Finite Group Databasebeta

← #17019606

Subgroups — #17019606

Order 4352.

656 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539709C211yesmaximal, characteristic
2#14538256C211yesmaximal, characteristic
3#14537865C211yesmaximal, characteristic, complemented
4#14528212C214yesmaximal, complemented
5#14528134C214yesmaximal, complemented
6#14527554C211yesmaximal, characteristic, complemented
7#14527469C211yesmaximal, characteristic, complemented
8#14527434C211yesmaximal, characteristic
9C2×C17×#475C211yesmaximal, characteristic, complemented
10C2×#10587628—21no—
11#11763359C2211yescharacteristic
12#11763183C2212yes—
13#11763159C2212yes—
14#11763031—22no—
15#11763019C2212yes—
16#11763011—22no—
17#11763009C2212yes—
18#11762948—22no—
19Q8×#3622—22no—
20Q8×#3622C2214yes—
21#11762862—22no—
22#11762862C2211yescharacteristic
23#11762862C2211yescharacteristic
24#11762833C2211yescharacteristic
25#11762808—21no—
26#11762559C2214yes—
27#11762556C2212yes—
28#11762555C2214yes—
29#11762552C2212yes—
30C2×C17×C4⋊Q8C2211yescharacteristic
31C2×C4×Q8×C17C2211yescharacteristic
32C17×#475—22no—
33C17×#475C2214yes—
34C2×C17×#106—21no—
35C2×C17×C4⋊C8C2211yescharacteristic
36C2×C17×#103—21no—
37C22×Q8×C17—21no—
38C22×Q8×C17D411yescharacteristic
39C2×#63252—21no—
40C22×C8×C17—21no—
41C17×C4⋊Q8—21no—
42C17×C4⋊Q8—21no—
43C17×C4⋊Q8—21no—
44C17×C4⋊Q8C2312yes—
45C4×Q8×C17—21no—
46C4×Q8×C17—21no—
47C4×Q8×C17D412yes—
48C4×Q8×C17C2312yes—
49C2×C4⋊C4×C17—21no—
50C2×C4⋊C4×C17—21no—
51C2×C4⋊C4×C17—21no—
52C2×C4⋊C4×C17D411yescharacteristic
53C2×C42×C17—21no—
54C2×C42×C17C2311yescharacteristic
55C2×#63231—22no—
56#10587628—44no—
57C2×#63230—24no—
58C2×#63230—22no—
59#10587616—21no—
60#10587616—21no—
61C2×#63223—21no—
62#10587561—21no—
63#10587560C2311yescharacteristic
64C4×D68—21no—
65C4×D68—21no—
66#10587555C2311yescharacteristic
67C17×#106—24no—
68C17×C4⋊C8—21no—
69C17×C4⋊C8C2312yes—
70C17×#103—24no—
71#10587506—24no—
72C22⋊#3622—21no—
73C22⋊#3622D412yes—
74#10587481—22no—
75#10587480—22no—
76C4⋊#3622—24no—
77C4⋊#3622—21no—
78C4⋊#3622D412yes—
79C4⋊#3622C2312yes—
80C4⋊#3622C2312yes—
81C4×#3622—24no—
82C4×#3622—21no—
83C4×#3622C2312yes—
84C2×Q8×C17—21no—
85C2×Q8×C17—22no—
86C2×Q8×C17—22no—
87C2×Q8×C17—41no—
88C2×Q8×C17—82no—
89C2×Q8×C17SD1614yescomplemented
90C2×Q8×C17C2×D412yes—
91C22×C4×C17—21no—
92C22×C4×C17—21no—
93C22×C4×C17—41no—
94C22×C4×C17C2×D411yescharacteristic
95C22×C4×C17C2×D411yescharacteristic
96C22×C4×C17#4111yescharacteristic
97C2×C22⋊D17—41no—
98#63252—42no—
99#63252—42no—
100C2×D68—21no—
101C2×D68—21no—
102C2×D68—21no—
103C2×C4×D17—41no—
104C2×Dic34—21no—
105C2×C8×C17—22no—
106C2×C8×C17—42no—
107C4⋊C4×C17—22no—
108C4⋊C4×C17—21no—
109C4⋊C4×C17—22no—
110C4⋊C4×C17—22no—
111C4⋊C4×C17—22no—
112C4⋊C4×C17—41no—
113C4⋊C4×C17—41no—
114C4⋊C4×C17C2×D412yes—
115C42×C17—22no—
116C42×C17—22no—
117C42×C17C2×D411yescharacteristic
