Finite Group Databasebeta

← #16930778

Subgroups — #16930778

Order 4352.

594 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763122C211yesmaximal, characteristic
2#14529620C211yesmaximal, characteristic
3#14528870C211yesmaximal, characteristic, complemented
4#14527351C211yesmaximal, characteristic
5#14526651C211yesmaximal, characteristic, complemented
6#14526632C211yesmaximal, characteristic, complemented
7C17×#2344C211yesmaximal, characteristic, complemented
8C2×#10587616C2211yescharacteristic
9#11763122—22no—
10#11763122—21no—
11#11763122—21no—
12C2×#10587478C412yescomplemented
13C2×#10587480C412yescomplemented
14C2×#10587477—21no—
15C2×#10587477—21no—
16C2×C22⋊#3622C2211yescharacteristic
17C2×C22⋊#3622C2211yescharacteristic
18#11762669C2211yescharacteristic
19#11762571—22no—
20#11762542—22no—
21C2×C17×#117—21no—
22C22×C4⋊C4×C17C2211yescharacteristic
23C17×#389—21no—
24C17×#387—22no—
25C17×#381—21no—
26C17×#380C2211yescharacteristic
27C4×C4⋊C4×C17—21no—
28C4×C22⋊C4×C17C2211yescharacteristic
29C23×C4×C17—21no—
30C23×C4×C17C2311yescharacteristic
31C22×C22⋊D17—21no—
32C22×D68C2×C411yescharacteristic
33C17×#117—42no—
34C17×#117—42no—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17—22no—
39C2×C4⋊C4×C17—21no—
40C2×C4⋊C4×C17C2×C412yes—
41C2×C4⋊C4×C17D412yes—
42C2×C4⋊C4×C17D411yescharacteristic
43C2×C4⋊C4×C17D411yescharacteristic
44C2×C22⋊C4×C17—21no—
45C2×C22⋊C4×C17D411yescharacteristic
46C2×C22⋊C4×C17D411yescharacteristic
47C2×C42×C17—22no—
48C2×C42×C17—21no—
49C2×C22⋊Dic17—21no—
50#10587616—21no—
51#10587616—21no—
52#10587616—21no—
53#10587616—21no—
54C2×#63227—21no—
55C2×C4⋊Dic17—21no—
56C2×C4⋊Dic17C2×C411yescharacteristic
57C22×#3622—21no—
58C22×#3622C2×C412yes—
59C17×#95—22no—
60C17×#95—22no—
61C17×#95—21no—
62C17×#95D412yes—
63#10587502—21no—
64C22⋊#3622—21no—
65C22⋊#3622—21no—
66C22⋊#3622—21no—
67C22⋊#3622—21no—
68#10587480—22no—
69#10587480—22no—
70#10587478—22no—
71#10587478—22no—
72#10587477—42no—
73#10587477—42no—
74C24×C17C2×D411yescharacteristic
75C23×D17—21no—
76C22×C4×C17—22no—
77C22×C4×C17—21no—
78C22×C4×C17—21no—
79C22×C4×C17—21no—
80C22×C4×C17—22no—
81C22×C4×C17—21no—
82C22×C4×C17—22no—
83C22×C4×C17—42no—
84C22×C4×C17—41no—
85C22×C4×C17—41no—
86C22×C4×C17—41no—
87C22×C4×C17—41no—
88C22×C4×C17C22×C411yescharacteristic
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17#4111yescharacteristic
93C22×C4×C17#4111yescharacteristic
94C2×C22⋊D17—41no—
95C2×C22⋊D17—41no—
96C2×C22⋊D17—41no—
97C2×C22⋊D17—41no—
98C2×C22⋊D17—41no—
99C2×C22⋊D17—41no—
100C22×Dic17—21no—
101C22×Dic17—41no—
102C2×D68—21no—
103C2×D68—21no—
104C2×D68—42no—
105C4⋊C4×C17—22no—
106C4⋊C4×C17—22no—
107C4⋊C4×C17—21no—
108C4⋊C4×C17—21no—
109C4⋊C4×C17—42no—
110C4⋊C4×C17—44no—
111C4⋊C4×C17C22⋊C412yes—
112C4⋊C4×C17C22⋊C412yes—
113C22⋊C4×C17—21no—
114C22⋊C4×C17—21no—
115C22⋊C4×C17—21no—
116C22⋊C4×C17—41no—
117C22⋊C4×C17—41no—
118C22⋊C4×C17D812yes—
119C42×C17—42no—
120C42×C17—44no—
121C22⋊Dic17—41no—
122C22⋊Dic17—41no—
123#63227—41no—
124#63227—41no—
125C4⋊Dic17—21no—
126C4⋊Dic17—21no—
127C4⋊Dic17—42no—
128C2×#3622—22no—
129C2×#3622—42no—
130C2×#3622—42no—
131C2×#3622—41no—
132#9260—171nomaximal, Sylow
133C23×C17—41no—
134C23×C17—41no—
135C23×C17C4×D411yescharacteristic
