Finite Group Databasebeta

← #17034879

Subgroups — #17034879

Order 4352.

886 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#63223C211yesmaximal, characteristic
2#14531191C211yesmaximal, characteristic
3#14528953C212yesmaximal, complemented
4#14528917C212yesmaximal, complemented
5#14528815C214yesmaximal
6C4×#10587478C212yesmaximal, complemented
7C4×#10587479C212yesmaximal, complemented
8C17×#2283C211yesmaximal, characteristic
9C22×#63223C2212yes—
10C2×#10587612—21no—
11C2×#10587478C2214yes—
12C2×#10587479C2214yes—
13C4×#63223C418yescomplemented
14C2×#10587474C2211yescharacteristic
15C2×#10587474C2211yescharacteristic
16C2×C4×#3622C2212yes—
17#11762679C2212yes—
18#11762675—21no—
19#11762669—21no—
20#11762669C2212yes—
21C4×C4⋊Dic17—21no—
22C4×C4⋊Dic17C2212yes—
23#11762571C2214yes—
24#11762571C2214yes—
25C22×C42×C17C2211yescharacteristic
26C17×#383—21no—
27C17×#382C2212yes—
28C17×#381—21no—
29C17×#380C2212yes—
30C4×C4⋊C4×C17C2211yescharacteristic
31C4×C4⋊C4×C17C2211yescharacteristic
32C4×C22⋊C4×C17—21no—
33C17×#376—21no—
34C23×C4×C17D411yescharacteristic
35C23×C4×C17D411yescharacteristic
36C23×C4×C17C2311yescharacteristic
37C2×C4⋊C4×C17C2312yes—
38C2×C4⋊C4×C17C2312yes—
39C2×C22⋊C4×C17—22no—
40C2×C42×C17—21no—
41C2×C42×C17—21no—
42C2×C42×C17C2×C414yes—
43C2×C42×C17D411yescharacteristic
44C2×C42×C17D411yescharacteristic
45C2×C42×C17Q814yes—
46C2×C42×C17C2311yescharacteristic
47C2×C42×C17C2311yescharacteristic
48C2×C22⋊Dic17—22no—
49#10587612—44no—
50C2×C4⋊Dic17—22no—
51C2×C4⋊Dic17—22no—
52C2×C4⋊Dic17C2314yes—
53C2×C4×Dic17—22no—
54C2×#63223—28no—
55C2×#63223C2×C418yes—
56C22×#3622C2312yes—
57C22×#3622C2312yes—
58C17×#95—22no—
59C17×#95—22no—
60C17×#95—22no—
61C17×#95—21no—
62C17×#95—21no—
63#10587502—22no—
64#10587502—22no—
65#10587479—28no—
66#10587478—28no—
67#10587474C2×C414yes—
68#10587474C2×C414yes—
69C4×#3622C2×C418yes—
70C24×C17C2×D411yescharacteristic
71C22×C4×C17—22no—
72C22×C4×C17—22no—
73C22×C4×C17—24no—
74C22×C4×C17—22no—
75C22×C4×C17—22no—
76C22×C4×C17—42no—
77C22×C4×C17C4⋊C414yes—
78C22×C4×C17C4⋊C414yes—
79C22×C4×C17C22×C412yes—
80C22×C4×C17C2×D411yescharacteristic
81C22×C4×C17C2×D411yescharacteristic
82C22×C4×C17C2×D411yescharacteristic
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D411yescharacteristic
85C22×C4×C17C2×Q812yes—
86C22×C4×C17C2411yescharacteristic
87C22×Dic17—24no—
88C4⋊C4×C17—24no—
89C4⋊C4×C17—24no—
90C22⋊C4×C17—44no—
91C42×C17—42no—
92C42×C17—42no—
93C42×C17C4⋊C414yes—
94C42×C17C4⋊C414yes—
95C42×C17C22×C412yes—
96C42×C17C22×C412yes—
97C42×C17C2×Q812yes—
98C42×C17C2×Q812yes—
99C22⋊Dic17—44no—
100C4⋊Dic17—24no—
101C4⋊Dic17—24no—
102C4⋊Dic17—44no—
103C4⋊Dic17—44no—
104C4×Dic17—44no—
105#63223—216no—
106C2×#3622—24no—
107C2×#3622C22×C414yes—
108C2×#3622C22×C414yes—
109C2×#3622#4118yes—
110#20819—171nomaximal, Sylow
111C23×C17—22no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—22no—
115C23×C17C2×C4⋊C412yes—
116C23×C17C22×D411yescharacteristic
117C2×C4×C17—22no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—22no—
121C2×C4×C17—22no—
122C2×C4×C17—24no—
123C2×C4×C17—24no—
124C2×C4×C17—28no—
125C2×C4×C17—44no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17—42no—
131C2×C4×C17—42no—
132C2×C4×C17C2×C4⋊C412yes—
133C2×C4×C17C2×C4⋊C412yes—
134C2×C4×C17C2×C4⋊C412yes—
135C2×C4×C17C2×C4⋊C412yes—
136C2×C4×C17C2×C4⋊C412yes—
137C2×C4×C17AGL(1,ℤ/8)12yes—
138C2×C4×C17#13712yes—
139C2×C4×C17C23×C411yescharacteristic
140C2×C4×C17C22×Q811yescharacteristic
141C2×C4×C17C2×#4114yes—
142C2×Dic17—44no—
143C2×Dic17—44no—
144C2×Dic17—44no—
145#3622—24no—
146#3622#11718yescomplemented
147C2×C4×M16—171no—
148#2285—171no—
149#2283D1711yescharacteristic
150#1847—172no—
151#1836—172no—
152#1809—174no—
153C4×#107—172no—
154C4×#106—172no—
155C22×C17—22no—
156C22×C17—22no—
157C22×C17—22no—
158C22×C17—42no—
159C22×C17—42no—
160C22×C17C22×C4⋊C411yescharacteristic
161C22×C17#53212yes—
162C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
163C22×C17C2×#13711yescharacteristic
164C22×C17#57512yes—
165C4×C17—22no—
166C4×C17—22no—
167C4×C17—22no—
168C4×C17—22no—
169C4×C17—82no—
170C4×C17—82no—
