Finite Group Databasebeta

← #16944815

Subgroups — #16944815

Order 4352.

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

normal subgroups only

This group has 436 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
1#14528517C211yesmaximal, characteristic
2#14527118C211yesmaximal, characteristic
3#14526808C212yesmaximal
4#14526784C212yesmaximal
5C17×#2611C211yesmaximal, characteristic
6C2×#10587612—21no—
7#11762866—21no—
8#11762576—21no—
9#11762576C2212yes—
10#11762535—21no—
11#11762533C2212yes—
12C2×C17×#125C411yescharacteristic, complemented
13C2×C17×#117C2211yescharacteristic
14C17×#400C411yescharacteristic, complemented
15C17×#399C2211yescharacteristic
16C17×#398C411yescharacteristic, complemented
17C17×#397C2211yescharacteristic
18C17×#389—21no—
19C17×#388—21no—
20C17×#387—21no—
21C17×#385—21no—
22C17×#376C411yescharacteristic, complemented
23C23×C4×C17D411yescharacteristic
24C17×#125—41no—
25C17×#125—41no—
26C17×#117—41no—
27C17×#117—41no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17D411yescharacteristic
39C2×C22⋊C4×C17C2311yescharacteristic
40C2×C42×C17—21no—
41C2×C42×C17C2×C411yescharacteristic
42C2×C42×C17D411yescharacteristic
43C2×C22⋊Dic17—22no—
44#10587612—24no—
45#10587612—42no—
46C2×C4⋊Dic17—41no—
47C2×C4×Dic17—41no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95C2×C411yescharacteristic
54C17×#95C2×C411yescharacteristic
55C22⋊#3622—22no—
56C4⋊#3622—41no—
57#10587474—41no—
58C24×C17C2×D411yescharacteristic
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—22no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22×C411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C22×Dic17—42no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—41no—
79C4⋊C4×C17—42no—
80C4⋊C4×C17—42no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—42no—
84C22⋊C4×C17—41no—
85C22⋊C4×C17—41no—
86C22⋊C4×C17—42no—
87C22⋊C4×C17—41no—
88C22⋊C4×C17—41no—
89C42×C17—21no—
90C42×C17—21no—
91C42×C17—21no—
92C42×C17—21no—
93C42×C17—42no—
94C22⋊Dic17—24no—
95C22⋊Dic17—42no—
96C4⋊Dic17—44no—
97C4×Dic17—44no—
98C2×#3622—42no—
99#11760—171nomaximal, Sylow
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17C2×C22⋊C411yescharacteristic
105C23×C17AGL(1,ℤ/8)11yescharacteristic
106C23×C17#13711yescharacteristic
107C2×C4×C17—22no—
108C2×C4×C17—22no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—22no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
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#9911yescharacteristic
136C2×C4×C17#9911yescharacteristic
137C2×C4×C17#10412yes—
138C2×Dic17—44no—
139C2×Dic17—82no—
140C2×Dic17—82no—
141#3622—82no—
142#2611D1711yescharacteristic
143#1740—171no—
144#1464—171no—
145#1352—172no—
146#1344—172no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17—21no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—81no—
156C22×C17C2×#9911yescharacteristic
157C22×C17#41711yescharacteristic
158C22×C17C2×#10411yescharacteristic
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—22no—
162C4×C17—42no—
163C4×C17—41no—
164C4×C17—41no—
165C4×C17—41no—
166C4×C17—41no—
167C4×C17—81no—
168C4×C17—81no—
169C4×C17—81no—
170Dic17—84no—
171C2×#125Dic1711yescharacteristic, complemented
172C2×#117—341no—
173C2×#117C2×D1711yescharacteristic
174#432—341no—
175#400Dic1711yescharacteristic, complemented
176#399C2×D1711yescharacteristic
177#398Dic1711yescharacteristic, complemented
178#397C2×D1711yescharacteristic
179#389—21no—
180#388—21no—
181#387—21no—
182#385—21no—
183#376Dic1711yescharacteristic, complemented
184#342—172no—
185#342—341no—
186#324—341no—
187#323—172no—
188C2×C17—21no—
189C2×C17—21no—
190C2×C17—41no—
191C2×C17—41no—
192C2×C17#150411yescharacteristic
193C2×C17#212311yescharacteristic
194C2×C17#212611yescharacteristic
195C23×C4C22⋊D1711yescharacteristic
196#125—41no—
197#125—41no—
198#117—41no—
199#117—41no—
200#117—344no—
201#117—682no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—681no—
207C2×C4⋊C4C2×Dic1711yescharacteristic
208C2×C4⋊C4C2×Dic1711yescharacteristic
209C2×C4⋊C4C2×Dic1711yescharacteristic
210C2×C4⋊C4C22⋊D1711yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—342no—
214C2×C22⋊C4C22⋊D1711yescharacteristic
215C2×C22⋊C4C22×D1711yescharacteristic
216C2×C42—21no—
217C2×C42—681no—
218C2×C42C2×Dic1711yescharacteristic
219C2×C42C22⋊D1711yescharacteristic
220C4⋊C8—681no—
221C22⋊C8—342no—
222#97—681no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95C2×Dic1711yescharacteristic
229#95C2×Dic1711yescharacteristic
230C17#1176011yescharacteristic, complemented, Sylow
231C24C2×C22⋊D1711yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—682no—
242C22×C4C22⋊Dic1711yescharacteristic
243C22×C4C22⋊Dic1711yescharacteristic
244C22×C4C22⋊Dic1711yescharacteristic
245C22×C4C22⋊Dic1711yescharacteristic
246C22×C4C22×Dic1711yescharacteristic
247C22×C4C2×C22⋊D1711yescharacteristic
248C2×C8—682no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—684no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—42no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—344no—
265C22⋊C4—682no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—42no—
271C42—684no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23#1058763511yescharacteristic
277C23#1058763711yescharacteristic
278C23C2×C22⋊Dic1711yescharacteristic
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—684no—
308C2×C4—1362no—
309C2×C4—1362no—
310C2×C4#1058750511yescharacteristic
311C2×C4#1058750511yescharacteristic
312C2×C4#1058750812yes—
313C8—1362no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22C2×#1058750511yescharacteristic
324C22#1176281011yescharacteristic
325C22C2×#1058750811yescharacteristic
326C4—21no—
327C4—21no—
328C4—22no—
329C4—42no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—1364no—
338C2—21no—
339C2—21no—
340C2—41no—
341C2—41no—
342C2#1452728811yescharacteristic
343C2#1453043811yescharacteristic
344C2#1453044911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2611C2
socleC22×C17C2×#99
prime radicalC24×C17C2×D4
cosocleC2×C22⋊C4×C17C23
hypercenter#2611D17
hypocenterC17#11760
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

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

NH ≅ G/Ntype
12. C2×C17×#125C4semidirect
14. C17×#400C4semidirect
16. C17×#398C4semidirect
22. C17×#376C4semidirect
171. C2×#125Dic17semidirect
175. #400Dic17semidirect
177. #398Dic17semidirect
183. #376Dic17semidirect
230. C17#11760semidirect

Maximal subgroups

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

subgroupindex
#145285172
#145271182
#145268082
#145267842
C17×#26112
#1176017