Finite Group Databasebeta

← #16956732

Subgroups — #16956732

Order 4352.

1710 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762675C211yesmaximal, characteristic
2C2×C4×C4⋊Dic17C211yesmaximal, characteristic
3#14527055C212yesmaximal
4#14527052C212yesmaximal
5#14527016C214yesmaximal
6#14527014C214yesmaximal
7C22×C4×Q8×C17C211yesmaximal, characteristic
8C22×C4⋊Dic17C2212yes—
9C22×C4⋊Dic17C2212yes—
10C22×C4×Dic17C2212yes—
11#11762675—24no—
12C4×C4⋊Dic17—24no—
13C2×#10587502C2218yes—
14C2×#10587502C2212yes—
15C2×#10587502C2212yes—
16C2×#10587502C2212yes—
17C2×#10587502C2218yes—
18C23×Q8×C17C2211yescharacteristic
19C2×C4×Q8×C17—24no—
20C2×C4×Q8×C17—24no—
21C2×C4×Q8×C17C418yescomplemented
22C22×C4⋊C4×C17C2212yes—
23C22×C4⋊C4×C17C2211yescharacteristic
24C22×C42×C17C2212yes—
25C22×C42×C17C2211yescharacteristic
26C22×Q8×C17—22no—
27C22×Q8×C17—22no—
28C22×Q8×C17—28no—
29C22×Q8×C17C2×C414yes—
30C23×C4×C17C2312yes—
31C23×C4×C17C2311yescharacteristic
32C23×C4×C17C2311yescharacteristic
33C23×C4×C17C2312yes—
34C23×C4×C17C2311yescharacteristic
35C23×Dic17C2314yes—
36C23×Dic17C2314yes—
37C4×Q8×C17—232no—
38C2×C4⋊C4×C17—24no—
39C2×C4⋊C4×C17—24no—
40C2×C4⋊C4×C17—22no—
41C2×C4⋊C4×C17—22no—
42C2×C4⋊C4×C17C2×C414yes—
43C2×C4⋊C4×C17C2×C418yes—
44C2×C42×C17—24no—
45C2×C42×C17—24no—
46C2×C42×C17C2×C418yes—
47C2×C42×C17C2×C414yes—
48C2×C42×C17D414yes—
49C2×C42×C17D414yes—
50C2×C4⋊Dic17—24no—
51C2×C4⋊Dic17—24no—
52C2×C4⋊Dic17—24no—
53C2×C4⋊Dic17—28no—
54C2×C4⋊Dic17Q818yescomplemented
55C2×C4×Dic17—24no—
56C2×C4×Dic17—44no—
57#10587502—24no—
58#10587502—24no—
59#10587502—24no—
60#10587502—216no—
61#10587502—48no—
62C24×C17C2411yescharacteristic
63C2×Q8×C17—216no—
64C2×Q8×C17—216no—
65C2×Q8×C17—28no—
66C2×Q8×C17—216no—
67C22×C4×C17—22no—
68C22×C4×C17—22no—
69C22×C4×C17—24no—
70C22×C4×C17—24no—
71C22×C4×C17—22no—
72C22×C4×C17—22no—
73C22×C4×C17—22no—
74C22×C4×C17—22no—
75C22×C4×C17—24no—
76C22×C4×C17—24no—
77C22×C4×C17C22×C414yes—
78C22×C4×C17C22×C412yes—
79C22×C4×C17C22×C412yes—
80C22×C4×C17C22×C414yes—
81C22×C4×C17C22×C412yes—
82C22×C4×C17C2×D412yes—
83C22×C4×C17C2×D412yes—
84C22×C4×C17C2×D412yes—
85C22×C4×C17C2×D412yes—
86C22×C4×C17C2×D412yes—
87C22×C4×C17C2×D412yes—
88C22×C4×C17C2×Q814yes—
89C22×C4×C17#4118yes—
90C22×C4×C17#4114yes—
91C22×Dic17—24no—
92C22×Dic17—28no—
93C22×Dic17—24no—
94C22×Dic17—28no—
95C22×Dic17—24no—
96C22×Dic17—28no—
97C22×Dic17—28no—
98C22×Dic17—24no—
99C22×Dic17—24no—
100C22×Dic17C2×Q818yes—
101C4⋊C4×C17—216no—
102C4⋊C4×C17—28no—
103C42×C17—216no—
104C42×C17C22⋊C4116yes—
105C4⋊Dic17—216no—
106C4⋊Dic17—48no—
107C4×Dic17—48no—
108#13925—171nomaximal, Sylow
109C23×C17C23×C411yescharacteristic
110C23×C17C22×D411yescharacteristic
111C23×C17C22×D411yescharacteristic
112C23×C17C22×Q812yes—
113C23×C17C2×#4114yes—
114C23×C17C2×#4112yes—
