Finite Group Databasebeta

← #16930367

Subgroups — #16930367

Order 4352.

684 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14534589C211yesmaximal, characteristic
2#14527355C212yesmaximal, complemented
3C2×#11762533C211yesmaximal, characteristic
4#14526651C212yesmaximal, complemented
5C17×#2315C211yesmaximal, characteristic
6C2×#10587610C2211yescharacteristic
7C2×C22⋊#3622C2212yes—
8#11762745C412yescomplemented
9#11762732—22no—
10#11762721C412yescomplemented
11#11762682—21no—
12#11762669C2211yescharacteristic
13#11762665—21no—
14#11762545—24no—
15#11762533—22no—
16C22×C22⋊C4×C17—21no—
17C22×C22⋊C4×C17C2211yescharacteristic
18C17×#385—24no—
19C17×#380—21no—
20C17×#380C2212yes—
21C25×C17D411yescharacteristic
22C23×C4×C17—22no—
23C23×C4×C17D411yescharacteristic
24C23×C4×C17C2311yescharacteristic
25C22×Dic34C2×C411yescharacteristic, complemented
26C2×C4⋊C4×C17—22no—
27C2×C4⋊C4×C17—22no—
28C2×C22⋊C4×C17—22no—
29C2×C22⋊C4×C17—22no—
30C2×C22⋊C4×C17—22no—
31C2×C22⋊C4×C17—22no—
32C2×C22⋊C4×C17—42no—
33C2×C22⋊C4×C17—42no—
34C2×C22⋊C4×C17C2×C412yes—
35C2×C22⋊C4×C17D412yes—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—22no—
39C2×C22⋊Dic17—22no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17C2×C411yescharacteristic, complemented
42#10587610—22no—
43#10587610—22no—
44C2×#63225—22no—
45C2×#63225—21no—
46C22×#3622—22no—
47C17×#95—22no—
48C17×#95D414yes—
49#10587502—22no—
50#10587502C2×C412yescomplemented
51C22⋊#3622—22no—
52C22⋊#3622—22no—
53C24×C17—21no—
54C24×C17—21no—
55C24×C17—21no—
56C24×C17—21no—
57C24×C17—81no—
58C24×C17—81no—
59C24×C17C22⋊C412yes—
60C24×C17SD1612yes—
61C24×C17Q1612yes—
62C24×C17C2×D411yescharacteristic
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—24no—
66C22×C4×C17—44no—
67C22×C4×C17—81no—
68C22×C4×C17—81no—
69C22×C4×C17—82no—
70C22×C4×C17—82no—
71C22×C4×C17C22⋊C412yes—
72C22×C4×C17C22×C411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C22×C4×C17C2×D412yes—
75C22×C4×C17#4112yes—
76C22×Dic17—21no—
77C22×Dic17—22no—
78C22×Dic17—21no—
79C2×Dic34—22no—
80C2×Dic34—81no—
81C4⋊C4×C17—82no—
82C4⋊C4×C17—82no—
83C22⋊C4×C17—24no—
84C22⋊C4×C17—24no—
85C22⋊C4×C17—44no—
86C22⋊C4×C17—42no—
87C22⋊C4×C17—42no—
88C22⋊Dic17—41no—
89C22⋊Dic17—41no—
90C22⋊Dic17—41no—
91C22⋊Dic17—41no—
92C22⋊Dic17—82no—
93C22⋊Dic17—82no—
94C4⋊Dic17—22no—
95C4⋊Dic17—81no—
96#63225—42no—
97#63225—82no—
98C2×#3622—42no—
99C2×#3622—82no—
100#9199—171nomaximal, Sylow
101C23×C17—22no—
102C23×C17—22no—
103C23×C17—22no—
104C23×C17—22no—
105C23×C17—22no—
106C23×C17—22no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—42no—
110C23×C17—42no—
111C23×C17—42no—
112C23×C17—42no—
113C23×C17—81no—
114C23×C17—81no—
115C23×C17—82no—
116C23×C17—82no—
117C23×C17—82no—
118C23×C17—82no—
119C23×C17#10314yes—
120C23×C17C2×C22⋊C411yescharacteristic
121C23×C17#12011yescharacteristic
122C23×C17#12311yescharacteristic
123C23×C17C2×SD1611yescharacteristic
124C23×C17C2×Q1611yescharacteristic
125C23×C17AGL(1,ℤ/8)11yescharacteristic
126C23×C17#13711yescharacteristic
127C2×C4×C17—22no—
128C2×C4×C17—42no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—42no—
132C2×C4×C17—42no—
133C2×C4×C17—42no—
134C2×C4×C17—44no—
135C2×C4×C17—44no—
136C2×C4×C17—44no—
137C2×C4×C17—81no—
138C2×C4×C17—88no—
139C2×C4×C17C4×D412yes—
140C2×C4×C17#12112yes—
141C2×Dic17—42no—
142C2×Dic17—42no—
143C2×Dic17—42no—
144C2×Dic17—42no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—42no—
148Dic34—82no—
149#3622—82no—
150#2626—171no—
151#2315D1711yescharacteristic
152#1520—172no—
153C2×#323—171no—
154#1289—172no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—24no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—44no—
167C22×C17—44no—
168C22×C17—44no—
169C22×C17—42no—
170C22×C17—42no—
171C22×C17—82no—
172C22×C17—82no—
173C22×C17—82no—
174C22×C17#38611yescharacteristic
175C22×C17C2×#10311yescharacteristic
176C22×C17#41711yescharacteristic
177C22×C17#45011yescharacteristic
178C22×C17#45111yescharacteristic
179C22×C17#48311yescharacteristic
