Finite Group Databasebeta

← #16929892

Subgroups — #16929892

Order 4352.

694 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762862C211yesmaximal, characteristic, complemented
2#14537115C211yesmaximal, characteristic
3C2×#11762542C212yesmaximal
4#14526663C212yesmaximal, complemented
5C17×#1434C211yesmaximal, characteristic, complemented
6C2×#10587615—21no—
7C2×#10587616C2211yescharacteristic
8#11762862—24no—
9C2×C4⋊#3622—21no—
10C2×C22⋊#3622C2212yes—
11C2×C4×#3622—21no—
12#11762711C411yescharacteristic, complemented
13#11762711C411yescharacteristic, complemented
14#11762707C411yescharacteristic, complemented
15#11762707C411yescharacteristic, complemented
16#11762700—21no—
17#11762700—21no—
18#11762688—21no—
19#11762669C2211yescharacteristic
20#11762542—24no—
21C22×C42×C17C2211yescharacteristic
22C2×C17×#95—21no—
23C2×C17×#95—21no—
24C2×C17×#95C2212yes—
25C23×C4×C17—22no—
26C23×C4×C17D411yescharacteristic
27C23×C4×C17D411yescharacteristic
28C23×C4×C17C2311yescharacteristic
29C22×C22⋊D17—21no—
30C22×D68C2×C411yescharacteristic
31C2×C42×C17—22no—
32C2×C42×C17—22no—
33C2×C42×C17C2×C411yescharacteristic
34C2×C42×C17C2×C411yescharacteristic
35C2×C42×C17D411yescharacteristic
36C2×C42×C17D411yescharacteristic
37C2×C22⋊Dic17—21no—
38#10587616—22no—
39#10587616—22no—
40#10587615—42no—
41#10587615—42no—
42C2×#63227—21no—
43C2×#63227—21no—
44C2×#63227—21no—
45C2×#63227—21no—
46C2×#63227—21no—
47C2×C4⋊Dic17C2×C411yescharacteristic
48C2×C4⋊Dic17C2×C411yescharacteristic
49C2×C4⋊Dic17C2×C411yescharacteristic
50C2×#63225—21no—
51C2×#63225—21no—
52C22×#3622—22no—
53C17×#95—24no—
54C17×#95—22no—
55C17×#95—22no—
56C17×#95—42no—
57#10587502—21no—
58#10587502—21no—
59#10587502—21no—
60C22⋊#3622—22no—
61C22⋊#3622D414yes—
62C4⋊#3622—24no—
63C4×#3622—24no—
64C24×C17C2×D411yescharacteristic
65C23×D17—21no—
66C22×C4×C17—22no—
67C22×C4×C17—22no—
68C22×C4×C17—22no—
69C22×C4×C17—22no—
70C22×C4×C17—24no—
71C22×C4×C17—41no—
72C22×C4×C17—41no—
73C22×C4×C17—41no—
74C22×C4×C17—42no—
75C22×C4×C17—42no—
76C22×C4×C17—42no—
77C22×C4×C17—42no—
78C22×C4×C17—42no—
79C22×C4×C17—42no—
80C22×C4×C17C22⋊C411yescharacteristic
81C22×C4×C17C22⋊C411yescharacteristic
82C22×C4×C17C22⋊C411yescharacteristic
83C22×C4×C17C22⋊C411yescharacteristic
84C22×C4×C17D812yes—
85C22×C4×C17SD1612yes—
86C22×C4×C17C22×C411yescharacteristic
87C22×C4×C17C2×D411yescharacteristic
88C22×C4×C17C2×D412yes—
89C22×C4×C17#4112yes—
90C2×C22⋊D17—42no—
91C2×C22⋊D17—41no—
92C2×C22⋊D17—41no—
93C2×C22⋊D17—42no—
94C22×Dic17—21no—
95C22×Dic17—21no—
96C22×Dic17—21no—
97C2×D68—22no—
98C2×D68—41no—
99C2×D68—41no—
100C42×C17—22no—
101C42×C17—22no—
102C42×C17—21no—
103C42×C17—21no—
104C42×C17—21no—
105C42×C17—21no—
106C22⋊Dic17—41no—
107C22⋊Dic17—41no—
108#63227—42no—
109#63227—42no—
110#63227—42no—
111#63227—42no—
112#63227—41no—
113#63227—41no—
114C4⋊Dic17—22no—
115C4⋊Dic17—41no—
116C4⋊Dic17—41no—
117#63225—42no—
118#63225—42no—
119C2×#3622—24no—
120C2×#3622—42no—
121C2×#3622—42no—
122#9114—171nomaximal, Sylow
123C23×C17—21no—
124C23×C17—21no—
125C23×C17—21no—
126C23×C17—21no—
127C23×C17C2×C22⋊C411yescharacteristic
128C23×C17#12112yes—
129C23×C17C2×D811yescharacteristic
