Finite Group Databasebeta

← #17951605

Subgroups — #17951605

Order 4352.

914 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 156 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
1D17⋊#407C212yesmaximal
2#14543687C214yesmaximal, complemented
3#14531163C211yesmaximal, characteristic
4C2×D17⋊M16—22no—
5#11763781—22no—
6#11763780—22no—
7#11763780C414yescomplemented
8#11763779C2214yes—
9#11763777—22no—
10#11763777C414yescomplemented
11C23×C4×D17C2211yescharacteristic
12C2×#10587612C411yescharacteristic
13#11762670C2212yes—
14#11762666—22no—
15#11762664C412yes—
16C4×#63227—22no—
17C2×C17×#117C411yescharacteristic
18C24×D17C2×C411yescharacteristic
19C23×C4×C17C2×C411yescharacteristic
20C23×Dic17C2311yescharacteristic
21C22×C4×D17—42no—
22C22×C4×D17—42no—
23C22×C4×D17C2×C412yes—
24C22×C4×D17C2×C412yes—
25C22×C4×D17D412yes—
26C22×C4×D17D411yescharacteristic
27C22×C4×D17D411yescharacteristic
28C22×C4×D17D412yes—
29C22×C4×D17Q811yescharacteristic
30C22×C4×D17Q811yescharacteristic
31C2×#63247—24no—
32D17⋊M16—44no—
33D17⋊M16—44no—
34C2×#63243—24no—
35C2×#63242—24no—
36C17×#117—42no—
37C2×C4⋊C4×C17—21no—
38C2×C22⋊C4×C17C2×C412yes—
39C2×C42×C17—21no—
40C2×C22⋊Dic17C2×C412yes—
41C2×#63227—24no—
42#10587612—42no—
43C2×C4⋊Dic17—21no—
44C2×C4×Dic17—21no—
45C22⋊#3624—24no—
46C22⋊#3624C2×C418yes—
47#10587543—24no—
48#10587543—24no—
49#10587502—24no—
50C24×C17C22×C411yescharacteristic
51C23×D17—44no—
52C23×D17—41no—
53C23×D17—41no—
54C23×D17C4212yes—
55C23×D17C22⋊C412yes—
56C23×D17C4⋊C411yescharacteristic
57C23×D17C4⋊C411yescharacteristic
58C22×C4×C17—22no—
59C22×C4×C17—22no—
60C22×C4×C17—22no—
61C22×C4×C17C4211yescharacteristic
62C22×C4×C17C4211yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C4⋊C411yescharacteristic
66C22×C4×C17C4⋊C411yescharacteristic
67C22×Dic17—22no—
68C22×Dic17—41no—
69C22×Dic17—41no—
70C22×Dic17C22×C412yes—
71C22×Dic17C2×D412yes—
72C22×Dic17C2×D411yescharacteristic
73C22×Dic17C2×Q811yescharacteristic
74C2×C4×D17—24no—
75C2×C4×D17—24no—
76C2×C4×D17—44no—
77C2×C4×D17—44no—
78C2×C4×D17—42no—
79C2×C4×D17—42no—
80C2×C4×D17—42no—
81C2×C4×D17—42no—
82C2×C4×D17—42no—
83C2×C4×D17—42no—
84C2×C4×D17C22⋊C412yes—
85C2×C4×D17C22⋊C412yes—
86C2×C4×D17C22⋊C412yes—
87C2×C4×D17C22⋊C412yes—
88C2×C4×D17C4⋊C412yes—
89C2×C4×D17C4⋊C412yes—
90C2×C4×D17C4⋊C412yes—
91C2×C4×D17C4⋊C412yes—
92#63247—44no—
93#63247—44no—
94C2×#3624—28no—
95#63243—48no—
96#63242—48no—
97C4⋊C4×C17—42no—
98C22⋊C4×C17—42no—
99C42×C17—42no—
100C22⋊Dic17—42no—
101#63227—48no—
102C4⋊Dic17—42no—
103C4×Dic17—42no—
104#8112—171nomaximal, Sylow
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—42no—
108C23×C17C2×C4211yescharacteristic
109C23×C17C2×C22⋊C411yescharacteristic
110C23×C17C2×C4⋊C411yescharacteristic
111C22×D17—22no—
112C22×D17—22no—
