Finite Group Databasebeta

← #16929366

Subgroups — #16929366

Order 4352.

740 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#11762745C211yesmaximal, characteristic
2#14531382C211yesmaximal, characteristic
3#14528875C212yesmaximal, complemented
4#14526664C212yesmaximal, complemented
5C2×C17×#323C211yesmaximal, characteristic
6C22×C4⋊Dic17C2211yescharacteristic
7C22×#63225—21no—
8C22×C22⋊Dic17—21no—
9#11762745—22no—
10#11762745C414yescomplemented
11#11762702—24no—
12#11762669C2212yes—
13#11762665—21no—
14C4×C22⋊Dic17—21no—
15C2×#10587502C2211yescharacteristic
16#11762572—24no—
17C22×C22⋊C4×C17C2211yescharacteristic
18C2×C17×C22⋊C8C2212yes—
19C17×#323—22no—
20C25×C17D411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17C2311yescharacteristic
23C23×Dic17—21no—
24C23×Dic17—21no—
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—24no—
31C2×C22⋊Dic17—22no—
32C2×C22⋊Dic17—22no—
33C2×C22⋊Dic17—22no—
34C2×C22⋊Dic17—44no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—22no—
37C2×C4⋊Dic17—22no—
38C2×C4⋊Dic17C2×C412yes—
39C2×C4⋊Dic17D414yes—
40C2×#63225—24no—
41C2×#63225—44no—
42C2×C4×Dic17—22no—
43C17×C22⋊C8—22no—
44C17×C22⋊C8—22no—
45#10587502—22no—
46#10587502—22no—
47#10587502—21no—
48#10587502C2×C412yes—
49C24×C17—21no—
50C24×C17—22no—
51C24×C17—21no—
52C24×C17—22no—
53C24×C17—42no—
54C24×C17—42no—
55C24×C17C22⋊C412yes—
56C24×C17C2×D411yescharacteristic
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—42no—
60C22×C4×C17—42no—
61C22×C4×C17C22⋊C412yes—
62C22×C4×C17C22×C411yescharacteristic
63C22×C4×C17C2×D412yes—
64C22×C4×C17C2×D411yescharacteristic
65C22×C4×C17#4112yes—
66C22×Dic17—24no—
67C22×Dic17—22no—
68C22×Dic17—22no—
69C22×Dic17—24no—
70C22×Dic17—42no—
71C22×Dic17—42no—
72C22×Dic17—42no—
73C22×Dic17—42no—
74C22×Dic17—41no—
75C22×Dic17—41no—
76C2×C8×C17—44no—
77C2×C8×C17—42no—
78C22⋊C4×C17—24no—
79C22⋊C4×C17—24no—
80C22⋊Dic17—48no—
81C22⋊Dic17—42no—
82C22⋊Dic17—42no—
83C22⋊Dic17—82no—
84C22⋊Dic17—82no—
85C4⋊Dic17—24no—
86C4⋊Dic17—42no—
87C4⋊Dic17—44no—
88#63225—48no—
89C4×Dic17—82no—
90#9024—171nomaximal, Sylow
91C23×C17—22no—
92C23×C17—22no—
93C23×C17—22no—
94C23×C17—22no—
95C23×C17—22no—
96C23×C17—42no—
97C23×C17—42no—
98C23×C17—48no—
99C23×C17—42no—
100C23×C17—42no—
101C23×C17—42no—
102C23×C17—42no—
103C23×C17—81no—
104C23×C17—81no—
105C23×C17—84no—
106C23×C17C2×C22⋊C411yescharacteristic
107C23×C17#12011yescharacteristic
108C23×C17#12311yescharacteristic
109C23×C17#13512yes—
110C23×C17#13712yes—
111C2×C4×C17—24no—
112C2×C4×C17—22no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—81no—
119C2×C4×C17C4×D412yes—
120C2×C4×C17#12112yes—
121C2×Dic17—24no—
122C2×Dic17—48no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—44no—
129C2×Dic17—44no—
130C2×Dic17—44no—
131C2×Dic17—82no—
132C2×Dic17—82no—
133C8×C17—82no—
134C2×#399—171no—
135#2312—171no—
136#1824—172no—
137C2×#323D1711yescharacteristic
138#1294—172no—
139C22×C17—24no—
140C22×C17—24no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—44no—
146C22×C17—42no—
147C22×C17—42no—
148C22×C17—42no—
149C22×C17—82no—
150C22×C17—84no—
151C22×C17—84no—
152C22×C17#38611yescharacteristic
153C22×C17#41611yescharacteristic
154C22×C17#41911yescharacteristic
155C22×C17#45212yes—
156C22×C17#45312yes—
157C22×C17#45812yes—
158C22×C17#48512yes—
159C4×C17—22no—
160C4×C17—22no—
161C4×C17—42no—
162Dic17—44no—
163Dic17—84no—
164Dic17—82no—
165C22×C4⋊C4—171no—
166C22×C4⋊C4—341no—
167C22×C22⋊C4—341no—
168C22×C22⋊C4C2×D1711yescharacteristic
