Finite Group Databasebeta

← #14533696

Subgroups — #14533696

Order 2176.

788 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 161 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×#10587570C212yesmaximal, complemented
2C2×#10587625C212yesmaximal, complemented
3C2×#10587567C212yesmaximal, complemented
4#11762726C211yesmaximal, characteristic, complemented
5#11762725C212yesmaximal, complemented
6#11762678C211yesmaximal, characteristic, complemented
7#11762668C212yesmaximal, complemented
8#11762667C212yesmaximal, complemented
9C17×#394C211yesmaximal, characteristic, complemented
10C22×D4×C17C2211yescharacteristic
11C22×C22⋊D17C2212yes—
12C22×C22⋊D17C2212yes—
13C23×Dic17C2212yes—
14C22×D68C2211yescharacteristic
15C22×C4×D17C2212yes—
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2212yes—
18C2×C22⋊C4×C17C2212yes—
19C2×C22⋊Dic17C2212yes—
20#10587625—28no—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×C4⋊Dic17C2211yescharacteristic
25C2×#63225C2212yes—
26C2×#63225C2212yes—
27C2×#63225C2212yes—
28C2×C4×Dic17C2211yescharacteristic
29#10587570—28no—
30#10587567—28no—
31C17×#95C2211yescharacteristic
32#10587502C2212yes—
33#10587502C2211yescharacteristic
34C24×C17C2312yes—
35C23×D17C2312yes—
36C2×D4×C17—24no—
37C2×D4×C17—41no—
38C22×C4×C17C2312yes—
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2311yescharacteristic
42C2×C22⋊D17—24no—
43C2×C22⋊D17—24no—
44C2×C22⋊D17—24no—
45C2×C22⋊D17—24no—
46C2×C22⋊D17—42no—
47C2×C22⋊D17D418yescomplemented
48C22×Dic17—24no—
49C22×Dic17—24no—
50C22×Dic17—24no—
51C22×Dic17C2312yes—
52C22×Dic17C2312yes—
53C22×Dic17C2312yes—
54C2×D68—24no—
55C2×D68—41no—
56C2×C4×D17—24no—
57C2×C4×D17—24no—
58C2×C4×D17—24no—
59C4⋊C4×C17—41no—
60C22⋊C4×C17—24no—
61C22⋊C4×C17—24no—
62C22⋊Dic17—24no—
63#63227—24no—
64#63227—24no—
65#63227—24no—
66C4⋊Dic17—41no—
67#63225—24no—
68#63225—24no—
69#63225—24no—
70C4×Dic17D414yescomplemented
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17C2×D414yes—
78C23×C17C2411yescharacteristic
79C22×D17—22no—
80C22×D17—22no—
81C22×D17—22no—
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—22no—
85C22×D17C2×D414yes—
86D4×C17—44no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17C2×D412yes—
97C22⋊D17—216no—
98C22⋊D17—48no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—48no—
106C2×Dic17C2×D414yes—
107C2×Dic17C2×D414yes—
108C2×Dic17#4114yes—
109D68—44no—
110C4×D17—48no—
111#2537—171nomaximal, Sylow
112C22×C17—24no—
113C22×C17—24no—
114C22×C17—24no—
115C22×C17—42no—
116C22×C17—42no—
117C22×C17—42no—
118C22×C17—42no—
119C22×C17C22×D411yescharacteristic
120C22×C17C22×D412yes—
121C22×C17C2×#4112yes—
122C22×C17H(5,𝔽2)11yescharacteristic
123C22×C172−1+411yescharacteristic
124C2×D17—24no—
125C2×D17—24no—
126C2×D17—24no—
127C2×D17—42no—
128C2×D17—42no—
129C2×D17—42no—
130C2×D17—42no—
131C4×C17—22no—
132C4×C17—42no—
133C4×C17—41no—
134C4×C17—41no—
135Dic17—42no—
136Dic17—42no—
137Dic17#12118yescomplemented
138C2×#121—172no—
139C2×#121—172no—
140C2×C4×D4—172no—
141#394—171no—
142#394—172no—
143#394D1711yescharacteristic, complemented
144#384—171no—
145#380—172no—
146#380—172no—
147C2×C17—44no—
