Finite Group Databasebeta

← #17189123

Subgroups — #17189123

Order 4352.

590 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538365C211yesmaximal, characteristic
2#14530992C212yesmaximal, complemented
3#14530986C211yesmaximal, characteristic, complemented
4#14530981C211yesmaximal, characteristic, complemented
5#14530975C212yesmaximal, complemented
6#14530969C211yesmaximal, characteristic, complemented
7#14530964C211yesmaximal, characteristic, complemented
8#14530566C212yesmaximal, complemented
9#14530489C212yesmaximal, complemented
10C2×C22⋊#63214C211yesmaximal, characteristic, complemented
11C2×C17×#427C211yesmaximal, characteristic, complemented
12C2×#10587590C2211yescharacteristic
13#11763320—21no—
14C22×#63214C2212yes—
15#11763050C2211yescharacteristic
16#11763050C2211yescharacteristic
17#11763045C2211yescharacteristic
18#11763045C2211yescharacteristic
19#11763038—21no—
20#11763038—21no—
21#11762847C2211yescharacteristic
22#11762806—21no—
23#11762636C2212yescomplemented
24#11762636C2212yescomplemented
25#11762634C2212yescomplemented
26#11762634C2212yescomplemented
27#11762612C2212yescomplemented
28#11762612C2212yescomplemented
29#11762609C2212yescomplemented
30#11762609C2212yescomplemented
31C22⋊#63214C2211yescharacteristic
32C22⋊#63214C2211yescharacteristic
33C22⋊#63214C2212yes—
34C23×C8×C17C2211yescharacteristic
35C2×C17×#117—21no—
36C17×#427—21no—
37C17×#427—21no—
38C17×#427C2212yes—
39C17×#427C2212yes—
40C2×C17×#107C2211yescharacteristic
41C2×C17×#107C2211yescharacteristic
42C2×C17×#106—21no—
43C23×C4×C17D411yescharacteristic
44C2×#63252—21no—
45C22×C8×C17—24no—
46C22×C8×C17D411yescharacteristic
47C22×C8×C17D411yescharacteristic
48C22×C8×C17D411yescharacteristic
49C22×C8×C17C2311yescharacteristic
50C22×C8×C17C2311yescharacteristic
51C22×C8×C17C2311yescharacteristic
52C17×#117—22no—
53C17×#117—22no—
54C17×#117—41no—
55C17×#117—41no—
56C2×C4⋊C4×C17—21no—
57C2×C4⋊C4×C17—21no—
58C2×C22⋊C4×C17—41no—
59C2×C42×C17—41no—
60#10587639—41no—
61#10587616—21no—
62#10587616—21no—
63#10587615—41no—
64C4×C22⋊D17—41no—
65C4×C22⋊D17—41no—
66#10587612—21no—
67#10587610—21no—
68#10587610—21no—
69C2×Dic68C2311yescharacteristic
70#10587590—24no—
71C2×D136C2311yescharacteristic
72C2×#63219—21no—
73C17×#107—21no—
74C17×#107—21no—
75C17×#107C2312yes—
76C17×#107C2312yes—
77C17×#106—22no—
78C17×#106—22no—
79#10587492—21no—
80#10587492—21no—
81#10587489C2311yescharacteristic
82#10587489C2311yescharacteristic
83#10587488—21no—
84#10587487—21no—
85#10587487—21no—
86C2×#63214—24no—
87C2×#63214C2312yes—
88C2×#63214C2312yes—
89C24×C17D811yescharacteristic
90C22×C4×C17—22no—
91C22×C4×C17—22no—
92C22×C4×C17—41no—
93C22×C4×C17—41no—
94C22×C4×C17D811yescharacteristic
95C22×C4×C17D811yescharacteristic
96C22×C4×C17D811yescharacteristic
97C22×C4×C17C2×D411yescharacteristic
98C22×C4×C17C2×D411yescharacteristic
99C22×C4×C17C2×D411yescharacteristic
100C2×C22⋊D17—41no—
101C2×C22⋊D17—41no—
102C2×C22⋊D17—41no—
103#63252—42no—
104#63252—42no—
105C2×D68—21no—
106C2×C4×D17—41no—
107C2×Dic34—21no—
108C2×C8×C17—22no—
109C2×C8×C17—22no—
110C2×C8×C17—22no—
111C2×C8×C17—22no—
112C2×C8×C17—22no—
113C2×C8×C17C2×D411yescharacteristic
114C2×C8×C17C2×D411yescharacteristic
115C2×C8×C17C2×D411yescharacteristic
116C2×C8×C17#4114yes—
117C2×C8×C17C2411yescharacteristic
118C4⋊C4×C17—22no—
119C4⋊C4×C17—22no—
