Finite Group Databasebeta

← #17953338

Subgroups — #17953338

Order 4352.

450 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 96 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#14544344C212yesmaximal, complemented
2#14544043C212yesmaximal
3#14543649C212yesmaximal, complemented
4C2×C8×C22⋊D17C211yesmaximal, characteristic
5#11763830C2214yes—
6#11763673—22no—
7#11763670—22no—
8#11763653—22no—
9#11763652—22no—
10C22×C8×D17C411yescharacteristic
11C2×C4×C22⋊D17C2211yescharacteristic
12C8×C22⋊D17—22no—
13C8×C22⋊D17—22no—
14C2×#10587485C2211yescharacteristic
15C2×C22⋊#3622C411yescharacteristic
16C2×#10587491C411yescharacteristic
17C2×C8×Dic17C2211yescharacteristic
18C23×C8×C17C411yescharacteristic
19C23×C4×C17C2×C411yescharacteristic
20C22×C22⋊D17Q811yescharacteristic
21C22×C4×D17C2×C411yescharacteristic
22C22×C8×C17—21no—
23C22×C8×C17—21no—
24C22×C8×C17—42no—
25C22×C8×C17C2×C411yescharacteristic
26C22×C8×C17C2×C411yescharacteristic
27C2×C22⋊Dic17D411yescharacteristic
28C4×C22⋊D17—22no—
29C4×C22⋊D17—22no—
30C2×#63227D411yescharacteristic
31C2×#63225Q811yescharacteristic
32C2×C4×Dic17C2311yescharacteristic
33C22×#3622C2×C411yescharacteristic
34C22×#3622C2×C411yescharacteristic
35C2×C8×D17—21no—
36C2×C8×D17—21no—
37C2×C8×D17—42no—
38C22⋊#3624—24no—
39#10587542—24no—
40C4⋊#3624—24no—
41C22⋊#3622—22no—
42#10587491—22no—
43#10587485—22no—
44C8×Dic17D412yes—
45C8×Dic17D412yes—
46C24×C17C4⋊C411yescharacteristic
47C23×D17C4⋊C411yescharacteristic
48C22×C4×C17—21no—
49C22×C4×C17—21no—
50C22×C4×C17—41no—
51C22×C4×C17—41no—
52C22×C4×C17C4⋊C411yescharacteristic
53C22×C4×C17C22×C411yescharacteristic
54C2×C22⋊D17—22no—
55C2×C22⋊D17—22no—
56C2×C22⋊D17—22no—
57C22×Dic17C4⋊C411yescharacteristic
58C22×Dic17C2×D411yescharacteristic
59C22×Dic17C2×Q811yescharacteristic
60C2×C4×D17—21no—
61C2×C4×D17—21no—
62C2×C4×D17—41no—
63C2×C4×D17—41no—
64C2×#3624—44no—
65C2×C8×C17—22no—
66C2×C8×C17—21no—
67C2×C8×C17—42no—
68C2×C8×C17—42no—
69C2×C8×C17—41no—
70C2×C8×C17C22⋊C411yescharacteristic
71C2×C8×C17C22⋊C411yescharacteristic
72C22⋊Dic17—22no—
73#63227—22no—
74#63225—22no—
75C4×Dic17C2×D412yes—
76C4×Dic17#4112yes—
77C2×#3622—21no—
78C2×#3622—41no—
79C2×#3622—41no—
80C2×#3622C22⋊C412yes—
81C8×D17—42no—
82C8×D17—42no—
83#8620—171nomaximal, Sylow
84C23×C17—21no—
85C23×C17—21no—
86C23×C17—41no—
87C23×C17—41no—
88C23×C17#10812yes—
89C23×C17C2×C4⋊C411yescharacteristic
90C22×D17—21no—
91C22×D17—21no—
92C22×D17—41no—
93C22×D17—41no—
94C22×D17#10812yes—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—42no—
99C2×C4×C17—41no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17C2×C22⋊C411yescharacteristic
103C2×C4×C17#11711yescharacteristic
104C22⋊D17—28no—
105C2×Dic17—21no—
106C2×Dic17—21no—
107C2×Dic17—21no—
108C2×Dic17—41no—
109C2×Dic17—41no—
110C2×Dic17#12112yes—
111C2×Dic17C22⋊Q812yes—
112C4×D17—42no—
113C4×D17—41no—
114C4×D17—41no—
115#3624—84no—
116C8×C17—21no—
117C8×C17—41no—
118C8×C17#10012yes—
119#3622—22no—
120#3622—41no—
121C2×C8×D4—171no—
122#1568—172no—
123#1466—172no—
124#1290—172no—
125C22×C17—21no—
126C22×C17—21no—
127C22×C17—22no—
128C22×C17—42no—
129C22×C17—42no—
130C22×C17—41no—
131C22×C17—41no—
132C22×C17#38011yescharacteristic
133C22×C17C2×#10811yescharacteristic
134C22×C17C2×#10811yescharacteristic
135C2×D17—22no—
136C2×D17—42no—
137C2×D17—41no—
138C2×D17—41no—
139C2×D17—42no—
140C4×C17—21no—
