Finite Group Databasebeta

← #17953603

Subgroups — #17953603

Order 4352.

514 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 108 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#14544641C212yesmaximal
2#14543662C212yesmaximal, complemented
3#14543662C212yesmaximal, complemented
4#14531190C211yesmaximal, characteristic
5C2×#10587544C2214yes—
6C2×#10587543—22no—
7#11763675—22no—
8C2×#10587578C411yescharacteristic
9C2×#10587578C411yescharacteristic
10C2×C42×D17C2211yescharacteristic
11#11762678C2211yescharacteristic
12#11762678C2211yescharacteristic
13#11762675C411yescharacteristic
14#11762668—22no—
15#11762660—22no—
16C17×#382C411yescharacteristic
17C22×C4×D17C2×C411yescharacteristic
18C22×C4×D17D412yes—
19C22×#3624—24no—
20C2×C4⋊C4×C17C2×C411yescharacteristic
21C2×C4⋊C4×C17C2×C411yescharacteristic
22C2×C42×C17C2×C411yescharacteristic
23C2×#63227—21no—
24C2×#63227—21no—
25C2×C4⋊Dic17C2×C411yescharacteristic
26C2×C4⋊Dic17C2×C411yescharacteristic
27C2×#63225—22no—
28C2×#63225—22no—
29C2×C4×Dic17—21no—
30C2×C4×Dic17—21no—
31C2×C4×Dic17D412yes—
32C2×C4×Dic17C2311yescharacteristic
33#10587578—42no—
34#10587578—42no—
35C42×D17—24no—
36#10587544—28no—
37#10587543—44no—
38C17×#95—22no—
39#10587502—22no—
40C23×D17C22⋊C411yescharacteristic
41C22×C4×C17—21no—
42C22×C4×C17—21no—
43C22×C4×C17C22⋊C412yes—
44C22×C4×C17C22×C411yescharacteristic
45C22×Dic17—21no—
46C22×Dic17—21no—
47C22×Dic17C22⋊C411yescharacteristic
48C22×Dic17C2×D412yes—
49C2×C4×D17—22no—
50C2×C4×D17—22no—
51C2×C4×D17—22no—
52C2×C4×D17—24no—
53C2×C4×D17—24no—
54C2×C4×D17—22no—
55C2×C4×D17—22no—
56C2×#3624—44no—
57C2×#3624—44no—
58C2×#3624—44no—
59C4⋊C4×C17—41no—
60C4⋊C4×C17—41no—
61C42×C17—22no—
62#63227—42no—
63#63227—42no—
64C4⋊Dic17—41no—
65C4⋊Dic17—41no—
66#63225—44no—
67#63225—42no—
68#63225—42no—
69C4×Dic17—24no—
70C4×Dic17—42no—
71C4×Dic17—42no—
72C4×Dic17#4114yes—
73#8728—171nomaximal, Sylow
74C23×C17C2×C22⋊C411yescharacteristic
75C22×D17—21no—
76C22×D17—21no—
77C22×D17—22no—
78C22×D17—22no—
79C22×D17#10112yes—
80C2×C4×C17—21no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—41no—
86C2×C4×C17—41no—
87C2×C4×C17—42no—
88C2×C4×C17—42no—
89C2×C4×C17#11712yes—
90C2×Dic17—21no—
91C2×Dic17—21no—
92C2×Dic17—41no—
93C2×Dic17—41no—
94C2×Dic17—42no—
95C2×Dic17—42no—
96C2×Dic17#10012yes—
97C2×Dic17#12314yes—
98C2×Dic17#13514yes—
99C2×Dic17AGL(1,ℤ/8)12yes—
100C2×Dic17#13712yes—
101C4×D17—28no—
102C4×D17—44no—
103C4×D17—44no—
104#3624—84no—
105#2285—171no—
106#1797—172no—
107#1296—172no—
108#1296—172no—
109C22×C17#38111yescharacteristic
110C22×C17C2×#10011yescharacteristic
111C22×C17C2×#10111yescharacteristic
112C22×C17#41612yes—
113C22×C17#41712yes—
114C2×D17—24no—
115C2×D17—22no—
116C2×D17—41no—
117C2×D17—41no—
118C2×D17—42no—
119C4×C17—22no—
120C4×C17—42no—
121C4×C17—81no—
122C4×C17—81no—
123Dic17—22no—
