Finite Group Databasebeta

← #17957428

Subgroups — #17957428

Order 4352.

1434 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1434 conjugacy classes of subgroups — too many to draw. Only its 180 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×#10587545C212yesmaximal, complemented
2C2×#11763779C212yesmaximal
3C2×#11763689C212yesmaximal, complemented
4C22×#10587564C211yesmaximal, characteristic
5C22×#63247C2214yes—
6C2×#10587545—24no—
7C2×#10587545—28no—
8#11763779—28no—
9C2×C22⋊#3624—24no—
10#11763689—24no—
11C23×Dic34C2211yescharacteristic
12C24×Dic17C2211yescharacteristic
13C2×#10587564—24no—
14C2×#10587564—24no—
15C2×#10587564—24no—
16C22×C4⋊Dic17C411yescharacteristic
17C22×#63225C411yescharacteristic
18C22×#63225C2211yescharacteristic
19C22×C22⋊Dic17C411yescharacteristic
20C22×C22⋊C4×C17C411yescharacteristic
21C25×C17C2×C411yescharacteristic
22C23×C4×C17C2×C411yescharacteristic
23C23×C4×C17C2×C411yescharacteristic
24C23×Dic17—24no—
25C23×Dic17—22no—
26C23×Dic17—22no—
27C23×Dic17—24no—
28C23×Dic17C2×C411yescharacteristic
29C23×Dic17C2×C411yescharacteristic
30C23×Dic17C2×C411yescharacteristic
31C23×Dic17D414yes—
32C23×Dic17C2311yescharacteristic
33C22×Dic34—24no—
34C22×Dic34—22no—
35C22×Dic34—22no—
36C22×Dic34—24no—
37C22×Dic34D414yes—
38C2×#63247—24no—
39C2×#63247—24no—
40C2×#63247—24no—
41C2×#63247—24no—
42C2×#63247—24no—
43C2×#63247D418yes—
44C22×#3624—24no—
45C2×C22⋊C4×C17—22no—
46C2×C22⋊C4×C17—22no—
47C2×C22⋊C4×C17—22no—
48C2×C22⋊Dic17—22no—
49C2×C22⋊Dic17—22no—
50C2×C22⋊Dic17—22no—
51C2×C4⋊Dic17—22no—
52C2×C4⋊Dic17—22no—
53C2×C4⋊Dic17—22no—
54C2×#63225—22no—
55C2×#63225—22no—
56C2×#63225—22no—
57C2×#63225—22no—
58C2×#63225—22no—
59C2×#63225D414yes—
60#10587564—232no—
61C22⋊#3624—216no—
62#10587545—216no—
63C24×C17—24no—
64C24×C17—21no—
65C24×C17—21no—
66C24×C17—24no—
67C24×C17—42no—
68C24×C17C22⋊C412yes—
69C24×C17C22×C411yescharacteristic
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17—24no—
75C22×C4×C17—42no—
76C22×C4×C17C22⋊C412yes—
77C22×C4×C17C22⋊C412yes—
78C22×Dic17—21no—
79C22×Dic17—21no—
80C22×Dic17—21no—
81C22×Dic17—21no—
82C22×Dic17—21no—
83C22×Dic17—21no—
84C22×Dic17—24no—
85C22×Dic17—24no—
86C22×Dic17—24no—
87C22×Dic17—28no—
88C22×Dic17—28no—
89C22×Dic17—24no—
90C22×Dic17—42no—
91C22×Dic17—44no—
92C22×Dic17—44no—
93C22×Dic17—42no—
94C22×Dic17—42no—
95C22×Dic17—42no—
96C22×Dic17—42no—
97C22×Dic17C22⋊C412yes—
98C22×Dic17C22⋊C412yes—
99C22×Dic17C22⋊C412yes—
100C22×Dic17C2×D412yes—
101C22×Dic17C2×D412yes—
102C22×Dic17C2×D414yes—
103C22×Dic17C2×D412yes—
104C22×Dic17#4114yes—
105C2×Dic34—24no—
106C2×Dic34—28no—
107C2×Dic34—28no—
108C2×Dic34—24no—
109C2×Dic34—42no—
110C2×Dic34—44no—
111C2×Dic34—44no—
112C2×Dic34—42no—
113C2×Dic34—42no—
114C2×Dic34—42no—
115#63247—28no—
116#63247—28no—
117#63247—44no—
118#63247—44no—
119C2×#3624—28no—
120C2×#3624—44no—
121C2×#3624—44no—
122C22⋊C4×C17—28no—
123C22⋊Dic17—28no—
124C4⋊Dic17—28no—
125#63225—28no—
126#63225—28no—
