Finite Group Databasebeta

← #16930375

Subgroups — #16930375

Order 4352.

612 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 111 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#14534589C211yesmaximal, characteristic
2#14531382C211yesmaximal, characteristic
3#14527357C212yesmaximal, complemented
4#14526652C212yesmaximal, complemented
5C2×C17×#323C211yesmaximal, characteristic
6C2×#10587610C2211yescharacteristic
7#11762745C412yescomplemented
8#11762732—22no—
9#11762721C412yescomplemented
10#11762702—24no—
11#11762682—21no—
12#11762669C2211yescharacteristic
13#11762669C2212yes—
14#11762665—21no—
15#11762665—21no—
16C4×C22⋊Dic17—21no—
17#11762547—24no—
18C22×C22⋊C4×C17C2211yescharacteristic
19C2×C17×C22⋊C8C2212yes—
20C17×#323—22no—
21C25×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17C2311yescharacteristic
24C22×Dic34C2×C411yescharacteristic
25C22×C8×C17—22no—
26C2×C22⋊C4×C17—22no—
27C2×C22⋊C4×C17—22no—
28C2×C22⋊C4×C17C2×C412yes—
29C2×C22⋊C4×C17D412yes—
30C2×C22⋊Dic17—21no—
31C2×C22⋊Dic17—22no—
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—22no—
34C2×C22⋊Dic17—22no—
35C2×C22⋊Dic17—22no—
36C2×C22⋊Dic17—22no—
37C2×C4⋊Dic17—22no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17C2×C411yescharacteristic
40#10587610—22no—
41#10587610—22no—
42C2×#63225—22no—
43C2×#63225—21no—
44C2×C4×Dic17—22no—
45C17×C22⋊C8—22no—
46C17×C22⋊C8—22no—
47#10587502—22no—
48#10587502—22no—
49#10587502C2×C412yes—
50#10587502D414yes—
51C24×C17—21no—
52C24×C17—21no—
53C24×C17—22no—
54C24×C17—22no—
55C24×C17—81no—
56C24×C17—81no—
57C24×C17C22⋊C412yes—
58C24×C17C2×D411yescharacteristic
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—81no—
62C22×C4×C17—81no—
63C22×C4×C17C22⋊C412yes—
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D412yes—
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17#4112yes—
68C22×Dic17—21no—
69C22×Dic17—22no—
70C22×Dic17—21no—
71C22×Dic17—24no—
72C2×Dic34—22no—
73C2×Dic34—81no—
74C2×C8×C17—42no—
75C2×C8×C17—82no—
76C22⋊C4×C17—24no—
77C22⋊C4×C17—24no—
78C22⋊Dic17—41no—
79C22⋊Dic17—41no—
80C22⋊Dic17—42no—
81C22⋊Dic17—42no—
82C22⋊Dic17—42no—
83C22⋊Dic17—82no—
84C22⋊Dic17—82no—
85C22⋊Dic17—82no—
86C22⋊Dic17—82no—
87C4⋊Dic17—22no—
88C4⋊Dic17—82no—
89C4⋊Dic17—81no—
90#63225—42no—
91#63225—82no—
92C4×Dic17—82no—
93#9201—171nomaximal, Sylow
94C23×C17—22no—
95C23×C17—22no—
96C23×C17—22no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17—42no—
100C23×C17—42no—
101C23×C17—42no—
102C23×C17—42no—
103C23×C17—42no—
104C23×C17—42no—
105C23×C17—81no—
106C23×C17—81no—
107C23×C17—84no—
108C23×C17—84no—
109C23×C17C2×C22⋊C411yescharacteristic
110C23×C17#12011yescharacteristic
111C23×C17#12311yescharacteristic
112C23×C17#13512yes—
113C23×C17#13712yes—
114C2×C4×C17—22no—
115C2×C4×C17—42no—
116C2×C4×C17—41no—
117C2×C4×C17—42no—
118C2×C4×C17—41no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×C4×C17—81no—
122C2×C4×C17C4×D412yes—
123C2×C4×C17#12112yes—
124C2×Dic17—42no—
125C2×Dic17—44no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129C2×Dic17—44no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—42no—
133C2×Dic17—44no—
134Dic34—82no—
135C8×C17—82no—
136#2626—171no—
137#2312—171no—
138#1520—172no—
139C2×#323D1711yescharacteristic
140#1289—172no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—42no—
148C22×C17—42no—
149C22×C17—44no—
150C22×C17—42no—
151C22×C17—42no—
152C22×C17—42no—
153C22×C17—82no—
154C22×C17—84no—
155C22×C17—84no—
156C22×C17#38611yescharacteristic
157C22×C17#41611yescharacteristic
158C22×C17#41911yescharacteristic
