Finite Group Databasebeta

← #17961567

Subgroups — #17961567

Order 4352.

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

normal subgroups only

This group has 638 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#14544618C212yesmaximal
2#14543882C214yesmaximal, complemented
3C2×#11762714C211yesmaximal, characteristic
4C2×#10587545—22no—
5C2×#10587545C2214yes—
6#11763779—22no—
7C23×Dic34C2211yescharacteristic
8C22×C4×Dic17C411yescharacteristic
9#11762714—22no—
10#11762714C412yes—
11#11762714C2212yes—
12C2×#10587502—22no—
13C22×C4⋊C4×C17C411yescharacteristic
14C23×C4×C17—21no—
15C23×C4×C17C2×C411yescharacteristic
16C23×Dic17—21no—
17C23×Dic17D412yes—
18C22×Dic34—24no—
19C22×Dic34—42no—
20C22×Dic34—41no—
21C22×Dic34—41no—
22C22×Dic34C2×C411yescharacteristic
23C22×Dic34D412yes—
24C22×Dic34C2311yescharacteristic
25C2×#63247—24no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17—41no—
28C2×C4⋊C4×C17—41no—
29C2×C4⋊C4×C17C2×C412yes—
30C2×C4×Dic17—21no—
31C2×C4×Dic17—42no—
32C2×C4×Dic17C2×C412yes—
33C22⋊#3624—42no—
34C22⋊#3624—42no—
35#10587545—28no—
36#10587545—44no—
37#10587502—24no—
38#10587502—24no—
39C24×C17C22⋊C411yescharacteristic
40C22×C4×C17—21no—
41C22×C4×C17—21no—
42C22×C4×C17—22no—
43C22×C4×C17—22no—
44C22×C4×C17—22no—
45C22×C4×C17—44no—
46C22×C4×C17—41no—
47C22×C4×C17—41no—
48C22×C4×C17C22⋊C411yescharacteristic
49C22×C4×C17C22×C411yescharacteristic
50C22×Dic17—24no—
51C22×Dic17—22no—
52C22×Dic17—22no—
53C22×Dic17—22no—
54C22×Dic17—24no—
55C22×Dic17—42no—
56C22×Dic17—42no—
57C22×Dic17—42no—
58C22×Dic17C22⋊C412yes—
59C22×Dic17C2×D412yes—
60C2×Dic34—24no—
61C2×Dic34—22no—
62C2×Dic34—24no—
63C2×Dic34—41no—
64C2×Dic34—41no—
65C2×Dic34—42no—
66C2×Dic34—41no—
67C2×Dic34—41no—
68C2×Dic34—48no—
69C2×Dic34—48no—
70C2×Dic34#4114yes—
71#63247—44no—
72#63247—44no—
73C2×#3624—44no—
74C4⋊C4×C17—42no—
75C4⋊C4×C17—42no—
76C4×Dic17—42no—
77C4×Dic17—42no—
78#12860—171nomaximal, Sylow
79C23×C17—22no—
80C23×C17—22no—
81C23×C17—21no—
82C23×C17—21no—
83C23×C17#9911yescharacteristic
84C23×C17#9911yescharacteristic
85C23×C17C2×C22⋊C411yescharacteristic
86C2×C4×C17—24no—
87C2×C4×C17—24no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—44no—
93C2×C4×C17—42no—
94C2×C4×C17—82no—
95C2×C4×C17—82no—
96C2×C4×C17#9912yes—
97C2×C4×C17#11712yes—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—24no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—44no—
105C2×Dic17—44no—
106C2×Dic17—48no—
107C2×Dic17—42no—
108C2×Dic17—42no—
109C2×Dic17#12314yes—
110Dic34—44no—
111Dic34—48no—
112Dic34—44no—
113#3624—84no—
114C2×#391—171no—
115#1784—172no—
116#1394—174no—
117C22×C17—24no—
118C22×C17—24no—
119C22×C17—21no—
120C22×C17—21no—
121C22×C17—21no—
122C22×C17—21no—
123C22×C17—21no—
124C22×C17—21no—
125C22×C17#35512yes—
126C22×C17#35612yes—
127C22×C17#38111yescharacteristic
128C22×C17C2×#9911yescharacteristic
129C22×C17C2×#9911yescharacteristic
130C4×C17—22no—
131C4×C17—22no—
