Finite Group Databasebeta

← #14532423

Subgroups — #14532423

Order 2176.

800 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C22×Q8×D17C211yesmaximal, characteristic
2C2×#10587574C212yesmaximal, complemented
3C2×#10587582C211yesmaximal, characteristic
4#11762730C211yesmaximal, characteristic, complemented
5#11762729C211yesmaximal, characteristic, complemented
6#11762728C211yesmaximal, characteristic, complemented
7#11762713C212yesmaximal, complemented
8#11762683C212yesmaximal
9#11762680C212yesmaximal
10#11762656C211yesmaximal, characteristic
11C17×#395C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C2×Q8×D17—24no—
14C2×Q8×D17—28no—
15C22×C4×D17C2212yes—
16C22×C4×D17C2211yescharacteristic
17C22×Dic34C2211yescharacteristic
18C22×Dic34C2211yescharacteristic
19C22×Dic34C2212yes—
20C2×C4⋊C4×C17C2211yescharacteristic
21C2×C4⋊C4×C17C2212yes—
22C2×#63227C2211yescharacteristic
23C2×#63227C2211yescharacteristic
24C2×#63227C2212yes—
25C2×C4⋊Dic17C2212yes—
26C2×#63225C2211yescharacteristic
27C2×#63225C2211yescharacteristic
28C2×#63225C2212yes—
29C2×#63225C2212yes—
30C2×C4×Dic17C2212yes—
31#10587582—24no—
32#10587574—28no—
33C17×#95C2212yes—
34C17×#95C2211yescharacteristic
35#10587502C2212yes—
36#10587502C2212yes—
37#10587502C2212yes—
38#10587502C2211yescharacteristic
39C23×D17C2311yescharacteristic
40C2×Q8×C17—22no—
41C2×Q8×C17—24no—
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C22×C4×C17C2312yes—
46C22×C4×C17C2312yes—
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2312yes—
50C22×Dic17C2312yes—
51C22×Dic17C2311yescharacteristic
52Q8×D17—232no—
53C2×C4×D17—24no—
54C2×C4×D17—24no—
55C2×C4×D17—24no—
56C2×C4×D17—24no—
57C2×C4×D17D414yes—
58C2×Dic34—22no—
59C2×Dic34—24no—
60C2×Dic34—24no—
61C2×Dic34—24no—
62C2×Dic34—24no—
63C2×Dic34—22no—
64C2×Dic34D418yes—
65C4⋊C4×C17—24no—
66C4⋊C4×C17—22no—
67#63227—22no—
68#63227—22no—
69#63227Q818yescomplemented
70C4⋊Dic17—24no—
71#63225—24no—
72#63225—22no—
73#63225—24no—
74#63225—22no—
75C4×Dic17—24no—
76C23×C17C2411yescharacteristic
77C22×D17—22no—
78C22×D17—21no—
79C22×D17—21no—
80C22×D17C2×D412yes—
81C22×D17C2×Q814yes—
82Q8×C17—28no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—21no—
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17C2×D412yes—
94C2×C4×C17C2×D414yes—
95C2×C4×C17C2×Q814yescomplemented
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—21no—
102C2×Dic17—21no—
103C2×Dic17—21no—
104C2×Dic17—21no—
105C2×Dic17C2×D414yes—
106C2×Dic17C2×D412yes—
107C2×Dic17C2×D414yes—
108C2×Dic17C2×Q814yescomplemented
109C4×D17—216no—
110C4×D17—28no—
111Dic34—28no—
112Dic34—216no—
113Dic34—44no—
114#2434—171nomaximal, Sylow
115C22×C17C22×D411yescharacteristic
116C22×C17C22×D412yes—
117C22×C17C22×Q812yes—
118C22×C172−1+411yescharacteristic
119C22×C172−1+411yescharacteristic
120C2×D17—22no—
121C2×D17—22no—
122C2×D17—24no—
123C2×D17—24no—
124C2×D17C4⋊Q814yescomplemented
125C4×C17—22no—
126C4×C17—24no—
127C4×C17—24no—
128C4×C17—41no—
129C4×C17—41no—
130Dic17—24no—
131Dic17—24no—
132Dic17—41no—
133Dic17—41no—
134Dic17#12014yes—
135C23×Q8—171no—
136C2×C4⋊Q8—172no—
137C2×C22⋊Q8—171no—
138#395—171no—
139#395—171no—
140#395—171no—
141#395D1711yescharacteristic, complemented
