Finite Group Databasebeta

← #16937561

Subgroups — #16937561

Order 4352.

414 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 75 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#14534449C211yesmaximal, characteristic
2#14528883C212yesmaximal
3#14526692C211yesmaximal, characteristic
4#14526636C212yesmaximal
5C17×#1820C211yesmaximal, characteristic
6#11762773C2212yes—
7#11762745—21no—
8#11762721—21no—
9#11762713—22no—
10#11762696—22no—
11C4×C22⋊Dic17C2212yes—
12#11762568—22no—
13#11762568—22no—
14C2×C17×#117C2211yescharacteristic
15C17×#381C2211yescharacteristic
16C17×#381C2211yescharacteristic
17C17×#342—21no—
18C17×#342—21no—
19C23×C4×C17C2311yescharacteristic
20C22×Dic34—21no—
21C17×#117—22no—
22C17×#117—22no—
23C2×C4⋊C4×C17Q812yescomplemented
24C2×C22⋊C4×C17—21no—
25C2×C22⋊C4×C17—21no—
26C2×C22⋊C4×C17D411yescharacteristic
27C2×C22⋊C4×C17D411yescharacteristic
28C2×C42×C17—21no—
29C2×C22⋊Dic17—22no—
30C2×C4⋊Dic17—21no—
31C2×#63225—22no—
32C2×C4×Dic17D412yes—
33C2×C4×Dic17D412yes—
34C2×#63223—22no—
35C17×#95—22no—
36C17×#95—22no—
37#10587502—21no—
38#10587502—22no—
39#10587502—21no—
40#10587502—22no—
41C22⋊#3622—22no—
42C22⋊#3622—22no—
43C24×C17C2×D411yescharacteristic
44C22×C4×C17—21no—
45C22×C4×C17—21no—
46C22×C4×C17—22no—
47C22×C4×C17—41no—
48C22×C4×C17—41no—
49C22×C4×C17—41no—
50C22×C4×C17C2×D412yes—
51C22×C4×C17C2×Q811yescharacteristic
52C22×C4×C17#4111yescharacteristic
53C22×C4×C17#4112yes—
54C22×Dic17—22no—
55C22×Dic17—22no—
56C2×Dic34—44no—
57C2×Dic34—81no—
58C4⋊C4×C17—24no—
59C22⋊C4×C17—41no—
60C22⋊C4×C17—41no—
61C22⋊C4×C17—41no—
62C22⋊C4×C17—41no—
63C22⋊C4×C17—41no—
64C22⋊C4×C17—41no—
65C42×C17—42no—
66C22⋊Dic17—44no—
67C4⋊Dic17—81no—
68#63225—44no—
69C4×Dic17—42no—
70C4×Dic17—42no—
71#63223—42no—
72#63223—42no—
73C2×#3622—42no—
74#10387—171nomaximal, Sylow
75C23×C17—21no—
76C23×C17—21no—
77C23×C17—42no—
78C23×C17#12011yescharacteristic
79C23×C17C22⋊Q811yescharacteristic
80C23×C17#12411yescharacteristic
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—41no—
84C2×C4×C17—41no—
85C2×C4×C17—42no—
86C2×C4×C17—42no—
87C2×C4×C17—42no—
88C2×C4×C17—42no—
89C2×C4×C17—41no—
90C2×C4×C17—41no—
91C2×C4×C17—82no—
92C2×C4×C17—81no—
93C2×C4×C17—82no—
94C2×C4×C17C22⋊Q812yes—
95C2×C4×C17#12412yes—
96C2×Dic17—24no—
97C2×Dic17—42no—
98C2×Dic17—42no—
99C2×Dic17—42no—
100C2×Dic17—42no—
101C2×Dic17—42no—
102Dic34—84no—
103#3622—82no—
104#2607—171no—
105#1826—172no—
106#1820D1711yescharacteristic
107#1306—171no—
108#1284—172no—
109C22×C17—22no—
110C22×C17—21no—
111C22×C17—21no—
112C22×C17—22no—
113C22×C17—22no—
114C22×C17—81no—
115C22×C17—81no—
116C22×C17#39311yescharacteristic
117C22×C17#45811yescharacteristic
118C22×C17#45811yescharacteristic
119C4×C17—22no—
120C4×C17—22no—
121C4×C17—81no—