171C4×C17#42814yes—
172C4×C17C2×#11712yes—
173C4×C17#53312yes—
174Dic17—84no—
175C22×M16—172no—
176C2×#117—341no—
177C22×C42C2×D1711yescharacteristic
178C2×#107—174no—
179C2×#106—174no—
180C4×M16—178no—
181C2×#97—171no—
182C2×#97—171no—
183C2×C4×C8—172no—
184#384—172no—
185#383—21no—
186#382—341no—
187#382C2×D1712yes—
188#381—21no—
189#380—172no—
190#380—341no—
191#380C2×D1712yes—
192C4×C4⋊C4—172no—
193C4×C4⋊C4—341no—
194C4×C4⋊C4C2×D1711yescharacteristic
195C4×C4⋊C4C2×D1711yescharacteristic
196C4×C22⋊C4—21no—
197#376—21no—
198#340—174no—
199#340—174no—
200C2×C17—42no—
201C2×C17#228511yescharacteristic
202C2×C17C2×#42811yescharacteristic
203C2×C17#291111yescharacteristic
204C2×C17#317112yes—
205C2×C17#317212yes—
206C23×C4C22⋊D1711yescharacteristic
207C23×C4C22⋊D1711yescharacteristic
208C23×C4C22×D1711yescharacteristic
209C2×M16—178no—
210C2×M16—348no—
211C22×C8—172no—
212C22×C8—172no—
213#117—684no—
214C2×C4⋊C4—174no—
215C2×C4⋊C4—342no—
216C2×C4⋊C4—342no—
217C2×C4⋊C4C22×D1712yes—
218C2×C4⋊C4C22×D1712yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—342no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—342no—
224C2×C42Dic3414yes—
225C2×C42C4×D1714yes—
226C2×C42C22⋊D1711yescharacteristic
227C2×C42C22⋊D1711yescharacteristic
228C2×C42C22×D1711yescharacteristic
229C2×C42C22×D1711yescharacteristic
230#107—348no—
231#106—348no—
232#97—174no—
233#97—174no—
234C4×C8—178no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—21no—
240#95—342no—
241#95—342no—
242C17#2081911yescharacteristic, complemented, Sylow
243C24C2×C22⋊D1711yescharacteristic
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—344no—
251C22×C4#6322514yes—
252C22×C4#6322514yes—
253C22×C4C2×Dic3412yes—
254C22×C4C2×C4×D1712yes—
255C22×C4C2×C22⋊D1711yescharacteristic
256C22×C4C2×C22⋊D1711yescharacteristic
257C22×C4C2×C22⋊D1711yescharacteristic
258C22×C4C2×C22⋊D1711yescharacteristic
259C22×C4C2×C22⋊D1711yescharacteristic
260C22×C4C23×D1711yescharacteristic
261M16—3416no—
262C2×C8—174no—
263C2×C8—178no—
264C2×C8—174no—
265C2×C8—344no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—344no—
269C4⋊C4—344no—
270C4⋊C4—684no—
271C4⋊C4—684no—
272C22⋊C4—44no—
273C22⋊C4—684no—
274C42—42no—
275C42—42no—
276C42—684no—
277C42#6322514yes—
278C42#6322514yes—
279C42C2×Dic3412yes—
280C42C2×Dic3412yes—
281C42C2×C4×D1712yes—
282C42C2×C4×D1712yes—
283C23—21no—
284C23—21no—
285C23—22no—
286C23—22no—
287C23C2×#6322512yes—
288C23C22×C22⋊D1711yescharacteristic
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—28no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—684no—
305C2×C4—684no—
306C2×C4—684no—
307C2×C4C2×#6322512yes—
308C2×C4C2×#6322512yes—
309C2×C4C2×#6322512yes—
310C2×C4C2×#6322512yes—
311C2×C4C2×#6322512yes—
312C2×C4#1058763512yes—
313C2×C4#1058763712yes—
314C2×C4C22×Dic3411yescharacteristic
315C2×C4C22×C4×D1711yescharacteristic
316C2×C4#41×D1714yes—
317C8—178no—
318C8—344no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—42no—
323C22—42no—
324C22C22×#6322511yescharacteristic
325C22#1176335512yes—
326C22C2×#1058763511yescharacteristic
327C22C2×#1058763711yescharacteristic
328C22#1176360712yes—
329C4—22no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—82no—
334C4—82no—
335C4—1364no—
336C4#1176285214yes—
337C4#117×D1712yes—
338C4#1176336212yes—
339C2—42no—
340C2#1453118711yescharacteristic
341C2C2×#1176285211yescharacteristic
342C2#1453737311yescharacteristic
343C2#1453938012yes—
344C2#1453939112yes—
subgroupG/H
centerC22×C4C2×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2283C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenter#2283D17
hypocenterC17#20819
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ #2283

Complements

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

NH ≅ G/Ntype
3. #14528953C2semidirect
4. #14528917C2semidirect
6. C4×#10587478C2semidirect
7. C4×#10587479C2semidirect
13. C4×#63223C4semidirect
146. #3622#117semidirect
242. C17#20819semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#632232
#145311912
#145289532
#145289172
#145288152
C4×#105874782
C4×#105874792
C17×#22832
#2081917