Finite Group Databasebeta

← #16925686

Subgroups — #16925686

Order 4352.

874 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C22×#10587562C211yesmaximal, characteristic
2#14528852C211yesmaximal, characteristic, complemented
3C2×#11762534C212yesmaximal, complemented
4C2×#11762532C212yesmaximal
5C17×#1817C211yesmaximal, characteristic, complemented
6C2×#10587562—22no—
7C2×#10587562—22no—
8C2×#10587562—24no—
9C2×#10587562—24no—
10C22×#63225C411yescharacteristic
11C22×#63225C411yescharacteristic
12C22×#63225C2211yescharacteristic
13C22×#63227C411yescharacteristic
14C22×#63227C411yescharacteristic
15C22×#63227C2211yescharacteristic
16C2×C22⋊#3622C2212yes—
17#11762564—22no—
18#11762534—24no—
19#11762532—24no—
20C22×C42×C17C2211yescharacteristic
21C2×C17×C22⋊C8C2212yes—
22C17×#336—22no—
23C24×D17C2×C411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17C2×C411yescharacteristic
26C23×C4×C17C2311yescharacteristic
27C23×Dic17C2×C411yescharacteristic
28C23×Dic17C2×C411yescharacteristic
29C23×Dic17C2×C411yescharacteristic
30C22×C8×C17—22no—
31C2×C42×C17—22no—
32C2×C42×C17—22no—
33C2×C42×C17D412yes—
34C2×C42×C17D412yes—
35C2×#63227—22no—
36C2×#63227—22no—
37C2×#63227—22no—
38C2×#63227—22no—
39C2×#63227—21no—
40C2×#63227—22no—
41C2×#63227—21no—
42C2×#63227—22no—
43C2×#63227—22no—
44C2×#63227D414yes—
45C2×#63225—22no—
46C2×#63225—22no—
47C2×#63225—21no—
48C2×#63225—22no—
49C2×#63225—21no—
50C2×#63225—22no—
51C2×#63225—22no—
52C2×#63225—22no—
53C2×#63225—22no—
54C2×#63225D414yes—
55C22×#3622—22no—
56#10587562—216no—
57#10587562—216no—
58C17×C22⋊C8—22no—
59C17×C22⋊C8—22no—
60C22⋊#3622—22no—
61C22⋊#3622—22no—
62C24×C17C22×C411yescharacteristic
63C23×D17—21no—
64C23×D17—21no—
65C23×D17—22no—
66C23×D17—22no—
67C23×D17—42no—
68C23×D17—42no—
69C23×D17C22⋊C412yes—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17—22no—
75C22×C4×C17—22no—
76C22×C4×C17—42no—
77C22×C4×C17—42no—
78C22×C4×C17C22⋊C412yes—
79C22×C4×C17C22⋊C412yes—
80C22×C4×C17C2×D411yescharacteristic
81C22×C4×C17C2×D412yes—
82C22×C4×C17C2×D412yes—
83C22×C4×C17C2×D411yescharacteristic
84C22×Dic17—21no—
85C22×Dic17—21no—
86C22×Dic17—21no—
87C22×Dic17—21no—
88C22×Dic17—21no—
89C22×Dic17—21no—
90C22×Dic17—22no—
91C22×Dic17—22no—
92C22×Dic17—42no—
93C22×Dic17—42no—
94C22×Dic17C22⋊C412yes—
95C22×Dic17C22⋊C412yes—
96C22×Dic17C22⋊C412yes—
97C2×C8×C17—44no—
98C2×C8×C17—42no—
99C42×C17—24no—
100C42×C17—24no—
101#63227—28no—
102#63227—28no—
103#63227—24no—
104#63227—24no—
105#63225—28no—
106#63225—28no—
107#63225—24no—
108#63225—24no—
109C2×#3622—44no—
110C2×#3622—42no—
111#8309—171nomaximal, Sylow
112C23×C17—22no—
113C23×C17—22no—
114C23×C17C2×C22⋊C411yescharacteristic
115C23×C17C2×C22⋊C411yescharacteristic
116C23×C17C2×C22⋊C411yescharacteristic