113C22×D17—24no—
114C22×D17—24no—
115C22×D17—24no—
116C22×D17—42no—
117C22×D17—41no—
118C22×D17—41no—
119C22×D17—42no—
120C22×D17—42no—
121C22×D17—42no—
122C22×D17—48no—
123C22×D17—48no—
124C22×D17#9512yes—
125C22×D17#9512yes—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—22no—
129C2×C4×C17—22no—
130C2×C4×C17—22no—
131C2×C4×C17—22no—
132C2×C4×C17—44no—
133C2×C4×C17—42no—
134C2×C4×C17#9511yescharacteristic
135C2×C4×C17#9511yescharacteristic
136C2×C4×C17#9511yescharacteristic
137C2×C4×C17#9511yescharacteristic
138C2×C4×C17#9912yes—
139C2×C4×C17#9912yes—
140C2×Dic17—22no—
141C2×Dic17—22no—
142C2×Dic17—42no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—44no—
146C2×Dic17—42no—
147C2×Dic17C2×C22⋊C412yes—
148C2×Dic17C2×C4⋊C412yes—
149C4×D17—24no—
150C4×D17—24no—
151C4×D17—24no—
152C4×D17—24no—
153C4×D17—42no—
154C4×D17—42no—
155C4×D17—42no—
156C4×D17—42no—
157#3624—48no—
158C2×#407—172no—
159#2280—171no—
160#1306—174no—
161C22×C17—22no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—22no—
165C22×C17—22no—
166C22×C17—42no—
167C22×C17—42no—
168C22×C17C2×#9511yescharacteristic
169C22×C17C2×#9911yescharacteristic
170C22×C17C2×#9911yescharacteristic
171C2×D17—22no—
172C2×D17—22no—
173C2×D17—22no—
174C2×D17—22no—
175C2×D17—22no—
176C2×D17—44no—
177C2×D17—44no—
178C2×D17—44no—
179C2×D17—48no—
180C2×D17—44no—
181C2×D17#34414yes—
182C4×C17—21no—
183C4×C17—21no—
184C4×C17—82no—
185C4×C17#34212yes—
186C4×C17#34212yes—
187Dic17—22no—
188Dic17—22no—
189Dic17—82no—
190C24×C4—171no—
191C22×M16—342no—
192C2×#117—171no—
193C2×#117#59511yescharacteristic
194#407—174no—
195#407—174no—
196#407—342no—
197#407—342no—
198C2×C22⋊C8—174no—
199C2×C22⋊C8—342no—
200#381—172no—
201#380—342no—
202#379—172no—
203C4×C22⋊C4—342no—
204C2×C17—21no—
205C2×C17—21no—
206C2×C17—41no—
207C2×C17—41no—
208C2×C17C2×#34211yescharacteristic
209C2×C17C2×#34412yes—
210D17—24no—
211D17—42no—
212C25—171no—
213C23×C4—171no—
214C23×C4—172no—
215C23×C4—172no—
216C23×C4—172no—
217C23×C4—171no—
218C23×C4—171no—
219C23×C4—171no—
220C23×C4—171no—
221C23×C4—172no—
222C23×C4—682no—
223C23×C4—682no—
224C23×C4C2×#59511yescharacteristic
225C2×M16—344no—
226C2×M16—344no—
227C2×M16—684no—
228C2×M16—684no—
229C22×C8—344no—
230#117—42no—
231#117—682no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—341no—
234C2×C22⋊C4—172no—
235C2×C22⋊C4—344no—
236C2×C22⋊C4C2×#59512yes—
237C2×C42—21no—
238C2×C42—341no—
239C22⋊C8—178no—
240C22⋊C8—344no—
241C22⋊C8—344no—
242C22⋊C8—344no—
243#95—344no—
244C17#811211yescharacteristic, complemented, Sylow
245C24—171no—
246C24—172no—
247C24—172no—
248C24—171no—
249C24—681no—
250C24—681no—
251C24—684no—
252C24C22×#59511yescharacteristic
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—172no—
257C22×C4—172no—
258C22×C4—171no—
259C22×C4—171no—
