Finite Group Databasebeta

← #16945452

Subgroups — #16945452

Order 4352.

452 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 107 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#14528772C211yesmaximal, characteristic
2#14527105C211yesmaximal, characteristic
3#14526808C214yesmaximal, complemented
4C17×#2487C211yesmaximal, characteristic
5#11762862—21no—
6C2×#10587505—21no—
7#11762669—21no—
8#11762576C2212yes—
9#11762533—21no—
10#11762533C2212yes—
11C2×C4×D4×C17C2211yescharacteristic
12C22×C4⋊C4×C17C411yescharacteristic, complemented
13C17×#400C412yescomplemented
14C17×#399C2212yes—
15C17×#389—21no—
16C17×#386—21no—
17C17×#385—21no—
18C17×#380—21no—
19C17×#380C411yescharacteristic, complemented
20C22×D4×C17D411yescharacteristic
21C23×C4×C17—21no—
22C23×C4×C17C2×C411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C4×D4×C17—41no—
25C4×D4×C17—41no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17—41no—
30C2×C4⋊C4×C17—41no—
31C2×C4⋊C4×C17C2×C412yes—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C22⋊C4×C17—21no—
34C2×C22⋊C4×C17—22no—
35C2×C22⋊C4×C17D411yescharacteristic
36C2×C22⋊C4×C17C2311yescharacteristic
37C2×C42×C17D411yescharacteristic
38C2×C22⋊Dic17—22no—
39C2×C4⋊Dic17—41no—
40C17×#95—22no—
41C17×#95—22no—
42C17×#95C2×C412yes—
43#10587505—42no—
44#10587502—41no—
45C22⋊#3622—22no—
46C4⋊#3622—41no—
47C4×#3622—41no—
48C24×C17C22⋊C411yescharacteristic
49C24×C17C2×D411yescharacteristic
50C2×D4×C17—41no—
51C2×D4×C17—41no—
52C2×D4×C17SD1612yes—
53C2×D4×C17Q1612yes—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—22no—
59C22×C4×C17—22no—
60C22×C4×C17—42no—
61C22×C4×C17—42no—
62C22×C4×C17—42no—
63C22×C4×C17—82no—
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22×C411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C22×Dic17—42no—
70C4⋊C4×C17—41no—
71C4⋊C4×C17—42no—
72C4⋊C4×C17—42no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—41no—
75C4⋊C4×C17—82no—
76C22⋊C4×C17—41no—
77C22⋊C4×C17—41no—
78C22⋊C4×C17—42no—
79C22⋊C4×C17—82no—
80C42×C17—21no—
81C42×C17—21no—
82C22⋊Dic17—42no—
83C22⋊Dic17—42no—
84C4⋊Dic17—82no—
85C2×#3622—42no—
86#11899—171nomaximal, Sylow
87C23×C17—21no—
88C23×C17—21no—
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—41no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17—41no—
95C23×C17#10312yes—
96C23×C17C2×C22⋊C411yescharacteristic
97C23×C17C2×SD1611yescharacteristic
98C23×C17C2×Q1611yescharacteristic
99D4×C17—42no—
100D4×C17—42no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—22no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—81no—
120C2×C4×C17—84no—
121C2×C4×C17#9911yescharacteristic
122C2×C4×C17#9911yescharacteristic
123C2×C4×C17#10312yes—
124C2×Dic17—82no—
125C2×Dic17—82no—
126C2×Dic17—82no—
127#3622—82no—
128#2487D1711yescharacteristic
129#1799—171no—
130#1460—171no—
131#1352—174no—
132C22×C17—21no—
133C22×C17—21no—
134C22×C17—21no—
135C22×C17—22no—
136C22×C17—21no—
137C22×C17—42no—
138C22×C17—41no—
139C22×C17—41no—
140C22×C17—42no—
141C22×C17—41no—
142C22×C17—41no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—81no—
146C22×C17C2×#9911yescharacteristic
147C22×C17C2×#10311yescharacteristic
148C22×C17#42111yescharacteristic
149C4×C17—21no—
150C4×C17—22no—
151C4×C17—41no—
