Finite Group Databasebeta

← #16950293

Subgroups — #16950293

Order 4352.

540 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C4×#10587621C211yesmaximal, characteristic
2C2×#11762600C212yesmaximal
3#14528772C211yesmaximal, characteristic
4#14526903C212yesmaximal
5C17×#1800C211yesmaximal, characteristic
6C2×#10587621C2211yescharacteristic
7C22×C4×Dic17C411yescharacteristic, complemented
8C2×#10587505C2212yes—
9#11762697—21no—
10#11762690—21no—
11#11762689C2211yescharacteristic
12#11762681C411yescharacteristic, complemented
13#11762670C411yescharacteristic, complemented
14#11762669—21no—
15C4×#63225—21no—
16C4×C22⋊Dic17—21no—
17C4×C22⋊Dic17C411yescharacteristic, complemented
18#11762600—24no—
19#11762576—21no—
20C2×C4×D4×C17C2211yescharacteristic
21C17×#423—21no—
22C2×C17×#100C2212yes—
23C17×#343—21no—
24C22×D4×C17C2311yescharacteristic
25C23×C4×C17C2×C411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C23×Dic17C2×C411yescharacteristic
28C23×Dic17C2×C411yescharacteristic
29C2×M16×C17—22no—
30C4×D4×C17—41no—
31C4×D4×C17—41no—
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17C2×C411yescharacteristic
34C2×C22⋊C4×C17D411yescharacteristic
35C2×C42×C17D411yescharacteristic
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—22no—
39C2×C22⋊Dic17C2×C411yescharacteristic
40C2×C22⋊Dic17C2×C411yescharacteristic
41#10587621—21no—
42#10587621—21no—
43#10587621D412yes—
44#10587621Q812yes—
45C2×C4⋊Dic17—41no—
46C2×#63225—21no—
47C2×#63225—21no—
48C2×C4×Dic17—21no—
49C2×C4×Dic17—21no—
50C2×C4×Dic17—21no—
51C2×C4×Dic17—41no—
52C2×C4×Dic17—41no—
53C17×C4⋊C8—41no—
54C17×C4⋊C8—41no—
55C17×#100—22no—
56C17×#100—22no—
57#10587505—22no—
58#10587505—22no—
59#10587502—21no—
60#10587502—21no—
61#10587502—21no—
62#10587502—21no—
63#10587502—21no—
64#10587502—41no—
65C24×C17C22×C411yescharacteristic
66C24×C17C2×D411yescharacteristic
67C2×D4×C17—41no—
68C2×D4×C17—41no—
69C2×D4×C17C2×D411yescharacteristic
70C2×D4×C17C2×Q811yescharacteristic
71C2×D4×C17#4111yescharacteristic
72C2×D4×C17#4111yescharacteristic
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17—21no—
77C22×C4×C17—42no—
78C22×C4×C17—42no—
79C22×C4×C17C22⋊C411yescharacteristic
80C22×C4×C17C22⋊C411yescharacteristic
81C22×C4×C17C22⋊C411yescharacteristic
82C22×C4×C17C22⋊C411yescharacteristic
83C22×C4×C17C2×D411yescharacteristic
84C22×Dic17—21no—
85C22×Dic17—21no—
86C22×Dic17—21no—
87C22×Dic17—21no—
88C22×Dic17—21no—
89C22×Dic17—42no—
90C22×Dic17—41no—
91C22×Dic17—41no—
92C22×Dic17—41no—
93C22×Dic17—42no—
94C22×Dic17C4⋊C412yescomplemented
95M16×C17—42no—
96M16×C17—42no—
97C2×C8×C17—42no—
98C4⋊C4×C17—41no—
99C22⋊C4×C17—41no—
100C22⋊C4×C17—41no—
101C42×C17—21no—
102C42×C17—21no—
103C22⋊Dic17—21no—
104C22⋊Dic17—22no—
105C22⋊Dic17—22no—
106C22⋊Dic17—42no—
107C22⋊Dic17—41no—
108C22⋊Dic17—42no—
109C22⋊Dic17—42no—
110C22⋊Dic17C4⋊C412yescomplemented
111C4⋊Dic17—82no—
112#63225—22no—
113#63225—22no—
114#63225—42no—
115C4×Dic17—42no—
116C4×Dic17—42no—
117C4×Dic17—41no—
118C4×Dic17—41no—
119#12794—171nomaximal, Sylow
