Finite Group Databasebeta

← #17189159

Subgroups — #17189159

Order 4352.

580 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763050C211yesmaximal, characteristic
2C2×#11762847C211yesmaximal, characteristic
3#14530993C214yesmaximal, complemented
4#14530987C212yesmaximal, complemented
5#14530982C212yesmaximal, complemented
6#14530572C212yesmaximal, complemented
7#14530505C212yesmaximal, complemented
8C17×#2112C211yesmaximal, characteristic
9C22×Dic68C2211yescharacteristic
10C2×#10587610—21no—
11C2×#10587612—21no—
12#11763050—21no—
13#11763050—21no—
14#11763050C2212yes—
15#11763050C2212yes—
16#11762847—21no—
17#11762847—21no—
18#11762847C2212yes—
19#11762847C2212yes—
20C2×#10587489C2211yescharacteristic
21C2×#10587489C2211yescharacteristic
22C2×#10587489C2211yescharacteristic
23C2×#10587488—21no—
24C2×#10587487—21no—
25#11762637C2214yescomplemented
26#11762635C2214yescomplemented
27#11762614C2214yescomplemented
28#11762614C2214yescomplemented
29C23×C8×C17C2211yescharacteristic
30C2×C17×M32C2212yes—
31C17×C22⋊C16C2212yes—
32C17×C22⋊C16C2212yes—
33C23×C4×C17D411yescharacteristic
34C22×Dic34—21no—
35C22×C8×C17—24no—
36C22×C8×C17D411yescharacteristic
37C22×C8×C17D411yescharacteristic
38C22×C8×C17D411yescharacteristic
39C22×C8×C17C2311yescharacteristic
40C22×C8×C17C2311yescharacteristic
41C22×C8×C17C2311yescharacteristic
42C2×C22⋊Dic17—41no—
43C2×C22⋊Dic17—41no—
44#10587612—22no—
45#10587612—22no—
46#10587612—41no—
47#10587612—41no—
48C2×C4⋊Dic17—21no—
49C2×C4⋊Dic17—21no—
50C2×C4⋊Dic17—21no—
51#10587610—22no—
52#10587610—22no—
53#10587610—41no—
54#10587610—41no—
55C2×#63225—41no—
56C2×C4×Dic17—41no—
57C2×Dic68—21no—
58C2×Dic68—24no—
59C2×Dic68C2312yes—
60C17×M32—24no—
61C2×C16×C17C2314yes—
62#10587489—21no—
63#10587489—21no—
64#10587489—21no—
65#10587489C2312yes—
66#10587489C2312yes—
67#10587489C2312yes—
68#10587488—22no—
69#10587488—22no—
70#10587487—22no—
71#10587487—22no—
72C24×C17D811yescharacteristic
73C22×C4×C17—22no—
74C22×C4×C17—22no—
75C22×C4×C17D811yescharacteristic
76C22×C4×C17D811yescharacteristic
77C22×C4×C17D811yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×C4×C17C2×D411yescharacteristic
80C22×C4×C17C2×D411yescharacteristic
81C22×Dic17—41no—
82C22×Dic17—41no—
83C22×Dic17—41no—
84C22×Dic17—41no—
85C2×Dic34—22no—
86C2×Dic34—42no—
87C2×Dic34—42no—
88C2×Dic34—41no—
89C2×C8×C17—22no—
90C2×C8×C17—22no—
91C2×C8×C17—22no—
92C2×C8×C17—22no—
93C2×C8×C17—22no—
94C2×C8×C17C2×D411yescharacteristic
95C2×C8×C17C2×D411yescharacteristic
96C2×C8×C17C2×D411yescharacteristic
97C2×C8×C17#4114yes—
98C2×C8×C17C2411yescharacteristic
99C22⋊Dic17—42no—
100C22⋊Dic17—42no—
101C22⋊Dic17—42no—
102C22⋊Dic17—42no—
103C4⋊Dic17—22no—
104C4⋊Dic17—22no—
105C4⋊Dic17—22no—
106C4⋊Dic17—41no—
107C4⋊Dic17—41no—
108C4⋊Dic17—41no—
109#63225—42no—
110#63225—42no—
111C4×Dic17—42no—
112C4×Dic17—42no—
113Dic68—24no—
114Dic68—42no—
115C16×C17—24no—
116#33649—171nomaximal, Sylow
117C23×C17—22no—
118C23×C17—22no—
119C23×C17—41no—
120C23×C17C2×D811yescharacteristic
121C23×C17C2×D811yescharacteristic
122C23×C17C2×D811yescharacteristic
123C2×C4×C17—22no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—22no—
129C2×C4×C17—22no—
130C2×C4×C17#12312yes—
131C2×C4×C17C2×D811yescharacteristic
132C2×C4×C17C2×D811yescharacteristic
133C2×C4×C17C2×D811yescharacteristic
134C2×C4×C17#13712yes—
135C2×C4×C17C22×D411yescharacteristic
136C2×Dic17—42no—
137C2×Dic17—42no—
138C2×Dic17—42no—
139C2×Dic17—42no—
140C2×Dic17—82no—
141C2×Dic17—82no—
142C2×Dic17—82no—
143C2×Dic17—82no—
144Dic34—42no—
145Dic34—42no—
146Dic34—82no—
147C8×C17—21no—
148C8×C17—21no—
149C8×C17#12312yes—
150C8×C17C2×#4112yes—
151C2×#467—171no—
152C2×#427—171no—
153#2237—174no—
154#2236—172no—
155#2235—172no—
156#2159—172no—
157#2141—172no—
158#2112D1711yescharacteristic
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—44no—
167C22×C17#48012yes—
168C22×C17#51012yes—
169C22×C17C22×D811yescharacteristic
170C4×C17—21no—
171C4×C17—21no—
172C4×C17#48012yes—
173C4×C17C2×#12311yescharacteristic
174C4×C17C2×#13711yescharacteristic
175Dic17—82no—
176Dic17—82no—
