Finite Group Databasebeta

← #16944618

Subgroups — #16944618

Order 4352.

448 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 107 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#14534605C211yesmaximal, characteristic
2#14528517C211yesmaximal, characteristic
3#14526809C214yesmaximal
4C17×#1458C211yesmaximal, characteristic
5C2×#10587555C411yescharacteristic, complemented
6C2×#10587612—21no—
7#11762748C412yescomplemented
8#11762745C2212yes—
9#11762729—21no—
10#11762721—21no—
11#11762707—21no—
12#11762675C411yescharacteristic, complemented
13#11762669—21no—
14#11762576—21no—
15#11762576C2212yes—
16C2×C17×#117C2211yescharacteristic
17C17×#431—21no—
18C17×#324—21no—
19C17×#323C2212yes—
20C23×C4×C17D411yescharacteristic
21C22×Dic34—21no—
22C17×#117—41no—
23C17×#117—41no—
24C2×C4⋊C4×C17C2×C411yescharacteristic
25C2×C4⋊C4×C17D411yescharacteristic
26C2×C22⋊C4×C17D411yescharacteristic
27C2×C22⋊C4×C17C2311yescharacteristic
28C2×C42×C17C2×C411yescharacteristic
29C2×C42×C17D411yescharacteristic
30C2×C22⋊Dic17—22no—
31C2×C22⋊Dic17—22no—
32#10587612—24no—
33#10587612—42no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—41no—
37C2×C4⋊Dic17C2×C412yes—
38C2×#63225—22no—
39C2×C4×Dic17—41no—
40#10587555—41no—
41#10587555—41no—
42C17×C4⋊C8—41no—
43C17×C22⋊C8—22no—
44C4×C8×C17—41no—
45#10587502—22no—
46#10587502—21no—
47#10587502—22no—
48#10587502C2×C412yes—
49C24×C17C2×D411yescharacteristic
50C22×C4×C17—21no—
51C22×C4×C17—21no—
52C22×C4×C17—22no—
53C22×C4×C17C22⋊C411yescharacteristic
54C22×C4×C17C22⋊C411yescharacteristic
55C22×C4×C17C22⋊C411yescharacteristic
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17SD1612yes—
58C22×C4×C17Q1612yes—
59C22×C4×C17C22×C411yescharacteristic
60C22×C4×C17C2×D411yescharacteristic
61C22×Dic17—22no—
62C22×Dic17—22no—
63C22×Dic17—42no—
64C2×Dic34—42no—
65C2×Dic34—82no—
66C2×C8×C17—42no—
67C4⋊C4×C17—41no—
68C4⋊C4×C17—41no—
69C22⋊C4×C17—41no—
70C22⋊C4×C17—41no—
71C42×C17—21no—
72C42×C17—21no—
73C42×C17—21no—
74C42×C17—21no—
75C22⋊Dic17—24no—
76C22⋊Dic17—42no—
77C22⋊Dic17—82no—
78C4⋊Dic17—44no—
79C4⋊Dic17—41no—
80C4⋊Dic17—41no—
81C4⋊Dic17—42no—
82C4⋊Dic17—42no—
83#63225—82no—
84C4×Dic17—44no—
85#11718—171nomaximal, Sylow
86C23×C17—21no—
87C23×C17—21no—
88C23×C17—41no—
89C23×C17—41no—
90C23×C17C2×C22⋊C411yescharacteristic
91C23×C17C2×SD1611yescharacteristic
92C23×C17C2×Q1611yescharacteristic
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—41no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17#9911yescharacteristic
108C2×C4×C17#9911yescharacteristic
109C2×C4×C17#10312yes—
110C2×C4×C17#10312yes—
111C2×C4×C17#10412yes—
112C2×Dic17—44no—
113C2×Dic17—42no—
114C2×Dic17—42no—
115C2×Dic17—42no—
116C2×Dic17—42no—
117C2×Dic17—42no—
118C2×Dic17—42no—
119C2×Dic17—82no—
120C2×Dic17—82no—
121Dic34—84no—
122C8×C17—82no—
123#2628—171no—
124#1740—171no—
125#1458D1711yescharacteristic
126#1352—174no—
127C22×C17—21no—
128C22×C17—21no—
129C22×C17—21no—
130C22×C17—21no—
131C22×C17—41no—
132C22×C17—41no—
133C22×C17—41no—
134C22×C17—41no—
135C22×C17—81no—
136C22×C17C2×#9911yescharacteristic
137C22×C17C2×#10311yescharacteristic
138C22×C17C2×#10411yescharacteristic
139C4×C17—22no—
140C4×C17—41no—
141C4×C17—41no—
142C4×C17—41no—
143C4×C17—41no—
144C4×C17#32812yes—
145C4×C17#32812yes—
