Finite Group Databasebeta

← #16953004

Subgroups — #16953004

Order 4352.

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

normal subgroups only

This group has 492 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#14528772C211yesmaximal, characteristic
2C4×#10587505C211yesmaximal, characteristic
3#14526909C212yesmaximal
4#14526909C212yesmaximal
5C17×#2566C211yesmaximal, characteristic
6C2×#10587505C2212yes—
7C2×#10587505C2212yes—
8#11762669—21no—
9C4×C22⋊Dic17—21no—
10#11762576—21no—
11#11762576—21no—
12C2×C17×#125C411yescharacteristic, complemented
13C2×C4×D4×C17C2211yescharacteristic
14C17×#396—21no—
15C17×#396—21no—
16C17×#390C2211yescharacteristic
17C17×#390C2211yescharacteristic
18C17×#385—21no—
19C17×#385—21no—
20C17×#384C411yescharacteristic, complemented
21C17×#382C411yescharacteristic, complemented
22C4×C4⋊C4×C17C411yescharacteristic, complemented
23C22×D4×C17C2311yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C17×#125—41no—
27C17×#125—41no—
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×C17C2×C411yescharacteristic
36C2×C4⋊C4×C17C2×C411yescharacteristic
37C2×C4⋊C4×C17C2×C411yescharacteristic
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17—21no—
42C2×C22⋊C4×C17D411yescharacteristic
43C2×C22⋊C4×C17D411yescharacteristic
44C2×C42×C17C2×C411yescharacteristic
45C2×C42×C17C2×C411yescharacteristic
46C2×C42×C17C2×C411yescharacteristic
47C2×C22⋊Dic17—22no—
48C2×C22⋊Dic17—22no—
49C2×C4⋊Dic17—41no—
50C2×C4×Dic17—41no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95—21no—
54#10587505—22no—
55#10587505—24no—
56#10587505Q814yes—
57#10587502—41no—
58#10587502—41no—
59C24×C17C2×D411yescharacteristic
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—81no—
74C22×C4×C17—81no—
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C22⋊C411yescharacteristic
78C22×C4×C17C22⋊C411yescharacteristic
79C22×C4×C17C22×C411yescharacteristic
80C22×Dic17—42no—
81C22×Dic17—42no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—42no—
86C4⋊C4×C17—42no—
87C4⋊C4×C17—41no—
88C4⋊C4×C17—41no—
89C4⋊C4×C17—81no—
90C4⋊C4×C17—81no—
91C22⋊C4×C17—41no—
92C22⋊C4×C17—41no—
93C22⋊C4×C17—42no—
94C22⋊C4×C17—42no—
95C22⋊C4×C17—81no—
96C22⋊C4×C17—81no—
97C42×C17—21no—
98C42×C17—21no—
99C42×C17—41no—
100C42×C17—41no—
101C22⋊Dic17—24no—
102C22⋊Dic17—42no—
103C22⋊Dic17—42no—
104C22⋊Dic17—42no—
105C4⋊Dic17—82no—
106C4×Dic17—82no—
107#13300—171nomaximal, Sylow
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—21no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17—41no—
116C23×C17C2×C22⋊C411yescharacteristic
117C23×C17C22⋊Q812yes—
118C23×C17#12411yescharacteristic
119C23×C17C4⋊Q811yescharacteristic
120D4×C17—42no—
121D4×C17—42no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—22no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—41no—
129C2×C4×C17—42no—
130C2×C4×C17—42no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
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—41no—
145C2×C4×C17—81no—
146C2×C4×C17—81no—
147C2×C4×C17#9911yescharacteristic
148C2×C4×C17#9911yescharacteristic
149C2×C4×C17C4×Q812yes—
150C2×Dic17—44no—
151C2×Dic17—82no—
152C2×Dic17—82no—
153C2×Dic17—82no—
154C2×Dic17—82no—
155C2×Dic17—82no—
156#2566D1711yescharacteristic
157#1799—171no—
158C4×#99—171no—
159#1393—172no—
160#1393—172no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—22no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—42no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17—41no—
172C22×C17—41no—
173C22×C17—81no—
174C22×C17—81no—
175C22×C17#39111yescharacteristic
176C22×C17C2×#9911yescharacteristic
177C22×C17#41011yescharacteristic
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—22no—
181C4×C17—42no—
182C4×C17—41no—
183C4×C17—41no—
184C4×C17—41no—
185C4×C17—81no—
186C4×C17—81no—
187C4×C17—81no—
188Dic17—84no—