117C23×C17#13112yes—
118C23×C17#13112yes—
119C22×D17—22no—
120C22×D17—24no—
121C22×D17—42no—
122C22×D17—42no—
123C22×D17—42no—
124C22×D17—41no—
125C22×D17—41no—
126C22×D17—42no—
127C22×D17—42no—
128C22×D17—42no—
129C22×D17—42no—
130C22×D17—84no—
131C22×D17—84no—
132C2×C4×C17—22no—
133C2×C4×C17—22no—
134C2×C4×C17—22no—
135C2×C4×C17—22no—
136C2×C4×C17—22no—
137C2×C4×C17—22no—
138C2×C4×C17—24no—
139C2×C4×C17—42no—
140C2×C4×C17—42no—
141C2×C4×C17—42no—
142C2×C4×C17—42no—
143C2×C4×C17—42no—
144C2×C4×C17—42no—
145C2×C4×C17#12012yes—
146C2×C4×C17#12012yes—
147C2×Dic17—22no—
148C2×Dic17—22no—
149C2×Dic17—22no—
150C2×Dic17—24no—
151C2×Dic17—42no—
152C2×Dic17—42no—
153C2×Dic17—42no—
154C2×Dic17—41no—
155C2×Dic17—41no—
156C2×Dic17—42no—
157C2×Dic17—42no—
158C2×Dic17—42no—
159C2×Dic17—42no—
160C8×C17—82no—
161#3622—82no—
162C22×#126—171no—
163#1817—171no—
164#1817D1711yescharacteristic, complemented
165C2×#324—172no—
166C2×#323—172no—
167C22×C17—24no—
168C22×C17—24no—
169C22×C17—21no—
170C22×C17—21no—
171C22×C17—21no—
172C22×C17—21no—
173C22×C17#37911yescharacteristic
174C22×C17#40812yes—
175C22×C17#40812yes—
176C22×C17#40811yescharacteristic
177C22×C17#40811yescharacteristic
178C22×C17#41012yes—
179C22×C17#41312yes—
180C2×D17—42no—
181C2×D17—42no—
182C2×D17—42no—
183C2×D17—84no—
184C2×D17—82no—
185C2×D17—82no—
186C2×D17—84no—
187C4×C17—22no—
188C4×C17—22no—
189C4×C17—42no—
190C4×C17—42no—
191Dic17—42no—
192Dic17—42no—
193Dic17—42no—
194C2×#126—344no—
195C2×#126—344no—
196C2×#126—342no—
197C2×#126—342no—
198C22×C4⋊C4—171no—
199C22×C4⋊C4—171no—
200C22×C4⋊C4—171no—
201C22×C22⋊C4—171no—
202C22×C22⋊C4—171no—
203C22×C22⋊C4—171no—
204C22×C42C2×D1711yescharacteristic
205C2×C22⋊C8—172no—
206C2×C22⋊C8C2×D1712yes—
207#336—22no—
208#336—342no—
209#324—344no—
210#323—344no—
211C2×C17—22no—
212C2×C17—22no—
213C2×C17#147312yes—
214C2×C17#147312yes—
215C2×C17#178011yescharacteristic
216C2×C17#178211yescharacteristic
217C2×C17#178511yescharacteristic
218D17—82no—
219C25—171no—
220C23×C4—171no—
221C23×C4—171no—
222C23×C4—171no—
223C23×C4C4×D1711yescharacteristic
224C23×C4C4×D1711yescharacteristic
225C23×C4C22×D1711yescharacteristic
226C22×C8—22no—
227C22×C8—342no—
228#126—3416no—
229#126—3416no—
230C2×C4⋊C4—174no—
231C2×C4⋊C4—342no—
232C2×C4⋊C4—342no—
233C2×C4⋊C4—342no—
234C2×C4⋊C4—342no—
235C2×C4⋊C4—341no—
236C2×C4⋊C4—342no—
237C2×C4⋊C4—341no—
238C2×C4⋊C4—342no—
239C2×C4⋊C4—342no—
240C2×C22⋊C4—174no—
241C2×C22⋊C4—342no—
242C2×C22⋊C4—342no—
243C2×C22⋊C4—342no—
244C2×C22⋊C4—342no—
245C2×C22⋊C4—341no—
246C2×C22⋊C4—342no—
247C2×C22⋊C4—341no—
248C2×C22⋊C4—342no—
249C2×C22⋊C4—342no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42D6812yes—
253C2×C42C22⋊D1712yes—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—342no—
