Finite Group Databasebeta

← #16946923

Subgroups — #16946923

Order 4352.

360 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 59 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#14537056C211yesmaximal, characteristic, complemented
2#14527821C212yesmaximal
3#14526810C212yesmaximal, complemented
4#14526728C211yesmaximal, characteristic, complemented
5C17×#2040C211yesmaximal, characteristic
6#11763125—22no—
7#11762751—22no—
8#11762748C2212yes—
9#11762730—22no—
10#11762729—21no—
11#11762675—21no—
12#11762543—22no—
13#11762535C2212yes—
14C17×#537C2211yescharacteristic
15C2×C17×#104—21no—
16C17×#417—21no—
17C17×#382C2211yescharacteristic
18C17×#343C2211yescharacteristic
19C2×#41×C17—21no—
20C22×Dic34—21no—
21C2×M16×C17—21no—
22C17×C22⋊Q8—41no—
23C17×#121—22no—
24C17×#121—41no—
25C2×C4⋊C4×C17D411yescharacteristic
26C2×C4⋊C4×C17C2311yescharacteristic
27C2×C42×C17D411yescharacteristic
28C2×C4⋊Dic17—21no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17D412yes—
31C2×#63225—22no—
32C17×#104—42no—
33C17×#103—41no—
34C17×#102—41no—
35C17×#95—22no—
36#10587503—42no—
37#10587502—21no—
38#10587502—22no—
39#10587502—22no—
40#10587502—22no—
41#10587502D412yes—
42C22⋊#3622—22no—
43#10587479—42no—
44#41×C17—41no—
45#41×C17—41no—
46#41×C17—41no—
47#41×C17—41no—
48C2×Q8×C17—21no—
49C2×D4×C17—21no—
50C2×D4×C17—22no—
51C2×D4×C17—41no—
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—41no—
55C22×C4×C17C2×D412yes—
56C22×C4×C17C2×D411yescharacteristic
57C22×Dic17—22no—
58C22×Dic17—22no—
59C2×Dic34—82no—
60C2×Dic34—81no—
61M16×C17—41no—
62M16×C17—41no—
63C2×C8×C17—41no—
64C4⋊C4×C17—22no—
65C4⋊C4×C17—41no—
66C4⋊C4×C17—41no—
67C22⋊C4×C17—42no—
68C42×C17—21no—
69C42×C17SD1612yes—
70C4⋊Dic17—42no—
71C4⋊Dic17—81no—
72C4⋊Dic17—81no—
73#63225—82no—
74C2×#3622—42no—
75#12185—171nomaximal, Sylow
76C23×C17—41no—
77C23×C17#12011yescharacteristic
78Q8×C17—41no—
79Q8×C17—41no—
80D4×C17—41no—
81D4×C17—41no—
82D4×C17—82no—
83D4×C17—81no—
84D4×C17—81no—
85C2×C4×C17—21no—
86C2×C4×C17—21no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—41no—
90C2×C4×C17—41no—
91C2×C4×C17—42no—
92C2×C4×C17—42no—
93C2×C4×C17—41no—
94C2×C4×C17—41no—
95C2×C4×C17—81no—
96C2×C4×C17—81no—
97C2×C4×C17C2×SD1611yescharacteristic
98C2×C4×C17#13711yescharacteristic
99C2×Dic17—42no—
100C2×Dic17—42no—
101C2×Dic17—42no—
102C2×Dic17—42no—
103C2×Dic17—42no—
104C2×Dic17—42no—
105Dic34—84no—
106C8×C17—81no—
107#3622—82no—
108#2844—171no—
109#2040D1711yescharacteristic
110#1616—172no—
111#1353—172no—
112#1324—171no—
113C22×C17—21no—
114C22×C17—21no—
115C22×C17—81no—
116C22×C17—81no—
117C22×C17—81no—
118C22×C17#44811yescharacteristic
119C22×C17#45411yescharacteristic
120C22×C17B(4,𝔽2)11yescharacteristic
121C4×C17—21no—
122C4×C17—21no—
123C4×C17—42no—
124C4×C17—42no—
125C4×C17—81no—
126C4×C17—81no—
127Dic17—82no—
128Dic17—82no—
129#537C2×D1711yescharacteristic
130#481—342no—
131C2×#104—21no—
132#417—21no—
133#400—172no—
134#400—342no—
135#395—342no—
136#395—341no—
137#382—341no—
138#382C2×D1711yescharacteristic
139#343C2×D1711yescharacteristic
140#327—342no—
141#324—172no—
142C2×C17—21no—
143C2×C17—21no—
144C2×C17—161no—
145C2×C17#159911yescharacteristic
146C2×C17#220111yescharacteristic
147C2×C17#220211yescharacteristic
148C2×#41—21no—
149C22×Q8—341no—
150C2×M16—21no—
151C22⋊Q8—41no—
152#121—22no—
153#121—41no—
154C2×C4⋊C4—172no—
155C2×C4⋊C4—341no—
156C2×C4⋊C4—341no—
157C2×C4⋊C4—342no—
158C2×C4⋊C4C22⋊D1711yescharacteristic
159C2×C4⋊C4C22×D1711yescharacteristic
160C2×C42C22⋊D1711yescharacteristic
161#106—682no—
162#104—42no—
163#103—41no—
164#102—41no—
165#102—682no—
166C22⋊C8—342no—
167#95—22no—
168#95—172no—
169#95—342no—
170#95—341no—
171#95—342no—
172#95—342no—
173C17#1218511yescharacteristic, complemented, Sylow
174#41—41no—
175#41—41no—
176#41—41no—
177#41—41no—
178C2×Q8—21no—
179C2×Q8—1362no—
180C2×Q8—1361no—
181C2×D4—21no—
182C2×D4—22no—
183C2×D4—41no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—41no—
187C22×C4—342no—
188C22×C4—342no—
189C22×C4D4×D1712yes—
190C22×C4C2×C22⋊D1711yescharacteristic
191M16—41no—
192M16—41no—
193C2×C8—41no—
194C2×C8—682no—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—682no—
199C4⋊C4—1362no—
200C4⋊C4—1361no—
201C4⋊C4—1361no—
202C22⋊C4—42no—
203C42—21no—
204C42#6322912yes—
205C23—41no—
206C23#1058762311yescharacteristic
207Q8—41no—
208Q8—41no—
209Q8—1364no—
210D4—41no—
211D4—41no—
212D4—82no—
213D4—81no—
214D4—81no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—682no—
228C2×C4—682no—
229C2×C4—682no—
230C2×C4—682no—
231C2×C4—682no—
232C2×C4—682no—
233C2×C4C2×#6322911yescharacteristic
234C2×C4#1058763711yescharacteristic
235C8—81no—
236C8—1362no—
237C22—21no—
238C22—21no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22#1176294711yescharacteristic
243C22#1176298111yescharacteristic
244C22#1176299511yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—81no—
250C4—81no—
251C4—1362no—
252C4—1362no—
253C2—21no—
254C2—21no—
255C2—161no—
256C2#1452772311yescharacteristic
257C2#1453079611yescharacteristic
258C2#1453080011yescharacteristic
subgroupG/H
centerC22#11762995
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2040C2
socleC22×C17B(4,𝔽2)
prime radicalC17×#537C22
hypercenter#2040D17
hypocenterC17#12185
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2040

Complements

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

NH ≅ G/Ntype
1. #14537056C2semidirect
3. #14526810C2semidirect
4. #14526728C2semidirect
173. C17#12185semidirect

Maximal subgroups

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

subgroupindex
#145370562
#145278212
#145268102
#145267282
C17×#20402
#1218517