177Dic17—82no—
178Dic17—82no—
179C22×Q16—171no—
180C23×C8C2×D1711yescharacteristic
181C2×C22⋊Q8—341no—
182C2×#117—341no—
183C2×M32C2×D1712yes—
184#467—172no—
185#467—172no—
186#467—341no—
187#467—341no—
188#427—172no—
189#427—172no—
190#427—341no—
191#427—341no—
192C2×#107—171no—
193C2×#107—171no—
194C2×#107—171no—
195C2×#106—341no—
196C2×#103—341no—
197#367—174no—
198#366—174no—
199#358—174no—
200#358—174no—
201C22⋊C16C2×D1712yes—
202C22⋊C16C2×D1712yes—
203C2×C17—22no—
204C2×C17—22no—
205C2×C17—41no—
206C2×C17C2×#48011yescharacteristic
207C2×C17C2×#51011yescharacteristic
208C2×C17#341411yescharacteristic
209C22×Q8—341no—
210C23×C4D6811yescharacteristic
211C2×Q16—172no—
212C2×Q16—341no—
213C2×Q16—344no—
214C22×C8—24no—
215C22×C8D6811yescharacteristic
216C22×C8D6811yescharacteristic
217C22×C8D6811yescharacteristic
218C22×C8C22×D1711yescharacteristic
219C22×C8C22×D1711yescharacteristic
220C22×C8C22×D1711yescharacteristic
221C22⋊Q8—342no—
222C22⋊Q8—342no—
223C22⋊Q8—681no—
224C22⋊Q8—681no—
225#117—342no—
226#117—342no—
227#117—681no—
228#117—681no—
229C2×C4⋊C4—341no—
230C2×C4⋊C4—341no—
231C2×C4⋊C4—341no—
232C2×C4⋊C4—681no—
233C2×C22⋊C4—681no—
234C2×C22⋊C4—681no—
235C2×C42—681no—
236M32—24no—
237C2×C16C22×D1714yes—
238#107—172no—
239#107—172no—
240#107—172no—
241#107—341no—
242#107—341no—
243#107—341no—
244#106—342no—
245#106—342no—
246#103—342no—
247#103—342no—
248C17#3364911yescharacteristic, complemented, Sylow
249C24D13611yescharacteristic
250C2×Q8—342no—
251C2×Q8—681no—
252C2×Q8—682no—
253C2×Q8—682no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—681no—
257C22×C4—681no—
258C22×C4—681no—
259C22×C4—681no—
260C22×C4D13611yescharacteristic
261C22×C4D13611yescharacteristic
262C22×C4D13611yescharacteristic
263C22×C4C2×D6811yescharacteristic
264C22×C4C2×D6811yescharacteristic
265C22×C4C2×D6811yescharacteristic
266Q16—344no—
267Q16—682no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8C2×D6811yescharacteristic
274C2×C8C2×D6811yescharacteristic
275C2×C8C2×D6811yescharacteristic
276C2×C8#6325414yes—
277C2×C8C23×D1711yescharacteristic
278C4⋊C4—342no—
279C4⋊C4—342no—
280C4⋊C4—342no—
281C4⋊C4—681no—
282C4⋊C4—681no—
283C4⋊C4—681no—
284C4⋊C4—682no—
285C4⋊C4—682no—
286C22⋊C4—682no—
287C22⋊C4—682no—
288C22⋊C4—682no—
289C22⋊C4—682no—
290C42—682no—
291C42—682no—
292C16—24no—
293C23—22no—
294C23—22no—
295C23—41no—
296C23C2×D13611yescharacteristic
297C23C2×D13611yescharacteristic
298C23C2×D13611yescharacteristic
299Q8—682no—
300Q8—682no—
301Q8—1362no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—682no—
310C2×C4—682no—
311C2×C4—682no—
312C2×C4—682no—
313C2×C4—1362no—
314C2×C4—1362no—
315C2×C4—1362no—
316C2×C4—1362no—
317C2×C4#1058757212yes—
318C2×C4C2×D13611yescharacteristic
319C2×C4C2×D13611yescharacteristic
320C2×C4C2×D13611yescharacteristic
321C2×C4#1058759512yes—
322C2×C4C22×D6811yescharacteristic
323C8—21no—
324C8—21no—
325C8#1058757212yes—
326C8C2×#6325412yes—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—44no—
335C22#1176312312yes—
336C22#1176325212yes—
337C22C22×D13611yescharacteristic
338C4—21no—
339C4—21no—
340C4—1362no—
341C4—1362no—
342C4—1362no—
343C4—1362no—
344C4#1176312312yes—
345C4C2×#1058757211yescharacteristic
346C4C2×#1058759511yescharacteristic
347C2—22no—
348C2—22no—
349C2—41no—
350C2C2×#1176312311yescharacteristic
351C2C2×#1176325211yescharacteristic
352C2#1454192011yescharacteristic
subgroupG/H
centerC22C22×D136
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC17×#2112C2
socleC22×C17C22×D8
prime radicalC24×C17D8
hypercenter#2112D17
hypocenterC17#33649
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
3. #14530993C2semidirect
4. #14530987C2semidirect
5. #14530982C2semidirect
6. #14530572C2semidirect
7. #14530505C2semidirect
25. #11762637C22semidirect
26. #11762635C22semidirect
27. #11762614C22semidirect
28. #11762614C22semidirect
248. C17#33649semidirect

Maximal subgroups

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

subgroupindex
C2×#117630502
C2×#117628472
#145309932
#145309872
#145309822
#145305722
#145305052
C17×#21122
#3364917