141C4×C17—41no—
142C4×C17C2×#10011yescharacteristic
143C4×C17#41311yescharacteristic
144Dic17—22no—
145Dic17—22no—
146Dic17—41no—
147C23×C8—171no—
148C23×C8#59511yescharacteristic
149C2×C4×D4—171no—
150C8×D4—342no—
151C8×D4—342no—
152#432—174no—
153C2×C4⋊C8—171no—
154C2×C22⋊C8—171no—
155C2×C22⋊C8—171no—
156C2×C4×C8—171no—
157#335—342no—
158#334—342no—
159#323—342no—
160#323—342no—
161C2×C17—21no—
162C2×C17—21no—
163C2×C17—42no—
164C2×C17#180011yescharacteristic
165C2×C17#181011yescharacteristic
166C2×C17#181211yescharacteristic
167D17—42no—
168C22×D4—171no—
169C23×C4—171no—
170C23×C4C2×#59511yescharacteristic
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—42no—
174C22×C8—171no—
175C22×C8—171no—
176C22×C8—341no—
177C22×C8—341no—
178C22×C8—682no—
179C22×C8C2×#59511yescharacteristic
180C22×C8C2×#59511yescharacteristic
181C4×D4—342no—
182C4×D4—342no—
183C2×C4⋊C4—171no—
184C2×C22⋊C4—171no—
185C2×C22⋊C4—171no—
186C2×C42—171no—
187C4⋊C8—342no—
188C4⋊C8—344no—
189C22⋊C8—342no—
190C22⋊C8—342no—
191C22⋊C8—344no—
192#97—344no—
193C4×C8—172no—
194C4×C8—172no—
195C17#862011yescharacteristic, complemented, Sylow
196C24—171no—
197C24C4⋊#59511yescharacteristic
198C2×D4—342no—
199C2×D4—342no—
200C2×D4—342no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—171no—
206C22×C4—171no—
207C22×C4—171no—
208C22×C4—341no—
209C22×C4—341no—
210C22×C4—681no—
211C22×C4—681no—
212C22×C4C4⋊#59511yescharacteristic
213C22×C4C22×#59511yescharacteristic
214C2×C8—22no—
215C2×C8—21no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—41no—
219C2×C8—172no—
220C2×C8—341no—
221C2×C8—682no—
222C2×C8—681no—
223C2×C8—682no—
224C2×C8—681no—
225C2×C8—684no—
226C2×C8C22⋊#59511yescharacteristic
227C2×C8C22⋊#59511yescharacteristic
228C4⋊C4—342no—
229C22⋊C4—342no—
230C22⋊C4—342no—
231C42—172no—
232C42—172no—
233C23—21no—
234C23—21no—
235C23—41no—
236C23—41no—
237C23—172no—
238C23—341no—
239C23—341no—
240C23—681no—
241C23—681no—
242C23#1058753512yes—
243C23C2×C4⋊#59511yescharacteristic
244D4—348no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—172no—
253C2×C4—172no—
254C2×C4—341no—
255C2×C4—341no—
256C2×C4—341no—
257C2×C4—682no—
258C2×C4—681no—
259C2×C4—681no—
260C2×C4—681no—
261C2×C4—681no—
262C2×C4#1058767711yescharacteristic
263C2×C4C2×C22⋊#59511yescharacteristic
264C8—21no—
265C8—41no—
266C8—342no—
267C8—681no—
268C8—1364no—
269C8#1058755312yes—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—342no—
278C22—682no—
279C22—682no—
280C22—681no—
281C22—681no—
282C22#1176375111yescharacteristic
283C22C2×#1058753511yescharacteristic
284C22D17⋊#10811yescharacteristic
285C4—21no—
286C4—41no—
287C4—342no—
288C4—342no—
289C4—681no—
290C4C2×#1058755311yescharacteristic
291C4#1176379511yescharacteristic
292C2—21no—
293C2—21no—
294C2—42no—
295C2—682no—
296C2#1454464811yescharacteristic
297C2#1454466311yescharacteristic
298C2#1454466611yescharacteristic
subgroupG/H
centerC22#11763751
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC23×C8×C17C4
socleC22×C17#380
prime radicalC22×C22⋊D17Q8
cosocleC2×C4×Dic17C23
hypercenterC23×C8#595
hypocenterC17#8620
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
1. #14544344C2semidirect
3. #14543649C2semidirect
195. C17#8620semidirect

Maximal subgroups

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

subgroupindex
#145443442
#145440432
#145436492
C2×C8×C22⋊D172
#862017