Finite Group Databasebeta

← #17958406

Subgroups — #17958406

Order 4352.

598 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 88 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#14544032C212yesmaximal
2#14543714C214yesmaximal, complemented
3C2×#11762716C211yesmaximal, characteristic
4#11763779—22no—
5#11763654—22no—
6#11763654C2214yes—
7C22×C4⋊Dic17C2211yescharacteristic
8C22×C4×Dic17C411yescharacteristic
9#11762716—22no—
10#11762716C412yes—
11#11762716C2212yes—
12C2×#10587502—22no—
13C23×Q8×C17C411yescharacteristic
14C22×Q8×C17—24no—
15C22×Q8×C17—21no—
16C22×Q8×C17—42no—
17C22×Q8×C17—41no—
18C22×Q8×C17—41no—
19C22×Q8×C17C2×C412yes—
20C23×C4×C17—21no—
21C23×C4×C17C2×C411yescharacteristic
22C23×Dic17—21no—
23C23×Dic17D412yes—
24C2×#63247—42no—
25C2×C4⋊Dic17—41no—
26C2×C4⋊Dic17—41no—
27C2×C4⋊Dic17C2×C411yescharacteristic
28C2×C4⋊Dic17D412yes—
29C2×C4⋊Dic17C2311yescharacteristic
30C2×C4×Dic17—21no—
31C2×C4×Dic17—42no—
32C2×C4×Dic17C2×C412yes—
33C22⋊#3624—24no—
34C22⋊#3624—42no—
35#10587502—24no—
36#10587502—24no—
37C24×C17C22⋊C411yescharacteristic
38C2×Q8×C17—24no—
39C2×Q8×C17—22no—
40C2×Q8×C17—48no—
41C2×Q8×C17—48no—
42C2×Q8×C17—42no—
43C2×Q8×C17—42no—
44C2×Q8×C17—42no—
45C2×Q8×C17—42no—
46C2×Q8×C17M1614yescomplemented
47C22×C4×C17—21no—
48C22×C4×C17—21no—
49C22×C4×C17—22no—
50C22×C4×C17—22no—
51C22×C4×C17—24no—
52C22×C4×C17—22no—
53C22×C4×C17—42no—
54C22×C4×C17—41no—
55C22×C4×C17—41no—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22×C411yescharacteristic
58C22×Dic17—24no—
59C22×Dic17—22no—
60C22×Dic17—22no—
61C22×Dic17—22no—
62C22×Dic17—42no—
63C22×Dic17—42no—
64C22×Dic17—44no—
65C22×Dic17C22⋊C412yes—
66C22×Dic17C2×D412yes—
67#63247—84no—
68C2×#3624—44no—
69C4⋊Dic17—41no—
70C4⋊Dic17—42no—
71C4⋊Dic17—41no—
72C4×Dic17—42no—
73C4×Dic17—42no—
74#11506—171nomaximal, Sylow
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—21no—
78C23×C17—21no—
79C23×C17#9911yescharacteristic
80C23×C17#9911yescharacteristic
81C23×C17C2×C22⋊C411yescharacteristic
82Q8×C17—44no—
83Q8×C17—48no—
84Q8×C17—44no—
85C2×C4×C17—24no—
86C2×C4×C17—24no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—21no—
91C2×C4×C17—24no—
92C2×C4×C17—44no—
93C2×C4×C17—44no—
94C2×C4×C17—42no—
95C2×C4×C17—42no—
96C2×C4×C17#10012yes—
97C2×C4×C17C2×M1612yes—
98C2×Dic17—24no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—48no—
102C2×Dic17—42no—
103C2×Dic17—44no—
104C2×Dic17—42no—
105C2×Dic17—42no—
106C2×Dic17—42no—
107C2×Dic17—82no—
108#3624—84no—
109C2×#391—171no—
110#1463—172no—
111#1319—174no—
112C22×C17—24no—
113C22×C17—24no—
114C22×C17—21no—
115C22×C17—21no—
116C22×C17—21no—
117C22×C17—21no—
118C22×C17—21no—
119C22×C17—21no—
120C22×C17#35412yes—
121C22×C17#35612yes—
122C22×C17#40711yescharacteristic
123C22×C17C2×#9911yescharacteristic
124C22×C17C2×#10011yescharacteristic
