Finite Group Databasebeta

← #17044765

Subgroups — #17044765

Order 4352.

946 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊#3622C211yesmaximal, characteristic
2C4×#10587612C211yesmaximal, characteristic
3#14529502C214yesmaximal, complemented
4#14528961C214yesmaximal
5#14528935C212yesmaximal, complemented
6#14528900C212yesmaximal, complemented
7C2×C17×#431C211yesmaximal, characteristic
8C23×#3622C2212yes—
9C2×#10587612C411yescharacteristic
10C2×#10587612C411yescharacteristic
11C2×C4⋊#3622—24no—
12C2×C4⋊#3622C2212yes—
13C2×C4⋊#3622C2212yes—
14C2×#10587485C2218yes—
15C4×C4⋊Dic17C412yes—
16C4×C4⋊Dic17C2212yes—
17C4×C22⋊Dic17C412yes—
18C4×C22⋊Dic17C2212yes—
19#11762655C412yes—
20C42×Dic17C2212yes—
21#11762563C2218yes—
22C22×C42×C17C2211yescharacteristic
23C17×#431—24no—
24C2×C17×C4⋊C8C2211yescharacteristic
25C2×C17×C4⋊C8C2211yescharacteristic
26C2×C17×C22⋊C8C2212yes—
27C2×C4×C8×C17C2212yes—
28C23×C4×C17C2×C411yescharacteristic
29C23×C4×C17C2×C411yescharacteristic
30C23×C4×C17C2311yescharacteristic
31C22×C8×C17C2312yes—
32C22×C8×C17C2312yes—
33C2×C42×C17C2×C411yescharacteristic
34C2×C42×C17C2×C411yescharacteristic
35C2×C42×C17D414yes—
36C2×C42×C17Q814yes—
37C2×C42×C17C2311yescharacteristic
38C2×C42×C17C2311yescharacteristic
39C2×C22⋊Dic17C2×C412yes—
40C2×C22⋊Dic17C2×C412yes—
41#10587612—24no—
42#10587612C818yescomplemented
43C2×C4⋊Dic17C2×C412yes—
44C2×C4⋊Dic17C2×C412yes—
45C2×C4×Dic17C2×C412yes—
46C2×C4×Dic17C2×C414yes—
47C2×C4×Dic17C2×C412yes—
48C2×C4×Dic17C2314yes—
49C22×#3622—24no—
50C22×#3622—44no—
51C22×#3622C2314yes—
52C17×C4⋊C8—22no—
53C17×C4⋊C8—22no—
54C17×C22⋊C8—24no—
55C4×C8×C17—24no—
56#10587502C2×C414yes—
57#10587502C2×C414yes—
58#10587485—48no—
59C4⋊#3622—24no—
60C4⋊#3622—24no—
61C24×C17C22×C411yescharacteristic
62C22×C4×C17—22no—
63C22×C4×C17—22no—
64C22×C4×C17—42no—
65C22×C4×C17C4⋊C414yes—
66C22×C4×C17C4⋊C414yes—
67C22×C4×C17C2×C814yes—
68C22×C4×C17M1614yes—
69C22×C4×C17C22×C411yescharacteristic
70C22×C4×C17C22×C411yescharacteristic
71C22×C4×C17C22×C411yescharacteristic
72C22×C4×C17C22×C411yescharacteristic
73C22×C4×C17C22×C411yescharacteristic
74C22×C4×C17C2×D412yes—
75C22×C4×C17C2×Q812yes—
76C22×C4×C17C2411yescharacteristic
77C22×Dic17C22×C414yes—
78C22×Dic17C22×C414yes—
79C2×C8×C17—22no—
80C2×C8×C17—22no—
81C2×C8×C17—24no—
82C2×C8×C17—24no—
83C42×C17C4⋊C414yes—
84C42×C17C4⋊C414yes—
85C42×C17C2×D412yes—
86C42×C17C2×D412yes—
87C42×C17C2×Q812yes—
88C42×C17C2×Q812yes—
89C22⋊Dic17—24no—
90C22⋊Dic17C2×C818yes—
91C4⋊Dic17—24no—
92C4⋊Dic17C2×C818yescomplemented
93C4×Dic17—24no—
94C4×Dic17—24no—
95C4×Dic17—24no—
96C4×Dic17—44no—
97C4×Dic17C2×C818yescomplemented
98C2×#3622—24no—
99C2×#3622—28no—
100C2×#3622—48no—
101#21972—171nomaximal, Sylow
102C23×C17—22no—
103C23×C17—22no—
104C23×C17C2×C4⋊C412yes—
