Finite Group Databasebeta

← #16927855

Subgroups — #16927855

Order 4352.

546 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 99 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#14531398C211yesmaximal, characteristic
2#14527111C211yesmaximal, characteristic
3#14526691C212yesmaximal, complemented
4#14526689C212yesmaximal, complemented
5C17×#1462C211yesmaximal, characteristic
6C22×C22⋊Dic17C411yescharacteristic
7C22×C22⋊Dic17C411yescharacteristic
8#11762773C2212yes—
9#11762684—21no—
10#11762684—21no—
11#11762684—21no—
12#11762684—21no—
13#11762669C411yescharacteristic
14#11762669C411yescharacteristic
15#11762669C2211yescharacteristic
16#11762669C2211yescharacteristic
17#11762533—21no—
18#11762533—21no—
19C22×C22⋊C4×C17C2211yescharacteristic
20C17×#407C2212yes—
21C17×#323—21no—
22C17×#323—21no—
23C25×C17C2×C411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17C2311yescharacteristic
26C23×Dic17—21no—
27C23×Dic17—21no—
28C2×M16×C17—22no—
29C2×C22⋊C4×C17—22no—
30C2×C22⋊C4×C17—22no—
31C2×C22⋊C4×C17D411yescharacteristic
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C22⋊C4×C17D411yescharacteristic
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—42no—
38C2×C22⋊Dic17—42no—
39C2×C22⋊Dic17C2×C411yescharacteristic
40C2×C22⋊Dic17C2×C411yescharacteristic
41C2×C22⋊Dic17C2×C411yescharacteristic
42C2×C22⋊Dic17C2×C411yescharacteristic
43C2×C4⋊Dic17—21no—
44C2×C4⋊Dic17—21no—
45C2×C4⋊Dic17—21no—
46C2×C4⋊Dic17—21no—
47C2×#63223—22no—
48C17×C22⋊C8—22no—
49C17×C22⋊C8—22no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54C22⋊#3622—22no—
55C22⋊#3622—22no—
56C24×C17—22no—
57C24×C17—22no—
58C24×C17—21no—
59C24×C17—21no—
60C24×C17—81no—
61C24×C17—81no—
62C24×C17C22⋊C411yescharacteristic
63C24×C17C22⋊C411yescharacteristic
64C24×C17C22×C411yescharacteristic
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—81no—
68C22×C4×C17—81no—
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×C4×C17#4112yes—
74C22×C4×C17#4112yes—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—21no—
78C22×Dic17—21no—
79C22×Dic17—42no—
80C22×Dic17—42no—
81C22×Dic17—81no—
82C22×Dic17—81no—
83C22×Dic17—81no—
84C22×Dic17—81no—
85M16×C17—82no—
86C2×C8×C17—42no—
87C22⋊C4×C17—22no—
88C22⋊C4×C17—22no—
89C22⋊C4×C17—22no—
90C22⋊C4×C17—22no—
91C22⋊Dic17—42no—
92C22⋊Dic17—42no—
93C22⋊Dic17—41no—
94C22⋊Dic17—41no—
95C22⋊Dic17—41no—
96C22⋊Dic17—41no—
97C4⋊Dic17—81no—
98C4⋊Dic17—81no—
99C4⋊Dic17—81no—
100C4⋊Dic17—81no—
101#63223—82no—
102C2×#3622—42no—
103#8756—171nomaximal, Sylow
104C23×C17—24no—
105C23×C17—24no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—21no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17—41no—
116C23×C17—41no—
117C23×C17—41no—
118C23×C17—81no—
119C23×C17—81no—
120C23×C17—81no—
121C23×C17—81no—
122C23×C17—84no—
123C23×C17—84no—
124C23×C17#10012yes—
125C23×C17#10112yes—
126C23×C17C2×C22⋊C411yescharacteristic
127C23×C17#11711yescharacteristic
128C23×C17#11711yescharacteristic
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—42no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—42no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—81no—
138C2×C4×C17—81no—
139C2×C4×C17#12312yes—
140C2×C4×C17#12312yes—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17—41no—
152C2×Dic17—41no—
153C2×Dic17—84no—
154C2×Dic17—84no—
155C8×C17—82no—
156#3622—82no—
157#2315—171no—
158#1462—171no—
159#1462D1711yescharacteristic
160#1305—172no—
161#1305—172no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—22no—
165C22×C17—22no—
166C22×C17—21no—
167C22×C17—21no—
168C22×C17—21no—
169C22×C17—21no—
170C22×C17—21no—
171C22×C17—21no—
172C22×C17—42no—
173C22×C17—42no—
174C22×C17—42no—
175C22×C17—42no—
176C22×C17—42no—
177C22×C17—42no—
178C22×C17—41no—
179C22×C17—41no—
180C22×C17—41no—
181C22×C17—41no—
182C22×C17—84no—
183C22×C17—84no—
184C22×C17#38111yescharacteristic
