Finite Group Databasebeta

← #16953479

Subgroups — #16953479

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×#11762634C211yesmaximal, characteristic, complemented
3C2×#11762609C211yesmaximal, characteristic, complemented
4C2×#11762609C211yesmaximal, characteristic, complemented
5C2×C22⋊#63214C211yesmaximal, characteristic, complemented
6#14526875C211yesmaximal, characteristic, complemented
7C17×#1808C211yesmaximal, characteristic, complemented
8C22×D136C2211yescharacteristic
9C2×#10587616—21no—
10C22×#63214C2211yescharacteristic
11C22×#63214C2211yescharacteristic
12#11763045—22no—
13#11763045—22no—
14C2×#10587489C2211yescharacteristic
15C2×#10587492—21no—
16#11762634—22no—
17#11762609—22no—
18#11762609—22no—
19C22⋊#63214—22no—
20C23×C8×C17C2211yescharacteristic
21C2×C17×#107C2211yescharacteristic
22C2×C17×#107C2211yescharacteristic
23C17×#380—21no—
24C17×#340—21no—
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×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17—21no—
38C2×C22⋊C4×C17—41no—
39#10587616—24no—
40#10587616—42no—
41C2×#63227—41no—
42C2×C4⋊Dic17—21no—
43C2×D136—21no—
44C2×D136—21no—
45C2×D136—21no—
46C2×D136—21no—
47C2×D136—21no—
48C2×D136D412yes—
49C17×#107—21no—
50C17×#107—21no—
51C17×#107—21no—
52C17×#107—21no—
53C17×#95—41no—
54#10587492—24no—
55#10587489—21no—
56#10587489D412yes—
57C2×#63214—21no—
58C2×#63214—21no—
59C2×#63214—21no—
60C2×#63214—21no—
61C2×#63214—21no—
62C2×#63214D412yescomplemented
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×C17—41no—
72C22×C4×C17—41no—
73C22×C4×C17D811yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C2×C22⋊D17—44no—
76C2×C22⋊D17—82no—
77C2×C22⋊D17—82no—
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
98C4⋊C4×C17—41no—
99C4⋊C4×C17—41no—
100C4⋊C4×C17—41no—
101C4⋊C4×C17—41no—
102C22⋊C4×C17—82no—
103#63227—44no—
104C4⋊Dic17—22no—
105C4⋊Dic17—41no—
106D136—22no—
107D136—22no—
108D136—41no—
109D136—41no—
110#63214—22no—
111#63214—41no—
112#13380—171nomaximal, Sylow
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17—21no—
118C23×C17—21no—
119C23×C17D1612yes—
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—81no—
136C2×C4×C17—81no—
137C2×C4×C17—81no—
138C2×C4×C17—81no—
139C2×C4×C17—81no—
140C2×C4×C17—81no—
141C2×C4×C17#12011yescharacteristic
142C2×C4×C17#12111yescharacteristic
143C2×C4×C17#12311yescharacteristic
144C2×C4×C17#13511yescharacteristic
145C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C17#13711yescharacteristic
147C22⋊D17—88no—
148C2×Dic17—42no—
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×#366—171no—
162C2×#357—171no—
163C2×#357—171no—
164C2×C22⋊C16—171no—
165#1808D1711yescharacteristic, complemented
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×D1611yescharacteristic
178C22×C17#50811yescharacteristic
179C22×C17#50911yescharacteristic
180C22×C17#51011yescharacteristic
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—161no—
190C4×C17—161no—
191C4×C17#39411yescharacteristic
192C4×C17#45211yescharacteristic
193C4×C17#46811yescharacteristic
194C4×C17#48211yescharacteristic
195Dic17—82no—
196C22×D8—171no—
197C23×C8C2×D1711yescharacteristic
198C2×#121—341no—
199C22×C16—171no—
200C22×C16—171no—
201#466—342no—
202#466—342no—
203C2×#107—171no—
204C2×#107C2×D1711yescharacteristic
205C2×#107C2×D1711yescharacteristic
206C2×#102—341no—
207#380—21no—
208#366—342no—
209#357—342no—
210#357—342no—
211C22⋊C16—342no—
212#340—21no—
213C2×C17—42no—
214C2×C17#203311yescharacteristic
215C2×C17#218411yescharacteristic
