Finite Group Databasebeta

← #16934844

Subgroups — #16934844

Order 4352.

650 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 103 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#14539541C211yesmaximal, characteristic
2C2×#11763140C211yesmaximal, characteristic, complemented
3#14533256C211yesmaximal, characteristic
4#14529768C211yesmaximal, characteristic, complemented
5#14528868C211yesmaximal, characteristic, complemented
6#14526630C211yesmaximal, characteristic, complemented
7C17×#1519C211yesmaximal, characteristic
8C2×#10587621—22no—
9C2×#10587610C2211yescharacteristic
10C22×C4⋊Dic17C2211yescharacteristic
11#11763140—22no—
12#11763140—22no—
13#11763139—21no—
14#11763125—21no—
15#11763036—22no—
16#11762960—21no—
17#11762947—21no—
18C2×#10587488—21no—
19C2×#10587487—21no—
20C2×C22⋊#3622C2211yescharacteristic
21#11762745—22no—
22#11762712—21no—
23#11762678—21no—
24C4×C22⋊Dic17C2211yescharacteristic
25#11762575—22no—
26#11762571—22no—
27C2×C17×#121C2211yescharacteristic
28C2×C17×C22⋊C8C2211yescharacteristic
29C17×#380C2211yescharacteristic
30C17×#327—22no—
31C22×D4×C17—21no—
32C22×D4×C17D411yescharacteristic
33C23×C4×C17C2311yescharacteristic
34C23×Dic17—21no—
35C22×Dic34D411yescharacteristic
36C22×C8×C17—21no—
37C17×#121—21no—
38C17×#121—21no—
39C17×#121—21no—
40C17×#121—21no—
41C2×C4⋊C4×C17—21no—
42C2×C4⋊C4×C17D411yescharacteristic
43C2×C22⋊C4×C17—21no—
44C2×C22⋊C4×C17—21no—
45C2×C22⋊Dic17—24no—
46C2×C22⋊Dic17—21no—
47C2×C22⋊Dic17D412yes—
48#10587621—44no—
49#10587621—44no—
50C2×#63229—22no—
51C2×C4⋊Dic17—21no—
52C2×C4⋊Dic17—21no—
53C2×C4⋊Dic17—21no—
54C2×C4⋊Dic17—21no—
55C2×C4⋊Dic17—21no—
56C2×C4⋊Dic17D412yes—
57C2×C4⋊Dic17D411yescharacteristic
58#10587610—21no—
59#10587610—21no—
60#10587610—21no—
61#10587610—21no—
62C2×#63225—24no—
63C2×#63225—21no—
64C2×C4×Dic17—21no—
65C22×#3622D412yes—
66C17×C22⋊C8—21no—
67C17×C22⋊C8D412yes—
68C17×#95—21no—
69#10587503—22no—
70#10587502—21no—
71C22⋊#3622—21no—
72C22⋊#3622—21no—
73#10587488—24no—
74#10587487—24no—
75#10587481—22no—
76#10587479—22no—
77C24×C17—21no—
78C24×C17C2×D411yescharacteristic
79C2×D4×C17—21no—
80C2×D4×C17—21no—
81C2×D4×C17—42no—
82C2×D4×C17—41no—
83C2×D4×C17—41no—
84C2×D4×C17—42no—
85C2×D4×C17—42no—
86C22×C4×C17—21no—
87C22×C4×C17—42no—
88C22×C4×C17—41no—
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17C2×D411yescharacteristic
94C22×C4×C17#4111yescharacteristic
95C22×Dic17—22no—
96C22×Dic17—22no—
97C22×Dic17—21no—
98C22×Dic17—21no—
99C22×Dic17—41no—
100C22×Dic17—41no—
101C22×Dic17—42no—
102C22×Dic17—42no—
103C2×Dic34—21no—
104C2×Dic34—21no—
105C2×Dic34—42no—
106C2×C8×C17—22no—
107C2×C8×C17—42no—
108C4⋊C4×C17—21no—
109C4⋊C4×C17—21no—
110C4⋊C4×C17—42no—
111C22⋊C4×C17—41no—
112C22⋊C4×C17—41no—
113C22⋊C4×C17—42no—
114C22⋊Dic17—22no—
115C22⋊Dic17—48no—
116C22⋊Dic17—41no—
117C22⋊Dic17—41no—
118C22⋊Dic17SD1614yescomplemented
119#63229—82no—
120C4⋊Dic17—21no—
121C4⋊Dic17—22no—
122C4⋊Dic17—21no—
123C4⋊Dic17—22no—
124C4⋊Dic17—41no—
125C4⋊Dic17—41no—
126#63225—48no—
127#63225—41no—
128#63225—41no—
