Finite Group Databasebeta

← #16934839

Subgroups — #16934839

Order 4352.

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

normal subgroups only

This group has 650 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#14539537C211yesmaximal, characteristic
2#14533251C211yesmaximal, characteristic
3#14529770C211yesmaximal, characteristic, complemented
4#14528870C211yesmaximal, characteristic, complemented
5#14527353C211yesmaximal, characteristic
6#14526630C211yesmaximal, characteristic, complemented
7C2×C17×#483C211yesmaximal, characteristic, complemented
8C2×#10587615—22no—
9C2×#10587616C2211yescharacteristic
10#11763137—21no—
11#11763127—21no—
12#11763040—22no—
13#11762959—21no—
14#11762948—21no—
15C2×C22⋊#3622C2211yescharacteristic
16#11762744—22no—
17#11762710—21no—
18#11762678—21no—
19#11762669C2211yescharacteristic
20C4×C22⋊Dic17C2211yescharacteristic
21#11762576—22no—
22#11762571—22no—
23#11762544—22no—
24C2×C17×C22⋊Q8C2211yescharacteristic
25C22×C4⋊C4×C17C2211yescharacteristic
26C17×#483—22no—
27C17×#483—22no—
28C2×C17×#106—21no—
29C2×C17×#103—21no—
30C2×C17×C22⋊C8C2211yescharacteristic
31C22×Q8×C17D411yescharacteristic
32C23×C4×C17—21no—
33C23×C4×C17C2311yescharacteristic
34C22×C22⋊D17—21no—
35C22×D68D411yescharacteristic
36C22×C8×C17—21no—
37C17×C22⋊Q8—21no—
38C17×C22⋊Q8—21no—
39C17×C22⋊Q8—21no—
40C17×C22⋊Q8—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×C17—21no—
47C2×C4⋊C4×C17D412yes—
48C2×C4⋊C4×C17D411yescharacteristic
49C2×C22⋊C4×C17—21no—
50C2×C22⋊Dic17—21no—
51C2×C22⋊Dic17D412yes—
52C2×#63230—22no—
53#10587616—21no—
54#10587616—21no—
55#10587616—21no—
56#10587616—21no—
57#10587615—44no—
58#10587615—44no—
59C2×#63227—24no—
60C2×#63227—21no—
61C2×C4⋊Dic17—21no—
62C2×C4⋊Dic17D411yescharacteristic
63C2×#63225—24no—
64C2×C4×Dic17—21no—
65C22×#3622D412yes—
66C17×#106—24no—
67C17×#103—24no—
68C17×C22⋊C8—21no—
69C17×C22⋊C8D412yes—
70#10587506—22no—
71#10587502—21no—
72#10587502—21no—
73C22⋊#3622—21no—
74C22⋊#3622—21no—
75#10587480—22no—
76#10587479—22no—
77C24×C17C2×D411yescharacteristic
78C23×D17—21no—
79C2×Q8×C17—21no—
80C2×Q8×C17—21no—
81C2×Q8×C17—42no—
82C22×C4×C17—22no—
83C22×C4×C17—21no—
84C22×C4×C17—21no—
85C22×C4×C17—42no—
86C22×C4×C17—41no—
87C22×C4×C17—41no—
88C22×C4×C17—42no—
89C22×C4×C17—42no—
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17C2×D411yescharacteristic
94C22×C4×C17C2×D411yescharacteristic
95C22×C4×C17#4111yescharacteristic
96C2×C22⋊D17—41no—
97C2×C22⋊D17—41no—
98C2×C22⋊D17—42no—
99C2×C22⋊D17—42no—
100C22×Dic17—22no—
101C22×Dic17—21no—
102C22×Dic17—41no—
103C2×D68—21no—
104C2×D68—21no—
105C2×D68—42no—
106C2×C8×C17—22no—
107C2×C8×C17—42no—
108C4⋊C4×C17—21no—
109C4⋊C4×C17—21no—
110C4⋊C4×C17—22no—
111C4⋊C4×C17—22no—
112C4⋊C4×C17—41no—
113C4⋊C4×C17—41no—
114C4⋊C4×C17—41no—
115C4⋊C4×C17—41no—
116C22⋊C4×C17—41no—
117C22⋊C4×C17—41no—
118C22⋊Dic17—22no—
119C22⋊Dic17—42no—
120C22⋊Dic17SD1614yescomplemented
121#63230—82no—
122#63227—48no—
123#63227—41no—
