Finite Group Databasebeta

← #16944539

Subgroups — #16944539

Order 4352.

452 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 115 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#14528523C211yesmaximal, characteristic
2#14527098C211yesmaximal, characteristic
3#14526790C214yesmaximal, complemented
4C17×#2621C211yesmaximal, characteristic
5C22×#63223—21no—
6#11762862—21no—
7#11762578—21no—
8#11762578C2212yes—
9#11762535C2212yes—
10#11762533—21no—
11C2×C17×C4⋊Q8C411yescharacteristic
12C2×C17×C22⋊Q8—21no—
13C2×C17×#117C2211yescharacteristic
14C17×#398—21no—
15C17×#398C2212yes—
16C17×#397—21no—
17C17×#397C412yes—
18C17×#391—21no—
19C17×#382C411yescharacteristic
20C22×Q8×C17—21no—
21C23×C4×C17D411yescharacteristic
22C17×C4⋊Q8—41no—
23C17×C4⋊Q8—41no—
24C17×C22⋊Q8—42no—
25C17×C22⋊Q8—42no—
26C17×#117—41no—
27C17×#117—41no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—22no—
31C2×C4⋊C4×C17C2×C412yes—
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C4⋊C4×C17C2311yescharacteristic
34C2×C22⋊C4×C17—22no—
35C2×C22⋊C4×C17C2×C411yescharacteristic
36C2×C22⋊C4×C17D411yescharacteristic
37C2×C42×C17C2×C411yescharacteristic
38C2×C42×C17D411yescharacteristic
39C2×#63223—24no—
40C2×#63223—22no—
41C2×#63223—41no—
42C2×#63223—42no—
43C22×#3622—41no—
44C17×#95—21no—
45C17×#95—22no—
46C17×#95—22no—
47C17×#95C2×C412yes—
48C22⋊#3622—22no—
49C4⋊#3622—41no—
50C4×#3622—41no—
51C24×C17C22⋊C411yescharacteristic
52C2×Q8×C17—41no—
53C2×Q8×C17—41no—
54C2×Q8×C17—42no—
55C2×Q8×C17—42no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—22no—
59C22×C4×C17—22no—
60C22×C4×C17—22no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17D812yes—
65C22×C4×C17SD1612yes—
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C4⋊C4×C17—41no—
70C4⋊C4×C17—41no—
71C4⋊C4×C17—42no—
72C4⋊C4×C17—42no—
73C4⋊C4×C17—42no—
74C4⋊C4×C17—42no—
75C4⋊C4×C17—42no—
76C22⋊C4×C17—41no—
77C22⋊C4×C17—41no—
78C22⋊C4×C17—44no—
79C42×C17—21no—
80C42×C17—21no—
81C42×C17—21no—
82C42×C17—21no—
83#63223—24no—
84#63223—44no—
85#63223—42no—
86C2×#3622—42no—
87C2×#3622—44no—
88C2×#3622—42no—
89#11700—171nomaximal, Sylow
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17#10212yes—
95C23×C17C2×C22⋊C411yescharacteristic
96Q8×C17—82no—
97Q8×C17—82no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C2×C4×C17—22no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17#9911yescharacteristic
119C2×C4×C17#9911yescharacteristic
120C2×C4×C17#10212yes—
121C2×C4×C17#10412yes—
122C2×C4×C17C2×D811yescharacteristic
123C2×C4×C17C2×SD1611yescharacteristic
124#3622—44no—
125#3622—82no—
126#2621D1711yescharacteristic
127#1742—171no—
128#1458—171no—
129#1346—174no—
130C22×C17—21no—
131C22×C17—21no—
132C22×C17—22no—
133C22×C17—42no—
134C22×C17—41no—
135C22×C17—41no—
136C22×C17—81no—
137C22×C17C2×#9911yescharacteristic
138C22×C17C2×#10211yescharacteristic
139C22×C17C2×#10411yescharacteristic
140C4×C17—21no—
141C4×C17—21no—
142C4×C17—41no—
143C4×C17—41no—
144C4×C17—41no—
145C4×C17—41no—
146C4×C17—82no—
147C4×C17—82no—
148C4×C17#32712yes—
149C4×C17#32712yes—
150C22×M16—341no—
151C2×C4⋊Q8Dic1711yescharacteristic
152C2×C22⋊Q8—21no—
153C2×#117C2×D1711yescharacteristic
154#431—341no—
155#398—21no—
156#398C2×D1712yes—
157#397—21no—
158#397Dic1712yes—
159#391—21no—
160#382Dic1711yescharacteristic
161#343—172no—
162#343—341no—
163#324—172no—
164#323—341no—
165C2×C17—21no—
166C2×C17—21no—
167C2×C17—42no—
168C2×C17C2×#32711yescharacteristic
169C2×C17#212311yescharacteristic
170C2×C17#213511yescharacteristic
171C22×Q8—21no—
172C23×C4C22⋊D1711yescharacteristic
173C2×M16—344no—
174C2×M16—342no—
175C2×M16—681no—
176C2×M16—682no—
177C22×C8—681no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182#117—41no—
183#117—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4C2×Dic1712yes—
188C2×C4⋊C4C22⋊D1711yescharacteristic
189C2×C4⋊C4C22×D1711yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4C2×Dic1711yescharacteristic
192C2×C22⋊C4C22⋊D1711yescharacteristic
193C2×C42C2×Dic1711yescharacteristic
194C2×C42C22⋊D1711yescharacteristic
195C4⋊C8—681no—
196C22⋊C8—342no—
197C4×C8—681no—
198#95—21no—
199#95—22no—
200#95—22no—
201#95C2×Dic1712yes—
202C17#1170011yescharacteristic, complemented, Sylow
203C24C22⋊Dic1711yescharacteristic
204C2×Q8—42no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—42no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4#6322812yes—
214C22×C4#6322912yes—
215C22×C4C22⋊Dic1711yescharacteristic
216C22×C4C22⋊Dic1711yescharacteristic
217C22×C4C22⋊Dic1711yescharacteristic
218C22×C4C22×Dic1711yescharacteristic
219C22×C4C2×C22⋊D1711yescharacteristic
220C22×C4C2×C22⋊D1711yescharacteristic
221M16—344no—
222M16—684no—
223M16—682no—
224C2×C8—682no—
225C2×C8—684no—
226C2×C8—682no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—44no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23#1058750312yes—
246C23C2×C22⋊Dic1711yescharacteristic
247Q8—82no—
248Q8—82no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4#1058750312yes—
270C2×C4#1058750511yescharacteristic
271C2×C4#1058750511yescharacteristic
272C2×C4#1058750812yes—
273C2×C4C2×#6322811yescharacteristic
274C2×C4C2×#6322911yescharacteristic
275C8—684no—
276C8—1362no—
277C22—21no—
278C22—21no—
279C22—22no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—81no—
284C22C2×#1058750511yescharacteristic
285C22C2×#1058750311yescharacteristic
286C22C2×#1058750811yescharacteristic
287C4—21no—
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—82no—
294C4—82no—
295C4#1176254312yes—
296C4#1176254312yes—
297C2—21no—
298C2—21no—
299C2—42no—
300C2C2×#1176254311yescharacteristic
301C2#1453043811yescharacteristic
302C2#1453047911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2621C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#2621D17
hypocenterC17#11700
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
3. #14526790C2semidirect
202. C17#11700semidirect

Maximal subgroups

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

subgroupindex
#145285232
#145270982
#145267902
C17×#26212
#1170017