Finite Group Databasebeta

← #16953487

Subgroups — #16953487

Order 4352.

498 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763045C211yesmaximal, characteristic
2C2×#11762637C211yesmaximal, characteristic, complemented
3C2×#11762611C211yesmaximal, characteristic, complemented
4C2×#11762611C211yesmaximal, characteristic, complemented
5#14528813C211yesmaximal, characteristic, complemented
6#14526876C211yesmaximal, characteristic, complemented
7C2×C17×C22⋊C16C211yesmaximal, characteristic, complemented
8C22×D136C2211yescharacteristic
9C2×#10587616—21no—
10#11763045—22no—
11#11763045—22no—
12C2×#10587489C2211yescharacteristic
13C2×#10587489C2211yescharacteristic
14C2×#10587489C2211yescharacteristic
15C2×#10587492—21no—
16#11762669—21no—
17#11762637—22no—
18#11762611—22no—
19#11762611—22no—
20#11762572—21no—
21C23×C8×C17C2211yescharacteristic
22C22×C16×C17C2211yescharacteristic
23C22×C16×C17C2211yescharacteristic
24C17×C22⋊C16—22no—
25C23×C4×C17D411yescharacteristic
26C22×C22⋊D17—41no—
27C22×D68—21no—
28C22×C8×C17—21no—
29C22×C8×C17—21no—
30C22×C8×C17—21no—
31C22×C8×C17—21no—
32C22×C8×C17—21no—
33C22×C8×C17—21no—
34C22×C8×C17D411yescharacteristic
35C22×C8×C17C2311yescharacteristic
36C2×C22⋊Dic17—41no—
37#10587616—24no—
38#10587616—42no—
39C2×#63227—41no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17—21no—
42C2×C4⋊Dic17—21no—
43C2×D136—21no—
44C2×D136—21no—
45C2×D136—21no—
46C2×D136—21no—
47C2×D136—21no—
48C2×D136D412yes—
49C2×C16×C17—21no—
50C2×C16×C17—21no—
51C2×C16×C17—21no—
52C2×C16×C17—21no—
53C2×C16×C17—21no—
54C2×C16×C17D412yescomplemented
55#10587502—41no—
56#10587492—24no—
57#10587489—21no—
58#10587489—21no—
59#10587489—21no—
60#10587489—21no—
61#10587489—21no—
62#10587489D412yes—
63C24×C17D811yescharacteristic
64C23×D17—41no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17D811yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C2×C22⋊D17—44no—
74C2×C22⋊D17—82no—
75C2×C22⋊D17—82no—
76C22×Dic17—41no—
77C22×Dic17—41no—
78C22×Dic17—41no—
79C2×D68—22no—
80C2×D68—41no—
81C2×D68—41no—
82C2×D68—41no—
83C2×D68—41no—
84C2×D68—41no—
85C2×C8×C17—21no—
86C2×C8×C17—21no—
87C2×C8×C17—21no—
88C2×C8×C17—41no—
89C2×C8×C17—41no—
90C2×C8×C17—41no—
91C2×C8×C17—41no—
92C2×C8×C17C2×D411yescharacteristic
93C2×C8×C17C2×D411yescharacteristic
94C2×C8×C17C2×D411yescharacteristic
95C2×C8×C17#4111yescharacteristic
96C2×C8×C17#4111yescharacteristic
97C2×C8×C17#4111yescharacteristic
98C22⋊Dic17—82no—
99#63227—44no—
100C4⋊Dic17—22no—
101C4⋊Dic17—41no—
102C4⋊Dic17—41no—
103C4⋊Dic17—41no—
104C4⋊Dic17—41no—
105C4⋊Dic17—41no—
106D136—22no—
107D136—22no—
108D136—41no—
109D136—41no—
110C16×C17—22no—
111C16×C17—41no—
112#13381—171nomaximal, Sylow
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17—21no—
118C23×C17—21no—
119C23×C17SD3212yes—
120C23×C17C2×D811yescharacteristic
121C22×D17—42no—
122C22×D17—81no—
123C22×D17—81no—
124C22×D17—81no—
125C22×D17—81no—
126C22×D17—81no—
