Finite Group Databasebeta

← #16953477

Subgroups — #16953477

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×#11763042C211yesmaximal, characteristic
2C2×#11762634C211yesmaximal, characteristic, complemented
3C2×#11762610C211yesmaximal, characteristic, complemented
4C2×#11762588C211yesmaximal, characteristic, complemented
5#14528810C211yesmaximal, characteristic, complemented
6#14526876C211yesmaximal, characteristic, complemented
7C2×C17×#357C211yesmaximal, characteristic, complemented
8C22×C8×D17C2211yescharacteristic
9C2×#10587624—21no—
10C22×#63214C2211yescharacteristic
11#11763042—22no—
12#11763042—22no—
13C2×#10587489C2211yescharacteristic
14C2×#10587489C2211yescharacteristic
15C2×#10587503—21no—
16#11762666—21no—
17#11762634—22no—
18#11762610—22no—
19#11762588—22no—
20#11762571—21no—
21C22×D8×C17C2211yescharacteristic
22C22×C16×C17C2211yescharacteristic
23C2×C17×#107C2211yescharacteristic
24C17×#357—22no—
25C22×D4×C17—21no—
26C22×C22⋊D17—41no—
27C22×C4×D17D411yescharacteristic
28C2×D8×C17—21no—
29C2×D8×C17—21no—
30C2×D8×C17—21no—
31C2×D8×C17—21no—
32C2×D8×C17—21no—
33C2×D8×C17D412yes—
34C22×C8×C17C2311yescharacteristic
35C2×C4⋊C4×C17—21no—
36C2×C22⋊Dic17—41no—
37#10587624—24no—
38#10587624—42no—
39C2×#63227—41no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17—21no—
42C22×#3622D411yescharacteristic
43C2×C8×D17—21no—
44C2×C8×D17—21no—
45C2×C8×D17—21no—
46C2×C8×D17—21no—
47C2×C8×D17—21no—
48C2×C8×D17—21no—
49C2×C16×C17—21no—
50C2×C16×C17—21no—
51C2×C16×C17—21no—
52C17×#107—21no—
53C17×#107—21no—
54#10587503—24no—
55#10587502—41no—
56#10587489—21no—
57#10587489—21no—
58#10587489—21no—
59#10587489D412yes—
60C2×#63214—21no—
61C2×#63214—21no—
62C2×#63214D412yescomplemented
63C24×C17—41no—
64C23×D17D811yescharacteristic
65C2×D4×C17—22no—
66C2×D4×C17—41no—
67C2×D4×C17—41no—
68C2×D4×C17—41no—
69C2×D4×C17—41no—
70C2×D4×C17—41no—
71C22×C4×C17—41no—
72C22×C4×C17C2×D411yescharacteristic
73C2×C22⋊D17—44no—
74C2×C22⋊D17—82no—
75C2×C22⋊D17—82no—
76C22×Dic17—41no—
77C22×Dic17—41no—
78C22×Dic17D811yescharacteristic
79C2×C4×D17—21no—
80C2×C4×D17—21no—
81C2×C4×D17—21no—
82C2×C4×D17—21no—
83C2×C4×D17—21no—
84C2×C4×D17—21no—
85D8×C17—22no—
86D8×C17—22no—
87D8×C17—41no—
88D8×C17—41no—
89C2×C8×C17C2×D411yescharacteristic
90C2×C8×C17C2×D411yescharacteristic
91C2×C8×C17C2×D411yescharacteristic
92C2×C8×C17#4111yescharacteristic
93C2×C8×C17#4111yescharacteristic
94C2×C8×C17#4111yescharacteristic
95C4⋊C4×C17—41no—
96C4⋊C4×C17—41no—
97C22⋊Dic17—44no—
98#63227—82no—
99C4⋊Dic17—22no—
100C4⋊Dic17—41no—
101C4⋊Dic17—41no—
102C4⋊Dic17—41no—
103C2×#3622—21no—
104C2×#3622—21no—
105C2×#3622—21no—
106C8×D17—41no—
107C8×D17—41no—
108C8×D17—41no—
109C8×D17—41no—
110C16×C17—41no—
111#63214—22no—
112#13380—171nomaximal, Sylow
113C23×C17—42no—
114C23×C17—81no—
115C23×C17—81no—
116C23×C17—81no—
117C23×C17—81no—
118C23×C17—81no—
119C23×C17—81no—
120C23×C17C2×D811yescharacteristic
121C22×D17—21no—
122C22×D17—21no—
123C22×D17—21no—
124C22×D17—21no—
