Finite Group Databasebeta

← #14536832

Subgroups — #14536832

Order 2176.

580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 125 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×#10587633C211yesmaximal, characteristic, complemented
2C2×#10587571C211yesmaximal, characteristic, complemented
3#11762727C212yesmaximal, complemented
4#11762724C212yesmaximal, complemented
5#11762722C212yesmaximal, complemented
6#11762720C211yesmaximal, characteristic, complemented
7#11762719C211yesmaximal, characteristic, complemented
8#11762718C211yesmaximal, characteristic, complemented
9#11762718C212yesmaximal, complemented
10#11762673C211yesmaximal, characteristic, complemented
11C17×#393C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×C22⋊D17C2211yescharacteristic
14C22×C22⋊D17C2212yes—
15C22×D68C2211yescharacteristic
16C22×D68C2212yes—
17C22×Dic34C2211yescharacteristic
18C2×C22⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2212yes—
20C2×C22⋊Dic17C2211yescharacteristic
21#10587633—24no—
22C2×#63227C2212yes—
23C2×#63227C2211yescharacteristic
24C2×#63227C2212yes—
25C2×#63227C2211yescharacteristic
26C2×#63227C2212yes—
27C2×#63227C2212yes—
28C2×#63227C2212yes—
29C2×#63225C2212yes—
30C2×C4×Dic17C2211yescharacteristic
31#10587571—24no—
32C17×#95C2211yescharacteristic
33C17×#95C2212yes—
34#10587502C2211yescharacteristic
35#10587502C2212yes—
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C23×D17C2312yes—
39C23×D17C2311yescharacteristic
40C2×Q8×C17—22no—
41C2×Q8×C17—42no—
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2312yes—
44C22×C4×C17C2312yes—
45C22×C4×C17C2311yescharacteristic
46C2×C22⋊D17—22no—
47C2×C22⋊D17—42no—
48C2×C22⋊D17—42no—
49C2×C22⋊D17—42no—
50C2×C22⋊D17—42no—
51C22×Dic17C2311yescharacteristic
52C22×Dic17C2312yes—
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2311yescharacteristic
55C2×D68—22no—
56C2×D68—42no—
57C2×D68—42no—
58C2×D68—42no—
59C2×D68—42no—
60C2×Dic34—22no—
61C2×Dic34—42no—
62C22⋊C4×C17—22no—
63C22⋊C4×C17—42no—
64C22⋊Dic17—22no—
65#63227—22no—
66#63227—22no—
67#63227—24no—
68#63227—24no—
69#63227—42no—
70#63227—42no—
71#63227—42no—
72#63225—42no—
73C4×Dic17D414yescomplemented
74C23×C17—21no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—21no—
78C23×C17—21no—
79C23×C17C2411yescharacteristic
80C22×D17—22no—
81C22×D17—21no—
82C22×D17—22no—
83C22×D17—21no—
84C22×D17—22no—
85C22×D17—22no—
86C22×D17—22no—
87C22×D17—22no—
88C22×D17—22no—
89C22×D17—22no—
90C22×D17—22no—
91C22×D17—21no—
92Q8×C17—44no—
93C2×C4×C17—22no—
94C2×C4×C17—21no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17C2×D412yes—
103C22⋊D17—44no—
104C22⋊D17—48no—
105C2×Dic17—22no—
106C2×Dic17—21no—
107C2×Dic17—21no—
108C2×Dic17—21no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17C2×D412yes—
114C2×Dic17C2×D412yes—
115D68—44no—
116D68—48no—
117Dic34—44no—
118#2819—171nomaximal, Sylow
119C22×C17—41no—
120C22×C17—41no—
121C22×C17—41no—
122C22×C17—42no—
123C22×C17—42no—
124C22×C17C22×D411yescharacteristic
125C22×C17H(5,𝔽2)12yes—
126C22×C17H(5,𝔽2)12yes—
127C22×C17H(5,𝔽2)11yescharacteristic
128C22×C172−1+411yescharacteristic
129C2×D17—42no—
130C2×D17—41no—
131C2×D17—41no—
132C2×D17—42no—
133C2×D17—42no—
134C2×D17—41no—
135C2×D17—42no—
136C2×D17—42no—
137C2×D17—42no—
138C2×D17—42no—
139C2×D17—42no—
140C2×D17—42no—
141C4×C17—22no—
142C4×C17—41no—
143C4×C17—42no—
144C4×C17—42no—
145Dic17—22no—
146Dic17—22no—
147Dic17—41no—
148Dic17—42no—
149C2×#124—171no—
150C2×#124—171no—
151#394—172no—
152#394—172no—
