Finite Group Databasebeta

← #14533746

Subgroups — #14533746

Order 2176.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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×#10587625C212yesmaximal, complemented
2C2×C4×C22⋊D17C211yesmaximal, characteristic, complemented
3#11762740C211yesmaximal, characteristic, complemented
4#11762735C211yesmaximal, characteristic, complemented
5#11762723C211yesmaximal, characteristic, complemented
6#11762717C211yesmaximal, characteristic, complemented
7#11762697C212yesmaximal, complemented
8#11762689C212yesmaximal, complemented
9#11762685C212yesmaximal, complemented
10C2×#10587502C211yesmaximal, characteristic, complemented
11C2×C17×#121C211yesmaximal, characteristic, complemented
12C22×D4×C17C2212yes—
13C22×D4×C17C2211yescharacteristic
14C23×C4×C17C2211yescharacteristic
15C22×C22⋊D17C2212yes—
16C22×C22⋊D17C2211yescharacteristic
17C23×Dic17C2212yes—
18C22×C4×D17C2211yescharacteristic
19C17×#121—24no—
20C2×C4⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2212yes—
22C2×C22⋊Dic17C2211yescharacteristic
23C2×C22⋊Dic17C2212yes—
24C2×C22⋊Dic17C2212yes—
25C2×C22⋊Dic17C2212yes—
26#10587625—28no—
27C4×C22⋊D17—24no—
28C2×#63227C2211yescharacteristic
29C2×#63227C2212yes—
30C2×#63225C2212yes—
31C2×#63225C2211yescharacteristic
32C2×C4×Dic17C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37#10587502C2211yescharacteristic
38C24×C17C2312yes—
39C24×C17C2311yescharacteristic
40C23×D17C2311yescharacteristic
41C2×D4×C17—24no—
42C2×D4×C17—24no—
43C2×D4×C17—24no—
44C2×D4×C17—22no—
45C2×D4×C17—42no—
46C22×C4×C17—42no—
47C22×C4×C17D414yescomplemented
48C22×C4×C17C2312yes—
49C22×C4×C17C2311yescharacteristic
50C22×C4×C17C2311yescharacteristic
51C2×C22⋊D17—22no—
52C2×C22⋊D17—22no—
53C2×C22⋊D17—24no—
54C2×C22⋊D17—24no—
55C2×C22⋊D17—42no—
56C2×C22⋊D17D414yescomplemented
57C22×Dic17—24no—
58C22×Dic17—24no—
59C22×Dic17—42no—
60C22×Dic17C2311yescharacteristic
61C22×Dic17C2312yes—
62C22×Dic17C2311yescharacteristic
63C22×Dic17C2311yescharacteristic
64C22×Dic17C2312yes—
65C2×C4×D17—22no—
66C2×C4×D17—42no—
67C4⋊C4×C17—22no—
68C22⋊C4×C17—24no—
69C22⋊Dic17—22no—
70C22⋊Dic17—24no—
71C22⋊Dic17—24no—
72C22⋊Dic17—42no—
73#63227—24no—
74#63227—22no—
75#63225—22no—
76#63225—24no—
77C4×Dic17—22no—
78C23×C17—22no—
79C23×C17—22no—
80C23×C17—22no—
81C23×C17—22no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—22no—
85C23×C17—22no—
86C23×C17—22no—
87C23×C17C2×D412yes—
88C23×C17C2×D412yes—
89C23×C17#4112yes—
90C23×C17C2411yescharacteristic
91C22×D17—21no—
92C22×D17—21no—
93C22×D17—22no—
94C22×D17—22no—
95C22×D17C2×D412yes—
96D4×C17—44no—
97D4×C17—48no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C2×C4×C17—44no—
104C2×C4×C17C2×D412yes—
105C2×C4×C17C2×D412yes—
106C22⋊D17—28no—
107C22⋊D17—48no—
108C2×Dic17—22no—
109C2×Dic17—21no—
110C2×Dic17—21no—
111C2×Dic17—22no—
112C2×Dic17—21no—
113C2×Dic17—22no—
114C2×Dic17—21no—
115C2×Dic17—22no—
116C2×Dic17—22no—
117C2×Dic17—22no—
118C2×Dic17—22no—
119C2×Dic17—48no—
120C2×Dic17C2×D412yes—
121C2×Dic17#4114yes—
122C4×D17—44no—
123#2543—171nomaximal, Sylow
124C22×C17—22no—
125C22×C17—22no—
126C22×C17—24no—
127C22×C17—24no—
128C22×C17—42no—
129C22×C17—42no—
130C22×C17—42no—
131C22×C17—42no—
132C22×C17—42no—
133C22×C17—42no—
134C22×C17—42no—
135C22×C17#12114yes—
136C22×C17C22×D411yescharacteristic
137C22×C17C22×D411yescharacteristic
138C22×C17C2×#4111yescharacteristic
139C22×C17C2×#4112yes—
140C22×C17H(5,𝔽2)12yes—
141C2×D17—22no—
142C2×D17—24no—
143C2×D17—41no—
144C2×D17—41no—
145C2×D17—42no—
146C4×C17—22no—
147C4×C17—22no—
148C4×C17—42no—
149Dic17—22no—
150Dic17—24no—
151Dic17—41no—
152Dic17—41no—
153Dic17—42no—
154C2×#121—172no—
