Finite Group Databasebeta

← #16930389

Subgroups — #16930389

Order 4352.

622 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 113 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#14534528C211yesmaximal, characteristic
2#14531401C211yesmaximal, characteristic, complemented
3#14527352C211yesmaximal, characteristic
4C2×#11762534C211yesmaximal, characteristic, complemented
5#14526652C211yesmaximal, characteristic, complemented
6#14526651C211yesmaximal, characteristic, complemented
7C17×#1519C211yesmaximal, characteristic
8C2×#10587621—21no—
9C2×#10587563—21no—
10C22×#63225C2211yescharacteristic
11C2×C22⋊#3622C2211yescharacteristic
12#11762740C412yescomplemented
13#11762732—22no—
14#11762723C412yescomplemented
15#11762706—22no—
16#11762704—22no—
17#11762689—21no—
18#11762669C2211yescharacteristic
19#11762669C2211yescharacteristic
20#11762667—21no—
21#11762543—22no—
22#11762534—22no—
23C2×C17×#121C2211yescharacteristic
24C2×C17×C22⋊C8C2211yescharacteristic
25C17×#380C2211yescharacteristic
26C17×#327—22no—
27C22×D4×C17—21no—
28C22×D4×C17C2×C411yescharacteristic
29C23×C4×C17C2311yescharacteristic
30C23×Dic17D411yescharacteristic
31C23×Dic17D411yescharacteristic
32C22×C8×C17—21no—
33C17×#121—21no—
34C17×#121—21no—
35C17×#121—21no—
36C17×#121—21no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17C2×C411yescharacteristic
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17—21no—
41C2×C22⋊Dic17—21no—
42C2×C22⋊Dic17—21no—
43C2×C22⋊Dic17—22no—
44C2×C22⋊Dic17—22no—
45#10587621—42no—
46#10587621—42no—
47C2×C4⋊Dic17—21no—
48C2×C4⋊Dic17—21no—
49C2×#63225—22no—
50C2×#63225—22no—
51C2×#63225—21no—
52C2×#63225—21no—
53C2×#63225—21no—
54C2×#63225—21no—
55C2×#63225—22no—
56C2×#63225C2×C412yes—
57C2×#63225D412yes—
58C2×C4×Dic17—22no—
59C22×#3622—21no—
60#10587563—42no—
61#10587563—42no—
62C17×C22⋊C8—21no—
63C17×C22⋊C8—21no—
64C17×#95D412yes—
65#10587502—22no—
66#10587502—22no—
67#10587502C2×C412yes—
68#10587502D412yes—
69C22⋊#3622—21no—
70C22⋊#3622—21no—
71C24×C17—21no—
72C24×C17C2×D411yescharacteristic
73C2×D4×C17—21no—
74C2×D4×C17—21no—
75C2×D4×C17—41no—
76C2×D4×C17—41no—
77C2×D4×C17—41no—
78C2×D4×C17—41no—
79C2×D4×C17—41no—
80C2×D4×C17—41no—
81C2×D4×C17—81no—
82C22×C4×C17—21no—
83C22×C4×C17—22no—
84C22×C4×C17—81no—
85C22×C4×C17C22×C411yescharacteristic
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17C2×D411yescharacteristic
88C22×C4×C17C2×D411yescharacteristic
89C22×C4×C17#4111yescharacteristic
90C22×C4×C17#4111yescharacteristic
91C22×Dic17—21no—
92C22×Dic17—21no—
93C22×Dic17—21no—
94C22×Dic17—21no—
95C22×Dic17—21no—
96C22×Dic17—21no—
97C22×Dic17—22no—
98C22×Dic17—22no—
99C22×Dic17—81no—
100C22×Dic17C22⋊C412yes—
101C22×Dic17C22⋊C412yes—
102C22×Dic17D812yes—
103C22×Dic17SD1612yes—
104C2×C8×C17—41no—
105C2×C8×C17—81no—
106C4⋊C4×C17—21no—
107C4⋊C4×C17—21no—
108C4⋊C4×C17—81no—
109C22⋊C4×C17—41no—
110C22⋊C4×C17—41no—
111C22⋊C4×C17—41no—
112C22⋊C4×C17—41no—
113C22⋊Dic17—41no—
114C22⋊Dic17—41no—
115C22⋊Dic17—41no—
116C22⋊Dic17—41no—
117C22⋊Dic17—44no—
118C22⋊Dic17—82no—
119C4⋊Dic17—81no—
120C4⋊Dic17—81no—
121#63225—22no—
122#63225—22no—
123#63225—22no—
124#63225—22no—
125#63225—44no—
126#63225—44no—
127#63225—82no—
128C4×Dic17—44no—
129C2×#3622—41no—
130C2×#3622—81no—
