Finite Group Databasebeta

← #16934848

Subgroups — #16934848

Order 4352.

552 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 99 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#14539556C211yesmaximal, characteristic
2C2×#11763137C211yesmaximal, characteristic
3#14533269C211yesmaximal, characteristic
4#14528868C211yesmaximal, characteristic
5#14527355C211yesmaximal, characteristic
6#14526632C211yesmaximal, characteristic
7C17×#1996C211yesmaximal, characteristic
8C2×#10587610C2211yescharacteristic
9C22×C4⋊Dic17C2211yescharacteristic
10#11763144—21no—
11#11763142—21no—
12#11763137—22no—
13#11763137—22no—
14#11763048—22no—
15#11762965—21no—
16#11762964—21no—
17C2×#10587479—21no—
18C2×#10587506—21no—
19C2×C22⋊#3622C2211yescharacteristic
20C2×C22⋊#3622C2211yescharacteristic
21#11762749—22no—
22#11762715—21no—
23#11762675—21no—
24#11762670—22no—
25#11762571—22no—
26#11762545—22no—
27C2×C17×C22⋊Q8C2211yescharacteristic
28C17×#380C2211yescharacteristic
29C4×C22⋊C4×C17C2211yescharacteristic
30C17×#342—22no—
31C22×Q8×C17D411yescharacteristic
32C23×C4×C17C2311yescharacteristic
33C23×Dic17—21no—
34C22×Dic34D411yescharacteristic
35C17×C22⋊Q8—21no—
36C17×C22⋊Q8—21no—
37C17×C22⋊Q8—21no—
38C17×C22⋊Q8—21no—
39C2×C4⋊C4×C17—21no—
40C2×C4⋊C4×C17—21no—
41C2×C4⋊C4×C17D411yescharacteristic
42C2×C22⋊C4×C17—21no—
43C2×C22⋊C4×C17—21no—
44C2×C22⋊C4×C17D412yes—
45C2×C42×C17—21no—
46C2×C22⋊Dic17—21no—
47C2×#63231—22no—
48C2×C4⋊Dic17—21no—
49C2×C4⋊Dic17—21no—
50C2×C4⋊Dic17—21no—
51C2×C4⋊Dic17—21no—
52C2×C4⋊Dic17—21no—
53C2×C4⋊Dic17D412yes—
54C2×C4⋊Dic17D411yescharacteristic
55#10587610—21no—
56#10587610—21no—
57#10587610—21no—
58#10587610—21no—
59C2×#63225—21no—
60C22×#3622—21no—
61C22×#3622D412yes—
62C17×#95—21no—
63C17×#95—21no—
64#10587506—22no—
65#10587506—24no—
66#10587502—24no—
67#10587502—24no—
68C22⋊#3622—21no—
69C22⋊#3622—21no—
70C22⋊#3622—21no—
71C22⋊#3622D412yes—
72#10587481—22no—
73#10587479—24no—
74#10587478—22no—
75C24×C17C2×D411yescharacteristic
76C2×Q8×C17—21no—
77C2×Q8×C17—21no—
78C2×Q8×C17—42no—
79C22×C4×C17—22no—
80C22×C4×C17—21no—
81C22×C4×C17—21no—
82C22×C4×C17—42no—
83C22×C4×C17—41no—
84C22×C4×C17C2×D411yescharacteristic
85C22×C4×C17C2×D411yescharacteristic
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17C2×D411yescharacteristic
88C22×C4×C17C2×D411yescharacteristic
89C22×C4×C17#4111yescharacteristic
90C22×Dic17—21no—
91C22×Dic17—22no—
92C22×Dic17—21no—
93C22×Dic17—41no—
94C22×Dic17—41no—
95C22×Dic17—42no—
96C22×Dic17—42no—
97C2×Dic34—21no—
98C2×Dic34—21no—
99C2×Dic34—42no—
100C4⋊C4×C17—21no—
101C4⋊C4×C17—21no—
102C4⋊C4×C17—42no—
103C4⋊C4×C17—41no—
104C4⋊C4×C17—41no—
105C22⋊C4×C17—22no—
106C22⋊C4×C17—22no—
107C22⋊C4×C17—41no—
108C22⋊C4×C17—41no—
109C22⋊C4×C17—42no—
110C42×C17—42no—
111C22⋊Dic17—41no—