148C2×C17C2×#12112yes—
149C2×C17#53711yescharacteristic
150C2×C17D4211yescharacteristic
151C2×C17#54712yes—
152C2×C17#55511yescharacteristic
153D17—44no—
154C22×D4—172no—
155C22×D4—172no—
156C22×D4—171no—
157C22×D4C2×D1711yescharacteristic
158C23×C4—172no—
159C23×C4—172no—
160#121—348no—
161#121—348no—
162C4×D4—348no—
163C2×C4⋊C4—172no—
164C2×C4⋊C4—172no—
165C2×C4⋊C4—172no—
166C2×C4⋊C4—171no—
167C2×C4⋊C4C2×D1711yescharacteristic
168C2×C22⋊C4—172no—
169C2×C22⋊C4—172no—
170C2×C22⋊C4—172no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4C2×D1712yes—
173C2×C22⋊C4C2×D1712yes—
174C2×C42—171no—
175#95—171no—
176#95—172no—
177#95C2×D1711yescharacteristic
178C17#253711yescharacteristic, complemented, Sylow
179C24—172no—
180C24C22×D1712yes—
181C2×D4—24no—
182C2×D4—41no—
183C2×D4—178no—
184C2×D4—344no—
185C2×D4—344no—
186C2×D4—344no—
187C2×D4—344no—
188C2×D4—344no—
189C2×D4—681no—
190C2×D4—682no—
191C22×C4—172no—
192C22×C4—172no—
193C22×C4—172no—
194C22×C4—344no—
195C22×C4—344no—
196C22×C4—344no—
197C22×C4—344no—
198C22×C4—344no—
199C22×C4—344no—
200C22×C4C22×D1712yes—
201C22×C4C22×D1711yescharacteristic
202C22×C4C22×D1711yescharacteristic
203C22×C4C22×D1711yescharacteristic
204C4⋊C4—41no—
205C4⋊C4—344no—
206C4⋊C4—344no—
207C4⋊C4—344no—
208C4⋊C4—681no—
209C22⋊C4—24no—
210C22⋊C4—24no—
211C22⋊C4—344no—
212C22⋊C4—344no—
213C22⋊C4—344no—
214C22⋊C4—344no—
215C42—174no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—174no—
223C23—342no—
224C23—342no—
225C23—342no—
226C23—342no—
227C23—342no—
228C23—342no—
229C23D4×D1714yes—
230C23C23×D1711yescharacteristic
231D4—44no—
232D4—3416no—
233D4—684no—
234D4—688no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—174no—
245C2×C4—174no—
246C2×C4—174no—
247C2×C4—342no—
248C2×C4—342no—
249C2×C4—342no—
250C2×C4—342no—
251C2×C4—342no—
252C2×C4—342no—
253C2×C4—688no—
254C2×C4—688no—
255C2×C4D4×D1712yes—
256C22—24no—
257C22—24no—
258C22—24no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—344no—
264C22—344no—
265C22—344no—
266C22—682no—
267C22—682no—
268C22—682no—
269C22—682no—
270C22C2×D4×D1712yes—
271C22C2×D4×D1711yescharacteristic
272C22#1058769111yescharacteristic
273C22#41×D1712yes—
274C22#1058769711yescharacteristic
275C4—22no—
276C4—42no—
277C4—41no—
278C4—41no—
279C4—178no—
280C4—682no—
281C4—682no—
282C2—44no—
283C2—684no—
284C2#121×D1712yes—
285C2#1176338611yescharacteristic
286C2#1176344111yescharacteristic
287C2#1176345712yes—
288C2#1176352111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
hypercenter#394D17
hypocenterC17#2537
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#10587570C2semidirect
2. C2×#10587625C2semidirect
3. C2×#10587567C2semidirect
4. #11762726C2semidirect
5. #11762725C2semidirect
6. #11762678C2semidirect
7. #11762668C2semidirect
8. #11762667C2semidirect
9. C17×#394C2semidirect
47. C2×C22⋊D17D4semidirect
70. C4×Dic17D4semidirect
137. Dic17#121semidirect
143. #394D17semidirect
178. C17#2537semidirect

Maximal subgroups

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

subgroupindex
C2×#105875702
C2×#105876252
C2×#105875672
#117627262
#117627252
#117626782
#117626682
#117626672
C17×#3942
#253717