131#9204—171nomaximal, Sylow
132C23×C17—41no—
133C23×C17—41no—
134C23×C17—42no—
135C23×C17—42no—
136C23×C17—41no—
137C23×C17—41no—
138C23×C17—41no—
139C23×C17C2×C22⋊C411yescharacteristic
140C23×C17#12011yescharacteristic
141C23×C17#12311yescharacteristic
142C23×C17C2×D811yescharacteristic
143C23×C17C2×SD1611yescharacteristic
144C23×C17AGL(1,ℤ/8)11yescharacteristic
145C23×C17#13711yescharacteristic
146D4×C17—81no—
147D4×C17—81no—
148D4×C17—81no—
149D4×C17—81no—
150D4×C17—81no—
151D4×C17—81no—
152C2×C4×C17—41no—
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—42no—
164C2×C4×C17C4×D411yescharacteristic
165C2×C4×C17C4×D411yescharacteristic
166C2×C4×C17#12111yescharacteristic
167C2×C4×C17#12111yescharacteristic
168C2×Dic17—22no—
169C2×Dic17—22no—
170C2×Dic17—22no—
171C2×Dic17—22no—
172C2×Dic17—22no—
173C2×Dic17—22no—
174C2×Dic17—22no—
175C2×Dic17—42no—
176C2×Dic17—42no—
177C2×Dic17—42no—
178C2×Dic17—42no—
179C2×Dic17—42no—
180C2×Dic17—42no—
181C2×Dic17—42no—
182C2×Dic17—42no—
183C2×Dic17—81no—
184C2×Dic17#10214yes—
185C8×C17—81no—
186#3622—81no—
187#2618—171no—
188#2316—171no—
189#1519—171no—
190#1519D1711yescharacteristic
191C2×#324—171no—
192#1289—171no—
193#1289—171no—
194C22×C17—21no—
195C22×C17—21no—
196C22×C17—21no—
197C22×C17—21no—
198C22×C17—21no—
199C22×C17—21no—
200C22×C17—41no—
201C22×C17—41no—
202C22×C17—41no—
203C22×C17—41no—
204C22×C17—82no—
205C22×C17—82no—
206C22×C17—81no—
207C22×C17—81no—
208C22×C17—81no—
209C22×C17#38611yescharacteristic
210C22×C17C2×#10211yescharacteristic
211C22×C17#41711yescharacteristic
212C22×C17#44711yescharacteristic
213C22×C17#44811yescharacteristic
214C22×C17#48011yescharacteristic
215C22×C17#48111yescharacteristic
216C4×C17—21no—
217C4×C17—21no—
218C4×C17—21no—
219C4×C17—21no—
220C4×C17—82no—
221C4×C17—81no—
222Dic17—24no—
223Dic17—42no—
224Dic17—82no—
225Dic17—82no—
226C2×#123—341no—
227C2×#121C2×D1711yescharacteristic
228C2×#117—341no—
229C22×C4⋊C4—171no—
230C2×C22⋊C8—171no—
231C2×C22⋊C8C2×D1711yescharacteristic
232#398—172no—
233#396—342no—
234#394—172no—
235#389—342no—
236#389—342no—
237#386—341no—
238#380—171no—
239#380—171no—
240#380—341no—
241#380C2×D1711yescharacteristic
242#327—22no—
243#327—342no—
244#324—342no—
245C2×C17—21no—
246C2×C17—21no—
247C2×C17—21no—
248C2×C17—21no—
249C2×C17—21no—
250C2×C17—21no—
251C2×C17—161no—
252C2×C17#187411yescharacteristic
253C2×C17#191211yescharacteristic
254C2×C17#191311yescharacteristic
255C22×D4—21no—
256C22×D4C4×D1711yescharacteristic
257C23×C4—171no—
258C23×C4—171no—
259C23×C4C22×D1711yescharacteristic
260C22×C8—21no—
261C22×C8—341no—
262#123—682no—
263#123—682no—
264#121—21no—
265#121—21no—
266#121—21no—
267#121—21no—
268#117—682no—
269#117—682no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—172no—
272C2×C4⋊C4—172no—
273C2×C4⋊C4—341no—
274C2×C4⋊C4—341no—
275C2×C4⋊C4—341no—
276C2×C4⋊C4—341no—
277C2×C4⋊C4—342no—
278C2×C4⋊C4—342no—
279C2×C4⋊C4—342no—
280C2×C4⋊C4—341no—
281C2×C4⋊C4—341no—
282C2×C4⋊C4C4×D1711yescharacteristic
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—342no—
286C2×C22⋊C4—341no—
287C2×C22⋊C4—341no—
288C2×C22⋊C4—342no—
289C2×C42—342no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—341no—
293C22⋊C8—341no—
294#95—172no—
295#95—172no—