124#63227—41no—
125C4⋊Dic17—21no—
126C4⋊Dic17—21no—
127C4⋊Dic17—42no—
128#63225—48no—
129C4×Dic17—42no—
130C2×#3622—22no—
131C2×#3622—42no—
132#9960—171nomaximal, Sylow
133C23×C17—41no—
134C23×C17—41no—
135C23×C17#12011yescharacteristic
136C23×C17#12111yescharacteristic
137C23×C17#12711yescharacteristic
138C23×C17C2×SD1612yes—
139C23×C17AGL(1,ℤ/8)12yes—
140C22×D17—42no—
141C22×D17—42no—
142C22×D17—41no—
143C22×D17—42no—
144Q8×C17—42no—
145Q8×C17—42no—
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—41no—
153C2×C4×C17—41no—
154C2×C4×C17—42no—
155C2×C4×C17—44no—
156C2×C4×C17—82no—
157C2×C4×C17—81no—
158C2×C4×C17—81no—
159C2×C4×C17—81no—
160C2×C4×C17—81no—
161C2×C4×C17#12011yescharacteristic
162C2×C4×C17#12011yescharacteristic
163C2×C4×C17#12111yescharacteristic
164C2×C4×C17#12111yescharacteristic
165C22⋊D17—82no—
166C22⋊D17—81no—
167C22⋊D17—81no—
168C2×Dic17—24no—
169C2×Dic17—42no—
170C2×Dic17—42no—
171C2×Dic17—41no—
172C2×Dic17—42no—
173C2×Dic17—81no—
174C2×Dic17—81no—
175C2×Dic17—81no—
176D68—42no—
177D68—81no—
178C8×C17—42no—
179#3622—42no—
180#3187—171no—
181C2×#483D1711yescharacteristic, complemented
182#2501—171no—
183#1996—171no—
184#1823—171no—
185#1519—171no—
186#1282—171no—
187C22×C17—21no—
188C22×C17—21no—
189C22×C17—21no—
190C22×C17—21no—
191C22×C17—21no—
192C22×C17—21no—
193C22×C17—42no—
194C22×C17—42no—
195C22×C17#39211yescharacteristic
196C22×C17#45012yes—
197C22×C17#46012yes—
198C22×C17#49211yescharacteristic
199C22×C17#49611yescharacteristic
200C2×D17—84no—
201C2×D17—81no—
202C2×D17—81no—
203C2×D17—81no—
204C4×C17—21no—
205C4×C17—21no—
206C4×C17—21no—
207C4×C17—81no—
208C4×C17—82no—
209C4×C17—81no—
210C4×C17#45812yes—
211Dic17—44no—
212Dic17—81no—
213Dic17—161no—
214C2×#123—342no—
215C2×C22⋊Q8C2×D1711yescharacteristic
216C2×#121—171no—
217C22×C4⋊C4C2×D1711yescharacteristic
218#483—22no—
219#483—22no—
220#483—341no—
221#481—341no—
222#465—342no—
223#450—341no—
224#448—341no—
225C2×#106—21no—
226C2×#103—21no—
227C2×C22⋊C8—171no—
228C2×C22⋊C8C2×D1711yescharacteristic
229#399—342no—
230#390—341no—
231#384—341no—
232#380—171no—
233C4×C22⋊C4—171no—
234#342—342no—
235#340—342no—
236#327—342no—
237C2×C17—22no—
238C2×C17—21no—
239C2×C17—22no—
240C2×C17—21no—
241C2×C17#200911yescharacteristic
242C2×C17#201411yescharacteristic
243C2×C17#202211yescharacteristic
244D17—161no—
245C22×Q8C22⋊D1711yescharacteristic
246C22×D4—171no—
247C22×D4—341no—
248C23×C4—21no—
249C23×C4C22×D1711yescharacteristic
250C2×SD16—342no—
251C22×C8—21no—
252C22×C8—172no—
253#123—684no—
254#123—684no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258C22⋊Q8—21no—
259#121—341no—
260#121—341no—
261#121—341no—
262#121—341no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—171no—
270C2×C4⋊C4—344no—
271C2×C4⋊C4—341no—
272C2×C4⋊C4C22⋊D1712yes—
273C2×C4⋊C4C22⋊D1711yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—172no—
276C2×C22⋊C4—344no—
277C2×C22⋊C4—341no—
278C2×C22⋊C4—341no—
279C2×C42—341no—
