Finite Group Databasebeta

← #16952991

Subgroups — #16952991

Order 4352.

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

normal subgroups only

This group has 488 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
1#14528776C211yesmaximal, characteristic
2C4×#10587505C211yesmaximal, characteristic
3#14526902C212yesmaximal
4#14526902C212yesmaximal
5C17×#2579C211yesmaximal, characteristic
6#11762833—21no—
7C2×#10587504C2212yes—
8C2×#10587505C2212yes—
9C4×C22⋊Dic17—21no—
10#11762578—21no—
11#11762576—21no—
12C2×C4×D4×C17C2211yescharacteristic
13C22×C4⋊C4×C17C411yescharacteristic, complemented
14C17×#398—21no—
15C17×#396—21no—
16C17×#387—21no—
17C17×#386C2211yescharacteristic
18C17×#386C2211yescharacteristic
19C17×#385—21no—
20C17×#385C411yescharacteristic, complemented
21C17×#385C411yescharacteristic, complemented
22C17×#380C411yescharacteristic, complemented
23C22×D4×C17C2311yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17C2×C411yescharacteristic
26C23×C4×C17C2×C411yescharacteristic
27C23×C4×C17D411yescharacteristic
28C4×D4×C17—41no—
29C4×D4×C17—41no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17—41no—
36C2×C4⋊C4×C17—41no—
37C2×C4⋊C4×C17D411yescharacteristic
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17C2×C411yescharacteristic
41C2×C22⋊C4×C17C2×C411yescharacteristic
42C2×C22⋊C4×C17C2×C411yescharacteristic
43C2×C22⋊C4×C17D411yescharacteristic
44C2×C42×C17D411yescharacteristic
45C2×C22⋊Dic17—22no—
46C2×C4×Dic17—41no—
47C2×#63223—22no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95—21no—
52C17×#95—21no—
53#10587505—22no—
54#10587505Q814yes—
55#10587504—24no—
56#10587502—41no—
57C4⋊#3622—41no—
58C4⋊#3622—41no—
59C24×C17C22×C411yescharacteristic
60C24×C17C2×D411yescharacteristic
61C2×D4×C17—41no—
62C2×D4×C17—41no—
63C2×D4×C17C2×Q812yes—
64C2×D4×C17#4112yes—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—42no—
74C22×C4×C17—41no—
75C22×C4×C17—41no—
76C22×C4×C17—81no—
77C22×C4×C17—81no—
78C22×C4×C17—81no—
79C22×C4×C17C22⋊C411yescharacteristic
80C22×C4×C17C22⋊C411yescharacteristic
81C22×C4×C17C22⋊C411yescharacteristic
82C22×C4×C17C22⋊C411yescharacteristic
83C22×C4×C17C2×D411yescharacteristic
84C22×Dic17—42no—
85C4⋊C4×C17—41no—
86C4⋊C4×C17—42no—
87C4⋊C4×C17—42no—
88C4⋊C4×C17—41no—
89C4⋊C4×C17—41no—
90C4⋊C4×C17—81no—
91C4⋊C4×C17—81no—
92C22⋊C4×C17—41no—
93C22⋊C4×C17—41no—
94C22⋊C4×C17—42no—
95C22⋊C4×C17—41no—
96C22⋊C4×C17—41no—
97C22⋊C4×C17—81no—
98C42×C17—21no—
99C42×C17—21no—
100C22⋊Dic17—24no—
101C22⋊Dic17—42no—
102C4×Dic17—82no—
103#63223—42no—
104#63223—42no—
105C2×#3622—42no—
106#13297—171nomaximal, Sylow
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17C2×C22⋊C411yescharacteristic
116C23×C17C4×Q812yes—
117C23×C17C22⋊Q812yes—
118D4×C17—42no—
119D4×C17—42no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—22no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—41no—
127C2×C4×C17—42no—
128C2×C4×C17—41no—
129C2×C4×C17—42no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—42no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17—41no—
144C2×C4×C17—81no—
145C2×C4×C17—81no—
146C2×C4×C17—81no—
147C2×C4×C17—82no—
148C2×C4×C17#9911yescharacteristic
149C2×C4×C17#9911yescharacteristic
150C2×C4×C17#12411yescharacteristic
151C2×C4×C17C4⋊Q811yescharacteristic
152C2×Dic17—44no—
153C2×Dic17—82no—
154C2×Dic17—82no—
155#3622—82no—
156#2579D1711yescharacteristic
157#1800—171no—
158C4×#99—171no—
159#1390—172no—
160#1390—172no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—22no—
164C22×C17—22no—
165C22×C17—42no—
166C22×C17—42no—
167C22×C17—42no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17—41no—
172C22×C17—81no—
173C22×C17#39111yescharacteristic
174C22×C17C2×#9911yescharacteristic