120C23×C17—21no—
121C23×C17—21no—
122C23×C17—21no—
123C23×C17—21no—
124C23×C17—41no—
125C23×C17—41no—
126C23×C17—41no—
127C23×C17—41no—
128C23×C17C2×C22⋊C411yescharacteristic
129C23×C17C2×C4⋊C411yescharacteristic
130C23×C17#11711yescharacteristic
131C23×C17#12111yescharacteristic
132C23×C17C22⋊Q811yescharacteristic
133D4×C17—41no—
134D4×C17—41no—
135D4×C17—41no—
136D4×C17—41no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—22no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—42no—
144C2×C4×C17—42no—
145C2×C4×C17—42no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—81no—
150C2×C4×C17#9911yescharacteristic
151C2×C4×C17#9911yescharacteristic
152C2×C4×C17#12111yescharacteristic
153C2×C4×C17C22⋊Q811yescharacteristic
154C2×Dic17—22no—
155C2×Dic17—22no—
156C2×Dic17—22no—
157C2×Dic17—42no—
158C2×Dic17—41no—
159C2×Dic17—41no—
160C2×Dic17—41no—
161C2×Dic17—42no—
162C2×Dic17—41no—
163C2×Dic17—41no—
164C2×Dic17—41no—
165C2×Dic17—42no—
166C2×Dic17—42no—
167C2×Dic17—41no—
168C2×Dic17—41no—
169C2×Dic17—41no—
170C2×Dic17—82no—
171C2×Dic17—82no—
172C2×Dic17—82no—
173C8×C17—82no—
174C4×#123—171no—
175C2×#352—172no—
176#1800D1711yescharacteristic
177#1799—171no—
178#1390—172no—
179C22×C17—21no—
180C22×C17—21no—
181C22×C17—21no—
182C22×C17—21no—
183C22×C17—21no—
184C22×C17—21no—
185C22×C17—41no—
186C22×C17—41no—
187C22×C17—41no—
188C22×C17—41no—
189C22×C17—42no—
190C22×C17—41no—
191C22×C17—41no—
192C22×C17—41no—
193C22×C17—41no—
194C22×C17—81no—
195C22×C17#38011yescharacteristic
196C22×C17C2×#9911yescharacteristic
197C22×C17#41011yescharacteristic
198C4×C17—21no—
199C4×C17—21no—
200C4×C17—21no—
201C4×C17—41no—
202C4×C17—41no—
203C4×C17—41no—
204C4×C17#35212yescomplemented
205Dic17—42no—
206Dic17—42no—
207Dic17—42no—
208Dic17—81no—
209Dic17—162no—
210C2×#123—171no—
211C2×C4×D4C2×D1711yescharacteristic
212C22×C42—171no—
213#423—21no—
214C2×#100C2×D1712yes—
215C2×#99—172no—
216#388—341no—
217#387—341no—
218#386—171no—
219#385—171no—
220#381—171no—
221#380—341no—
222C4×C4⋊C4—341no—
223C4×C22⋊C4—171no—
224C4×C22⋊C4—341no—
225#352—344no—
226#343—21no—
227#342—341no—
228C2×C17—21no—
229C2×C17—21no—
230C2×C17—41no—
231C2×C17—41no—
232C2×C17—41no—
233C2×C17—41no—
234C2×C17#179911yescharacteristic
235C2×C17C2×#35211yescharacteristic
236C2×C17#212011yescharacteristic
237C22×D4C22×D1711yescharacteristic
238C23×C4—171no—
239C23×C4—171no—
240C23×C4C4×D1711yescharacteristic
241C23×C4D6811yescharacteristic
242C2×M16—22no—
243#123—172no—
244#123—172no—
245#123—341no—
246#123—341no—
247C4×D4—41no—
248C4×D4—41no—
249C2×C4⋊C4—341no—
250C2×C4⋊C4—341no—
251C2×C4⋊C4—681no—
252C2×C4⋊C4C22⋊D1711yescharacteristic
253C2×C22⋊C4—171no—
254C2×C22⋊C4—171no—
255C2×C22⋊C4—341no—
256C2×C22⋊C4—341no—
257C2×C22⋊C4—342no—
258C2×C22⋊C4C4×D1711yescharacteristic
259C2×C22⋊C4D6811yescharacteristic
260C2×C42—341no—
261C2×C42—341no—
262C2×C42—341no—
263C2×C42—681no—
264C2×C42—681no—
265C2×C42C22⋊D1711yescharacteristic
266C4⋊C8—41no—
267C4⋊C8—41no—
268#100—22no—
269#100—22no—
270#99—342no—
271#99—342no—
272#95—341no—
