Finite Group Databasebeta

← #17981865

Subgroups — #17981865

Order 4352.

666 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 97 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#14545951C212yesmaximal, complemented
2#14545806C214yesmaximal, complemented
3C22×#10587543C211yesmaximal, characteristic
4C2×#10587670—22no—
5C22⋊#63241—22no—
6C22⋊#63241C2214yescomplemented
7C23×#3624C411yescharacteristic
8C23×#3624C2211yescharacteristic
9C2×#10587543—22no—
10C2×#10587543—22no—
11C2×#10587543C412yes—
12C2×#10587543C2212yes—
13C23×C4×D17C411yescharacteristic
14C24×D17C811yescharacteristic
15C23×C4×C17C811yescharacteristic
16C23×Dic17C2×C411yescharacteristic
17C22×C4×D17—21no—
18C22×C4×D17—21no—
19C22×C4×D17—21no—
20C22×C4×D17—21no—
21C22×C4×D17—42no—
22C22×C4×D17—42no—
23C22×C4×D17C812yes—
24C22×C4×D17C2×C412yes—
25C22×#3624—21no—
26C22×#3624—21no—
27C22×#3624—42no—
28C22×#3624—42no—
29C22×#3624C2×C412yes—
30C22×#3624C2×C411yescharacteristic
31C22×#3624D412yes—
32C22×#3624D412yes—
33C22×#3624C2311yescharacteristic
34#10587670—44no—
35C2×#63241—24no—
36#10587543—24no—
37#10587543—24no—
38C24×C17C2×C811yescharacteristic
39C23×D17—21no—
40C23×D17—21no—
41C23×D17—24no—
42C23×D17—42no—
43C23×D17—42no—
44C23×D17C2×C812yes—
45C22×C4×C17—21no—
46C22×C4×C17—21no—
47C22×C4×C17—42no—
48C22×C4×C17C2×C812yes—
49C22×Dic17—22no—
50C22×Dic17—22no—
51C22×Dic17C22⋊C411yescharacteristic
52C22×Dic17C22⋊C411yescharacteristic
53C22×Dic17C2×C811yescharacteristic
54C22×Dic17M1611yescharacteristic
55C22×Dic17M1611yescharacteristic
56C22×Dic17C22×C411yescharacteristic
57C2×C4×D17—22no—
58C2×C4×D17—22no—
59C2×C4×D17—22no—
60C2×C4×D17—22no—
61C2×C4×D17—44no—
62C2×C4×D17—44no—
63C2×C4×D17—42no—
64C2×C4×D17—42no—
65C2×C4×D17—42no—
66C2×C4×D17—42no—
67C2×C4×D17—44no—
68C2×C4×D17—42no—
69C2×C4×D17—42no—
70C2×#3624—24no—
71C2×#3624—22no—
72C2×#3624—41no—
73C2×#3624—42no—
74C2×#3624—41no—
75C2×#3624—42no—
76C2×#3624—42no—
77C2×#3624—42no—
78C2×#3624C22⋊C412yes—
79C2×#3624C2×D412yes—
80#63241—44no—
81#11418—171nomaximal, Sylow
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—22no—
85C23×C17C22⋊C811yescharacteristic
86C23×C17C22⋊C811yescharacteristic
87C23×C17C22×C811yescharacteristic
88C22×D17—24no—
89C22×D17—24no—
90C22×D17—22no—
91C22×D17—22no—
92C22×D17—42no—
93C22×D17—416no—
94C22×D17—42no—
95C22×D17—42no—
96C22×D17—42no—
97C22×D17—42no—
98C22×D17—42no—
99C22×D17M3214yescomplemented
100C2×C4×C17—22no—
101C2×C4×C17—42no—
102C2×C4×C17—42no—
103C2×C4×C17—42no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—21no—
108C2×Dic17—21no—
109C2×Dic17—21no—
110C2×Dic17—21no—
111C2×Dic17C22⋊C811yescharacteristic
112C2×Dic17C22⋊C811yescharacteristic
113C2×Dic17#9912yes—
114C2×Dic17#10012yes—
115C2×Dic17C2×C22⋊C411yescharacteristic
116C2×Dic17C2×M1611yescharacteristic
117C4×D17—44no—
118C4×D17—44no—
119C4×D17—44no—
120C4×D17—42no—
121C4×D17—42no—
122#3624—22no—
123#3624—22no—
124#3624—42no—
125C22×C22⋊C8—171no—
126#2112—172no—
127#1314—174no—
128C22×C17—28no—
129C22×C17—21no—
130C22×C17—21no—
131C22×C17—21no—
132C22×C17—21no—
133C22×C17—21no—
134C22×C17—21no—
135C22×C17#32312yes—
136C22×C17#34912yes—
137C22×C17C2×C22⋊C811yescharacteristic
138C22×C17C2×M3212yes—
139C2×D17—24no—
140C2×D17—48no—
141C2×D17—48no—
142C2×D17—44no—
143C2×D17—42no—
144C2×D17—42no—
145C2×D17—44no—
146C4×C17—42no—
147Dic17—21no—
148Dic17—21no—
149Dic17#32312yes—
