Finite Group Databasebeta

← #16963473

Subgroups — #16963473

Order 4352.

1558 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14527033C214yesmaximal, complemented
2#14527025C212yesmaximal
3#14527021C214yesmaximal, complemented
4#14527016C212yesmaximal, complemented
5#14527011C212yesmaximal, complemented
6C2×C17×#390C211yesmaximal, characteristic
7C22×C22⋊Dic17C2212yes—
8C22×C22⋊Dic17C2214yes—
9C22×C22⋊Dic17C2212yes—
10C22×C4×Dic17C2212yes—
11C2×#10587502C2214yes—
12C2×#10587502C2214yes—
13C2×#10587502C2212yes—
14C2×#10587502C2212yes—
15C2×#10587502C2214yes—
16C2×#10587502C2212yes—
17C23×D4×C17C2211yescharacteristic
18C22×C4⋊C4×C17C2211yescharacteristic
19C22×C22⋊C4×C17C2214yes—
20C22×C42×C17C2211yescharacteristic
21C17×#390C418yescomplemented
22C25×C17C2312yes—
23C22×D4×C17—24no—
24C22×D4×C17—42no—
25C22×D4×C17—41no—
26C22×D4×C17—41no—
27C22×D4×C17C2×C414yes—
28C23×C4×C17C2312yes—
29C23×C4×C17C2312yes—
30C23×C4×C17C2311yescharacteristic
31C23×Dic17C2314yes—
32C23×Dic17C2314yes—
33C2×C4⋊C4×C17—41no—
34C2×C4⋊C4×C17—41no—
35C2×C4⋊C4×C17C2×C414yes—
36C2×C22⋊C4×C17—44no—
37C2×C22⋊C4×C17—44no—
38C2×C22⋊C4×C17C2×C4116yes—
39C2×C42×C17—42no—
40C2×C42×C17C2×C414yes—
41C2×C22⋊Dic17—28no—
42C2×C22⋊Dic17—28no—
43C2×C22⋊Dic17—42no—
44C2×C22⋊Dic17—44no—
45C2×C22⋊Dic17—44no—
46C2×C22⋊Dic17—42no—
47C2×C22⋊Dic17—44no—
48C2×C4×Dic17—44no—
49C2×C4×Dic17—42no—
50#10587502—28no—
51#10587502—28no—
52#10587502—44no—
53#10587502—42no—
54#10587502—42no—
55#10587502—42no—
56C24×C17—24no—
57C24×C17—24no—
58C24×C17—24no—
59C24×C17—24no—
60C24×C17—22no—
61C24×C17—22no—
62C24×C17—24no—
63C24×C17—24no—
64C24×C17C22×C414yes—
65C24×C17C2411yescharacteristic
66C2×D4×C17—28no—
67C2×D4×C17—44no—
68C2×D4×C17—44no—
69C2×D4×C17—48no—
70C2×D4×C17—48no—
71C22×C4×C17—22no—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17—24no—
75C22×C4×C17—22no—
76C22×C4×C17—22no—
77C22×C4×C17—24no—
78C22×C4×C17—22no—
79C22×C4×C17—24no—
80C22×C4×C17—22no—
81C22×C4×C17—24no—
82C22×C4×C17C22×C414yes—
83C22×C4×C17C22×C414yes—
84C22×C4×C17C22×C412yes—
85C22×C4×C17#4114yes—
86C22×Dic17—24no—
87C22×Dic17—24no—
88C22×Dic17—24no—
89C22×Dic17—24no—
90C22×Dic17—24no—
91C22×Dic17—24no—
92C22×Dic17—24no—
93C22×Dic17—24no—
94C22×Dic17—24no—
95C22×Dic17—24no—
96C22×Dic17—24no—
97C22×Dic17—24no—
98C22×Dic17—24no—
99C22×Dic17#4118yes—
100C4⋊C4×C17—44no—
101C22⋊C4×C17—416no—
102C42×C17—44no—
103C22⋊Dic17—48no—
104C22⋊Dic17—416no—
105C22⋊Dic17—48no—
106C4×Dic17—48no—
107#14503—171nomaximal, Sylow
108C23×C17—24no—
109C23×C17—28no—
110C23×C17—28no—
111C23×C17—24no—
112C23×C17—24no—
113C23×C17—44no—
114C23×C17—44no—
115C23×C17—44no—
116C23×C17—44no—
117C23×C17—44no—
118C23×C17—44no—
119C23×C17—42no—
120C23×C17—42no—
121C23×C17—44no—
122C23×C17—44no—
123C23×C17—44no—
124C23×C17—44no—
125C23×C17—44no—
126C23×C17—42no—