136C23×C17#12011yescharacteristic
137C23×C17#12111yescharacteristic
138C23×C17C2×D811yescharacteristic
139C23×C17#13511yescharacteristic
140C23×C17AGL(1,ℤ/8)11yescharacteristic
141C23×C17AGL(1,ℤ/8)11yescharacteristic
142C22×D17—41no—
143C22×D17—41no—
144C22×D17—41no—
145C22×D17—81no—
146C22×D17—81no—
147C2×C4×C17—22no—
148C2×C4×C17—22no—
149C2×C4×C17—22no—
150C2×C4×C17—22no—
151C2×C4×C17—22no—
152C2×C4×C17—22no—
153C2×C4×C17—41no—
154C2×C4×C17—41no—
155C2×C4×C17—41no—
156C2×C4×C17—41no—
157C2×C4×C17—41no—
158C2×C4×C17—42no—
159C2×C4×C17—42no—
160C2×C4×C17—41no—
161C2×C4×C17—41no—
162C2×C4×C17—41no—
163C2×C4×C17—41no—
164C2×C4×C17—42no—
165C2×C4×C17—42no—
166C2×C4×C17—41no—
167C2×C4×C17—42no—
168C2×C4×C17—41no—
169C2×C4×C17—42no—
170C2×C4×C17—42no—
171C2×C4×C17—41no—
172C2×C4×C17C2×C22⋊C411yescharacteristic
173C2×C4×C17C4×D411yescharacteristic
174C2×C4×C17#12111yescharacteristic
175C2×C4×C17#12311yescharacteristic
176C22⋊D17—81no—
177C22⋊D17—81no—
178C22⋊D17—81no—
179C22⋊D17—81no—
180C2×Dic17—41no—
181C2×Dic17—41no—
182C2×Dic17—41no—
183C2×Dic17—81no—
184C2×Dic17—81no—
185C2×Dic17—81no—
186D68—42no—
187D68—81no—
188#3622—42no—
189#3622—81no—
190C2×#480—171no—
191#2344D1711yescharacteristic, complemented
192#1964—171no—
193#1823—171no—
194#1519—171no—
195#1289—171no—
196#1282—171no—
197C22×C17—21no—
198C22×C17—21no—
199C22×C17—21no—
200C22×C17—21no—
201C22×C17—21no—
202C22×C17—21no—
203C22×C17—41no—
204C22×C17—41no—
205C22×C17—41no—
206C22×C17—41no—
207C22×C17#38611yescharacteristic
208C22×C17C4×D811yescharacteristic
209C22×C17#44211yescharacteristic
210C22×C17#44711yescharacteristic
211C22×C17#44911yescharacteristic
212C22×C17#45911yescharacteristic
213C22×C17#46311yescharacteristic
214C2×D17—81no—
215C2×D17—81no—
216C2×D17—81no—
217C2×D17—161no—
218C2×D17—161no—
219C4×C17—21no—
220C4×C17—21no—
221C4×C17—21no—
222C4×C17—42no—
223C4×C17—42no—
224C4×C17—42no—
225C4×C17—82no—
226C4×C17—81no—
227C4×C17#41012yes—
228Dic17—81no—
229Dic17—161no—
230C2×#121—171no—
231C2×#117—21no—
232C22×C4⋊C4C2×D1711yescharacteristic
233#480—342no—
234#480—341no—
235#480—341no—
236C2×#107—172no—
237C2×#102—172no—
238C2×#99—341no—
239C2×#99—341no—
240C2×C22⋊C8—171no—
241C2×C22⋊C8—171no—
242#389—21no—
243#387—22no—
244#381—21no—
245#380—171no—
246#380C2×D1711yescharacteristic
247C4×C4⋊C4—21no—
248C4×C22⋊C4C2×D1711yescharacteristic
249#340—342no—
250#327—342no—
251C2×C17—21no—
252C2×C17—21no—
253C2×C17—21no—
254C2×C17—21no—
255C2×C17—21no—
256C2×C17—21no—
257C2×C17#187911yescharacteristic
258C2×C17#189911yescharacteristic
259C2×C17#190511yescharacteristic
260D17—161no—
261C22×D4—171no—
262C22×D4—341no—
263C23×C4—21no—
264C23×C4C22×D1711yescharacteristic
265C22×C8—172no—
266C22×C8—341no—
267#121—341no—
268#121—341no—
269#121—341no—
270#121—341no—
271#117—42no—
272#117—42no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—171no—
279C2×C4⋊C4—341no—
280C2×C4⋊C4C4×D1712yes—
281C2×C4⋊C4D6812yes—
282C2×C4⋊C4C22⋊D1711yescharacteristic
283C2×C4⋊C4C22⋊D1711yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—341no—
286C2×C22⋊C4—341no—
287C2×C22⋊C4C22⋊D1711yescharacteristic
288C2×C22⋊C4C22⋊D1711yescharacteristic
289C2×C42—22no—
290C2×C42—21no—
291#107—342no—
292#107—342no—
293#102—342no—
294#102—342no—
