Finite Group Databasebeta

← #16952989

Subgroups — #16952989

Order 4352.

488 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 91 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#14534128C211yesmaximal, characteristic
2#14528774C211yesmaximal, characteristic
3#99×Dic17C211yesmaximal, characteristic
4#14526903C211yesmaximal, characteristic
5#14526902C211yesmaximal, characteristic
6#14526901C211yesmaximal, characteristic
7C17×#1390C211yesmaximal, characteristic
8C2×D4×Dic17C2211yescharacteristic
9C22×C4⋊Dic17C411yescharacteristic, complemented
10#11762832—21no—
11C2×#10587504C2211yescharacteristic
12C2×#10587505C2211yescharacteristic
13#11762741—21no—
14#11762732—21no—
15#11762696—21no—
16#11762689C2211yescharacteristic
17#11762684—21no—
18#11762684C411yescharacteristic, complemented
19#11762682C411yescharacteristic, complemented
20#11762667C411yescharacteristic, complemented
21C22⋊C4×Dic17—21no—
22#11762577—21no—
23#11762575—21no—
24C2×C17×#100C2211yescharacteristic
25C2×C17×#99C2211yescharacteristic
26C17×#386C2211yescharacteristic
27C22×D4×C17C2311yescharacteristic
28C23×C4×C17C2×C411yescharacteristic
29C23×Dic17C2×C411yescharacteristic
30C23×Dic17C2×C411yescharacteristic
31C23×Dic17D411yescharacteristic
32C2×M16×C17—21no—
33C2×C22⋊C4×C17—21no—
34C2×C22⋊C4×C17—21no—
35C2×C22⋊C4×C17C2×C411yescharacteristic
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17C2×C411yescharacteristic
39C2×C22⋊Dic17C2×C411yescharacteristic
40C2×C22⋊Dic17D411yescharacteristic
41D4×Dic17—41no—
42D4×Dic17—41no—
43C2×C4⋊Dic17—21no—
44C2×C4⋊Dic17—21no—
45C2×C4⋊Dic17—21no—
46C2×C4⋊Dic17—41no—
47C2×C4⋊Dic17—41no—
48C2×C4⋊Dic17D411yescharacteristic
49C2×#63225—21no—
50C2×#63225—21no—
51C2×C4×Dic17—41no—
52C2×C4×Dic17D411yescharacteristic
53C2×#63223—21no—
54C17×#100—21no—
55C17×#100—21no—
56C17×#99—21no—
57C17×#99Q812yes—
58C17×#95—21no—
59#10587505—21no—
60#10587505Q812yes—
61#10587504—21no—
62#10587504—21no—
63#10587502—21no—
64#10587502—21no—
65#10587502—21no—
66#10587502—21no—
67#10587502—41no—
68#10587485—41no—
69#10587485—41no—
70C24×C17C22×C411yescharacteristic
71C24×C17C2×D411yescharacteristic
72C2×D4×C17—41no—
73C2×D4×C17—41no—
74C2×D4×C17C2×Q811yescharacteristic
75C2×D4×C17C2×Q811yescharacteristic
76C2×D4×C17#4111yescharacteristic
77C2×D4×C17#4111yescharacteristic
78C22×C4×C17—21no—
79C22×C4×C17—21no—
80C22×C4×C17—42no—
81C22×C4×C17—41no—
82C22×C4×C17—41no—
83C22×C4×C17C2×D411yescharacteristic
84C22×Dic17—21no—
85C22×Dic17—21no—
86C22×Dic17—21no—
87C22×Dic17—21no—
88C22×Dic17—21no—
89C22×Dic17—21no—
90C22×Dic17—41no—
91C22×Dic17—41no—
92C22×Dic17—81no—
93C22×Dic17—81no—
94C22×Dic17—81no—
95C22×Dic17C22⋊C411yescharacteristic
96C22×Dic17C22⋊C411yescharacteristic
97C22×Dic17C22⋊C411yescharacteristic
98C22×Dic17C22⋊C411yescharacteristic
99M16×C17—41no—
100M16×C17—41no—
101C2×C8×C17—41no—
102C22⋊C4×C17—22no—
103C22⋊C4×C17—42no—
104C22⋊C4×C17—41no—
105C22⋊C4×C17—41no—
106C22⋊Dic17—22no—
107C22⋊Dic17—41no—
108C22⋊Dic17—41no—
109C22⋊Dic17—41no—
110C22⋊Dic17—41no—
111C22⋊Dic17—81no—
112C4⋊Dic17—41no—
113C4⋊Dic17—42no—
114C4⋊Dic17—41no—
115C4⋊Dic17—41no—
116C4⋊Dic17—81no—
117#63225—42no—
118#63225—81no—
119C4×Dic17—21no—