125C4×C17—22no—
126C4×C17—22no—
127C4×C17—44no—
128Dic17—44no—
129Dic17—42no—
130Dic17—42no—
131C23×Q8#59511yescharacteristic
132C22×C4⋊C4—171no—
133C22×C42—171no—
134#407—342no—
135#391—172no—
136#391—172no—
137#391—342no—
138C2×#95—342no—
139#324—174no—
140#324—342no—
141C2×C17—21no—
142C2×C17—22no—
143C2×C17—22no—
144C2×C17—21no—
145C2×C17#146211yescharacteristic
146C2×C17C2×#35411yescharacteristic
147C2×C17C2×#35611yescharacteristic
148C22×Q8—24no—
149C22×Q8—21no—
150C22×Q8—42no—
151C22×Q8—41no—
152C22×Q8—41no—
153C22×Q8C2×#59512yes—
154C23×C4—21no—
155C23×C4—172no—
156C23×C4—341no—
157C23×C4C2×#59511yescharacteristic
158C2×M16—682no—
159C2×C4⋊C4—172no—
160C2×C4⋊C4—171no—
161C2×C4⋊C4—171no—
162C2×C4⋊C4—681no—
163C2×C4⋊C4—681no—
164C2×C42—172no—
165C2×C42—341no—
166C2×C42—682no—
167C22⋊C8—344no—
168C22⋊C8—682no—
169#95—344no—
170#95—344no—
171C17#1150611yescharacteristic, complemented, Sylow
172C24C22⋊#59511yescharacteristic
173C2×Q8—24no—
174C2×Q8—22no—
175C2×Q8—48no—
176C2×Q8—48no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8#6324714yescomplemented
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—172no—
192C22×C4—172no—
193C22×C4—344no—
194C22×C4—342no—
195C22×C4—342no—
196C22×C4—342no—
197C22×C4—682no—
198C22×C4—682no—
199C22×C4—684no—
200C22×C4C22⋊#59511yescharacteristic
201C22×C4C22×#59511yescharacteristic
202M16—1364no—
203C2×C8—684no—
204C4⋊C4—682no—
205C4⋊C4—681no—
206C4⋊C4—681no—
207C42—682no—
208C42—682no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23#1058755111yescharacteristic
214C23#1058755111yescharacteristic
215C23C2×C22⋊#59511yescharacteristic
216Q8—44no—
217Q8—44no—
218Q8—48no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—344no—
231C2×C4—342no—
232C2×C4—342no—
233C2×C4—688no—
234C2×C4—682no—
235C2×C4—684no—
236C2×C4—682no—
237C2×C4—682no—
238C2×C4—682no—
239C2×C4—1362no—
240C2×C4#1058755312yes—
241C2×C4C2×#6324712yes—
242C8—1364no—
243C22—24no—
244C22—24no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22#1176371512yes—
252C22#1176371912yes—
253C22#1176377911yescharacteristic
254C22C2×#1058755111yescharacteristic
255C22C2×#1058755311yescharacteristic
256C4—22no—
257C4—22no—
258C4—44no—
259C4—684no—
260C4—682no—
261C4—682no—
262C2—21no—
263C2—22no—
264C2—22no—
265C2—21no—
266C2#1454402911yescharacteristic
267C2C2×#1176371511yescharacteristic
268C2C2×#1176371911yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC23×Q8×C17C4
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊Dic17C23
hypercenterC23×Q8#595
hypocenterC17#11506
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. #14543714C2semidirect
46. C2×Q8×C17M16semidirect
171. C17#11506semidirect
181. C2×Q8#63247semidirect

Maximal subgroups

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

subgroupindex
#145440322
#145437142
C2×#117627162
#1150617