105C23×C17C22×C812yes—
106C23×C17C2×M1612yes—
107C23×C17C23×C411yescharacteristic
108C2×C4×C17—22no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
112C2×C4×C17—22no—
113C2×C4×C17—24no—
114C2×C4×C17—44no—
115C2×C4×C17C4⋊C8116yescomplemented
116C2×C4×C17C2×C4⋊C412yes—
117C2×C4×C17C2×C4⋊C412yes—
118C2×C4×C17C2×C4⋊C412yes—
119C2×C4×C17C2×C4⋊C412yes—
120C2×C4×C17C2×C4⋊C412yes—
121C2×C4×C17C22×C812yes—
122C2×C4×C17C22×C812yes—
123C2×C4×C17C2×M1612yes—
124C2×C4×C17C2×M1612yes—
125C2×C4×C17C22×D411yescharacteristic
126C2×C4×C17C22×Q811yescharacteristic
127C2×C4×C17H(5,𝔽2)12yes—
128C2×C4×C172−1+412yes—
129C2×Dic17—24no—
130C2×Dic17—24no—
131C2×Dic17—24no—
132C2×Dic17—24no—
133C2×Dic17—24no—
134C2×Dic17C22×C818yes—
135C8×C17—42no—
136C8×C17—42no—
137#3622—44no—
138C2×#431D1711yescharacteristic
139C22×C4⋊C8—171no—
140C4×#117—171no—
141#1939—174no—
142#1850—174no—
143#1842—172no—
144#1831—172no—
145C22×C17—22no—
146C22×C17—22no—
147C22×C17—24no—
148C22×C17—24no—
149C22×C17C2×C4⋊C814yes—
150C22×C17C22×C4⋊C411yescharacteristic
151C22×C17#52012yes—
152C22×C17C23×C811yescharacteristic
153C22×C17C22×M1611yescharacteristic
154C22×C17#56812yes—
155C4×C17—22no—
156C4×C17—22no—
157C4×C17—22no—
158C4×C17C2×C4⋊C814yes—
159C4×C17C2×C4⋊C814yes—
160C4×C17#53511yescharacteristic
161C4×C17#53611yescharacteristic
162C4×C17#54412yes—
163Dic17—28no—
164Dic17—44no—
165C23×C8—172no—
166C2×#117—171no—
167C2×#117—171no—
168C22×C42C2×D1711yescharacteristic
169#431—24no—
170C2×C4⋊C8—172no—
171C2×C4⋊C8—172no—
172C2×C4⋊C8—178no—
173C2×C4⋊C8—344no—
174C2×C4⋊C8C2×D1711yescharacteristic
175C2×C4⋊C8C2×D1711yescharacteristic
176C2×C22⋊C8C2×D1712yes—
177C2×C4×C8C2×D1712yes—
178C4×C4⋊C4—172no—
179C4×C4⋊C4—172no—
180C4×C22⋊C4—172no—
181C4×C22⋊C4—172no—
182#376—172no—
183C43—172no—
184#336—178no—
185C2×C17—24no—
186C2×C17#231711yescharacteristic
187C2×C17C22×C4⋊C811yescharacteristic
188C2×C17#290511yescharacteristic
189C2×C17#298812yes—
190C2×C17#298912yes—
191C23×C4C4×D1711yescharacteristic
192C23×C4C4×D1711yescharacteristic
193C23×C4C22×D1711yescharacteristic
194C22×C8—174no—
195C22×C8—344no—
196C22×C8—684no—
197C22×C8C22×D1712yes—
198C22×C8C22×D1712yes—
199#117—178no—
200#117—344no—
201C2×C4⋊C4—172no—
202C2×C4⋊C4—172no—
203C2×C22⋊C4—172no—
204C2×C22⋊C4—172no—
205C2×C42—172no—
206C2×C42—172no—
207C2×C42—174no—
208C2×C42—174no—
209C2×C42Dic3414yes—
210C2×C42C4×D1711yescharacteristic
211C2×C42C4×D1711yescharacteristic
212C2×C42C22⋊D1714yes—
213C2×C42C22×D1711yescharacteristic
214C2×C42C22×D1711yescharacteristic
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—344no—
218C4⋊C8—344no—
219C4⋊C8—688no—
220C22⋊C8—24no—
221C4×C8—24no—
222#95—174no—
223#95—174no—
224C17#2197211yescharacteristic, complemented, Sylow
225C24C2×C4×D1711yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—174no—
