Finite Group Databasebeta

← #16930369

Subgroups — #16930369

Order 4352.

560 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 113 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#14534599C211yesmaximal, characteristic
2#14531401C211yesmaximal, characteristic
3#14527356C211yesmaximal, characteristic
4C2×#11762534C211yesmaximal, characteristic
5#14526652C211yesmaximal, characteristic
6#14526651C211yesmaximal, characteristic
7C17×#1520C211yesmaximal, characteristic
8C2×#10587563—21no—
9C22×#63225C2211yescharacteristic
10C2×C22⋊#3622C2211yescharacteristic
11#11762747C412yescomplemented
12#11762742—22no—
13#11762728C412yescomplemented
14#11762706—22no—
15#11762704—22no—
16#11762681—21no—
17#11762670—21no—
18#11762669C2211yescharacteristic
19#11762669C2211yescharacteristic
20#11762667—21no—
21#11762546—22no—
22#11762534—22no—
23C2×C17×C22⋊Q8C2211yescharacteristic
24C2×C17×C22⋊C8C2211yescharacteristic
25C17×#380C2211yescharacteristic
26C17×#328—22no—
27C22×Q8×C17C2×C411yescharacteristic
28C23×C4×C17C2311yescharacteristic
29C23×Dic17D411yescharacteristic
30C23×Dic17D411yescharacteristic
31C22×C8×C17—21no—
32C17×C22⋊Q8—21no—
33C17×C22⋊Q8—21no—
34C17×C22⋊Q8—21no—
35C17×C22⋊Q8—21no—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17C2×C411yescharacteristic
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17—21no—
41C2×C22⋊Dic17—21no—
42C2×C22⋊Dic17—21no—
43C2×C4⋊Dic17—21no—
44C2×C4⋊Dic17—21no—
45C2×#63225—22no—
46C2×#63225—22no—
47C2×#63225—21no—
48C2×#63225—21no—
49C2×#63225—21no—
50C2×#63225—21no—
51C2×#63225—22no—
52C2×#63225C2×C412yes—
53C2×#63225D412yes—
54C2×C4×Dic17—22no—
55C22×#3622—21no—
56#10587563—42no—
57#10587563—42no—
58C17×C22⋊C8—21no—
59C17×C22⋊C8—21no—
60C17×#95D412yes—
61#10587502—22no—
62#10587502—22no—
63#10587502—22no—
64#10587502—22no—
65#10587502C2×C412yes—
66#10587502D412yes—
67C22⋊#3622—21no—
68C22⋊#3622—21no—
69C24×C17C2×D411yescharacteristic
70C2×Q8×C17—21no—
71C2×Q8×C17—21no—
72C2×Q8×C17—81no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—22no—
76C22×C4×C17—81no—
77C22×C4×C17C22×C411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×C4×C17C2×D411yescharacteristic
80C22×C4×C17C2×D411yescharacteristic
81C22×C4×C17#4111yescharacteristic
82C22×C4×C17#4111yescharacteristic
83C22×Dic17—21no—
84C22×Dic17—21no—
85C22×Dic17—21no—
86C22×Dic17—21no—
87C22×Dic17—21no—
88C22×Dic17—21no—
89C22×Dic17—22no—
90C22×Dic17—22no—
91C22×Dic17—81no—
92C22×Dic17C22⋊C412yes—
93C22×Dic17C22⋊C412yes—
94C22×Dic17SD1612yes—
95C22×Dic17Q1612yes—
96C2×C8×C17—41no—
97C2×C8×C17—81no—
98C4⋊C4×C17—21no—
99C4⋊C4×C17—21no—
100C4⋊C4×C17—41no—
101C4⋊C4×C17—41no—
102C4⋊C4×C17—81no—
103C22⋊C4×C17—41no—
104C22⋊C4×C17—41no—
105C22⋊C4×C17—41no—
106C22⋊C4×C17—41no—
107C22⋊Dic17—41no—
108C22⋊Dic17—41no—
109C22⋊Dic17—41no—
110C22⋊Dic17—41no—
111C4⋊Dic17—81no—
112C4⋊Dic17—81no—
113#63225—22no—
114#63225—22no—
115#63225—22no—
116#63225—22no—
117#63225—44no—
118#63225—82no—
119#63225—82no—
120C4×Dic17—44no—