127C22×D17—81no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17#12011yescharacteristic
136C2×C4×C17#12111yescharacteristic
137C2×C4×C17#12311yescharacteristic
138C2×C4×C17#13511yescharacteristic
139C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C17#13711yescharacteristic
141C22⋊D17—88no—
142C2×Dic17—42no—
143C2×Dic17—81no—
144C2×Dic17—81no—
145C2×Dic17—81no—
146C2×Dic17—81no—
147C2×Dic17—81no—
148C2×Dic17—81no—
149C2×Dic17—81no—
150C2×Dic17—81no—
151D68—42no—
152D68—42no—
153D68—81no—
154D68—81no—
155C8×C17—41no—
156C8×C17#12111yescharacteristic
157C8×C17#12111yescharacteristic
158C8×C17#12311yescharacteristic
159C8×C17#12411yescharacteristic
160C2×#466—171no—
161C2×#367—171no—
162C2×#357—171no—
163C2×#357—171no—
164C2×C22⋊C16D1711yescharacteristic, complemented
165#1808—171no—
166#1380—171no—
167C22×C17—22no—
168C22×C17—22no—
169C22×C17—22no—
170C22×C17—41no—
171C22×C17—41no—
172C22×C17—41no—
173C22×C17—41no—
174C22×C17#44711yescharacteristic
175C22×C17#46611yescharacteristic
176C22×C17#48011yescharacteristic
177C22×C17C2×SD3211yescharacteristic
178C22×C17#50811yescharacteristic
179C22×C17#50911yescharacteristic
180C22×C17#50911yescharacteristic
181C2×D17—82no—
182C2×D17—82no—
183C2×D17—82no—
184C2×D17—161no—
185C2×D17—161no—
186C2×D17—161no—
187C2×D17—161no—
188C4×C17—41no—
189C4×C17#39411yescharacteristic
190C4×C17#45211yescharacteristic
191C4×C17#46811yescharacteristic
192C4×C17#48211yescharacteristic
193Dic17—82no—
194Dic17—161no—
195Dic17—161no—
196C22×D8—171no—
197C23×C8C2×D1711yescharacteristic
198C2×#121—341no—
199C22×C16C2×D1711yescharacteristic
200C22×C16C2×D1711yescharacteristic
201#466—342no—
202#466—342no—
203C2×#107—171no—
204C2×#107—171no—
205C2×#107—171no—
206C2×#102—341no—
207#380—341no—
208#367—342no—
209#357—342no—
210#357—342no—
211C22⋊C16—22no—
212#340—341no—
213C2×C17—42no—
214C2×C17#203311yescharacteristic
215C2×C17#218611yescharacteristic
216C2×C17#218711yescharacteristic
217C2×C17#221711yescharacteristic
218C2×C17#221811yescharacteristic
219C2×C17#223311yescharacteristic
220C2×C17#223411yescharacteristic
221D17—162no—
222C22×D4—341no—
223C22×D4—681no—
224C23×C4D6811yescharacteristic
225C2×D8—172no—
226C2×D8—341no—
227C2×D8—341no—
228C2×D8—341no—
229C2×D8—341no—
230C2×D8—341no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8D6811yescharacteristic
238C22×C8C22×D1711yescharacteristic
239#121—344no—
240#121—682no—
241C2×C4⋊C4—341no—
242C2×C4⋊C4—341no—
243C2×C4⋊C4—341no—
244C2×C22⋊C4—681no—
245C2×C22⋊C4—681no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—21no—
251C2×C16C22⋊D1712yescomplemented
252#107—172no—
253#107—341no—
254#107—341no—
255#107—341no—
256#107—341no—
257#107—341no—
258#102—344no—
259#95—681no—
260C17#1338111yescharacteristic, complemented, Sylow
261C24—681no—
262C24D13611yescharacteristic
263C2×D4—342no—
264C2×D4—684no—
265C2×D4—681no—
266C2×D4—681no—
267C2×D4—681no—
268C2×D4—681no—
269C2×D4—681no—
270C2×D4—1362no—
271C2×D4—1362no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—681no—
