Finite Group Databasebeta

← #16944885

Subgroups — #16944885

Order 4352.

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

normal subgroups only

This group has 544 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
1C4×#10587505C211yesmaximal, characteristic, complemented
2#14527122C211yesmaximal, characteristic, complemented
3#14526784C212yesmaximal, complemented
4#14526781C212yesmaximal, complemented
5C17×#2535C211yesmaximal, characteristic
6C2×#10587505—21no—
7#11762773—21no—
8C4×C22⋊Dic17—21no—
9#11762576C2212yes—
10#11762535—21no—
11#11762533C2212yes—
12C2×C17×#121—21no—
13C2×C4×D4×C17C411yescharacteristic, complemented
14C2×C4×D4×C17C2211yescharacteristic
15C17×#397C411yescharacteristic, complemented
16C17×#397C2211yescharacteristic
17C17×#392C411yescharacteristic, complemented
18C17×#392C2211yescharacteristic
19C17×#386—21no—
20C17×#386—21no—
21C17×#382C411yescharacteristic, complemented
22C17×#380—21no—
23C22×D4×C17—21no—
24C22×D4×C17—21no—
25C22×D4×C17C2×C411yescharacteristic
26C22×D4×C17D411yescharacteristic
27C23×C4×C17—21no—
28C23×C4×C17D411yescharacteristic
29C23×C4×C17D411yescharacteristic
30C17×#121—44no—
31C4×D4×C17—41no—
32C4×D4×C17—41no—
33C4×D4×C17—41no—
34C4×D4×C17—41no—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17C2×C411yescharacteristic
37C2×C4⋊C4×C17D411yescharacteristic
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17—21no—
42C2×C22⋊C4×C17—21no—
43C2×C22⋊C4×C17C2×C411yescharacteristic
44C2×C22⋊C4×C17C2311yescharacteristic
45C2×C42×C17C2×C411yescharacteristic
46C2×C22⋊Dic17—22no—
47C2×C4×Dic17—41no—
48C2×#63223—41no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95—21no—
52C17×#95C2×C411yescharacteristic
53C17×#95C2×C411yescharacteristic
54#10587505—24no—
55#10587502—41no—
56C22⋊#3622—22no—
57C22⋊#3622—41no—
58C24×C17—21no—
59C24×C17C22⋊C411yescharacteristic
60C24×C17C2×D411yescharacteristic
61C2×D4×C17—22no—
62C2×D4×C17—22no—
63C2×D4×C17—41no—
64C2×D4×C17—41no—
65C2×D4×C17—42no—
66C2×D4×C17—42no—
67C2×D4×C17—42no—
68C2×D4×C17—42no—
69C2×D4×C17—42no—
70C2×D4×C17—81no—
71C2×D4×C17—81no—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17—21no—
77C22×C4×C17—21no—
78C22×C4×C17—21no—
79C22×C4×C17—42no—
80C22×C4×C17—42no—
81C22×C4×C17—42no—
82C22×C4×C17—81no—
83C22×C4×C17—81no—
84C22×C4×C17C22⋊C411yescharacteristic
85C22×C4×C17C22⋊C411yescharacteristic
86C22×C4×C17C22⋊C411yescharacteristic
87C22×C4×C17C22×C411yescharacteristic
88C22×C4×C17C2×D411yescharacteristic
89C22×Dic17—42no—
90C4⋊C4×C17—41no—
91C4⋊C4×C17—41no—
92C4⋊C4×C17—42no—
93C22⋊C4×C17—41no—
94C22⋊C4×C17—41no—
95C22⋊C4×C17—42no—
96C22⋊C4×C17—42no—
97C22⋊C4×C17—41no—
98C22⋊C4×C17—41no—
99C22⋊C4×C17—81no—
100C22⋊C4×C17—81no—
101C42×C17—21no—
102C42×C17—21no—
103C22⋊Dic17—24no—
104C22⋊Dic17—42no—
105C4×Dic17—82no—
106#63223—82no—
107C2×#3622—42no—
108#11774—171nomaximal, Sylow
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—22no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—41no—
116C23×C17—41no—
117C23×C17—41no—
118C23×C17—41no—
119C23×C17—41no—
120C23×C17—41no—
121C23×C17—41no—
122C23×C17—41no—
123C23×C17—41no—
124C23×C17—41no—
125C23×C17C2×C22⋊C411yescharacteristic
126C23×C17AGL(1,ℤ/8)12yes—
127D4×C17—44no—
128D4×C17—42no—
129D4×C17—42no—
130D4×C17—84no—
131D4×C17—84no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—41no—
138C2×C4×C17—42no—
139C2×C4×C17—42no—
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—41no—
152C2×C4×C17—81no—
153C2×C4×C17—81no—
154C2×C4×C17—84no—
155C2×C4×C17#9911yescharacteristic
156C2×C4×C17#9911yescharacteristic
157C2×C4×C17#10412yes—
158C2×Dic17—44no—
159C2×Dic17—82no—
160C2×Dic17—82no—
161#3622—82no—
162#2535D1711yescharacteristic
163C4×#99—171no—
164#1465—171no—
165#1344—172no—
166#1343—172no—
167C22×C17—21no—
168C22×C17—21no—
169C22×C17—22no—
170C22×C17—42no—
171C22×C17—42no—
172C22×C17—42no—
173C22×C17—42no—
174C22×C17—41no—
175C22×C17—41no—
176C22×C17—42no—
177C22×C17—41no—
178C22×C17—41no—
179C22×C17—41no—
