Finite Group Databasebeta

← #16943787

Subgroups — #16943787

Order 4352.

442 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 95 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#14527125C211yesmaximal, characteristic
2#14527122C211yesmaximal, characteristic
3#14526732C212yesmaximal, complemented
4#14526714C212yesmaximal, complemented
5C17×#2322C211yesmaximal, characteristic
6#11762866—21no—
7#11762773—21no—
8#11762535—21no—
9#11762533—21no—
10#11762533C2212yes—
11#11762533C2212yes—
12C2×C17×#123C411yescharacteristic
13C2×C4×D4×C17C2211yescharacteristic
14C22×C4⋊C4×C17C411yescharacteristic
15C17×#389—21no—
16C17×#388—21no—
17C17×#387—21no—
18C17×#386C411yescharacteristic
19C17×#385—21no—
20C17×#385C2211yescharacteristic
21C17×#381C2211yescharacteristic
22C4×C22⋊C4×C17C411yescharacteristic
23C22×D4×C17C2×C411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17C2×C411yescharacteristic
26C23×C4×C17C2×C411yescharacteristic
27C23×C4×C17D411yescharacteristic
28C17×#123—22no—
29C17×#123—22no—
30C4×D4×C17—41no—
31C4×D4×C17—41no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17—41no—
37C2×C4⋊C4×C17—41no—
38C2×C4⋊C4×C17D411yescharacteristic
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17C2×C411yescharacteristic
42C2×C22⋊C4×C17C2×C411yescharacteristic
43C2×C22⋊C4×C17D411yescharacteristic
44C2×C22⋊C4×C17C2311yescharacteristic
45C2×C42×C17—21no—
46C2×C42×C17D411yescharacteristic
47C2×#63223—41no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95—21no—
52C17×#95—21no—
53C22⋊#3622—22no—
54C22⋊#3622—22no—
55C22⋊#3622—41no—
56C4⋊#3622—41no—
57#10587474—41no—
58C24×C17C22⋊C411yescharacteristic
59C24×C17C22×C411yescharacteristic
60C2×D4×C17—21no—
61C2×D4×C17—21no—
62C2×D4×C17—41no—
63C2×D4×C17—41no—
64C22×C4×C17—21no—
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—41no—
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×C17M1612yescomplemented
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D411yescharacteristic
85C4⋊C4×C17—24no—
86C4⋊C4×C17—41no—
87C4⋊C4×C17—42no—
88C4⋊C4×C17—41no—
89C4⋊C4×C17—41no—
90C4⋊C4×C17—81no—
91C22⋊C4×C17—24no—
92C22⋊C4×C17—21no—
93C22⋊C4×C17—41no—
94C22⋊C4×C17—41no—
95C22⋊C4×C17—41no—
96C22⋊C4×C17—81no—
97C22⋊C4×C17M1612yescomplemented
98C42×C17—21no—
99C42×C17—21no—
100C42×C17—81no—
101#63223—82no—
102C2×#3622—42no—
103C2×#3622—42no—
104#11528—171nomaximal, Sylow
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—41no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17C2×C22⋊C411yescharacteristic
115C23×C17C2×M1611yescharacteristic
116C23×C17C2×M1611yescharacteristic
117D4×C17—42no—
118D4×C17—42no—
119C2×C4×C17—22no—
120C2×C4×C17—22no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—41no—
128C2×C4×C17—42no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—42no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—42no—
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
150#3622—82no—
151#3622—82no—
152#2322D1711yescharacteristic
153#1466—171no—
154#1465—171no—
155#1326—172no—
156#1317—172no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—42no—
164C22×C17—42no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—81no—
171C22×C17#40711yescharacteristic
172C22×C17C2×#9911yescharacteristic
173C22×C17#41311yescharacteristic
174C4×C17—21no—
175C4×C17—21no—
176C4×C17—42no—
177C4×C17—41no—
178C4×C17—41no—
179C4×C17—41no—
180C4×C17—41no—
181C4×C17—42no—
182C4×C17—81no—
183C4×C17—81no—
184C2×#123Dic1711yescharacteristic
185C2×C4×D4C2×D1711yescharacteristic
