Finite Group Databasebeta

← #17953327

Subgroups — #17953327

Order 4352.

486 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 106 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#14544648C212yesmaximal
2#14544338C212yesmaximal, complemented
3#14543654C212yesmaximal, complemented
4C2×#11762872C211yesmaximal, characteristic
5#11763812C2214yes—
6C2×#10587553—22no—
7#11763689—22no—
8#11763671—22no—
9#11763670—22no—
10C23×#3622C411yescharacteristic
11C22×C8×D17C411yescharacteristic
12C2×C4×C22⋊D17C2211yescharacteristic
13#11762872—22no—
14#11762872—22no—
15C2×#10587485C2211yescharacteristic
16C2×#10587491C411yescharacteristic
17C2×C8×Dic17C2211yescharacteristic
18C2×C17×C22⋊C8C411yescharacteristic
19C23×C4×C17C2×C411yescharacteristic
20C22×C22⋊D17D411yescharacteristic
21C22×C4×D17C2×C411yescharacteristic
22C2×#63247—24no—
23C22×C8×C17C2×C411yescharacteristic
24C22×C8×C17C2×C411yescharacteristic
25C2×C22⋊Dic17Q811yescharacteristic
26C4×C22⋊D17—22no—
27C4×C22⋊D17—22no—
28C2×#63227Q811yescharacteristic
29C2×#63225D411yescharacteristic
30C2×C4×Dic17C2311yescharacteristic
31C22×#3622—21no—
32C22×#3622—21no—
33C22×#3622—42no—
34C22×#3622C2×C411yescharacteristic
35C22×#3622C2×C411yescharacteristic
36C2×C8×D17—21no—
37C2×C8×D17—21no—
38C2×C8×D17—42no—
39#10587553—44no—
40#10587544—24no—
41C4⋊#3624—24no—
42C17×C22⋊C8—22no—
43#10587491—22no—
44#10587485—22no—
45C8×Dic17D412yes—
46C8×Dic17D412yes—
47C24×C17C4⋊C411yescharacteristic
48C23×D17C4⋊C411yescharacteristic
49C22×C4×C17—21no—
50C22×C4×C17—21no—
51C22×C4×C17—41no—
52C22×C4×C17—41no—
53C22×C4×C17C4⋊C411yescharacteristic
54C22×C4×C17C22×C411yescharacteristic
55C2×C22⋊D17—22no—
56C2×C22⋊D17—22no—
57C2×C22⋊D17SD1614yes—
58C22×Dic17C4⋊C411yescharacteristic
59C22×Dic17C2×D411yescharacteristic
60C22×Dic17C2×Q811yescharacteristic
61C2×C4×D17—21no—
62C2×C4×D17—21no—
63C2×C4×D17—41no—
64C2×C4×D17—41no—
65#63247—44no—
66#63247—44no—
67C2×#3624—44no—
68C2×C8×C17—21no—
69C2×C8×C17—41no—
70C2×C8×C17—41no—
71C2×C8×C17C22⋊C412yes—
72C22⋊Dic17—22no—
73#63227—22no—
74#63225—22no—
75C4×Dic17C2×D412yes—
76C4×Dic17#4112yes—
77C2×#3622—22no—
78C2×#3622—21no—
79C2×#3622—42no—
80C2×#3622—42no—
81C2×#3622—41no—
82C2×#3622C22⋊C411yescharacteristic
83C2×#3622C22⋊C411yescharacteristic
84C8×D17—42no—
85C8×D17—42no—
86#8614—171nomaximal, Sylow
87C23×C17—21no—
88C23×C17—21no—
89C23×C17—41no—
90C23×C17—41no—
91C23×C17#10612yes—
92C23×C17C2×C4⋊C411yescharacteristic
93C22×D17—21no—
94C22×D17—21no—
95C22×D17—41no—
96C22×D17—41no—
97C22×D17#10612yes—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—42no—
102C2×C4×C17—41no—
103C2×C4×C17—41no—
104C2×C4×C17—41no—
105C2×C4×C17C2×C22⋊C411yescharacteristic
106C2×C4×C17#11711yescharacteristic
107C22⋊D17—28no—
108C2×Dic17—21no—
109C2×Dic17—41no—
110C2×Dic17—41no—
111C2×Dic17#12112yes—
112C2×Dic17C22⋊Q812yes—
113C2×Dic17C2×SD1612yes—
114C2×Dic17#13512yes—
115C4×D17—42no—
116C4×D17—41no—
117C4×D17—41no—
118#3624—84no—
119C8×C17—22no—
120C8×C17—41no—
121#3622—21no—
122#3622—41no—
123#3622#10012yescomplemented
124C2×C8×D4—171no—
125#1800—172no—
126#1566—172no—
127#1292—172no—
128C22×C17—21no—
129C22×C17—21no—
130C22×C17—22no—
131C22×C17—42no—
132C22×C17—42no—
133C22×C17—41no—
134C22×C17—41no—
135C22×C17#38011yescharacteristic
136C22×C17C2×#10611yescharacteristic
137C22×C17#42711yescharacteristic
138C2×D17—22no—
139C2×D17—42no—
140C2×D17—41no—
141C2×D17—41no—
