Finite Group Databasebeta

← #14534527

Subgroups — #14534527

Order 2176.

632 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#10587580C212yesmaximal, complemented
2C2×#10587581C211yesmaximal, characteristic
3C2×C4⋊C4×D17C211yesmaximal, characteristic, complemented
4#11762740C211yesmaximal, characteristic, complemented
5#11762733C212yesmaximal, complemented
6#11762731C212yesmaximal, complemented
7#11762727C211yesmaximal, characteristic, complemented
8#11762726C211yesmaximal, characteristic, complemented
9#11762687C212yesmaximal, complemented
10#11762668C211yesmaximal, characteristic, complemented
11C17×#398C211yesmaximal, characteristic, complemented
12C22×D68C2212yes—
13C22×D68C2211yescharacteristic
14C22×C4×D17C2211yescharacteristic
15C22×C4×D17C2211yescharacteristic
16C22×C4×D17C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C4⋊C4×C17C2212yes—
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×C4⋊Dic17C2211yescharacteristic
27C2×#63225C2211yescharacteristic
28C2×#63225C2211yescharacteristic
29C2×#63225C2212yes—
30C2×#63225C2212yes—
31#10587581—24no—
32#10587580—28no—
33C4⋊C4×D17—24no—
34C17×#95C2211yescharacteristic
35C17×#95C2212yes—
36#10587502C2211yescharacteristic
37#10587502C2212yes—
38C23×D17C2312yes—
39C23×D17C2311yescharacteristic
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2312yes—
42C22×C4×C17C2312yes—
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2312yes—
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2311yescharacteristic
49C2×D68—24no—
50C2×D68—24no—
51C2×D68—24no—
52C2×D68—22no—
53C2×D68—42no—
54C2×C4×D17—22no—
55C2×C4×D17—22no—
56C2×C4×D17—22no—
57C2×C4×D17—42no—
58C2×C4×D17—42no—
59C2×C4×D17D414yes—
60C2×C4×D17D414yescomplemented
61C4⋊C4×C17—22no—
62C4⋊C4×C17—24no—
63C4⋊C4×C17—22no—
64#63227—24no—
65#63227—24no—
66#63227—24no—
67#63227—24no—
68#63227—42no—
69#63227—42no—
70C4⋊Dic17—22no—
71#63225—22no—
72#63225—22no—
73#63225—24no—
74#63225—42no—
75C23×C17C2411yescharacteristic
76C22×D17—22no—
77C22×D17—22no—
78C22×D17—22no—
79C22×D17—22no—
80C22×D17—22no—
81C22×D17—22no—
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—22no—
85C22×D17C2×D412yes—
86C22×D17C2×D412yes—
87C22×D17#4112yes—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—22no—
91C2×C4×C17—21no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17C2×D412yes—
100C2×C4×C17C2×D412yes—
101C2×Dic17—21no—
102C2×Dic17—21no—
103C2×Dic17—22no—
104C2×Dic17—21no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17C2×D412yes—
110C2×Dic17C2×D412yes—
111D68—44no—
112D68—48no—
113C4×D17—28no—
114C4×D17—44no—
115C4×D17—44no—
116#2618—171nomaximal, Sylow
117C22×C17C22×D411yescharacteristic
118C22×C17C22×D411yescharacteristic
119C22×C17C2×#4111yescharacteristic
120C22×C17H(5,𝔽2)12yes—
121C22×C172−1+412yes—
122C2×D17—22no—
123C2×D17—22no—
124C2×D17—24no—
125C2×D17—24no—
126C2×D17—42no—
127C2×D17—42no—
128C2×D17—42no—
129C2×D17—42no—
130C2×D17—42no—
131C2×D17—42no—
132C2×D17—42no—
133C2×D17#12114yes—
134C4×C17—22no—
135C4×C17—22no—
136C4×C17—42no—
137C4×C17—41no—
138C4×C17—42no—
139Dic17—22no—
140Dic17—22no—
141Dic17—41no—
142Dic17—42no—
143C2×#123—172no—
144C2×#121—171no—
145C22×C4⋊C4—171no—
146#398—171no—
147#398D1711yescharacteristic, complemented
148#396—172no—
149#396—172no—
150#394—171no—
151#394—171no—
152#386—172no—
