Finite Group Databasebeta

← #16931720

Subgroups — #16931720

Order 4352.

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

normal subgroups only

This group has 514 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#14527465C212yesmaximal, complemented
4#14526687C212yesmaximal, complemented
5C17×#2563C211yesmaximal, characteristic
6#11762887—24no—
7#11762862C2212yes—
8#11762833C2212yes—
9C2×#10587504—21no—
10#11762773—22no—
11#11762578—21no—
12#11762558—24no—
13C2×C17×C4⋊Q8C2211yescharacteristic
14C2×C4×D4×C17C2211yescharacteristic
15C17×#393C414yes—
16C17×#390—21no—
17C17×#382—21no—
18C17×#380—22no—
19C4×C4⋊C4×C17C2211yescharacteristic
20C22×Q8×C17C2×C412yes—
21C22×D4×C17D411yescharacteristic
22C23×C4×C17D412yes—
23C17×C4⋊Q8—22no—
24C17×C4⋊Q8—22no—
25C4×D4×C17—22no—
26C4×D4×C17—22no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C22⋊C4×C17—24no—
32C2×C22⋊C4×C17C2×C412yes—
33C2×C42×C17—21no—
34C2×C42×C17C2311yescharacteristic
35C2×#63223—24no—
36C2×#63223—22no—
37C17×#95—24no—
38C17×#95C2×C412yes—
39#10587504—42no—
40C22⋊#3622—24no—
41C22⋊#3622—22no—
42C4⋊#3622—22no—
43C4⋊#3622—22no—
44C4⋊#3622D414yes—
45C4×#3622—22no—
46C24×C17C22⋊C412yes—
47C2×Q8×C17—24no—
48C2×Q8×C17—82no—
49C2×D4×C17—22no—
50C2×D4×C17—21no—
51C2×D4×C17—22no—
52C2×D4×C17—21no—
53C22×C4×C17—22no—
54C22×C4×C17—22no—
55C22×C4×C17—22no—
56C22×C4×C17—22no—
57C22×C4×C17—44no—
58C22×C4×C17C22⋊C412yes—
59C22×C4×C17C22×C411yescharacteristic
60C22×C4×C17C2×D411yescharacteristic
61C22×C4×C17C2×D411yescharacteristic
62C4⋊C4×C17—22no—
63C4⋊C4×C17—42no—
64C4⋊C4×C17—41no—
65C4⋊C4×C17—42no—
66C4⋊C4×C17—41no—
67C22⋊C4×C17—24no—
68C22⋊C4×C17—84no—
69C42×C17—41no—
70C42×C17—41no—
71C42×C17C2×D412yes—
72C42×C17#4112yes—
73#63223—42no—
74#63223—42no—
75#63223—84no—
76C2×#3622—24no—
77C2×#3622—42no—
78#9399—171nomaximal, Sylow
79C23×C17—22no—
80C23×C17—22no—
81C23×C17—22no—
82C23×C17—42no—
83C23×C17—42no—
84C23×C17C2×C22⋊C411yescharacteristic
85Q8×C17—84no—
86D4×C17—42no—
87D4×C17—42no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—41no—
91C2×C4×C17—44no—
92C2×C4×C17—42no—
93C2×C4×C17—42no—
94C2×C4×C17—44no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17—82no—
99C2×C4×C17C4×D412yes—
100C2×C4×C17#12011yescharacteristic
101C2×C4×C17#12112yes—
102C2×C4×C17#12311yescharacteristic
103C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
104C2×C4×C17#13712yes—
105C2×C4×C17#13711yescharacteristic
106#3622—44no—
107#3622—82no—
108#2975—171no—
109#2563D1711yescharacteristic
110#1800—171no—
111#1548—172no—
112#1303—172no—
113C22×C17—21no—
114C22×C17—21no—
115C22×C17—44no—
116C22×C17—42no—
117C22×C17—42no—
118C22×C17—42no—
119C22×C17—82no—
120C22×C17#38611yescharacteristic
121C22×C17#41711yescharacteristic
122C22×C17#41911yescharacteristic
123C4×C17—21no—
124C4×C17—22no—
125C4×C17—42no—
126C4×C17—82no—
127C4×C17—82no—
128C4×C17#45512yes—
129C4×C17#46812yes—
130C4×C17#47012yes—
131C2×C4⋊Q8C2×D1711yescharacteristic
