Finite Group Databasebeta

← #16938088

Subgroups — #16938088

Order 4352.

398 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 93 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#14538583C211yesmaximal, characteristic
2#14528795C211yesmaximal, characteristic
3#14527442C211yesmaximal, characteristic, complemented
4#14527349C211yesmaximal, characteristic, complemented
5#14527104C211yesmaximal, characteristic
6#14526644C211yesmaximal, characteristic, complemented
7C17×#1570C211yesmaximal, characteristic, complemented
8#11763366C2211yescharacteristic
9C2×C4×D68C2211yescharacteristic
10#11763110—22no—
11#11763100—21no—
12#11763090—21no—
13#11762862C2211yescharacteristic
14#11762851—21no—
15#11762815—21no—
16C2×#10587492—21no—
17#11762675C2211yescharacteristic
18#11762572—21no—
19#11762557—21no—
20#11762554—21no—
21#11762547—22no—
22#11762532—22no—
23C17×#431C2211yescharacteristic
24C17×#423C2211yescharacteristic
25C2×C17×C4⋊C8C2211yescharacteristic
26C17×#334—22no—
27C22×D68D411yescharacteristic
28C22×C4×D17—21no—
29C2×M16×C17—21no—
30C22×C8×C17—21no—
31C2×C42×C17C2311yescharacteristic
32C2×#63227—21no—
33#10587612—21no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17D411yescharacteristic
36#10587610—21no—
37#10587610—21no—
38C4×D68—21no—
39C4×D68—21no—
40C4×D68—21no—
41C4×D68—21no—
42#10587556—21no—
43#10587555D411yescharacteristic
44#10587555D411yescharacteristic
45C4×Dic34—21no—
46C4×Dic34—21no—
47C17×C4⋊C8—21no—
48C17×C4⋊C8—21no—
49C17×C4⋊C8—21no—
50C17×C4⋊C8—21no—
51C17×C4⋊C8D412yes—
52C17×C22⋊C8D412yes—
53C4×C8×C17—21no—
54#10587502—21no—
55#10587502—21no—
56C22⋊#3622—21no—
57#10587492—22no—
58#10587492—22no—
59#10587492—42no—
60#10587489—22no—
61#10587488—22no—
62C4⋊#3622—21no—
63C4×#3622—21no—
64C23×D17—21no—
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17#4111yescharacteristic
68C22×Dic17—21no—
69C22×Dic17—41no—
70C2×D68—21no—
71C2×D68—21no—
72C2×D68—21no—
73C2×D68—21no—
74C2×D68—21no—
75C2×D68SD1612yes—
76C2×C4×D17—41no—
77C2×C4×D17—41no—
78C2×C4×D17—82no—
79C2×Dic34—21no—
80C2×Dic34—21no—
81M16×C17—41no—
82M16×C17—41no—
83C2×C8×C17—22no—
84C2×C8×C17—22no—
85C2×C8×C17—41no—
86C2×C8×C17—41no—
87C2×C8×C17—41no—
88C42×C17C2×D411yescharacteristic
89C42×C17C2×D411yescharacteristic
90C42×C17#4111yescharacteristic
91C42×C17#4111yescharacteristic
92C22⋊Dic17—41no—
93#63227—41no—
94#63227—41no—
95C4⋊Dic17—21no—
96C4⋊Dic17—21no—
97C4⋊Dic17—21no—
98C4⋊Dic17—21no—
99C4⋊Dic17—21no—
100C4⋊Dic17—21no—
101C4⋊Dic17—41no—
102C4⋊Dic17—41no—
103#63225—41no—
104#63225—41no—
105C4×Dic17—41no—
106C2×#3622—41no—
107#10468—171nomaximal, Sylow
108C23×C17#12111yescharacteristic
109C22×D17—22no—
110C22×D17—41no—
111C22×D17—41no—
112C22×D17—81no—
113C22×D17—81no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17#12011yescharacteristic
117C2×C4×C17#12111yescharacteristic
118C2×C4×C17#12111yescharacteristic
119C2×C4×C17#12311yescharacteristic
120C2×C4×C17#12311yescharacteristic
121C2×C4×C17#12411yescharacteristic
122C2×C4×C17C2×SD1611yescharacteristic
123C2×C4×C17#13511yescharacteristic
124C2×C4×C17#13711yescharacteristic
125C2×C4×C17#13711yescharacteristic
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—81no—
130C2×Dic17—81no—
131C2×Dic17—81no—
132C2×Dic17—81no—
133D68—22no—
134D68—22no—
135D68—41no—
136D68—41no—
137C4×D17—82no—
138C4×D17—82no—
139Dic34—81no—
140Dic34—81no—
141C8×C17—42no—
142C8×C17—42no—
143C8×C17—81no—
144#3622—81no—
145#3077—171no—
146#1804—171no—
147#1570D1711yescharacteristic, complemented
148#1542—171no—
149#1518—171no—
150#1460—171no—
151#1286—171no—
152C22×C17—21no—
153C22×C17—21no—
154C22×C17#39411yescharacteristic
155C22×C17#46511yescharacteristic
156C22×C17#47011yescharacteristic
157C2×D17—42no—
158C2×D17—82no—
159C2×D17—81no—
160C2×D17—81no—
161C2×D17—82no—
162C4×C17—21no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—21no—
166C4×C17#44811yescharacteristic
167C4×C17#45211yescharacteristic
168C4×C17#48311yescharacteristic
