacteristic
212C4×C17#49711yescharacteristic
213Dic17—82no—
214Dic17—82no—
215C22×Q16—341no—
216C2×C4⋊Q8—171no—
217C2×#127—171no—
218C2×#124—341no—
219C2×C4×Q8C2×D1711yescharacteristic
220#486—342no—
221#486—342no—
222#462—342no—
223#462—342no—
224C2×C4⋊C8—171no—
225C2×C4⋊C8C2×D1711yescharacteristic
226C2×#103—341no—
227C2×#102—342no—
228C2×C4×C8C2×D1711yescharacteristic
229#394—342no—
230#390—342no—
231#390—341no—
232C4×C4⋊C4—171no—
233#340—342no—
234#331—342no—
235Q8⋊C8—22no—
236C2×C17#164012yes—
237C2×C17#164112yes—
238C2×C17#199911yescharacteristic
239C2×C17#200411yescharacteristic
240C2×C17#201511yescharacteristic
241D17—162no—
242C22×Q8—341no—
243C22×Q8D6811yescharacteristic
244C22×D4—172no—
245C22×D4—341no—
246C22×D4—341no—
247C2×Q16—344no—
248C2×Q16—682no—
249C2×Q16—682no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—341no—
253C4⋊Q8—172no—
254C4⋊Q8—341no—
255C4⋊Q8—341no—
256C4⋊Q8—341no—
257#127—341no—
258#127—341no—
259#127—341no—
260#127—341no—
261#124—682no—
262#124—682no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—172no—
269C2×C4⋊C4—341no—
270C2×C4⋊C4—341no—
271C2×C4⋊C4D6811yescharacteristic
272C2×C22⋊C4—344no—
273C2×C22⋊C4—344no—
274C2×C42—21no—
275C2×C42—341no—
276C2×C42C22×D1711yescharacteristic
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—172no—
280C4⋊C8—341no—
281#103—344no—
282#102—348no—
283C4×C8—21no—
284C4×C8C22⋊D1712yes—
285#95—341no—
286C17#1006311yescharacteristic, complemented, Sylow
287C24—342no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—342no—
294C2×Q8—681no—
295C2×Q8—681no—
296C2×Q8—681no—
297C2×Q8—681no—
298C2×Q8—681no—
299C2×Q8D13612yes—
300C2×D4—342no—
301C2×D4—342no—
302C2×D4—684no—
303C2×D4—681no—
304C2×D4—681no—
305C2×D4—681no—
306C2×D4—681no—
307C2×D4—681no—
308C2×D4—681no—
309C2×D4—682no—
310C2×D4—682no—
311C2×D4—681no—
312C2×D4—681no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—342no—
316C22×C4—681no—
317C22×C4C2×D6811yescharacteristic
318C22×C4D4×D1711yescharacteristic
319C22×C4D4×D1711yescharacteristic
320Q16—688no—
321C2×C8—22no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—342no—
326C2×C8—681no—
327C2×C8—681no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—342no—
333C4⋊C4—342no—
334C4⋊C4—342no—
335C4⋊C4—342no—
336C4⋊C4—681no—
337C4⋊C4—681no—
338C22⋊C4—688no—
339C22⋊C4—1364no—
340C42—41no—
341C42—41no—
342C42—681no—
343C42—681no—
344C42#6325211yescharacteristic
345C42D4×D1711yescharacteristic
346C42D4×D1711yescharacteristic
347C42C2×C22⋊D1711yescharacteristic
348C23—682no—
349C23—682no—
350C23—682no—
351C23—684no—
352C23—684no—
353C23#1058756811yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—41no—
357Q8—41no—
358Q8—682no—
359Q8—682no—
360Q8—1361no—
361Q8—1361no—
362D4—684no—
363D4—1362no—
364D4—1361no—
365D4—1361no—
366D4—1361no—
367D4—1362no—
368D4—1361no—
369D4—1361no—
370D4—1361no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—682no—
378C2×C4—682no—
379C2×C4—682no—
380C2×C4—682no—
381C2×C4—1361no—
382C2×C4—1361no—
383C2×C4#1058756811yescharacteristic
384C2×C4#1058757011yescharacteristic
385C2×C4#1058757011yescharacteristic
386C2×C4C2×D13611yescharacteristic
387C2×C4#1058759011yescharacteristic
388C2×C4#1058759411yescharacteristic
389C2×C4#1058759411yescharacteristic
390C2×C4#1058760212yes—
391C2×C4#1058760412yes—
392C2×C4#1058761611yescharacteristic
393C2×C4#1058762311yescharacteristic
394C2×C4#1058762611yescharacteristic
395C8—42no—
396C8—81no—
397C8—682no—
398C22—1362no—
399C22—1362no—
400C22—1364no—
401C22—1364no—
402C22—1362no—
403C22#1176271811yescharacteristic
404C22#1176294511yescharacteristic
405C22#1176295611yescharacteristic
406C22#1176297412yes—
407C22#1176298812yes—
408C4—21no—
409C4—21no—
410C4—42no—
411C4—81no—
412C4—1362no—
413C4—1362no—
414C4#1176294511yescharacteristic
415C4#1176295611yescharacteristic
416C4#1176303412yes—
417C4#1176304511yescharacteristic
418C4#1176306211yescharacteristic
419C4#1176319011yescharacteristic
420C4#1176320211yescharacteristic
421C2—2722no—
422C2#1452797212yes—
423C2#1452797912yes—
424C2#1452978211yescharacteristic
425C2#1452981211yescharacteristic
426C2#1452987911yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×Q8⋊C8C2
socleC23×C17#120
prime radicalC2×#10587560C22
hypercenterC2×Q8⋊C8D17
hypocenterC17#10063
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#11763019C2semidirect
4. #14529735C2semidirect
6. #14526624C2semidirect
7. C2×C17×Q8⋊C8C2semidirect
190. C2×Q8⋊C8D17semidirect
286. C17#10063semidirect

Maximal subgroups

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

subgroupindex
C2×#117631562
C2×#117630192
#145337832
#145297352
C2×#117625522
#145266242
C2×C17×Q8⋊C82
#1006317