Finite Group Databasebeta

← #16934776

Subgroups — #16934776

Order 4352.

610 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 105 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#14539541C211yesmaximal, characteristic
2#14533258C211yesmaximal, characteristic, complemented
3#14529621C211yesmaximal, characteristic
4#14528874C211yesmaximal, characteristic, complemented
5#14527357C211yesmaximal, characteristic, complemented
6#14526630C211yesmaximal, characteristic, complemented
7C2×C17×#481C211yesmaximal, characteristic
8C2×#10587610C2211yescharacteristic
9#11763139—21no—
10#11763125—21no—
11#11763036—22no—
12#11762960—21no—
13#11762947—21no—
14C2×#10587505—22no—
15C2×C22⋊#3622C2211yescharacteristic
16#11762740—22no—
17#11762714—21no—
18#11762682—22no—
19#11762678—21no—
20#11762669C2211yescharacteristic
21C4×C22⋊Dic17C2211yescharacteristic
22#11762571—22no—
23#11762547—22no—
24C2×C17×#121C2211yescharacteristic
25C22×C4⋊C4×C17C2211yescharacteristic
26C17×#481—22no—
27C17×#481—22no—
28C2×C17×#106—21no—
29C2×C17×#102—21no—
30C2×C17×C22⋊C8C2211yescharacteristic
31C22×D4×C17—21no—
32C22×D4×C17D411yescharacteristic
33C23×C4×C17—21no—
34C23×C4×C17C2311yescharacteristic
35C22×Dic34D411yescharacteristic
36C22×C8×C17—21no—
37C17×#121—21no—
38C17×#121—21no—
39C17×#121—21no—
40C17×#121—21no—
41C2×C4⋊C4×C17—21no—
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17—21no—
44C2×C4⋊C4×C17—21no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17D412yes—
47C2×C4⋊C4×C17D411yescharacteristic
48C2×C22⋊C4×C17—21no—
49C2×C22⋊Dic17—21no—
50C2×C22⋊Dic17—21no—
51C2×C22⋊Dic17D412yes—
52C2×#63229—22no—
53C2×C4⋊Dic17—21no—
54C2×C4⋊Dic17D411yescharacteristic
55#10587610—21no—
56#10587610—21no—
57#10587610—21no—
58#10587610—21no—
59C2×#63225—24no—
60C2×#63225—21no—
61C2×C4×Dic17—21no—
62C22×#3622D412yes—
63C17×#106—24no—
64C17×#102—24no—
65C17×C22⋊C8—21no—
66C17×C22⋊C8D412yes—
67#10587505—44no—
68#10587503—22no—
69#10587502—24no—
70#10587502—21no—
71#10587502—21no—
72C22⋊#3622—21no—
73C22⋊#3622—21no—
74#10587481—22no—
75#10587479—22no—
76C24×C17—21no—
77C24×C17C2×D411yescharacteristic
78C2×D4×C17—21no—
79C2×D4×C17—21no—
80C2×D4×C17—42no—
81C2×D4×C17—41no—
82C2×D4×C17—41no—
83C2×D4×C17—42no—
84C2×D4×C17—42no—
85C22×C4×C17—22no—
86C22×C4×C17—21no—
87C22×C4×C17—42no—
88C22×C4×C17—41no—
89C22×C4×C17—41no—
90C22×C4×C17—82no—
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17C2×D411yescharacteristic
94C22×C4×C17C2×D411yescharacteristic
95C22×C4×C17C2×D411yescharacteristic
96C22×C4×C17#4111yescharacteristic
97C22×Dic17—22no—
98C22×Dic17—21no—
99C22×Dic17—21no—
100C22×Dic17—41no—
101C2×Dic34—21no—
102C2×Dic34—21no—
103C2×Dic34—42no—
104C2×C8×C17—22no—
105C2×C8×C17—42no—
106C4⋊C4×C17—21no—
107C4⋊C4×C17—22no—
108C4⋊C4×C17—21no—
109C4⋊C4×C17—22no—
110C4⋊C4×C17—41no—
111C4⋊C4×C17—41no—
112C22⋊C4×C17—41no—
113C22⋊C4×C17—41no—
114C22⋊Dic17—22no—
115C22⋊Dic17—41no—
116C22⋊Dic17—41no—
117C22⋊Dic17—42no—