150Dic17C2×#9911yescharacteristic
151Dic17C2×#10011yescharacteristic
152C24×C4—171no—
153C23×C8—171no—
154C23×C8—171no—
155C2×M32—342no—
156C2×C22⋊C8—172no—
157C2×C22⋊C8—172no—
158C2×C22⋊C8—342no—
159C2×C22⋊C8—342no—
160C22⋊C16—174no—
161C22⋊C16—342no—
162C2×C17—22no—
163C2×C17—22no—
164C2×C17—22no—
165C2×C17C2×#32311yescharacteristic
166C2×C17#211211yescharacteristic
167C2×C17C2×#34911yescharacteristic
168D17—44no—
169C25—171no—
170C23×C4—171no—
171C23×C4—172no—
172C23×C4—172no—
173C23×C4—341no—
174C23×C4—341no—
175C23×C4—341no—
176C23×C4—341no—
177C23×C4—682no—
178C23×C4—682no—
179C23×C4#363311yescharacteristic
180C22×C8—172no—
181C22×C8—171no—
182C22×C8—172no—
183C22×C8—171no—
184C22×C8—172no—
185C22×C8—341no—
186C22×C8—341no—
187C22×C8—682no—
188C22×C8—682no—
189M32—684no—
190C2×C16—344no—
191C22⋊C8—344no—
192C22⋊C8—344no—
193C17#1141811yescharacteristic, complemented, Sylow
194C24—172no—
195C24—344no—
196C24—341no—
197C24—341no—
198C24—682no—
199C24—682no—
200C24C2×#363311yescharacteristic
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—42no—
204C22×C4—171no—
205C22×C4—171no—
206C22×C4—171no—
207C22×C4—171no—
208C22×C4—171no—
209C22×C4—171no—
210C22×C4—342no—
211C22×C4—342no—
212C22×C4—342no—
213C22×C4—342no—
214C22×C4—342no—
215C22×C4—342no—
216C22×C4—682no—
217C22×C4—682no—
218C22×C4—682no—
219C22×C4—682no—
220C22×C4—684no—
221C22×C4—684no—
222C22×C4—682no—
223C22×C4—682no—
224C22×C4—684no—
225C22×C4C2×#363312yes—
226C2×C8—172no—
227C2×C8—172no—
228C2×C8—344no—
229C2×C8—342no—
230C2×C8—681no—
231C2×C8—682no—
232C2×C8—681no—
233C2×C8—682no—
234C2×C8—682no—
235C2×C8—682no—
236C16—684no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—174no—
241C23—344no—
242C23—344no—
243C23—342no—
244C23—342no—
245C23—682no—
246C23—682no—
247C23—682no—
248C23—6816no—
249C23—682no—
250C23—682no—
251C23—682no—
252C23C22⋊#363311yescharacteristic
253C23C22⋊#363311yescharacteristic
254C23C22×#363311yescharacteristic
255C2×C4—22no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—172no—
260C2×C4—172no—
261C2×C4—171no—
262C2×C4—171no—
263C2×C4—171no—
264C2×C4—171no—
265C2×C4—341no—
266C2×C4—341no—
267C2×C4—341no—
268C2×C4—341no—
269C2×C4—342no—
270C2×C4—342no—
271C2×C4—342no—
272C2×C4—682no—
273C2×C4—682no—
274C2×C4—684no—
275C2×C4—684no—
276C2×C4—684no—
277C8—342no—
278C8—342no—
279C8—682no—
280C22—28no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—344no—
288C22—684no—
289C22—688no—
290C22—688no—
291C22—684no—
292C22—682no—
293C22—682no—
294C22#1176389412yes—
295C22#1176390812yes—
296C22C2×C22⋊#363311yescharacteristic
297C22#1176392812yes—
298C4—42no—
299C4—171no—
300C4—171no—
301C4—172no—
302C4—341no—
303C4—341no—
304C2—22no—
305C2—22no—
306C2—22no—
307C2—684no—
308C2C2×#1176389411yescharacteristic
309C2#1454595311yescharacteristic
310C2C2×#1176390811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#3633
commutatorC23×C17C22×C8
FrattiniC23C22×#3633
FittingC23×C4×C17C8
socleC22×C17C2×C22⋊C8
prime radicalC24×D17C8
cosocleC22×#3624C23
hypercenterC23×C4#3633
hypocenterC17#11418
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. #14545951C2semidirect
2. #14545806C2semidirect
6. C22⋊#63241C22semidirect
99. C22×D17M32semidirect
193. C17#11418semidirect

Maximal subgroups

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

subgroupindex
#145459512
#145458062
C22×#105875432
#1141817