Finite Group Databasebeta

← #16937473

Subgroups — #16937473

Order 4352.

508 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 97 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#14533790C211yesmaximal, characteristic, complemented
2#14529620C211yesmaximal, characteristic
3#14527353C211yesmaximal, characteristic
4C2×#11762544C211yesmaximal, characteristic
5#14526664C211yesmaximal, characteristic, complemented
6#14526652C211yesmaximal, characteristic, complemented
7C17×#1282C211yesmaximal, characteristic, complemented
8C2×#10587564—21no—
9C2×#10587616C2211yescharacteristic
10C2×#10587477—21no—
11C2×#10587477—21no—
12#11762741—21no—
13#11762730—21no—
14#11762715—21no—
15#11762707—21no—
16#11762684—21no—
17#11762669C2211yescharacteristic
18#11762669C2211yescharacteristic
19C2×#10587502C2211yescharacteristic
20#11762544—21no—
21#11762544—21no—
22#11762544—21no—
23#11762544—21no—
24C2×C17×C22⋊C8C2211yescharacteristic
25C2×C17×C22⋊C8C2211yescharacteristic
26C4×C22⋊C4×C17C2211yescharacteristic
27C23×C4×C17C2311yescharacteristic
28C22×C22⋊D17—21no—
29C23×Dic17—21no—
30C22×D68Q811yescharacteristic, complemented
31C22×Dic34—21no—
32C22×C8×C17—21no—
33C22×C8×C17—21no—
34C2×C22⋊C4×C17D411yescharacteristic
35C2×C22⋊C4×C17D411yescharacteristic
36C2×C42×C17—21no—
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—21no—
39#10587616—21no—
40#10587616—21no—
41#10587616—21no—
42#10587616—21no—
43C2×#63227—21no—
44C2×C4⋊Dic17—21no—
45C2×C4⋊Dic17—21no—
46C2×C4⋊Dic17—21no—
47C2×C4⋊Dic17—21no—
48C2×C4⋊Dic17—21no—
49C2×C4⋊Dic17Q811yescharacteristic, complemented
50C2×#63225—21no—
51C2×#63225—21no—
52#10587564—42no—
53#10587564—42no—
54C17×C22⋊C8—21no—
55C17×C22⋊C8—21no—
56C17×C22⋊C8—21no—
57C17×C22⋊C8—21no—
58C17×#95—21no—
59#10587502—21no—
60#10587502—21no—
61#10587502—21no—
62#10587502—21no—
63#10587502D411yescharacteristic
64#10587502D411yescharacteristic
65#10587502D411yescharacteristic
66#10587502D411yescharacteristic
67#10587477—42no—
68#10587477—42no—
69C24×C17C2×D411yescharacteristic
70C23×D17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—81no—
74C22×C4×C17C2×D411yescharacteristic
75C22×C4×C17C2×D411yescharacteristic
76C22×C4×C17C2×Q811yescharacteristic
77C22×C4×C17#4111yescharacteristic
78C22×C4×C17#4111yescharacteristic
79C22×C4×C17#4111yescharacteristic
80C2×C22⋊D17—41no—
81C2×C22⋊D17—41no—
82C2×C22⋊D17—41no—
83C2×C22⋊D17—41no—
84C2×C22⋊D17—41no—
85C2×C22⋊D17—41no—
86C22×Dic17—21no—
87C22×Dic17—21no—
88C22×Dic17—21no—
89C22×Dic17—21no—
90C22×Dic17—21no—
91C22×Dic17—42no—
92C22×Dic17—41no—
93C22×Dic17—41no—
94C22×Dic17—41no—
95C22×Dic17—81no—
96C2×D68—21no—
97C2×D68—21no—
98C2×D68—81no—
99C2×Dic34—42no—
100C2×Dic34—42no—
101C2×Dic34—81no—
102C2×C8×C17—41no—
103C2×C8×C17—41no—
104C2×C8×C17—81no—
105C2×C8×C17—81no—
106C22⋊C4×C17—21no—
107C22⋊C4×C17—21no—
108C22⋊C4×C17D812yes—
109C22⋊C4×C17SD1612yes—
110C42×C17—81no—
111C22⋊Dic17—41no—
112C22⋊Dic17—41no—
113C22⋊Dic17—41no—
114C22⋊Dic17—41no—
115#63227—41no—
116#63227—41no—
117C4⋊Dic17—21no—
118C4⋊Dic17—21no—
119C4⋊Dic17—42no—
120C4⋊Dic17—42no—
121C4⋊Dic17—81no—
122C4⋊Dic17—81no—
123C4⋊Dic17—81no—
124#63225—42no—
125#63225—42no—
126#10374—171nomaximal, Sylow
127C23×C17—41no—
128C23×C17—41no—
129C23×C17#12011yescharacteristic
130C23×C17C22⋊Q811yescharacteristic