118C42×C17C2×D411yescharacteristic
119C42×C17#4111yescharacteristic
120C42×C17C2411yescharacteristic
121#63231—44no—
122#63230—44no—
123#63230—84no—
124#63227—41no—
125#63227—41no—
126C4⋊Dic17—21no—
127C4⋊Dic17—21no—
128#63223—42no—
129C2×#3622—22no—
130C2×#3622—22no—
131C2×#3622C2×D414yes—
132#19397—171nomaximal, Sylow
133C23×C17#12111yescharacteristic
134C22×D17—41no—
135Q8×C17—22no—
136Q8×C17—24no—
137Q8×C17—44no—
138Q8×C17—82no—
139Q8×C17C2×SD1614yes—
140C2×C4×C17—22no—
141C2×C4×C17—24no—
142C2×C4×C17—22no—
143C2×C4×C17—22no—
144C2×C4×C17—41no—
145C2×C4×C17—42no—
146C2×C4×C17—81no—
147C2×C4×C17—82no—
148C2×C4×C17#12111yescharacteristic
149C2×C4×C17#12111yescharacteristic
150C2×C4×C17#12111yescharacteristic
151C2×C4×C17C2×SD1612yes—
152C2×C4×C17#13512yes—
153C2×C4×C17C22×D411yescharacteristic
154C2×C4×C17C22×D411yescharacteristic
155C2×C4×C17C2×#4111yescharacteristic
156C22⋊D17—82no—
157C2×Dic17—41no—
158D68—41no—
159D68—41no—
160D68—42no—
161D68—81no—
162C4×D17—82no—
163Dic34—41no—
164Dic34—41no—
165C8×C17—42no—
166#3622—24no—
167#3622—42no—
168#3622—42no—
169#3206—171no—
170C2×#475D1711yescharacteristic, complemented
171#3037—171no—
172#2982—171no—
173#1680—174no—
174#1666—174no—
175#1570—171no—
176#1549—171no—
177#1540—171no—
178C22×C17—21no—
179C22×C17—21no—
180C22×C17#46512yes—
181C22×C17C2×#12111yescharacteristic
182C2×D17—81no—
183C2×D17—81no—
184C2×D17—81no—
185C4×C17—21no—
186C4×C17—24no—
187C4×C17—21no—
188C4×C17—21no—
189C4×C17—42no—
190C4×C17—82no—
191C4×C17#46512yes—
192C4×C17C22×SD1611yescharacteristic
193C4×C17C2×#13511yescharacteristic
194C4×C17#57712yes—
195Dic17—81no—
196C2×#137—341no—
197#532—171no—
198C2×C4⋊Q8C2×D1711yescharacteristic
199C2×C4×Q8C2×D1711yescharacteristic
200#492—172no—
201#487—172no—
202#475—22no—
203#475C2×D1714yes—
204#464—342no—
205#462—172no—
206#461—342no—
207#460—172no—
208#448—342no—
209C8×Q8—174no—
210C8×Q8—342no—
211#431—171no—
212#431—171no—
213#431—342no—
214C2×#106—21no—
215#423—171no—
216C2×C4⋊C8C2×D1711yescharacteristic
217#417—341no—
218C2×#103—21no—
219#334—174no—
220#332—174no—
221#332—172no—
222#331—172no—
223C2×C17—22no—
224C2×C17#304311yescharacteristic
225C2×C17C2×#46511yescharacteristic
226C2×C17#306911yescharacteristic
227D17—161no—
228C2×#41—341no—
229C22×Q8—21no—
230C22×Q8C22⋊D1711yescharacteristic
231#137—684no—
232C2×Q16—342no—
233C2×SD16—344no—
234C2×SD16—342no—
235C2×M16—341no—
236C22×C8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—171no—
241C4⋊Q8C22×D1712yes—
242#127—171no—
243#124—341no—
244#121—341no—
245#121—341no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×Q8C22⋊D1712yes—
249C4×Q8C22×D1712yes—
250C4×D4—341no—
251C4×D4—341no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4C22⋊D1711yescharacteristic
256C2×C42—21no—
257C2×C42C22×D1711yescharacteristic
258#106—24no—
259C4⋊C8—21no—
260C4⋊C8—172no—
261C4⋊C8—172no—
