Finite Group Databasebeta

← #17958125

Subgroups — #17958125

Order 4352.

726 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 98 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#14544010C212yesmaximal
2#14543711C212yesmaximal, complemented
3#14543710C212yesmaximal, complemented
4D17×#381C211yesmaximal, characteristic
5#11763777—22no—
6#11763653—22no—
7#11763652C2214yes—
8C23×C4×D17C411yescharacteristic
9C2×C22⋊C4×D17C411yescharacteristic
10C2×C22⋊C4×D17C2211yescharacteristic
11#11762672C2211yescharacteristic
12#11762671—22no—
13#11762670C411yescharacteristic
14#11762670C2211yescharacteristic
15#95×D17—22no—
16C17×#381C411yescharacteristic
17C24×D17C2×C411yescharacteristic
18C23×C4×C17C2×C411yescharacteristic
19C23×Dic17C2×C411yescharacteristic
20C22×C4×D17—21no—
21C22×C4×D17—21no—
22C22×C4×D17—22no—
23C22×C4×D17—21no—
24C22×C4×D17—41no—
25C22×C4×D17—41no—
26C22×C4×D17—42no—
27C22×C4×D17C814yescomplemented
28C22×C4×D17D412yes—
29C2×#63242—42no—
30C2×C22⋊C4×C17C2×C411yescharacteristic
31C2×C22⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊Dic17C2×C411yescharacteristic
33C2×C22⋊Dic17C2311yescharacteristic
34C2×#63227—21no—
35C2×#63227D412yes—
36C22⋊C4×D17—41no—
37C22⋊C4×D17—41no—
38C22⋊C4×D17—42no—
39C22⋊#3624—24no—
40#10587543—42no—
41C17×#95—22no—
42#10587502—22no—
43#10587502—22no—
44#10587502—22no—
45C24×C17C22×C411yescharacteristic
46C23×D17—21no—
47C23×D17—21no—
48C23×D17—22no—
49C23×D17—44no—
50C23×D17C22⋊C411yescharacteristic
51C23×D17C22⋊C411yescharacteristic
52C23×D17C2×C812yes—
53C23×D17M1612yes—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—41no—
58C22×C4×C17—41no—
59C22×C4×C17C22⋊C412yes—
60C22×C4×C17C2×C812yescomplemented
61C22×Dic17—21no—
62C22×Dic17—21no—
63C22×Dic17—21no—
64C22×Dic17—41no—
65C22×Dic17—41no—
66C22×Dic17C2×C812yescomplemented
67C22×Dic17C2×D412yes—
68C2×C4×D17—24no—
69C2×C4×D17—24no—
70C2×C4×D17—22no—
71C2×C4×D17—22no—
72C2×C4×D17—42no—
73C2×C4×D17—42no—
74C2×C4×D17—44no—
75C2×C4×D17—44no—
76C2×C4×D17—41no—
77C2×C4×D17—41no—
78C2×C4×D17—42no—
79C2×C4×D17—44no—
80C2×C4×D17—44no—
81C2×C4×D17—44no—
82C2×C4×D17—44no—
83C2×C4×D17—42no—
84C2×#3624—44no—
85#63242—84no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C22⋊Dic17—41no—
89C22⋊Dic17—41no—
90#63227—42no—
91#63227—42no—
92C2×#1317—171nomaximal, Sylow
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—42no—
96C23×C17C2×C22⋊C411yescharacteristic
97C23×C17C22×C811yescharacteristic
98C23×C17C2×M1611yescharacteristic
99C22×D17—24no—
100C22×D17—21no—
101C22×D17—21no—
102C22×D17—22no—
103C22×D17—22no—
104C22×D17—22no—
105C22×D17—22no—
106C22×D17—44no—
107C22×D17—48no—
108C22×D17—44no—
109C22×D17—44no—
110C22×D17—44no—
111C22×D17—42no—
112C22×D17C22⋊C812yes—
113C22×D17C22⋊C812yes—
114C22×D17#9911yescharacteristic
115C22×D17#9911yescharacteristic
116C22×D17#10012yes—
117C2×C4×C17—22no—
118C2×C4×C17—22no—
119C2×C4×C17—42no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17—41no—
126C2×Dic17—22no—
127C2×Dic17—22no—
128C2×Dic17—42no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—42no—
132C2×Dic17—42no—
133C2×Dic17—42no—
134C2×Dic17—41no—
135C4×D17—48no—
136C4×D17—44no—
137C4×D17—44no—
138C4×D17—44no—
139C4×D17—82no—
140C4×D17—84no—
141#3624—84no—
142C2×#381—171no—
143C2×#323—172no—
144#1317—172no—
145#1317—172no—
146C22×C17—22no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—42no—
152C22×C17—42no—
153C22×C17C2×C22⋊C811yescharacteristic
154C22×C17C2×#9911yescharacteristic
155C22×C17C2×#10011yescharacteristic
156C2×D17—22no—
157C2×D17—24no—
158C2×D17—24no—
159C2×D17—22no—
160C2×D17—44no—
161C2×D17—44no—
162C2×D17—42no—
163C2×D17—42no—
164C2×D17—44no—
165C2×D17—44no—
166C2×D17—42no—
167C2×D17#32312yes—
168C2×D17#32312yes—
169C2×D17#35214yes—
170C4×C17—42no—