115C23×C17H(5,𝔽2)12yes—
116C23×C172−1+412yes—
117Q8×C17—232no—
118C2×C4×C17—24no—
119C2×C4×C17—24no—
120C2×C4×C17—24no—
121C2×C4×C17—24no—
122C2×C4×C17—24no—
123C2×C4×C17—24no—
124C2×C4×C17—28no—
125C2×C4×C17—28no—
126C2×C4×C17—22no—
127C2×C4×C17—22no—
128C2×C4×C17—24no—
129C2×C4×C17—24no—
130C2×C4×C17—28no—
131C2×C4×C17—42no—
132C2×C4×C17—42no—
133C2×C4×C17—44no—
134C2×C4×C17C2×C22⋊C414yes—
135C2×C4×C17C2×C22⋊C414yes—
136C2×C4×C17C2×C22⋊C414yes—
137C2×C4×C17#11718yes—
138C2×C4×C17C4×Q818yes—
139C2×C4×C17C22⋊Q818yes—
140C2×C4×C17#12414yes—
141C2×C4×C17C4⋊Q814yes—
142C2×Dic17—28no—
143C2×Dic17—216no—
144C2×Dic17—44no—
145C2×Dic17—44no—
146C2×Dic17—48no—
147C2×Dic17—44no—
148C2×Dic17—44no—
149C2×Dic17—44no—
150C2×Dic17—48no—
151C2×Dic17—48no—
152C22×C4×Q8D1711yescharacteristic
153C2×#382—171no—
154C2×C4×C4⋊C4—171no—
155#1446—172no—
156#1445—172no—
157#1436—174no—
158C4⋊#95—174no—
159C22×C17C22×C22⋊C411yescharacteristic
160C22×C17C2×#11712yes—
161C22×C17C2×C4×Q812yes—
162C22×C17#52012yes—
163C22×C17C2×C22⋊Q812yes—
164C22×C17#52512yes—
165C22×C17#52512yes—
166C22×C17C2×#12411yescharacteristic
167C22×C17C2×C4⋊Q811yescharacteristic
168C22×C17#53511yescharacteristic
169C22×C17#53611yescharacteristic
170C22×C17#53712yes—
171C22×C17#55414yes—
172C22×C17#55614yes—
173C22×C17#55714yes—
174C22×C17#55914yes—
175C4×C17—24no—
176C4×C17—24no—
177C4×C17—28no—
178C4×C17—44no—
179C4×C17#39118yes—
180Dic17—48no—
181Dic17—84no—
182C23×Q8C2×D1711yescharacteristic
183C2×C4×Q8—24no—
184C2×C4×Q8—24no—
185C2×C4×Q8Dic1718yescomplemented
186C22×C4⋊C4—172no—
187C22×C4⋊C4—172no—
188C22×C4⋊C4C2×D1712yes—
189C22×C4⋊C4C2×D1711yescharacteristic
190C22×C42—172no—
191C22×C42C2×D1712yes—
192C22×C42C2×D1711yescharacteristic
193#382—344no—
194C4×C4⋊C4—344no—
195C2×#95—178no—
196C2×#95—172no—
197C2×#95—172no—
198C2×#95—172no—
199C2×#95—178no—
200C2×C17#228011yescharacteristic
201C2×C17C2×#39111yescharacteristic
202C2×C17#231111yescharacteristic
203C2×C17#235514yes—
204C2×C17#258012yes—
205C2×C17#258112yes—
206C2×C17#258412yes—
207C2×C17#259212yes—
208C22×Q8—28no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8C2×Dic1714yes—
212C23×C4—174no—
213C23×C4—174no—
214C23×C4C22×D1712yes—
215C23×C4C22×D1712yes—
216C23×C4C22×D1711yescharacteristic
217C23×C4C22×D1711yescharacteristic
218C23×C4C22×D1711yescharacteristic
219C4×Q8—232no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—178no—
225C2×C4⋊C4—344no—
226C2×C4⋊C4—344no—
227C2×C4⋊C4—348no—
228C2×C4⋊C4—344no—
229C2×C4⋊C4C2×Dic1714yes—
230C2×C4⋊C4C2×Dic1718yes—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—344no—
234C2×C42—684no—
235C2×C42C2×Dic1718yes—
236C2×C42C2×Dic1714yes—
237C2×C42C22⋊D1714yes—
238C2×C42C22⋊D1714yes—
239#95—3416no—
240#95—344no—
241#95—344no—
242#95—344no—
243#95—688no—
244C17#1392511yescharacteristic, complemented, Sylow