159C22×C17#45212yes—
160C22×C17#48512yes—
161C4×C17—22no—
162C4×C17—22no—
163C4×C17—42no—
164Dic17—82no—
165Dic17—84no—
166Dic17—81no—
167Dic17—81no—
168C2×C22⋊Q8—171no—
169C22×C22⋊C4C2×D1711yescharacteristic
170C2×C22⋊C8C2×D1712yes—
171#399—172no—
172#396—342no—
173#393—172no—
174#388—344no—
175#385—341no—
176#380—172no—
177#380—171no—
178#379—341no—
179#379—341no—
180C4×C22⋊C4—341no—
181#328—344no—
182#323—22no—
183C2×C17—21no—
184C2×C17—22no—
185C2×C17—21no—
186C2×C17—22no—
187C2×C17—82no—
188C2×C17#187311yescharacteristic
189C2×C17#191111yescharacteristic
190C2×C17#191311yescharacteristic
191C25D6811yescharacteristic
192C22×Q8—171no—
193C23×C4D6811yescharacteristic
194C23×C4C22×D1711yescharacteristic
195C22×C8—22no—
196C22⋊Q8—342no—
197C22⋊Q8—342no—
198C2×C4⋊C4—171no—
199C2×C4⋊C4—341no—
200C2×C4⋊C4—342no—
201C2×C4⋊C4—341no—
202C2×C4⋊C4—342no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—341no—
206C2×C22⋊C4—342no—
207C2×C22⋊C4—341no—
208C2×C22⋊C4—342no—
209C2×C22⋊C4—342no—
210C2×C22⋊C4—342no—
211C2×C22⋊C4—342no—
212C2×C22⋊C4C4×D1712yes—
213C2×C22⋊C4C22⋊D1712yes—
214C2×C42—342no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217#95—172no—
218#95—174no—
219#95—342no—
220#95—342no—
221C17#920111yescharacteristic, complemented, Sylow
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—81no—
227C24—81no—
228C24#6322712yes—
229C24C2×D6811yescharacteristic
230C2×Q8—342no—
231C2×Q8—1361no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4—344no—
237C22×C4—342no—
238C22×C4—341no—
239C22×C4—341no—
240C22×C4#6322712yes—
241C22×C4C2×C4×D1711yescharacteristic
242C22×C4D4×D1712yes—
243C22×C4#6325412yes—
244C22×C4C2×C22⋊D1711yescharacteristic
245C2×C8—42no—
246C2×C8—82no—
247C4⋊C4—342no—
248C4⋊C4—682no—
249C4⋊C4—1362no—
250C4⋊C4—1362no—
251C4⋊C4—1361no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—681no—
255C22⋊C4—681no—
256C22⋊C4—682no—
257C22⋊C4—682no—
258C22⋊C4—682no—
259C22⋊C4—1362no—
260C22⋊C4—1362no—
261C22⋊C4—1362no—
262C22⋊C4—1362no—
263C42—1362no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—81no—
276C23—81no—
277C23—84no—
278C23—84no—
279C23#1058756811yescharacteristic
280C23#1058757211yescharacteristic
281C23#1058759012yes—
282C23#1058759512yes—
283C23C2×#6322711yescharacteristic
284Q8—1362no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—81no—
293C2×C4—684no—
294C2×C4—682no—
295C2×C4—684no—
296C2×C4—682no—
297C2×C4—682no—
298C2×C4—682no—
299C2×C4—684no—
300C2×C4—682no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4#1058756712yes—
304C2×C4#1058762512yes—
305C8—82no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22—84no—
321C22#1176268611yescharacteristic
322C22#1176280611yescharacteristic
323C22#1176281911yescharacteristic
324C22#1176297312yes—
325C22#1176315112yes—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—1364no—
330C4—1362no—
331C4—1361no—
332C4—1361no—
333C2—22no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—82no—
338C2#1452909411yescharacteristic
339C2#1452931411yescharacteristic
340C2#1452932411yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC2×C17×#323C2
socleC22×C17#386
prime radicalC25×C17D4
cosocleC23×C4×C17C23
hypercenterC2×#323D17
hypocenterC17#9201
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #14527357C2semidirect
4. #14526652C2semidirect
7. #11762745C4semidirect
9. #11762721C4semidirect
221. C17#9201semidirect

Maximal subgroups

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

subgroupindex
#145345892
#145313822
#145273572
#145266522
C2×C17×#3232
#920117