260C22×C4—172no—
261C22×C4—172no—
262C22×C4—172no—
263C22×C4—172no—
264C22×C4—172no—
265C22×C4—172no—
266C22×C4—172no—
267C22×C4—172no—
268C22×C4—344no—
269C22×C4—344no—
270C22×C4—342no—
271C22×C4—684no—
272C22×C4—684no—
273C22×C4—682no—
274C22×C4—682no—
275C22×C4—682no—
276C22×C4—682no—
277C22×C4—682no—
278C22×C4—682no—
279C22×C4—681no—
280C22×C4—681no—
281C22×C4C4×#59511yescharacteristic
282C22×C4C4×#59511yescharacteristic
283C22×C4C4⋊#59511yescharacteristic
284C22×C4C4⋊#59511yescharacteristic
285C22×C4C22⋊#59511yescharacteristic
286C22×C4C22⋊#59511yescharacteristic
287M16—688no—
288M16—684no—
289M16—684no—
290C2×C8—348no—
291C2×C8—688no—
292C4⋊C4—42no—
293C4⋊C4—682no—
294C22⋊C4—42no—
295C22⋊C4—688no—
296C22⋊C4—682no—
297C42—42no—
298C42—682no—
299C23—21no—
300C23—21no—
301C23—42no—
302C23—172no—
303C23—172no—
304C23—344no—
305C23—344no—
306C23—344no—
307C23—342no—
308C23—342no—
309C23—682no—
310C23—681no—
311C23—681no—
312C23—682no—
313C23—682no—
314C23—682no—
315C23—688no—
316C23—688no—
317C23C2×C4×#59511yescharacteristic
318C23C2×C4⋊#59511yescharacteristic
319C23C2×C22⋊#59511yescharacteristic
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—172no—
329C2×C4—172no—
330C2×C4—342no—
331C2×C4—342no—
332C2×C4—344no—
333C2×C4—344no—
334C2×C4—344no—
335C2×C4—344no—
336C2×C4—682no—
337C2×C4—682no—
338C2×C4—682no—
339C2×C4—682no—
340C2×C4—681no—
341C2×C4—682no—
342C2×C4—681no—
343C2×C4—684no—
344C2×C4—682no—
345C2×C4#1058753912yes—
346C2×C4(C2×#595)⋊C411yescharacteristic
347C2×C4(C2×#595)⋊C411yescharacteristic
348C2×C4(C2×#595)⋊C411yescharacteristic
349C2×C4(C2×#595)⋊C411yescharacteristic
350C2×C4#1058755112yes—
351C8—688no—
352C22—22no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—22no—
357C22—42no—
358C22—42no—
359C22—174no—
360C22—342no—
361C22—342no—
362C22—342no—
363C22—342no—
364C22—342no—
365C22—684no—
366C22—684no—
367C22—684no—
368C22—688no—
369C22—684no—
370C22C2×(C2×#595)⋊C411yescharacteristic
371C22C2×#1058755111yescharacteristic
372C22C2×#1058753911yescharacteristic
373C4—21no—
374C4—21no—
375C4—82no—
376C4—342no—
377C4—342no—
378C4—1362no—
379C4#1176368712yes—
380C4#1176368712yes—
381C2—21no—
382C2—21no—
383C2—41no—
384C2—41no—
385C2—344no—
386C2—682no—
387C2C2×#1176368711yescharacteristic
388C2#1454449212yes—
subgroupG/H
centerC2×C4(C2×#595)⋊C4
commutatorC23×C17C2×C42
FrattiniC24C22×#595
FittingC2×C17×#117C4
socleC22×C17C2×#95
prime radicalC24×D17C2×C4
cosocleC23×Dic17C23
hypercenterC2×#117#595
hypocenterC17#8112
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #14543687C2semidirect
7. #11763780C4semidirect
10. #11763777C4semidirect
244. C17#8112semidirect

Maximal subgroups

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

subgroupindex
D17⋊#4072
#145436872
#145311632
#811217