Finite Group Databasebeta

← #14534261

Subgroups — #14534261

Order 2176.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10587625C211yesmaximal, characteristic
2C22×C4⋊Dic17C211yesmaximal, characteristic, complemented
3#11762751C212yesmaximal, complemented
4#11762725C212yesmaximal, complemented
5#11762702C212yesmaximal, complemented
6#11762701C212yesmaximal, complemented
7#11762689C211yesmaximal, characteristic, complemented
8#11762688C211yesmaximal, characteristic, complemented
9#11762686C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic17C211yesmaximal, characteristic, complemented
11C17×#386C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C23×C4×C17C2211yescharacteristic
14C22×C22⋊D17C2211yescharacteristic
15C22×C22⋊D17C2211yescharacteristic
16C23×Dic17C2211yescharacteristic
17C23×Dic17C2211yescharacteristic
18C2×C22⋊C4×C17C2212yes—
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊Dic17C2211yescharacteristic
21C2×C22⋊Dic17C2211yescharacteristic
22C2×C22⋊Dic17C2212yes—
23#10587625—24no—
24C2×#63227C2212yes—
25C2×#63227C2211yescharacteristic
26C2×#63227C2212yes—
27C2×C4⋊Dic17—24no—
28C2×C4⋊Dic17C2212yes—
29C2×C4⋊Dic17C2212yes—
30C2×#63225C2211yescharacteristic
31C2×C4×Dic17C2212yes—
32C17×#95C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C24×C17C2311yescharacteristic
39C23×D17C2311yescharacteristic
40C2×D4×C17—22no—
41C2×D4×C17—22no—
42C2×D4×C17—22no—
43C22×C4×C17—22no—
44C22×C4×C17—24no—
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2312yes—
47C22×C4×C17C2312yes—
48C2×C22⋊D17—22no—
49C2×C22⋊D17—22no—
50C2×C22⋊D17—22no—
51C2×C22⋊D17—22no—
52C2×C22⋊D17—42no—
53C22×Dic17—22no—
54C22×Dic17—42no—
55C22×Dic17—42no—
56C22×Dic17D414yes—
57C22×Dic17C2312yes—
58C22×Dic17C2311yescharacteristic
59C22×Dic17C2312yes—
60C22×Dic17C2311yescharacteristic
61C22×Dic17C2311yescharacteristic
62C22⋊C4×C17—22no—
63C22⋊C4×C17—24no—
64C22⋊Dic17—22no—
65C22⋊Dic17—22no—
66C22⋊Dic17—42no—
67#63227—24no—
68#63227—22no—
69#63227—42no—
70C4⋊Dic17—24no—
71C4⋊Dic17—24no—
72#63225—22no—
73C4×Dic17—42no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—21no—
77C23×C17—22no—
78C23×C17—21no—
79C23×C17—21no—
80C23×C17—22no—
81C23×C17C2×D412yes—
82C23×C17#4112yes—
83C23×C17#4112yes—
84C23×C17C2411yescharacteristic
85C22×D17—21no—
86C22×D17—21no—
87C22×D17—21no—
88C22×D17—22no—
89C22×D17—22no—
90D4×C17—44no—
91C2×C4×C17—28no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—21no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—22no—
100C2×C4×C17#4114yes—
101C22⋊D17—44no—
102C22⋊D17—44no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—21no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—21no—
109C2×Dic17—21no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—44no—
115C2×Dic17—44no—
116C2×Dic17C2×D412yes—
117C2×Dic17C2×D412yes—
118#2591—171nomaximal, Sylow
119C22×C17—22no—
120C22×C17—22no—
121C22×C17—24no—
122C22×C17—24no—
123C22×C17—41no—
124C22×C17—41no—
125C22×C17—41no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17#12414yes—
129C22×C17C22×D411yescharacteristic
130C22×C17C2×#4111yescharacteristic
131C22×C17C2×#4112yes—
132C22×C17C2×#4111yescharacteristic
133C22×C17H(5,𝔽2)12yes—
134C2×D17—41no—
135C2×D17—41no—
136C2×D17—41no—
137C2×D17—42no—
138C2×D17—42no—
139C4×C17—24no—
140C4×C17—41no—
141C4×C17—42no—
142Dic17—22no—
143Dic17—22no—
144Dic17—41no—
145Dic17—42no—
146Dic17—42no—
147C2×#121—171no—
148C22×C4⋊C4—171no—
149#400—172no—
150#394—172no—
151#388—172no—