295#99—682no—
296#99—682no—
297C22⋊C8—341no—
298C22⋊C8—341no—
299C22⋊C8—341no—
300C22⋊C8—341no—
301#95—22no—
302#95—22no—
303#95—21no—
304#95—341no—
305#95C22⋊D1712yes—
306C17#926011yescharacteristic, complemented, Sylow
307C24—341no—
308C24C2×C22⋊D1711yescharacteristic
309C2×D4—341no—
310C2×D4—341no—
311C2×D4—682no—
312C2×D4—681no—
313C2×D4—681no—
314C2×D4—681no—
315C2×D4—681no—
316C2×D4—681no—
317C2×D4—681no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—341no—
331C22×C4—681no—
332C22×C4C2×C4×D1711yescharacteristic
333C22×C4C2×D6811yescharacteristic
334C22×C4#6325211yescharacteristic
335C22×C4#6325211yescharacteristic
336C22×C4C2×C22⋊D1711yescharacteristic
337C22×C4C2×C22⋊D1711yescharacteristic
338C2×C8—342no—
339C2×C8—682no—
340C2×C8—682no—
341C2×C8—681no—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—42no—
347C4⋊C4—44no—
348C4⋊C4—341no—
349C4⋊C4—341no—
350C4⋊C4—682no—
351C4⋊C4#6322712yes—
352C4⋊C4#6322712yes—
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—21no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—681no—
359C22⋊C4—681no—
360C22⋊C4—681no—
361C22⋊C4—681no—
362C22⋊C4#6322812yes—
363C42—42no—
364C42—44no—
365C23—41no—
366C23—41no—
367C23—681no—
368C23—681no—
369C23—681no—
370C23—1361no—
371C23—1361no—
372C23C4×C22⋊D1711yescharacteristic
373C23#1058761611yescharacteristic
374C23C2×#6322811yescharacteristic
375C23#1058761811yescharacteristic
376C23#1058763511yescharacteristic
377C23#1058763611yescharacteristic
378C23#1058763911yescharacteristic
379D4—682no—
380D4—1361no—
381D4—1361no—
382D4—1361no—
383D4—1361no—
384D4—1361no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—681no—
411C2×C4—681no—
412C2×C4—681no—
413C2×C4—1361no—
414C2×C4—1361no—
415C2×C4—1361no—
416C2×C4C2×#6322711yescharacteristic
417C2×C4C4×C22⋊D1711yescharacteristic
418C2×C4#1058761511yescharacteristic
419C2×C4#1058761611yescharacteristic
420C8—682no—
421C8—1361no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—2721no—
433C22—2721no—
434C22—1361no—
435C22—1361no—
436C22—1361no—
437C22#1176268811yescharacteristic
438C22C4×#6322811yescharacteristic
439C22#1176291911yescharacteristic
440C22#1176294311yescharacteristic
441C22#1176295411yescharacteristic
442C22#1176300411yescharacteristic
443C22#1176302411yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—82no—
451C4—81no—
452C4—2721no—
453C4—1361no—
454C4#1176278212yes—
455C2—21no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—2721no—
462C2#1452913211yescharacteristic
463C2#1452924211yescharacteristic
464C2#1452927811yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2344C2
socleC22×C17#386
prime radicalC2×#10587616C22
cosocleC23×C4×C17C23
hypercenter#2344D17
hypocenterC17#9260
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2344

Complements

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

NH ≅ G/Ntype
3. #14528870C2semidirect
5. #14526651C2semidirect
6. #14526632C2semidirect
7. C17×#2344C2semidirect
12. C2×#10587478C4semidirect
13. C2×#10587480C4semidirect
191. #2344D17semidirect
306. C17#9260semidirect

Maximal subgroups

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

subgroupindex
C2×#117631222
#145296202
#145288702
#145273512
#145266512
#145266322
C17×#23442
#926017