127C23×C17—44no—
128C23×C17—42no—
129C23×C17C23×C411yescharacteristic
130C23×C17C2×#4112yes—
131C23×C17C2×#4114yes—
132C23×C17H(5,𝔽2)12yes—
133C23×C17H(5,𝔽2)11yescharacteristic
134C23×C17H(5,𝔽2)11yescharacteristic
135C23×C17H(5,𝔽2)14yes—
136D4×C17—416no—
137C2×C4×C17—22no—
138C2×C4×C17—22no—
139C2×C4×C17—24no—
140C2×C4×C17—24no—
141C2×C4×C17—24no—
142C2×C4×C17—24no—
143C2×C4×C17—24no—
144C2×C4×C17—28no—
145C2×C4×C17—24no—
146C2×C4×C17—44no—
147C2×C4×C17—44no—
148C2×C4×C17—42no—
149C2×C4×C17—44no—
150C2×C4×C17—42no—
151C2×C4×C17#11714yes—
152C2×Dic17—28no—
153C2×Dic17—28no—
154C2×Dic17—28no—
155C2×Dic17—44no—
156C2×Dic17—44no—
157C2×Dic17—44no—
158C2×Dic17—44no—
159C2×Dic17—44no—
160C2×Dic17—44no—
161C2×Dic17—44no—
162C2×Dic17—44no—
163C2×Dic17—44no—
164C2×Dic17—44no—
165C2×Dic17—44no—
166C2×#390D1711yescharacteristic
167#1440—174no—
168#1438—172no—
169#1437—174no—
170#1436—172no—
171#1434—172no—
172C22×C17—44no—
173C22×C17—48no—
174C22×C17—48no—
175C22×C17—44no—
176C22×C17—44no—
177C22×C17—84no—
178C22×C17—84no—
179C22×C17—84no—
180C22×C17—84no—
181C22×C17C2×#11711yescharacteristic
182C22×C17C4×#4112yes—
183C22×C17#51811yescharacteristic
184C22×C17#51811yescharacteristic
185C22×C17#51812yes—
186C22×C17#53814yes—
187C22×C17#53812yes—
188C22×C17#53812yes—
189C22×C17#54014yes—
190C22×C17#55112yes—
191C22×C17#55314yes—
192C22×C17#55512yes—
193C22×C17#55914yes—
194C22×C17#56012yes—
195C22×C17#56012yes—
196C4×C17—24no—
197C4×C17—44no—
198C4×C17—44no—
199Dic17—48no—
200Dic17—84no—
201C23×D4C2×D1711yescharacteristic
202C22×C4⋊C4C2×D1711yescharacteristic
203C22×C22⋊C4—172no—
204C22×C22⋊C4—174no—
205C22×C22⋊C4—172no—
206C22×C22⋊C4C2×D1714yes—
207C22×C42—172no—
208C22×C42C2×D1711yescharacteristic
209#390Dic1718yescomplemented
210C2×#95—174no—
211C2×#95—174no—
212C2×#95—172no—
213C2×#95—172no—
214C2×#95—174no—
215C2×#95—172no—
216C2×C17—84no—
217C2×C17#231211yescharacteristic
218C2×C17#233312yes—
219C2×C17#235712yes—
220C2×C17#237211yescharacteristic
221C2×C17#238011yescharacteristic
222C2×C17#279714yes—
223C2×C17#280314yes—
224C25C22×D1712yes—
225C22×D4—24no—
226C22×D4—42no—
227C22×D4—41no—
228C22×D4—41no—
229C22×D4C2×Dic1714yes—
230C23×C4—174no—
231C23×C4—174no—
232C23×C4C22×D1712yes—
233C23×C4C22×D1712yes—
234C23×C4C22×D1711yescharacteristic
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4C2×Dic1714yes—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—348no—
241C2×C22⋊C4—348no—
242C2×C22⋊C4—682no—
243C2×C22⋊C4—684no—
244C2×C22⋊C4—684no—
245C2×C22⋊C4—682no—
246C2×C22⋊C4—684no—
247C2×C22⋊C4C2×Dic17116yes—
248C2×C42—42no—
249C2×C42—684no—
250C2×C42—682no—
251C2×C42C2×Dic1714yes—
252#95—348no—
253#95—348no—
254#95—682no—
255#95—682no—
256#95—684no—
257#95—682no—
258C17#1450311yescharacteristic, complemented, Sylow
259C24—24no—
260C24—24no—
261C24—24no—
262C24—22no—