175C22×C17#41011yescharacteristic
176C4×C17—21no—
177C4×C17—21no—
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—41no—
181C4×C17—42no—
182C4×C17—41no—
183C4×C17—41no—
184C4×C17—81no—
185C4×C17—81no—
186C4×C17—81no—
187Dic17—84no—
188C2×C4×D4C2×D1711yescharacteristic
189C22×C4⋊C4Dic1711yescharacteristic, complemented
190#423—341no—
191C2×#100—172no—
192C2×#99—172no—
193#398—21no—
194#396—21no—
195#387—21no—
196#386C2×D1711yescharacteristic
197#386C2×D1711yescharacteristic
198#385—21no—
199#385Dic1711yescharacteristic, complemented
200#385Dic1711yescharacteristic, complemented
201#380Dic1711yescharacteristic, complemented
202C4×C22⋊C4—341no—
203#343—341no—
204#342—341no—
205C2×C17—21no—
206C2×C17—21no—
207C2×C17—42no—
208C2×C17—42no—
209C2×C17#199611yescharacteristic
210C2×C17#212011yescharacteristic
211C2×C17#212111yescharacteristic
212C22×D4C22×D1711yescharacteristic
213C23×C4C2×Dic1711yescharacteristic
214C23×C4C2×Dic1711yescharacteristic
215C23×C4C2×Dic1711yescharacteristic
216C23×C4C22⋊D1711yescharacteristic
217C2×M16—342no—
218C4×D4—41no—
219C4×D4—41no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4C22⋊D1711yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—342no—
231C2×C22⋊C4C2×Dic1711yescharacteristic
232C2×C22⋊C4C2×Dic1711yescharacteristic
233C2×C22⋊C4C2×Dic1711yescharacteristic
234C2×C22⋊C4C22⋊D1711yescharacteristic
235C2×C42—681no—
236C2×C42C22⋊D1711yescharacteristic
237C4⋊C8—681no—
238C4⋊C8—681no—
239#100—344no—
240#99—174no—
241#99—342no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—681no—
248C17#1329711yescharacteristic, complemented, Sylow
249C24C22×Dic1711yescharacteristic
250C24C2×C22⋊D1711yescharacteristic
251C2×D4—41no—
252C2×D4—41no—
253C2×D4Q8×D1712yes—
254C2×D4#6325612yes—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—682no—
270C22×C4C22⋊Dic1711yescharacteristic
271C22×C4C22⋊Dic1711yescharacteristic
272C22×C4C22⋊Dic1711yescharacteristic
273C22×C4C22⋊Dic1711yescharacteristic
274C22×C4C2×C22⋊D1711yescharacteristic
275M16—682no—
276M16—682no—
277C2×C8—682no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—81no—
284C4⋊C4—81no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—81no—
291C22⋊C4—344no—
292C22⋊C4—682no—
293C42—21no—
294C42—21no—
295C42—1362no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23Q8×Dic1712yes—
305C23#1058763212yes—
306C23C2×C22⋊Dic1711yescharacteristic
307D4—42no—
308D4—42no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—684no—
338C2×C4—1362no—
339C2×C4—1362no—
340C2×C4#1058750511yescharacteristic
341C2×C4#1058750511yescharacteristic
342C2×C4#1058763011yescharacteristic
343C2×C4#1058763311yescharacteristic
344C8—1362no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—81no—
357C22#1176271611yescharacteristic
358C22C2×#1058750511yescharacteristic
359C22#1176278311yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—1364no—
372C2—21no—
373C2—21no—
374C2—42no—
375C2—42no—
376C2#1452977011yescharacteristic
377C2#1453042511yescharacteristic
378C2#1453043011yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC24×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#2579C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#2579D17
hypocenterC17#13297
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2579

Complements

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

NH ≅ G/Ntype
13. C22×C4⋊C4×C17C4semidirect
20. C17×#385C4semidirect
21. C17×#385C4semidirect
22. C17×#380C4semidirect
189. C22×C4⋊C4Dic17semidirect
199. #385Dic17semidirect
200. #385Dic17semidirect
201. #380Dic17semidirect
248. C17#13297semidirect

Maximal subgroups

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

subgroupindex
#145287762
C4×#105875052
#145269022
C17×#25792
#1329717