Finite Group Databasebeta

← #16944805

Subgroups — #16944805

Order 4352.

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

normal subgroups only

This group has 486 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#14528517C211yesmaximal, characteristic, complemented
2#14527118C211yesmaximal, characteristic, complemented
3#14526805C212yesmaximal, complemented
4#14526781C212yesmaximal
5C17×#2610C211yesmaximal, characteristic
6C2×#10587612—21no—
7#11762866—21no—
8#11762576—21no—
9#11762576C2212yes—
10#11762535—21no—
11#11762533C2212yes—
12C2×C17×#124C411yescharacteristic, complemented
13C2×C17×#117C2211yescharacteristic
14C17×#395C411yescharacteristic, complemented
15C17×#394C411yescharacteristic, complemented
16C17×#393C2211yescharacteristic
17C17×#392C2211yescharacteristic
18C17×#390—21no—
19C17×#388—21no—
20C17×#388—21no—
21C17×#386—21no—
22C17×#376C411yescharacteristic, complemented
23C22×Q8×C17C2×C411yescharacteristic
24C22×D4×C17—21no—
25C22×D4×C17C2×C411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C17×#124—41no—
28C17×#124—41no—
29C17×#117—41no—
30C17×#117—41no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C22⋊C4×C17—21no—
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17D411yescharacteristic
41C2×C22⋊C4×C17C2311yescharacteristic
42C2×C42×C17—21no—
43C2×C42×C17C2×C411yescharacteristic
44C2×C42×C17D411yescharacteristic
45C2×C22⋊Dic17—22no—
46#10587612—24no—
47#10587612—42no—
48C2×C4⋊Dic17—41no—
49C2×C4×Dic17—41no—
50C17×#95—21no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95C2×C411yescharacteristic
54C17×#95C2×C411yescharacteristic
55C22⋊#3622—22no—
56C4⋊#3622—41no—
57#10587474—41no—
58C24×C17—21no—
59C24×C17C2×D411yescharacteristic
60C2×Q8×C17—41no—
61C2×Q8×C17—81no—
62C2×D4×C17—41no—
63C2×D4×C17—42no—
64C2×D4×C17—41no—
65C2×D4×C17—41no—
66C2×D4×C17—81no—
67C2×D4×C17—81no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—22no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C22⋊C411yescharacteristic
78C22×C4×C17C22⋊C411yescharacteristic
79C22×C4×C17C22⋊C411yescharacteristic
80C22×C4×C17C22×C411yescharacteristic
81C22×C4×C17C2×D411yescharacteristic
82C22×Dic17—42no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—81no—
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—21no—
96C42×C17—21no—
97C42×C17—21no—
98C42×C17—21no—
99C42×C17—42no—
100C22⋊Dic17—24no—
101C22⋊Dic17—42no—
102C4⋊Dic17—44no—
103C4×Dic17—44no—
104C2×#3622—42no—
105#11758—171nomaximal, Sylow
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17—41no—
116C23×C17—41no—
117C23×C17C2×C22⋊C411yescharacteristic
118C23×C17AGL(1,ℤ/8)11yescharacteristic
119C23×C17#13711yescharacteristic
120Q8×C17—82no—
121D4×C17—82no—
122D4×C17—82no—
123D4×C17—82no—
124C2×C4×C17—22no—
125C2×C4×C17—22no—
126C2×C4×C17—22no—
127C2×C4×C17—22no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—22no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
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#9911yescharacteristic
150C2×C4×C17#9911yescharacteristic
151C2×C4×C17#10412yes—
152C2×Dic17—44no—
153C2×Dic17—82no—
154C2×Dic17—82no—
155#3622—82no—
156#2610D1711yescharacteristic
157#1740—171no—
158#1464—171no—
159#1351—172no—
160#1343—172no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—81no—
170C22×C17—81no—
171C22×C17—81no—
172C22×C17—81no—
173C22×C17—81no—
174C22×C17—81no—
175C22×C17—81no—
176C22×C17—81no—
177C22×C17C2×#9911yescharacteristic
178C22×C17#41711yescharacteristic
179C22×C17C2×#10411yescharacteristic
180C4×C17—21no—
181C4×C17—21no—
182C4×C17—22no—
183C4×C17—42no—
184C4×C17—41no—
185C4×C17—41no—
186C4×C17—41no—
187C4×C17—41no—
188C4×C17—81no—
189C4×C17—81no—
190Dic17—84no—
191C2×#124Dic1711yescharacteristic, complemented