131C23×C17#12411yescharacteristic
132C23×C17C2×D811yescharacteristic
133C23×C17C2×SD1611yescharacteristic
134C23×C17AGL(1,ℤ/8)11yescharacteristic
135C23×C17#13711yescharacteristic
136C22×D17—41no—
137C22×D17—41no—
138C22×D17—41no—
139C22×D17—81no—
140C22×D17—81no—
141C2×C4×C17—22no—
142C2×C4×C17—22no—
143C2×C4×C17—41no—
144C2×C4×C17—41no—
145C2×C4×C17—41no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—41no—
151C2×C4×C17—41no—
152C2×C4×C17—41no—
153C2×C4×C17C22⋊Q811yescharacteristic
154C2×C4×C17C22⋊Q811yescharacteristic
155C2×C4×C17#12411yescharacteristic
156C2×C4×C17#12411yescharacteristic
157C22⋊D17—81no—
158C22⋊D17—81no—
159C22⋊D17—81no—
160C22⋊D17—81no—
161C2×Dic17—22no—
162C2×Dic17—41no—
163C2×Dic17—41no—
164C2×Dic17—41no—
165C2×Dic17—41no—
166C2×Dic17—41no—
167C2×Dic17—41no—
168C2×Dic17—41no—
169C2×Dic17—41no—
170C2×Dic17—41no—
171C2×Dic17—41no—
172C2×Dic17—41no—
173C2×Dic17—41no—
174C2×Dic17—41no—
175C2×Dic17—41no—
176C2×Dic17—82no—
177C2×Dic17—82no—
178C2×Dic17—81no—
179C2×Dic17—81no—
180C2×Dic17—81no—
181D68—81no—
182D68—81no—
183Dic34—84no—
184C8×C17—81no—
185C8×C17—81no—
186#2549—171no—
187#1964—171no—
188#1519—171no—
189C2×#327—171no—
190#1294—171no—
191#1289—171no—
192#1282D1711yescharacteristic, complemented
193C22×C17—21no—
194C22×C17—21no—
195C22×C17—21no—
196C22×C17—21no—
197C22×C17—21no—
198C22×C17—21no—
199C22×C17—41no—
200C22×C17—41no—
201C22×C17—41no—
202C22×C17—41no—
203C22×C17#39311yescharacteristic
204C22×C17#44711yescharacteristic
205C22×C17#44811yescharacteristic
206C22×C17#47411yescharacteristic
207C22×C17#47611yescharacteristic
208C22×C17#48611yescharacteristic
209C22×C17#48911yescharacteristic
210C2×D17—81no—
211C2×D17—81no—
212C2×D17—81no—
213C2×D17—161no—
214C2×D17—161no—
215C4×C17—21no—
216C4×C17—21no—
217C4×C17—21no—
218C4×C17—21no—
219C4×C17—42no—
220C4×C17—42no—
221Dic17—42no—
222Dic17—81no—
223Dic17—81no—
224Dic17—81no—
225Dic17—81no—
226Dic17—161no—
227C2×C22⋊Q8—341no—
228C2×#121—171no—
229C2×#99—341no—
230C2×#99—341no—
231C2×C22⋊C8C2×D1711yescharacteristic
232C2×C22⋊C8C2×D1711yescharacteristic
233#398—341no—
234#395—341no—
235#391—341no—
236#389—341no—
237#385—341no—
238#380—171no—
239#380—171no—
240C4×C22⋊C4C2×D1711yescharacteristic
241C2×#95—171no—
242#327—341no—
243#327—341no—
244#327—341no—
245#327—341no—
246C2×C17—21no—
247C2×C17—21no—
248C2×C17—21no—
249C2×C17—21no—
250C2×C17—21no—
251C2×C17—21no—
252C2×C17#202311yescharacteristic
253C2×C17#203111yescharacteristic
254C2×C17#203211yescharacteristic
255D17—161no—
256C22×Q8—341no—
257C22×D4—171no—
258C22×D4—341no—
259C23×C4—341no—
260C23×C4C22×D1711yescharacteristic
261C22×C8—21no—
262C22×C8—21no—
263C22⋊Q8—682no—
264C22⋊Q8—682no—
265#121—341no—
266#121—341no—
267#121—341no—
268#121—341no—
269C2×C4⋊C4—171no—
270C2×C4⋊C4—341no—
271C2×C4⋊C4—341no—
272C2×C4⋊C4—341no—
273C2×C4⋊C4—341no—
274C2×C4⋊C4—341no—
275C2×C4⋊C4—341no—
276C2×C4⋊C4—341no—
277C2×C22⋊C4—341no—
278C2×C22⋊C4—341no—
279C2×C22⋊C4—341no—
280C2×C22⋊C4D6811yescharacteristic
281C2×C22⋊C4D6811yescharacteristic
282C2×C42—21no—
283#99—682no—
284#99—682no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289#95—21no—
290#95—171no—
291#95—171no—
292#95—171no—
293#95—171no—
294#95—341no—
295#95—341no—
296#95—341no—
297#95—341no—
298C17#1037411yescharacteristic, complemented, Sylow
