Finite Group Databasebeta

← #16945207

Subgroups — #16945207

Order 4352.

432 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 87 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#14528861C211yesmaximal, characteristic, complemented
2#14527098C211yesmaximal, characteristic, complemented
3#14526805C212yesmaximal, complemented
4#14526781C212yesmaximal
5C17×#2610C211yesmaximal, characteristic
6#11762862—21no—
7#11762672—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—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95C2×C411yescharacteristic
50C17×#95C2×C411yescharacteristic
51#10587502—41no—
52#10587502—41no—
53C22⋊#3622—22no—
54C4⋊#3622—41no—
55C4×#3622—41no—
56C24×C17—21no—
57C24×C17C2×D411yescharacteristic
58C2×Q8×C17—41no—
59C2×Q8×C17—81no—
60C2×D4×C17—41no—
61C2×D4×C17—42no—
62C2×D4×C17—42no—
63C2×D4×C17—81no—
64C2×D4×C17—81no—
65C22×C4×C17—21no—
66C22×C4×C17—22no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—41no—
72C22×C4×C17C22⋊C411yescharacteristic
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22×C411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×Dic17—42no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—81no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—41no—
84C22⋊C4×C17—42no—
85C22⋊C4×C17—42no—
86C22⋊C4×C17—42no—
87C22⋊C4×C17—41no—
88C22⋊C4×C17—41no—
89C22⋊C4×C17—81no—
90C22⋊C4×C17—81no—
91C42×C17—21no—
92C42×C17—21no—
93C42×C17—21no—
94C42×C17—21no—
95C42×C17—42no—
96C22⋊Dic17—42no—
97C22⋊Dic17—42no—
98C2×#3622—42no—
99#11844—171nomaximal, Sylow
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17C2×C22⋊C411yescharacteristic
112C23×C17#13512yes—
113Q8×C17—82no—
114D4×C17—84no—
115D4×C17—82no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—22no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17#9911yescharacteristic
141C2×C4×C17#9911yescharacteristic
142C2×Dic17—82no—
143C2×Dic17—82no—
144C2×Dic17—82no—
145#3622—82no—
146#2610D1711yescharacteristic
147#1820—171no—
148#1458—171no—
149#1351—172no—
150#1343—172no—
151C22×C17—21no—
152C22×C17—21no—
153C22×C17—22no—
154C22×C17—42no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—81no—
158C22×C17—81no—
159C22×C17—81no—
160C22×C17—81no—
161C22×C17—81no—
162C22×C17—81no—
163C22×C17—81no—
164C22×C17—81no—
165C22×C17C2×#9911yescharacteristic
166C22×C17#41611yescharacteristic
167C22×C17#42111yescharacteristic
168C4×C17—21no—
169C4×C17—21no—
170C4×C17—42no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17—41no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17—81no—
177C4×C17—81no—
178Dic17—162no—
179C2×#124Dic1711yescharacteristic, complemented
180C2×#117C2×D1711yescharacteristic
181#431—341no—
182#395Dic1711yescharacteristic, complemented
183#394Dic1711yescharacteristic, complemented
184#393C2×D1711yescharacteristic
185#392C2×D1711yescharacteristic
186#390—21no—
187#388—21no—
188#388—21no—
189#386—21no—
190#381—341no—
191#376Dic1711yescharacteristic, complemented
192#342—172no—
193#342—341no—
194#324—341no—
195#323—172no—
196C2×C17—21no—
197C2×C17—21no—
198C2×C17—42no—
199C2×C17—161no—
200C2×C17#151611yescharacteristic
201C2×C17#212211yescharacteristic
202C2×C17#212611yescharacteristic
203C22×Q8C2×Dic1711yescharacteristic
204C22×D4—21no—
205C22×D4C2×Dic1711yescharacteristic
206C23×C4C22⋊D1711yescharacteristic
207#124—41no—
208#124—41no—
209#117—41no—
210#117—41no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C2×Dic1711yescharacteristic
213C2×C4⋊C4C22⋊D1711yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—342no—
221C2×C22⋊C4C22⋊D1711yescharacteristic
222C2×C22⋊C4C22×D1711yescharacteristic
223C2×C42—21no—
224C2×C42C2×Dic1711yescharacteristic
225C2×C42C22⋊D1711yescharacteristic
226C4⋊C8—681no—
227C22⋊C8—342no—
228C4×C8—681no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—681no—
233#95—681no—
234#95C2×Dic1711yescharacteristic
235#95C2×Dic1711yescharacteristic
236C17#1184411yescharacteristic, complemented, Sylow
237C24—21no—
238C24C2×C22⋊D1711yescharacteristic
239C2×Q8—41no—
240C2×Q8—81no—
241C2×D4—41no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—81no—
245C2×D4—81no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—682no—
254C22×C4C22⋊Dic1711yescharacteristic
255C22×C4C22⋊Dic1711yescharacteristic
256C22×C4C22⋊Dic1711yescharacteristic
257C22×C4C22⋊Dic1711yescharacteristic
258C22×C4C22×Dic1711yescharacteristic
259C22×C4C2×C22⋊D1711yescharacteristic
260C2×C8—682no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—81no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—81no—
272C22⋊C4—81no—
273C22⋊C4—682no—
274C22⋊C4—682no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—42no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23#1058763612yes—
292C23C2×C22⋊Dic1711yescharacteristic
293Q8—82no—
294D4—84no—
295D4—82no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—1362no—
321C2×C4—1362no—
322C2×C4—1362no—
323C2×C4#1058750511yescharacteristic
324C2×C4#1058750511yescharacteristic
325C8—1362no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22C2×#1058750511yescharacteristic
341C22#1176280511yescharacteristic
342C22#1176282611yescharacteristic
343C4—21no—
344C4—21no—
345C4—42no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4—2722no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—161no—
358C2#1452734011yescharacteristic
359C2#1453043411yescharacteristic
360C2#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#11844
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. #14528861C2semidirect
2. #14527098C2semidirect
3. #14526805C2semidirect
12. C2×C17×#124C4semidirect
14. C17×#395C4semidirect
15. C17×#394C4semidirect
22. C17×#376C4semidirect
179. C2×#124Dic17semidirect
182. #395Dic17semidirect
183. #394Dic17semidirect
191. #376Dic17semidirect
236. C17#11844semidirect

Maximal subgroups

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

subgroupindex
#145288612
#145270982
#145268052
#145267812
C17×#26102
#1184417