Finite Group Databasebeta

← #17955970

Subgroups — #17955970

Order 4352.

920 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 126 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#14544640C211yesmaximal, characteristic, complemented
2#14544510C211yesmaximal, characteristic, complemented
3C2×#11763652C214yesmaximal, complemented
4C2×#11762712C211yesmaximal, characteristic
5C2×#10587544—21no—
6C2×C22⋊#3624C2212yes—
7C2×C22⋊#3624C2212yes—
8C2×#10587542—21no—
9#11763675—21no—
10#11763675—21no—
11#11763652—28no—
12C22×C4⋊Dic17C411yescharacteristic
13C22×C22⋊Dic17C412yes—
14C22×C22⋊Dic17C2212yes—
15C22×C4×Dic17C2211yescharacteristic
16#11762712—21no—
17#11762712—21no—
18#11762712—21no—
19#11762712—21no—
20C23×D4×C17C411yescharacteristic
21C25×C17C2×C412yes—
22C22×D4×C17—24no—
23C22×D4×C17—21no—
24C22×D4×C17—21no—
25C22×D4×C17—42no—
26C22×D4×C17—42no—
27C23×C4×C17C2×C411yescharacteristic
28C23×Dic17C2×C411yescharacteristic
29C23×Dic17C2×C412yes—
30C23×Dic17C2311yescharacteristic
31C22×#3624—22no—
32C22×#3624—22no—
33C2×C22⋊Dic17—22no—
34C2×C22⋊Dic17—22no—
35C2×C22⋊Dic17—22no—
36C2×C22⋊Dic17—22no—
37C2×C22⋊Dic17—24no—
38C2×C22⋊Dic17—24no—
39C2×C22⋊Dic17—44no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17—21no—
42C2×C4⋊Dic17—42no—
43C2×C4×Dic17—42no—
44C2×C4×Dic17D411yescharacteristic
45C2×C4×Dic17D411yescharacteristic
46C2×C4×Dic17D411yescharacteristic
47C2×C4×Dic17D411yescharacteristic
48C22⋊#3624—22no—
49C22⋊#3624—24no—
50C22⋊#3624D414yescomplemented
51#10587544—42no—
52#10587542—42no—
53C24×C17—22no—
54C24×C17—22no—
55C24×C17—24no—
56C24×C17—24no—
57C24×C17—42no—
58C24×C17—42no—
59C24×C17—42no—
60C24×C17—42no—
61C24×C17M1614yes—
62C24×C17C22×C411yescharacteristic
63C2×D4×C17—24no—
64C2×D4×C17—24no—
65C2×D4×C17—416no—
66C2×D4×C17—42no—
67C2×D4×C17—42no—
68C2×D4×C17—42no—
69C2×D4×C17—42no—
70C22×C4×C17—22no—
71C22×C4×C17—22no—
72C22×C4×C17—22no—
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C22⋊C411yescharacteristic
75C22×Dic17—22no—
76C22×Dic17—22no—
77C22×Dic17—22no—
78C22×Dic17—24no—
79C22×Dic17—41no—
80C22×Dic17—41no—
81C22×Dic17—41no—
82C22×Dic17—42no—
83C22×Dic17—42no—
84C22×Dic17C22⋊C411yescharacteristic
85C22×Dic17C22⋊C411yescharacteristic
86C22×Dic17C2×D411yescharacteristic
87C22×Dic17C2×D411yescharacteristic
88C22×Dic17C2×D412yes—
89C22×Dic17#4112yes—
90C2×#3624—24no—
91C2×#3624—42no—
92C2×#3624—42no—
93C2×#3624—42no—
94C2×#3624—42no—
95C2×#3624—42no—
96C22⋊Dic17—44no—
97C22⋊Dic17—44no—
98C22⋊Dic17—44no—
99C22⋊Dic17—42no—
100C22⋊Dic17—42no—
101C4⋊Dic17—42no—
102C4⋊Dic17—42no—
103C4×Dic17—22no—
104C4×Dic17—22no—
105C4×Dic17—41no—
106C4×Dic17—41no—
107#9653—171nomaximal, Sylow
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—21no—
112C23×C17—24no—
113C23×C17—28no—
114C23×C17—44no—
115C23×C17—44no—
116C23×C17—44no—