262C4⋊C8—172no—
263C4⋊C8—344no—
264C4⋊C8—341no—
265C4⋊C8C22×D1712yes—
266#103—24no—
267#103—344no—
268#103—342no—
269#102—342no—
270C22⋊C8—172no—
271C22⋊C8—341no—
272C4×C8—172no—
273C4×C8—344no—
274C4×C8—341no—
275C17#1939711yescharacteristic, complemented, Sylow
276#41—682no—
277#41—682no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—82no—
283C2×Q8—341no—
284C2×Q8#6323014yescomplemented
285C2×Q8C2×C22⋊D1712yes—
286C2×D4—341no—
287C2×D4—341no—
288C2×D4—341no—
289C2×D4—681no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—681no—
294C22×C4D4×D1711yescharacteristic
295C22×C4#6325411yescharacteristic
296C22×C4C2×C22⋊D1711yescharacteristic
297Q16—684no—
298SD16—684no—
299SD16—1364no—
300M16—682no—
301C2×C8—22no—
302C2×C8—42no—
303C2×C8—174no—
304C2×C8—342no—
305C2×C8—342no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—341no—
314C4⋊C4—341no—
315C4⋊C4C2×C22⋊D1712yes—
316C22⋊C4—681no—
317C22⋊C4—681no—
318C42—22no—
319C42—22no—
320C42D4×D1711yescharacteristic
321C42#6325411yescharacteristic
322C42C2×C22⋊D1711yescharacteristic
323C42C23×D1711yescharacteristic
324C23—681no—
325C23#1058762511yescharacteristic
326Q8—24no—
327Q8—22no—
328Q8—44no—
329Q8—82no—
330Q8—681no—
331Q8—681no—
332Q8C2×#6323014yes—
333D4—681no—
334D4—681no—
335D4—682no—
336D4—1361no—
337D4—1362no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—681no—
347C2×C4—1362no—
348C2×C4#1058762511yescharacteristic
349C2×C4#1058762511yescharacteristic
350C2×C4#1058762511yescharacteristic
351C2×C4C2×#6323012yes—
352C2×C4#1058763612yes—
353C2×C4C2×D4×D1711yescharacteristic
354C2×C4C2×#6325411yescharacteristic
355C2×C4C22×C22⋊D1711yescharacteristic
356C8—42no—
357C8—344no—
358C8—682no—
359C8—682no—
360C22—21no—
361C22—21no—
362C22—1361no—
363C22—1361no—
364C22—1361no—
365C22#1176304012yes—
366C22C2×#1058762511yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—24no—
371C4—42no—
372C4—82no—
373C4—1361no—
374C4#1176304012yes—
375C4C22×#6323011yescharacteristic
376C4C2×#1058763611yescharacteristic
377C4#1176361512yes—
378C2—22no—
379C2—2721no—
380C2#1453830211yescharacteristic
381C2C2×#1176304011yescharacteristic
382C2#1453851811yescharacteristic
subgroupG/H
centerC22C2×#10587625
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC2×C17×#475C2
socleC22×C17C2×#121
prime radical#11763359C22
hypercenterC2×#475D17
hypocenterC17#19397
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. #14537865C2semidirect
4. #14528212C2semidirect
5. #14528134C2semidirect
6. #14527554C2semidirect
7. #14527469C2semidirect
9. C2×C17×#475C2semidirect
89. C2×Q8×C17SD16semidirect
170. C2×#475D17semidirect
275. C17#19397semidirect
284. C2×Q8#63230semidirect

Maximal subgroups

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

subgroupindex
#145397092
#145382562
#145378652
#145282122
#145281342
#145275542
#145274692
#145274342
C2×C17×#4752
#1939717