216C2×C17#218911yescharacteristic
217C2×C17#221511yescharacteristic
218C2×C17#222011yescharacteristic
219C2×C17#223211yescharacteristic
220C2×C17#223411yescharacteristic
221D17—162no—
222C22×D4—341no—
223C22×D4—681no—
224C23×C4C22⋊D1711yescharacteristic
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×C8C22⋊D1711yescharacteristic
238C22×C8C22×D1711yescharacteristic
239#121—344no—
240#121—682no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—341no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—681no—
246C2×C16—172no—
247C2×C16—341no—
248C2×C16—341no—
249C2×C16—341no—
250C2×C16—341no—
251C2×C16—341no—
252#107—21no—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—172no—
257#107—341no—
258#102—344no—
259#95—41no—
260C17#1338011yescharacteristic, complemented, Sylow
261C24—681no—
262C24#6322811yescharacteristic
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—41no—
279C22×C4—41no—
280C22×C4—681no—
281C22×C4#6322811yescharacteristic
282C22×C4C2×C22⋊D1711yescharacteristic
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×C8#6325211yescharacteristic
296C2×C8D4×D1711yescharacteristic
297C2×C8D4×D1711yescharacteristic
298C2×C8D4×D1711yescharacteristic
299C2×C8#6325411yescharacteristic
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—342no—
305C4⋊C4—681no—
306C22⋊C4—82no—
307C22⋊C4—684no—
308C16—342no—
309C16—681no—
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#1058749712yes—
324C23C2×#6322811yescharacteristic
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—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—682no—
344C2×C4—1361no—
345C2×C4—1361no—
346C2×C4#1058761511yescharacteristic
347C2×C4#1058762311yescharacteristic
348C2×C4#1058762511yescharacteristic
349C2×C4#1058762811yescharacteristic
350C2×C4#1058763511yescharacteristic
351C2×C4#1058763611yescharacteristic
352C8—41no—
353C8#1058756911yescharacteristic
354C8#1058757011yescharacteristic
355C8#1058757011yescharacteristic
356C8#1058757111yescharacteristic
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#1176294211yescharacteristic
372C22#1176304311yescharacteristic
373C22#1176312211yescharacteristic
374C22C2×#1058749711yescharacteristic
375C22#1176323911yescharacteristic
376C22#1176324411yescharacteristic
377C22#1176325011yescharacteristic
378C4—41no—
379C4—161no—
380C4—161no—
381C4—1362no—
382C4#1176272411yescharacteristic
383C4#1176296911yescharacteristic
384C4#1176305711yescharacteristic
385C4#1176313311yescharacteristic
386C2—42no—
387C2—2722no—
388C2#1452996611yescharacteristic
389C2#1453069511yescharacteristic
390C2#1453072211yescharacteristic
391C2#1453086811yescharacteristic
392C2#1453089911yescharacteristic
393C2#1453096411yescharacteristic
394C2#1453097511yescharacteristic
subgroupG/H
centerC23C2×#63228
commutator, cosocleC22×C8×C17C23
FrattiniC22×C8C22×D17
FittingC17×#1808C2
socleC23×C17C2×D8
prime radicalC2×#11763045C2
hypercenter#1808D17
hypocenterC17#13380
LCSG ▷ C22×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ #1808

Complements

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

NH ≅ G/Ntype
2. C2×#11762634C2semidirect
3. C2×#11762609C2semidirect
4. C2×#11762609C2semidirect
5. C2×C22⋊#63214C2semidirect
6. #14526875C2semidirect
7. C17×#1808C2semidirect
62. C2×#63214D4semidirect
165. #1808D17semidirect
260. C17#13380semidirect

Maximal subgroups

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

subgroupindex
C2×#117630452
C2×#117626342
C2×#117626092
C2×C22⋊#632142
#145268752
C17×#18082
#1338017