Finite Group Databasebeta

← #14533782

Subgroups — #14533782

Order 2176.

676 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 151 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×#10587626C211yesmaximal, characteristic
2C2×#10587571C212yesmaximal, complemented
3C2×#10587573C211yesmaximal, characteristic, complemented
4#11762727C211yesmaximal, characteristic, complemented
5#11762726C211yesmaximal, characteristic, complemented
6#11762723C211yesmaximal, characteristic, complemented
7#11762709C212yesmaximal, complemented
8#11762709C212yesmaximal, complemented
9#11762709C212yesmaximal, complemented
10#11762660C211yesmaximal, characteristic, complemented
11C17×#394C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C22×C22⋊D17C2212yes—
14C22×C22⋊D17C2212yes—
15C22×D68C2211yescharacteristic
16C22×Dic34C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C22⋊C4×C17C2212yes—
19C2×C22⋊C4×C17C2212yes—
20C2×C22⋊Dic17C2212yes—
21C2×C22⋊Dic17C2212yes—
22#10587626—24no—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×#63225C2211yescharacteristic
28C2×#63225C2212yes—
29C2×#63225C2211yescharacteristic
30C2×C4×Dic17C2211yescharacteristic
31C2×C4×Dic17C2211yescharacteristic
32C2×C4×Dic17C2212yes—
33C2×C4×Dic17C2211yescharacteristic
34#10587573—24no—
35#10587571—28no—
36C17×#95C2211yescharacteristic
37#10587502C2211yescharacteristic
38C24×C17C2312yes—
39C23×D17C2312yes—
40C2×D4×C17—22no—
41C2×D4×C17—42no—
42C22×C4×C17C2312yes—
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C22×C4×C17C2311yescharacteristic
46C2×C22⋊D17—24no—
47C2×C22⋊D17—24no—
48C2×C22⋊D17—24no—
49C2×C22⋊D17—24no—
50C2×C22⋊D17—24no—
51C2×C22⋊D17—24no—
52C22×Dic17C2312yes—
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2311yescharacteristic
55C22×Dic17C2311yescharacteristic
56C22×Dic17C2311yescharacteristic
57C2×D68—22no—
58C2×D68—42no—
59C2×Dic34—22no—
60C2×Dic34—22no—
61C2×Dic34D414yescomplemented
62C4⋊C4×C17—22no—
63C22⋊C4×C17—24no—
64C22⋊C4×C17—42no—
65C22⋊Dic17—24no—
66C22⋊Dic17—42no—
67#63227—24no—
68#63227—24no—
69#63227—42no—
70#63227—42no—
71#63225—22no—
72#63225—22no—
73#63225—24no—
74C4×Dic17—24no—
75C4×Dic17—22no—
76C4×Dic17—22no—
77C4×Dic17D414yes—
78C23×C17—22no—
79C23×C17—22no—
80C23×C17—22no—
81C23×C17—22no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—22no—
85C23×C17C2411yescharacteristic
86C22×D17—22no—
87C22×D17—22no—
88C22×D17—22no—
89C22×D17—22no—
90C22×D17—22no—
91C22×D17—22no—
92C22×D17—22no—
93D4×C17—44no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—21no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17C2×D412yes—
103C2×C4×C17C2×D412yes—
104C22⋊D17—48no—
105C22⋊D17—48no—
106C2×Dic17—21no—
107C2×Dic17—21no—
108C2×Dic17—22no—
109C2×Dic17—21no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17C2×D412yes—
114C2×Dic17C2×D412yes—
115C2×Dic17C2×D412yes—
116C2×Dic17C2×D412yes—
117C2×Dic17#4112yes—
118C2×Dic17#4114yes—
119D68—44no—
120Dic34—28no—
121#2548—171nomaximal, Sylow
122C22×C17—42no—
123C22×C17—42no—
124C22×C17—42no—
125C22×C17—42no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17—42no—
129C22×C17C22×D411yescharacteristic
130C22×C17C22×D411yescharacteristic
131C22×C17C2×#4111yescharacteristic
132C22×C17C2×#4112yes—
133C22×C17H(5,𝔽2)12yes—
134C2×D17—42no—
135C2×D17—42no—
136C2×D17—42no—
137C2×D17—42no—
138C2×D17—42no—
139C2×D17—42no—
140C2×D17—42no—
141C4×C17—22no—
142C4×C17—22no—
143C4×C17—41no—
144C4×C17—42no—
145Dic17—22no—
146Dic17—41no—
147Dic17—42no—
148Dic17#12114yes—
149Dic17#12414yescomplemented
