Finite Group Databasebeta

← #14534405

Subgroups — #14534405

Order 2176.

582 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 145 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×#10587562C211yesmaximal, characteristic, complemented
2#11762740C211yesmaximal, characteristic, complemented
3#11762735C211yesmaximal, characteristic, complemented
4#11762723C211yesmaximal, characteristic, complemented
5#11762717C211yesmaximal, characteristic
6#11762709C212yesmaximal, complemented
7#11762703C212yesmaximal, complemented
8#11762688C212yesmaximal, complemented
9#11762682C212yesmaximal, complemented
10#11762655C211yesmaximal, characteristic, complemented
11C2×C4×D4×C17C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C23×C4×C17C2212yes—
14C22×C22⋊D17C2212yes—
15C4×D4×C17—24no—
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2212yes—
18C2×C42×C17C2211yescharacteristic
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21C2×C22⋊Dic17C2212yes—
22C2×#63227C2211yescharacteristic
23C2×#63227C2211yescharacteristic
24C2×#63227C2211yescharacteristic
25C2×#63227C2212yes—
26C2×#63225C2211yescharacteristic
27C2×#63225C2211yescharacteristic
28C2×#63225C2211yescharacteristic
29C2×#63225C2212yes—
30C2×C4×Dic17C2212yes—
31#10587562—24no—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2211yescharacteristic
36#10587502C2211yescharacteristic
37C24×C17C2312yes—
38C23×D17C2311yescharacteristic
39C2×D4×C17—24no—
40C2×D4×C17—22no—
41C22×C4×C17—24no—
42C22×C4×C17—42no—
43C22×C4×C17—42no—
44C22×C4×C17C2312yes—
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2311yescharacteristic
47C22×C4×C17C2311yescharacteristic
48C2×C22⋊D17—24no—
49C2×C22⋊D17—24no—
50C2×C22⋊D17—42no—
51C22×Dic17C2312yes—
52C22×Dic17C2311yescharacteristic
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2311yescharacteristic
55C22×Dic17C2312yes—
56C4⋊C4×C17—22no—
57C22⋊C4×C17—24no—
58C42×C17D414yescomplemented
59C22⋊Dic17—24no—
60C22⋊Dic17—24no—
61C22⋊Dic17—42no—
62#63227—22no—
63#63227—22no—
64#63227—22no—
65#63227—42no—
66#63225—22no—
67#63225—22no—
68#63225—22no—
69#63225—42no—
70C4×Dic17—24no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17#4114yes—
78C23×C17C2411yescharacteristic
79C22×D17—21no—
80C22×D17—22no—
81C22×D17—22no—
82C22×D17—21no—
83C22×D17—21no—
84D4×C17—28no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—48no—
93C2×C4×C17C2×D412yes—
94C2×C4×C17C2×D412yes—
95C2×C4×C17C2×D412yes—
96C22⋊D17—48no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—21no—
100C2×Dic17—21no—
101C2×Dic17—21no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—21no—
107C2×Dic17—22no—
108C2×Dic17—21no—
109C2×Dic17#4114yes—
110#2603—171nomaximal, Sylow
111C22×C17—24no—
112C22×C17—24no—
113C22×C17—24no—
114C22×C17—42no—
115C22×C17—42no—
116C22×C17—42no—
117C22×C17—42no—
118C22×C17C22×D411yescharacteristic
119C22×C17C2×#4112yes—
120C22×C17C2×#4112yes—
121C22×C17H(5,𝔽2)11yescharacteristic
122C22×C17H(5,𝔽2)11yescharacteristic
123C2×D17—41no—
124C2×D17—41no—
125C2×D17—41no—
126C2×D17—42no—
127C2×D17—42no—
128C4×C17—22no—
129C4×C17—22no—
130C4×C17—22no—
131C4×C17—42no—
132Dic17—24no—
133Dic17—41no—
134Dic17—41no—
135Dic17—41no—
136Dic17—42no—
137C2×#126—171no—
138C2×C4×D4D1711yescharacteristic, complemented
139#398—171no—
140#397—171no—
141#394—171no—
142#392—171no—
