Finite Group Databasebeta

← #16929912

Subgroups — #16929912

Order 4352.

674 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762866C211yesmaximal, characteristic, complemented
2#14537115C211yesmaximal, characteristic
3C2×#11762542C212yesmaximal
4#14526663C212yesmaximal, complemented
5C17×#1434C211yesmaximal, characteristic, complemented
6C2×#10587615—21no—
7C2×#10587616C2211yescharacteristic
8#11762866—24no—
9C2×C4⋊#3622—21no—
10C2×C22⋊#3622C2212yes—
11C2×#10587474—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—44no—
41C2×#63227—21no—
42C2×#63227—21no—
43C2×#63227—21no—
44C2×#63227—21no—
45C2×#63227—21no—
46C2×C4⋊Dic17C2×C411yescharacteristic
47C2×C4⋊Dic17C2×C411yescharacteristic
48C2×C4⋊Dic17C2×C411yescharacteristic
49C2×#63225—21no—
50C2×#63225—21no—
51C22×#3622—22no—
52C17×#95—24no—
53C17×#95—22no—
54C17×#95—22no—
55C17×#95—42no—
56#10587502—21no—
57#10587502—21no—
58#10587502—21no—
59C22⋊#3622—22no—
60C22⋊#3622D414yes—
61C4⋊#3622—24no—
62#10587474—24no—
63C24×C17C2×D411yescharacteristic
64C23×D17—21no—
65C22×C4×C17—22no—
66C22×C4×C17—22no—
67C22×C4×C17—21no—
68C22×C4×C17—22no—
69C22×C4×C17—21no—
70C22×C4×C17—22no—
71C22×C4×C17—24no—
72C22×C4×C17—41no—
73C22×C4×C17—41no—
74C22×C4×C17—41no—
75C22×C4×C17—42no—
76C22×C4×C17—42no—
77C22×C4×C17—42no—
78C22×C4×C17—42no—
79C22×C4×C17—42no—
80C22×C4×C17—42no—
81C22×C4×C17C22⋊C411yescharacteristic
82C22×C4×C17C22⋊C411yescharacteristic
83C22×C4×C17C22⋊C411yescharacteristic
84C22×C4×C17C22⋊C411yescharacteristic
85C22×C4×C17C22×C411yescharacteristic
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17C2×D412yes—
88C22×C4×C17#4112yes—
89C2×C22⋊D17—42no—
90C2×C22⋊D17—41no—
91C2×C22⋊D17—42no—
92C2×C22⋊D17—41no—
93C22×Dic17—21no—
94C22×Dic17—21no—
95C22×Dic17—21no—
96C2×D68—22no—
97C2×D68—41no—
98C2×D68—41no—
99C42×C17—22no—
100C42×C17—22no—
101C42×C17—21no—
102C42×C17—21no—
103C42×C17—21no—
104C42×C17—21no—
105C22⋊Dic17—42no—
106#63227—42no—
107#63227—42no—
108#63227—42no—
109#63227—42no—
110#63227—41no—
111#63227—41no—
112C4⋊Dic17—22no—
113C4⋊Dic17—41no—
114C4⋊Dic17—41no—
115#63225—42no—
116#63225—42no—
117C2×#3622—24no—
118C2×#3622—42no—
119C2×#3622—42no—
120#9118—171nomaximal, Sylow
121C23×C17—21no—
122C23×C17—21no—
123C23×C17—21no—
124C23×C17—21no—
125C23×C17C2×C22⋊C411yescharacteristic
126C23×C17#12112yes—
127C23×C17AGL(1,ℤ/8)11yescharacteristic
128C23×C17AGL(1,ℤ/8)12yes—
129C23×C17#13711yescharacteristic
130C22×D17—41no—
131C22×D17—41no—
132C22×D17—41no—
133C22×D17—41no—
134C22×D17—41no—
135C22×D17—41no—
136C22×D17—41no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
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—21no—
150C2×C4×C17—21no—
151C2×C4×C17—21no—
152C2×C4×C17—42no—
153C2×C4×C17—42no—
154C2×C4×C17—44no—
155C2×C4×C17—42no—
156C2×C4×C17—42no—
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#10412yes—
165C2×C4×C17C4×D412yes—
166C2×C4×C17#12011yescharacteristic
167C2×C4×C17#12311yescharacteristic
168C22⋊D17—82no—
169C22⋊D17—82no—
170C2×Dic17—41no—
171C2×Dic17—41no—
172C2×Dic17—41no—
173C2×Dic17—41no—
174C2×Dic17—41no—
175C2×Dic17—41no—
176C2×Dic17—41no—
177C2×Dic17—41no—
178C2×Dic17—41no—
179D68—82no—
180#3622—44no—
181C2×#432—171no—
182#2854—171no—
183C2×#327—172no—
184#1434D1711yescharacteristic, complemented
185#1294—172no—
186C22×C17—22no—
187C22×C17—22no—
188C22×C17—22no—
189C22×C17—21no—
190C22×C17—21no—
191C22×C17—21no—
192C22×C17—21no—
193C22×C17—22no—
194C22×C17#38611yescharacteristic
195C22×C17#41011yescharacteristic
196C22×C17#41011yescharacteristic
197C22×C17#41711yescharacteristic
198C22×C17#41811yescharacteristic
199C22×C17C2×#10411yescharacteristic
200C22×C17#42111yescharacteristic
201C22×C17#46812yes—
202C22×C17#46912yes—
203C2×D17—81no—
204C2×D17—81no—
205C2×D17—81no—
206C2×D17—81no—
207C2×D17—81no—
208C2×D17—81no—
209C2×D17—81no—
210C4×C17—22no—
211C4×C17—21no—
212C4×C17—22no—
213C4×C17—21no—
214C4×C17—41no—
215C4×C17—41no—
216C4×C17—41no—
217C4×C17—84no—
218Dic17—81no—
219Dic17—81no—
220Dic17—81no—
221C2×#123—341no—
222C2×#121—171no—
223C22×C42C2×D1711yescharacteristic
224#432—344no—
225C2×C4⋊C8—341no—
226C2×C22⋊C8—172no—