245C24C23×D1711yescharacteristic
246C2×Q8—216no—
247C2×Q8—216no—
248C2×Q8—216no—
249C2×Q8—28no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—178no—
261C22×C4—348no—
262C22×C4—344no—
263C22×C4—348no—
264C22×C4—344no—
265C22×C4—348no—
266C22×C4—348no—
267C22×C4—344no—
268C22×C4—344no—
269C22×C4—344no—
270C22×C4#6325218yes—
271C22×C4Q8×D1714yes—
272C22×C4#6325614yes—
273C22×C4C22×Dic1714yes—
274C22×C4C22×Dic1714yes—
275C22×C4C22×Dic1712yes—
276C22×C4C22×Dic1712yes—
277C22×C4C22×Dic1712yes—
278C22×C4C2×C22⋊D1712yes—
279C22×C4C2×C22⋊D1712yes—
280C22×C4C2×C22⋊D1712yes—
281C22×C4C2×C22⋊D1712yes—
282C22×C4C2×C22⋊D1712yes—
283C22×C4C2×C22⋊D1712yes—
284C4⋊C4—216no—
285C4⋊C4—28no—
286C4⋊C4—3416no—
287C4⋊C4—688no—
288C42—216no—
289C42—688no—
290C42C22⋊Dic17116yes—
291C23C2×#6325214yes—
292C23C2×Q8×D1712yes—
293C23C2×#6325612yes—
294C23#1058769612yes—
295C23#1058769712yes—
296C23C23×Dic1711yescharacteristic
297C23C22×C22⋊D1711yescharacteristic
298C23C22×C22⋊D1711yescharacteristic
299Q8—232no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—28no—
305C2×C4—28no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—28no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—3416no—
317C2×C4—348no—
318C2×C4—688no—
319C2×C4—684no—
320C2×C4—684no—
321C2×C4—688no—
322C2×C4—688no—
323C2×C4—684no—
324C2×C4—684no—
325C2×C4—684no—
326C2×C4#1058761218yes—
327C2×C4#1058763014yes—
328C2×C4Q8×Dic1718yes—
329C2×C4#1058763218yes—
330C2×C4#1058763314yes—
331C2×C4C2×C22⋊Dic1714yes—
332C2×C4C2×C22⋊Dic1714yes—
333C2×C4C2×C22⋊Dic1714yes—
334C22C22×C22⋊Dic1711yescharacteristic
335C22C2×#1058761212yes—
336C22C2×Q8×Dic1712yes—
337C22#1176329212yes—
338C22C2×#1058763212yes—
339C22#1176332012yes—
340C22#1176332012yes—
341C22C2×#1058763311yescharacteristic
342C22C2×#1058763011yescharacteristic
343C22#1176337311yescharacteristic
344C22#1176338011yescharacteristic
345C22#1176338712yes—
346C22#1176351914yes—
347C22#1176353214yes—
348C22#1176353714yes—
349C22#1176354414yes—
350C4—24no—
351C4—28no—
352C4—24no—
353C4—44no—
354C4—688no—
355C4—1364no—
356C4#1176271618yes—
357C2#1453116711yescharacteristic
358C2C2×#1176271611yescharacteristic
359C2#1453137611yescharacteristic
360C2#1453174114yes—
361C2#1453414212yes—
362C2#1453415012yes—
363C2#1453417212yes—
364C2#1453427212yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC22×C4×Q8×C17C2
socle, prime radical, cosocleC24×C17C24
hypercenterC22×C4×Q8D17
hypocenterC17#13925
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
21. C2×C4×Q8×C17C4semidirect
54. C2×C4⋊Dic17Q8semidirect
185. C2×C4×Q8Dic17semidirect
244. C17#13925semidirect

Maximal subgroups

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

subgroupindex
C2×#117626752
C2×C4×C4⋊Dic172
#145270552
#145270522
#145270162
#145270142
C22×C4×Q8×C172
#1392517