Finite Group Databasebeta

← #16921635

Subgroups — #16921635

Order 4352.

338 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 71 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#14527122C211yesmaximal, characteristic
3C17×#1955C211yesmaximal, characteristic
4C2×#10587505C412yes—
5#11762773—21no—
6#11762669—21no—
7#11762576—21no—
8#11762535C412yescomplemented
9#11762533—21no—
10C2×C4×D4×C17C2211yescharacteristic
11C2×C17×C4⋊C8C411yescharacteristic, complemented
12C2×C17×C22⋊C8—21no—
13C2×C17×C22⋊C8—21no—
14C2×C17×#97C411yescharacteristic, complemented
15C22×D4×C17C2×C411yescharacteristic
16C23×C4×C17D411yescharacteristic
17C23×C4×C17D411yescharacteristic
18C22×C8×C17—21no—
19C22×C8×C17—21no—
20C4×D4×C17—41no—
21C4×D4×C17—41no—
22C2×C4⋊C4×C17C2×C411yescharacteristic
23C2×C22⋊C4×C17D411yescharacteristic
24C2×C22⋊C4×C17Q811yescharacteristic
25C2×C42×C17C2×C411yescharacteristic
26C2×C22⋊Dic17—22no—
27C2×C4⋊Dic17—41no—
28C2×#63223—41no—
29C17×C4⋊C8—41no—
30C17×C22⋊C8—42no—
31C17×C22⋊C8—42no—
32C17×#97—41no—
33#10587505—42no—
34#10587502—41no—
35C22⋊#3622—22no—
36C22⋊#3622—41no—
37C24×C17C22⋊C411yescharacteristic
38C24×C17C4⋊C411yescharacteristic
39C2×D4×C17—21no—
40C2×D4×C17—21no—
41C2×D4×C17—41no—
42C2×D4×C17—41no—
43C22×C4×C17—21no—
44C22×C4×C17—21no—
45C22×C4×C17—21no—
46C22×C4×C17—21no—
47C22×C4×C17—41no—
48C22×C4×C17—41no—
49C22×C4×C17—41no—
50C22×C4×C17—41no—
51C22×C4×C17C4211yescharacteristic
52C22×C4×C17C22⋊C411yescharacteristic
53C22×C4×C17C22⋊C411yescharacteristic
54C22×C4×C17C4⋊C411yescharacteristic
55C22×C4×C17C4⋊C411yescharacteristic
56C22×Dic17—42no—
57C2×C8×C17—22no—
58C2×C8×C17—41no—
59C2×C8×C17—41no—
60C2×C8×C17—41no—
61C2×C8×C17—41no—
62C2×C8×C17—41no—
63C4⋊C4×C17—41no—
64C22⋊C4×C17—41no—
65C22⋊C4×C17—41no—
66C42×C17—21no—
67C42×C17—21no—
68C22⋊Dic17—42no—
69C22⋊Dic17—42no—
70C4⋊Dic17—81no—
71C4⋊Dic17—81no—
72#63223—81no—
73#63223—81no—
74C2×#3622—42no—
75#7322—171nomaximal, Sylow
76C23×C17—21no—
77C23×C17—21no—
78C23×C17—21no—
79C23×C17—21no—
80C23×C17—21no—
81C23×C17—41no—
82C23×C17—41no—
83C23×C17—41no—
84C23×C17—41no—
85C23×C17#9511yescharacteristic
86C23×C17#10011yescharacteristic
87C23×C17#10111yescharacteristic
88D4×C17—42no—
89D4×C17—42no—
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—41no—
99C2×C4×C17—41no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17—41no—
103C2×C4×C17—41no—
104C2×C4×C17—41no—
105C2×C4×C17—81no—
106C2×C4×C17—81no—
107C2×C4×C17#9911yescharacteristic
108C2×C4×C17#9911yescharacteristic
109C2×Dic17—82no—
110C2×Dic17—82no—
111C2×Dic17—82no—
112C8×C17—42no—
113C8×C17—81no—
114#3622—82no—
115#1955D1711yescharacteristic
116#1799—171no—
117#1465—171no—
118C22×C17—21no—
119C22×C17—21no—
120C22×C17—21no—
121C22×C17—21no—
122C22×C17—41no—
123C22×C17—41no—
124C22×C17—42no—
125C22×C17—41no—
126C22×C17—41no—
127C22×C17—41no—
128C22×C17—41no—
129C22×C17—41no—
130C22×C17—81no—
131C22×C17—81no—
132C22×C17#34211yescharacteristic
133C22×C17#34311yescharacteristic
134C22×C17#34411yescharacteristic
135C4×C17—21no—
136C4×C17—41no—
137C4×C17—41no—
138C4×C17—41no—
139C4×C17—41no—
140C4×C17#35311yescharacteristic