120C4×Dic17—21no—
121C4×Dic17—82no—
122#63223—41no—
123#63223—41no—
124C2×#3622—41no—
125#13297—171nomaximal, Sylow
126C23×C17—21no—
127C23×C17—21no—
128C23×C17—21no—
129C23×C17—21no—
130C23×C17—41no—
131C23×C17—41no—
132C23×C17—41no—
133C23×C17—41no—
134C23×C17C2×C22⋊C411yescharacteristic
135C23×C17C4×Q811yescharacteristic
136C23×C17C4×Q811yescharacteristic
137C23×C17C22⋊Q811yescharacteristic
138C23×C17C22⋊Q811yescharacteristic
139D4×C17—41no—
140D4×C17—41no—
141D4×C17—41no—
142D4×C17—41no—
143C2×C4×C17—21no—
144C2×C4×C17—21no—
145C2×C4×C17—22no—
146C2×C4×C17—41no—
147C2×C4×C17—42no—
148C2×C4×C17—41no—
149C2×C4×C17—42no—
150C2×C4×C17—42no—
151C2×C4×C17—41no—
152C2×C4×C17—41no—
153C2×C4×C17—41no—
154C2×C4×C17—81no—
155C2×C4×C17—81no—
156C2×C4×C17#12411yescharacteristic
157C2×C4×C17C4⋊Q811yescharacteristic
158C2×Dic17—21no—
159C2×Dic17—21no—
160C2×Dic17—21no—
161C2×Dic17—41no—
162C2×Dic17—42no—
163C2×Dic17—41no—
164C2×Dic17—41no—
165C2×Dic17—41no—
166C2×Dic17—42no—
167C2×Dic17—41no—
168C2×Dic17—41no—
169C2×Dic17—41no—
170C2×Dic17—41no—
171C2×Dic17—41no—
172C2×Dic17—41no—
173C2×Dic17—81no—
174C2×Dic17—81no—
175C2×Dic17—81no—
176C2×Dic17—81no—
177C2×Dic17—81no—
178C2×Dic17—82no—
179C2×Dic17#9911yescharacteristic
180C2×Dic17#9911yescharacteristic
181C8×C17—81no—
182#3622—81no—
183#2579—171no—
184#1800—171no—
185C4×#99—171no—
186#1390—171no—
187#1390—171no—
188#1390—171no—
189#1390D1711yescharacteristic
190C22×C17—21no—
191C22×C17—21no—
192C22×C17—21no—
193C22×C17—21no—
194C22×C17—21no—
195C22×C17—21no—
196C22×C17—41no—
197C22×C17—41no—
198C22×C17—41no—
199C22×C17—41no—
200C22×C17—42no—
201C22×C17—41no—
202C22×C17—41no—
203C22×C17—41no—
204C22×C17—41no—
205C22×C17—81no—
206C22×C17#39111yescharacteristic
207C22×C17C2×#9911yescharacteristic
208C22×C17#41011yescharacteristic
209C4×C17—21no—
210C4×C17—21no—
211C4×C17—42no—
212C4×C17—81no—
213C4×C17—82no—
214Dic17—21no—
215Dic17—21no—
216Dic17—41no—
217Dic17—41no—
218Dic17—41no—
219Dic17—81no—
220Dic17—82no—
221Dic17—81no—
222C2×C4×D4—171no—
223C22×C4⋊C4—171no—
224#423—341no—
225C2×#100—171no—
226C2×#100C2×D1711yescharacteristic
227C2×#99—171no—
228C2×#99C2×D1711yescharacteristic
229#398—341no—
230#396—341no—
231#387—341no—
232#386—171no—
233#386C2×D1711yescharacteristic
234#385—171no—
235#385—171no—
236#385—341no—
237#380—171no—
238C4×C22⋊C4—341no—
239#343—341no—
240#342—341no—
241C2×C17—21no—
242C2×C17—21no—
243C2×C17—41no—
244C2×C17—41no—
245C2×C17—41no—
246C2×C17—41no—
247C2×C17#199611yescharacteristic
248C2×C17#212011yescharacteristic
249C2×C17#212111yescharacteristic
250C22×D4C22×D1711yescharacteristic
251C23×C4—171no—
252C23×C4—171no—
253C23×C4—171no—
254C23×C4C4×D1711yescharacteristic
255C2×M16—21no—
256C2×M16—341no—
257C4×D4—681no—
258C4×D4—681no—
259C2×C4⋊C4—171no—
260C2×C4⋊C4—341no—
261C2×C4⋊C4—341no—
262C2×C4⋊C4—341no—
263C2×C4⋊C4—341no—
264C2×C4⋊C4—341no—
265C2×C4⋊C4—681no—
266C2×C4⋊C4—681no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—171no—
270C2×C22⋊C4—171no—
271C2×C22⋊C4—171no—
272C2×C22⋊C4—341no—
273C2×C22⋊C4—341no—
274C2×C22⋊C4C4×D1711yescharacteristic
275C2×C42—171no—
276C2×C42—681no—
277C4⋊C8—681no—
278C4⋊C8—681no—
279#100—21no—
280#100—21no—