127#10968—171nomaximal, Sylow
128C23×C17—22no—
129C23×C17—22no—
130C23×C17—24no—
131C23×C17—24no—
132C23×C17—22no—
133C23×C17—24no—
134C23×C17—22no—
135C23×C17—24no—
136C23×C17—416no—
137C23×C17—42no—
138C23×C17—42no—
139C23×C17—42no—
140C23×C17—42no—
141C23×C17#10114yescomplemented
142C23×C17C2×C22⋊C411yescharacteristic
143C23×C17C2×C22⋊C411yescharacteristic
144C23×C17C2×C22⋊C411yescharacteristic
145C23×C17C4×D414yes—
146C2×C4×C17—22no—
147C2×C4×C17—22no—
148C2×C4×C17—24no—
149C2×C4×C17—42no—
150C2×C4×C17—42no—
151C2×C4×C17—42no—
152C2×C4×C17—42no—
153C2×C4×C17—42no—
154C2×Dic17—22no—
155C2×Dic17—22no—
156C2×Dic17—22no—
157C2×Dic17—24no—
158C2×Dic17—24no—
159C2×Dic17—42no—
160C2×Dic17—42no—
161C2×Dic17—42no—
162C2×Dic17—42no—
163C2×Dic17—42no—
164C2×Dic17—42no—
165C2×Dic17—48no—
166C2×Dic17—48no—
167C2×Dic17—42no—
168C2×Dic17#12012yes—
169C2×Dic17#12012yes—
170C2×Dic17#12014yes—
171C2×Dic17#12114yes—
172C2×Dic17#12114yes—
173C2×Dic17#12114yes—
174C2×Dic17#12312yes—
175C2×Dic17#12312yes—
176C2×Dic17#12412yes—
177C2×Dic17#12712yes—
178Dic34—48no—
179Dic34—48no—
180#3624—48no—
181C22×C22⋊Q8—171no—
182C22×#101—172no—
183C2×#407—172no—
184C2×#343—172no—
185C22×C17—28no—
186C22×C17—21no—
187C22×C17—21no—
188C22×C17—21no—
189C22×C17—21no—
190C22×C17—28no—
191C22×C17—24no—
192C22×C17—48no—
193C22×C17—48no—
194C22×C17—44no—
195C22×C17—42no—
196C22×C17—42no—
197C22×C17#37911yescharacteristic
198C22×C17#38612yes—
199C22×C17#38612yes—
200C22×C17#39012yes—
201C22×C17C2×#10112yes—
202C22×C17#41312yes—
203C4×C17—42no—
204C4×C17—42no—
205Dic17—42no—
206Dic17—42no—
207Dic17—42no—
208Dic17#39214yes—
209Dic17#39414yes—
210C23×Q8—171no—
211C24×C4—171no—
212C22×M16—174no—
213C2×C22⋊Q8—344no—
214C2×C22⋊Q8—344no—
215C2×C22⋊Q8—344no—
216C22×C4⋊C4—171no—
217C22×C4⋊C4—171no—
218C22×C4⋊C4—171no—
219C22×C22⋊C4—171no—
220C22×C22⋊C4#59511yescharacteristic
221C2×#101—348no—
222C2×#101—344no—
223#407—348no—
224C2×C22⋊C8—344no—
225#343—344no—
226C2×C17—22no—
227C2×C17—22no—
228C2×C17—44no—
229C2×C17#143811yescharacteristic
230C2×C17#178411yescharacteristic
231C2×C17#178511yescharacteristic
232C2×C17#190114yes—
233C25C2×#59511yescharacteristic
234C22×Q8—174no—
235C22×Q8—342no—
236C22×Q8—342no—
237C22×Q8—344no—
238C22×Q8—344no—
239C23×C4—171no—
240C23×C4—174no—
241C23×C4—171no—
242C23×C4—171no—
243C23×C4—171no—
244C23×C4—342no—
245C23×C4—342no—
246C23×C4—344no—
247C23×C4—344no—
248C23×C4C2×#59511yescharacteristic
249C23×C4C2×#59511yescharacteristic
250C2×M16—178no—
251C2×M16—344no—
252C2×M16—344no—
253C2×M16—344no—
254C2×M16—344no—
255C2×M16—344no—
256C22×C8—344no—
257C22⋊Q8—3432no—
258C2×C4⋊C4—174no—
259C2×C4⋊C4—342no—
260C2×C4⋊C4—342no—
261C2×C4⋊C4—342no—
262C2×C4⋊C4—342no—
263C2×C4⋊C4—342no—
264C2×C4⋊C4—342no—
265C2×C4⋊C4—342no—
266C2×C4⋊C4—342no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—342no—
271C2×C22⋊C4—342no—
272C2×C22⋊C4—342no—
273#101—3416no—
274C22⋊C8—3416no—
275C17#1096811yescharacteristic, complemented, Sylow