141C4×C17#35411yescharacteristic
142Dic17—162no—
143C2×C4×D4C2×D1711yescharacteristic
144C2×C4⋊C8Dic1711yescharacteristic, complemented
145C2×#99—172no—
146#407—341no—
147C2×C22⋊C8—21no—
148C2×C22⋊C8—21no—
149C2×#97Dic1711yescharacteristic, complemented
150#380—341no—
151#342—341no—
152#324—172no—
153#323—341no—
154C2×C17—21no—
155C2×C17—21no—
156C2×C17—41no—
157C2×C17—41no—
158C2×C17—81no—
159C2×C17#130511yescharacteristic
160C2×C17#139311yescharacteristic
161C2×C17#139511yescharacteristic
162C22×D4C4×D1711yescharacteristic
163C23×C4C22⋊D1711yescharacteristic
164C23×C4C22⋊D1711yescharacteristic
165C2×M16—681no—
166C22×C8—21no—
167C22×C8—21no—
168C4×D4—41no—
169C4×D4—41no—
170C2×C4⋊C4—681no—
171C2×C4⋊C4C4×D1711yescharacteristic
172C2×C22⋊C4—342no—
173C2×C22⋊C4Dic3411yescharacteristic
174C2×C22⋊C4D6811yescharacteristic
175C2×C42C2×Dic1711yescharacteristic
176C4⋊C8—41no—
177#99—682no—
178C22⋊C8—42no—
179C22⋊C8—42no—
180C22⋊C8—342no—
181C22⋊C8—681no—
182#97—41no—
183#95—681no—
184C17#732211yescharacteristic, complemented, Sylow
185C24#6322511yescharacteristic
186C24#6322711yescharacteristic
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—41no—
190C2×D4—41no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—682no—
200C22×C4C4×Dic1711yescharacteristic
201C22×C4#6322511yescharacteristic
202C22×C4C4⋊Dic1711yescharacteristic
203C22×C4#6322711yescharacteristic
204C22×C4C22⋊Dic1711yescharacteristic
205M16—1361no—
206M16—1361no—
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—682no—
214C4⋊C4—41no—
215C4⋊C4—1361no—
216C4⋊C4—1361no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—682no—
220C22⋊C4—682no—
221C42—21no—
222C42—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23#1058749411yescharacteristic
233C23#1058749511yescharacteristic
234C23#1058750211yescharacteristic
235D4—42no—
236D4—42no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—1362no—
255C2×C4—1362no—
256C2×C4—1362no—
257C2×C4#1058750511yescharacteristic
258C2×C4#1058750511yescharacteristic
259C8—42no—
260C8—81no—
261C8—1362no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22—81no—
276C22#1176257611yescharacteristic
277C22#1176257711yescharacteristic
278C22#1176258111yescharacteristic
279C4—21no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—2722no—
285C4#1176260211yescharacteristic
286C4#1176260411yescharacteristic
287C2—21no—
288C2—21no—
289C2—41no—
290C2—41no—
291C2—81no—
292C2#1452668911yescharacteristic
293C2#1452690911yescharacteristic
294C2#1452691311yescharacteristic
subgroupG/H
centerC22#11762576
commutatorC22×C4×C17C42
FrattiniC2×C4×D4C2×D17
FittingC17×#1955C2
socleC22×C17#342
prime radicalC22×D4×C17C2×C4
cosocleC2×C4×D4×C17C22
hypercenter#1955D17
hypocenterC17#7322
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1955

Complements

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

NH ≅ G/Ntype
8. #11762535C4semidirect
11. C2×C17×C4⋊C8C4semidirect
14. C2×C17×#97C4semidirect
144. C2×C4⋊C8Dic17semidirect
149. C2×#97Dic17semidirect
184. C17#7322semidirect

Maximal subgroups

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

subgroupindex
#145287722
#145271222
C17×#19552
#732217