Finite Group Databasebeta

← #14533719

Subgroups — #14533719

Order 2176.

760 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 157 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×#10587571C211yesmaximal, characteristic, complemented
2C2×#10587582C212yesmaximal, complemented
3C2×#10587570C212yesmaximal, complemented
4C2×#10587579C211yesmaximal, characteristic, complemented
5#11762744C211yesmaximal, characteristic, complemented
6#11762719C211yesmaximal, characteristic, complemented
7#11762718C211yesmaximal, characteristic, complemented
8#11762710C212yesmaximal, complemented
9#11762709C212yesmaximal, complemented
10#11762660C211yesmaximal, characteristic, complemented
11C17×#399C211yesmaximal, characteristic, complemented
12C22×C22⋊D17C2211yescharacteristic
13C22×C22⋊D17C2212yes—
14C22×D68C2211yescharacteristic
15C22×D68C2212yes—
16C22×C4×D17C2212yes—
17C22×Dic34C2211yescharacteristic
18C2×C4⋊C4×C17C2212yes—
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2212yes—
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊Dic17C2211yescharacteristic
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2211yescharacteristic
26C2×#63227C2212yes—
27C2×#63227C2212yes—
28C2×#63227C2211yescharacteristic
29C2×#63225C2212yes—
30C2×#63225C2212yes—
31C2×C4×Dic17C2212yes—
32C2×C4×Dic17C2211yescharacteristic
33#10587582—28no—
34#10587579—24no—
35#10587571—24no—
36#10587570—28no—
37C17×#95C2211yescharacteristic
38#10587502C2211yescharacteristic
39C24×C17C2311yescharacteristic
40C23×D17C2311yescharacteristic
41C23×D17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2312yes—
44C22×C4×C17C2311yescharacteristic
45C22×C4×C17C2312yes—
46C2×C22⋊D17—24no—
47C2×C22⋊D17—24no—
48C2×C22⋊D17—24no—
49C2×C22⋊D17—22no—
50C2×C22⋊D17—42no—
51C22×Dic17C2311yescharacteristic
52C22×Dic17C2312yes—
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2311yescharacteristic
55C2×D68—24no—
56C2×D68—22no—
57C2×D68—24no—
58C2×D68—24no—
59C2×D68—42no—
60C2×C4×D17—24no—
61C2×C4×D17—24no—
62C2×C4×D17—24no—
63C2×Dic34—22no—
64C2×Dic34—24no—
65C4⋊C4×C17—22no—
66C4⋊C4×C17—24no—
67C22⋊C4×C17—24no—
68C22⋊C4×C17—22no—
69C22⋊Dic17—22no—
70#63227—22no—
71#63227—24no—
72#63227—24no—
73#63227—22no—
74#63227—24no—
75#63227—24no—
76#63225—24no—
77#63225D418yescomplemented
78C4×Dic17—42no—
79C4×Dic17D414yescomplemented
80C23×C17—21no—
81C23×C17—22no—
82C23×C17—22no—
83C23×C17—21no—
84C23×C17—21no—
85C23×C17C2411yescharacteristic
86C22×D17—22no—
87C22×D17—22no—
88C22×D17—22no—
89C22×D17—22no—
90C22×D17—22no—
91C22×D17—22no—
92C22×D17—22no—
93C22×D17—21no—
94C22×D17—22no—
95C22×D17—21no—
96C22×D17—21no—
97C22×D17#4114yes—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—22no—
104C2×C4×C17—22no—
105C2×C4×C17—22no—
106C2×C4×C17C2×D412yes—
107C2×C4×C17C2×D414yes—
108C22⋊D17—48no—
109C22⋊D17—44no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—21no—
113C2×Dic17—21no—
114C2×Dic17—21no—
115C2×Dic17—22no—
116C2×Dic17C2×D412yes—
117C2×Dic17C2×D414yes—
118C2×Dic17C2×D412yes—
119C2×Dic17C2×D414yes—
120D68—28no—
121D68—48no—
122C4×D17—48no—
123Dic34—44no—
124#2540—171nomaximal, Sylow
125C22×C17—41no—
126C22×C17—41no—
127C22×C17—41no—
128C22×C17—42no—
129C22×C17—42no—
130C22×C17C22×D411yescharacteristic
131C22×C17C22×D412yes—
132C22×C17C2×#4112yes—
133C22×C17H(5,𝔽2)11yescharacteristic
134C22×C172−1+411yescharacteristic
135C2×D17—24no—
136C2×D17—24no—
137C2×D17—24no—
138C2×D17—42no—
139C2×D17—42no—
140C2×D17—42no—
141C2×D17—42no—
142C2×D17—41no—
143C2×D17—41no—
144C2×D17—41no—
145C2×D17—42no—
146C2×D17—42no—
147C4×C17—22no—
148C4×C17—24no—
149C4×C17—41no—
150C4×C17—42no—
151Dic17—22no—
152Dic17—24no—
153Dic17—41no—
154Dic17#12714yes—
155C2×#124—171no—
156C2×C22⋊Q8—172no—