124Dic17—81no—
125Dic17—81no—
126Dic17#48214yes—
127Dic17#48514yes—
128C2×#117—171no—
129C2×#117—171no—
130C22×C42—171no—
131C2×C4⋊C8—174no—
132C2×C22⋊C8—342no—
133#384—171no—
134#384—171no—
135#382—171no—
136#382#59511yescharacteristic
137#380—342no—
138C4×C4⋊C4—342no—
139#336—342no—
140C2×C17#154311yescharacteristic
141C2×C17#155212yes—
142C2×C17#156011yescharacteristic
143C2×C17#182111yescharacteristic
144C2×C17#182512yes—
145D17—42no—
146C23×C4—172no—
147C23×C4—171no—
148C22×C8—344no—
149#117—682no—
150#117—682no—
151C2×C4⋊C4—171no—
152C2×C4⋊C4—171no—
153C2×C4⋊C4—342no—
154C2×C4⋊C4—342no—
155C2×C4⋊C4C2×#59511yescharacteristic
156C2×C4⋊C4C2×#59511yescharacteristic
157C2×C22⋊C4—341no—
158C2×C22⋊C4—341no—
159C2×C42—171no—
160C2×C42—172no—
161C2×C42—344no—
162C2×C42—341no—
163C2×C42—341no—
164C2×C42C2×#59511yescharacteristic
165C4⋊C8—348no—
166C22⋊C8—684no—
167#95—22no—
168#95—342no—
169C17#872811yescharacteristic, complemented, Sylow
170C24—171no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—172no—
174C22×C4—171no—
175C22×C4—342no—
176C22×C4—342no—
177C22×C4—344no—
178C22×C4—344no—
179C22×C4—342no—
180C22×C4—342no—
181C22×C4—342no—
182C22×C4—341no—
183C22×C4—341no—
184C22×C4C22⋊#59512yes—
185C22×C4C22×#59511yescharacteristic
186C2×C8—684no—
187C2×C8—684no—
188C2×C8—684no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—681no—
192C4⋊C4—681no—
193C4⋊C4—684no—
194C4⋊C4—682no—
195C4⋊C4—682no—
196C22⋊C4—682no—
197C22⋊C4—682no—
198C42—22no—
199C42—174no—
200C42—344no—
201C42—682no—
202C42—682no—
203C23—172no—
204C23—341no—
205C23—341no—
206C23—342no—
207C23—342no—
208C23C2×C22⋊#59511yescharacteristic
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—174no—
219C2×C4—174no—
220C2×C4—172no—
221C2×C4—172no—
222C2×C4—172no—
223C2×C4—348no—
224C2×C4—341no—
225C2×C4—341no—
226C2×C4—684no—
227C2×C4—684no—
228C2×C4—681no—
229C2×C4—681no—
230C2×C4—682no—
231C2×C4—682no—
232C2×C4#1058767712yes—
233C8—1364no—
234C22—342no—
235C22—344no—
236C22—682no—
237C22—681no—
238C22—681no—
239C22#1176375211yescharacteristic
240C22D17⋊#10011yescharacteristic
241C22D17⋊#10111yescharacteristic
242C22#1176379912yes—
243C22#1176380012yes—
244C4—22no—
245C4—42no—
246C4—81no—
247C4—81no—
248C4—174no—
249C4—174no—
250C4—342no—
251C4—1361no—
252C4—1361no—
253C2—682no—
254C2#1454428811yescharacteristic
255C2#1454430612yes—
256C2#1454432211yescharacteristic
257C2#1454468411yescharacteristic
258C2#1454468912yes—
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#382C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#382#595
hypocenterC17#8728
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
2. #14543662C2semidirect
3. #14543662C2semidirect
169. C17#8728semidirect

Maximal subgroups

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

subgroupindex
#145446412
#145436622
#145311902
#872817