118C22⋊Dic17SD1614yescomplemented
119#63229—82no—
120C4⋊Dic17—21no—
121C4⋊Dic17—21no—
122C4⋊Dic17—42no—
123#63225—41no—
124#63225—41no—
125#63225—84no—
126C4×Dic17—42no—
127C2×#3622—22no—
128C2×#3622—42no—
129#9951—171nomaximal, Sylow
130C23×C17—41no—
131C23×C17—41no—
132C23×C17—41no—
133C23×C17—42no—
134C23×C17—81no—
135C23×C17—81no—
136C23×C17#12011yescharacteristic
137C23×C17#12111yescharacteristic
138C23×C17#12711yescharacteristic
139C23×C17C2×SD1612yes—
140C23×C17#13712yes—
141D4×C17—42no—
142D4×C17—42no—
143D4×C17—82no—
144D4×C17—81no—
145D4×C17—81no—
146C2×C4×C17—22no—
147C2×C4×C17—22no—
148C2×C4×C17—22no—
149C2×C4×C17—41no—
150C2×C4×C17—41no—
151C2×C4×C17—41no—
152C2×C4×C17—44no—
153C2×C4×C17—41no—
154C2×C4×C17—42no—
155C2×C4×C17—42no—
156C2×C4×C17—82no—
157C2×C4×C17—82no—
158C2×C4×C17#12011yescharacteristic
159C2×C4×C17#12011yescharacteristic
160C2×C4×C17#12111yescharacteristic
161C2×C4×C17#12111yescharacteristic
162C2×Dic17—24no—
163C2×Dic17—42no—
164C2×Dic17—42no—
165C2×Dic17—42no—
166C2×Dic17—42no—
167C2×Dic17—41no—
168C2×Dic17—41no—
169C2×Dic17—81no—
170C2×Dic17—81no—
171C2×Dic17—81no—
172Dic34—42no—
173Dic34—81no—
174C8×C17—42no—
175#3622—42no—
176#3187—171no—
177C2×#481D1711yescharacteristic
178#2502—171no—
179#1964—171no—
180#1824—171no—
181#1520—171no—
182#1282—171no—
183C22×C17—21no—
184C22×C17—21no—
185C22×C17—21no—
186C22×C17—21no—
187C22×C17—21no—
188C22×C17—21no—
189C22×C17—42no—
190C22×C17—42no—
191C22×C17—81no—
192C22×C17—81no—
193C22×C17—81no—
194C22×C17—162no—
195C22×C17#39211yescharacteristic
196C22×C17#44812yes—
197C22×C17#46112yes—
198C22×C17#49211yescharacteristic
199C22×C17#49711yescharacteristic
200C4×C17—21no—
201C4×C17—21no—
202C4×C17—21no—
203C4×C17—82no—
204C4×C17—81no—
205C4×C17#45812yes—
206Dic17—44no—
207Dic17—81no—
208Dic17—81no—
209Dic17—161no—
210C2×C22⋊Q8—171no—
211C2×#121C2×D1711yescharacteristic
212C22×C4⋊C4C2×D1711yescharacteristic
213#483—341no—
214#481—22no—
215#481—22no—
216#481—341no—
217#465—342no—
218#450—341no—
219#448—341no—
220C2×#106—21no—
221C2×#102—21no—
222C2×#99—342no—
223C2×C22⋊C8—171no—
224C2×C22⋊C8C2×D1711yescharacteristic
225#398—342no—
226#391—341no—
227#385—342no—
228#384—341no—
229#380—171no—
230C4×C22⋊C4—171no—
231#340—342no—
232#328—342no—
233C2×C17—22no—
234C2×C17—21no—
235C2×C17—22no—
236C2×C17—21no—
237C2×C17—161no—
238C2×C17#200811yescharacteristic
239C2×C17#201311yescharacteristic
240C2×C17#202211yescharacteristic
241C22×Q8—171no—
242C22×D4—21no—
243C22×D4C22⋊D1711yescharacteristic
244C23×C4—21no—
245C23×C4C22×D1711yescharacteristic
246C2×SD16—342no—
247C22×C8—21no—
248C22×C8—172no—
249C22⋊Q8—341no—
250C22⋊Q8—341no—
251C22⋊Q8—341no—
252C22⋊Q8—341no—
253#121—21no—
254#121—21no—
255#121—21no—
256#121—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—171no—
263C2×C4⋊C4—341no—
264C2×C4⋊C4—341no—
265C2×C4⋊C4—344no—
266C2×C4⋊C4C22⋊D1712yes—
267C2×C4⋊C4C22⋊D1711yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—172no—
270C2×C22⋊C4—341no—
