Finite Group Databasebeta

← #14534343

Subgroups — #14534343

Order 2176.

604 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 139 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×D4×Dic17C211yesmaximal, characteristic, complemented
2C2×#10587563C211yesmaximal, characteristic, complemented
3#11762749C211yesmaximal, characteristic, complemented
4#11762745C211yesmaximal, characteristic, complemented
5#11762723C211yesmaximal, characteristic, complemented
6#11762703C212yesmaximal, complemented
7#11762689C212yesmaximal, complemented
8#11762689C212yesmaximal, complemented
9#11762681C212yesmaximal, complemented
10#11762676C211yesmaximal, characteristic, complemented
11C17×#392C211yesmaximal, characteristic
12C22×D4×C17C2211yescharacteristic
13C22×D4×C17C2212yes—
14C23×Dic17C2211yescharacteristic
15C23×Dic17C2212yes—
16C2×C22⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2212yes—
18C2×C22⋊Dic17C2211yescharacteristic
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21C2×C22⋊Dic17C2212yes—
22C2×C22⋊Dic17C2212yes—
23D4×Dic17—24no—
24C2×C4⋊Dic17C2211yescharacteristic
25C2×#63225C2211yescharacteristic
26C2×#63225C2211yescharacteristic
27C2×#63225C2212yes—
28C2×C4×Dic17C2211yescharacteristic
29C2×C4×Dic17C2211yescharacteristic
30#10587563—24no—
31C17×#95C2211yescharacteristic
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C24×C17C2311yescharacteristic
38C24×C17C2312yes—
39C2×D4×C17—24no—
40C2×D4×C17—24no—
41C2×D4×C17—24no—
42C2×D4×C17—22no—
43C2×D4×C17—42no—
44C22×C4×C17C2312yes—
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2311yescharacteristic
47C22×Dic17—22no—
48C22×Dic17—22no—
49C22×Dic17—22no—
50C22×Dic17—24no—
51C22×Dic17—42no—
52C22×Dic17—42no—
53C22×Dic17C2312yes—
54C22×Dic17C2312yes—
55C22×Dic17C2311yescharacteristic
56C22×Dic17C2311yescharacteristic
57C22×Dic17C2311yescharacteristic
58C22×Dic17C2311yescharacteristic
59C22⋊C4×C17—22no—
60C22⋊C4×C17—42no—
61C22⋊Dic17—24no—
62C22⋊Dic17—22no—
63C22⋊Dic17—24no—
64C22⋊Dic17—24no—
65C22⋊Dic17—42no—
66C4⋊Dic17—22no—
67#63225—22no—
68#63225—22no—
69#63225—42no—
70C4×Dic17—22no—
71C4×Dic17D414yescomplemented
72C23×C17—21no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—21no—
76C23×C17—22no—
77C23×C17—22no—
78C23×C17—22no—
79C23×C17—22no—
80C23×C17—22no—
81C23×C17—22no—
82C23×C17#4114yes—
83C23×C17#4112yes—
84C23×C17C2411yescharacteristic
85D4×C17—28no—
86D4×C17—48no—
87C2×C4×C17—22no—
88C2×C4×C17—21no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17C2×D412yes—
94C2×Dic17—28no—
95C2×Dic17—21no—
96C2×Dic17—21no—
97C2×Dic17—21no—
98C2×Dic17—21no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—21no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—48no—
109C2×Dic17C2×D412yes—
110C2×Dic17C2×D412yes—
111C2×Dic17#4112yes—
112#2598—171nomaximal, Sylow
113C22×C17—22no—
114C22×C17—24no—
115C22×C17—24no—
116C22×C17—24no—
117C22×C17—24no—
118C22×C17—41no—
119C22×C17—42no—
120C22×C17—42no—
121C22×C17—41no—
122C22×C17—42no—
123C22×C17—42no—
124C22×C17—42no—
125C22×C17C22×D411yescharacteristic
126C22×C17C2×#4111yescharacteristic
127C22×C17C2×#4112yes—
128C22×C17C2×#4111yescharacteristic
129C22×C17H(5,𝔽2)12yes—
130C4×C17—22no—
131C4×C17—41no—
132C4×C17—42no—
133Dic17—22no—
134Dic17—22no—
135Dic17—22no—
136Dic17—41no—
137Dic17—42no—
138Dic17—42no—
