Finite Group Databasebeta

← #16945100

Subgroups — #16945100

Order 4352.

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

normal subgroups only

This group has 446 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#14528772C211yesmaximal, characteristic
2#14527122C211yesmaximal, characteristic
3#14526805C212yesmaximal, complemented
4#14526784C212yesmaximal, complemented
5C17×#2398C211yesmaximal, characteristic
6C2×#10587505—21no—
7#11762773—21no—
8#11762669—21no—
9#11762576C2212yes—
10#11762535—21no—
11#11762533C2212yes—
12C2×C4×Q8×C17C411yescharacteristic, complemented
13C2×C4×D4×C17C2211yescharacteristic
14C17×#400—21no—
15C17×#397C411yescharacteristic, complemented
16C17×#397C2211yescharacteristic
17C17×#394—21no—
18C17×#393C411yescharacteristic, complemented
19C17×#393C2211yescharacteristic
20C4×C22⋊C4×C17—21no—
21C4×C22⋊C4×C17—21no—
22C17×#376C411yescharacteristic, complemented
23C22×Q8×C17C2×C411yescharacteristic
24C22×D4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C4×Q8×C17—41no—
28C4×Q8×C17—41no—
29C4×D4×C17—41no—
30C4×D4×C17—41no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C22⋊C4×C17—21no—
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17C2×C411yescharacteristic
39C2×C22⋊C4×C17C2311yescharacteristic
40C2×C42×C17—21no—
41C2×C42×C17—21no—
42C2×C42×C17C2×C411yescharacteristic
43C2×C22⋊Dic17—22no—
44C2×C4⋊Dic17—41no—
45C2×#63223—41no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95C2×C411yescharacteristic
52C17×#95C2×C411yescharacteristic
53#10587505—42no—
54#10587502—41no—
55C22⋊#3622—22no—
56C22⋊#3622—41no—
57C24×C17C22⋊C411yescharacteristic
58C24×C17C2×D411yescharacteristic
59C2×Q8×C17—42no—
60C2×Q8×C17—41no—
61C2×D4×C17—21no—
62C2×D4×C17—21no—
63C2×D4×C17—41no—
64C2×D4×C17—41no—
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—42no—
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C22⋊C411yescharacteristic
78C22×C4×C17C22×C411yescharacteristic
79C22×C4×C17C2×D411yescharacteristic
80C22×Dic17—42no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—42no—
84C22⋊C4×C17—41no—
85C22⋊C4×C17—41no—
86C22⋊C4×C17—42no—
87C22⋊C4×C17—42no—
88C22⋊C4×C17—42no—
89C22⋊C4×C17—41no—
90C22⋊C4×C17—41no—
91C42×C17—21no—
92C42×C17—21no—
93C42×C17—42no—
94C42×C17—41no—
95C42×C17—41no—
96C22⋊Dic17—42no—
97C22⋊Dic17—42no—
98C4⋊Dic17—82no—
99#63223—82no—
100C2×#3622—42no—
101#11823—171nomaximal, Sylow
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17C2×C22⋊C411yescharacteristic
112C23×C17AGL(1,ℤ/8)11yescharacteristic
113C23×C17#13711yescharacteristic
114Q8×C17—42no—
115Q8×C17—42no—
116D4×C17—42no—
117D4×C17—42no—
118C2×C4×C17—22no—
119C2×C4×C17—22no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17—41no—
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—81no—
143C2×C4×C17—81no—
144C2×C4×C17#9911yescharacteristic
145C2×C4×C17#9911yescharacteristic
146C2×Dic17—82no—
147C2×Dic17—82no—
148C2×Dic17—82no—
149#3622—82no—
150#2398D1711yescharacteristic
151#1799—171no—
152#1465—171no—
153#1351—172no—
154#1344—172no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—42no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—81no—
168C22×C17—81no—
169C22×C17C2×#9911yescharacteristic
170C22×C17#41711yescharacteristic
171C22×C17#42111yescharacteristic
172C4×C17—21no—
173C4×C17—42no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17—41no—
177C4×C17—41no—
178C4×C17—81no—
179C4×C17—81no—
180C4×C17—81no—
181C4×C17#35412yes—
182Dic17—162no—
183C2×C4×Q8Dic1711yescharacteristic, complemented
184C2×C4×D4C2×D1711yescharacteristic
