Finite Group Databasebeta

← #16929329

Subgroups — #16929329

Order 4352.

836 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 135 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#14531382C211yesmaximal, characteristic
2#14528719C212yesmaximal, complemented
3C2×#11762533C211yesmaximal, characteristic
4#14526663C212yesmaximal, complemented
5C2×C17×#393C211yesmaximal, characteristic
6C2×#10587506—24no—
7C2×C22⋊#3622C2212yes—
8#11762702—24no—
9#11762669C2212yes—
10#11762665—21no—
11C4×C22⋊Dic17—21no—
12#11762533—22no—
13C23×Q8×C17C2211yescharacteristic
14C22×C22⋊C4×C17—21no—
15C22×C22⋊C4×C17C2211yescharacteristic
16C17×#393—22no—
17C17×#393C414yescomplemented
18C2×C17×#95—21no—
19C2×C17×#95C2211yescharacteristic
20C25×C17D411yescharacteristic
21C22×Q8×C17—21no—
22C22×Q8×C17—22no—
23C22×Q8×C17—22no—
24C22×Q8×C17—44no—
25C22×Q8×C17C2×C412yes—
26C23×C4×C17—21no—
27C23×C4×C17—21no—
28C23×C4×C17D411yescharacteristic
29C23×C4×C17C2311yescharacteristic
30C2×C22⋊C4×C17—22no—
31C2×C22⋊C4×C17—22no—
32C2×C22⋊C4×C17—24no—
33C2×C22⋊C4×C17—44no—
34C2×C22⋊C4×C17C2×C412yes—
35C2×C22⋊C4×C17D412yes—
36C2×C22⋊Dic17—22no—
37C2×C22⋊Dic17—22no—
38C2×C22⋊Dic17—22no—
39C2×C4⋊Dic17D414yes—
40C2×C4×Dic17—22no—
41C22×#3622—22no—
42C17×#95—24no—
43C17×#95—21no—
44C17×#95C2×C412yes—
45#10587506—48no—
46#10587502—22no—
47#10587502—22no—
48C22⋊#3622—22no—
49C22⋊#3622—22no—
50C24×C17—21no—
51C24×C17—22no—
52C24×C17—21no—
53C24×C17—22no—
54C24×C17—42no—
55C24×C17—42no—
56C24×C17C22⋊C412yes—
57C24×C17C2×D411yescharacteristic
58C2×Q8×C17—28no—
59C2×Q8×C17—24no—
60C2×Q8×C17—44no—
61C2×Q8×C17—44no—
62C2×Q8×C17—44no—
63C2×Q8×C17—48no—
64C2×Q8×C17—42no—
65C22×C4×C17—24no—
66C22×C4×C17—22no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—22no—
70C22×C4×C17—42no—
71C22×C4×C17—42no—
72C22×C4×C17—41no—
73C22×C4×C17—41no—
74C22×C4×C17—42no—
75C22×C4×C17—42no—
76C22×C4×C17—42no—
77C22×C4×C17—42no—
78C22×C4×C17C22⋊C412yes—
79C22×C4×C17C22×C411yescharacteristic
80C22×C4×C17C2×D412yes—
81C22×C4×C17C2×D411yescharacteristic
82C22×C4×C17#4112yes—
83C22×Dic17—24no—
84C22⋊C4×C17—24no—
85C22⋊C4×C17—24no—
86C22⋊C4×C17—48no—
87C22⋊Dic17—42no—
88C22⋊Dic17—42no—
89C22⋊Dic17—82no—
90C22⋊Dic17—82no—
91C4⋊Dic17—44no—
92C4×Dic17—82no—
93C2×#3622—44no—
94C2×#3622—42no—
95#9016—171nomaximal, Sylow
96C23×C17—22no—
97C23×C17—22no—
98C23×C17—22no—
99C23×C17—22no—
100C23×C17—22no—
101C23×C17—42no—
102C23×C17—42no—
103C23×C17—48no—
104C23×C17—42no—
105C23×C17—42no—
106C23×C17—42no—
107C23×C17—42no—
108C23×C17—81no—
109C23×C17—81no—
110C23×C17—84no—
111C23×C17C2×C22⋊C411yescharacteristic
112C23×C17#12011yescharacteristic
113C23×C17#12311yescharacteristic
114C23×C17#13512yes—
115C23×C17#13712yes—
116Q8×C17—28no—
117Q8×C17—48no—
118Q8×C17—44no—
119C2×C4×C17—24no—
120C2×C4×C17—24no—
121C2×C4×C17—22no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×C4×C17—44no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—44no—
130C2×C4×C17—42no—
131C2×C4×C17—42no—
132C2×C4×C17—42no—
133C2×C4×C17—42no—
134C2×C4×C17—81no—
135C2×C4×C17—84no—
136C2×C4×C17C4×D412yes—
137C2×C4×C17#12112yes—
138C2×Dic17—44no—
139C2×Dic17—44no—
140C2×Dic17—44no—
141#3622—82no—
142C2×#393D1711yescharacteristic
143#2312—171no—
144#1787—172no—
145C2×#323—171no—
146#1294—172no—
147C22×C17—24no—
148C22×C17—24no—
149C22×C17—21no—
150C22×C17—21no—
151C22×C17—21no—
152C22×C17—21no—
153C22×C17—44no—
154C22×C17—42no—
155C22×C17—42no—
156C22×C17—42no—
157C22×C17—82no—
158C22×C17—84no—
159C22×C17—84no—
160C22×C17#38611yescharacteristic
161C22×C17#41611yescharacteristic
162C22×C17#41911yescharacteristic
163C22×C17#45212yes—
