Finite Group Databasebeta

← #14537113

Subgroups — #14537113

Order 2176.

688 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 169 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×#10587615C211yesmaximal, characteristic, complemented
2C2×#10587625C211yesmaximal, characteristic
3C22×C4×Dic17C211yesmaximal, characteristic, complemented
4#11762709C212yesmaximal, complemented
5#11762704C212yesmaximal, complemented
6#11762703C212yesmaximal, complemented
7#11762699C212yesmaximal, complemented
8#11762689C211yesmaximal, characteristic, complemented
9#11762686C211yesmaximal, characteristic, complemented
10#11762667C211yesmaximal, characteristic, complemented
11C2×C17×#123C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C23×C4×C17C2211yescharacteristic
14C22×C22⋊D17C2211yescharacteristic
15C22×C22⋊D17C2211yescharacteristic
16C23×Dic17C2211yescharacteristic
17C23×Dic17C2211yescharacteristic
18C17×#123—24no—
19C2×C4⋊C4×C17C2212yes—
20C2×C22⋊C4×C17C2212yes—
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊Dic17C2212yes—
23C2×C22⋊Dic17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25#10587625—24no—
26#10587615—24no—
27C2×#63227C2211yescharacteristic
28C2×#63227C2212yes—
29C2×#63227C2212yes—
30C2×#63225C2212yes—
31C2×#63225C2211yescharacteristic
32C2×#63225C2212yes—
33C2×C4×Dic17—24no—
34C2×C4×Dic17C2212yes—
35C2×C4×Dic17C2212yes—
36#10587502C2212yes—
37#10587502C2212yes—
38#10587502C2212yes—
39C24×C17C2311yescharacteristic
40C24×C17C2311yescharacteristic
41C23×D17C2311yescharacteristic
42C2×D4×C17—22no—
43C2×D4×C17—22no—
44C2×D4×C17—22no—
45C22×C4×C17—22no—
46C22×C4×C17—22no—
47C22×C4×C17D414yescomplemented
48C22×C4×C17C2311yescharacteristic
49C22×C4×C17C2312yes—
50C22×C4×C17C2312yes—
51C2×C22⋊D17—22no—
52C2×C22⋊D17—22no—
53C2×C22⋊D17—22no—
54C2×C22⋊D17—22no—
55C2×C22⋊D17—24no—
56C22×Dic17—22no—
57C22×Dic17—24no—
58C22×Dic17—24no—
59C22×Dic17D414yes—
60C22×Dic17C2311yescharacteristic
61C22×Dic17C2312yes—
62C22×Dic17C2311yescharacteristic
63C22×Dic17C2312yes—
64C22×Dic17C2311yescharacteristic
65C4⋊C4×C17—24no—
66C22⋊C4×C17—22no—
67C22⋊C4×C17—24no—
68C22⋊Dic17—22no—
69C22⋊Dic17—22no—
70C22⋊Dic17—24no—
71#63227—24no—
72#63227—22no—
73#63227—24no—
74#63225—24no—
75#63225—22no—
76#63225—24no—
77C4×Dic17—24no—
78C4×Dic17—24no—
79C23×C17—22no—
80C23×C17—22no—
81C23×C17—21no—
82C23×C17—21no—
83C23×C17—22no—
84C23×C17—22no—
85C23×C17—21no—
86C23×C17C2×D412yes—
87C23×C17C2×D412yes—
88C23×C17#4112yes—
89C23×C17C2411yescharacteristic
90C22×D17—21no—
91C22×D17—21no—
92C22×D17—21no—
93C22×D17—22no—
94C22×D17—22no—
95D4×C17—44no—
96C2×C4×C17—28no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—22no—
104C2×C4×C17—22no—
105C2×C4×C17C2×D414yes—
106C22⋊D17—44no—
107C22⋊D17—44no—
108C2×Dic17—28no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—28no—
113C2×Dic17—21no—
114C2×Dic17—22no—
115C2×Dic17—21no—
116C2×Dic17—22no—
117C2×Dic17—21no—
118C2×Dic17—22no—
119C2×Dic17C2×D412yes—
120C2×Dic17C2×D412yes—
121C2×Dic17#4114yes—
122C2×Dic17#4114yes—
123#2854—171nomaximal, Sylow
124C22×C17—22no—
125C22×C17—22no—
126C22×C17—24no—
127C22×C17—24no—
128C22×C17—41no—
129C22×C17—41no—
130C22×C17—41no—
131C22×C17—42no—
132C22×C17—42no—
133C22×C17#12114yes—
134C22×C17C22×D411yescharacteristic
135C22×C17C22×D411yescharacteristic
136C22×C17C2×#4111yescharacteristic
137C22×C17C2×#4112yes—
138C22×C17C2×#4112yes—
139C2×D17—41no—
140C2×D17—41no—
141C2×D17—41no—
142C2×D17—42no—
143C2×D17—42no—
144C4×C17—24no—
145C4×C17—41no—
146C4×C17—42no—
147Dic17—22no—
148Dic17—24no—
149Dic17—41no—
150Dic17—42no—
151Dic17#12314yescomplemented
152C2×#123—171no—
153C2×#123D1711yescharacteristic, complemented