185C2×#99—341no—
186#407—341no—
187#400—21no—
188#397Dic1711yescharacteristic, complemented
189#397C2×D1711yescharacteristic
190#394—21no—
191#393Dic1711yescharacteristic, complemented
192#393C2×D1711yescharacteristic
193#380—341no—
194C4×C22⋊C4—21no—
195C4×C22⋊C4—21no—
196#376Dic1711yescharacteristic, complemented
197#342—172no—
198#324—341no—
199#323—172no—
200C2×C17—21no—
201C2×C17—21no—
202C2×C17—41no—
203C2×C17—41no—
204C2×C17—81no—
205C2×C17#151311yescharacteristic
206C2×C17#212011yescharacteristic
207C2×C17C2×#35411yescharacteristic
208C22×Q8C2×Dic1711yescharacteristic
209C22×D4C22⋊D1711yescharacteristic
210C23×C4C22⋊D1711yescharacteristic
211C23×C4C22⋊D1711yescharacteristic
212C2×M16—681no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×D4—41no—
216C4×D4—41no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—681no—
219C2×C4⋊C4C2×Dic1711yescharacteristic
220C2×C4⋊C4C22⋊D1711yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—342no—
226C2×C22⋊C4C2×Dic1711yescharacteristic
227C2×C22⋊C4C22×D1711yescharacteristic
228C2×C42—21no—
229C2×C42—21no—
230C2×C42C2×Dic1711yescharacteristic
231#99—682no—
232C22⋊C8—342no—
233C22⋊C8—681no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—681no—
240#95C2×Dic1711yescharacteristic
241#95C2×Dic1711yescharacteristic
242C17#1182311yescharacteristic, complemented, Sylow
243C24C22⋊Dic1711yescharacteristic
244C24C2×C22⋊D1711yescharacteristic
245C2×Q8—42no—
246C2×Q8—41no—
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—42no—
260C22×C4—42no—
261C22×C4—682no—
262C22×C4C22⋊Dic1711yescharacteristic
263C22×C4C22⋊Dic1711yescharacteristic
264C22×C4C22⋊Dic1711yescharacteristic
265C22×C4C22×Dic1711yescharacteristic
266C22×C4C2×C22⋊D1711yescharacteristic
267M16—1362no—
268C2×C8—682no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—1362no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—682no—
281C22⋊C4—682no—
282C42—21no—
283C42—21no—
284C42—42no—
285C42—41no—
286C42—41no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23#1058763511yescharacteristic
297C23#1058763711yescharacteristic
298C23C2×C22⋊Dic1711yescharacteristic
299Q8—42no—
300Q8—42no—
301D4—42no—
302D4—42no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—1362no—
330C2×C4—1362no—
331C2×C4—1362no—
332C2×C4#1058750511yescharacteristic
333C2×C4#1058750511yescharacteristic
334C8—1362no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—81no—
348C22—81no—
349C22C2×#1058750511yescharacteristic
350C22#1176281011yescharacteristic
351C22#1176282611yescharacteristic
352C4—21no—
353C4—41no—
354C4—42no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—2722no—
362C4#1176260412yes—
363C2—21no—
364C2—21no—
365C2—41no—
366C2—41no—
367C2—81no—
368C2#1452732711yescharacteristic
369C2#1453042511yescharacteristic
370C2C2×#1176260411yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2398C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#2398D17
hypocenterC17#11823
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2398

Complements

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

NH ≅ G/Ntype
3. #14526805C2semidirect
4. #14526784C2semidirect
12. C2×C4×Q8×C17C4semidirect
15. C17×#397C4semidirect
18. C17×#393C4semidirect
22. C17×#376C4semidirect
183. C2×C4×Q8Dic17semidirect
188. #397Dic17semidirect
191. #393Dic17semidirect
196. #376Dic17semidirect
242. C17#11823semidirect

Maximal subgroups

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

subgroupindex
#145287722
#145271222
#145268052
#145267842
C17×#23982
#1182317