112C22⋊Dic17—41no—
113#63231—82no—
114C4⋊Dic17—21no—
115C4⋊Dic17—22no—
116C4⋊Dic17—21no—
117C4⋊Dic17—22no—
118C4⋊Dic17—41no—
119C4⋊Dic17—41no—
120#63225—41no—
121#63225—41no—
122C2×#3622—22no—
123C2×#3622—22no—
124C2×#3622—42no—
125C2×#3622—42no—
126#9961—171nomaximal, Sylow
127C23×C17—41no—
128C23×C17—41no—
129C23×C17#12011yescharacteristic
130C23×C17#12111yescharacteristic
131C23×C17#12711yescharacteristic
132C23×C17#13512yes—
133C23×C17#13712yes—
134Q8×C17—42no—
135Q8×C17—42no—
136C2×C4×C17—22no—
137C2×C4×C17—22no—
138C2×C4×C17—22no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—42no—
143C2×C4×C17—42no—
144C2×C4×C17—41no—
145C2×C4×C17—41no—
146C2×C4×C17—42no—
147C2×C4×C17—81no—
148C2×C4×C17—81no—
149C2×C4×C17—81no—
150C2×C4×C17—81no—
151C2×C4×C17—81no—
152C2×C4×C17#12011yescharacteristic
153C2×C4×C17#12011yescharacteristic
154C2×C4×C17#12111yescharacteristic
155C2×C4×C17#12111yescharacteristic
156C2×Dic17—42no—
157C2×Dic17—42no—
158C2×Dic17—41no—
159C2×Dic17—41no—
160C2×Dic17—42no—
161C2×Dic17—44no—
162C2×Dic17—82no—
163C2×Dic17—81no—
164C2×Dic17—81no—
165Dic34—42no—
166Dic34—81no—
167#3622—42no—
168#3622—42no—
169#3189—171no—
170C2×#483—171no—
171#2503—171no—
172#1996D1711yescharacteristic
173#1823—171no—
174#1520—171no—
175#1282—171no—
176C22×C17—21no—
177C22×C17—21no—
178C22×C17—21no—
179C22×C17—21no—
180C22×C17—21no—
181C22×C17—21no—
182C22×C17—42no—
183C22×C17—42no—
184C22×C17#39211yescharacteristic
185C22×C17#45212yes—
186C22×C17#46412yes—
187C22×C17#49511yescharacteristic
188C22×C17#49711yescharacteristic
189C4×C17—21no—
190C4×C17—21no—
191C4×C17—21no—
192C4×C17—82no—
193C4×C17—81no—
194C4×C17—81no—
195C4×C17—161no—
196C4×C17#45812yes—
197Dic17—81no—
198Dic17—82no—
199Dic17—81no—
200C2×C22⋊Q8—171no—
201C2×C22⋊Q8C2×D1711yescharacteristic
202C22×C4⋊C4—171no—
203#484—341no—
204#484—341no—
205#483—342no—
206#483—342no—
207#467—342no—
208#451—341no—
209#451—341no—
210C2×#106—341no—
211C2×#103—341no—
212C2×C22⋊C8—171no—
213C2×C22⋊C8—171no—
214#400—342no—
215#391—341no—
216#382—341no—
217#381—342no—
218#380C2×D1711yescharacteristic
219C4×C22⋊C4C2×D1711yescharacteristic
220#342—22no—
221#340—342no—
222#328—342no—
223C2×C17—22no—
224C2×C17—21no—
225C2×C17—22no—
226C2×C17—21no—
227C2×C17#200911yescharacteristic
228C2×C17#201511yescharacteristic
229C2×C17#202211yescharacteristic
230C22×Q8—171no—
231C22×Q8D6811yescharacteristic
232C23×C4—341no—
233C23×C4C22×D1711yescharacteristic
234C2×Q16—342no—
235C22×C8—172no—
236C22×C8—341no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—341no—
242C22⋊Q8—341no—
243C22⋊Q8—341no—
244C22⋊Q8—341no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—172no—
248C2×C4⋊C4—171no—
249C2×C4⋊C4—341no—
250C2×C4⋊C4—341no—
251C2×C4⋊C4—341no—
252C2×C4⋊C4—341no—
253C2×C4⋊C4—341no—