154C2×#121—171no—
155C22×C42—171no—
156#390—172no—
157#389—172no—
158#388—172no—
159#388—172no—
160#386—171no—
161#386—171no—
162#380—171no—
163C2×C17—24no—
164C2×C17—81no—
165C2×C17C2×#12111yescharacteristic
166C2×C17C2×#12311yescharacteristic
167C2×C17#52511yescharacteristic
168C2×C17#52912yes—
169C2×C17#53212yes—
170D17—81no—
171C22×D4—171no—
172C22×D4—171no—
173C22×D4C2×D1711yescharacteristic
174C23×C4—171no—
175C23×C4—171no—
176C23×C4C2×D1711yescharacteristic
177#123—24no—
178#123—344no—
179#121—344no—
180C2×C4⋊C4—171no—
181C2×C4⋊C4—172no—
182C2×C4⋊C4—172no—
183C2×C4⋊C4C2×D1712yes—
184C2×C22⋊C4—171no—
185C2×C22⋊C4—171no—
186C2×C22⋊C4—172no—
187C2×C22⋊C4—171no—
188C2×C22⋊C4—172no—
189C2×C22⋊C4—172no—
190C2×C22⋊C4C2×D1711yescharacteristic
191C2×C22⋊C4C2×D1712yes—
192C2×C42—172no—
193C2×C42—172no—
194C2×C42—344no—
195#95—172no—
196#95—172no—
197#95—172no—
198C17#285411yescharacteristic, complemented, Sylow
199C24—171no—
200C24C22×D1711yescharacteristic
201C24C22×D1711yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—342no—
206C2×D4—344no—
207C2×D4—342no—
208C2×D4—342no—
209C2×D4—342no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—171no—
213C22×C4—174no—
214C22×C4—171no—
215C22×C4—172no—
216C22×C4—172no—
217C22×C4—171no—
218C22×C4—344no—
219C22×C4—344no—
220C22×C4—342no—
221C22×C4C22⋊D1714yescomplemented
222C22×C4C22×D1712yes—
223C22×C4C22×D1711yescharacteristic
224C22×C4C22×D1712yes—
225C4⋊C4—24no—
226C4⋊C4—342no—
227C4⋊C4—344no—
228C4⋊C4—344no—
229C22⋊C4—22no—
230C22⋊C4—24no—
231C22⋊C4—342no—
232C22⋊C4—344no—
233C22⋊C4—344no—
234C22⋊C4—342no—
235C22⋊C4—342no—
236C22⋊C4—344no—
237C42—344no—
238C42—344no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23—21no—
246C23—341no—
247C23—341no—
248C23—342no—
249C23—342no—
250C23—341no—
251C23D4×D1712yes—
252C23#6325412yes—
253C23C2×C22⋊D1712yes—
254C23C23×D1711yescharacteristic
255D4—44no—
256D4—684no—
257D4—684no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—28no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—172no—
268C2×C4—174no—
269C2×C4—172no—
270C2×C4—174no—
271C2×C4—348no—
272C2×C4—348no—
273C2×C4—342no—
274C2×C4—342no—
275C2×C4—342no—
276C2×C4—341no—
277C2×C4—341no—
278C2×C4—342no—
279C2×C4—342no—
280C2×C4—342no—
281C2×C4—341no—
282C2×C4C2×C22⋊D1714yes—
283C22—22no—
284C22—22no—
285C22—24no—
286C22—24no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—42no—
292C22—681no—
293C22—681no—
294C22—681no—
295C22—682no—
296C22—682no—
297C22#1058762514yes—
298C22C2×D4×D1711yescharacteristic
299C22C2×#6325411yescharacteristic
300C22#41×D1712yes—
301C22#41×D1712yes—
302C22C22×C22⋊D1711yescharacteristic
303C4—24no—
304C4—41no—
305C4—42no—
306C4—174no—
307C4—342no—
308C4—344no—
309C4—681no—
310C4—682no—
311C2—24no—
312C2—81no—
313C2—1361no—
314C2C2×#1058762511yescharacteristic
315C2#123×D1711yescharacteristic
316C2#1176331711yescharacteristic
317C2#1176333612yes—
318C2#1176335512yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#123C2
prime radicalC2×#10587625C2
hypercenterC2×#123D17
hypocenterC17#2854
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#10587615C2semidirect
3. C22×C4×Dic17C2semidirect
4. #11762709C2semidirect
5. #11762704C2semidirect
6. #11762703C2semidirect
7. #11762699C2semidirect
8. #11762689C2semidirect
9. #11762686C2semidirect
10. #11762667C2semidirect
11. C2×C17×#123C2semidirect
47. C22×C4×C17D4semidirect
151. Dic17#123semidirect
153. C2×#123D17semidirect
198. C17#2854semidirect
221. C22×C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876152
C2×#105876252
C22×C4×Dic172
#117627092
#117627042
#117627032
#117626992
#117626892
#117626862
#117626672
C2×C17×#1232
#285417