132C2×C4×D4C2×D1711yescharacteristic
133#436—344no—
134#431—172no—
135#423—172no—
136C2×#100—341no—
137#407—342no—
138#393Dic1714yes—
139#390—21no—
140#382—21no—
141#380—22no—
142C4×C4⋊C4C2×D1711yescharacteristic
143#343—341no—
144#333—344no—
145C2×C17—21no—
146C2×C17—21no—
147C2×C17—82no—
148C2×C17#188511yescharacteristic
149C2×C17#189411yescharacteristic
150C2×C17#191411yescharacteristic
151C22×Q8C2×Dic1712yes—
152C22×D4C22⋊D1711yescharacteristic
153C23×C4C22⋊D1712yes—
154C2×M16—344no—
155C2×M16—342no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4×D4—22no—
159C4×D4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C22⋊D1711yescharacteristic
164C2×C22⋊C4—24no—
165C2×C22⋊C4C2×Dic1712yes—
166C2×C42—21no—
167C2×C42C22×D1711yescharacteristic
168C4⋊C8—174no—
169C4⋊C8—342no—
170C4⋊C8—342no—
171#100—682no—
172C22⋊C8—344no—
173C22⋊C8—342no—
174C4×C8—342no—
175#95—24no—
176#95C2×Dic1712yes—
177C17#939911yescharacteristic, complemented, Sylow
178C24C22⋊Dic1712yes—
179C2×Q8—24no—
180C2×Q8—82no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—21no—
184C2×D4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—44no—
190C22×C4C22⋊Dic1712yes—
191C22×C4C22×Dic1711yescharacteristic
192C22×C4C2×C22⋊D1711yescharacteristic
193C22×C4C2×C22⋊D1711yescharacteristic
194M16—682no—
195M16—682no—
196M16—1364no—
197C2×C8—344no—
198C2×C8—682no—
199C4⋊C4—22no—
200C4⋊C4—42no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C22⋊C4—24no—
205C22⋊C4—84no—
206C42—41no—
207C42—41no—
208C42D4×D1712yes—
209C42#6325412yes—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×C22⋊Dic1711yescharacteristic
216Q8—84no—
217D4—42no—
218D4—42no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—82no—
230C2×C4D4×Dic1712yes—
231C2×C4#1058762111yescharacteristic
232C2×C4#1058762311yescharacteristic
233C2×C4#1058762512yes—
234C2×C4#1058762812yes—
235C2×C4#1058763511yescharacteristic
236C2×C4#1058763711yescharacteristic
237C8—684no—
238C8—1362no—
239C22—21no—
240C22—21no—
241C22—44no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—82no—
246C22#1176268911yescharacteristic
247C22#1176281011yescharacteristic
248C22#1176281811yescharacteristic
249C4—22no—
250C4—21no—
251C4—42no—
252C4—82no—
253C4—82no—
254C4#1176298612yes—
255C4#1176305712yes—
256C4#1176306512yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#1452917311yescharacteristic
261C2#1452921811yescharacteristic
262C2#1452933211yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2563C2
socleC22×C17#386
prime radicalC22×D4×C17D4
cosocleC2×C42×C17C23
hypercenter#2563D17
hypocenterC17#9399
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2563

Complements

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

NH ≅ G/Ntype
3. #14527465C2semidirect
4. #14526687C2semidirect
177. C17#9399semidirect

Maximal subgroups

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

subgroupindex
#145378252
#145287762
#145274652
#145266872
C17×#25632
#939917