Finite Group Databasebeta

← #16935302

Subgroups — #16935302

Order 4352.

584 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#14539230C211yesmaximal, characteristic
2#14538815C212yesmaximal, complemented
3#14527182C212yesmaximal, complemented
4#14526615C211yesmaximal, characteristic, complemented
5C17×#1985C211yesmaximal, characteristic, complemented
6C2×#10587590—22no—
7#11763359C2212yes—
8#11763203—22no—
9#11763198—22no—
10#11763127—22no—
11#11763094—22no—
12#11763009—22no—
13#11762948—22no—
14C4×#63230—24no—
15#11762862C2212yes—
16#11762536—24no—
17C2×C4×Q8×C17C2211yescharacteristic
18C2×C17×C4⋊C8C2211yescharacteristic
19C2×C17×#97C2211yescharacteristic
20C17×#336—22no—
21C22×Q8×C17D411yescharacteristic
22C2×#63252—22no—
23C22×C8×C17—21no—
24C22×C8×C17D412yes—
25C4×Q8×C17—22no—
26C4×Q8×C17—22no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17D411yescharacteristic
29C2×C42×C17—21no—
30C2×C42×C17C2311yescharacteristic
31C2×#63230—24no—
32#10587616—22no—
33#10587616—22no—
34C2×Dic68—22no—
35#10587590—44no—
36#10587590—44no—
37C2×D136—22no—
38C2×#63219—24no—
39#10587561—22no—
40#10587560D412yes—
41C4×D68—22no—
42C4×D68—22no—
43#10587555D412yes—
44C17×C4⋊C8—22no—
45C17×#97—22no—
46#10587506—24no—
47C22⋊#3622—22no—
48#10587480—24no—
49#10587479—24no—
50C4⋊#3622—22no—
51C4×#3622D414yes—
52C2×Q8×C17—22no—
53C2×Q8×C17—42no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17C2×D411yescharacteristic
57C22×C4×C17C2×D411yescharacteristic
58C22×C4×C17#4111yescharacteristic
59C2×C22⋊D17—42no—
60#63252—44no—
61#63252—44no—
62C2×D68—22no—
63C2×D68—22no—
64C2×D68—22no—
65C2×C4×D17—42no—
66C2×Dic34—22no—
67C2×C8×C17—22no—
68C2×C8×C17—22no—
69C2×C8×C17—41no—
70C2×C8×C17—42no—
71C2×C8×C17SD1614yescomplemented
72C4⋊C4×C17—22no—
73C4⋊C4×C17—42no—
74C42×C17—41no—
75C42×C17—41no—
76C42×C17C2×D412yes—
77C42×C17C2×D412yes—
78#63230—48no—
79#63227—42no—
80#63227—42no—
81C4⋊Dic17—22no—
82C4⋊Dic17—22no—
83C2×#3622—24no—
84Dic68—44no—
85D136—44no—
86#63219—48no—
87#10034—171nomaximal, Sylow
88C23×C17#12111yescharacteristic
89C22×D17—42no—
90Q8×C17—24no—
91Q8×C17—42no—
92C2×C4×C17—22no—
93C2×C4×C17—41no—
94C2×C4×C17—41no—
95C2×C4×C17—41no—
96C2×C4×C17—41no—
97C2×C4×C17—42no—
98C2×C4×C17—41no—
99C2×C4×C17#12012yes—
100C2×C4×C17#12011yescharacteristic
101C2×C4×C17#12112yes—
102C2×C4×C17#12711yescharacteristic
103C2×C4×C17C2×SD1612yes—
104C2×C4×C17AGL(1,ℤ/8)12yes—
105C22⋊D17—84no—
106C2×Dic17—42no—
107D68—44no—
108D68—44no—
109D68—44no—
110C4×D17—82no—
111C4×D17—82no—
112Dic34—44no—
113C8×C17—24no—
114C8×C17—81no—
115#3622—44no—
116#3150—171no—
117#3103—172no—
118#1985D1711yescharacteristic, complemented
119#1480—172no—
120#1275—171no—
121C22×C17—21no—
122C22×C17—21no—
123C22×C17#39211yescharacteristic
124C22×C17#46012yes—
125C2×D17—82no—
126C2×D17—82no—
127C2×D17—82no—
128C4×C17—22no—
129C4×C17—22no—
130C4×C17—42no—
131C4×C17—81no—
132C4×C17#45012yes—
133C4×C17#49211yescharacteristic
134C4×C17#49611yescharacteristic
135Dic17—82no—
136C2×#135—342no—
137#532—172no—
138C2×C4×Q8C2×D1711yescharacteristic