120C4⋊C4×C17—41no—
121C4⋊C4×C17—41no—
122C22⋊C4×C17—42no—
123C22⋊C4×C17—42no—
124C42×C17—42no—
125C42×C17—42no—
126C22⋊Dic17—41no—
127C22⋊Dic17—41no—
128C22⋊Dic17—41no—
129#63227—41no—
130#63227—41no—
131C4⋊Dic17—21no—
132C4⋊Dic17—21no—
133#63225—41no—
134#63225—41no—
135C4×Dic17—41no—
136Dic68—22no—
137D136—22no—
138#63219—42no—
139#63214—22no—
140#63214—22no—
141#33645—171nomaximal, Sylow
142C23×C17—22no—
143C23×C17—22no—
144C23×C17—41no—
145C23×C17C2×D811yescharacteristic
146C23×C17C2×D811yescharacteristic
147C23×C17C2×D811yescharacteristic
148C22×D17—41no—
149C2×C4×C17—22no—
150C2×C4×C17—21no—
151C2×C4×C17—21no—
152C2×C4×C17—21no—
153C2×C4×C17—21no—
154C2×C4×C17—22no—
155C2×C4×C17—22no—
156C2×C4×C17—42no—
157C2×C4×C17—42no—
158C2×C4×C17—82no—
159C2×C4×C17—82no—
160C2×C4×C17#12312yes—
161C2×C4×C17C2×D811yescharacteristic
162C2×C4×C17C2×D811yescharacteristic
163C2×C4×C17C2×D811yescharacteristic
164C2×C4×C17#13712yes—
165C2×C4×C17C22×D411yescharacteristic
166C22⋊D17—82no—
167C22⋊D17—82no—
168C22⋊D17—82no—
169C2×Dic17—41no—
170C2×Dic17—41no—
171C2×Dic17—41no—
172D68—41no—
173D68—41no—
174C4×D17—82no—
175Dic34—41no—
176Dic34—41no—
177C8×C17—21no—
178C8×C17—21no—
179C8×C17#12312yes—
180C8×C17C2×#4112yes—
181#3050—171no—
182C2×#427D1711yescharacteristic, complemented
183#2237—172no—
184#2236—171no—
185#2235—171no—
186#2234—172no—
187#2233—171no—
188#2232—171no—
189#2158—172no—
190#2138—172no—
191C2×C22⋊C16—171no—
192C22×C17—21no—
193C22×C17—21no—
194C22×C17—21no—
195C22×C17—21no—
196C22×C17—21no—
197C22×C17—21no—
198C22×C17—21no—
199C22×C17—44no—
200C22×C17#48012yes—
201C22×C17#50812yes—
202C22×C17C22×D811yescharacteristic
203C2×D17—81no—
204C2×D17—81no—
205C2×D17—81no—
206C4×C17—21no—
207C4×C17—21no—
208C4×C17—82no—
209C4×C17—82no—
210C4×C17#48012yes—
211C4×C17C2×#12311yescharacteristic
212C4×C17C2×#13711yescharacteristic
213Dic17—81no—
214Dic17—81no—
215Dic17—81no—
216C2×#135—171no—
217C23×C8C2×D1711yescharacteristic
218#525—341no—
219C2×#117—21no—
220C22×C16—172no—
221#467—171no—
222#467—171no—
223#466—171no—
224#466—171no—
225#465—341no—
226#465—341no—
227#427—21no—
228#427—21no—
229#427—171no—
230#427C2×D1712yes—
231#427C2×D1712yes—
232C2×#107C2×D1711yescharacteristic
233C2×#107C2×D1711yescharacteristic
234C2×#106—21no—
235#416—341no—
236#367—172no—
237#367—172no—
238#366—172no—
239#366—172no—
240#358—172no—
241#358—172no—
242#357—172no—
243#357—172no—
244C22⋊C16—171no—
245C22⋊C16—171no—
246C22⋊C16—172no—
247C2×C17—22no—
248C2×C17—22no—
249C2×C17—41no—
250C2×C17C2×#48011yescharacteristic
251C2×C17C2×#50811yescharacteristic
252C2×C17#341411yescharacteristic
253D17—161no—
254C2×#41—341no—
255C23×C4C22⋊D1711yescharacteristic
256#135—344no—
257C2×Q16—171no—
258C2×SD16—341no—
259C2×D8—171no—
260C22×C8—24no—
261C22×C8C22⋊D1711yescharacteristic
262C22×C8C22⋊D1711yescharacteristic
263C22×C8C22⋊D1711yescharacteristic
264C22×C8C22×D1711yescharacteristic
265C22×C8C22×D1711yescharacteristic
266C22×C8C22×D1711yescharacteristic
267#123—681no—
268C22⋊Q8—341no—
269C22⋊Q8—341no—
270#121—341no—
271#121—341no—
272#120—681no—
273C4×D4—681no—
274C4×D4—681no—
275#117—22no—
276#117—22no—
277#117—41no—
278#117—41no—
279#117—341no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C22⋊C4—41no—
283C2×C42—41no—
284C2×C16—172no—
285C2×C16—172no—
286C2×C16—344no—
287#107—21no—
288#107—21no—
289#107—171no—