142C2×D17—42no—
143C4×C17—21no—
144C4×C17—41no—
145C4×C17C2×#10011yescharacteristic
146C4×C17#41311yescharacteristic
147Dic17—22no—
148Dic17—41no—
149Dic17#46514yes—
150C23×C8—171no—
151C23×C8—171no—
152C2×C4×D4—171no—
153C8×D4—342no—
154C8×D4—342no—
155#423—174no—
156C2×C4⋊C8—171no—
157C2×#100—342no—
158C2×C22⋊C8—171no—
159C2×C22⋊C8#59511yescharacteristic
160C2×C4×C8—171no—
161#343—342no—
162#334—342no—
163#334—342no—
164C2×C17—21no—
165C2×C17—21no—
166C2×C17—42no—
167C2×C17#180011yescharacteristic
168C2×C17#180711yescharacteristic
169C2×C17#181311yescharacteristic
170D17—42no—
171C22×D4—171no—
172C23×C4—171no—
173C23×C4C2×#59511yescharacteristic
174C2×M16—344no—
175C22×C8—171no—
176C22×C8—171no—
177C22×C8—341no—
178C22×C8—341no—
179C22×C8—341no—
180C22×C8—341no—
181C22×C8—682no—
182C22×C8—682no—
183C22×C8C2×#59511yescharacteristic
184C22×C8C2×#59511yescharacteristic
185C4×D4—342no—
186C4×D4—342no—
187C2×C4⋊C4—171no—
188C2×C22⋊C4—171no—
189C2×C22⋊C4—171no—
190C2×C42—171no—
191C4⋊C8—342no—
192C4⋊C8—344no—
193C4⋊C8—344no—
194#100—684no—
195C22⋊C8—22no—
196C22⋊C8—342no—
197C4×C8—172no—
198C4×C8—172no—
199C17#861411yescharacteristic, complemented, Sylow
200C24—171no—
201C24C4⋊#59511yescharacteristic
202C2×D4—174no—
203C2×D4—342no—
204C2×D4—342no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—171no—
210C22×C4—171no—
211C22×C4—171no—
212C22×C4—341no—
213C22×C4—341no—
214C22×C4—681no—
215C22×C4—681no—
216C22×C4C4⋊#59511yescharacteristic
217C22×C4C22×#59511yescharacteristic
218M16—684no—
219M16—684no—
220C2×C8—21no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—171no—
224C2×C8—171no—
225C2×C8—342no—
226C2×C8—341no—
227C2×C8—682no—
228C2×C8—682no—
229C2×C8—682no—
230C2×C8—682no—
231C2×C8—681no—
232C2×C8—684no—
233C2×C8C22⋊#59512yes—
234C4⋊C4—342no—
235C22⋊C4—342no—
236C22⋊C4—342no—
237C42—172no—
238C42—172no—
239C23—21no—
240C23—21no—
241C23—41no—
242C23—41no—
243C23—172no—
244C23—341no—
245C23—341no—
246C23—681no—
247C23—681no—
248C23#1058753312yes—
249C23C2×C4⋊#59511yescharacteristic
250D4—348no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—172no—
259C2×C4—172no—
260C2×C4—172no—
261C2×C4—172no—
262C2×C4—341no—
263C2×C4—682no—
264C2×C4—681no—
265C2×C4—681no—
266C2×C4—681no—
267C2×C4—681no—
268C2×C4#1058767711yescharacteristic
269C2×C4C2×C22⋊#59511yescharacteristic
270C8—22no—
271C8—41no—
272C8—172no—
273C8—341no—
274C8—681no—
275C8—1364no—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—342no—
284C22—682no—
285C22—682no—
286C22—681no—
287C22—681no—
288C22#1176375111yescharacteristic
289C22C2×#1058753311yescharacteristic
290C22#1176381911yescharacteristic
291C4—21no—
292C4—41no—
293C4—174no—
294C4—342no—
295C4—681no—
296C4D17⋊#10011yescharacteristic
297C4#1176379611yescharacteristic
298C2—21no—
299C2—21no—
300C2—42no—
301C2—682no—
302C2#1454464911yescharacteristic
303C2#1454466011yescharacteristic
304C2#1454466911yescharacteristic
subgroupG/H
centerC22#11763751
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C22⋊C8C4
socleC22×C17#380
prime radicalC22×C22⋊D17D4
cosocleC2×C4×Dic17C23
hypercenterC2×C22⋊C8#595
hypocenterC17#8614
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #14544338C2semidirect
3. #14543654C2semidirect
123. #3622#100semidirect
199. C17#8614semidirect

Maximal subgroups

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

subgroupindex
#145446482
#145443382
#145436542
C2×#117628722
#861417