142#391—172no—
143#385—172no—
144#385—172no—
145C4×C22⋊C4—171no—
146C2×C17C2×#12011yescharacteristic
147C2×C17C2×C4⋊Q811yescharacteristic
148C2×C17#53611yescharacteristic
149C2×C17D4×Q812yes—
150C2×C17D4×Q812yes—
151D17—24no—
152C22×Q8—172no—
153C22×Q8—171no—
154C22×Q8—171no—
155C22×Q8—344no—
156C22×Q8—348no—
157C22×Q8C2×D1711yescharacteristic
158C23×C4—171no—
159C23×C4—172no—
160C4⋊Q8—348no—
161C22⋊Q8—344no—
162C2×C4⋊C4—171no—
163C2×C4⋊C4—171no—
164C2×C4⋊C4—172no—
165C2×C4⋊C4—172no—
166C2×C4⋊C4—172no—
167C2×C4⋊C4C2×D1711yescharacteristic
168C2×C4⋊C4C2×D1712yes—
169C2×C22⋊C4—172no—
170C2×C22⋊C4—171no—
171C2×C22⋊C4—171no—
172C2×C42—172no—
173#95—171no—
174#95—172no—
175#95—172no—
176#95—172no—
177#95C2×D1711yescharacteristic
178#95C2×D1712yes—
179C17#243411yescharacteristic, complemented, Sylow
180C24—171no—
181C2×Q8—22no—
182C2×Q8—24no—
183C2×Q8—178no—
184C2×Q8—344no—
185C2×Q8—344no—
186C2×Q8—344no—
187C2×Q8—342no—
188C2×Q8—344no—
189C2×Q8—3432no—
190C2×Q8—342no—
191C22×C4—171no—
192C22×C4—172no—
193C22×C4—172no—
194C22×C4—174no—
195C22×C4—171no—
196C22×C4—171no—
197C22×C4—344no—
198C22×C4—344no—
199C22×C4—344no—
200C22×C4—344no—
201C22×C4C22×D1711yescharacteristic
202C22×C4C22×D1711yescharacteristic
203C22×C4C22×D1711yescharacteristic
204C22×C4C22×D1712yes—
205C22×C4C22×D1712yes—
206C4⋊C4—22no—
207C4⋊C4—24no—
208C4⋊C4—342no—
209C4⋊C4—344no—
210C4⋊C4—344no—
211C4⋊C4—342no—
212C4⋊C4—344no—
213C22⋊C4—178no—
214C22⋊C4—342no—
215C22⋊C4—342no—
216C42—344no—
217C23—172no—
218C23—174no—
219C23—342no—
220C23—341no—
221C23—341no—
222C23C23×D1711yescharacteristic
223Q8—28no—
224Q8—3416no—
225Q8—348no—
226Q8—684no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—174no—
238C2×C4—172no—
239C2×C4—174no—
240C2×C4—174no—
241C2×C4—348no—
242C2×C4—3416no—
243C2×C4—342no—
244C2×C4—342no—
245C2×C4—342no—
246C2×C4—342no—
247C2×C4—341no—
248C2×C4—341no—
249C2×C4—342no—
250C2×C4—341no—
251C2×C4—341no—
252C2×C4D4×D1712yes—
253C2×C4D4×D1714yes—
254C2×C4Q8×D1714yescomplemented
255C22—174no—
256C22—344no—
257C22—344no—
258C22—342no—
259C22—342no—
260C22C2×D4×D1712yes—
261C22C2×D4×D1711yescharacteristic
262C22C2×Q8×D1712yes—
263C22#1058769411yescharacteristic
264C22#1058769711yescharacteristic
265C4—24no—
266C4—22no—
267C4—24no—
268C4—41no—
269C4—41no—
270C4—174no—
271C4—344no—
272C4—344no—
273C4—681no—
274C4—681no—
275C2—344no—
276C2#120×D1711yescharacteristic
277C2C4⋊Q8×D1711yescharacteristic
278C2#1176338211yescharacteristic
279C2#1176347212yes—
280C2#1176347412yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#395C2
prime radicalC23×D17C23
hypercenter#395D17
hypocenterC17#2434
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#10587574C2semidirect
4. #11762730C2semidirect
5. #11762729C2semidirect
6. #11762728C2semidirect
7. #11762713C2semidirect
11. C17×#395C2semidirect
69. #63227Q8semidirect
95. C2×C4×C17C2×Q8semidirect
108. C2×Dic17C2×Q8semidirect
124. C2×D17C4⋊Q8semidirect
141. #395D17semidirect
179. C17#2434semidirect
254. C2×C4Q8×D17semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D172
C2×#105875742
C2×#105875822
#117627302
#117627292
#117627282
#117627132
#117626832
#117626802
#117626562
C17×#3952
#243417