186C22×C4⋊C4Dic1711yescharacteristic
187#432—341no—
188#407—341no—
189#389—21no—
190#388—21no—
191#387—21no—
192#386Dic1711yescharacteristic
193#385—21no—
194#385C2×D1711yescharacteristic
195#381C2×D1711yescharacteristic
196C4×C22⋊C4Dic1711yescharacteristic
197#324—341no—
198#323—172no—
199#323—172no—
200#323—341no—
201C2×C17—21no—
202C2×C17—21no—
203C2×C17—41no—
204C2×C17—41no—
205C2×C17—81no—
206C2×C17#146511yescharacteristic
207C2×C17#212111yescharacteristic
208C2×C17#212311yescharacteristic
209C22×D4C2×Dic1711yescharacteristic
210C23×C4C2×Dic1711yescharacteristic
211C23×C4C2×Dic1711yescharacteristic
212C23×C4C2×Dic1711yescharacteristic
213C23×C4C22⋊D1711yescharacteristic
214C2×M16—681no—
215#123—22no—
216#123—22no—
217C4×D4—41no—
218C4×D4—41no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4C22⋊D1711yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4C2×Dic1711yescharacteristic
229C2×C22⋊C4C2×Dic1711yescharacteristic
230C2×C22⋊C4C22⋊D1711yescharacteristic
231C2×C22⋊C4C22×D1711yescharacteristic
232C2×C42—21no—
233C2×C42C22⋊D1711yescharacteristic
234C4⋊C8—681no—
235C22⋊C8—342no—
236C22⋊C8—342no—
237C22⋊C8—681no—
238#97—681no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244C17#1152811yescharacteristic, complemented, Sylow
245C24C22⋊Dic1711yescharacteristic
246C24C22×Dic1711yescharacteristic
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4#6322312yescomplemented
267C22×C4C22⋊Dic1711yescharacteristic
268C22×C4C22⋊Dic1711yescharacteristic
269C22×C4C22⋊Dic1711yescharacteristic
270C22×C4C2×C22⋊D1711yescharacteristic
271C22×C4C2×C22⋊D1711yescharacteristic
272M16—1362no—
273C2×C8—682no—
274C2×C8—682no—
275C4⋊C4—24no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—81no—
281C22⋊C4—24no—
282C22⋊C4—21no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—81no—
287C22⋊C4#6322312yescomplemented
288C42—21no—
289C42—21no—
290C42—81no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23C2×#6322311yescharacteristic
301C23C2×#6322311yescharacteristic
302C23C2×C22⋊Dic1711yescharacteristic
303D4—42no—
304D4—42no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
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—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4#1058750511yescharacteristic
335C2×C4#1058750511yescharacteristic
336C8—1362no—
337C8—1362no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—42no—
345C22—42no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—81no—
352C22#1176277311yescharacteristic
353C22C2×#1058750511yescharacteristic
354C22#1176279311yescharacteristic
355C4—21no—
356C4—21no—
357C4—42no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—42no—
363C4—81no—
364C4—81no—
365C2—21no—
366C2—21no—
367C2—41no—
368C2—41no—
369C2—81no—
370C2#1452712211yescharacteristic
371C2#1453043011yescharacteristic
372C2#1453043811yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC24×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2322C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C22⋊C4×C17C23
hypercenter#2322D17
hypocenterC17#11528
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2322

Complements

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

NH ≅ G/Ntype
3. #14526732C2semidirect
4. #14526714C2semidirect
82. C22×C4×C17M16semidirect
97. C22⋊C4×C17M16semidirect
244. C17#11528semidirect
266. C22×C4#63223semidirect
287. C22⋊C4#63223semidirect

Maximal subgroups

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

subgroupindex
#145271252
#145271222
#145267322
#145267142
C17×#23222
#1152817