164C22×C17#45512yes—
165C22×C17B(4,𝔽2)12yes—
166C22×C17#48512yes—
167C4×C17—22no—
168C4×C17—22no—
169C4×C17—42no—
170C4×C17—44no—
171C4×C17—82no—
172Dic17—84no—
173C23×Q8C2×D1711yescharacteristic
174C22×C22⋊C4—21no—
175C22×C22⋊C4C2×D1711yescharacteristic
176C2×#103—344no—
177C2×C22⋊C8—172no—
178#393—22no—
179#393Dic1714yescomplemented
180#388—344no—
181#380—172no—
182#379—341no—
183C4×C22⋊C4—341no—
184C2×#95—21no—
185C2×#95C2×D1711yescharacteristic
186#323—342no—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17—22no—
190C2×C17—82no—
191C2×C17#160614yes—
192C2×C17#187311yescharacteristic
193C2×C17#188511yescharacteristic
194C2×C17#188911yescharacteristic
195C25C22⋊D1711yescharacteristic
196C22×Q8—21no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—44no—
200C22×Q8C2×Dic1712yes—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4C22⋊D1711yescharacteristic
204C23×C4C22×D1711yescharacteristic
205C22×C8—342no—
206C2×C4⋊C4—174no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—342no—
212C2×C22⋊C4—342no—
213C2×C22⋊C4—342no—
214C2×C22⋊C4C2×Dic1712yes—
215C2×C22⋊C4C22⋊D1712yes—
216C2×C42—342no—
217#103—688no—
218C22⋊C8—342no—
219C22⋊C8—342no—
220#95—24no—
221#95—21no—
222#95—342no—
223#95—342no—
224#95C2×Dic1712yes—
225C17#901611yescharacteristic, complemented, Sylow
226C24—21no—
227C24—22no—
228C24—21no—
229C24—22no—
230C24—42no—
231C24—42no—
232C24C22⋊Dic1712yes—
233C24C2×C22⋊D1711yescharacteristic
234C2×Q8—28no—
235C2×Q8—24no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—48no—
240C2×Q8—42no—
241C22×C4—24no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—344no—
255C22×C4C22⋊Dic1712yes—
256C22×C4D4×D1712yes—
257C22×C4#6325412yes—
258C22×C4C22×Dic1711yescharacteristic
259C22×C4C2×C22⋊D1711yescharacteristic
260C2×C8—684no—
261C2×C8—682no—
262C4⋊C4—684no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—48no—
266C22⋊C4—682no—
267C22⋊C4—682no—
268C22⋊C4—1362no—
269C22⋊C4—1362no—
270C42—1362no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—42no—
277C23—42no—
278C23—48no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—81no—
284C23—81no—
285C23—84no—
286C23#1058762111yescharacteristic
287C23#1058762311yescharacteristic
288C23#1058762812yes—
289C23#1058763612yes—
290C23C2×C22⋊Dic1711yescharacteristic
291Q8—28no—
292Q8—48no—
293Q8—44no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—84no—
311C2×C4—684no—
312C2×C4—684no—
313C2×C4—684no—
314C2×C4D4×Dic1712yes—
315C2×C4#1058762512yes—
316C8—1362no—
317C22—24no—
318C22—24no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—44no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—82no—
328C22—84no—
329C22—84no—
330C22#1176268911yescharacteristic
331C22#1176280511yescharacteristic
332C22#1176281811yescharacteristic
333C22#1176296912yes—
334C22#1176298612yes—
335C22#1176299512yes—
336C22#1176314712yes—
337C4—22no—
338C4—22no—
339C4—42no—
340C4—44no—
341C4—82no—
342C4—1364no—
343C2—21no—
344C2—21no—
345C2—22no—
346C2—82no—
347C2#1452776214yes—
348C2#1452909711yescharacteristic
349C2#1452917311yescharacteristic
350C2#1452919311yescharacteristic
subgroupG/H
centerC23#10587623
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC2×C17×#393C2
socleC23×C17#120
prime radicalC25×C17D4
cosocleC23×C4×C17C23
hypercenterC2×#393D17
hypocenterC17#9016
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
2. #14528719C2semidirect
4. #14526663C2semidirect
17. C17×#393C4semidirect
179. #393Dic17semidirect
225. C17#9016semidirect

Maximal subgroups

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

subgroupindex
#145313822
#145287192
C2×#117625332
#145266632
C2×C17×#3932
#901617