280#106—24no—
281#106—342no—
282#103—24no—
283#103—342no—
284#102—342no—
285C22⋊C8—21no—
286C22⋊C8—341no—
287C22⋊C8—341no—
288C22⋊C8C22⋊D1712yes—
289#95—341no—
290#95—341no—
291C17#996011yescharacteristic, complemented, Sylow
292C24—341no—
293C24D4×D1711yescharacteristic
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—42no—
297C2×D4—341no—
298C2×D4—341no—
299C2×D4—682no—
300C2×D4—682no—
301C2×D4—682no—
302C2×D4—681no—
303C2×D4—681no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—342no—
313C22×C4—341no—
314C22×C4—681no—
315C22×C4D4×D1711yescharacteristic
316C22×C4D4×D1711yescharacteristic
317C22×C4#6325411yescharacteristic
318C22×C4C2×C22⋊D1711yescharacteristic
319C22×C4C2×C22⋊D1711yescharacteristic
320C22×C4C2×C22⋊D1711yescharacteristic
321SD16—1362no—
322C2×C8—22no—
323C2×C8—42no—
324C2×C8—342no—
325C2×C8—682no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—341no—
335C4⋊C4—341no—
336C4⋊C4—682no—
337C4⋊C4—688no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—174no—
341C22⋊C4—342no—
342C22⋊C4—688no—
343C22⋊C4—681no—
344C22⋊C4—681no—
345C22⋊C4—682no—
346C42—682no—
347C23—41no—
348C23—41no—
349C23—682no—
350C23—682no—
351C23—682no—
352C23—681no—
353C23SD16×D1712yes—
354C23#1058760012yes—
355C23#1058762311yescharacteristic
356C23#1058762511yescharacteristic
357C23#1058762611yescharacteristic
358Q8—42no—
359Q8—42no—
360D4—682no—
361D4—1361no—
362D4—1362no—
363D4—1361no—
364D4—1361no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—44no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—344no—
381C2×C4—682no—
382C2×C4—682no—
383C2×C4—682no—
384C2×C4—681no—
385C2×C4—1361no—
386C2×C4—1361no—
387C2×C4—1361no—
388C2×C4#1058762311yescharacteristic
389C2×C4#1058762411yescharacteristic
390C2×C4#1058762511yescharacteristic
391C2×C4#1058763911yescharacteristic
392C8—42no—
393C8—682no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—42no—
401C22—42no—
402C22—1364no—
403C22—1361no—
404C22—1361no—
405C22—1361no—
406C22#1176271711yescharacteristic
407C22#1176295812yes—
408C22#1176300712yes—
409C22#1176318011yescharacteristic
410C22#1176319511yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—81no—
415C4—82no—
416C4—81no—
417C4—2721no—
418C4—684no—
419C4—1361no—
420C4#1176299812yes—
421C2—21no—
422C2—22no—
423C2—22no—
424C2—21no—
425C2—2721no—
426C2#1452984211yescharacteristic
427C2#1452987011yescharacteristic
428C2#1452991211yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC2×C17×#483C2
socleC22×C17#392
prime radicalC2×#10587616C22
hypercenterC2×#483D17
hypocenterC17#9960
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
3. #14529770C2semidirect
4. #14528870C2semidirect
6. #14526630C2semidirect
7. C2×C17×#483C2semidirect
120. C22⋊Dic17SD16semidirect
181. C2×#483D17semidirect
291. C17#9960semidirect

Maximal subgroups

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

subgroupindex
#145395372
#145332512
#145297702
#145288702
#145273532
#145266302
C2×C17×#4832
#996017