Finite Group Databasebeta

← #16931730

Subgroups — #16931730

Order 4352.

534 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 123 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#14537825C211yesmaximal, characteristic
2#14528776C211yesmaximal, characteristic
3#14527456C212yesmaximal, complemented
4#14526687C212yesmaximal, complemented
5C17×#2547C211yesmaximal, characteristic
6#11762887—24no—
7#11762862C2212yes—
8#11762833C2212yes—
9C2×#10587504—21no—
10#11762773—22no—
11#11762578—21no—
12#11762555—24no—
13C2×C17×C4⋊Q8C2211yescharacteristic
14C2×C17×#124—21no—
15C2×C4×D4×C17C2211yescharacteristic
16C17×#397C414yes—
17C17×#391—21no—
18C17×#380—22no—
19C4×C4⋊C4×C17C2211yescharacteristic
20C22×Q8×C17—21no—
21C22×D4×C17D411yescharacteristic
22C23×C4×C17D412yes—
23C17×C4⋊Q8—22no—
24C17×C4⋊Q8—22no—
25C17×#124—44no—
26C4×D4×C17—22no—
27C4×D4×C17—22no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17C2×C412yes—
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C22⋊C4×C17—24no—
33C2×C22⋊C4×C17C2×C412yes—
34C2×C42×C17—21no—
35C2×C42×C17C2311yescharacteristic
36C2×#63223—24no—
37C2×#63223—22no—
38C17×#95—24no—
39C17×#95C2×C412yes—
40#10587504—42no—
41C22⋊#3622—24no—
42C22⋊#3622—22no—
43C4⋊#3622—22no—
44C4⋊#3622—22no—
45C4⋊#3622D414yes—
46C4×#3622—22no—
47C24×C17C22⋊C412yes—
48C2×Q8×C17—42no—
49C2×Q8×C17—42no—
50C2×Q8×C17—41no—
51C2×Q8×C17—41no—
52C2×D4×C17—22no—
53C2×D4×C17—22no—
54C2×D4×C17—21no—
55C2×D4×C17—21no—
56C22×C4×C17—22no—
57C22×C4×C17—22no—
58C22×C4×C17—22no—
59C22×C4×C17—22no—
60C22×C4×C17—44no—
61C22×C4×C17C22⋊C412yes—
62C22×C4×C17C22×C411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C4⋊C4×C17—22no—
66C4⋊C4×C17—24no—
67C4⋊C4×C17—42no—
68C4⋊C4×C17—41no—
69C4⋊C4×C17—41no—
70C22⋊C4×C17—24no—
71C22⋊C4×C17—48no—
72C42×C17—42no—
73C42×C17C2×D412yes—
74C42×C17#4112yes—
75#63223—42no—
76#63223—42no—
77#63223—84no—
78C2×#3622—24no—
79C2×#3622—42no—
80#9401—171nomaximal, Sylow
81C23×C17—22no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—42no—
85C23×C17—42no—
86C23×C17C2×C22⋊C411yescharacteristic
87Q8×C17—82no—
88Q8×C17—82no—
89D4×C17—42no—
90D4×C17—42no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—41no—
94C2×C4×C17—44no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17—42no—
100C2×C4×C17—42no—
101C2×C4×C17—42no—
102C2×C4×C17—82no—
103C2×C4×C17C4×D412yes—
104C2×C4×C17#12011yescharacteristic
105C2×C4×C17#12112yes—
106C2×C4×C17#12311yescharacteristic
107C2×C4×C17AGL(1,ℤ/8)12yes—
108C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
109C2×C4×C17#13711yescharacteristic
110#3622—44no—
111#3622—82no—
112#2975—171no—
113#2547D1711yescharacteristic
114#1800—171no—
115#1546—172no—
116#1303—172no—
117C22×C17—21no—
118C22×C17—21no—
119C22×C17—44no—
120C22×C17—42no—
121C22×C17—42no—
122C22×C17—42no—
123C22×C17—82no—
124C22×C17#38611yescharacteristic
125C22×C17#41711yescharacteristic
126C22×C17#41811yescharacteristic
127C4×C17—21no—
128C4×C17—22no—