271C2×C22⋊C4—341no—
272C2×C42—341no—
273#106—24no—
274#106—342no—
275#103—342no—
276#102—24no—
277#102—342no—
278#99—684no—
279C22⋊C8—21no—
280C22⋊C8—341no—
281C22⋊C8—341no—
282C22⋊C8C22⋊D1712yes—
283#95—344no—
284#95—341no—
285#95—341no—
286C17#995111yescharacteristic, complemented, Sylow
287C24—21no—
288C24D4×D1711yescharacteristic
289C2×Q8—341no—
290C2×Q8—341no—
291C2×Q8—682no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—82no—
305C22×C4—342no—
306C22×C4—341no—
307C22×C4—341no—
308C22×C4—681no—
309C22×C4D4×D1711yescharacteristic
310C22×C4D4×D1711yescharacteristic
311C22×C4#6325411yescharacteristic
312C22×C4C2×C22⋊D1711yescharacteristic
313C22×C4C2×C22⋊D1711yescharacteristic
314C22×C4C2×C22⋊D1711yescharacteristic
315SD16—1362no—
316C2×C8—22no—
317C2×C8—42no—
318C2×C8—342no—
319C2×C8—682no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—341no—
327C4⋊C4—341no—
328C4⋊C4—682no—
329C4⋊C4—681no—
330C4⋊C4—681no—
331C4⋊C4—1364no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—174no—
335C22⋊C4—342no—
336C22⋊C4—681no—
337C22⋊C4—681no—
338C22⋊C4—682no—
339C42—682no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—42no—
344C23—81no—
345C23—81no—
346C23SD16×D1712yes—
347C23#1058760112yes—
348C23#1058762311yescharacteristic
349C23#1058762511yescharacteristic
350C23#1058762611yescharacteristic
351Q8—682no—
352Q8—1361no—
353D4—42no—
354D4—42no—
355D4—81no—
356D4—82no—
357D4—81no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—44no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—344no—
371C2×C4—682no—
372C2×C4—682no—
373C2×C4—682no—
374C2×C4—682no—
375C2×C4—681no—
376C2×C4—681no—
377C2×C4—1361no—
378C2×C4—1361no—
379C2×C4—1361no—
380C2×C4#1058762311yescharacteristic
381C2×C4#1058762411yescharacteristic
382C2×C4#1058762511yescharacteristic
383C2×C4#1058763911yescharacteristic
384C8—42no—
385C8—682no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—42no—
393C22—42no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—162no—
398C22#1176271711yescharacteristic
399C22#1176294612yes—
400C22#1176301212yes—
401C22#1176318011yescharacteristic
402C22#1176320011yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—82no—
407C4—81no—
408C4—2721no—
409C4—684no—
410C4—1361no—
411C4—1361no—
412C4#1176299812yes—
413C2—21no—
414C2—22no—
415C2—22no—
416C2—21no—
417C2—161no—
418C2#1452983511yescharacteristic
419C2#1452986511yescharacteristic
420C2#1452991211yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC2×C17×#481C2
socleC22×C17#392
prime radicalC2×C17×#121C22
hypercenterC2×#481D17
hypocenterC17#9951
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
2. #14533258C2semidirect
4. #14528874C2semidirect
5. #14527357C2semidirect
6. #14526630C2semidirect
118. C22⋊Dic17SD16semidirect
286. C17#9951semidirect

Maximal subgroups

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

subgroupindex
#145395412
#145332582
#145296212
#145288742
#145273572
#145266302
C2×C17×#4812
#995117