157C2×#121—172no—
158C2×C4×D4—171no—
159#399—171no—
160#399D1711yescharacteristic, complemented
161#392—171no—
162#392—171no—
163#390—172no—
164#390—172no—
165C4×C4⋊C4—171no—
166C2×C17—81no—
167C2×C17#52511yescharacteristic
168C2×C17C2×#12711yescharacteristic
169C2×C17#53711yescharacteristic
170C2×C17#54612yes—
171C2×C17#54812yes—
172D17—44no—
173D17—81no—
174C22×Q8—171no—
175C22×D4—171no—
176C22×D4—172no—
177C22×D4—171no—
178C22×D4—172no—
179C23×C4—172no—
180#124—344no—
181C22⋊Q8—348no—
182#121—348no—
183C4×D4—344no—
184C2×C4⋊C4—172no—
185C2×C4⋊C4—172no—
186C2×C4⋊C4C2×D1711yescharacteristic
187C2×C4⋊C4C2×D1712yes—
188C2×C22⋊C4—171no—
189C2×C22⋊C4—172no—
190C2×C22⋊C4—172no—
191C2×C22⋊C4—172no—
192C2×C22⋊C4—172no—
193C2×C22⋊C4—171no—
194C2×C22⋊C4—171no—
195C2×C22⋊C4C2×D1712yes—
196C2×C22⋊C4C2×D1711yescharacteristic
197C2×C42—171no—
198C2×C42—172no—
199#95—171no—
200#95C2×D1711yescharacteristic
201C17#254011yescharacteristic, complemented, Sylow
202C24—172no—
203C24—171no—
204C24C22×D1711yescharacteristic
205C2×Q8—344no—
206C2×Q8—342no—
207C2×D4—344no—
208C2×D4—344no—
209C2×D4—344no—
210C2×D4—344no—
211C2×D4—342no—
212C2×D4—344no—
213C2×D4—342no—
214C2×D4—344no—
215C2×D4—682no—
216C2×D4—682no—
217C22×C4—171no—
218C22×C4—172no—
219C22×C4—171no—
220C22×C4—171no—
221C22×C4—344no—
222C22×C4—344no—
223C22×C4—344no—
224C22×C4C22×D1711yescharacteristic
225C22×C4C22×D1712yes—
226C22×C4C22×D1711yescharacteristic
227C22×C4C22×D1712yes—
228C4⋊C4—22no—
229C4⋊C4—24no—
230C4⋊C4—178no—
231C4⋊C4—344no—
232C22⋊C4—24no—
233C22⋊C4—22no—
234C22⋊C4—344no—
235C22⋊C4—344no—
236C22⋊C4—344no—
237C22⋊C4—342no—
238C22⋊C4—342no—
239C22⋊C4—344no—
240C22⋊C4—342no—
241C42—174no—
242C42—682no—
243C23—21no—
244C23—22no—
245C23—21no—
246C23—22no—
247C23—21no—
248C23—174no—
249C23—342no—
250C23—342no—
251C23—342no—
252C23—342no—
253C23—342no—
254C23—342no—
255C23—341no—
256C23—342no—
257C23—341no—
258C23—342no—
259C23—341no—
260C23C23×D1711yescharacteristic
261Q8—684no—
262D4—348no—
263D4—688no—
264D4—684no—
265D4—688no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—172no—
275C2×C4—174no—
276C2×C4—172no—
277C2×C4—174no—
278C2×C4—341no—
279C2×C4—342no—
280C2×C4—341no—
281C2×C4—342no—
282C2×C4—341no—
283C2×C4—342no—
284C2×C4—688no—
285C2×C4D4×D1712yes—
286C2×C4D4×D1714yes—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22—344no—
293C22—344no—
294C22—344no—
295C22—682no—
296C22—682no—
297C22—682no—
298C22—682no—
299C22—682no—
300C22—681no—
301C22—681no—
302C22—681no—
303C22—682no—
304C22C2×D4×D1712yes—
305C22C2×D4×D1711yescharacteristic
306C22#1058769411yescharacteristic
307C22#41×D1712yes—
308C22#1058769611yescharacteristic
309C4—22no—
310C4—24no—
311C4—42no—
312C4—41no—
313C4—174no—
314C4—342no—
315C4—344no—
316C4—681no—
317C2—81no—
318C2—684no—
319C2—1361no—
320C2#1176331711yescharacteristic
321C2#127×D1711yescharacteristic
322C2#1176338911yescharacteristic
323C2#1176345112yes—
324C2#1176346912yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#399C2
hypercenter#399D17
hypocenterC17#2540
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#10587571C2semidirect
2. C2×#10587582C2semidirect
3. C2×#10587570C2semidirect
4. C2×#10587579C2semidirect
5. #11762744C2semidirect
6. #11762719C2semidirect
7. #11762718C2semidirect
8. #11762710C2semidirect
9. #11762709C2semidirect
10. #11762660C2semidirect
11. C17×#399C2semidirect
77. #63225D4semidirect
79. C4×Dic17D4semidirect
160. #399D17semidirect
201. C17#2540semidirect

Maximal subgroups

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

subgroupindex
C2×#105875712
C2×#105875822
C2×#105875702
C2×#105875792
#117627442
#117627192
#117627182
#117627102
#117627092
#117626602
C17×#3992
#254017