281#100—341no—
282#100—341no—
283#99—21no—
284#99—172no—
285#99—341no—
286#99Dic3412yes—
287#95—21no—
288#95—341no—
289#95—341no—
290#95—341no—
291#95—341no—
292#95—681no—
293C17#1329711yescharacteristic, complemented, Sylow
294C24C2×C4×D1711yescharacteristic
295C24D4×D1711yescharacteristic
296C2×D4—41no—
297C2×D4—41no—
298C2×D4C2×Dic3411yescharacteristic
299C2×D4#6325211yescharacteristic
300C2×D4#6325411yescharacteristic
301C2×D4Q8×D1711yescharacteristic
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—171no—
308C22×C4—171no—
309C22×C4—171no—
310C22×C4—171no—
311C22×C4—341no—
312C22×C4—341no—
313C22×C4—341no—
314C22×C4—341no—
315C22×C4—341no—
316C22×C4—341no—
317C22×C4—681no—
318C22×C4—681no—
319C22×C4—1361no—
320C22×C4—1361no—
321C22×C4—1361no—
322C22×C4D4×D1711yescharacteristic
323M16—41no—
324M16—41no—
325M16—681no—
326M16—681no—
327C2×C8—41no—
328C2×C8—681no—
329C4⋊C4—681no—
330C4⋊C4—682no—
331C4⋊C4—681no—
332C4⋊C4—681no—
333C4⋊C4—682no—
334C4⋊C4—1361no—
335C4⋊C4—1361no—
336C22⋊C4—22no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—342no—
341C22⋊C4—681no—
342C22⋊C4—681no—
343C22⋊C4—681no—
344C22⋊C4—681no—
345C22⋊C4—1361no—
346C42—341no—
347C42—341no—
348C42—1362no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23C4×Dic3411yescharacteristic
358C23#1058756411yescharacteristic
359C23C22⋊C4×D1711yescharacteristic
360C23#1058757311yescharacteristic
361C23#1058758211yescharacteristic
362D4—41no—
363D4—41no—
364D4—41no—
365D4—41no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—171no—
380C2×C4—171no—
381C2×C4—341no—
382C2×C4—341no—
383C2×C4—341no—
384C2×C4—681no—
385C2×C4—682no—
386C2×C4—681no—
387C2×C4—681no—
388C2×C4—681no—
389C2×C4—682no—
390C2×C4—681no—
391C2×C4—681no—
392C2×C4—681no—
393C2×C4—681no—
394C2×C4—681no—
395C2×C4—681no—
396C2×C4—1361no—
397C2×C4—1361no—
398C2×C4—1361no—
399C2×C4—1361no—
400C2×C4—1361no—
401C2×C4—1362no—
402C2×C4#1058757111yescharacteristic
403C2×C4#1058757411yescharacteristic
404C8—81no—
405C8—1361no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—42no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—81no—
422C22#1176271311yescharacteristic
423C22#99×D1711yescharacteristic
424C22#1176278111yescharacteristic
425C4—21no—
426C4—21no—
427C4—42no—
428C4—81no—
429C4—82no—
430C4—341no—
431C4—341no—
432C4—681no—
433C4—681no—
434C4—681no—
435C4—1361no—
436C4—1362no—
437C4—1361no—
438C2—21no—
439C2—21no—
440C2—41no—
441C2—41no—
442C2—41no—
443C2—41no—
444C2#1452976711yescharacteristic
445C2#1453042211yescharacteristic
446C2#1453042911yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC24×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#1390C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#1390D17
hypocenterC17#13297
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1390

Complements

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

NH ≅ G/Ntype
9. C22×C4⋊Dic17C4semidirect
18. #11762684C4semidirect
19. #11762682C4semidirect
20. #11762667C4semidirect
293. C17#13297semidirect

Maximal subgroups

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

subgroupindex
#145341282
#145287742
#99×Dic172
#145269032
#145269022
#145269012
C17×#13902
#1329717