257C22⋊C8—342no—
258C17#830911yescharacteristic, complemented, Sylow
259C24—172no—
260C24—341no—
261C24—341no—
262C24—342no—
263C24—342no—
264C24—682no—
265C24—682no—
266C24C2×C4×D1711yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—172no—
276C22×C4—172no—
277C22×C4—172no—
278C22×C4—341no—
279C22×C4—341no—
280C22×C4—341no—
281C22×C4—342no—
282C22×C4—342no—
283C22×C4—341no—
284C22×C4—341no—
285C22×C4—341no—
286C22×C4—682no—
287C22×C4—682no—
288C22×C4#6322712yes—
289C22×C4#6322712yes—
290C22×C4C2×D6811yescharacteristic
291C22×C4D4×D1712yes—
292C22×C4D4×D1712yes—
293C22×C4C2×C22⋊D1711yescharacteristic
294C2×C8—44no—
295C2×C8—42no—
296C2×C8—684no—
297C2×C8—682no—
298C4⋊C4—348no—
299C4⋊C4—348no—
300C4⋊C4—344no—
301C4⋊C4—344no—
302C22⋊C4—348no—
303C22⋊C4—348no—
304C22⋊C4—344no—
305C22⋊C4—344no—
306C42—24no—
307C42—24no—
308C23—22no—
309C23—22no—
310C23—342no—
311C23—344no—
312C23—682no—
313C23—682no—
314C23—682no—
315C23—682no—
316C23—682no—
317C23—681no—
318C23—681no—
319C23—682no—
320C23—682no—
321C23—1364no—
322C23—1364no—
323C23C22⋊C4×D1711yescharacteristic
324C23C22⋊C4×D1711yescharacteristic
325C23#1058758712yes—
326C23#1058759312yes—
327C23C2×#6322711yescharacteristic
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—344no—
342C2×C4—342no—
343C2×C4—342no—
344C2×C4—342no—
345C2×C4—682no—
346C2×C4—682no—
347C2×C4—682no—
348C2×C4—681no—
349C2×C4—681no—
350C2×C4—682no—
351C2×C4—682no—
352C2×C4—682no—
353C2×C4—682no—
354C2×C4#1058756812yes—
355C2×C4#1058762312yes—
356C8—82no—
357C8—1362no—
358C22—24no—
359C22—24no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—682no—
365C22—682no—
366C22—682no—
367C22—1364no—
368C22—1364no—
369C22—1362no—
370C22—1362no—
371C22#1176266411yescharacteristic
372C22#1176277412yes—
373C22#1176277412yes—
374C22#1176277511yescharacteristic
375C22#1176277711yescharacteristic
376C22#1176278212yes—
377C22#1176279112yes—
378C4—22no—
379C4—22no—
380C4—42no—
381C4—42no—
382C4—682no—
383C4—682no—
384C4—682no—
385C2—22no—
386C2—22no—
387C2—1362no—
388C2#1452715112yes—
389C2#1452715412yes—
390C2#1452869511yescharacteristic
391C2#1452870111yescharacteristic
392C2#1452871011yescharacteristic
subgroupG/H
centerC23C2×#63227
commutatorC24×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1817C2
socleC23×C17C2×C22⋊C4
prime radicalC24×D17C2×C4
cosocleC23×C4×C17C23
hypercenter#1817D17
hypocenterC17#8309
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. #14528852C2semidirect
3. C2×#11762534C2semidirect
5. C17×#1817C2semidirect
164. #1817D17semidirect
258. C17#8309semidirect

Maximal subgroups

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

subgroupindex
C22×#105875622
#145288522
C2×#117625342
C2×#117625322
C17×#18172
#830917