276C24—21no—
277C24—21no—
278C24—24no—
279C24—24no—
280C24—42no—
281C24C22⋊#59512yes—
282C24C22×#59511yescharacteristic
283C2×Q8—344no—
284C2×Q8—348no—
285C2×Q8—348no—
286C2×Q8—344no—
287C2×Q8—682no—
288C2×Q8—684no—
289C2×Q8—682no—
290C2×Q8—684no—
291C2×Q8—682no—
292C2×Q8—682no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—174no—
300C22×C4—174no—
301C22×C4—172no—
302C22×C4—172no—
303C22×C4—172no—
304C22×C4—172no—
305C22×C4—172no—
306C22×C4—172no—
307C22×C4—344no—
308C22×C4—348no—
309C22×C4—348no—
310C22×C4—344no—
311C22×C4—344no—
312C22×C4—344no—
313C22×C4—341no—
314C22×C4—341no—
315C22×C4—341no—
316C22×C4—341no—
317C22×C4—341no—
318C22×C4—341no—
319C22×C4—682no—
320C22×C4—684no—
321C22×C4—682no—
322C22×C4—684no—
323C22×C4—682no—
324C22×C4—682no—
325C22×C4—682no—
326C22×C4C22⋊#59512yes—
327C22×C4C22⋊#59512yes—
328M16—348no—
329M16—348no—
330M16—684no—
331M16—684no—
332C2×C8—348no—
333C2×C8—684no—
334C2×C8—684no—
335C4⋊C4—348no—
336C4⋊C4—348no—
337C4⋊C4—348no—
338C22⋊C4—28no—
339C22⋊C4—348no—
340C23—22no—
341C23—22no—
342C23—24no—
343C23—24no—
344C23—22no—
345C23—24no—
346C23—22no—
347C23—24no—
348C23—416no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23#1058754514yescomplemented
354C23D4×#59514yes—
355C23C2×C22⋊#59511yescharacteristic
356C23C2×C22⋊#59511yescharacteristic
357C23C2×C22⋊#59511yescharacteristic
358Q8—688no—
359Q8—688no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—174no—
369C2×C4—172no—
370C2×C4—174no—
371C2×C4—172no—
372C2×C4—174no—
373C2×C4—172no—
374C2×C4—172no—
375C2×C4—174no—
376C2×C4—172no—
377C2×C4—172no—
378C2×C4—342no—
379C2×C4—342no—
380C2×C4—342no—
381C2×C4—344no—
382C2×C4—344no—
383C2×C4—688no—
384C2×C4—688no—
385C2×C4—682no—
386C2×C4—682no—
387C2×C4—682no—
388C2×C4—682no—
389C2×C4—682no—
390C2×C4—682no—
391C2×C4—682no—
392C8—688no—
393C22—28no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—28no—
399C22—24no—
400C22—48no—
401C22—48no—
402C22—42no—
403C22—44no—
404C22—42no—
405C22#1176375011yescharacteristic
406C22#1176375712yes—
407C22#1176375712yes—
408C22#1176376112yes—
409C22C2×#1058754512yes—
410C22#1176379612yes—
411C4—42no—
412C4—42no—
413C4—174no—
414C4—174no—
415C4—682no—
416C4—682no—
417C4—682no—
418C2—22no—
419C2—22no—
420C2—44no—
421C2#1454396611yescharacteristic
422C2#1454461811yescharacteristic
423C2#1454462111yescharacteristic
424C2#1454481114yes—
subgroupG/H
centerC23C2×C22⋊#595
commutatorC24×C17C22×C4
FrattiniC24C22×#595
FittingC22×C22⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC25×C17C2×C4
cosocleC23×Dic17C23
hypercenterC22×C22⋊C4#595
hypocenterC17#10968
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#10587545C2semidirect
3. C2×#11763689C2semidirect
141. C23×C17#101semidirect
275. C17#10968semidirect
353. C23#10587545semidirect

Maximal subgroups

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

subgroupindex
C22×#105875452
C2×#117637792
C2×#117636892
C22×#105875642
#1096817