125C22×D17—21no—
126C22×D17—21no—
127C22×D17D1612yes—
128D4×C17—42no—
129D4×C17—42no—
130D4×C17—81no—
131D4×C17—81no—
132C2×C4×C17—81no—
133C2×C4×C17—81no—
134C2×C4×C17—81no—
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—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—42no—
146C2×Dic17—81no—
147C2×Dic17—81no—
148C2×Dic17—81no—
149C2×Dic17—81no—
150C2×Dic17—81no—
151C4×D17—41no—
152C4×D17—41no—
153C4×D17—41no—
154C4×D17—41no—
155C8×C17#12111yescharacteristic
156C8×C17#12111yescharacteristic
157C8×C17#12311yescharacteristic
158C8×C17#12411yescharacteristic
159#3622—41no—
160C2×#466—171no—
161C2×#366—171no—
162C2×#357—171no—
163C2×#357D1711yescharacteristic, complemented
164C2×C22⋊C16—171no—
165#1808—171no—
166#1380—171no—
167C22×C17—82no—
168C22×C17—82no—
169C22×C17—82no—
170C22×C17—161no—
171C22×C17—161no—
172C22×C17—161no—
173C22×C17—161no—
174C22×C17#44711yescharacteristic
175C22×C17#46611yescharacteristic
176C22×C17#48011yescharacteristic
177C22×C17C2×D1611yescharacteristic
178C22×C17#50811yescharacteristic
179C22×C17#50911yescharacteristic
180C22×C17#51011yescharacteristic
181C2×D17—22no—
182C2×D17—22no—
183C2×D17—22no—
184C2×D17—41no—
185C2×D17—41no—
186C2×D17—41no—
187C2×D17—41no—
188C4×C17—161no—
189C4×C17#39411yescharacteristic
190C4×C17#45211yescharacteristic
191C4×C17#46811yescharacteristic
192C4×C17#48211yescharacteristic
193Dic17—41no—
194Dic17—82no—
195Dic17—161no—
196C22×D8C2×D1711yescharacteristic
197C23×C8—171no—
198C2×#121—341no—
199C22×C16—171no—
200C22×C16C2×D1711yescharacteristic
201#466—342no—
202#466—342no—
203C2×#107—171no—
204C2×#107—171no—
205C2×#107C2×D1711yescharacteristic
206C2×#102—341no—
207#380—341no—
208#366—342no—
209#357—22no—
210#357—342no—
211C22⋊C16—342no—
212#340—341no—
213C2×C17—162no—
214C2×C17#203311yescharacteristic
215C2×C17#218411yescharacteristic
216C2×C17#218911yescharacteristic
217C2×C17#221511yescharacteristic
218C2×C17#222011yescharacteristic
219C2×C17#223211yescharacteristic
220C2×C17#223411yescharacteristic
221D17—42no—
222C22×D4—21no—
223C22×D4—681no—
224C23×C4—171no—
225C2×D8—21no—
226C2×D8—21no—
227C2×D8—21no—
228C2×D8—21no—
229C2×D8—21no—
230C2×D8C22⋊D1712yes—
231C22×C8—171no—
232C22×C8—341no—
233C22×C8—341no—
234C22×C8—341no—
235C22×C8—341no—
236C22×C8—341no—
237C22×C8—341no—
238C22×C8C22×D1711yescharacteristic
239#121—344no—
240#121—682no—
241C2×C4⋊C4—21no—
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—172no—
250C2×C16—341no—
251C2×C16—341no—
252#107—21no—
253#107—21no—
254#107—172no—
255#107—341no—
256#107—341no—
257#107—341no—
258#102—344no—
259#95—681no—
260C17#1338011yescharacteristic, complemented, Sylow
261C24—41no—
262C24—171no—
263C2×D4—22no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—684no—
270C2×D4—1362no—
271C2×D4—1362no—
272C22×C4—41no—
273C22×C4—171no—
274C22×C4—341no—
275C22×C4—341no—
276C22×C4—341no—
277C22×C4—341no—
278C22×C4—341no—
279C22×C4—341no—
280C22×C4—681no—
281C22×C4—681no—
282C22×C4D4×D1711yescharacteristic
283D8—22no—