169C2×C22⋊C8C2×D1712yes—
170#399—174no—
171#399—342no—
172#388—344no—
173#380—172no—
174#379—341no—
175C4×C22⋊C4—341no—
176C2×#95—171no—
177#340—344no—
178#323—22no—
179C2×C17—21no—
180C2×C17—21no—
181C2×C17—22no—
182C2×C17—82no—
183C2×C17#161814yes—
184C2×C17#187311yescharacteristic
185C2×C17#188711yescharacteristic
186C2×C17#189011yescharacteristic
187C25D6811yescharacteristic
188C23×C4—341no—
189C23×C4—341no—
190C23×C4D6811yescharacteristic
191C23×C4C22×D1711yescharacteristic
192C22×C8—22no—
193C2×C4⋊C4—172no—
194C2×C4⋊C4—174no—
195C2×C4⋊C4—342no—
196C2×C4⋊C4—342no—
197C2×C4⋊C4—341no—
198C2×C4⋊C4—344no—
199C2×C4⋊C4—684no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—342no—
203C2×C22⋊C4—344no—
204C2×C22⋊C4—342no—
205C2×C22⋊C4—342no—
206C2×C22⋊C4—684no—
207C2×C22⋊C4C4×D1712yes—
208C2×C22⋊C4C22⋊D1712yes—
209C2×C42—342no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212#95—172no—
213#95—342no—
214#95—342no—
215#95—341no—
216C17#902411yescharacteristic, complemented, Sylow
217C24—21no—
218C24—22no—
219C24—21no—
220C24—22no—
221C24—42no—
222C24—42no—
223C24#6322712yes—
224C24C2×D6811yescharacteristic
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—344no—
230C22×C4—342no—
231C22×C4—344no—
232C22×C4—342no—
233C22×C4—682no—
234C22×C4—682no—
235C22×C4—682no—
236C22×C4—682no—
237C22×C4—681no—
238C22×C4—681no—
239C22×C4#6322712yes—
240C22×C4C2×C4×D1711yescharacteristic
241C22×C4D4×D1712yes—
242C22×C4#6325412yes—
243C22×C4C2×C22⋊D1711yescharacteristic
244C2×C8—44no—
245C2×C8—42no—
246C4⋊C4—344no—
247C4⋊C4—684no—
248C4⋊C4—688no—
249C4⋊C4—682no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—688no—
253C22⋊C4—682no—
254C22⋊C4—682no—
255C22⋊C4—1362no—
256C22⋊C4—1362no—
257C42—1362no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23—48no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—81no—
271C23—81no—
272C23—84no—
273C23#1058756811yescharacteristic
274C23#1058757211yescharacteristic
275C23#1058759012yes—
276C23#1058759512yes—
277C23C2×#6322711yescharacteristic
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—81no—
286C2×C4—344no—
287C2×C4—688no—
288C2×C4—682no—
289C2×C4—682no—
290C2×C4—682no—
291C2×C4—682no—
292C2×C4—684no—
293C2×C4—684no—
294C2×C4—684no—
295C2×C4—682no—
296C2×C4—1362no—
297C2×C4—1362no—
298C2×C4#1058756712yes—
299C2×C4#1058762512yes—
300C8—82no—
301C22—24no—
302C22—24no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—44no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—82no—
312C22—84no—
313C22—84no—
314C22#1176268611yescharacteristic
315C22#1176280611yescharacteristic
316C22#1176281911yescharacteristic
317C22#1176297312yes—
318C22#1176297912yes—
319C22#1176300012yes—
320C22#1176315112yes—
321C4—22no—
322C4—22no—
323C4—42no—
324C4—684no—
325C4—1364no—
326C4—1362no—
327C2—21no—
328C2—21no—
329C2—22no—
330C2—82no—
331C2#1452783414yes—
332C2#1452909411yescharacteristic
333C2#1452918011yescharacteristic
334C2#1452919511yescharacteristic
subgroupG/H
centerC23#10587568
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC2×C17×#323C2
socleC23×C17#120
prime radicalC25×C17D4
cosocleC23×C4×C17C23
hypercenterC2×#323D17
hypocenterC17#9024
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #14528875C2semidirect
4. #14526664C2semidirect
10. #11762745C4semidirect
216. C17#9024semidirect

Maximal subgroups

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

subgroupindex
C2×#117627452
#145313822
#145288752
#145266642
C2×C17×#3232
#902417