192C2×#117—341no—
193C2×#117C2×D1711yescharacteristic
194#432—341no—
195#395Dic1711yescharacteristic, complemented
196#394Dic1711yescharacteristic, complemented
197#393C2×D1711yescharacteristic
198#392C2×D1711yescharacteristic
199#390—21no—
200#388—21no—
201#388—21no—
202#386—21no—
203#376Dic1711yescharacteristic, complemented
204#342—172no—
205#342—341no—
206#324—341no—
207#323—172no—
208C2×C17—21no—
209C2×C17—21no—
210C2×C17—41no—
211C2×C17—41no—
212C2×C17—161no—
213C2×C17#150411yescharacteristic
214C2×C17#212211yescharacteristic
215C2×C17#212611yescharacteristic
216C22×Q8C2×Dic1711yescharacteristic
217C22×D4—21no—
218C22×D4C2×Dic1711yescharacteristic
219C23×C4C22⋊D1711yescharacteristic
220#124—41no—
221#124—41no—
222#117—41no—
223#117—41no—
224#117—344no—
225#117—682no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—681no—
228C2×C4⋊C4C2×Dic1711yescharacteristic
229C2×C4⋊C4C22⋊D1711yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—342no—
237C2×C22⋊C4C22⋊D1711yescharacteristic
238C2×C22⋊C4C22×D1711yescharacteristic
239C2×C42—21no—
240C2×C42—681no—
241C2×C42C2×Dic1711yescharacteristic
242C2×C42C22⋊D1711yescharacteristic
243C4⋊C8—681no—
244C22⋊C8—342no—
245#97—681no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95C2×Dic1711yescharacteristic
250#95C2×Dic1711yescharacteristic
251C17#1175811yescharacteristic, complemented, Sylow
252C24—21no—
253C24C2×C22⋊D1711yescharacteristic
254C2×Q8—41no—
255C2×Q8—81no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—81no—
261C2×D4—81no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—682no—
271C22×C4C22⋊Dic1711yescharacteristic
272C22×C4C22⋊Dic1711yescharacteristic
273C22×C4C22⋊Dic1711yescharacteristic
274C22×C4C22⋊Dic1711yescharacteristic
275C22×C4C22×Dic1711yescharacteristic
276C22×C4C2×C22⋊D1711yescharacteristic
277C2×C8—682no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—81no—
281C4⋊C4—684no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C22⋊C4—344no—
292C22⋊C4—682no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—42no—
298C42—684no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23#1058763511yescharacteristic
311C23#1058763711yescharacteristic
312C23C2×C22⋊Dic1711yescharacteristic
313Q8—82no—
314D4—82no—
315D4—82no—
316D4—82no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
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—684no—
343C2×C4—1362no—
344C2×C4—1362no—
345C2×C4#1058750511yescharacteristic
346C2×C4#1058750511yescharacteristic
347C2×C4#1058750812yes—
348C8—1362no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22C2×#1058750511yescharacteristic
366C22#1176281011yescharacteristic
367C22C2×#1058750811yescharacteristic
368C4—21no—
369C4—21no—
370C4—22no—
371C4—42no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—81no—
377C4—81no—
378C4—1364no—
379C2—21no—
380C2—21no—
381C2—41no—
382C2—41no—
383C2—161no—
384C2#1452728811yescharacteristic
385C2#1453043411yescharacteristic
386C2#1453044911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2610C2
socleC22×C17C2×#99
prime radicalC17×#392C22
cosocleC2×C22⋊C4×C17C23
hypercenter#2610D17
hypocenterC17#11758
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2610

Complements

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

NH ≅ G/Ntype
1. #14528517C2semidirect
2. #14527118C2semidirect
3. #14526805C2semidirect
12. C2×C17×#124C4semidirect
14. C17×#395C4semidirect
15. C17×#394C4semidirect
22. C17×#376C4semidirect
191. C2×#124Dic17semidirect
195. #395Dic17semidirect
196. #394Dic17semidirect
203. #376Dic17semidirect
251. C17#11758semidirect

Maximal subgroups

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

subgroupindex
#145285172
#145271182
#145268052
#145267812
C17×#26102
#1175817