189Dic17—162no—
190C2×#125Dic1711yescharacteristic, complemented
191C2×C4×D4C2×D1711yescharacteristic
192C2×#99—172no—
193C2×#99—172no—
194#396—21no—
195#396—21no—
196#390C2×D1711yescharacteristic
197#390C2×D1711yescharacteristic
198#385—21no—
199#385—21no—
200#384Dic1711yescharacteristic, complemented
201#382Dic1711yescharacteristic, complemented
202#380—341no—
203C4×C4⋊C4Dic1711yescharacteristic, complemented
204C4×C22⋊C4—341no—
205#342—341no—
206#342—341no—
207C2×C17—21no—
208C2×C17—21no—
209C2×C17—42no—
210C2×C17—41no—
211C2×C17—41no—
212C2×C17#199611yescharacteristic
213C2×C17#212611yescharacteristic
214C2×C17#212711yescharacteristic
215C22×D4C22×D1711yescharacteristic
216C23×C4C22⋊D1711yescharacteristic
217C23×C4C22⋊D1711yescharacteristic
218#125—41no—
219#125—41no—
220C4×D4—41no—
221C4×D4—41no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—681no—
228C2×C4⋊C4C2×Dic1711yescharacteristic
229C2×C4⋊C4C2×Dic1711yescharacteristic
230C2×C4⋊C4C2×Dic1711yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—342no—
236C2×C22⋊C4—342no—
237C2×C22⋊C4C22⋊D1711yescharacteristic
238C2×C22⋊C4C22⋊D1711yescharacteristic
239C2×C42—681no—
240C2×C42C2×Dic1711yescharacteristic
241C2×C42C2×Dic1711yescharacteristic
242C2×C42C2×Dic1711yescharacteristic
243#99—174no—
244#99—342no—
245#99—344no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—681no—
250#95—681no—
251C17#1330011yescharacteristic, complemented, Sylow
252C24C2×C22⋊D1711yescharacteristic
253C24C2×C22⋊D1711yescharacteristic
254C2×D4—41no—
255C2×D4—41no—
256C2×D4Q8×D1712yes—
257C2×D4#6325612yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—682no—
269C22×C4—682no—
270C22×C4C22⋊Dic1711yescharacteristic
271C22×C4C22⋊Dic1711yescharacteristic
272C22×C4C22⋊Dic1711yescharacteristic
273C22×C4C22⋊Dic1711yescharacteristic
274C22×C4C22×Dic1711yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—81no—
283C4⋊C4—81no—
284C4⋊C4—1362no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C22⋊C4—344no—
292C22⋊C4—682no—
293C22⋊C4—682no—
294C22⋊C4—682no—
295C42—21no—
296C42—21no—
297C42—41no—
298C42—41no—
299C42—1362no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23#1058763011yescharacteristic
309C23#1058763212yes—
310C23#1058763311yescharacteristic
311C23C2×C22⋊Dic1711yescharacteristic
312D4—42no—
313D4—42no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
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—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—684no—
340C2×C4—1362no—
341C2×C4—1362no—
342C2×C4—1362no—
343C2×C4—1362no—
344C2×C4—1362no—
345C2×C4#1058750511yescharacteristic
346C2×C4#1058750511yescharacteristic
347C2×C4Q8×Dic1712yes—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—42no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22#1176271611yescharacteristic
363C22C2×#1058750511yescharacteristic
364C22#1176278311yescharacteristic
365C4—21no—
366C4—21no—
367C4—22no—
368C4—42no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4—2722no—
376C4—1364no—
377C2—21no—
378C2—21no—
379C2—41no—
380C2—42no—
381C2—41no—
382C2#1452977011yescharacteristic
383C2#1453044911yescharacteristic
384C2#1453045311yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#2566C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#2566D17
hypocenterC17#13300
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2566

Complements

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

NH ≅ G/Ntype
12. C2×C17×#125C4semidirect
20. C17×#384C4semidirect
21. C17×#382C4semidirect
22. C4×C4⋊C4×C17C4semidirect
190. C2×#125Dic17semidirect
200. #384Dic17semidirect
201. #382Dic17semidirect
203. C4×C4⋊C4Dic17semidirect
251. C17#13300semidirect

Maximal subgroups

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

subgroupindex
#145287722
C4×#105875052
#145269092
C17×#25662
#1330017