Finite Group Databasebeta

← #16945192

Subgroups — #16945192

Order 4352.

506 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14528858C211yesmaximal, characteristic
2C2×#11762533C211yesmaximal, characteristic
3#14526808C212yesmaximal, complemented
4#14526784C212yesmaximal, complemented
5C17×#2482C211yesmaximal, characteristic
6C2×C22⋊#3622—21no—
7#11762665—21no—
8#11762576—21no—
9#11762576C2212yes—
10#11762533—21no—
11#11762533C2212yes—
12C2×C17×#117C411yescharacteristic, complemented
13C22×C22⋊C4×C17C2211yescharacteristic
14C17×#399C411yescharacteristic, complemented
15C17×#399C2211yescharacteristic
16C17×#397C411yescharacteristic, complemented
17C17×#397C2211yescharacteristic
18C17×#388—21no—
19C17×#388—21no—
20C17×#385—21no—
21C17×#381C411yescharacteristic, complemented
22C17×#379—21no—
23C25×C17D411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C17×#117—41no—
27C17×#117—41no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17—21no—
33C2×C22⋊C4×C17—21no—
34C2×C22⋊C4×C17—21no—
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—41no—
39C2×C22⋊C4×C17—41no—
40C2×C22⋊C4×C17C2×C411yescharacteristic
41C2×C22⋊C4×C17D411yescharacteristic
42C2×C22⋊C4×C17D411yescharacteristic
43C2×C22⋊C4×C17C2311yescharacteristic
44C2×C42×C17—21no—
45C2×C22⋊Dic17—22no—
46C2×C22⋊Dic17—41no—
47C2×C22⋊Dic17—41no—
48C22×#3622—41no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95—21no—
52C17×#95C2×C411yescharacteristic
53C17×#95C2×C411yescharacteristic
54C22⋊#3622—22no—
55C22⋊#3622—41no—
56C24×C17—21no—
57C24×C17—21no—
58C24×C17—22no—
59C24×C17—41no—
60C24×C17—42no—
61C24×C17—81no—
62C24×C17C22⋊C411yescharacteristic
63C24×C17C22⋊C411yescharacteristic
64C24×C17C2×D411yescharacteristic
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—42no—
74C22×C4×C17—81no—
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C22×C411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×Dic17—42no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—81no—
83C4⋊C4×C17—81no—
84C22⋊C4×C17—41no—
85C22⋊C4×C17—41no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C22⋊C4×C17—42no—
89C22⋊C4×C17—42no—
90C22⋊C4×C17—41no—
91C22⋊C4×C17—41no—
92C22⋊C4×C17—42no—
93C22⋊C4×C17—81no—
94C22⋊C4×C17—81no—
95C42×C17—42no—
96C22⋊Dic17—42no—
97C22⋊Dic17—42no—
98C22⋊Dic17—82no—
99C22⋊Dic17—82no—
100C2×#3622—42no—
101C2×#3622—82no—
102#11841—171nomaximal, Sylow
103C23×C17—22no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—41no—
112C23×C17—42no—
113C23×C17—42no—
114C23×C17—41no—
115C23×C17—41no—
116C23×C17—41no—
117C23×C17—41no—
118C23×C17—42no—
119C23×C17—42no—
120C23×C17—81no—
121C23×C17—81no—
122C23×C17—84no—
123C23×C17—84no—
124C23×C17#9911yescharacteristic
125C23×C17#9911yescharacteristic
126C23×C17C2×C22⋊C411yescharacteristic
127C23×C17#13512yes—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—42no—
133C2×C4×C17—42no—
134C2×C4×C17—42no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17—41no—
144C2×C4×C17—41no—
145C2×C4×C17—41no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—41no—
151C2×C4×C17—81no—
152C2×Dic17—82no—
153C2×Dic17—82no—
154C2×Dic17—82no—
155#3622—82no—
156#2482D1711yescharacteristic
157#1819—171no—
158C2×#323—171no—
159#1352—172no—
160#1344—172no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—21no—
167C22×C17—22no—
168C22×C17—21no—
169C22×C17—42no—
170C22×C17—42no—
171C22×C17—42no—
172C22×C17—42no—
173C22×C17—41no—
174C22×C17—41no—
175C22×C17—41no—
176C22×C17—41no—
177C22×C17—41no—
178C22×C17—81no—
179C22×C17—84no—
180C22×C17—81no—
181C22×C17—84no—
182C22×C17#35212yes—
183C22×C17C2×#9911yescharacteristic
184C22×C17#41611yescharacteristic
185C22×C17#42111yescharacteristic
186C4×C17—41no—
187C4×C17—41no—
188C4×C17—41no—