152C4×C17—41no—
153C4×C17—41no—
154C4×C17—82no—
155C4×C17—82no—
156C4×C17#35212yes—
157Dic17—162no—
158C2×C4×D4C2×D1711yescharacteristic
159C22×C4⋊C4Dic1711yescharacteristic, complemented
160#431—341no—
161C2×#99—341no—
162#400Dic1712yescomplemented
163#399C2×D1712yes—
164#389—21no—
165#386—21no—
166#385—21no—
167#380—21no—
168#380—341no—
169#380Dic1711yescharacteristic, complemented
170#342—172no—
171#323—172no—
172#323—341no—
173C2×C17—21no—
174C2×C17—21no—
175C2×C17—41no—
176C2×C17—42no—
177C2×C17—41no—
178C2×C17#152011yescharacteristic
179C2×C17C2×#35211yescharacteristic
180C2×C17#212811yescharacteristic
181C22×D4C22⋊D1711yescharacteristic
182C23×C4—21no—
183C23×C4C2×Dic1711yescharacteristic
184C23×C4C22⋊D1711yescharacteristic
185C4×D4—41no—
186C4×D4—41no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—681no—
193C2×C4⋊C4C2×Dic1712yes—
194C2×C4⋊C4C2×Dic1711yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—342no—
198C2×C22⋊C4C22⋊D1711yescharacteristic
199C2×C22⋊C4C22×D1711yescharacteristic
200C2×C42C22⋊D1711yescharacteristic
201C4⋊C8—681no—
202#99—682no—
203C22⋊C8—342no—
204C4×C8—681no—
205#95—22no—
206#95—22no—
207#95—681no—
208#95C2×Dic1712yes—
209C17#1189911yescharacteristic, complemented, Sylow
210C24C22⋊Dic1711yescharacteristic
211C24C2×C22⋊D1711yescharacteristic
212C2×D4—41no—
213C2×D4—41no—
214C2×D4#6323012yes—
215C2×D4#6323112yes—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—82no—
226C22×C4—682no—
227C22×C4C22⋊Dic1711yescharacteristic
228C22×C4C22⋊Dic1711yescharacteristic
229C22×C4C22⋊Dic1711yescharacteristic
230C22×C4C22×Dic1711yescharacteristic
231C22×C4C2×C22⋊D1711yescharacteristic
232C2×C8—682no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—82no—
239C4⋊C4—1362no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—42no—
243C22⋊C4—82no—
244C22⋊C4—682no—
245C22⋊C4—682no—
246C42—21no—
247C42—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23#1058750612yes—
257C23C2×#6323011yescharacteristic
258C23C2×#6323111yescharacteristic
259C23C2×C22⋊Dic1711yescharacteristic
260D4—42no—
261D4—42no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—84no—
282C2×C4—1362no—
283C2×C4—1362no—
284C2×C4—1362no—
285C2×C4#1058750511yescharacteristic
286C2×C4#1058750511yescharacteristic
287C2×C4#1058750612yes—
288C8—1362no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#1058750511yescharacteristic
304C22C2×#1058750611yescharacteristic
305C22#1176282611yescharacteristic
306C4—21no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4—2722no—
314C4#1176259912yes—
315C2—21no—
316C2—21no—
317C2—41no—
318C2—42no—
319C2—41no—
320C2#1452735611yescharacteristic
321C2C2×#1176259911yescharacteristic
322C2#1453045611yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2487C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#2487D17
hypocenterC17#11899
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
3. #14526808C2semidirect
12. C22×C4⋊C4×C17C4semidirect
13. C17×#400C4semidirect
19. C17×#380C4semidirect
159. C22×C4⋊C4Dic17semidirect
162. #400Dic17semidirect
169. #380Dic17semidirect
209. C17#11899semidirect

Maximal subgroups

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

subgroupindex
#145287722
#145271052
#145268082
C17×#24872
#1189917