290#107—171no—
291#107C22×D1712yes—
292#107C22×D1712yes—
293#106—22no—
294#106—22no—
295#106—341no—
296#103—341no—
297#103—341no—
298#102—341no—
299#102—341no—
300C17#3364511yescharacteristic, complemented, Sylow
301C24#6322811yescharacteristic
302#41—682no—
303#41—682no—
304C2×Q8—341no—
305C2×D4—341no—
306C2×D4—681no—
307C2×D4—681no—
308C2×D4—681no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—681no—
314C22×C4#6322811yescharacteristic
315C22×C4#6322811yescharacteristic
316C22×C4#6322811yescharacteristic
317C22×C4C2×C22⋊D1711yescharacteristic
318C22×C4C2×C22⋊D1711yescharacteristic
319C22×C4C2×C22⋊D1711yescharacteristic
320Q16—342no—
321SD16—682no—
322D8—342no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8#6325214yes—
329C2×C8C2×C22⋊D1711yescharacteristic
330C2×C8C2×C22⋊D1711yescharacteristic
331C2×C8C2×C22⋊D1711yescharacteristic
332C2×C8C23×D1711yescharacteristic
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—341no—
338C4⋊C4—341no—
339C4⋊C4—681no—
340C4⋊C4—681no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—681no—
344C22⋊C4—681no—
345C22⋊C4—681no—
346C22⋊C4—681no—
347C22⋊C4—681no—
348C42—42no—
349C42—42no—
350C42—681no—
351C16—342no—
352C16—342no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—681no—
357C23C2×#6322811yescharacteristic
358C23C2×#6322811yescharacteristic
359C23C2×#6322811yescharacteristic
360Q8—681no—
361Q8—681no—
362D4—681no—
363D4—681no—
364D4—1362no—
365D4—1362no—
366D4—1362no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—681no—
379C2×C4—681no—
380C2×C4—681no—
381C2×C4—1362no—
382C2×C4#1058761512yes—
383C2×C4C2×#6322811yescharacteristic
384C2×C4C2×#6322811yescharacteristic
385C2×C4C2×#6322811yescharacteristic
386C2×C4#1058762812yes—
387C2×C4C22×C22⋊D1711yescharacteristic
388C8—21no—
389C8—21no—
390C8#1058761512yes—
391C8C2×#6325212yes—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—1361no—
401C22—1361no—
402C22—1361no—
403C22#1176312212yes—
404C22#1176323912yes—
405C22C22×#6322811yescharacteristic
406C4—21no—
407C4—21no—
408C4—82no—
409C4—82no—
410C4—1361no—
411C4—1361no—
412C4—1361no—
413C4#1176312212yes—
414C4C2×#1058761511yescharacteristic
415C4C2×#1058762811yescharacteristic
416C2—22no—
417C2—22no—
418C2—41no—
419C2—2721no—
420C2C2×#1176312211yescharacteristic
421C2C2×#1176323911yescharacteristic
422C2#1454191811yescharacteristic
subgroupG/H
centerC2×C4C2×#63228
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC2×C17×#427C2
socleC22×C17C22×D8
prime radical#14538365C2
hypercenterC2×#427D17
hypocenterC17#33645
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. #14530992C2semidirect
3. #14530986C2semidirect
4. #14530981C2semidirect
5. #14530975C2semidirect
6. #14530969C2semidirect
7. #14530964C2semidirect
8. #14530566C2semidirect
9. #14530489C2semidirect
10. C2×C22⋊#63214C2semidirect
11. C2×C17×#427C2semidirect
23. #11762636C22semidirect
24. #11762636C22semidirect
25. #11762634C22semidirect
26. #11762634C22semidirect
27. #11762612C22semidirect
28. #11762612C22semidirect
29. #11762609C22semidirect
30. #11762609C22semidirect
182. C2×#427D17semidirect
300. C17#33645semidirect

Maximal subgroups

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

subgroupindex
#145383652
#145309922
#145309862
#145309812
#145309752
#145309692
#145309642
#145305662
#145304892
C2×C22⋊#632142
C2×C17×#4272
#3364517