139#497—342no—
140#496—342no—
141#481—342no—
142#475—342no—
143#460—342no—
144#448—342no—
145C4×SD16—344no—
146#431—172no—
147C2×C4⋊C8C2×D1711yescharacteristic
148C2×#97C2×D1711yescharacteristic
149#336—22no—
150D4⋊C8—344no—
151C2×C17—22no—
152C2×C17#201011yescharacteristic
153C2×C17#201411yescharacteristic
154C2×C17#201711yescharacteristic
155D17—162no—
156C2×#41—342no—
157C22×Q8D6811yescharacteristic
158#135—684no—
159#135—684no—
160C2×Q16—342no—
161C2×SD16—344no—
162C2×SD16—344no—
163C2×D8—342no—
164C22×C8—21no—
165C22×C8C22⋊D1712yes—
166C4⋊Q8—172no—
167#127—172no—
168#124—342no—
169#121—342no—
170#121—342no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×D4—342no—
174C4×D4—342no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4D6811yescharacteristic
177C2×C42—21no—
178C2×C42C22×D1711yescharacteristic
179#106—344no—
180C4⋊C8—22no—
181C4⋊C8—342no—
182#103—344no—
183#102—344no—
184C22⋊C8—342no—
185#97—22no—
186C4×C8—174no—
187C17#1003411yescharacteristic, complemented, Sylow
188#41—684no—
189#41—684no—
190C2×Q8—22no—
191C2×Q8—42no—
192C2×Q8—342no—
193C2×D4—342no—
194C2×D4—342no—
195C2×D4—342no—
196C2×D4—682no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—682no—
200C22×C4C2×D6811yescharacteristic
201C22×C4#6325211yescharacteristic
202C22×C4C2×C22⋊D1711yescharacteristic
203Q16—684no—
204SD16—688no—
205SD16—688no—
206D8—684no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—344no—
212C2×C8#6323014yescomplemented
213C4⋊C4—22no—
214C4⋊C4—42no—
215C4⋊C4—342no—
216C4⋊C4—342no—
217C22⋊C4—682no—
218C22⋊C4—682no—
219C42—41no—
220C42—41no—
221C42D4×D1712yes—
222C42D4×D1712yes—
223C23—682no—
224C23#1058761611yescharacteristic
225Q8—24no—
226Q8—42no—
227Q8—684no—
228D4—684no—
229D4—684no—
230D4—684no—
231D4—1364no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—682no—
240C2×C4—1362no—
241C2×C4—1362no—
242C2×C4#1058756812yes—
243C2×C4#1058757012yes—
244C2×C4#1058762311yescharacteristic
245C2×C4#1058762611yescharacteristic
246C2×C4C2×#6323012yes—
247C2×C4#1058763512yes—
248C8—24no—
249C8—81no—
250C8—684no—
251C22—21no—
252C22—21no—
253C22—1362no—
254C22—1362no—
255C22—1362no—
256C22#1176271811yescharacteristic
257C22#1176300912yes—
258C4—22no—
259C4—22no—
260C4—42no—
261C4—81no—
262C4—1362no—
263C4#1176295912yes—
264C4#1176318311yescharacteristic
265C4#1176319911yescharacteristic
266C2—22no—
267C2—2722no—
268C2#1452985111yescharacteristic
269C2#1452987111yescharacteristic
270C2#1452988811yescharacteristic
subgroupG/H
centerC2×C4#10587626
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1985C2
socleC22×C17#392
prime radical#14539230C2
hypercenter#1985D17
hypocenterC17#10034
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. #14538815C2semidirect
3. #14527182C2semidirect
4. #14526615C2semidirect
5. C17×#1985C2semidirect
71. C2×C8×C17SD16semidirect
118. #1985D17semidirect
187. C17#10034semidirect
212. C2×C8#63230semidirect

Maximal subgroups

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

subgroupindex
#145392302
#145388152
#145271822
#145266152
C17×#19852
#1003417