130C23×C17C2×SD1611yescharacteristic
131C23×C17#13512yes—
132C22×D17—41no—
133C22×D17—41no—
134C22×D17—41no—
135C22×D17—41no—
136C22×D17—41no—
137C22×D17—41no—
138C22×D17—41no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—21no—
145C2×C4×C17—21no—
146C2×C4×C17—21no—
147C2×C4×C17—21no—
148C2×C4×C17—21no—
149C2×C4×C17—24no—
150C2×C4×C17—24no—
151C2×C4×C17—21no—
152C2×C4×C17—21no—
153C2×C4×C17—21no—
154C2×C4×C17—42no—
155C2×C4×C17—42no—
156C2×C4×C17—44no—
157C2×C4×C17—44no—
158C2×C4×C17—44no—
159C2×C4×C17—44no—
160C2×C4×C17—82no—
161C2×C4×C17—82no—
162C2×C4×C17—82no—
163C2×C4×C17—82no—
164C2×C4×C17#10212yes—
165C2×C4×C17#10212yes—
166C2×C4×C17#10412yes—
167C2×C4×C17C4×D412yes—
168C2×C4×C17#12011yescharacteristic
169C2×C4×C17#12311yescharacteristic
170C22⋊D17—82no—
171C22⋊D17—82no—
172C2×Dic17—41no—
173C2×Dic17—41no—
174C2×Dic17—41no—
175C2×Dic17—41no—
176C2×Dic17—41no—
177C2×Dic17—41no—
178C2×Dic17—41no—
179C2×Dic17—41no—
180C2×Dic17—41no—
181D68—82no—
182#3622—44no—
183C2×#431—171no—
184#2854—171no—
185C2×#327—172no—
186#1434D1711yescharacteristic, complemented
187#1294—172no—
188C22×C17—22no—
189C22×C17—22no—
190C22×C17—22no—
191C22×C17—21no—
192C22×C17—21no—
193C22×C17—21no—
194C22×C17—21no—
195C22×C17—22no—
196C22×C17#38611yescharacteristic
197C22×C17#41011yescharacteristic
198C22×C17#41011yescharacteristic
199C22×C17C2×#10211yescharacteristic
200C22×C17#41611yescharacteristic
201C22×C17C2×#10411yescharacteristic
202C22×C17#42111yescharacteristic
203C22×C17#46512yes—
204C22×C17#46612yes—
205C2×D17—81no—
206C2×D17—81no—
207C2×D17—81no—
208C2×D17—81no—
209C2×D17—81no—
210C2×D17—81no—
211C2×D17—81no—
212C4×C17—22no—
213C4×C17—22no—
214C4×C17—22no—
215C4×C17—21no—
216C4×C17—22no—
217C4×C17—21no—
218C4×C17—41no—
219C4×C17—84no—
220Dic17—81no—
221Dic17—81no—
222Dic17—81no—
223C2×#123—341no—
224C2×#121—171no—
225C22×C42C2×D1711yescharacteristic
226#431—344no—
227C2×C4⋊C8—341no—
228C2×C22⋊C8—172no—
229C2×C4×C8—341no—
230#390—171no—
231#390—171no—
232#389—171no—
233#389—171no—
234#388—341no—
235#388—341no—
236#386—341no—
237#380—171no—
238C2×#95—21no—
239C2×#95—21no—
240C2×#95C2×D1712yes—
241#327—344no—
242C2×C17—22no—
243C2×C17—22no—
244C2×C17#150011yescharacteristic
245C2×C17#150211yescharacteristic
246C2×C17#151711yescharacteristic
247C2×C17#152111yescharacteristic
248C2×C17#187111yescharacteristic
249C2×C17#189311yescharacteristic
250C2×C17#189911yescharacteristic
251D17—161no—
252C22×D4—171no—
253C22×D4—341no—
254C23×C4—22no—
255C23×C4C22⋊D1711yescharacteristic
256C23×C4C22⋊D1711yescharacteristic
257C23×C4C22×D1711yescharacteristic
258C22×C8—342no—
259#123—682no—
260#123—682no—
261#121—342no—
262#121—342no—
263C2×C4⋊C4—171no—
264C2×C4⋊C4—171no—
265C2×C4⋊C4—171no—
266C2×C4⋊C4—341no—
267C2×C4⋊C4—341no—
268C2×C22⋊C4—341no—
269C2×C22⋊C4—341no—
270C2×C22⋊C4—341no—
271C2×C22⋊C4—341no—
272C2×C22⋊C4—341no—
273C2×C22⋊C4—341no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42C4×D1711yescharacteristic
277C2×C42C4×D1711yescharacteristic
278C2×C42D6811yescharacteristic
279C2×C42D6811yescharacteristic
280C4⋊C8—344no—
281C22⋊C8—174no—
282C22⋊C8—342no—
283C4×C8—344no—
284#95—24no—
285#95—22no—
286#95—22no—
287#95—42no—