122C4×C17—81no—
123C4×C17—82no—
124C4×C17—161no—
125Dic17—44no—
126Dic17—82no—
127C2×#117C2×D1711yescharacteristic
128#407—172no—
129#399—341no—
130#393—341no—
131#391—342no—
132#387—342no—
133#381C2×D1711yescharacteristic
134#381C2×D1711yescharacteristic
135C4×C22⋊C4—172no—
136#342—21no—
137#342—21no—
138#339—342no—
139#339—342no—
140C2×C17—21no—
141C2×C17—21no—
142C2×C17—41no—
143C2×C17—41no—
144C2×C17#203011yescharacteristic
145C2×C17#203112yes—
146C22×Q8—341no—
147C23×C4C22×D1711yescharacteristic
148C2×M16—342no—
149#117—22no—
150#117—22no—
151C2×C4⋊C4—342no—
152C2×C4⋊C4—341no—
153C2×C4⋊C4Dic3412yescomplemented
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—342no—
157C2×C22⋊C4C22⋊D1711yescharacteristic
158C2×C22⋊C4C22⋊D1711yescharacteristic
159C2×C42—21no—
160C2×C42—172no—
161C2×C42—172no—
162C22⋊C8—342no—
163C22⋊C8—342no—
164#95—22no—
165#95—22no—
166#95—341no—
167#95—342no—
168#95—341no—
169#95—342no—
170C17#1038711yescharacteristic, complemented, Sylow
171C24C2×C22⋊D1711yescharacteristic
172C2×Q8—684no—
173C2×Q8—1361no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—342no—
181C22×C4—342no—
182C22×C4C2×Dic3411yescharacteristic
183C22×C4#6325211yescharacteristic
184C22×C4D4×D1712yes—
185C22×C4#6325412yes—
186M16—682no—
187M16—682no—
188C2×C8—682no—
189C4⋊C4—24no—
190C4⋊C4—684no—
191C4⋊C4—1361no—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C22⋊C4—41no—
197C22⋊C4—41no—
198C22⋊C4—684no—
199C42—42no—
200C42—682no—
201C42—682no—
202C23—21no—
203C23—21no—
204C23—42no—
205C23#1058761011yescharacteristic
206C23#1058762211yescharacteristic
207C23#1058762311yescharacteristic
208Q8—1364no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—82no—
220C2×C4—81no—
221C2×C4—82no—
222C2×C4—344no—
223C2×C4—682no—
224C2×C4—682no—
225C2×C4—682no—
226C2×C4—682no—
227C2×C4—682no—
228C2×C4#1058756412yes—
229C2×C4#1058757112yes—
230C8—1362no—
231C22—22no—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—22no—
236C22—81no—
237C22—81no—
238C22#1176272111yescharacteristic
239C22#1176299811yescharacteristic
240C22#1176299811yescharacteristic
241C4—22no—
242C4—22no—
243C4—81no—
244C4—81no—
245C4—82no—
246C4—161no—
247C4—684no—
248C4—1362no—
249C2—21no—
250C2—21no—
251C2—41no—
252C2—41no—
253C2#1452995111yescharacteristic
254C2#1452995812yes—
subgroupG/H
centerC22#11762721
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#1820C2
socleC22×C17#393
prime radicalC24×C17C2×D4
hypercenter#1820D17
hypocenterC17#10387
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
23. C2×C4⋊C4×C17Q8semidirect
153. C2×C4⋊C4Dic34semidirect
170. C17#10387semidirect

Maximal subgroups

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

subgroupindex
#145344492
#145288832
#145266922
#145266362
C17×#18202
#1038717