Finite Group Databasebeta

← #16944683

Subgroups — #16944683

Order 4352.

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

normal subgroups only

This group has 442 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
1C4×#10587505C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic
3#14526808C212yesmaximal, complemented
4#14526784C212yesmaximal, complemented
5C17×#2633C211yesmaximal, characteristic
6C2×#10587505—21no—
7C2×C22⋊#3622—21no—
8C4×C22⋊Dic17—21no—
9#11762576C2212yes—
10#11762535—21no—
11#11762533C2212yes—
12C2×C17×#125C411yescharacteristic, complemented
13C2×C4×D4×C17C2211yescharacteristic
14C17×#399C411yescharacteristic, complemented
15C17×#399C2211yescharacteristic
16C17×#398—21no—
17C17×#397C411yescharacteristic, complemented
18C17×#397C2211yescharacteristic
19C17×#394—21no—
20C17×#385—21no—
21C17×#385—21no—
22C17×#383C411yescharacteristic, complemented
23C22×D4×C17D411yescharacteristic
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×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C4⋊C4×C17D411yescharacteristic
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17C2×C411yescharacteristic
42C2×C22⋊C4×C17C2311yescharacteristic
43C2×C42×C17C2×C411yescharacteristic
44C2×C22⋊Dic17—22no—
45C2×C4×Dic17—41no—
46C22×#3622—41no—
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×D4×C17—22no—
60C2×D4×C17—22no—
61C2×D4×C17—41no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
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—81no—
71C22×C4×C17—81no—
72C22×C4×C17C22⋊C411yescharacteristic
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22×C411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C22×Dic17—42no—
78C4⋊C4×C17—41no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—42no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—81no—
85C4⋊C4×C17—81no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C22⋊C4×C17—42no—
89C22⋊C4×C17—42no—
90C22⋊C4×C17—81no—
91C22⋊C4×C17—81no—
92C42×C17—21no—
93C42×C17—21no—
94C22⋊Dic17—24no—
95C22⋊Dic17—42no—
96C4×Dic17—82no—
97C2×#3622—42no—
98C2×#3622—82no—
99#11732—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—
111D4×C17—44no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—41no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
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—81no—
137C2×C4×C17—81no—
138C2×C4×C17#9911yescharacteristic
139C2×C4×C17#9911yescharacteristic
140C2×Dic17—44no—
141C2×Dic17—82no—
142C2×Dic17—82no—
143#3622—82no—
144#2633D1711yescharacteristic
145C4×#99—171no—
146#1459—171no—
147#1352—172no—
148#1344—172no—
149C22×C17—21no—
150C22×C17—21no—
151C22×C17—22no—
152C22×C17—22no—
153C22×C17—42no—
154C22×C17—42no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—81no—
160C22×C17—81no—
161C22×C17C2×#9911yescharacteristic
162C22×C17#41611yescharacteristic
163C22×C17C2×#10411yescharacteristic
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—41no—
167C4×C17—41no—
168C4×C17—41no—
169C4×C17—41no—
170C4×C17—81no—
171C4×C17—81no—
172C4×C17—81no—
173C4×C17—81no—
174Dic17—84no—
175C2×#125Dic1711yescharacteristic, complemented
176C2×C4×D4C2×D1711yescharacteristic
177C2×#99—341no—
178C2×C22⋊C8—341no—
179#399Dic1711yescharacteristic, complemented
180#399C2×D1711yescharacteristic
181#398—21no—
182#397Dic1711yescharacteristic, complemented
183#397C2×D1711yescharacteristic
184#394—21no—
185#385—21no—
186#385—21no—
187#383Dic1711yescharacteristic, complemented
188C4×C22⋊C4—341no—
189#342—172no—
190#324—341no—
191#323—172no—
192C2×C17—21no—
193C2×C17—21no—
194C2×C17—42no—
195C2×C17—42no—
196C2×C17#150111yescharacteristic
197C2×C17#212311yescharacteristic
198C2×C17#212711yescharacteristic
199C22×D4C22⋊D1711yescharacteristic
200C23×C4C22⋊D1711yescharacteristic
201C23×C4C22⋊D1711yescharacteristic
202C22×C8—681no—
203#125—41no—
204#125—41no—
205C4×D4—41no—
206C4×D4—41no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C2×Dic1711yescharacteristic
212C2×C4⋊C4C2×Dic1711yescharacteristic
213C2×C4⋊C4C22⋊D1711yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—342no—
219C2×C22⋊C4C2×Dic1711yescharacteristic
220C2×C22⋊C4C22×D1711yescharacteristic
221C2×C42—681no—
222C2×C42C2×Dic1711yescharacteristic
223#99—344no—
224C22⋊C8—342no—
225C22⋊C8—681no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—681no—
231#95C2×Dic1711yescharacteristic
232#95C2×Dic1711yescharacteristic
233C17#1173211yescharacteristic, complemented, Sylow
234C24C22⋊Dic1711yescharacteristic
235C24C2×C22⋊D1711yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4—682no—
250C22×C4C22⋊Dic1711yescharacteristic
251C22×C4C22⋊Dic1711yescharacteristic
252C22×C4C22⋊Dic1711yescharacteristic
253C22×C4C22×Dic1711yescharacteristic
254C22×C4C2×C22⋊D1711yescharacteristic
255C2×C8—682no—
256C2×C8—1362no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—81no—
270C22⋊C4—81no—
271C22⋊C4—344no—
272C22⋊C4—682no—
273C42—21no—
274C42—21no—
275C42—1362no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23#1058750812yes—
285C23#1058763612yes—
286C23C2×C22⋊Dic1711yescharacteristic
287D4—44no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
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—81no—
313C2×C4—81no—
314C2×C4—684no—
315C2×C4—1362no—
316C2×C4—1362no—
317C2×C4#1058750511yescharacteristic
318C2×C4#1058750511yescharacteristic
319C8—1362no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—42no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22—81no—
332C22C2×#1058750511yescharacteristic
333C22#1176280511yescharacteristic
334C22C2×#1058750811yescharacteristic
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—1364no—
346C2—21no—
347C2—21no—
348C2—42no—
349C2—42no—
350C2#1452727511yescharacteristic
351C2#1453043811yescharacteristic
352C2#1453045311yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2633C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#2633D17
hypocenterC17#11732
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2633

Complements

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

NH ≅ G/Ntype
3. #14526808C2semidirect
4. #14526784C2semidirect
12. C2×C17×#125C4semidirect
14. C17×#399C4semidirect
17. C17×#397C4semidirect
22. C17×#383C4semidirect
175. C2×#125Dic17semidirect
179. #399Dic17semidirect
182. #397Dic17semidirect
187. #383Dic17semidirect
233. C17#11732semidirect

Maximal subgroups

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

subgroupindex
C4×#105875052
#145271022
#145268082
#145267842
C17×#26332
#1173217