189C4×C17—41no—
190C4×C17—81no—
191C4×C17—81no—
192C4×C17—81no—
193C4×C17—81no—
194Dic17—162no—
195C2×#117Dic1711yescharacteristic, complemented
196C22×C22⋊C4C2×D1711yescharacteristic
197C2×C22⋊C8—341no—
198#399Dic1711yescharacteristic, complemented
199#399C2×D1711yescharacteristic
200#397Dic1711yescharacteristic, complemented
201#397C2×D1711yescharacteristic
202#388—21no—
203#388—21no—
204#385—21no—
205#381Dic1711yescharacteristic, complemented
206#379—21no—
207#379—341no—
208#342—172no—
209#342—341no—
210#323—172no—
211#323—341no—
212C2×C17—21no—
213C2×C17—21no—
214C2×C17—22no—
215C2×C17—42no—
216C2×C17—41no—
217C2×C17—81no—
218C2×C17#151611yescharacteristic
219C2×C17C2×#35211yescharacteristic
220C2×C17#212711yescharacteristic
221C25C22⋊D1711yescharacteristic
222C23×C4C2×Dic1711yescharacteristic
223C23×C4C22⋊D1711yescharacteristic
224C22×C8—681no—
225#117—41no—
226#117—41no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C2×Dic1711yescharacteristic
230C2×C4⋊C4C2×Dic1711yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—342no—
240C2×C22⋊C4—681no—
241C2×C22⋊C4—681no—
242C2×C22⋊C4C2×Dic1711yescharacteristic
243C2×C22⋊C4C22⋊D1711yescharacteristic
244C2×C22⋊C4C22⋊D1711yescharacteristic
245C2×C22⋊C4C22×D1711yescharacteristic
246C2×C42—21no—
247C22⋊C8—342no—
248C22⋊C8—681no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95C2×Dic1711yescharacteristic
253#95C2×Dic1711yescharacteristic
254C17#1184111yescharacteristic, complemented, Sylow
255C24—21no—
256C24—21no—
257C24—22no—
258C24—41no—
259C24—42no—
260C24—81no—
261C24C22⋊Dic1711yescharacteristic
262C24C22⋊Dic1711yescharacteristic
263C24C2×C22⋊D1711yescharacteristic
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—42no—
273C22×C4—81no—
274C22×C4—682no—
275C22×C4C22⋊Dic1711yescharacteristic
276C22×C4C22⋊Dic1711yescharacteristic
277C22×C4C22×Dic1711yescharacteristic
278C22×C4C2×C22⋊D1711yescharacteristic
279C2×C8—682no—
280C2×C8—1362no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—81no—
284C4⋊C4—81no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—42no—
294C22⋊C4—81no—
295C22⋊C4—81no—
296C22⋊C4—682no—
297C22⋊C4—682no—
298C22⋊C4—1362no—
299C22⋊C4—1362no—
300C42—42no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—22no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—81no—
319C23—81no—
320C23—84no—
321C23—84no—
322C23#1058750511yescharacteristic
323C23#1058750511yescharacteristic
324C23#1058763612yes—
325C23C2×C22⋊Dic1711yescharacteristic
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—1362no—
351C2×C4—1362no—
352C2×C4—1362no—
353C8—1362no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—81no—
372C22—84no—
373C22—81no—
374C22—84no—
375C22#1176259912yes—
376C22C2×#1058750511yescharacteristic
377C22#1176280511yescharacteristic
378C22#1176282611yescharacteristic
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—2722no—
388C2—21no—
389C2—21no—
390C2—22no—
391C2—42no—
392C2—41no—
393C2—81no—
394C2#1452734011yescharacteristic
395C2C2×#1176259911yescharacteristic
396C2#1453045311yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2482C2
socleC22×C17C2×#99
prime radicalC25×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#2482D17
hypocenterC17#11841
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
3. #14526808C2semidirect
4. #14526784C2semidirect
12. C2×C17×#117C4semidirect
14. C17×#399C4semidirect
16. C17×#397C4semidirect
21. C17×#381C4semidirect
195. C2×#117Dic17semidirect
198. #399Dic17semidirect
200. #397Dic17semidirect
205. #381Dic17semidirect
254. C17#11841semidirect

Maximal subgroups

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

subgroupindex
#145288582
C2×#117625332
#145268082
#145267842
C17×#24822
#1184117