153#380—171no—
154C2×C17C2×#12111yescharacteristic
155C2×C17#53411yescharacteristic
156C2×C17#53611yescharacteristic
157C2×C17#53712yes—
158C2×C17#53912yes—
159D17—24no—
160D17—82no—
161C22×D4—172no—
162C22×D4—171no—
163C23×C4—171no—
164C23×C4—171no—
165C23×C4—171no—
166#123—348no—
167#121—344no—
168C2×C4⋊C4—171no—
169C2×C4⋊C4—171no—
170C2×C4⋊C4—171no—
171C2×C4⋊C4—172no—
172C2×C4⋊C4—172no—
173C2×C4⋊C4—344no—
174C2×C4⋊C4C2×D1711yescharacteristic
175C2×C4⋊C4C2×D1711yescharacteristic
176C2×C4⋊C4C2×D1712yes—
177C2×C22⋊C4—172no—
178C2×C22⋊C4—172no—
179C2×C22⋊C4—172no—
180C2×C22⋊C4—172no—
181C2×C22⋊C4—172no—
182C2×C22⋊C4—172no—
183#95—171no—
184#95—172no—
185#95C2×D1711yescharacteristic
186#95C2×D1712yes—
187C17#261811yescharacteristic, complemented, Sylow
188C24—171no—
189C24—172no—
190C2×D4—344no—
191C2×D4—344no—
192C2×D4—344no—
193C2×D4—342no—
194C2×D4—682no—
195C22×C4—171no—
196C22×C4—171no—
197C22×C4—174no—
198C22×C4—174no—
199C22×C4—172no—
200C22×C4—171no—
201C22×C4—342no—
202C22×C4—342no—
203C22×C4—342no—
204C22×C4—682no—
205C22×C4—682no—
206C22×C4C22×D1712yes—
207C22×C4C22×D1712yes—
208C22×C4C22×D1711yescharacteristic
209C22×C4C22×D1711yescharacteristic
210C22×C4C22×D1711yescharacteristic
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—24no—
214C4⋊C4—342no—
215C4⋊C4—342no—
216C4⋊C4—344no—
217C4⋊C4—342no—
218C4⋊C4—682no—
219C22⋊C4—344no—
220C22⋊C4—344no—
221C22⋊C4—344no—
222C22⋊C4—344no—
223C22⋊C4—682no—
224C22⋊C4—682no—
225C23—172no—
226C23—172no—
227C23—172no—
228C23—342no—
229C23—342no—
230C23—342no—
231C23—342no—
232C23—342no—
233C23—342no—
234C23—342no—
235C23—342no—
236C23—342no—
237C23C23×D1711yescharacteristic
238D4—684no—
239D4—688no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—172no—
252C2×C4—172no—
253C2×C4—348no—
254C2×C4—341no—
255C2×C4—341no—
256C2×C4—342no—
257C2×C4—342no—
258C2×C4—341no—
259C2×C4—342no—
260C2×C4—342no—
261C2×C4—342no—
262C2×C4—684no—
263C2×C4—684no—
264C2×C4D4×D1712yes—
265C2×C4D4×D1712yes—
266C22—174no—
267C22—342no—
268C22—342no—
269C22—344no—
270C22—344no—
271C22—682no—
272C22—682no—
273C22—682no—
274C22—682no—
275C22—682no—
276C22—682no—
277C22—682no—
278C22C2×D4×D1711yescharacteristic
279C22C2×D4×D1711yescharacteristic
280C22#1058769412yes—
281C22#41×D1711yescharacteristic
282C22#1058769612yes—
283C4—22no—
284C4—22no—
285C4—42no—
286C4—41no—
287C4—42no—
288C4—342no—
289C4—342no—
290C4—681no—
291C4—682no—
292C2—344no—
293C2—1362no—
294C2#121×D1711yescharacteristic
295C2#1176337011yescharacteristic
296C2#1176338411yescharacteristic
297C2#1176338912yes—
298C2#1176340712yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#398C2
prime radicalC2×#10587581C2
hypercenter#398D17
hypocenterC17#2618
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#10587580C2semidirect
3. C2×C4⋊C4×D17C2semidirect
4. #11762740C2semidirect
5. #11762733C2semidirect
6. #11762731C2semidirect
7. #11762727C2semidirect
8. #11762726C2semidirect
9. #11762687C2semidirect
10. #11762668C2semidirect
11. C17×#398C2semidirect
60. C2×C4×D17D4semidirect
147. #398D17semidirect
187. C17#2618semidirect

Maximal subgroups

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

subgroupindex
C2×#105875802
C2×#105875812
C2×C4⋊C4×D172
#117627402
#117627332
#117627312
#117627272
#117627262
#117626872
#117626682
C17×#3982
#261817