279C22×C4—681no—
280C22×C4—681no—
281C22×C4D13611yescharacteristic
282C22×C4C2×D6811yescharacteristic
283D8—342no—
284D8—342no—
285D8—681no—
286D8—681no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8#6325211yescharacteristic
295C2×C8D4×D1711yescharacteristic
296C2×C8D4×D1711yescharacteristic
297C2×C8#6325411yescharacteristic
298C2×C8#6325411yescharacteristic
299C2×C8C2×C22⋊D1711yescharacteristic
300C4⋊C4—342no—
301C4⋊C4—681no—
302C4⋊C4—681no—
303C4⋊C4—681no—
304C4⋊C4—681no—
305C4⋊C4—681no—
306C22⋊C4—684no—
307C22⋊C4—1362no—
308C16—22no—
309C16—41no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—682no—
317C23—1361no—
318C23—1361no—
319C23—1361no—
320C23—1361no—
321C23—1361no—
322C23—1361no—
323C23#1058747112yes—
324C23C2×D13611yescharacteristic
325D4—682no—
326D4—682no—
327D4—1368no—
328D4—1361no—
329D4—1361no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—682no—
338C2×C4—1361no—
339C2×C4—1361no—
340C2×C4—1361no—
341C2×C4—1361no—
342C2×C4—1361no—
343C2×C4—1361no—
344C2×C4—1361no—
345C2×C4—1361no—
346C2×C4#1058756811yescharacteristic
347C2×C4#1058757211yescharacteristic
348C2×C4#1058759011yescharacteristic
349C2×C4#1058759411yescharacteristic
350C2×C4#1058759511yescharacteristic
351C2×C4#1058761611yescharacteristic
352C8—41no—
353C8#1058756911yescharacteristic
354C8#1058757111yescharacteristic
355C8#1058762411yescharacteristic
356C8#1058762511yescharacteristic
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—2721no—
365C22—2721no—
366C22—2721no—
367C22—2721no—
368C22—1362no—
369C22—1362no—
370C22—1362no—
371C22#1176294511yescharacteristic
372C22#1176304511yescharacteristic
373C22#1176312311yescharacteristic
374C22C2×#1058747111yescharacteristic
375C22#1176324111yescharacteristic
376C22#1176324711yescharacteristic
377C22#1176324711yescharacteristic
378C4—41no—
379C4—2721no—
380C4—2721no—
381C4—1362no—
382C4#1176272511yescharacteristic
383C4#1176297311yescharacteristic
384C4#1176305511yescharacteristic
385C4#1176313411yescharacteristic
386C2—42no—
387C2—2722no—
388C2#1452996711yescharacteristic
389C2#1453070911yescharacteristic
390C2#1453071511yescharacteristic
391C2#1453088011yescharacteristic
392C2#1453088711yescharacteristic
393C2#1453097011yescharacteristic
394C2#1453097611yescharacteristic
subgroupG/H
centerC23C2×D136
commutator, cosocleC22×C8×C17C23
FrattiniC22×C8C22×D17
FittingC2×C17×C22⋊C16C2
socleC23×C17C2×D8
prime radicalC2×#11763045C2
hypercenterC2×C22⋊C16D17
hypocenterC17#13381
LCSG ▷ C22×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×#11762637C2semidirect
3. C2×#11762611C2semidirect
4. C2×#11762611C2semidirect
5. #14528813C2semidirect
6. #14526876C2semidirect
7. C2×C17×C22⋊C16C2semidirect
54. C2×C16×C17D4semidirect
164. C2×C22⋊C16D17semidirect
251. C2×C16C22⋊D17semidirect
260. C17#13381semidirect

Maximal subgroups

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

subgroupindex
C2×#117630452
C2×#117626372
C2×#117626112
#145288132
#145268762
C2×C17×C22⋊C162
#1338117