273#95—341no—
274#95—341no—
275#95—341no—
276#95—341no—
277#95—681no—
278C17#1279411yescharacteristic, complemented, Sylow
279C24C2×C4×D1711yescharacteristic
280C24C2×D6811yescharacteristic
281C2×D4—41no—
282C2×D4—41no—
283C2×D4D4×D1711yescharacteristic
284C2×D4#6325411yescharacteristic
285C2×D4Q8×D1711yescharacteristic
286C2×D4#6325611yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—172no—
294C22×C4—341no—
295C22×C4—341no—
296C22×C4—341no—
297C22×C4—341no—
298C22×C4—341no—
299C22×C4—682no—
300C22×C4—681no—
301C22×C4—681no—
302C22×C4—681no—
303C22×C4—682no—
304C22×C4#6322711yescharacteristic
305C22×C4#6322711yescharacteristic
306C22×C4#6322711yescharacteristic
307C22×C4#6322711yescharacteristic
308C22×C4C2×C22⋊D1711yescharacteristic
309M16—42no—
310M16—42no—
311C2×C8—42no—
312C4⋊C4—41no—
313C4⋊C4—342no—
314C4⋊C4—342no—
315C4⋊C4—682no—
316C4⋊C4—1362no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—172no—
320C22⋊C4—341no—
321C22⋊C4—342no—
322C22⋊C4—342no—
323C22⋊C4—682no—
324C22⋊C4—681no—
325C22⋊C4—682no—
326C22⋊C4—682no—
327C42—21no—
328C42—21no—
329C42—682no—
330C42—682no—
331C42—681no—
332C42—681no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23C4⋊C4×D1711yescharacteristic
342C23#1058757811yescharacteristic
343C23#1058758111yescharacteristic
344C23#1058758311yescharacteristic
345C23C2×#6322711yescharacteristic
346D4—41no—
347D4—41no—
348D4—41no—
349D4—41no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—342no—
364C2×C4—342no—
365C2×C4—342no—
366C2×C4—681no—
367C2×C4—681no—
368C2×C4—682no—
369C2×C4—681no—
370C2×C4—682no—
371C2×C4—681no—
372C2×C4—681no—
373C2×C4—681no—
374C2×C4—682no—
375C2×C4—682no—
376C2×C4—681no—
377C2×C4—681no—
378C2×C4—681no—
379C2×C4—1362no—
380C2×C4—1362no—
381C2×C4—1362no—
382C2×C4#1058747711yescharacteristic
383C2×C4#1058747711yescharacteristic
384C2×C4#1058762411yescharacteristic
385C2×C4#1058763211yescharacteristic
386C8—82no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22#1176266611yescharacteristic
404C22C2×#1058747711yescharacteristic
405C22#1176278211yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—2722no—
413C4—682no—
414C4—682no—
415C4—682no—
416C4—1361no—
417C4#1176260012yescomplemented
418C2—21no—
419C2—21no—
420C2—41no—
421C2—41no—
422C2—41no—
423C2—41no—
424C2#1452876911yescharacteristic
425C2C2×#1176260011yescharacteristic
426C2#1453042611yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC24×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#1800C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#1800D17
hypocenterC17#12794
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
7. C22×C4×Dic17C4semidirect
12. #11762681C4semidirect
13. #11762670C4semidirect
17. C4×C22⋊Dic17C4semidirect
94. C22×Dic17C4⋊C4semidirect
110. C22⋊Dic17C4⋊C4semidirect
204. C4×C17#352semidirect
278. C17#12794semidirect
417. C4#11762600semidirect

Maximal subgroups

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

subgroupindex
C4×#105876212
C2×#117626002
#145287722
#145269032
C17×#18002
#1279417