Finite Group Databasebeta

← #16945002

Subgroups — #16945002

Order 4352.

402 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 87 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#14528861C211yesmaximal, characteristic
2#14527118C211yesmaximal, characteristic
3#14526784C214yesmaximal
4C17×#2621C211yesmaximal, characteristic
5#11762866—21no—
6#11762672—21no—
7#11762576—21no—
8#11762576C2212yes—
9#11762535—21no—
10#11762533C2212yes—
11C2×C17×C4⋊Q8C411yescharacteristic, complemented
12C2×C17×C22⋊Q8—21no—
13C2×C17×#117C2211yescharacteristic
14C17×#398—21no—
15C17×#398C412yescomplemented
16C17×#397—21no—
17C17×#397C2212yes—
18C17×#391—21no—
19C17×#382C411yescharacteristic, complemented
20C22×Q8×C17—21no—
21C23×C4×C17D411yescharacteristic
22C17×C4⋊Q8—41no—
23C17×C4⋊Q8—41no—
24C17×C22⋊Q8—44no—
25C17×#117—41no—
26C17×#117—41no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17C2×C412yes—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17—22no—
34C2×C22⋊C4×C17D411yescharacteristic
35C2×C22⋊C4×C17C2311yescharacteristic
36C2×C42×C17C2×C411yescharacteristic
37C2×C42×C17D411yescharacteristic
38C2×C22⋊Dic17—22no—
39C17×#95—21no—
40C17×#95—22no—
41C17×#95—22no—
42C17×#95C2×C412yes—
43#10587502—41no—
44#10587502—41no—
45C22⋊#3622—22no—
46C4⋊#3622—41no—
47#10587474—41no—
48C24×C17C2×D411yescharacteristic
49C2×Q8×C17—42no—
50C2×Q8×C17—42no—
51C2×Q8×C17—41no—
52C2×Q8×C17—41no—
53C22×C4×C17—22no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—22no—
57C22×C4×C17—22no—
58C22×C4×C17—41no—
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22×C411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C22×Dic17—42no—
66C4⋊C4×C17—41no—
67C4⋊C4×C17—41no—
68C4⋊C4×C17—42no—
69C4⋊C4×C17—42no—
70C4⋊C4×C17—44no—
71C4⋊C4×C17—42no—
72C22⋊C4×C17—41no—
73C22⋊C4×C17—41no—
74C22⋊C4×C17—44no—
75C42×C17—21no—
76C42×C17—21no—
77C42×C17—21no—
78C42×C17—21no—
79C22⋊Dic17—42no—
80C22⋊Dic17—42no—
81C2×#3622—42no—
82#11801—171nomaximal, Sylow
83C23×C17—21no—
84C23×C17—21no—
85C23×C17—41no—
86C23×C17—41no—
87C23×C17C2×C22⋊C411yescharacteristic
88C23×C17AGL(1,ℤ/8)12yes—
89Q8×C17—82no—
90Q8×C17—82no—
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—21no—
99C2×C4×C17—21no—
100C2×C4×C17—42no—
101C2×C4×C17—42no—
102C2×C4×C17—41no—
103C2×C4×C17—41no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17#9911yescharacteristic
113C2×C4×C17#9911yescharacteristic
114C2×Dic17—82no—
115C2×Dic17—82no—
116C2×Dic17—82no—
117#3622—82no—
118#2621D1711yescharacteristic
119#1820—171no—
120#1464—171no—
121#1344—174no—
122C22×C17—21no—
123C22×C17—21no—
124C22×C17—22no—
125C22×C17—42no—
126C22×C17—41no—
127C22×C17—41no—
128C22×C17—81no—
129C22×C17C2×#9911yescharacteristic
130C22×C17#41811yescharacteristic
131C22×C17#42111yescharacteristic
132C4×C17—21no—
133C4×C17—21no—
134C4×C17—41no—
135C4×C17—41no—
136C4×C17—41no—
137C4×C17—41no—
138C4×C17—41no—
139C4×C17—82no—
140C4×C17—82no—
141Dic17—162no—
142C2×C4⋊Q8Dic1711yescharacteristic, complemented
143C2×C22⋊Q8—21no—
144C2×#117C2×D1711yescharacteristic
145#432—341no—
146#398—21no—
147#398Dic1712yescomplemented
148#397—21no—
149#397C2×D1712yes—
150#391—21no—
151#382Dic1711yescharacteristic, complemented
152#381—341no—
153#342—172no—
154#342—341no—
155#324—341no—
156#323—172no—
157C2×C17—21no—
158C2×C17—21no—
159C2×C17—42no—
160C2×C17#151011yescharacteristic
161C2×C17#212311yescharacteristic
162C2×C17#212911yescharacteristic
163C22×Q8—21no—
164C23×C4C22⋊D1711yescharacteristic
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C22⋊Q8—44no—
168#117—41no—
169#117—41no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4C2×Dic1712yes—
174C2×C4⋊C4C2×Dic1711yescharacteristic
175C2×C4⋊C4C22⋊D1711yescharacteristic
176C2×C22⋊C4—22no—
177C2×C22⋊C4—342no—
178C2×C22⋊C4C22⋊D1711yescharacteristic
179C2×C22⋊C4C22×D1711yescharacteristic
180C2×C42C2×Dic1711yescharacteristic
181C2×C42C22⋊D1711yescharacteristic
182C4⋊C8—681no—
183C22⋊C8—342no—
184#97—681no—
185#95—21no—
186#95—22no—
187#95—22no—
188#95—681no—
189#95—681no—
190#95C2×Dic1712yes—
191C17#1180111yescharacteristic, complemented, Sylow
192C24C2×C22⋊D1711yescharacteristic
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—41no—
196C2×Q8—41no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—41no—
203C22×C4—682no—
204C22×C4C22⋊Dic1711yescharacteristic
205C22×C4C22⋊Dic1711yescharacteristic
206C22×C4C22⋊Dic1711yescharacteristic
207C22×C4C22⋊Dic1711yescharacteristic
208C22×C4C22×Dic1711yescharacteristic
209C22×C4C2×C22⋊D1711yescharacteristic
210C2×C8—682no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C4⋊C4—42no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—44no—
220C22⋊C4—682no—
221C22⋊C4—682no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23#1058761812yes—
231C23C2×C22⋊Dic1711yescharacteristic
232Q8—82no—
233Q8—82no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—1362no—
256C2×C4—1362no—
257C2×C4—1362no—
258C2×C4#1058750511yescharacteristic
259C2×C4#1058750511yescharacteristic
260C8—1362no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22C2×#1058750511yescharacteristic
269C22#1176281411yescharacteristic
270C22#1176282611yescharacteristic
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—82no—
279C4—82no—
280C4—2722no—
281C2—21no—
282C2—21no—
283C2—42no—
284C2#1452731311yescharacteristic
285C2#1453043811yescharacteristic
286C2#1453045911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2621C2
socleC22×C17C2×#99
prime radicalC24×C17C2×D4
cosocleC2×C22⋊C4×C17C23
hypercenter#2621D17
hypocenterC17#11801
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
11. C2×C17×C4⋊Q8C4semidirect
15. C17×#398C4semidirect
19. C17×#382C4semidirect
142. C2×C4⋊Q8Dic17semidirect
147. #398Dic17semidirect
151. #382Dic17semidirect
191. C17#11801semidirect

Maximal subgroups

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

subgroupindex
#145288612
#145271182
#145267842
C17×#26212
#1180117