Finite Group Databasebeta

← #17951282

Subgroups — #17951282

Order 4352.

874 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 148 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#14543666C214yesmaximal, complemented
3D17×#407C211yesmaximal, characteristic
4C2×D17⋊M16—22no—
5#11763781C414yes—
6#11763780—22no—
7#11763780C414yes—
8#11763779—22no—
9#11763777—22no—
10#11763777C2214yes—
11C23×C4×D17C2211yescharacteristic
12C2×M16×D17C412yes—
13#11762773C411yescharacteristic
14#11762772C2212yes—
15#11762771—22no—
16C22⋊C8×D17—22no—
17C17×#407C411yescharacteristic
18C24×D17Q811yescharacteristic
19C23×C4×C17C2×C411yescharacteristic
20C23×Dic17D411yescharacteristic
21C22×C4×D17—44no—
22C22×C4×D17C2×C411yescharacteristic
23C22×C4×D17C2×C412yes—
24C22×C4×D17C2×C412yes—
25C22×C4×D17D412yes—
26C22×C4×D17D411yescharacteristic
27C22×C4×D17D412yes—
28C22×C4×D17Q811yescharacteristic
29C22×C4×D17C2311yescharacteristic
30C2×#63247—24no—
31D17⋊M16—48no—
32C2×#63243—24no—
33C2×#63242—24no—
34C2×M16×C17C2×C412yes—
35C2×#63223C2×C412yes—
36M16×D17—44no—
37C2×#63218—22no—
38C2×C8×D17—22no—
39C22⋊#3624—24no—
40C22⋊#3624—24no—
41#10587543—24no—
42#10587543C2×C418yes—
43C17×C22⋊C8—22no—
44C22⋊#3622—22no—
45#10587491—24no—
46C24×C17C4⋊C411yescharacteristic
47C23×D17—42no—
48C23×D17—44no—
49C23×D17C4⋊C412yes—
50C23×D17C4⋊C411yescharacteristic
51C23×D17C4⋊C412yes—
52C23×D17C2×Q811yescharacteristic
53C22×C4×C17—42no—
54C22×C4×C17C4212yes—
55C22×C4×C17C22⋊C412yes—
56C22×C4×C17C4⋊C411yescharacteristic
57C22×C4×C17C22×C411yescharacteristic
58C22×Dic17—42no—
59C22×Dic17C22⋊C412yes—
60C22×Dic17C22⋊C412yes—
61C22×Dic17C4⋊C411yescharacteristic
62C22×Dic17C2×D411yescharacteristic
63C2×C4×D17—28no—
64C2×C4×D17—24no—
65C2×C4×D17—44no—
66C2×C4×D17—42no—
67C2×C4×D17—44no—
68C2×C4×D17—42no—
69C2×C4×D17—44no—
70C2×C4×D17—42no—
71C2×C4×D17C4211yescharacteristic
72C2×C4×D17C4211yescharacteristic
73C2×C4×D17C22⋊C411yescharacteristic
74C2×C4×D17C22⋊C412yes—
75C2×C4×D17C22⋊C412yes—
76C2×C4×D17C22⋊C411yescharacteristic
77C2×C4×D17C4⋊C412yes—
78C2×C4×D17C4⋊C412yes—
79C2×C4×D17C22×C412yes—
80C2×C4×D17C2×D412yes—
81#63247—48no—
82C2×#3624—28no—
83#63243—48no—
84#63242—48no—
85M16×C17—42no—
86C2×C8×C17—22no—
87#63223—42no—
88C2×#3622—22no—
89#63218—44no—
90C8×D17—44no—
91#8055—171nomaximal, Sylow
92C23×C17—41no—
93C23×C17—42no—
94C23×C17#9512yes—
95C23×C17C2×C4⋊C411yescharacteristic
96C22×D17—24no—
97C22×D17—22no—
98C22×D17—24no—
99C22×D17—24no—
100C22×D17—24no—
101C22×D17—42no—
102C22×D17—41no—
103C22×D17—42no—
104C22×D17—42no—
105C22×D17—42no—
106C22×D17—416no—
107C22×D17#9511yescharacteristic
108C22×D17#9511yescharacteristic
109C22×D17C2×C4⋊C412yes—
110C2×C4×C17—22no—
111C2×C4×C17—24no—
112C2×C4×C17—42no—
113C2×C4×C17—41no—
114C2×C4×C17#9512yes—
115C2×C4×C17C2×C4211yescharacteristic
116C2×C4×C17C2×C22⋊C411yescharacteristic
117C2×Dic17—24no—
118C2×Dic17—22no—
119C2×Dic17—42no—
120C2×Dic17—41no—
121C2×Dic17#9511yescharacteristic
122C2×Dic17#9511yescharacteristic
123C2×Dic17C2×C22⋊C412yes—
124C4×D17—24no—
125C4×D17—22no—
126C4×D17—24no—
127C4×D17—22no—
128C4×D17—22no—
129C4×D17—24no—
130C4×D17—42no—
131C4×D17—42no—
132C4×D17—42no—
133C4×D17#9914yes—
134#3624—48no—
135C8×C17—42no—
136#3622—42no—
137C2×#407—171no—
138C2×#407—172no—
139#1298—174no—
140C22×C17—22no—
141C22×C17—22no—
142C22×C17—22no—
143C22×C17—22no—
144C22×C17—21no—
145C22×C17—44no—
146C22×C17C2×#9511yescharacteristic
147C2×D17—22no—
148C2×D17—22no—
149C2×D17—22no—
150C2×D17—22no—
151C2×D17—21no—
152C2×D17—48no—
153C2×D17—44no—
154C2×D17—44no—