230C22×C4—174no—
231C22×C4C8×D1714yes—
232C22×C4#6321814yes—
233C22×C4#6322514yes—
234C22×C4#6322514yes—
235C22×C4C2×Dic3412yes—
236C22×C4C2×C4×D1711yescharacteristic
237C22×C4C2×C4×D1711yescharacteristic
238C22×C4C2×C4×D1711yescharacteristic
239C22×C4C2×C4×D1711yescharacteristic
240C22×C4C2×C4×D1711yescharacteristic
241C22×C4C2×C22⋊D1712yes—
242C22×C4C23×D1711yescharacteristic
243C2×C8—24no—
244C2×C8—24no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—348no—
248C2×C8—344no—
249C2×C8—688no—
250C4⋊C4—178no—
251C4⋊C4—344no—
252C22⋊C4—178no—
253C22⋊C4—344no—
254C42—178no—
255C42—344no—
256C42—344no—
257C42—344no—
258C42—684no—
259C42#6322514yes—
260C42#6322514yes—
261C42C2×Dic3412yes—
262C42C2×Dic3412yes—
263C42C2×C22⋊D1712yes—
264C42C2×C22⋊D1712yes—
265C23—22no—
266C23—22no—
267C23C2×C8×D1712yes—
268C23C2×#6321812yes—
269C23C2×#6322512yes—
270C23C22×C4×D1711yescharacteristic
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—44no—
278C2×C4—178no—
279C2×C4—344no—
280C2×C4—344no—
281C2×C4—344no—
282C2×C4—344no—
283C2×C4—344no—
284C2×C4#10587485116yescomplemented
285C2×C4C2×C8×D1712yes—
286C2×C4C2×C8×D1712yes—
287C2×C4C2×#6321812yes—
288C2×C4C2×#6321812yes—
289C2×C4C2×#6322512yes—
290C2×C4C2×#6322512yes—
291C2×C4C2×#6322512yes—
292C2×C4C2×#6322512yes—
293C2×C4C2×#6322512yes—
294C2×C4C22×Dic3411yescharacteristic
295C2×C4#1058769612yes—
296C2×C4#1058769712yes—
297C2×C4C22×C22⋊D1711yescharacteristic
298C8—42no—
299C8—42no—
300C8—684no—
301C22—22no—
302C22—22no—
303C22—24no—
304C22—24no—
305C22C2×#1058748514yes—
306C22C22×#6322511yescharacteristic
307C22#1176328912yes—
308C22C22×C8×D1711yescharacteristic
309C22C22×#6321811yescharacteristic
310C22#1176357512yes—
311C4—22no—
312C4—22no—
313C4—22no—
314C4—348no—
315C4—684no—
316C4C2×#1058748514yes—
317C4C2×#1058748514yes—
318C4#1176337311yescharacteristic
319C4#1176338011yescharacteristic
320C4#1176343712yes—
321C2—24no—
322C2#1453140411yescharacteristic
323C2C22×#1058748511yescharacteristic
324C2#1453734711yescharacteristic
325C2#1453789212yes—
326C2#1453789712yes—
subgroupG/H
center, FrattiniC22×C4C23×D17
commutatorC22×C17C23×C8
FittingC2×C17×#431C2
socleC23×C17C23×C4
prime radicalC24×C17C22×C4
cosocleC22×C4×C17C24
hypercenterC2×#431D17
hypocenterC17#21972
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C2×#431

Complements

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

NH ≅ G/Ntype
3. #14529502C2semidirect
5. #14528935C2semidirect
6. #14528900C2semidirect
42. #10587612C8semidirect
92. C4⋊Dic17C2×C8semidirect
97. C4×Dic17C2×C8semidirect
115. C2×C4×C17C4⋊C8semidirect
224. C17#21972semidirect
284. C2×C4#10587485semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#36222
C4×#105876122
#145295022
#145289612
#145289352
#145289002
C2×C17×#4312
#2197217