129C4×Dic17—42no—
130C2×#3622—22no—
131C2×#3622—42no—
132#9960—171nomaximal, Sylow
133C23×C17—41no—
134C23×C17—41no—
135C23×C17—42no—
136C23×C17—42no—
137C23×C17—41no—
138C23×C17—42no—
139C23×C17#12011yescharacteristic
140C23×C17#12111yescharacteristic
141C23×C17#12711yescharacteristic
142C23×C17C2×SD1612yes—
143C23×C17AGL(1,ℤ/8)12yes—
144D4×C17—42no—
145D4×C17—81no—
146D4×C17—82no—
147D4×C17—81no—
148D4×C17—81no—
149C2×C4×C17—22no—
150C2×C4×C17—22no—
151C2×C4×C17—22no—
152C2×C4×C17—41no—
153C2×C4×C17—41no—
154C2×C4×C17—41no—
155C2×C4×C17—41no—
156C2×C4×C17—42no—
157C2×C4×C17—81no—
158C2×C4×C17—81no—
159C2×C4×C17—81no—
160C2×C4×C17#12011yescharacteristic
161C2×C4×C17#12011yescharacteristic
162C2×C4×C17#12111yescharacteristic
163C2×C4×C17#12111yescharacteristic
164C2×Dic17—24no—
165C2×Dic17—42no—
166C2×Dic17—42no—
167C2×Dic17—41no—
168C2×Dic17—41no—
169C2×Dic17—42no—
170C2×Dic17—44no—
171C2×Dic17—82no—
172C2×Dic17—81no—
173C2×Dic17—81no—
174C2×Dic17—81no—
175C2×Dic17—81no—
176Dic34—42no—
177Dic34—42no—
178C8×C17—42no—
179#3622—42no—
180#3187—171no—
181C2×#483—171no—
182#2501—171no—
183#1996—171no—
184#1823—171no—
185#1519D1711yescharacteristic
186#1282—171no—
187C22×C17—21no—
188C22×C17—21no—
189C22×C17—21no—
190C22×C17—21no—
191C22×C17—21no—
192C22×C17—21no—
193C22×C17—42no—
194C22×C17—42no—
195C22×C17—84no—
196C22×C17—81no—
197C22×C17—81no—
198C22×C17—81no—
199C22×C17#39211yescharacteristic
200C22×C17#45012yes—
201C22×C17#46012yes—
202C22×C17#49211yescharacteristic
203C22×C17#49611yescharacteristic
204C4×C17—21no—
205C4×C17—21no—
206C4×C17—21no—
207C4×C17—81no—
208C4×C17—161no—
209C4×C17#45812yes—
210Dic17—44no—
211Dic17—81no—
212Dic17—82no—
213Dic17—81no—
214C2×#123—342no—
215C2×C22⋊Q8—171no—
216C2×#121C2×D1711yescharacteristic
217C22×C4⋊C4—171no—
218#483—342no—
219#483—341no—
220#483—342no—
221#481—341no—
222#465—342no—
223#450—341no—
224#448—341no—
225C2×#106—341no—
226C2×#103—341no—
227C2×C22⋊C8—171no—
228C2×C22⋊C8C2×D1711yescharacteristic
229#399—342no—
230#390—341no—
231#384—341no—
232#380C2×D1711yescharacteristic
233C4×C22⋊C4—171no—
234#342—342no—
235#340—342no—
236#327—22no—
237C2×C17—22no—
238C2×C17—21no—
239C2×C17—22no—
240C2×C17—21no—
241C2×C17—161no—
242C2×C17#200911yescharacteristic
243C2×C17#201411yescharacteristic
244C2×C17#202211yescharacteristic
245C22×Q8—171no—
246C22×D4—21no—
247C22×D4D6811yescharacteristic
248C23×C4—341no—
249C23×C4C22×D1711yescharacteristic
250C2×SD16—342no—
251C22×C8—21no—
252C22×C8—172no—
253#123—684no—
254#123—684no—
255C22⋊Q8—341no—
256C22⋊Q8—341no—
257C22⋊Q8—341no—
258C22⋊Q8—341no—
259#121—21no—
260#121—21no—
261#121—21no—
262#121—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—172no—
265C2×C4⋊C4—171no—
266C2×C4⋊C4—341no—
267C2×C4⋊C4—341no—
268C2×C4⋊C4—341no—
269C2×C4⋊C4—341no—
270C2×C4⋊C4—344no—
271C2×C4⋊C4—341no—
272C2×C4⋊C4—341no—
273C2×C4⋊C4D6811yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—172no—
277C2×C22⋊C4—344no—
278C2×C22⋊C4—341no—
279C2×C42—341no—
280#106—342no—
281#106—344no—
282#103—342no—