117C23×C17—44no—
118C23×C17—42no—
119C23×C17—42no—
120C23×C17—42no—
121C23×C17—42no—
122C23×C17—42no—
123C23×C17—42no—
124C23×C17—44no—
125C23×C17—84no—
126C23×C17—84no—
127C23×C17—84no—
128C23×C17—84no—
129C23×C17C2×C22⋊C411yescharacteristic
130C23×C17C4×D412yes—
131C23×C17C2×M1612yes—
132C23×C17#13112yes—
133D4×C17—48no—
134D4×C17—48no—
135C2×C4×C17—28no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17#9912yes—
141C2×C4×C17#10012yes—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—42no—
152C2×Dic17—42no—
153C2×Dic17—44no—
154C2×Dic17—42no—
155C2×Dic17—42no—
156C2×Dic17—42no—
157C2×Dic17—84no—
158C2×Dic17#10012yes—
159C2×Dic17#12112yes—
160C2×Dic17#12411yescharacteristic
161C2×Dic17#12711yescharacteristic
162#3624—44no—
163#3624—82no—
164C2×#390—171no—
165#1797—171no—
166#1749—171no—
167C2×#323—174no—
168C22×C17—22no—
169C22×C17—22no—
170C22×C17—22no—
171C22×C17—22no—
172C22×C17—22no—
173C22×C17—22no—
174C22×C17—44no—
175C22×C17—44no—
176C22×C17—44no—
177C22×C17—88no—
178C22×C17—88no—
179C22×C17—82no—
180C22×C17—82no—
181C22×C17—82no—
182C22×C17—82no—
183C22×C17#39011yescharacteristic
184C22×C17#40711yescharacteristic
185C22×C17#40811yescharacteristic
186C22×C17C2×#9911yescharacteristic
187C22×C17#41011yescharacteristic
188C22×C17C2×#10011yescharacteristic
189C22×C17C2×#10011yescharacteristic
190C22×C17#43514yes—
191C4×C17—22no—
192C4×C17—22no—
193Dic17—22no—
194Dic17—21no—
195Dic17—21no—
196Dic17—22no—
197Dic17—41no—
198Dic17—82no—
199C23×D4#59511yescharacteristic
200C22×C4⋊C4—171no—
201C22×C22⋊C4—172no—
202C22×C22⋊C4—172no—
203C22×C42—171no—
204C2×C4⋊C8—341no—
205C2×C22⋊C8—172no—
206C2×C22⋊C8—172no—
207C2×#97—341no—
208#390—341no—
209#390—341no—
210#390—341no—
211#390—341no—
212#336—341no—
213#336—341no—
214#323—348no—
215C2×C17—24no—
216C2×C17—84no—
217C2×C17#146211yescharacteristic
218C2×C17#146611yescharacteristic
219C2×C17#146911yescharacteristic
220C2×C17#147111yescharacteristic
221C2×C17#195511yescharacteristic
222C2×C17#196411yescharacteristic
223C2×C17#196511yescharacteristic
224C25C2×#59512yes—
225C22×D4—24no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4—42no—
229C22×D4—42no—
230C23×C4—171no—
231C23×C4—171no—
232C23×C4—172no—
233C23×C4C2×#59511yescharacteristic
234C22×C8—342no—
235C22×C8—342no—
236C2×C4⋊C4—341no—
237C2×C4⋊C4—341no—
238C2×C4⋊C4—682no—
239C2×C22⋊C4—344no—
240C2×C22⋊C4—342no—
241C2×C22⋊C4—342no—
242C2×C22⋊C4—342no—
243C2×C22⋊C4—342no—
244C2×C22⋊C4—344no—
245C2×C22⋊C4—684no—
246C2×C42—171no—
247C2×C42—171no—
248C2×C42—171no—
249C2×C42—171no—
250C2×C42—682no—
251C4⋊C8—682no—
252C22⋊C8—174no—
253C22⋊C8—342no—
254C22⋊C8—344no—
255#97—682no—
256C17#965311yescharacteristic, complemented, Sylow
257C24—22no—
258C24—22no—
259C24—24no—
260C24—24no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24#6324714yes—