153#393—171no—
154#393—172no—
155#393D1711yescharacteristic, complemented
156#392—172no—
157#392—171no—
158#392—171no—
159#382—171no—
160C2×C17—81no—
161C2×C17#53511yescharacteristic
162C2×C17#53511yescharacteristic
163C2×C17#53711yescharacteristic
164C2×C17H(3,𝔽4)12yes—
165C2×C17#56212yes—
166D17—82no—
167D17—81no—
168C22×Q8—171no—
169C22×Q8C2×D1711yescharacteristic
170C22×D4—171no—
171C22×D4—172no—
172C22×D4—171no—
173C22×D4—172no—
174#124—344no—
175#124—344no—
176C2×C4⋊C4—172no—
177C2×C22⋊C4—171no—
178C2×C22⋊C4—172no—
179C2×C22⋊C4—171no—
180C2×C22⋊C4—172no—
181C2×C22⋊C4—171no—
182C2×C22⋊C4—172no—
183C2×C22⋊C4—172no—
184C2×C22⋊C4—172no—
185C2×C22⋊C4C2×D1711yescharacteristic
186C2×C22⋊C4C2×D1712yes—
187C2×C42—171no—
188#95—172no—
189#95—171no—
190#95—172no—
191#95C2×D1711yescharacteristic
192#95C2×D1712yes—
193C17#281911yescharacteristic, complemented, Sylow
194C24—171no—
195C24—172no—
196C24C22×D1711yescharacteristic
197C2×Q8—22no—
198C2×Q8—42no—
199C2×Q8—342no—
200C2×Q8—682no—
201C2×D4—342no—
202C2×D4—342no—
203C2×D4—682no—
204C2×D4—682no—
205C2×D4—682no—
206C2×D4—682no—
207C2×D4—682no—
208C2×D4—682no—
209C2×D4—682no—
210C2×D4—682no—
211C22×C4—171no—
212C22×C4—171no—
213C22×C4—172no—
214C22×C4—171no—
215C22×C4C22×D1712yes—
216C22×C4C22×D1712yes—
217C22×C4C22×D1711yescharacteristic
218C22×C4C22×D1711yescharacteristic
219C4⋊C4—682no—
220C22⋊C4—22no—
221C22⋊C4—42no—
222C22⋊C4—342no—
223C22⋊C4—344no—
224C22⋊C4—344no—
225C22⋊C4—342no—
226C22⋊C4—342no—
227C22⋊C4—682no—
228C22⋊C4—682no—
229C22⋊C4—682no—
230C42—174no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—342no—
237C23—341no—
238C23—342no—
239C23—342no—
240C23—342no—
241C23—341no—
242C23—342no—
243C23—342no—
244C23—342no—
245C23—341no—
246C23—342no—
247C23—342no—
248C23C23×D1711yescharacteristic
249Q8—44no—
250Q8—684no—
251D4—684no—
252D4—688no—
253D4—684no—
254D4—688no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—172no—
265C2×C4—172no—
266C2×C4—341no—
267C2×C4—341no—
268C2×C4—342no—
269C2×C4—341no—
270C2×C4—342no—
271C2×C4—342no—
272C2×C4—342no—
273C2×C4—342no—
274C2×C4D4×D1712yes—
275C22—42no—
276C22—42no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—682no—
281C22—682no—
282C22—682no—
283C22—682no—
284C22—682no—
285C22—682no—
286C22—682no—
287C22—681no—
288C22—681no—
289C22—682no—
290C22—682no—
291C22—681no—
292C22C2×D4×D1711yescharacteristic
293C22#1058769112yes—
294C22#1058769411yescharacteristic
295C22#1058769611yescharacteristic
296C22#1058769612yes—
297C4—22no—
298C4—42no—
299C4—42no—
300C4—41no—
301C4—342no—
302C4—342no—
303C4—682no—
304C4—681no—
305C2—81no—
306C2—1362no—
307C2—1361no—
308C2#1176337411yescharacteristic
309C2#1176337711yescharacteristic
310C2#1176338911yescharacteristic
311C2#1176355112yes—
312C2#1176355512yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#393C2
hypercenter#393D17
hypocenterC17#2819
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#10587633C2semidirect
2. C2×#10587571C2semidirect
3. #11762727C2semidirect
4. #11762724C2semidirect
5. #11762722C2semidirect
6. #11762720C2semidirect
7. #11762719C2semidirect
8. #11762718C2semidirect
9. #11762718C2semidirect
10. #11762673C2semidirect
11. C17×#393C2semidirect
73. C4×Dic17D4semidirect
155. #393D17semidirect
193. C17#2819semidirect

Maximal subgroups

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

subgroupindex
C2×#105876332
C2×#105875712
#117627272
#117627242
#117627222
#117627202
#117627192
#117627182
#117626732
C17×#3932
#281917