Finite Group Databasebeta

← #16930343

Subgroups — #16930343

Order 4352.

458 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 119 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#14537422C211yesmaximal, characteristic
2#14527469C212yesmaximal, complemented
3#14527098C211yesmaximal, characteristic
4#14526642C212yesmaximal, complemented
5C17×#2628C211yesmaximal, characteristic
6#11762870—24no—
7#11762862C2212yes—
8#11762833C2212yes—
9#11762773—21no—
10C4×#63223—21no—
11#11762555—24no—
12#11762535—21no—
13#11762533—21no—
14C2×C17×C4⋊Q8C2211yescharacteristic
15C2×C17×#117C2211yescharacteristic
16C17×#400C412yes—
17C17×#399C412yes—
18C17×#395—21no—
19C17×#393—21no—
20C17×#389—21no—
21C17×#382C2211yescharacteristic
22C17×#380—21no—
23C22×Q8×C17—21no—
24C23×C4×C17D411yescharacteristic
25C17×C4⋊Q8—22no—
26C17×C4⋊Q8—22no—
27C17×#117—22no—
28C17×#117—22no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—22no—
32C2×C4⋊C4×C17C2×C412yes—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17D411yescharacteristic
35C2×C22⋊C4×C17—22no—
36C2×C22⋊C4×C17C2×C411yescharacteristic
37C2×C22⋊C4×C17D411yescharacteristic
38C2×C42×C17D411yescharacteristic
39C2×C42×C17C2311yescharacteristic
40C2×#63223—22no—
41C17×#95—22no—
42C17×#95—22no—
43C17×#95—21no—
44C17×#95C2×C412yes—
45C22⋊#3622—24no—
46C22⋊#3622—22no—
47C4⋊#3622—22no—
48C4⋊#3622—22no—
49C4⋊#3622D414yes—
50#10587474—22no—
51C4×#3622—22no—
52C4×#3622—22no—
53C24×C17C22⋊C411yescharacteristic
54C2×Q8×C17—42no—
55C2×Q8×C17—82no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—22no—
59C22×C4×C17—22no—
60C22×C4×C17—22no—
61C22×C4×C17—22no—
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22×C411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C4⋊C4×C17—22no—
69C4⋊C4×C17—22no—
70C4⋊C4×C17—24no—
71C4⋊C4×C17—41no—
72C4⋊C4×C17—41no—
73C4⋊C4×C17—42no—
74C4⋊C4×C17—82no—
75C22⋊C4×C17—22no—
76C22⋊C4×C17—22no—
77C22⋊C4×C17—82no—
78C42×C17—21no—
79C42×C17—21no—
80C42×C17C2×D412yes—
81C42×C17#4112yes—
82#63223—42no—
83#63223—42no—
84C2×#3622—24no—
85C2×#3622—42no—
86#9194—171nomaximal, Sylow
87C23×C17—21no—
88C23×C17—21no—
89C23×C17—21no—
90C23×C17—41no—
91C23×C17—41no—
92C23×C17C2×C22⋊C411yescharacteristic
93Q8×C17—84no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—41no—
100C2×C4×C17—41no—
101C2×C4×C17—42no—
102C2×C4×C17—42no—
103C2×C4×C17—41no—
104C2×C4×C17—41no—
105C2×C4×C17—42no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17C4×D412yes—
112C2×C4×C17#12011yescharacteristic
113C2×C4×C17#12112yes—
114C2×C4×C17#12311yescharacteristic
115C2×C4×C17#13512yes—
116C2×C4×C17#13712yes—
117#3622—44no—
118#3622—82no—
119#2921—171no—
120#2628D1711yescharacteristic
121#1549—172no—
122#1458—171no—
123#1286—172no—
124C22×C17—21no—
125C22×C17—21no—
126C22×C17—42no—
127C22×C17—41no—
128C22×C17—41no—
129C22×C17—41no—
130C22×C17—81no—
131C22×C17#38611yescharacteristic
132C22×C17#41611yescharacteristic
133C22×C17#41911yescharacteristic
134C4×C17—22no—
135C4×C17—22no—
136C4×C17—41no—
137C4×C17—41no—
138C4×C17—41no—
139C4×C17—82no—
140C4×C17—82no—
141C4×C17#45212yes—
142C4×C17#48512yes—
143C2×C4⋊Q8C2×D1711yescharacteristic
144C2×#117C2×D1711yescharacteristic
145#433—344no—
146#431—172no—
147#423—172no—
148#407—341no—
149C4×M16—341no—
150#400Dic1712yes—
151#399Dic1712yes—
152#395—21no—
153#393—21no—
154#389—21no—
155#382C2×D1711yescharacteristic
156#380—21no—
157#332—344no—
158#324—341no—
159#323—341no—
160C2×C17—21no—
161C2×C17—21no—
162C2×C17—81no—
163C2×C17#187311yescharacteristic
164C2×C17#191011yescharacteristic
165C2×C17#191511yescharacteristic
166C22×Q8—21no—
167C23×C4C22⋊D1711yescharacteristic
168C2×M16—342no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171#117—22no—
172#117—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C2×Dic1712yes—
177C2×C4⋊C4C2×Dic1711yescharacteristic
178C2×C4⋊C4C22⋊D1711yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4C2×Dic1711yescharacteristic
181C2×C22⋊C4C22⋊D1711yescharacteristic
182C2×C42C22⋊D1711yescharacteristic
183C2×C42C22×D1711yescharacteristic
184C4⋊C8—174no—
185C4⋊C8—342no—
186C4⋊C8—342no—
187C22⋊C8—344no—
188C22⋊C8—342no—
189#97—342no—
190C4×C8—342no—
191C4×C8—342no—
192#95—22no—
193#95—22no—
194#95—21no—
195#95C2×Dic1712yes—
196C17#919411yescharacteristic, complemented, Sylow
197C24C22⋊Dic1711yescharacteristic
198C2×Q8—42no—
199C2×Q8—82no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4C22⋊Dic1711yescharacteristic
207C22×C4C22⋊Dic1711yescharacteristic
208C22×C4C22⋊Dic1711yescharacteristic
209C22×C4C22×Dic1711yescharacteristic
210C22×C4C2×C22⋊D1711yescharacteristic
211C22×C4C2×C22⋊D1711yescharacteristic
212M16—682no—
213M16—682no—
214C2×C8—344no—
215C2×C8—682no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—24no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—82no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—82no—
226C42—21no—
227C42—21no—
228C42D4×D1712yes—
229C42#6325412yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23C2×C22⋊Dic1711yescharacteristic
236Q8—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4D4×Dic1712yes—
255C2×C4#1058762111yescharacteristic
256C2×C4#1058762311yescharacteristic
257C2×C4#1058762512yes—
258C2×C4#1058762812yes—
259C2×C4#1058763612yes—
260C8—684no—
261C8—1362no—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22#1176268911yescharacteristic
270C22#1176280511yescharacteristic
271C22#1176281811yescharacteristic
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—82no—
278C4—82no—
279C4#1176296912yes—
280C4#1176314712yes—
281C2—21no—
282C2—21no—
283C2—81no—
284C2#1452909711yescharacteristic
285C2#1452931211yescharacteristic
286C2#1452933711yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2628C2
socleC22×C17#386
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#2628D17
hypocenterC17#9194
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2628

Complements

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

NH ≅ G/Ntype
2. #14527469C2semidirect
4. #14526642C2semidirect
196. C17#9194semidirect

Maximal subgroups

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

subgroupindex
#145374222
#145274692
#145270982
#145266422
C17×#26282
#919417