Finite Group Databasebeta

← #16935268

Subgroups — #16935268

Order 4352.

512 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 83 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#14540094C211yesmaximal, characteristic
2#14538906C212yesmaximal, complemented
3#14527394C212yesmaximal, complemented
4#14526654C211yesmaximal, characteristic, complemented
5C17×#1985C211yesmaximal, characteristic, complemented
6#11763359C2212yes—
7#11763122—22no—
8#11763104—22no—
9#11763090—22no—
10#11763045—22no—
11#11763009—22no—
12#11763005—22no—
13#11762948—22no—
14#11762907—24no—
15#11762866C2212yes—
16#11762548—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×D136—22no—
35#10587561—22no—
36#10587560D412yes—
37C4×D68—22no—
38C4×D68—22no—
39#10587555D412yes—
40C17×C4⋊C8—22no—
41C17×#97—22no—
42#10587506—24no—
43C22⋊#3622—22no—
44#10587492—24no—
45#10587489—22no—
46#10587480—24no—
47#10587478—24no—
48C4⋊#3622—22no—
49#10587474D414yes—
50C2×Q8×C17—22no—
51C2×Q8×C17—42no—
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17C2×D411yescharacteristic
55C22×C4×C17C2×D411yescharacteristic
56C22×C4×C17#4111yescharacteristic
57C2×C22⋊D17—42no—
58#63252—84no—
59C2×D68—22no—
60C2×D68—22no—
61C2×D68—22no—
62C2×C4×D17—42no—
63C2×Dic34—22no—
64C2×C8×C17—22no—
65C2×C8×C17—22no—
66C2×C8×C17—41no—
67C2×C8×C17—41no—
68C2×C8×C17—42no—
69C4⋊C4×C17—22no—
70C4⋊C4×C17—42no—
71C42×C17—41no—
72C42×C17—41no—
73C42×C17C2×D412yes—
74C42×C17C2×D412yes—
75#63230—48no—
76#63227—42no—
77#63227—42no—
78C4⋊Dic17—22no—
79C4⋊Dic17—22no—
80C2×#3622—24no—
81D136—82no—
82#10028—171nomaximal, Sylow
83C23×C17#12111yescharacteristic
84C22×D17—42no—
85Q8×C17—24no—
86Q8×C17—42no—
87C2×C4×C17—22no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—22no—
91C2×C4×C17—41no—
92C2×C4×C17—41no—
93C2×C4×C17—41no—
94C2×C4×C17—41no—
95C2×C4×C17—42no—
96C2×C4×C17#12012yes—
97C2×C4×C17#12011yescharacteristic
98C2×C4×C17#12112yes—
99C2×C4×C17#12711yescharacteristic
100C22⋊D17—84no—
101C2×Dic17—42no—
102D68—44no—
103D68—42no—
104D68—42no—
105D68—82no—
106C4×D17—84no—
107Dic34—82no—
108C8×C17—42no—
109C8×C17—42no—
110#3622—44no—
111#3245—171no—
112#3112—172no—
113#1985D1711yescharacteristic, complemented
114#1530—172no—
115#1290—171no—
116C22×C17—21no—
117C22×C17—21no—
118C22×C17#39211yescharacteristic
119C22×C17#47212yes—
120C2×D17—82no—
121C2×D17—82no—
122C2×D17—82no—
123C4×C17—21no—
124C4×C17—22no—
125C4×C17—22no—
126C4×C17—42no—
127C4×C17—81no—
128C4×C17#45511yescharacteristic
129C4×C17#45611yescharacteristic
130Dic17—82no—
131#532—172no—
132C2×C4×Q8C2×D1711yescharacteristic
133#480—342no—
134#477—342no—
135#474—342no—
136#466—342no—
137#460—342no—
138#459—342no—
139#448—342no—
140#440—344no—
141#432—172no—
142C2×C4⋊C8C2×D1711yescharacteristic
143C2×#97C2×D1711yescharacteristic
144#336—22no—
145#329—344no—
146C2×C17—22no—
147C2×C17#200711yescharacteristic
148C2×C17#201011yescharacteristic
149C2×C17#201811yescharacteristic
150D17—162no—
151C2×#41—342no—
152C22×Q8D6811yescharacteristic
153C2×SD16—344no—
154C2×D8—342no—
155C22×C8—21no—
156C22×C8C22⋊D1712yes—
157C4⋊Q8—172no—
158#127—172no—
159#124—342no—
160#121—342no—
161#121—342no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×D4—342no—
165C4×D4—342no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4D6811yescharacteristic
168C2×C42—21no—
169C2×C42C22×D1711yescharacteristic
170#107—344no—
171#107—342no—
172C4⋊C8—22no—
173C4⋊C8—342no—
174#103—344no—
175#102—344no—
176#102—344no—
177C22⋊C8—342no—
178#97—22no—
179#97—174no—
180C17#1002811yescharacteristic, complemented, Sylow
181#41—1364no—
182C2×Q8—22no—
183C2×Q8—42no—
184C2×Q8—342no—
185C2×D4—342no—
186C2×D4—342no—
187C2×D4—342no—
188C2×D4—682no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—682no—
192C22×C4C2×D6811yescharacteristic
193C22×C4#6325211yescharacteristic
194C22×C4C2×C22⋊D1711yescharacteristic
195SD16—688no—
196D8—1362no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—344no—
203C4⋊C4—22no—
204C4⋊C4—42no—
205C4⋊C4—342no—
206C4⋊C4—342no—
207C22⋊C4—682no—
208C22⋊C4—682no—
209C42—41no—
210C42—41no—
211C42D4×D1712yes—
212C42D4×D1712yes—
213C23—682no—
214C23#1058761611yescharacteristic
215Q8—24no—
216Q8—42no—
217Q8—1362no—
218D4—684no—
219D4—682no—
220D4—682no—
221D4—1362no—
222D4—1364no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—682no—
233C2×C4—1364no—
234C2×C4#1058756812yes—
235C2×C4#1058757012yes—
236C2×C4#1058762311yescharacteristic
237C2×C4#1058762611yescharacteristic
238C8—42no—
239C8—42no—
240C8—684no—
241C22—21no—
242C22—21no—
243C22—1362no—
244C22—1362no—
245C22—1362no—
246C22#1176271811yescharacteristic
247C22#1176307912yes—
248C4—21no—
249C4—22no—
250C4—22no—
251C4—42no—
252C4—81no—
253C4—1362no—
254C4#1176298611yescharacteristic
255C4#1176299111yescharacteristic
256C2—22no—
257C2—2722no—
258C2#1452983011yescharacteristic
259C2#1452985111yescharacteristic
260C2#1452989311yescharacteristic
subgroupG/H
centerC22#11762718
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1985C2
socleC22×C17#392
prime radical#14540094C2
hypercenter#1985D17
hypocenterC17#10028
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. #14538906C2semidirect
3. #14527394C2semidirect
4. #14526654C2semidirect
5. C17×#1985C2semidirect
113. #1985D17semidirect
180. C17#10028semidirect

Maximal subgroups

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

subgroupindex
#145400942
#145389062
#145273942
#145266542
C17×#19852
#1002817