146Dic17—82no—
147Dic17—84no—
148Dic17—82no—
149C2×C4⋊Q8—171no—
150C2×#117—341no—
151C2×#117C2×D1711yescharacteristic
152#431—21no—
153#400—172no—
154#399—172no—
155#395—341no—
156#393—341no—
157#389—341no—
158#382—171no—
159#380—341no—
160#342—172no—
161#342—341no—
162#324—21no—
163#323C2×D1712yes—
164C2×C17—21no—
165C2×C17—21no—
166C2×C17—41no—
167C2×C17—41no—
168C2×C17C2×#32811yescharacteristic
169C2×C17#212311yescharacteristic
170C2×C17#212811yescharacteristic
171C22×Q8—341no—
172C23×C4D6811yescharacteristic
173C4⋊Q8—681no—
174C4⋊Q8—681no—
175#117—41no—
176#117—41no—
177#117—344no—
178#117—682no—
179C2×C4⋊C4—172no—
180C2×C4⋊C4—341no—
181C2×C4⋊C4—341no—
182C2×C4⋊C4—342no—
183C2×C4⋊C4—681no—
184C2×C4⋊C4C4×D1711yescharacteristic
185C2×C4⋊C4C22⋊D1711yescharacteristic
186C2×C22⋊C4—342no—
187C2×C22⋊C4—342no—
188C2×C22⋊C4D6811yescharacteristic
189C2×C22⋊C4C22×D1711yescharacteristic
190C2×C42—681no—
191C2×C42C4×D1711yescharacteristic
192C2×C42C22⋊D1711yescharacteristic
193C4⋊C8—41no—
194C22⋊C8—22no—
195C4×C8—41no—
196#95—172no—
197#95—342no—
198#95—341no—
199#95—342no—
200C17#1171811yescharacteristic, complemented, Sylow
201C24C2×D6811yescharacteristic
202C2×Q8—682no—
203C2×Q8—1362no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—342no—
208C22×C4—342no—
209C22×C4—682no—
210C22×C4#6321912yes—
211C22×C4Dic6812yes—
212C22×C4#6322711yescharacteristic
213C22×C4#6322711yescharacteristic
214C22×C4#6322711yescharacteristic
215C22×C4#6322711yescharacteristic
216C22×C4C2×C4×D1711yescharacteristic
217C22×C4C2×C22⋊D1711yescharacteristic
218C2×C8—42no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—682no—
222C4⋊C4—684no—
223C4⋊C4—681no—
224C4⋊C4—681no—
225C4⋊C4—682no—
226C4⋊C4—1362no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—344no—
230C22⋊C4—682no—
231C22⋊C4—1362no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—684no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23C2×#6321911yescharacteristic
242C23C2×Dic6811yescharacteristic
243C23C2×#6322711yescharacteristic
244Q8—1364no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—684no—
260C2×C4—682no—
261C2×C4—682no—
262C2×C4—682no—
263C2×C4—682no—
264C2×C4—682no—
265C2×C4—682no—
266C2×C4—1362no—
267C2×C4—1362no—
268C2×C4#1058747711yescharacteristic
269C2×C4#1058747711yescharacteristic
270C2×C4#1058748712yes—
271C2×C4#1058748712yes—
272C2×C4#1058749612yes—
273C8—82no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22C2×#1058747711yescharacteristic
284C22C2×#1058748711yescharacteristic
285C22C2×#1058749611yescharacteristic
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—1362no—
292C4—1364no—
293C4—1362no—
294C4#1176254712yes—
295C4#1176254712yes—
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2C2×#1176254711yescharacteristic
301C2#1453043911yescharacteristic
302C2#1453045511yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1458C2
socleC22×C17C2×#99
prime radicalC24×C17C2×D4
cosocleC2×C22⋊C4×C17C23
hypercenter#1458D17
hypocenterC17#11718
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
5. C2×#10587555C4semidirect
7. #11762748C4semidirect
12. #11762675C4semidirect
200. C17#11718semidirect

Maximal subgroups

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

subgroupindex
#145346052
#145285172
#145268092
C17×#14582
#1171817