155C2×D17—48no—
156C2×D17#34212yes—
157C2×D17#43014yes—
158C4×C17—22no—
159C4×C17—21no—
160C4×C17C2×#9912yes—
161Dic17—22no—
162Dic17—21no—
163Dic17#34212yes—
164C24×C4—171no—
165C22×M16—172no—
166C22×M16—342no—
167#407—174no—
168#407—174no—
169#407—171no—
170#407—172no—
171#407—342no—
172#407—342no—
173#407—342no—
174#407#59511yescharacteristic
175C2×C22⋊C8—174no—
176C2×C22⋊C8—342no—
177C2×C22⋊C8—342no—
178C2×C17—42no—
179C2×C17—41no—
180C2×C17#173812yes—
181C2×C17C2×#34211yescharacteristic
182D17—22no—
183D17—24no—
184D17—41no—
185C25—171no—
186C23×C4—171no—
187C23×C4—172no—
188C23×C4—171no—
189C23×C4—172no—
190C23×C4—172no—
191C23×C4—171no—
192C23×C4—171no—
193C23×C4—172no—
194C23×C4—171no—
195C23×C4—684no—
196C23×C4C2×#59511yescharacteristic
197C2×M16—172no—
198C2×M16—342no—
199C2×M16—344no—
200C2×M16—344no—
201C2×M16—688no—
202C2×M16—684no—
203C2×M16C2×#59512yes—
204C22×C8—342no—
205C22×C8—344no—
206C22⋊C8—22no—
207C22⋊C8—178no—
208C22⋊C8—344no—
209C22⋊C8—344no—
210C22⋊C8—344no—
211C22⋊C8—344no—
212C22⋊C8—342no—
213C17#805511yescharacteristic, complemented, Sylow
214C24—171no—
215C24—172no—
216C24—171no—
217C24—172no—
218C24—682no—
219C24—684no—
220C24C4⋊#59511yescharacteristic
221C22×C4—42no—
222C22×C4—171no—
223C22×C4—171no—
224C22×C4—172no—
225C22×C4—171no—
226C22×C4—172no—
227C22×C4—171no—
228C22×C4—171no—
229C22×C4—171no—
230C22×C4—172no—
231C22×C4—172no—
232C22×C4—172no—
233C22×C4—172no—
234C22×C4—172no—
235C22×C4—172no—
236C22×C4—344no—
237C22×C4—348no—
238C22×C4—682no—
239C22×C4—684no—
240C22×C4—684no—
241C22×C4—682no—
242C22×C4—684no—
243C22×C4—682no—
244C22×C4—682no—
245C22×C4C4×#59512yes—
246C22×C4C4⋊#59511yescharacteristic
247C22×C4C22⋊#59512yes—
248C22×C4C22×#59511yescharacteristic
249M16—42no—
250M16—684no—
251M16—688no—
252M16—688no—
253M16—682no—
254C2×C8—22no—
255C2×C8—348no—
256C2×C8—342no—
257C2×C8—684no—
258C2×C8—688no—
259C23—41no—
260C23—42no—
261C23—171no—
262C23—172no—
263C23—171no—
264C23—342no—
265C23—344no—
266C23—344no—
267C23—344no—
268C23—344no—
269C23—681no—
270C23—682no—
271C23—682no—
272C23—682no—
273C23—682no—
274C23—6816no—
275C23(C2×#595)⋊C412yes—
276C23C2×C4⋊#59511yescharacteristic
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—171no—
282C2×C4—174no—
283C2×C4—172no—
284C2×C4—171no—
285C2×C4—342no—
286C2×C4—344no—
287C2×C4—344no—
288C2×C4—342no—
289C2×C4—344no—
290C2×C4—342no—
291C2×C4—344no—
292C2×C4—342no—
293C2×C4—682no—
294C2×C4—682no—
295C2×C4—682no—
296C2×C4—681no—
297C2×C4—682no—
298C2×C4(C2×#595)⋊C412yes—
299C2×C4C2×C4×#59511yescharacteristic
300C2×C4C2×C22⋊#59511yescharacteristic
301C8—42no—
302C8—688no—
303C8—682no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—21no—
308C22—22no—
309C22—44no—
310C22—174no—
311C22—172no—
312C22—342no—
313C22—342no—
314C22—342no—
315C22—342no—
316C22—341no—
317C22—688no—
318C22—684no—
319C22—684no—
320C22—688no—
321C22C2×(C2×#595)⋊C411yescharacteristic
322C4—21no—
323C4—22no—
324C4—172no—
325C4—342no—
326C4—341no—
327C4D17⋊#9912yes—
328C2—41no—
329C2—42no—
330C2—344no—
331C2—342no—
332C2—681no—
333C2#1454447712yes—
334C2#1454448111yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#595
commutatorC2×C4×C17C2×C42
FrattiniC22×C4C22×#595
FittingC17×#407C4
socleC22×C17C2×#95
prime radicalC24×D17Q8
cosocleC22×C4×D17C23
hypercenter#407#595
hypocenterC17#8055
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #14543666C2semidirect
213. C17#8055semidirect

Maximal subgroups

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

subgroupindex
D17⋊#4072
#145436662
D17×#4072
#805517