180C22×C17#48411yescharacteristic
181C4×C17—22no—
182C4×C17—22no—
183C4×C17—42no—
184C4×C17—84no—
185Dic17—82no—
186Dic17—81no—
187Dic17—81no—
188C2×C22⋊Q8—171no—
189C22×C22⋊C4—21no—
190C22×C22⋊C4C2×D1711yescharacteristic
191C2×C22⋊C8—172no—
192#399—172no—
193#396—342no—
194#393—172no—
195#385—24no—
196#385—341no—
197#380—21no—
198#380—171no—
199#380C2×D1712yes—
200#379—341no—
201#328—344no—
202#323—342no—
203C2×C17—21no—
204C2×C17—21no—
205C2×C17—21no—
206C2×C17—21no—
207C2×C17—21no—
208C2×C17—21no—
209C2×C17—44no—
210C2×C17#187211yescharacteristic
211C2×C17#191111yescharacteristic
212C2×C17#191311yescharacteristic
213C25C22⋊D1711yescharacteristic
214C22×Q8—171no—
215C23×C4—22no—
216C23×C4C22⋊D1711yescharacteristic
217C23×C4C22×D1711yescharacteristic
218C22×C8—342no—
219C22⋊Q8—342no—
220C22⋊Q8—342no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—171no—
224C2×C4⋊C4—341no—
225C2×C4⋊C4—341no—
226C2×C4⋊C4—342no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—341no—
234C2×C22⋊C4—341no—
235C2×C22⋊C4—342no—
236C2×C22⋊C4—342no—
237C2×C22⋊C4C4×D1712yes—
238C2×C22⋊C4D6812yes—
239C22⋊C8—342no—
240C22⋊C8—342no—
241#95—22no—
242#95—172no—
243#95—342no—
244#95C22⋊D1714yes—
245C17#919911yescharacteristic, complemented, Sylow
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—81no—
251C24—81no—
252C24#6322712yes—
253C24#6322912yes—
254C24#6323112yes—
255C24C2×C22⋊D1711yescharacteristic
256C2×Q8—342no—
257C2×Q8—1361no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—44no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—342no—
267C22×C4—341no—
268C22×C4—341no—
269C22×C4#6322712yes—
270C22×C4C2×C4×D1711yescharacteristic
271C22×C4C2×D6811yescharacteristic
272C22×C4#6325212yes—
273C22×C4C2×C22⋊D1712yes—
274C2×C8—682no—
275C2×C8—1362no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—342no—
279C4⋊C4—682no—
280C4⋊C4—1362no—
281C4⋊C4—1361no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—681no—
288C22⋊C4—681no—
289C22⋊C4—681no—
290C22⋊C4—681no—
291C22⋊C4—1362no—
292C22⋊C4—1362no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—41no—
300C23—41no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—81no—
306C23—81no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23#1058748114yes—
312C23C2×#6322711yescharacteristic
313C23#1058761511yescharacteristic
314C23#1058761811yescharacteristic
315C23C2×#6322911yescharacteristic
316C23#1058762811yescharacteristic
317C23C2×#6323111yescharacteristic
318C23#1058763911yescharacteristic
319Q8—1362no—
320C2×C4—22no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—81no—
331C2×C4—88no—
332C2×C4—682no—
333C2×C4—682no—
334C2×C4—682no—
335C2×C4—682no—
336C2×C4—681no—
337C2×C4—681no—
338C2×C4—682no—
339C2×C4C4×C22⋊D1712yes—
340C2×C4#1058761612yes—
341C8—1362no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—24no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22#1176268811yescharacteristic
362C22C2×#1058748111yescharacteristic
363C22#1176280811yescharacteristic
364C22#1176296011yescharacteristic
365C22#1176296511yescharacteristic
366C22#1176313911yescharacteristic
367C22#1176314411yescharacteristic
368C4—22no—
369C4—22no—
370C4—42no—
371C4—84no—
372C4—1362no—
373C4—1361no—
374C4—1361no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#1452909111yescharacteristic
383C2#1452931611yescharacteristic
384C2#1452932611yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2315C2
socleC22×C17#386
prime radicalC25×C17D4
cosocleC23×C4×C17C23
hypercenter#2315D17
hypocenterC17#9199
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2315

Complements

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

NH ≅ G/Ntype
2. #14527355C2semidirect
4. #14526651C2semidirect
8. #11762745C4semidirect
10. #11762721C4semidirect
25. C22×Dic34C2×C4semidirect
41. C2×C4⋊Dic17C2×C4semidirect
50. #10587502C2×C4semidirect
245. C17#9199semidirect

Maximal subgroups

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

subgroupindex
#145345892
#145273552
C2×#117625332
#145266512
C17×#23152
#919917