169C4×C17#48511yescharacteristic
170Dic17—81no—
171Dic17—81no—
172Dic17—81no—
173Dic17—161no—
174#533—171no—
175C2×C4×D4—171no—
176#478—342no—
177#476—341no—
178#474—341no—
179#431—171no—
180#431C2×D1711yescharacteristic
181#428—341no—
182#423C2×D1711yescharacteristic
183C2×C4⋊C8C2×D1711yescharacteristic
184#418—341no—
185C2×#102—341no—
186#382—171no—
187#340—341no—
188#334—22no—
189#333—341no—
190#332—341no—
191#328—342no—
192#323—342no—
193C2×C17—21no—
194C2×C17—21no—
195C2×C17#203511yescharacteristic
196C2×C17#204811yescharacteristic
197C2×C17#205311yescharacteristic
198D17—82no—
199C22×D4—171no—
200C23×C4—341no—
201C2×M16—21no—
202C22×C8—21no—
203C4⋊Q8—171no—
204C4⋊Q8—171no—
205#125—341no—
206C22⋊Q8—341no—
207C22⋊Q8—341no—
208C4×Q8—341no—
209C4×Q8—341no—
210C4×D4—341no—
211C4×D4—341no—
212C4×D4—341no—
213C4×D4—341no—
214#117—341no—
215C2×C4⋊C4—171no—
216C2×C4⋊C4—341no—
217C2×C22⋊C4—341no—
218C2×C42C22×D1711yescharacteristic
219#107—342no—
220#106—342no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—341no—
226C4⋊C8C22⋊D1712yes—
227#102—342no—
228#102—342no—
229#102—682no—
230C22⋊C8—341no—
231C22⋊C8D6812yes—
232C4×C8—21no—
233C4×C8—341no—
234#95—341no—
235#95—341no—
236C17#1046811yescharacteristic, complemented, Sylow
237C24—341no—
238C2×Q8—341no—
239C2×Q8—341no—
240C2×D4—172no—
241C2×D4—341no—
242C2×D4—341no—
243C2×D4—341no—
244C2×D4—341no—
245C2×D4—341no—
246C22×C4—341no—
247C22×C4—681no—
248C22×C4—681no—
249C22×C4—681no—
250C22×C4—1362no—
251C22×C4C2×D6811yescharacteristic
252C22×C4D4×D1711yescharacteristic
253C22×C4#6325611yescharacteristic
254M16—41no—
255M16—41no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—681no—
262C4⋊C4—341no—
263C4⋊C4—341no—
264C4⋊C4—341no—
265C4⋊C4—341no—
266C4⋊C4—341no—
267C4⋊C4—341no—
268C4⋊C4—681no—
269C4⋊C4—681no—
270C4⋊C4—681no—
271C4⋊C4—681no—
272C22⋊C4—681no—
273C22⋊C4—681no—
274C22⋊C4—681no—
275C42—681no—
276C42#6325211yescharacteristic
277C42D4×D1711yescharacteristic
278C42#6325611yescharacteristic
279C42C2×C22⋊D1711yescharacteristic
280C23—342no—
281C23—681no—
282C23—681no—
283C23—1361no—
284C23—1361no—
285C23#1058758111yescharacteristic
286Q8—1361no—
287Q8—1361no—
288D4—342no—
289D4—342no—
290D4—681no—
291D4—681no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—681no—
295C2×C4—681no—
296C2×C4—681no—
297C2×C4—1362no—
298C2×C4—1362no—
299C2×C4—1361no—
300C2×C4—1361no—
301C2×C4—1361no—
302C2×C4—1361no—
303C2×C4#1058758011yescharacteristic
304C2×C4#1058758011yescharacteristic
305C2×C4#1058758111yescharacteristic
306C2×C4SD16×D1711yescharacteristic
307C2×C4#1058760111yescharacteristic
308C2×C4#1058760111yescharacteristic
309C2×C4#1058760211yescharacteristic
310C2×C4#1058761611yescharacteristic
311C2×C4#1058762311yescharacteristic
312C2×C4#1058763311yescharacteristic
313C8—42no—
314C8—42no—
315C8—81no—
316C8—1361no—
317C22—21no—
318C22—21no—
319C22—682no—
320C22—1362no—
321C22—1362no—
322C22—1361no—
323C22—1361no—
324C22#1176272611yescharacteristic
325C22#1176303911yescharacteristic
326C22#1176306811yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—2721no—
332C4—1361no—
333C4—1361no—
334C4—1361no—
335C4#1176294611yescharacteristic
336C4#1176296811yescharacteristic
337C4#1176314111yescharacteristic
338C4#1176315311yescharacteristic
339C2—21no—
340C2—21no—
341C2—1362no—
342C2#1452998211yescharacteristic
343C2#1453005911yescharacteristic
344C2#1453009011yescharacteristic
subgroupG/H
centerC22#11762726
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1570C2
socleC22×C17#394
prime radicalC22×D68D4
hypercenter#1570D17
hypocenterC17#10468
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
3. #14527442C2semidirect
4. #14527349C2semidirect
6. #14526644C2semidirect
7. C17×#1570C2semidirect
147. #1570D17semidirect
236. C17#10468semidirect

Maximal subgroups

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

subgroupindex
#145385832
#145287952
#145274422
#145273492
#145271042
#145266442
C17×#15702
#1046817