129C4×C17—42no—
130C4×C17—82no—
131C4×C17—82no—
132C4×C17#45512yes—
133C4×C17#46812yes—
134C4×C17#46912yes—
135C2×C4⋊Q8C2×D1711yescharacteristic
136C2×#124—21no—
137C2×C4×D4C2×D1711yescharacteristic
138#436—344no—
139#431—172no—
140#423—172no—
141C2×#100—341no—
142#407—342no—
143#397Dic1714yes—
144#391—21no—
145#380—22no—
146C4×C4⋊C4C2×D1711yescharacteristic
147#343—341no—
148#332—344no—
149C2×C17—21no—
150C2×C17—21no—
151C2×C17—82no—
152C2×C17#188511yescharacteristic
153C2×C17#189511yescharacteristic
154C2×C17#191511yescharacteristic
155C22×Q8—21no—
156C22×D4C22⋊D1711yescharacteristic
157C23×C4C22⋊D1712yes—
158C2×M16—344no—
159C2×M16—342no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162#124—44no—
163C4×D4—22no—
164C4×D4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4C2×Dic1712yes—
168C2×C4⋊C4C22⋊D1711yescharacteristic
169C2×C22⋊C4—24no—
170C2×C22⋊C4C2×Dic1712yes—
171C2×C42—21no—
172C2×C42C22×D1711yescharacteristic
173C4⋊C8—174no—
174C4⋊C8—342no—
175C4⋊C8—342no—
176#100—682no—
177C22⋊C8—344no—
178C22⋊C8—342no—
179C4×C8—342no—
180#95—24no—
181#95C2×Dic1712yes—
182C17#940111yescharacteristic, complemented, Sylow
183C24C22⋊Dic1712yes—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—44no—
197C22×C4C22⋊Dic1712yes—
198C22×C4C22×Dic1711yescharacteristic
199C22×C4C2×C22⋊D1711yescharacteristic
200C22×C4C2×C22⋊D1711yescharacteristic
201M16—682no—
202M16—682no—
203M16—1364no—
204C2×C8—344no—
205C2×C8—682no—
206C4⋊C4—22no—
207C4⋊C4—24no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C22⋊C4—24no—
212C22⋊C4—48no—
213C42—42no—
214C42D4×D1712yes—
215C42#6325412yes—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23C2×C22⋊Dic1711yescharacteristic
222Q8—82no—
223Q8—82no—
224D4—42no—
225D4—42no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—82no—
238C2×C4#1058761812yes—
239C2×C4D4×Dic1712yes—
240C2×C4#1058762111yescharacteristic
241C2×C4#1058762311yescharacteristic
242C2×C4#1058762512yes—
243C2×C4#1058763511yescharacteristic
244C2×C4#1058763711yescharacteristic
245C8—684no—
246C8—1362no—
247C22—21no—
248C22—21no—
249C22—44no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—82no—
254C22#1176268911yescharacteristic
255C22#1176281011yescharacteristic
256C22#1176281411yescharacteristic
257C4—22no—
258C4—21no—
259C4—42no—
260C4—82no—
261C4—82no—
262C4#1176298612yes—
263C4#1176305312yes—
264C4#1176306012yes—
265C2—21no—
266C2—21no—
267C2—82no—
268C2#1452917311yescharacteristic
269C2#1452922311yescharacteristic
270C2#1452933711yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2547C2
socleC22×C17#386
prime radicalC22×D4×C17D4
cosocleC2×C42×C17C23
hypercenter#2547D17
hypocenterC17#9401
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
3. #14527456C2semidirect
4. #14526687C2semidirect
182. C17#9401semidirect

Maximal subgroups

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

subgroupindex
#145378252
#145287762
#145274562
#145266872
C17×#25472
#940117