Finite Group Databasebeta

← #16944663

Subgroups — #16944663

Order 4352.

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

normal subgroups only

This group has 470 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
1C4×#10587505C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic
3#14526805C212yesmaximal, complemented
4#14526784C212yesmaximal, complemented
5C17×#2398C211yesmaximal, characteristic
6C2×#10587505—21no—
7C2×C22⋊#3622—21no—
8C4×C22⋊Dic17—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—
45C22×#3622—41no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95C2×C411yescharacteristic
52C17×#95C2×C411yescharacteristic
53#10587505—24no—
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—22no—
62C2×D4×C17—22no—
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—42no—
73C22×C4×C17—42no—
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C22×C411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×Dic17—42no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—42no—
83C22⋊C4×C17—41no—
84C22⋊C4×C17—41no—
85C22⋊C4×C17—42no—
86C22⋊C4×C17—42no—
87C22⋊C4×C17—42no—
88C22⋊C4×C17—42no—
89C42×C17—21no—
90C42×C17—21no—
91C42×C17—42no—
92C42×C17—41no—
93C42×C17—41no—
94C22⋊Dic17—24no—
95C22⋊Dic17—42no—
96C4×Dic17—82no—
97C2×#3622—42no—
98C2×#3622—82no—
99#11728—171nomaximal, Sylow
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17#10412yes—
109C23×C17C2×C22⋊C411yescharacteristic
110C23×C17#13512yes—
111Q8×C17—42no—
112Q8×C17—42no—
113D4×C17—44no—
114C2×C4×C17—22no—
115C2×C4×C17—22no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
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—81no—
139C2×C4×C17—81no—
140C2×C4×C17#9911yescharacteristic
141C2×C4×C17#9911yescharacteristic
142C2×Dic17—44no—
143C2×Dic17—82no—
144C2×Dic17—82no—
145#3622—82no—
146#2398D1711yescharacteristic
147C4×#99—171no—
148#1459—171no—
149#1351—172no—
150#1344—172no—
151C22×C17—21no—
152C22×C17—21no—
153C22×C17—22no—
154C22×C17—22no—
155C22×C17—42no—
156C22×C17—42no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—81no—
162C22×C17—81no—
163C22×C17C2×#9911yescharacteristic
164C22×C17#41611yescharacteristic
165C22×C17C2×#10411yescharacteristic
166C4×C17—21no—
167C4×C17—42no—
168C4×C17—41no—
169C4×C17—41no—
170C4×C17—41no—
171C4×C17—41no—
172C4×C17—81no—
173C4×C17—81no—
174C4×C17—81no—
175C4×C17#35412yes—
176Dic17—84no—
177C2×C4×Q8Dic1711yescharacteristic, complemented
178C2×C4×D4C2×D1711yescharacteristic
179C2×#99—341no—
180C2×C22⋊C8—341no—
181#400—21no—
182#397Dic1711yescharacteristic, complemented
183#397C2×D1711yescharacteristic
184#394—21no—
185#393Dic1711yescharacteristic, complemented
186#393C2×D1711yescharacteristic
187C4×C22⋊C4—21no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—341no—
190#376Dic1711yescharacteristic, complemented
191#342—172no—
192#324—341no—
193#323—172no—
194C2×C17—21no—
195C2×C17—21no—
196C2×C17—42no—
197C2×C17—42no—
198C2×C17#150111yescharacteristic
199C2×C17#212011yescharacteristic
200C2×C17C2×#35411yescharacteristic
201C22×Q8C2×Dic1711yescharacteristic
202C22×D4C22⋊D1711yescharacteristic
203C23×C4C22⋊D1711yescharacteristic
204C23×C4C22⋊D1711yescharacteristic
205C22×C8—681no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×D4—41no—
209C4×D4—41no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C2×Dic1711yescharacteristic
212C2×C4⋊C4C22⋊D1711yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—342no—
218C2×C22⋊C4C2×Dic1711yescharacteristic
219C2×C22⋊C4C22×D1711yescharacteristic
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—681no—
223C2×C42C2×Dic1711yescharacteristic
224#99—344no—
225C22⋊C8—342no—
226C22⋊C8—681no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—681no—
233#95C2×Dic1711yescharacteristic
234#95C2×Dic1711yescharacteristic
235C17#1172811yescharacteristic, complemented, Sylow
236C24C22⋊Dic1711yescharacteristic
237C24C2×C22⋊D1711yescharacteristic
238C2×Q8—42no—
239C2×Q8—41no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—682no—
254C22×C4C22⋊Dic1711yescharacteristic
255C22×C4C22⋊Dic1711yescharacteristic
256C22×C4C22⋊Dic1711yescharacteristic
257C22×C4C22×Dic1711yescharacteristic
258C22×C4C2×C22⋊D1711yescharacteristic
259C2×C8—682no—
260C2×C8—1362no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—344no—
271C22⋊C4—682no—
272C42—21no—
273C42—21no—
274C42—42no—
275C42—41no—
276C42—41no—
277C42—1362no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#1058750812yes—
287C23#1058763612yes—
288C23C2×C22⋊Dic1711yescharacteristic
289Q8—42no—
290Q8—42no—
291D4—44no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—684no—
319C2×C4—1362no—
320C2×C4—1362no—
321C2×C4#1058750511yescharacteristic
322C2×C4#1058750511yescharacteristic
323C8—1362no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22—81no—
336C22C2×#1058750511yescharacteristic
337C22#1176280511yescharacteristic
338C22C2×#1058750811yescharacteristic
339C4—21no—
340C4—41no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—1364no—
349C4#1176260412yes—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2#1452727511yescharacteristic
355C2#1453042511yescharacteristic
356C2C2×#1176260411yescharacteristic
subgroupG/H
centerC2×C4#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#11728
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ 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
177. C2×C4×Q8Dic17semidirect
182. #397Dic17semidirect
185. #393Dic17semidirect
190. #376Dic17semidirect
235. C17#11728semidirect

Maximal subgroups

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

subgroupindex
C4×#105875052
#145271022
#145268052
#145267842
C17×#23982
#1172817