121C2×#3622—41no—
122C2×#3622—81no—
123#9200—171nomaximal, Sylow
124C23×C17—41no—
125C23×C17—41no—
126C23×C17C2×C22⋊C411yescharacteristic
127C23×C17#12011yescharacteristic
128C23×C17#12311yescharacteristic
129C23×C17C2×SD1611yescharacteristic
130C23×C17C2×Q1611yescharacteristic
131C23×C17AGL(1,ℤ/8)11yescharacteristic
132C23×C17#13711yescharacteristic
133Q8×C17—81no—
134Q8×C17—81no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—42no—
142C2×C4×C17—42no—
143C2×C4×C17—42no—
144C2×C4×C17—42no—
145C2×C4×C17—41no—
146C2×C4×C17—41no—
147C2×C4×C17—42no—
148C2×C4×C17C4×D411yescharacteristic
149C2×C4×C17C4×D411yescharacteristic
150C2×C4×C17#12111yescharacteristic
151C2×C4×C17#12111yescharacteristic
152C2×Dic17—22no—
153C2×Dic17—22no—
154C2×Dic17—22no—
155C2×Dic17—22no—
156C2×Dic17—22no—
157C2×Dic17—22no—
158C2×Dic17—22no—
159C2×Dic17—42no—
160C2×Dic17—42no—
161C2×Dic17—42no—
162C2×Dic17—42no—
163C2×Dic17—42no—
164C2×Dic17—42no—
165C2×Dic17—42no—
166C2×Dic17—42no—
167C2×Dic17—81no—
168C2×Dic17#10314yes—
169C8×C17—81no—
170#3622—81no—
171#2627—171no—
172#2316—171no—
173#1520—171no—
174#1520D1711yescharacteristic
175C2×#324—171no—
176#1289—171no—
177#1289—171no—
178C22×C17—21no—
179C22×C17—21no—
180C22×C17—21no—
181C22×C17—21no—
182C22×C17—21no—
183C22×C17—21no—
184C22×C17—41no—
185C22×C17—41no—
186C22×C17—41no—
187C22×C17—41no—
188C22×C17#38611yescharacteristic
189C22×C17C2×#10311yescharacteristic
190C22×C17#41711yescharacteristic
191C22×C17#45011yescharacteristic
192C22×C17#45111yescharacteristic
193C22×C17#48311yescharacteristic
194C22×C17#48411yescharacteristic
195C4×C17—21no—
196C4×C17—21no—
197C4×C17—21no—
198C4×C17—21no—
199C4×C17—82no—
200C4×C17—81no—
201C4×C17—81no—
202Dic17—24no—
203Dic17—42no—
204Dic17—82no—
205Dic17—82no—
206C2×C22⋊Q8C2×D1711yescharacteristic
207C2×#117—341no—
208C22×C4⋊C4—171no—
209C2×C22⋊C8—171no—
210C2×C22⋊C8C2×D1711yescharacteristic
211#400—172no—
212#398—342no—
213#395—172no—
214#389—342no—
215#389—342no—
216#385—341no—
217#381—341no—
218#380—171no—
219#380—171no—
220#380—341no—
221#380C2×D1711yescharacteristic
222#328—22no—
223#328—342no—
224#324—342no—
225C2×C17—21no—
226C2×C17—21no—
227C2×C17—21no—
228C2×C17—21no—
229C2×C17—21no—
230C2×C17—21no—
231C2×C17#187211yescharacteristic
232C2×C17#191211yescharacteristic
233C2×C17#191311yescharacteristic
234C22×Q8C4×D1711yescharacteristic
235C23×C4—171no—
236C23×C4—171no—
237C23×C4C22×D1711yescharacteristic
238C22×C8—21no—
239C22×C8—341no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244#117—682no—
245#117—682no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—172no—
249C2×C4⋊C4—172no—
250C2×C4⋊C4—341no—
251C2×C4⋊C4—341no—
252C2×C4⋊C4—341no—
253C2×C4⋊C4—341no—
254C2×C4⋊C4—342no—
255C2×C4⋊C4—342no—
256C2×C4⋊C4—342no—
257C2×C4⋊C4—341no—
258C2×C4⋊C4—341no—
259C2×C4⋊C4C4×D1711yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—341no—
263C2×C22⋊C4—341no—
264C2×C42—342no—
265C22⋊C8—21no—
266C22⋊C8—21no—
267C22⋊C8—341no—
268C22⋊C8—341no—