150C2×#127—171no—
151C2×#124—172no—
152C2×C4×Q8—171no—
153#394—171no—
154#394—171no—
155#394—171no—
156#394D1711yescharacteristic, complemented
157#390—172no—
158#390—172no—
159#390—172no—
160C4×C4⋊C4—171no—
161C2×C17—82no—
162C2×C17C2×#12111yescharacteristic
163C2×C17C2×#12411yescharacteristic
164C2×C17#53511yescharacteristic
165C2×C17#55012yes—
166C2×C17#55912yes—
167D17—82no—
168C22×Q8—171no—
169C22×D4—172no—
170C22×D4—172no—
171C22×D4—171no—
172C22×D4C2×D1711yescharacteristic
173#127—344no—
174#124—348no—
175C4×Q8—344no—
176C2×C4⋊C4—171no—
177C2×C4⋊C4—171no—
178C2×C4⋊C4—172no—
179C2×C4⋊C4C2×D1711yescharacteristic
180C2×C22⋊C4—172no—
181C2×C22⋊C4—172no—
182C2×C22⋊C4—172no—
183C2×C22⋊C4—172no—
184C2×C22⋊C4—172no—
185C2×C22⋊C4—172no—
186C2×C22⋊C4C2×D1712yes—
187C2×C22⋊C4C2×D1712yes—
188C2×C42—171no—
189C2×C42—171no—
190C2×C42—171no—
191C2×C42—172no—
192#95—171no—
193#95C2×D1711yescharacteristic
194C17#254811yescharacteristic, complemented, Sylow
195C24—172no—
196C24C22×D1712yes—
197C2×Q8—174no—
198C2×Q8—342no—
199C2×Q8—342no—
200C2×D4—22no—
201C2×D4—42no—
202C2×D4—344no—
203C2×D4—344no—
204C2×D4—344no—
205C2×D4—344no—
206C2×D4—344no—
207C2×D4—344no—
208C2×D4—342no—
209C2×D4—682no—
210C22×C4—171no—
211C22×C4—171no—
212C22×C4—171no—
213C22×C4—172no—
214C22×C4—171no—
215C22×C4C22×D1712yes—
216C22×C4C22×D1711yescharacteristic
217C22×C4C22×D1711yescharacteristic
218C22×C4C22×D1711yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—342no—
221C4⋊C4—344no—
222C4⋊C4—342no—
223C22⋊C4—24no—
224C22⋊C4—42no—
225C22⋊C4—344no—
226C22⋊C4—344no—
227C22⋊C4—344no—
228C22⋊C4—682no—
229C22⋊C4—682no—
230C22⋊C4—682no—
231C42—174no—
232C42—342no—
233C42—342no—
234C42—344no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—342no—
243C23—342no—
244C23—342no—
245C23—342no—
246C23—342no—
247C23—342no—
248C23—342no—
249C23C23×D1711yescharacteristic
250Q8—348no—
251D4—44no—
252D4—684no—
253D4—688no—
254D4—688no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—172no—
264C2×C4—172no—
265C2×C4—172no—
266C2×C4—174no—
267C2×C4—172no—
268C2×C4—172no—
269C2×C4—341no—
270C2×C4—341no—
271C2×C4—342no—
272C2×C4—341no—
273C2×C4—342no—
274C2×C4—342no—
275C2×C4—342no—
276C2×C4D4×D1712yes—
277C2×C4D4×D1712yes—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—682no—
286C22—682no—
287C22—682no—
288C22—682no—
289C22—682no—
290C22—682no—
291C22—682no—
292C22C2×D4×D1711yescharacteristic
293C22C2×D4×D1711yescharacteristic
294C22#1058769112yes—
295C22#41×D1712yes—
296C22#41×D1711yescharacteristic
297C4—22no—
298C4—22no—
299C4—42no—
300C4—41no—
301C4—174no—
302C4—174no—
303C4—342no—
304C4—682no—
305C4—681no—
306C2—82no—
307C2—1362no—
308C2#121×D1711yescharacteristic
309C2#124×D1711yescharacteristic
310C2#1176337411yescharacteristic
311C2#1176347712yes—
312C2#1176354312yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radicalC2×#10587626C2
hypercenter#394D17
hypocenterC17#2548
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#10587571C2semidirect
3. C2×#10587573C2semidirect
4. #11762727C2semidirect
5. #11762726C2semidirect
6. #11762723C2semidirect
7. #11762709C2semidirect
8. #11762709C2semidirect
9. #11762709C2semidirect
10. #11762660C2semidirect
11. C17×#394C2semidirect
61. C2×Dic34D4semidirect
149. Dic17#124semidirect
156. #394D17semidirect
194. C17#2548semidirect

Maximal subgroups

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

subgroupindex
C2×#105876262
C2×#105875712
C2×#105875732
#117627272
#117627262
#117627232
#117627092
#117626602
C17×#3942
#254817