299C24—341no—
300C24C2×D6811yescharacteristic
301C2×Q8—682no—
302C2×Q8—682no—
303C2×Q8—1361no—
304C2×D4—341no—
305C2×D4—341no—
306C2×D4—681no—
307C2×D4—681no—
308C2×D4—681no—
309C2×D4—681no—
310C2×D4—681no—
311C2×D4—681no—
312C2×D4—1361no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—81no—
316C22×C4—341no—
317C22×C4—341no—
318C22×C4—341no—
319C22×C4—341no—
320C22×C4—341no—
321C22×C4—682no—
322C22×C4—681no—
323C22×C4—681no—
324C22×C4—681no—
325C22×C4—1361no—
326C22×C4#6325211yescharacteristic
327C22×C4#6325211yescharacteristic
328C22×C4D4×D1711yescharacteristic
329C22×C4D4×D1711yescharacteristic
330C22×C4#6325411yescharacteristic
331C22×C4Q8×D1711yescharacteristic
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—81no—
335C2×C8—81no—
336C4⋊C4—341no—
337C4⋊C4—341no—
338C4⋊C4—682no—
339C4⋊C4—682no—
340C4⋊C4—682no—
341C4⋊C4—682no—
342C4⋊C4—1361no—
343C4⋊C4—1361no—
344C4⋊C4—1361no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—681no—
348C22⋊C4—681no—
349C22⋊C4—681no—
350C22⋊C4—681no—
351C22⋊C4—681no—
352C22⋊C4—681no—
353C22⋊C4#6321912yes—
354C22⋊C4D13612yes—
355C42—81no—
356C23—41no—
357C23—41no—
358C23—681no—
359C23—681no—
360C23—681no—
361C23—1361no—
362C23—1361no—
363C23#1058756111yescharacteristic
364C23#1058756811yescharacteristic
365C23#1058758311yescharacteristic
366C23C2×#6321911yescharacteristic
367C23C2×D13611yescharacteristic
368C23#1058759411yescharacteristic
369C23#1058759511yescharacteristic
370Q8—1364no—
371D4—1361no—
372D4—1361no—
373D4—1361no—
374D4—1361no—
375D4—1361no—
376D4—1361no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—342no—
390C2×C4—681no—
391C2×C4—681no—
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—681no—
402C2×C4—681no—
403C2×C4—681no—
404C2×C4—1361no—
405C2×C4—1361no—
406C2×C4—1361no—
407C2×C4—1362no—
408C2×C4—1362no—
409C2×C4#1058757111yescharacteristic
410C2×C4#1058757111yescharacteristic
411C2×C4#1058758211yescharacteristic
412C2×C4#1058758211yescharacteristic
413C8—81no—
414C8—81no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—2721no—
426C22—2721no—
427C22—1361no—
428C22—1361no—
429C22—1361no—
430C22#1176272211yescharacteristic
431C22#1176294511yescharacteristic
432C22#1176295011yescharacteristic
433C22#1176309011yescharacteristic
434C22#1176310011yescharacteristic
435C22#1176315611yescharacteristic
436C22#1176316911yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—42no—
443C4—2721no—
444C4—682no—
445C4—1361no—
446C4—1361no—
447C4—1361no—
448C4—1361no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—2721no—
456C2#1452992111yescharacteristic
457C2#1452995911yescharacteristic
458C2#1452996211yescharacteristic
subgroupG/H
centerC22#11762722
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#1282C2
socleC22×C17#393
prime radicalC2×#10587616C22
hypercenter#1282D17
hypocenterC17#10374
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
1. #14533790C2semidirect
5. #14526664C2semidirect
6. #14526652C2semidirect
7. C17×#1282C2semidirect
30. C22×D68Q8semidirect
49. C2×C4⋊Dic17Q8semidirect
192. #1282D17semidirect
298. C17#10374semidirect

Maximal subgroups

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

subgroupindex
#145337902
#145296202
#145273532
C2×#117625442
#145266642
#145266522
C17×#12822
#1037417