283#103—344no—
284#102—342no—
285C22⋊C8—21no—
286C22⋊C8—341no—
287C22⋊C8—341no—
288C22⋊C8C22⋊D1712yes—
289#95—21no—
290#95—341no—
291C17#996011yescharacteristic, complemented, Sylow
292C24—21no—
293C24D4×D1711yescharacteristic
294C2×Q8—341no—
295C2×Q8—341no—
296C2×Q8—682no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—41no—
304C22×C4—21no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—342no—
308C22×C4—342no—
309C22×C4—341no—
310C22×C4—341no—
311C22×C4—681no—
312C22×C4—682no—
313C22×C4—682no—
314C22×C4—681no—
315C22×C4C2×D6811yescharacteristic
316C22×C4#6325211yescharacteristic
317C22×C4D4×D1711yescharacteristic
318C22×C4D4×D1711yescharacteristic
319C22×C4D4×D1711yescharacteristic
320C22×C4C2×C22⋊D1711yescharacteristic
321SD16—1362no—
322C2×C8—22no—
323C2×C8—42no—
324C2×C8—342no—
325C2×C8—682no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—42no—
329C4⋊C4—341no—
330C4⋊C4—342no—
331C4⋊C4—342no—
332C4⋊C4—341no—
333C4⋊C4—688no—
334C4⋊C4—681no—
335C4⋊C4—681no—
336C4⋊C4—681no—
337C4⋊C4—681no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—42no—
341C22⋊C4—174no—
342C22⋊C4—342no—
343C22⋊C4—688no—
344C22⋊C4—681no—
345C22⋊C4—681no—
346C42—682no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23#1058756811yescharacteristic
354C23#1058757011yescharacteristic
355C23#1058759712yes—
356C23SD16×D1712yes—
357C23#1058762611yescharacteristic
358Q8—682no—
359Q8—682no—
360D4—42no—
361D4—81no—
362D4—82no—
363D4—81no—
364D4—81no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—344no—
377C2×C4—682no—
378C2×C4—682no—
379C2×C4—684no—
380C2×C4—681no—
381C2×C4—681no—
382C2×C4—682no—
383C2×C4—1362no—
384C2×C4—1361no—
385C2×C4—1361no—
386C2×C4—1361no—
387C2×C4—1361no—
388C2×C4#1058756811yescharacteristic
389C2×C4#1058757011yescharacteristic
390C2×C4#1058761611yescharacteristic
391C2×C4#1058762311yescharacteristic
392C8—42no—
393C8—682no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—42no—
401C22—42no—
402C22—84no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#1176271811yescharacteristic
407C22#1176296112yes—
408C22#1176300812yes—
409C22#1176318011yescharacteristic
410C22#1176319711yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—81no—
415C4—161no—
416C4—684no—
417C4—1361no—
418C4—1362no—
419C4—1361no—
420C4#1176300012yes—
421C2—21no—
422C2—22no—
423C2—22no—
424C2—21no—
425C2—161no—
426C2#1452984511yescharacteristic
427C2#1452987311yescharacteristic
428C2#1452991411yescharacteristic
subgroupG/H
centerC22#11762718
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#1519C2
socleC22×C17#392
prime radicalC2×C17×#121C22
hypercenter#1519D17
hypocenterC17#9960
LCSG ▷ C23×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. C2×#11763140C2semidirect
4. #14529768C2semidirect
5. #14528868C2semidirect
6. #14526630C2semidirect
118. C22⋊Dic17SD16semidirect
291. C17#9960semidirect

Maximal subgroups

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

subgroupindex
#145395412
C2×#117631402
#145332562
#145297682
#145288682
#145266302
C17×#15192
#996017