Finite Group Databasebeta

← #16950499

Subgroups — #16950499

Order 4352.

418 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587469C211yesmaximal, characteristic
2C2×#11762636C211yesmaximal, characteristic, complemented
3C2×#11762634C211yesmaximal, characteristic, complemented
4#14528810C211yesmaximal, characteristic
5#14528810C211yesmaximal, characteristic
6#14526876C211yesmaximal, characteristic, complemented
7C17×#1380C211yesmaximal, characteristic, complemented
8C22×C8×D17C2211yescharacteristic
9C2×#10587469—22no—
10C2×#10587469—22no—
11C2×#10587469C414yescomplemented
12C22×#63214C2211yescharacteristic
13C2×#10587489C2211yescharacteristic
14C2×#10587489C2211yescharacteristic
15#11762666—21no—
16#11762666—21no—
17#11762636—22no—
18#11762634—22no—
19#11762571—21no—
20#11762571—21no—
21C22×C16×C17C2211yescharacteristic
22C2×C17×#107C2211yescharacteristic
23C2×C17×#107C2211yescharacteristic
24C22×C4×D17D411yescharacteristic
25C22×C8×C17C2311yescharacteristic
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17—21no—
28C2×#63227—41no—
29C2×#63227—41no—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—21no—
32C22×#3622D411yescharacteristic
33C2×C8×D17—21no—
34C2×C8×D17—21no—
35C2×C8×D17—21no—
36C2×C8×D17—21no—
37C2×C8×D17C2×C412yes—
38C2×C8×D17Q812yes—
39C2×C16×C17—21no—
40C2×C16×C17—21no—
41C2×C16×C17C2×C412yes—
42C17×#107—21no—
43C17×#107—21no—
44C17×#107—21no—
45C17×#107—21no—
46#10587502—41no—
47#10587502—41no—
48#10587489—21no—
49#10587489—21no—
50#10587489—21no—
51#10587489—21no—
52C2×#63214C2×C412yes—
53C2×#63214D412yes—
54C2×#63214D412yes—
55#10587469—28no—
56C23×D17D811yescharacteristic
57C22×C4×C17—41no—
58C22×C4×C17—41no—
59C22×C4×C17C2×D411yescharacteristic
60C22×Dic17—41no—
61C22×Dic17—41no—
62C22×Dic17D811yescharacteristic
63C2×C4×D17—21no—
64C2×C4×D17—21no—
65C2×C4×D17—21no—
66C2×C4×D17—21no—
67C2×C4×D17C4⋊C412yes—
68C2×C4×D17Q1612yes—
69C2×C8×C17C22×C411yescharacteristic
70C2×C8×C17C2×D411yescharacteristic
71C2×C8×C17C2×D411yescharacteristic
72C2×C8×C17C2×Q811yescharacteristic
73C2×C8×C17#4111yescharacteristic
74C2×C8×C17#4111yescharacteristic
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—41no—
79#63227—82no—
80#63227—82no—
81C4⋊Dic17—41no—
82C4⋊Dic17—41no—
83C4⋊Dic17—41no—
84C4⋊Dic17—41no—
85C2×#3622—21no—
86C2×#3622—21no—
87C2×#3622C4⋊C412yes—
88C8×D17—22no—
89C8×D17—22no—
90C8×D17—22no—
91C8×D17—22no—
92C16×C17—22no—
93#63214C22⋊C414yescomplemented
94#12836—171nomaximal, Sylow
95C23×C17C2×D811yescharacteristic
96C22×D17—21no—
97C22×D17—21no—
98C22×D17—21no—
99C22×D17—21no—
100C22×D17—21no—
101C22×D17—21no—
102C22×D17#10712yes—
103C2×C4×C17—81no—
104C2×C4×C17—81no—
105C2×C4×C17—81no—
106C2×C4×C17—81no—
107C2×C4×C17—81no—
108C2×C4×C17—81no—
109C2×C4×C17C2×C4⋊C411yescharacteristic
110C2×C4×C17#12111yescharacteristic
111C2×C4×C17#12111yescharacteristic
112C2×C4×C17C2×Q1611yescharacteristic
113C2×C4×C17#13511yescharacteristic
114C2×C4×C17#13511yescharacteristic
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—81no—
118C2×Dic17—81no—
119C2×Dic17—81no—
120C2×Dic17—81no—
121C2×Dic17—81no—
122C2×Dic17—81no—
123C2×Dic17#10712yes—
124C4×D17—22no—
125C4×D17—22no—
126C4×D17—22no—
127C4×D17—22no—
128C8×C17C2×C22⋊C411yescharacteristic
129C8×C17#11711yescharacteristic
130C8×C17C22⋊Q811yescharacteristic
131C8×C17C22⋊Q811yescharacteristic
132#3622—22no—
133C22×M32—171no—
134C2×#367—171no—
135C2×#366—171no—
136#1808—171no—
137#1808—171no—
138#1380—171no—
139#1380D1711yescharacteristic, complemented
140C22×C17C2×#10711yescharacteristic
141C22×C17#46611yescharacteristic
142C22×C17#46611yescharacteristic
143C22×C17#50911yescharacteristic
144C22×C17#50911yescharacteristic
145C22×C17#51011yescharacteristic
146C22×C17#51011yescharacteristic
147C2×D17—22no—
148C2×D17—22no—
149C2×D17—22no—
150C2×D17—41no—
151C2×D17—41no—
152C2×D17—41no—
153C2×D17—41no—
154C4×C17—161no—
155C4×C17—161no—
156C4×C17#38011yescharacteristic
157C4×C17#42711yescharacteristic
158C4×C17#46711yescharacteristic
159C4×C17#46711yescharacteristic
160Dic17—22no—
161Dic17—161no—
162Dic17—161no—
163C23×C8—171no—