284D8—22no—
285D8—41no—
286D8—41no—
287C2×C8—341no—
288C2×C8—341no—
289C2×C8—341no—
290C2×C8—681no—
291C2×C8—681no—
292C2×C8—681no—
293C2×C8—681no—
294C2×C8#6325211yescharacteristic
295C2×C8#6325211yescharacteristic
296C2×C8D4×D1711yescharacteristic
297C2×C8D4×D1711yescharacteristic
298C2×C8#6325411yescharacteristic
299C2×C8C2×C22⋊D1711yescharacteristic
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—342no—
303C4⋊C4—681no—
304C4⋊C4—681no—
305C4⋊C4—681no—
306C22⋊C4—684no—
307C22⋊C4—1362no—
308C16—41no—
309C16—342no—
310C23—42no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—172no—
318C23—341no—
319C23—341no—
320C23—341no—
321C23—341no—
322C23—341no—
323C23—341no—
324C23D8×D1711yescharacteristic
325D4—42no—
326D4—42no—
327D4—81no—
328D4—81no—
329D4—1368no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—341no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—681no—
337C2×C4—681no—
338C2×C4—681no—
339C2×C4—681no—
340C2×C4—682no—
341C2×C4—1361no—
342C2×C4—1361no—
343C2×C4—1361no—
344C2×C4—1361no—
345C2×C4—1361no—
346C2×C4#1058756911yescharacteristic
347C2×C4#1058757011yescharacteristic
348C2×C4#1058759711yescharacteristic
349C2×C4#1058760111yescharacteristic
350C2×C4#1058760211yescharacteristic
351C2×C4#1058762311yescharacteristic
352C8—681no—
353C8#1058757011yescharacteristic
354C8#1058757111yescharacteristic
355C8#1058761511yescharacteristic
356C8#1058762511yescharacteristic
357C22—82no—
358C22—82no—
359C22—82no—
360C22—161no—
361C22—161no—
362C22—161no—
363C22—161no—
364C22—342no—
365C22—342no—
366C22—342no—
367C22—681no—
368C22—681no—
369C22—681no—
370C22—681no—
371C22#1176294111yescharacteristic
372C22#1176304411yescharacteristic
373C22#1176312111yescharacteristic
374C22D16×D1711yescharacteristic
375C22#1176324011yescharacteristic
376C22#1176324311yescharacteristic
377C22#1176324811yescharacteristic
378C4—161no—
379C4—2721no—
380C4—681no—
381C4—1362no—
382C4#1176272411yescharacteristic
383C4#1176296811yescharacteristic
384C4#1176305411yescharacteristic
385C4#1176313211yescharacteristic
386C2—162no—
387C2—682no—
388C2#1452996411yescharacteristic
389C2#1453069411yescharacteristic
390C2#1453072111yescharacteristic
391C2#1453086911yescharacteristic
392C2#1453089611yescharacteristic
393C2#1453096311yescharacteristic
394C2#1453097411yescharacteristic
subgroupG/H
centerC23D8×D17
commutator, cosocleC22×C8×C17C23
FrattiniC22×C8C22×D17
FittingC2×C17×#357C2
socleC23×C17C2×D8
prime radicalC2×#11763042C2
hypercenterC2×#357D17
hypocenterC17#13380
LCSG ▷ C22×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#11762634C2semidirect
3. C2×#11762610C2semidirect
4. C2×#11762588C2semidirect
5. #14528810C2semidirect
6. #14526876C2semidirect
7. C2×C17×#357C2semidirect
62. C2×#63214D4semidirect
163. C2×#357D17semidirect
260. C17#13380semidirect

Maximal subgroups

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

subgroupindex
C2×#117630422
C2×#117626342
C2×#117626102
C2×#117625882
#145288102
#145268762
C2×C17×#3572
#1338017