132C4×C17—82no—
133Dic17—24no—
134Dic17—44no—
135Dic17—42no—
136C23×Q8—171no—
137C22×C4⋊C4#59511yescharacteristic
138C22×C42—171no—
139C2×#101—174no—
140C2×#101—342no—
141#407—342no—
142#391—172no—
143#391—172no—
144#391—342no—
145C2×#95—342no—
146C2×C17—21no—
147C2×C17—22no—
148C2×C17—22no—
149C2×C17—21no—
150C2×C17#181911yescharacteristic
151C2×C17C2×#35511yescharacteristic
152C2×C17C2×#35611yescharacteristic
153C22×Q8—172no—
154C22×Q8—171no—
155C22×Q8—171no—
156C22×Q8—344no—
157C22×Q8—681no—
158C22×Q8—681no—
159C22×Q8—682no—
160C23×C4—21no—
161C23×C4—172no—
162C23×C4—341no—
163C23×C4C2×#59511yescharacteristic
164C2×M16—344no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4C2×#59512yes—
169C2×C42—172no—
170C2×C42—341no—
171C2×C42—682no—
172#101—348no—
173#101—684no—
174C22⋊C8—682no—
175C22⋊C8—682no—
176#95—344no—
177#95—344no—
178C17#1286011yescharacteristic, complemented, Sylow
179C24C22⋊#59511yescharacteristic
180C2×Q8—174no—
181C2×Q8—344no—
182C2×Q8—342no—
183C2×Q8—344no—
184C2×Q8—682no—
185C2×Q8—681no—
186C2×Q8—681no—
187C2×Q8—681no—
188C2×Q8—681no—
189C2×Q8—688no—
190C2×Q8—688no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—44no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—172no—
200C22×C4—172no—
201C22×C4—344no—
202C22×C4—344no—
203C22×C4—342no—
204C22×C4—342no—
205C22×C4—342no—
206C22×C4—682no—
207C22×C4—682no—
208C22×C4—682no—
209C22×C4C22⋊#59511yescharacteristic
210C22×C4C22×#59511yescharacteristic
211M16—684no—
212M16—684no—
213C2×C8—684no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C42—682no—
217C42—682no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23#1058755111yescharacteristic
223C23#1058755111yescharacteristic
224C23C2×C22⋊#59511yescharacteristic
225Q8—684no—
226Q8—688no—
227Q8—684no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—174no—
239C2×C4—342no—
240C2×C4—342no—
241C2×C4—344no—
242C2×C4—342no—
243C2×C4—342no—
244C2×C4—342no—
245C2×C4—684no—
246C2×C4—684no—
247C2×C4—688no—
248C2×C4—682no—
249C2×C4—682no—
250C2×C4#1058753912yes—
251C2×C4#1058767712yes—
252C8—1364no—
253C22—24no—
254C22—24no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22#1176371712yes—
262C22#1176371912yes—
263C22#1176375211yescharacteristic
264C22C2×#1058755111yescharacteristic
265C22C2×#1058753911yescharacteristic
266C4—22no—
267C4—22no—
268C4—82no—
269C4—344no—
270C4—684no—
271C4—682no—
272C2—21no—
273C2—22no—
274C2—22no—
275C2—21no—
276C2#1454467911yescharacteristic
277C2C2×#1176371711yescharacteristic
278C2C2×#1176371911yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC22×C4⋊C4×C17C4
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC22×Dic34C23
hypercenterC22×C4⋊C4#595
hypocenterC17#12860
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #14543882C2semidirect
178. C17#12860semidirect

Maximal subgroups

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

subgroupindex
#145446182
#145438822
C2×#117627142
#1286017