227C2×#97—341no—
228#390—171no—
229#390—171no—
230#389—171no—
231#389—171no—
232#388—341no—
233#388—341no—
234#386—341no—
235#380—171no—
236C2×#95—21no—
237C2×#95—21no—
238C2×#95C2×D1712yes—
239#327—344no—
240C2×C17—22no—
241C2×C17—22no—
242C2×C17#150511yescharacteristic
243C2×C17#150911yescharacteristic
244C2×C17#151211yescharacteristic
245C2×C17#151411yescharacteristic
246C2×C17#187111yescharacteristic
247C2×C17#189511yescharacteristic
248C2×C17#189911yescharacteristic
249D17—161no—
250C22×D4—171no—
251C22×D4—341no—
252C23×C4—22no—
253C23×C4C22⋊D1711yescharacteristic
254C23×C4C22⋊D1711yescharacteristic
255C23×C4C22×D1711yescharacteristic
256C22×C8—342no—
257#123—684no—
258#121—342no—
259#121—342no—
260C2×C4⋊C4—171no—
261C2×C4⋊C4—171no—
262C2×C4⋊C4—171no—
263C2×C4⋊C4—341no—
264C2×C4⋊C4—341no—
265C2×C22⋊C4—341no—
266C2×C22⋊C4—341no—
267C2×C22⋊C4—341no—
268C2×C22⋊C4—341no—
269C2×C22⋊C4—341no—
270C2×C22⋊C4—341no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42C4×D1711yescharacteristic
274C2×C42C4×D1711yescharacteristic
275C2×C42D6811yescharacteristic
276C2×C42D6811yescharacteristic
277C4⋊C8—344no—
278C22⋊C8—174no—
279C22⋊C8—342no—
280#97—344no—
281#95—24no—
282#95—22no—
283#95—22no—
284#95—42no—
285#95—341no—
286#95—341no—
287#95—341no—
288C17#911811yescharacteristic, complemented, Sylow
289C24—341no—
290C24C2×C22⋊D1711yescharacteristic
291C2×D4—342no—
292C2×D4—681no—
293C2×D4—681no—
294C2×D4—681no—
295C2×D4—682no—
296C2×D4—681no—
297C2×D4—682no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—341no—
315C22×C4—341no—
316C22×C4—341no—
317C22×C4#6322711yescharacteristic
318C22×C4#6322711yescharacteristic
319C22×C4#6322711yescharacteristic
320C22×C4#6322711yescharacteristic
321C22×C4C2×C4×D1711yescharacteristic
322C22×C4C2×D6811yescharacteristic
323C22×C4D4×D1712yes—
324C22×C4#6325412yes—
325C2×C8—344no—
326C2×C8—682no—
327C2×C8—682no—
328C4⋊C4—342no—
329C4⋊C4—681no—
330C4⋊C4—681no—
331C4⋊C4—682no—
332C4⋊C4—682no—
333C22⋊C4—682no—
334C22⋊C4—682no—
335C22⋊C4—682no—
336C22⋊C4—682no—
337C22⋊C4—682no—
338C22⋊C4—681no—
339C22⋊C4—681no—
340C42—22no—
341C42—22no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—681no—
351C23—681no—
352C23—681no—
353C23—681no—
354C23—681no—
355C23—681no—
356C23—681no—
357C23C2×#6322711yescharacteristic
358C23#1058761811yescharacteristic
359C23#1058762512yes—
360C23#1058762811yescharacteristic
361C23#1058763512yes—
362D4—1362no—
363D4—1362no—
364D4—1362no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—681no—
393C2×C4—681no—
394C2×C4—681no—
395C2×C4—681no—
396C2×C4—681no—
397C2×C4—681no—
398C2×C4—681no—
399C2×C4—681no—
400C2×C4—681no—
401C2×C4#1058747612yes—
402C2×C4#1058756712yes—
403C2×C4#1058756811yescharacteristic
404C2×C4#1058757211yescharacteristic
405C8—684no—
406C22—22no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—1361no—
415C22—1361no—
416C22—1361no—
417C22—1361no—
418C22—1361no—
419C22—1361no—
420C22—1361no—
421C22#1176268611yescharacteristic
422C22#1176278211yescharacteristic
423C22#1176278211yescharacteristic
424C22#1176280811yescharacteristic
425C22#1176281311yescharacteristic
426C22C2×#1058747611yescharacteristic
427C22#1176282711yescharacteristic
428C22#1176305712yes—
429C22#1176306012yes—
430C4—22no—
431C4—21no—
432C4—22no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—84no—
438C4—1361no—
439C4—1361no—
440C4—1361no—
441C2—22no—
442C2—22no—
443C2—2721no—
444C2#1452729111yescharacteristic
445C2#1452730711yescharacteristic
446C2#1452732011yescharacteristic
447C2#1452733011yescharacteristic
448C2#1452908411yescharacteristic
449C2#1452922011yescharacteristic
450C2#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#9118
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×#11762866C2semidirect
4. #14526663C2semidirect
5. C17×#1434C2semidirect
12. #11762711C4semidirect
13. #11762711C4semidirect
14. #11762707C4semidirect
15. #11762707C4semidirect
184. #1434D17semidirect
288. C17#9118semidirect

Maximal subgroups

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

subgroupindex
C2×#117628662
#145371152
C2×#117625422
#145266632
C17×#14342
#911817