266C24C22×#59511yescharacteristic
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—416no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—171no—
278C22×C4—171no—
279C22×C4—171no—
280C22×C4—172no—
281C22×C4—172no—
282C22×C4—171no—
283C22×C4—342no—
284C22×C4—342no—
285C22×C4—342no—
286C22×C4—344no—
287C22×C4—681no—
288C22×C4—681no—
289C22×C4—681no—
290C22×C4—682no—
291C22×C4—682no—
292C22×C4C22⋊#59511yescharacteristic
293C22×C4C22⋊#59511yescharacteristic
294C2×C8—344no—
295C2×C8—682no—
296C2×C8—682no—
297C2×C8—682no—
298C2×C8—682no—
299C2×C8—682no—
300C4⋊C4—682no—
301C4⋊C4—682no—
302C22⋊C4—684no—
303C22⋊C4—682no—
304C22⋊C4—682no—
305C22⋊C4—684no—
306C22⋊C4—684no—
307C42—342no—
308C42—342no—
309C42—681no—
310C42—681no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—24no—
316C23—28no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—44no—
328C23—84no—
329C23—84no—
330C23—84no—
331C23—84no—
332C23#1058767812yes—
333C23D4×#59512yes—
334C23C2×#6324712yes—
335C23C2×C22⋊#59511yescharacteristic
336D4—48no—
337D4—48no—
338C2×C4—28no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—172no—
344C2×C4—171no—
345C2×C4—171no—
346C2×C4—172no—
347C2×C4—341no—
348C2×C4—341no—
349C2×C4—341no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—341no—
353C2×C4—341no—
354C2×C4—341no—
355C2×C4—341no—
356C2×C4—682no—
357C2×C4—682no—
358C2×C4—684no—
359C2×C4—682no—
360C2×C4—682no—
361C2×C4—682no—
362C2×C4—1364no—
363C2×C4#1058755112yes—
364C2×C4#1058755312yes—
365C8—684no—
366C8—1362no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—44no—
374C22—44no—
375C22—44no—
376C22—88no—
377C22—88no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22#1176376111yescharacteristic
383C22#1176377911yescharacteristic
384C22#1176378311yescharacteristic
385C22C2×#1058755111yescharacteristic
386C22#1176378911yescharacteristic
387C22C2×#1058755311yescharacteristic
388C22D17⋊#10011yescharacteristic
389C22#1176384014yes—
390C4—22no—
391C4—22no—
392C4—342no—
393C4—341no—
394C4—341no—
395C4—342no—
396C4—681no—
397C4—1362no—
398C2—24no—
399C2—84no—
400C2#1454402911yescharacteristic
401C2#1454404411yescharacteristic
402C2#1454405411yescharacteristic
403C2#1454405911yescharacteristic
404C2#1454490811yescharacteristic
405C2#1454492311yescharacteristic
406C2#1454492611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC24×C17C22×C4
FrattiniC24C22×#595
Fitting, prime radicalC23×D4×C17C4
socleC23×C17C2×C22⋊C4
cosocleC23×Dic17C23
hypercenterC23×D4#595
hypocenterC17#9653
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #14544640C2semidirect
2. #14544510C2semidirect
3. C2×#11763652C2semidirect
50. C22⋊#3624D4semidirect
256. C17#9653semidirect

Maximal subgroups

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

subgroupindex
#145446402
#145445102
C2×#117636522
C2×#117627122
#965317