296#95—342no—
297#95—342no—
298#95C22⋊D1712yes—
299C17#920411yescharacteristic, complemented, Sylow
300C24—21no—
301C24D4×D1711yescharacteristic
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—81no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—81no—
314C22×C4—172no—
315C22×C4—172no—
316C22×C4—172no—
317C22×C4—172no—
318C22×C4—341no—
319C22×C4—341no—
320C22×C4—341no—
321C22×C4—341no—
322C22×C4—341no—
323C22×C4—341no—
324C22×C4—342no—
325C22×C4—342no—
326C22×C4—1361no—
327C22×C4C2×C4×D1711yescharacteristic
328C22×C4#6325211yescharacteristic
329C22×C4D4×D1711yescharacteristic
330C22×C4D4×D1711yescharacteristic
331C22×C4#6325411yescharacteristic
332C22×C4C2×C22⋊D1711yescharacteristic
333C2×C8—41no—
334C2×C8—81no—
335C2×C8—681no—
336C2×C8—1361no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—81no—
340C4⋊C4—342no—
341C4⋊C4—342no—
342C4⋊C4—342no—
343C4⋊C4—342no—
344C4⋊C4—684no—
345C4⋊C4—684no—
346C4⋊C4—1362no—
347C4⋊C4—1361no—
348C4⋊C4—1361no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—684no—
354C22⋊C4—681no—
355C22⋊C4—681no—
356C22⋊C4—681no—
357C22⋊C4—681no—
358C22⋊C4—1362no—
359C42—684no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23C22⋊C4×D1711yescharacteristic
368C23#1058756911yescharacteristic
369C23D8×D1711yescharacteristic
370C23#1058759711yescharacteristic
371C23SD16×D1711yescharacteristic
372C23#1058760111yescharacteristic
373C23#1058762311yescharacteristic
374D4—81no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4—81no—
379D4—81no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—174no—
393C2×C4—342no—
394C2×C4—342no—
395C2×C4—342no—
396C2×C4—342no—
397C2×C4—342no—
398C2×C4—342no—
399C2×C4—342no—
400C2×C4—682no—
401C2×C4—682no—
402C2×C4—682no—
403C2×C4—682no—
404C2×C4—682no—
405C2×C4—682no—
406C2×C4—682no—
407C2×C4—682no—
408C2×C4—1361no—
409C2×C4#1058756711yescharacteristic
410C2×C4#1058757011yescharacteristic
411C2×C4C4×C22⋊D1711yescharacteristic
412C2×C4#1058762411yescharacteristic
413C8—81no—
414C8—1361no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—82no—
426C22—82no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22#1176268511yescharacteristic
431C22#102×D1711yescharacteristic
432C22#1176280711yescharacteristic
433C22#1176294111yescharacteristic
434C22#1176294611yescharacteristic
435C22#1176312111yescharacteristic
436C22#1176312611yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—82no—
442C4—81no—
443C4—344no—
444C4—682no—
445C4—1362no—
446C4—1362no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—161no—
454C2#1452909811yescharacteristic
455C2#1452931811yescharacteristic
456C2#1452932311yescharacteristic
subgroupG/H
centerC22#11762685
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1519C2
socleC22×C17#386
prime radicalC2×C17×#121C22
cosocleC23×C4×C17C23
hypercenter#1519D17
hypocenterC17#9204
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
2. #14531401C2semidirect
4. C2×#11762534C2semidirect
5. #14526652C2semidirect
6. #14526651C2semidirect
12. #11762740C4semidirect
14. #11762723C4semidirect
299. C17#9204semidirect

Maximal subgroups

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

subgroupindex
#145345282
#145314012
#145273522
C2×#117625342
#145266522
#145266512
C17×#15192
#920417