263C24—22no—
264C24—24no—
265C24—24no—
266C24—24no—
267C24C22×Dic1714yes—
268C24C23×D1711yescharacteristic
269C2×D4—28no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—48no—
273C2×D4—48no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—178no—
286C22×C4—344no—
287C22×C4—344no—
288C22×C4—344no—
289C22×C4—344no—
290C22×C4—344no—
291C22×C4—344no—
292C22×C4—344no—
293C22×C4—344no—
294C22×C4—344no—
295C22×C4—344no—
296C22×C4—344no—
297C22×C4—344no—
298C22×C4—344no—
299C22×C4#6325214yes—
300C22×C4C22×Dic1714yes—
301C22×C4C22×Dic1714yes—
302C22×C4C22×Dic1712yes—
303C4⋊C4—44no—
304C22⋊C4—416no—
305C22⋊C4—688no—
306C22⋊C4—6816no—
307C22⋊C4—688no—
308C42—44no—
309C42—688no—
310C23—24no—
311C23—28no—
312C23—28no—
313C23—24no—
314C23—24no—
315C23—42no—
316C23—42no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—42no—
323C23—42no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23—44no—
328C23—44no—
329C23—44no—
330C23—44no—
331C23C2×#6325212yes—
332C23#1058769111yescharacteristic
333C23#1058769114yes—
334C23#1058769111yescharacteristic
335C23#1058769112yes—
336C23#41×D1714yes—
337C23C23×Dic1711yescharacteristic
338D4—416no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—28no—
347C2×C4—24no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—348no—
354C2×C4—348no—
355C2×C4—348no—
356C2×C4—684no—
357C2×C4—684no—
358C2×C4—684no—
359C2×C4—684no—
360C2×C4—684no—
361C2×C4—684no—
362C2×C4—684no—
363C2×C4—684no—
364C2×C4—684no—
365C2×C4—684no—
366C2×C4—684no—
367C2×C4#1058761214yes—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—48no—
372C22—48no—
373C22—84no—
374C22—84no—
375C22—84no—
376C22—84no—
377C22C2×#1058761211yescharacteristic
378C22#41×Dic1712yes—
379C22#1176328411yescharacteristic
380C22#1176328411yescharacteristic
381C22#1176328412yes—
382C22#1176339114yes—
383C22#1176339712yes—
384C22#1176339712yes—
385C22#1176340914yes—
386C22#1176348912yes—
387C22#1176350914yes—
388C22#1176352112yes—
389C22#1176354214yes—
390C22#1176354612yes—
391C22#1176354612yes—
392C4—24no—
393C4—44no—
394C4—44no—
395C4—688no—
396C4—1364no—
397C2—84no—
398C2#1453138211yescharacteristic
399C2#1453155112yes—
400C2#1453175712yes—
401C2#1453189611yescharacteristic
402C2#1453196111yescharacteristic
403C2#1453663814yes—
404C2#1453671314yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#390C2
socle, cosocleC24×C17C24
prime radicalC23×D4×C17C22
hypercenterC2×#390D17
hypocenterC17#14503
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. #14527033C2semidirect
3. #14527021C2semidirect
4. #14527016C2semidirect
5. #14527011C2semidirect
21. C17×#390C4semidirect
209. #390Dic17semidirect
258. C17#14503semidirect

Maximal subgroups

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

subgroupindex
#145270332
#145270252
#145270212
#145270162
#145270112
C2×C17×#3902
#1450317