143#390—172no—
144#388—172no—
145#386—172no—
146#385—172no—
147#376—171no—
148C2×C17—44no—
149C2×C17#52511yescharacteristic
150C2×C17#53211yescharacteristic
151C2×C17#53511yescharacteristic
152C2×C17#55112yes—
153C2×C17#55312yes—
154D17—81no—
155C22×D4—172no—
156C22×D4C2×D1711yescharacteristic
157C23×C4C2×D1712yes—
158#126—344no—
159C4×D4—24no—
160C2×C4⋊C4—171no—
161C2×C4⋊C4—171no—
162C2×C4⋊C4—171no—
163C2×C4⋊C4—172no—
164C2×C4⋊C4C2×D1711yescharacteristic
165C2×C22⋊C4—172no—
166C2×C22⋊C4—172no—
167C2×C22⋊C4—171no—
168C2×C22⋊C4—171no—
169C2×C22⋊C4—171no—
170C2×C22⋊C4—172no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4C2×D1712yes—
173C2×C42—172no—
174C2×C42C2×D1711yescharacteristic
175#95—172no—
176#95—171no—
177#95—171no—
178#95—172no—
179#95—172no—
180C17#260311yescharacteristic, complemented, Sylow
181C24—171no—
182C24C22×D1712yes—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—344no—
186C2×D4—344no—
187C2×D4—682no—
188C22×C4—24no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—172no—
192C22×C4—172no—
193C22×C4—171no—
194C22×C4—171no—
195C22×C4—171no—
196C22×C4C22×D1712yes—
197C22×C4C22×D1711yescharacteristic
198C22×C4C22×D1711yescharacteristic
199C22×C4C22×D1711yescharacteristic
200C4⋊C4—22no—
201C4⋊C4—342no—
202C4⋊C4—342no—
203C4⋊C4—342no—
204C4⋊C4—682no—
205C22⋊C4—24no—
206C22⋊C4—344no—
207C22⋊C4—344no—
208C22⋊C4—342no—
209C22⋊C4—342no—
210C22⋊C4—342no—
211C22⋊C4—682no—
212C22⋊C4—682no—
213C42—344no—
214C42C22⋊D1714yescomplemented
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—342no—
222C23—341no—
223C23—341no—
224C23—342no—
225C23—341no—
226C23#6325214yes—
227C23C23×D1711yescharacteristic
228D4—28no—
229D4—688no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—48no—
238C2×C4—174no—
239C2×C4—342no—
240C2×C4—342no—
241C2×C4—342no—
242C2×C4—342no—
243C2×C4—342no—
244C2×C4—341no—
245C2×C4—341no—
246C2×C4—342no—
247C2×C4—342no—
248C2×C4—341no—
249C2×C4—341no—
250C2×C4—341no—
251C2×C4C2×C22⋊D1712yes—
252C2×C4C2×C22⋊D1712yes—
253C2×C4C2×C22⋊D1712yes—
254C22—24no—
255C22—24no—
256C22—24no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—682no—
262C22—682no—
263C22—681no—
264C22—681no—
265C22—681no—
266C22C2×#6325212yes—
267C22#1058769111yescharacteristic
268C22#1058769111yescharacteristic
269C22#41×D1712yes—
270C22C22×C22⋊D1711yescharacteristic
271C4—22no—
272C4—22no—
273C4—22no—
274C4—42no—
275C4—344no—
276C4—682no—
277C4—681no—
278C4—681no—
279C4—681no—
280C2—44no—
281C2—1361no—
282C2#1176332011yescharacteristic
283C2#1176335511yescharacteristic
284C2#1176337611yescharacteristic
285C2#1176348812yes—
286C2#1176350812yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C4×D4×C17C2
prime radical#11762717C2
hypercenterC2×C4×D4D17
hypocenterC17#2603
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#10587562C2semidirect
2. #11762740C2semidirect
3. #11762735C2semidirect
4. #11762723C2semidirect
6. #11762709C2semidirect
7. #11762703C2semidirect
8. #11762688C2semidirect
9. #11762682C2semidirect
10. #11762655C2semidirect
11. C2×C4×D4×C17C2semidirect
58. C42×C17D4semidirect
138. C2×C4×D4D17semidirect
180. C17#2603semidirect
214. C42C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105875622
#117627402
#117627352
#117627232
#117627172
#117627092
#117627032
#117626882
#117626822
#117626552
C2×C4×D4×C172
#260317