139C2×C4×D4—171no—
140C2×#117—171no—
141#400—171no—
142#399—171no—
143#394—171no—
144#392D1711yescharacteristic
145#388—172no—
146#386—172no—
147#386—172no—
148#385—172no—
149#383—171no—
150C2×C17—42no—
151C2×C17—44no—
152C2×C17#52511yescharacteristic
153C2×C17#52511yescharacteristic
154C2×C17#53511yescharacteristic
155C2×C17#55112yes—
156C2×C17#55312yes—
157C22×D4C2×D1711yescharacteristic
158C22×D4C2×D1712yes—
159C23×C4—171no—
160C23×C4—172no—
161C4×D4—344no—
162#117—344no—
163C2×C4⋊C4—171no—
164C2×C4⋊C4—171no—
165C2×C4⋊C4—171no—
166C2×C4⋊C4—172no—
167C2×C22⋊C4—171no—
168C2×C22⋊C4—172no—
169C2×C22⋊C4—172no—
170C2×C22⋊C4—172no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4C2×D1711yescharacteristic
173C2×C22⋊C4C2×D1712yes—
174C2×C42—171no—
175C2×C42—171no—
176#95—172no—
177#95—172no—
178#95—171no—
179#95—172no—
180#95—172no—
181#95C2×D1711yescharacteristic
182C17#259811yescharacteristic, complemented, Sylow
183C24C22×D1711yescharacteristic
184C24C22×D1712yes—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—22no—
189C2×D4—42no—
190C22×C4—171no—
191C22×C4—171no—
192C22×C4—171no—
193C22×C4—172no—
194C22×C4—171no—
195C22×C4—172no—
196C22×C4—342no—
197C22×C4—342no—
198C22×C4—344no—
199C22×C4—342no—
200C22×C4—682no—
201C22×C4—682no—
202C22×C4C22×D1712yes—
203C22×C4C22×D1711yescharacteristic
204C22×C4C22×D1711yescharacteristic
205C4⋊C4—342no—
206C4⋊C4—342no—
207C4⋊C4—342no—
208C4⋊C4—682no—
209C22⋊C4—22no—
210C22⋊C4—42no—
211C22⋊C4—344no—
212C22⋊C4—342no—
213C22⋊C4—344no—
214C22⋊C4—344no—
215C22⋊C4—682no—
216C42—174no—
217C42—342no—
218C23—21no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—21no—
228C23#6325414yes—
229C23#6325412yes—
230C23C23×D1711yescharacteristic
231D4—28no—
232D4—48no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—172no—
240C2×C4—172no—
241C2×C4—172no—
242C2×C4—348no—
243C2×C4—341no—
244C2×C4—341no—
245C2×C4—341no—
246C2×C4—342no—
247C2×C4—341no—
248C2×C4—341no—
249C2×C4—342no—
250C2×C4—342no—
251C2×C4—342no—
252C2×C4—342no—
253C2×C4—342no—
254C2×C4—342no—
255C2×C4—342no—
256C2×C4—688no—
257C2×C4D4×D1712yes—
258C22—24no—
259C22—24no—
260C22—24no—
261C22—24no—
262C22—22no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—41no—
267C22—42no—
268C22—42no—
269C22—41no—
270C22C2×D4×D1711yescharacteristic
271C22C2×#6325411yescharacteristic
272C22C2×#6325412yes—
273C22#1058769112yes—
274C22#41×D1711yescharacteristic
275C4—22no—
276C4—42no—
277C4—41no—
278C4—342no—
279C4—342no—
280C4—342no—
281C4—682no—
282C4—681no—
283C4—682no—
284C2—42no—
285C2—44no—
286C2#1176331511yescharacteristic
287C2#1176331911yescharacteristic
288C2#1176337211yescharacteristic
289C2#1176349012yes—
290C2#1176350212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
Fitting, prime radicalC17×#392C2
hypercenter#392D17
hypocenterC17#2598
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic17C2semidirect
2. C2×#10587563C2semidirect
3. #11762749C2semidirect
4. #11762745C2semidirect
5. #11762723C2semidirect
6. #11762703C2semidirect
7. #11762689C2semidirect
8. #11762689C2semidirect
9. #11762681C2semidirect
10. #11762676C2semidirect
71. C4×Dic17D4semidirect
182. C17#2598semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic172
C2×#105875632
#117627492
#117627452
#117627232
#117627032
#117626892
#117626812
#117626762
C17×#3922
#259817