254C2×C4⋊C4—341no—
255C2×C4⋊C4D6811yescharacteristic
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—341no—
259C2×C22⋊C4C22⋊D1712yes—
260C2×C42—21no—
261#107—342no—
262#106—344no—
263#103—342no—
264#103—344no—
265#103—342no—
266C22⋊C8—172no—
267C22⋊C8—341no—
268C22⋊C8—341no—
269C22⋊C8—341no—
270#95—21no—
271#95—21no—
272#95—344no—
273#95—344no—
274C17#996111yescharacteristic, complemented, Sylow
275C24C2×C22⋊D1711yescharacteristic
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—42no—
279C2×Q8—341no—
280C2×Q8—341no—
281C2×Q8—682no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—341no—
288C22×C4—341no—
289C22×C4—342no—
290C22×C4—681no—
291C22×C4—681no—
292C22×C4—682no—
293C22×C4—682no—
294C22×C4C2×D6811yescharacteristic
295C22×C4#6325211yescharacteristic
296C22×C4D4×D1711yescharacteristic
297C22×C4D4×D1711yescharacteristic
298C22×C4D4×D1711yescharacteristic
299C22×C4D4×D1711yescharacteristic
300Q16—1362no—
301C2×C8—342no—
302C2×C8—342no—
303C2×C8—682no—
304C2×C8—682no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—342no—
311C4⋊C4—341no—
312C4⋊C4—342no—
313C4⋊C4—341no—
314C4⋊C4—681no—
315C4⋊C4—681no—
316C4⋊C4—681no—
317C4⋊C4—681no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—42no—
323C22⋊C4—681no—
324C22⋊C4—681no—
325C42—42no—
326C23—41no—
327C23—41no—
328C23#1058761611yescharacteristic
329C23#1058762311yescharacteristic
330C23#1058762611yescharacteristic
331C23#1058762812yes—
332C23#1058763612yes—
333Q8—42no—
334Q8—42no—
335Q8—682no—
336Q8—1361no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—682no—
354C2×C4—682no—
355C2×C4—684no—
356C2×C4—681no—
357C2×C4—681no—
358C2×C4—682no—
359C2×C4—1361no—
360C2×C4—1362no—
361C2×C4—1361no—
362C2×C4#1058756811yescharacteristic
363C2×C4#1058756811yescharacteristic
364C2×C4#1058757011yescharacteristic
365C2×C4#1058757011yescharacteristic
366C8—682no—
367C8—682no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#1176271811yescharacteristic
377C22#1176296912yes—
378C22#1176303112yes—
379C22#1176319111yescharacteristic
380C22#1176320111yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—81no—
386C4—81no—
387C4—161no—
388C4—1361no—
389C4—1362no—
390C4—1361no—
391C4#1176300012yes—
392C2—22no—
393C2—21no—
394C2—21no—
395C2—22no—
396C2#1452984311yescharacteristic
397C2#1452987711yescharacteristic
398C2#1452991411yescharacteristic
subgroupG/H
centerC22#11762718
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#1996C2
socleC22×C17#392
prime radicalC24×C17C2×D4
hypercenter#1996D17
hypocenterC17#9961
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
274. C17#9961semidirect

Maximal subgroups

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

subgroupindex
#145395562
C2×#117631372
#145332692
#145288682
#145273552
#145266322
C17×#19962
#996117