152#388—172no—
153#386—171no—
154#386—171no—
155#386—171no—
156#386D1711yescharacteristic, complemented
157C4×C22⋊C4—171no—
158C2×C17—24no—
159C2×C17—81no—
160C2×C17#52511yescharacteristic
161C2×C17C2×#12411yescharacteristic
162C2×C17#53411yescharacteristic
163C2×C17#55312yes—
164C2×C17#55312yes—
165D17—81no—
166C22×D4—171no—
167C22×D4—171no—
168C22×D4C2×D1711yescharacteristic
169C23×C4—171no—
170C23×C4—171no—
171C23×C4C2×D1711yescharacteristic
172#121—344no—
173C2×C4⋊C4—172no—
174C2×C4⋊C4—172no—
175C2×C4⋊C4—171no—
176C2×C4⋊C4—344no—
177C2×C22⋊C4—171no—
178C2×C22⋊C4—171no—
179C2×C22⋊C4—171no—
180C2×C22⋊C4—172no—
181C2×C22⋊C4—172no—
182C2×C22⋊C4—172no—
183C2×C22⋊C4C2×D1711yescharacteristic
184C2×C22⋊C4C2×D1712yes—
185C2×C42—172no—
186#95—172no—
187#95—172no—
188#95—172no—
189#95—172no—
190#95C2×D1712yes—
191C17#259111yescharacteristic, complemented, Sylow
192C24—171no—
193C24C22×D1711yescharacteristic
194C24C22×D1711yescharacteristic
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—342no—
199C2×D4—342no—
200C2×D4—342no—
201C2×D4—342no—
202C2×D4—682no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—171no—
206C22×C4—171no—
207C22×C4—174no—
208C22×C4—171no—
209C22×C4—172no—
210C22×C4—172no—
211C22×C4—342no—
212C22×C4—682no—
213C22×C4—682no—
214C22×C4C22×D1711yescharacteristic
215C22×C4C22×D1712yes—
216C22×C4C22×D1712yes—
217C4⋊C4—344no—
218C4⋊C4—344no—
219C4⋊C4—342no—
220C22⋊C4—22no—
221C22⋊C4—24no—
222C22⋊C4—342no—
223C22⋊C4—342no—
224C22⋊C4—342no—
225C22⋊C4—344no—
226C22⋊C4—682no—
227C22⋊C4—682no—
228C42—682no—
229C23—22no—
230C23—22no—
231C23—21no—
232C23—21no—
233C23—22no—
234C23—21no—
235C23—22no—
236C23—341no—
237C23—341no—
238C23—342no—
239C23—341no—
240C23—342no—
241C23#6325212yes—
242C23D4×D1712yes—
243C23#6325412yes—
244C23C23×D1711yescharacteristic
245D4—44no—
246D4—684no—
247D4—684no—
248C2×C4—28no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—172no—
258C2×C4—172no—
259C2×C4—342no—
260C2×C4—342no—
261C2×C4—342no—
262C2×C4—341no—
263C2×C4—341no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—341no—
267C2×C4—342no—
268C2×C4—342no—
269C2×C4—342no—
270C2×C4—684no—
271C2×C4—684no—
272C2×C4#6325414yes—
273C22—24no—
274C22—24no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—682no—
283C22—682no—
284C22—681no—
285C22—681no—
286C22—681no—
287C22#1058757114yes—
288C22C2×#6325211yescharacteristic
289C22C2×D4×D1711yescharacteristic
290C22C2×#6325412yes—
291C22C2×#6325411yescharacteristic
292C22#1058769612yes—
293C4—24no—
294C4—42no—
295C4—41no—
296C4—342no—
297C4—342no—
298C4—682no—
299C4—681no—
300C4—682no—
301C2—24no—
302C2—81no—
303C2—1361no—
304C2#1176331511yescharacteristic
305C2C2×#1058757111yescharacteristic
306C2#1176337011yescharacteristic
307C2#1176350512yes—
308C2#1176351112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#386C2
prime radicalC2×#10587625C2
hypercenter#386D17
hypocenterC17#2591
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Dic17C2semidirect
3. #11762751C2semidirect
4. #11762725C2semidirect
5. #11762702C2semidirect
6. #11762701C2semidirect
7. #11762689C2semidirect
8. #11762688C2semidirect
9. #11762686C2semidirect
10. C22⋊C4×Dic17C2semidirect
11. C17×#386C2semidirect
156. #386D17semidirect
191. C17#2591semidirect

Maximal subgroups

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

subgroupindex
C2×#105876252
C22×C4⋊Dic172
#117627512
#117627252
#117627022
#117627012
#117626892
#117626882
#117626862
C22⋊C4×Dic172
C17×#3862
#259117