269#95—172no—
270#95—172no—
271#95—342no—
272#95—342no—
273#95—342no—
274#95—342no—
275#95C22⋊D1712yes—
276C17#920011yescharacteristic, complemented, Sylow
277C24D4×D1711yescharacteristic
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—81no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—81no—
285C22×C4—172no—
286C22×C4—172no—
287C22×C4—172no—
288C22×C4—172no—
289C22×C4—341no—
290C22×C4—341no—
291C22×C4—341no—
292C22×C4—341no—
293C22×C4—341no—
294C22×C4—341no—
295C22×C4—342no—
296C22×C4—342no—
297C22×C4—1361no—
298C22×C4C2×C4×D1711yescharacteristic
299C22×C4#6325211yescharacteristic
300C22×C4D4×D1711yescharacteristic
301C22×C4D4×D1711yescharacteristic
302C22×C4#6325411yescharacteristic
303C22×C4C2×C22⋊D1711yescharacteristic
304C2×C8—41no—
305C2×C8—81no—
306C2×C8—681no—
307C2×C8—1361no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—81no—
313C4⋊C4—342no—
314C4⋊C4—342no—
315C4⋊C4—342no—
316C4⋊C4—342no—
317C4⋊C4—684no—
318C4⋊C4—1362no—
319C4⋊C4—1362no—
320C4⋊C4—1361no—
321C4⋊C4—1361no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—681no—
327C22⋊C4—681no—
328C22⋊C4—681no—
329C22⋊C4—681no—
330C42—684no—
331C23—41no—
332C23—41no—
333C23C22⋊C4×D1711yescharacteristic
334C23#1058756911yescharacteristic
335C23SD16×D1711yescharacteristic
336C23#1058760011yescharacteristic
337C23Q16×D1711yescharacteristic
338C23#1058760411yescharacteristic
339C23#1058762311yescharacteristic
340Q8—81no—
341Q8—81no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—174no—
356C2×C4—342no—
357C2×C4—342no—
358C2×C4—342no—
359C2×C4—342no—
360C2×C4—342no—
361C2×C4—342no—
362C2×C4—342no—
363C2×C4—682no—
364C2×C4—682no—
365C2×C4—682no—
366C2×C4—682no—
367C2×C4—682no—
368C2×C4—682no—
369C2×C4—682no—
370C2×C4—682no—
371C2×C4—1361no—
372C2×C4#1058756711yescharacteristic
373C2×C4#1058757011yescharacteristic
374C2×C4C4×C22⋊D1711yescharacteristic
375C2×C4#1058762411yescharacteristic
376C8—81no—
377C8—1361no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22#1176268511yescharacteristic
389C22#103×D1711yescharacteristic
390C22#1176280911yescharacteristic
391C22#1176295811yescharacteristic
392C22#1176296311yescharacteristic
393C22#1176313811yescharacteristic
394C22#1176314311yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—344no—
403C4—682no—
404C4—1362no—
405C4—1362no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2—21no—
411C2—21no—
412C2#1452908811yescharacteristic
413C2#1452931811yescharacteristic
414C2#1452932311yescharacteristic
subgroupG/H
centerC22#11762685
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1520C2
socleC22×C17#386
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#1520D17
hypocenterC17#9200
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
11. #11762747C4semidirect
13. #11762728C4semidirect
276. C17#9200semidirect

Maximal subgroups

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

subgroupindex
#145345992
#145314012
#145273562
C2×#117625342
#145266522
#145266512
C17×#15202
#920017