171C4×C17—42no—
172C4×C17—81no—
173Dic17—42no—
174Dic17—42no—
175Dic17—81no—
176C24×C4—171no—
177C22×C22⋊C4—171no—
178C22×C22⋊C4—171no—
179C2×C22⋊C8—342no—
180#381—171no—
181#381—171no—
182#381—171no—
183#381—342no—
184#381#59511yescharacteristic
185C2×#95—342no—
186#323—174no—
187#323—342no—
188C2×C17—21no—
189C2×C17—21no—
190C2×C17—41no—
191C2×C17—41no—
192C2×C17C2×#32311yescharacteristic
193C2×C17C2×#35212yes—
194D17—22no—
195D17—42no—
196D17#131714yescomplemented
197C25—171no—
198C23×C4—171no—
199C23×C4—174no—
200C23×C4—172no—
201C23×C4—342no—
202C23×C4—341no—
203C23×C4—341no—
204C23×C4—341no—
205C23×C4—681no—
206C23×C4—681no—
207C23×C4—682no—
208C23×C4C2×#59511yescharacteristic
209C22×C8—682no—
210C2×C22⋊C4—171no—
211C2×C22⋊C4—172no—
212C2×C22⋊C4—171no—
213C2×C22⋊C4—341no—
214C2×C22⋊C4—681no—
215C2×C22⋊C4—681no—
216C2×C22⋊C4—682no—
217C2×C22⋊C4C2×#59511yescharacteristic
218C2×C22⋊C4C2×#59511yescharacteristic
219C22⋊C8—344no—
220C22⋊C8—682no—
221#95—22no—
222#95—342no—
223#95—342no—
224#95—342no—
225C17C2×#131711yescharacteristic, complemented, Sylow
226C24—171no—
227C24—172no—
228C24—172no—
229C24—171no—
230C24—341no—
231C24—341no—
232C24—342no—
233C24—684no—
234C24C22×#59511yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—172no—
241C22×C4—172no—
242C22×C4—342no—
243C22×C4—344no—
244C22×C4—344no—
245C22×C4—342no—
246C22×C4—341no—
247C22×C4—341no—
248C22×C4—341no—
249C22×C4—681no—
250C22×C4—681no—
251C22×C4—682no—
252C22×C4—684no—
253C22×C4—684no—
254C22×C4—682no—
255C22×C4—682no—
256C22×C4—684no—
257C22×C4—684no—
258C22×C4—684no—
259C22×C4—682no—
260C22×C4—684no—
261C22×C4—681no—
262C22×C4—681no—
263C22×C4#6324212yescomplemented
264C22×C4C22⋊#59512yes—
265C2×C8—684no—
266C2×C8—1364no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—682no—
270C22⋊C4—682no—
271C22⋊C4—681no—
272C22⋊C4—681no—
273C23—21no—
274C23—21no—
275C23—42no—
276C23—171no—
277C23—172no—
278C23—171no—
279C23—172no—
280C23—172no—
281C23—344no—
282C23—342no—
283C23—342no—
284C23—342no—
285C23—342no—
286C23—341no—
287C23—341no—
288C23—682no—
289C23—684no—
290C23—684no—
291C23—684no—
292C23—688no—
293C23—684no—
294C23C2×#6324211yescharacteristic
295C23D17⋊M1611yescharacteristic
296C23C2×C22⋊#59511yescharacteristic
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—342no—
307C2×C4—342no—
308C2×C4—684no—
309C2×C4—684no—
310C2×C4—684no—
311C2×C4—688no—
312C2×C4—682no—
313C2×C4—681no—
314C2×C4—681no—
315C2×C4—682no—
316C2×C4—682no—
317C2×C4—682no—
318C2×C4—681no—
319C2×C4—1362no—
320C2×C4—1364no—
321C8—1364no—
322C22—22no—
323C22—21no—
324C22—21no—
325C22—41no—
326C22—41no—
327C22—42no—
328C22—42no—
329C22—174no—
330C22—172no—
331C22—172no—
332C22—344no—
333C22—344no—
334C22—342no—
335C22—342no—
336C22—682no—
337C22—682no—
338C22—684no—
339C22—684no—
340C22—684no—
341C22—684no—
342C22—682no—
343C22#1176377711yescharacteristic
344C22D17⋊#9911yescharacteristic
345C22D17⋊#10011yescharacteristic
346C4—42no—
347C4—42no—
348C4—81no—
349C4—682no—
350C4—682no—
351C4—1361no—
352C2—21no—
353C2—21no—
354C2—41no—
355C2—41no—
356C2—174no—
357C2—342no—
358C2—682no—
359C2#1454401011yescharacteristic
360C2D17⋊#35212yes—
subgroupG/H
centerC22D17⋊#99
commutatorC23×C17C22×C8
FrattiniC24C22×#595
FittingC17×#381C4
socleC22×C17C2×#99
prime radicalC24×D17C2×C4
cosocleC2×C22⋊Dic17C23
hypercenter#381#595
hypocenterC17C2×#1317
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #14543711C2semidirect
3. #14543710C2semidirect
27. C22×C4×D17C8semidirect
60. C22×C4×C17C2×C8semidirect
66. C22×Dic17C2×C8semidirect
196. D17#1317semidirect
225. C17C2×#1317semidirect
263. C22×C4#63242semidirect

Maximal subgroups

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

subgroupindex
#145440102
#145437112
#145437102
D17×#3812
C2×#131717