164C2×M32—174no—
165C2×M32—342no—
166C2×M32—342no—
167C22×C16—171no—
168C22×C16C2×D1711yescharacteristic
169C2×#107—171no—
170C2×#107—171no—
171C2×#107C2×D1711yescharacteristic
172C2×#107C2×D1711yescharacteristic
173#380—341no—
174#380—341no—
175#367—342no—
176#366—342no—
177#340—341no—
178#340—341no—
179C2×C17#180811yescharacteristic
180C2×C17#215911yescharacteristic
181C2×C17#215911yescharacteristic
182C2×C17#221811yescharacteristic
183C2×C17#221911yescharacteristic
184C2×C17#222011yescharacteristic
185C2×C17#222011yescharacteristic
186D17—42no—
187C23×C4—171no—
188C22×C8—171no—
189C22×C8—172no—
190C22×C8—172no—
191C22×C8—341no—
192C22×C8—341no—
193C22×C8—341no—
194C22×C8—341no—
195C22×C8C22×D1711yescharacteristic
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—341no—
199C2×C4⋊C4—341no—
200C2×C22⋊C4—681no—
201C2×C22⋊C4—681no—
202M32—348no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—172no—
206C2×C16—172no—
207C2×C16—172no—
208C2×C16C4×D1712yes—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—341no—
214#107—341no—
215#107—341no—
216#107—341no—
217#95—681no—
218#95—681no—
219C17#1283611yescharacteristic, complemented, Sylow
220C24—171no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—171no—
224C22×C4—172no—
225C22×C4—172no—
226C22×C4—341no—
227C22×C4—341no—
228C22×C4—341no—
229C22×C4—341no—
230C22×C4—681no—
231C22×C4—681no—
232C22×C4D4×D1711yescharacteristic
233C2×C8—172no—
234C2×C8—341no—
235C2×C8—341no—
236C2×C8—342no—
237C2×C8—342no—
238C2×C8—342no—
239C2×C8—342no—
240C2×C8C2×C4×D1711yescharacteristic
241C2×C8#6325211yescharacteristic
242C2×C8#6325211yescharacteristic
243C2×C8D4×D1711yescharacteristic
244C2×C8D4×D1711yescharacteristic
245C2×C8Q8×D1711yescharacteristic
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—681no—
251C4⋊C4—681no—
252C4⋊C4—681no—
253C4⋊C4—681no—
254C22⋊C4—1362no—
255C22⋊C4—1362no—
256C16—22no—
257C16—174no—
258C23—172no—
259C23—341no—
260C23—341no—
261C23—341no—
262C23—341no—
263C23—341no—
264C23—341no—
265C23D8×D1711yescharacteristic
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—172no—
273C2×C4—341no—
274C2×C4—341no—
275C2×C4—342no—
276C2×C4—342no—
277C2×C4—342no—
278C2×C4—342no—
279C2×C4—1361no—
280C2×C4—1361no—
281C2×C4—1361no—
282C2×C4—1361no—
283C2×C4—1361no—
284C2×C4—1361no—
285C2×C4#1058757011yescharacteristic
286C2×C4#1058757011yescharacteristic
287C2×C4C4⋊C4×D1711yescharacteristic
288C2×C4#1058760211yescharacteristic
289C2×C4#1058760211yescharacteristic
290C2×C4Q16×D1711yescharacteristic
291C8—342no—
292C8#1058755811yescharacteristic
293C8C22⋊C4×D1711yescharacteristic
294C8#1058758211yescharacteristic
295C8#1058758211yescharacteristic
296C22—342no—
297C22—342no—
298C22—342no—
299C22—681no—
300C22—681no—
301C22—681no—
302C22—681no—
303C22#107×D1711yescharacteristic
304C22#1176304411yescharacteristic
305C22#1176304411yescharacteristic
306C22#1176324311yescharacteristic
307C22#1176324611yescharacteristic
308C22#1176324811yescharacteristic
309C22#1176325111yescharacteristic
310C4—161no—
311C4—161no—
312C4—2721no—
313C4—2721no—
314C4—342no—
315C4#1176266811yescharacteristic
316C4#1176284411yescharacteristic
317C4#1176304911yescharacteristic
318C4#1176304911yescharacteristic
319C2—682no—
320C2#1452881211yescharacteristic
321C2#1453056911yescharacteristic
322C2#1453057111yescharacteristic
323C2#1453088611yescharacteristic
324C2#1453089111yescharacteristic
325C2#1453089611yescharacteristic
326C2#1453090011yescharacteristic
subgroupG/H
centerC23D8×D17
commutatorC2×C8×C17C22×C4
FrattiniC22×C8C22×D17
FittingC17×#1380C2
socleC23×C17C2×D8
prime radicalC23×D17D8
cosocleC22×C8×C17C23
hypercenter#1380D17
hypocenterC17#12836
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
2. C2×#11762636C2semidirect
3. C2×#11762634C2semidirect
6. #14526876C2semidirect
7. C17×#1380C2semidirect
11. C2×#10587469C4semidirect
93. #63214C22⋊C4semidirect
139. #1380D17semidirect
219. C17#12836semidirect

Maximal subgroups

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

subgroupindex
C22×#105874692
C2×#117626362
C2×#117626342
#145288102
#145268762
C17×#13802
#1283617