Finite Group Databasebeta

← #16952982

Subgroups — #16952982

Order 4352.

488 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 111 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#14528776C211yesmaximal, characteristic
2C4×#10587505C211yesmaximal, characteristic
3#14526902C214yesmaximal
4C17×#2487C211yesmaximal, characteristic
5#11762833—21no—
6C2×#10587504C2212yes—
7C2×#10587505C2212yes—
8C4×C22⋊Dic17—21no—
9#11762578—21no—
10#11762576—21no—
11C2×C4×D4×C17C2211yescharacteristic
12C22×C4⋊C4×C17C411yescharacteristic, complemented
13C17×#400—21no—
14C17×#399—21no—
15C17×#389—21no—
16C17×#386C2212yes—
17C17×#385C412yescomplemented
18C17×#380—21no—
19C17×#380C411yescharacteristic, complemented
20C22×D4×C17C2311yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17C2×C412yes—
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—21no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17—41no—
31C2×C4⋊C4×C17—41no—
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17—22no—
34C2×C22⋊C4×C17C2×C412yes—
35C2×C22⋊C4×C17C2×C411yescharacteristic
36C2×C22⋊C4×C17D411yescharacteristic
37C2×C42×C17D411yescharacteristic
38C2×C22⋊Dic17—22no—
39C2×C4×Dic17—41no—
40C2×#63223—22no—
41C17×#95—22no—
42C17×#95—21no—
43C17×#95—22no—
44#10587505—22no—
45#10587505Q814yes—
46#10587504—24no—
47#10587502—41no—
48C4⋊#3622—41no—
49C4⋊#3622—41no—
50C24×C17C22×C411yescharacteristic
51C24×C17C2×D411yescharacteristic
52C2×D4×C17—41no—
53C2×D4×C17—41no—
54C2×D4×C17C2×Q812yes—
55C2×D4×C17#4112yes—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—22no—
61C22×C4×C17—22no—
62C22×C4×C17—42no—
63C22×C4×C17—42no—
64C22×C4×C17—42no—
65C22×C4×C17—82no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×Dic17—42no—
72C4⋊C4×C17—41no—
73C4⋊C4×C17—42no—
74C4⋊C4×C17—42no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—82no—
78C22⋊C4×C17—41no—
79C22⋊C4×C17—41no—
80C22⋊C4×C17—42no—
81C22⋊C4×C17—82no—
82C42×C17—21no—
83C42×C17—21no—
84C22⋊Dic17—24no—
85C22⋊Dic17—42no—
86C4×Dic17—82no—
87#63223—42no—
88#63223—42no—
89C2×#3622—42no—
90#13295—171nomaximal, Sylow
91C23×C17—21no—
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—41no—
96C23×C17—41no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17C2×C22⋊C411yescharacteristic
100C23×C17C4×Q812yes—
101C23×C17C22⋊Q812yes—
102D4×C17—42no—
103D4×C17—42no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17—81no—
123C2×C4×C17—84no—
124C2×C4×C17#9911yescharacteristic
125C2×C4×C17#9911yescharacteristic
126C2×C4×C17#12411yescharacteristic
127C2×C4×C17C4⋊Q811yescharacteristic
128C2×Dic17—44no—
129C2×Dic17—82no—
130C2×Dic17—82no—
131#3622—82no—
132#2487D1711yescharacteristic
133#1800—171no—
134C4×#99—171no—
135#1390—174no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—22no—
139C22×C17—22no—
140C22×C17—42no—
141C22×C17—42no—
142C22×C17—42no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17—41no—
147C22×C17—81no—
148C22×C17#39111yescharacteristic
149C22×C17C2×#9911yescharacteristic
150C22×C17#41011yescharacteristic
151C4×C17—21no—
152C4×C17—21no—
153C4×C17—21no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—41no—
157C4×C17—82no—
158C4×C17—82no—
159C4×C17#35212yes—
160Dic17—84no—
161C2×C4×D4C2×D1711yescharacteristic
162C22×C4⋊C4Dic1711yescharacteristic, complemented
163#423—341no—
164C2×#100—172no—
165C2×#99—172no—
166#400—21no—
167#399—21no—
168#389—21no—
169#386C2×D1712yes—
170#385Dic1712yescomplemented
171#380—21no—
172#380Dic1711yescharacteristic, complemented
173C4×C22⋊C4—341no—
174#343—341no—
175#342—341no—
176C2×C17—21no—
177C2×C17—21no—
178C2×C17—42no—
179C2×C17—42no—
180C2×C17#199611yescharacteristic
181C2×C17C2×#35211yescharacteristic
182C2×C17#212211yescharacteristic
183C22×D4C22×D1711yescharacteristic
184C23×C4C2×Dic1711yescharacteristic
185C23×C4C2×Dic1712yes—
186C23×C4C22⋊D1711yescharacteristic
187C2×M16—342no—
188C4×D4—41no—
189C4×D4—41no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4C22⋊D1711yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—342no—
199C2×C22⋊C4C2×Dic1712yes—
200C2×C22⋊C4C2×Dic1711yescharacteristic
201C2×C22⋊C4C22⋊D1711yescharacteristic
202C2×C42—681no—
203C2×C42C22⋊D1711yescharacteristic
204C4⋊C8—681no—
205C4⋊C8—681no—
206#100—344no—
207#99—174no—
208#99—342no—
209#95—22no—
210#95—21no—
211#95—22no—
212#95—681no—
213C17#1329511yescharacteristic, complemented, Sylow
214C24C22×Dic1711yescharacteristic
215C24C2×C22⋊D1711yescharacteristic
216C2×D4—41no—
217C2×D4—41no—
218C2×D4Q8×D1712yes—
219C2×D4#6325612yes—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—82no—
230C22×C4—682no—
231C22×C4C22⋊Dic1711yescharacteristic
232C22×C4C22⋊Dic1711yescharacteristic
233C22×C4C22⋊Dic1711yescharacteristic
234C22×C4C22⋊Dic1711yescharacteristic
235C22×C4C2×C22⋊D1711yescharacteristic
236M16—682no—
237M16—682no—
238C2×C8—682no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—82no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—82no—
249C22⋊C4—344no—
250C22⋊C4—682no—
251C42—21no—
252C42—21no—
253C42—1362no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23Q8×Dic1712yes—
263C23#1058763212yes—
264C23C2×C22⋊Dic1711yescharacteristic
265D4—42no—
266D4—42no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—81no—
286C2×C4—84no—
287C2×C4—684no—
288C2×C4—1362no—
289C2×C4—1362no—
290C2×C4#1058750511yescharacteristic
291C2×C4#1058750511yescharacteristic
292C2×C4#1058763011yescharacteristic
293C2×C4#1058763311yescharacteristic
294C8—1362no—
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#1176271611yescharacteristic
308C22C2×#1058750511yescharacteristic
309C22#1176278311yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—82no—
317C4—82no—
318C4—1364no—
319C4#1176259912yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—42no—
324C2#1452977011yescharacteristic
325C2C2×#1176259911yescharacteristic
326C2#1453043411yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC24×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#2487C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#2487D17
hypocenterC17#13295
LCSG ▷ C24×C17 ▷ C22×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
12. C22×C4⋊C4×C17C4semidirect
17. C17×#385C4semidirect
19. C17×#380C4semidirect
162. C22×C4⋊C4Dic17semidirect
170. #385Dic17semidirect
172. #380Dic17semidirect
213. C17#13295semidirect

Maximal subgroups

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

subgroupindex
#145287762
C4×#105875052
#145269022
C17×#24872
#1329517