185C22×C17C2×#10011yescharacteristic
186C22×C17C2×#10111yescharacteristic
187C4×C17—22no—
188C4×C17—22no—
189C4×C17—41no—
190C4×C17—41no—
191Dic17—81no—
192Dic17—81no—
193Dic17—81no—
194Dic17—81no—
195C22×C22⋊C4—171no—
196C22×C22⋊C4—171no—
197C22×C22⋊C4C2×D1711yescharacteristic
198#407—172no—
199#407C2×D1712yes—
200#385—341no—
201#385—341no—
202#385—341no—
203#385—341no—
204#380—171no—
205#380—171no—
206#380—171no—
207#380—171no—
208#323—21no—
209#323—21no—
210#323—341no—
211#323—341no—
212C2×C17—21no—
213C2×C17—21no—
214C2×C17—41no—
215C2×C17—41no—
216C2×C17—42no—
217C2×C17—42no—
218C2×C17#182111yescharacteristic
219C2×C17#182811yescharacteristic
220C2×C17#182911yescharacteristic
221C25C4×D1711yescharacteristic
222C23×C4—341no—
223C23×C4—341no—
224C23×C4C4×D1711yescharacteristic
225C23×C4C22×D1711yescharacteristic
226C2×M16—22no—
227C2×M16—342no—
228C2×C4⋊C4—341no—
229C2×C4⋊C4—341no—
230C2×C4⋊C4—341no—
231C2×C4⋊C4—341no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—171no—
235C2×C22⋊C4—171no—
236C2×C22⋊C4—171no—
237C2×C22⋊C4—171no—
238C2×C22⋊C4—341no—
239C2×C22⋊C4—341no—
240C2×C22⋊C4—682no—
241C2×C22⋊C4—682no—
242C2×C22⋊C4D6811yescharacteristic
243C2×C22⋊C4D6811yescharacteristic
244C2×C22⋊C4C22⋊D1711yescharacteristic
245C2×C22⋊C4C22⋊D1711yescharacteristic
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—342no—
249C22⋊C8—342no—
250#95—341no—
251#95—341no—
252#95—341no—
253#95—341no—
254C17#875611yescharacteristic, complemented, Sylow
255C24—22no—
256C24—22no—
257C24—21no—
258C24—21no—
259C24—81no—
260C24—81no—
261C24#6322711yescharacteristic
262C24#6322711yescharacteristic
263C24C2×C4×D1711yescharacteristic
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—341no—
269C22×C4—341no—
270C22×C4—341no—
271C22×C4—341no—
272C22×C4—682no—
273C22×C4—682no—
274C22×C4—1361no—
275C22×C4—1361no—
276C22×C4—1361no—
277C22×C4—1361no—
278C22×C4#6322711yescharacteristic
279C22×C4#6322711yescharacteristic
280C22×C4C2×D6811yescharacteristic
281C22×C4#6325412yes—
282C22×C4#6325412yes—
283C22×C4C2×C22⋊D1711yescharacteristic
284M16—82no—
285M16—1362no—
286C2×C8—42no—
287C2×C8—682no—
288C4⋊C4—1361no—
289C4⋊C4—1361no—
290C4⋊C4—1361no—
291C4⋊C4—1361no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—682no—
297C22⋊C4—682no—
298C22⋊C4—681no—
299C22⋊C4—681no—
300C22⋊C4—681no—
301C22⋊C4—681no—
302C23—24no—
303C23—24no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—84no—
321C23—84no—
322C23#1058749412yes—
323C23#1058749512yes—
324C23#1058756311yescharacteristic
325C23#1058756311yescharacteristic
326C23C2×#6322711yescharacteristic
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—681no—
338C2×C4—681no—
339C2×C4—681no—
340C2×C4—681no—
341C2×C4—681no—
342C2×C4—681no—
343C2×C4—681no—
344C2×C4—681no—
345C2×C4—681no—
346C2×C4—681no—
347C2×C4—681no—
348C2×C4—681no—
349C2×C4—1364no—
350C2×C4—1364no—
351C2×C4#1058757212yes—
352C2×C4#1058762112yes—
353C8—82no—
354C8—1362no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—84no—
376C22—84no—
377C22#1176267011yescharacteristic
378C22C2×#1058749411yescharacteristic
379C22C2×#1058749511yescharacteristic
380C4—22no—
381C4—22no—
382C4—41no—
383C4—41no—
384C4—1361no—
385C4—1361no—
386C4—1361no—
387C4—1361no—
388C2—21no—
389C2—21no—
390C2—41no—
391C2—41no—
392C2—42no—
393C2—42no—
394C2#1452886211yescharacteristic
395C2#1452889011yescharacteristic
396C2#1452889311yescharacteristic
subgroupG/H
centerC22#11762670
commutatorC24×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1462C2
socleC22×C17#381
prime radicalC25×C17C2×C4
cosocleC23×C4×C17C23
hypercenter#1462D17
hypocenterC17#8756
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
3. #14526691C2semidirect
4. #14526689C2semidirect
254. C17#8756semidirect

Maximal subgroups

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

subgroupindex
#145313982
#145271112
#145266912
#145266892
C17×#14622
#875617