155C2×#121D1711yescharacteristic, complemented
156C2×C4×D4—171no—
157#398—171no—
158#397—171no—
159#394—171no—
160#392—171no—
161#388—172no—
162#386—172no—
163#386—172no—
164C2×#95—171no—
165C2×C17—24no—
166C2×C17—82no—
167C2×C17C2×#12111yescharacteristic
168C2×C17#53211yescharacteristic
169C2×C17#53411yescharacteristic
170C2×C17#54612yes—
171C2×C17#55312yes—
172D17—42no—
173C22×D4—171no—
174C22×D4—172no—
175C22×D4C2×D1712yes—
176C22×D4C2×D1711yescharacteristic
177C23×C4—172no—
178C23×C4—171no—
179C23×C4C2×D1711yescharacteristic
180#121—24no—
181#121—348no—
182C4×D4—344no—
183C2×C4⋊C4—171no—
184C2×C4⋊C4—172no—
185C2×C4⋊C4C2×D1711yescharacteristic
186C2×C22⋊C4—171no—
187C2×C22⋊C4—172no—
188C2×C22⋊C4—172no—
189C2×C22⋊C4—171no—
190C2×C22⋊C4—172no—
191C2×C22⋊C4—172no—
192C2×C22⋊C4C2×D1712yes—
193C2×C42—171no—
194#95—172no—
195#95—171no—
196#95—171no—
197#95—172no—
198#95—172no—
199C17#254311yescharacteristic, complemented, Sylow
200C24—171no—
201C24C22×D1712yes—
202C24C22×D1711yescharacteristic
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—24no—
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—174no—
209C2×D4—344no—
210C2×D4—342no—
211C2×D4—342no—
212C2×D4—344no—
213C2×D4—682no—
214C22×C4—42no—
215C22×C4—172no—
216C22×C4—171no—
217C22×C4—172no—
218C22×C4—171no—
219C22×C4—171no—
220C22×C4—344no—
221C22×C4—344no—
222C22×C4—342no—
223C22×C4—682no—
224C22×C4—682no—
225C22×C4C22⋊D1714yescomplemented
226C22×C4C22×D1712yes—
227C22×C4C22×D1711yescharacteristic
228C22×C4C22×D1711yescharacteristic
229C4⋊C4—22no—
230C4⋊C4—344no—
231C4⋊C4—342no—
232C22⋊C4—24no—
233C22⋊C4—344no—
234C22⋊C4—344no—
235C22⋊C4—344no—
236C22⋊C4—342no—
237C22⋊C4—342no—
238C22⋊C4—682no—
239C42—342no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—172no—
250C23—341no—
251C23—342no—
252C23—342no—
253C23—341no—
254C23D4×D1712yes—
255C23#6325412yes—
256C23C2×C22⋊D1712yes—
257C23C23×D1711yescharacteristic
258D4—44no—
259D4—48no—
260D4—348no—
261D4—688no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—44no—
268C2×C4—172no—
269C2×C4—174no—
270C2×C4—342no—
271C2×C4—341no—
272C2×C4—342no—
273C2×C4—341no—
274C2×C4—342no—
275C2×C4—342no—
276C2×C4—342no—
277C2×C4—342no—
278C2×C4—342no—
279C2×C4—341no—
280C2×C4—341no—
281C2×C4—684no—
282C2×C4—688no—
283C2×C4C2×C22⋊D1712yes—
284C2×C4C2×C22⋊D1712yes—
285C22—24no—
286C22—22no—
287C22—22no—
288C22—24no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—342no—
297C22—344no—
298C22—682no—
299C22—681no—
300C22—681no—
301C22#1058762414yes—
302C22C2×D4×D1711yescharacteristic
303C22C2×#6325411yescharacteristic
304C22#1058769112yes—
305C22#41×D1712yes—
306C22C22×C22⋊D1711yescharacteristic
307C4—22no—
308C4—22no—
309C4—42no—
310C4—342no—
311C4—344no—
312C4—681no—
313C4—681no—
314C4—682no—
315C2—24no—
316C2—82no—
317C2—682no—
318C2C2×#1058762411yescharacteristic
319C2#1176335511yescharacteristic
320C2#1176336811yescharacteristic
321C2#1176344812yes—
322C2#1176350212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#121C2
hypercenterC2×#121D17
hypocenterC17#2543
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#10587625C2semidirect
2. C2×C4×C22⋊D17C2semidirect
3. #11762740C2semidirect
4. #11762735C2semidirect
5. #11762723C2semidirect
6. #11762717C2semidirect
7. #11762697C2semidirect
8. #11762689C2semidirect
9. #11762685C2semidirect
10. C2×#10587502C2semidirect
11. C2×C17×#121C2semidirect
47. C22×C4×C17D4semidirect
56. C2×C22⋊D17D4semidirect
155. C2×#121D17semidirect
199. C17#2543semidirect
225. C22×C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876252
C2×C4×C22⋊D172
#117627402
#117627352
#117627232
#117627172
#117626972
#117626892
#117626852
C2×#105875022
C2×C17×#1212
#254317