288#95—341no—
289#95—341no—
290#95—341no—
291C17#911411yescharacteristic, complemented, Sylow
292C24—341no—
293C24C2×C22⋊D1711yescharacteristic
294C2×D4—342no—
295C2×D4—681no—
296C2×D4—681no—
297C2×D4—681no—
298C2×D4—682no—
299C2×D4—681no—
300C2×D4—682no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—341no—
316C22×C4—341no—
317C22×C4—341no—
318C22×C4#6322711yescharacteristic
319C22×C4#6322711yescharacteristic
320C22×C4#6322711yescharacteristic
321C22×C4#6322711yescharacteristic
322C22×C4#6322812yes—
323C22×C4#6323012yes—
324C22×C4C2×C4×D1711yescharacteristic
325C22×C4C2×D6811yescharacteristic
326C22×C4D4×D1712yes—
327C22×C4#6325412yes—
328C2×C8—344no—
329C2×C8—682no—
330C2×C8—682no—
331C4⋊C4—342no—
332C4⋊C4—681no—
333C4⋊C4—681no—
334C4⋊C4—682no—
335C4⋊C4—682no—
336C22⋊C4—682no—
337C22⋊C4—681no—
338C22⋊C4—681no—
339C22⋊C4—682no—
340C22⋊C4—682no—
341C22⋊C4—682no—
342C22⋊C4—681no—
343C22⋊C4—681no—
344C42—22no—
345C42—22no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—681no—
355C23—681no—
356C23—681no—
357C23—681no—
358C23—681no—
359C23—681no—
360C23—681no—
361C23C2×#6322711yescharacteristic
362C23C2×#6322811yescharacteristic
363C23#1058762512yes—
364C23C2×#6323011yescharacteristic
365C23#1058763612yes—
366D4—1362no—
367D4—1362no—
368D4—1362no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—681no—
395C2×C4—681no—
396C2×C4—681no—
397C2×C4—681no—
398C2×C4—681no—
399C2×C4—681no—
400C2×C4—681no—
401C2×C4—681no—
402C2×C4—681no—
403C2×C4#1058747612yes—
404C2×C4#1058748012yes—
405C2×C4#1058748012yes—
406C2×C4#1058756712yes—
407C2×C4#1058756811yescharacteristic
408C2×C4#1058757211yescharacteristic
409C8—684no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—1361no—
419C22—1361no—
420C22—1361no—
421C22—1361no—
422C22—1361no—
423C22—1361no—
424C22—1361no—
425C22#1176268611yescharacteristic
426C22#1176278211yescharacteristic
427C22#1176278211yescharacteristic
428C22C2×#1058748011yescharacteristic
429C22#1176280311yescharacteristic
430C22C2×#1058747611yescharacteristic
431C22#1176282711yescharacteristic
432C22#1176304012yes—
433C22#1176304312yes—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—21no—
438C4—22no—
439C4—21no—
440C4—41no—
441C4—84no—
442C4—1361no—
443C4—1361no—
444C4—1361no—
445C2—22no—
446C2—22no—
447C2—2721no—
448C2#1452726811yescharacteristic
449C2#1452727811yescharacteristic
450C2#1452734311yescharacteristic
451C2#1452735911yescharacteristic
452C2#1452908411yescharacteristic
453C2#1452921011yescharacteristic
454C2#1452924011yescharacteristic
subgroupG/H
centerC23C2×#63227
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1434C2
socleC23×C17C2×C22⋊C4
prime radicalC2×#10587616C22
cosocleC23×C4×C17C23
hypercenter#1434D17
hypocenterC17#9114
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
1. C2×#11762862C2semidirect
4. #14526663C2semidirect
5. C17×#1434C2semidirect
12. #11762711C4semidirect
13. #11762711C4semidirect
14. #11762707C4semidirect
15. #11762707C4semidirect
186. #1434D17semidirect
291. C17#9114semidirect

Maximal subgroups

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

subgroupindex
C2×#117628622
#145371152
C2×#117625422
#145266632
C17×#14342
#911417