180C22×C17—81no—
181C22×C17—81no—
182C22×C17—81no—
183C22×C17—81no—
184C22×C17—81no—
185C22×C17—81no—
186C22×C17C2×#9911yescharacteristic
187C22×C17#41811yescharacteristic
188C22×C17C2×#10411yescharacteristic
189C4×C17—21no—
190C4×C17—21no—
191C4×C17—22no—
192C4×C17—41no—
193C4×C17—41no—
194C4×C17—41no—
195C4×C17—81no—
196C4×C17—81no—
197C4×C17—81no—
198Dic17—84no—
199C2×#121—21no—
200C2×C4×D4Dic1711yescharacteristic, complemented
201C2×C4×D4C2×D1711yescharacteristic
202C2×#99—341no—
203#407—341no—
204#397Dic1711yescharacteristic, complemented
205#397C2×D1711yescharacteristic
206#392Dic1711yescharacteristic, complemented
207#392C2×D1711yescharacteristic
208#386—21no—
209#386—21no—
210#382Dic1711yescharacteristic, complemented
211#380—21no—
212C4×C22⋊C4—341no—
213#342—172no—
214#324—341no—
215#323—172no—
216C2×C17—21no—
217C2×C17—21no—
218C2×C17—42no—
219C2×C17—81no—
220C2×C17—82no—
221C2×C17#150711yescharacteristic
222C2×C17#212011yescharacteristic
223C2×C17#212711yescharacteristic
224C22×D4—21no—
225C22×D4—21no—
226C22×D4C2×Dic1711yescharacteristic
227C22×D4C22⋊D1711yescharacteristic
228C23×C4—21no—
229C23×C4C22⋊D1711yescharacteristic
230C23×C4C22⋊D1711yescharacteristic
231C2×M16—681no—
232#121—44no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4C2×Dic1711yescharacteristic
239C2×C4⋊C4C22⋊D1711yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—342no—
246C2×C22⋊C4C2×Dic1711yescharacteristic
247C2×C22⋊C4C22×D1711yescharacteristic
248C2×C42—681no—
249C2×C42C2×Dic1711yescharacteristic
250#99—344no—
251C22⋊C8—342no—
252C22⋊C8—681no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—681no—
257#95C2×Dic1711yescharacteristic
258#95C2×Dic1711yescharacteristic
259C17#1177411yescharacteristic, complemented, Sylow
260C24—21no—
261C24C22⋊Dic1711yescharacteristic
262C24C2×C22⋊D1711yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—81no—
273C2×D4—81no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—682no—
287C22×C4C22⋊Dic1711yescharacteristic
288C22×C4C22⋊Dic1711yescharacteristic
289C22×C4C22⋊Dic1711yescharacteristic
290C22×C4C22×Dic1711yescharacteristic
291C22×C4C2×C22⋊D1711yescharacteristic
292M16—1362no—
293C2×C8—682no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—81no—
304C22⋊C4—81no—
305C22⋊C4—344no—
306C22⋊C4—682no—
307C42—21no—
308C42—21no—
309C42—1362no—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23#1058761812yes—
327C23C2×C22⋊Dic1711yescharacteristic
328D4—44no—
329D4—42no—
330D4—42no—
331D4—84no—
332D4—84no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
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—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—84no—
356C2×C4—684no—
357C2×C4—1362no—
358C2×C4—1362no—
359C2×C4#1058750511yescharacteristic
360C2×C4#1058750511yescharacteristic
361C2×C4#1058750812yes—
362C8—1362no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—42no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22C2×#1058750511yescharacteristic
383C22#1176281411yescharacteristic
384C22C2×#1058750811yescharacteristic
385C4—22no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—1364no—
395C2—21no—
396C2—21no—
397C2—42no—
398C2—81no—
399C2—82no—
400C2#1452730011yescharacteristic
401C2#1453042511yescharacteristic
402C2#1453045311yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
Fitting, prime radicalC17×#2535C2
socleC22×C17C2×#99
cosocleC2×C22⋊C4×C17C23
hypercenter#2535D17
hypocenterC17#11774
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. C4×#10587505C2semidirect
2. #14527122C2semidirect
3. #14526784C2semidirect
4. #14526781C2semidirect
13. C2×C4×D4×C17C4semidirect
15. C17×#397C4semidirect
17. C17×#392C4semidirect
21. C17×#382C4semidirect
200. C2×C4×D4Dic17semidirect
204. #397Dic17semidirect
206. #392Dic17semidirect
210. #382Dic17semidirect
259. C17#11774semidirect

Maximal subgroups

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

subgroupindex
C4×#105875052
#145271222
#145267842
#145267812
C17×#25352
#1177417