Finite Group Databasebeta

← #17958323

Subgroups — #17958323

Order 4352.

460 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 86 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#14544021C212yesmaximal
2#14543727C212yesmaximal, complemented
3#14543714C212yesmaximal, complemented
4#14531491C211yesmaximal, characteristic
5#11763777—22no—
6#11763654C2214yes—
7#11763653—22no—
8C2×#10587633C411yescharacteristic
9C2×C4×D68C2211yescharacteristic
10#11762716C2211yescharacteristic
11#11762710C411yescharacteristic
12#11762700—22no—
13#11762680—22no—
14#11762678C2211yescharacteristic
15#11762655C411yescharacteristic
16C2×C4×Q8×C17C411yescharacteristic
17C22×Q8×C17C2×C411yescharacteristic
18C22×D68C2×C411yescharacteristic
19C22×C4×D17D412yes—
20C2×#63242—42no—
21C4×Q8×C17—42no—
22C2×C4⋊C4×C17—21no—
23C2×C4⋊C4×C17C2×C411yescharacteristic
24C2×C42×C17—21no—
25C2×C42×C17C2×C411yescharacteristic
26#10587633—22no—
27#10587633C814yescomplemented
28C2×#63227—22no—
29C2×#63227—22no—
30C2×#63227D412yes—
31C2×C4⋊Dic17C2311yescharacteristic
32C2×#63225—22no—
33C2×C4×Dic17C2×C411yescharacteristic
34C2×C4×Dic17C2×C411yescharacteristic
35C4×D68—42no—
36C22⋊#3624—24no—
37#10587543—42no—
38#10587502—21no—
39#10587502—21no—
40#10587502—22no—
41C23×D17C22⋊C412yes—
42C2×Q8×C17—21no—
43C2×Q8×C17—41no—
44C2×Q8×C17—41no—
45C2×Q8×C17C2×C812yescomplemented
46C22×C4×C17—21no—
47C22×C4×C17—21no—
48C22×C4×C17C22⋊C411yescharacteristic
49C22×C4×C17C22⋊C411yescharacteristic
50C22×C4×C17C22×C411yescharacteristic
51C22×Dic17—21no—
52C22×Dic17—21no—
53C22×Dic17C2×D412yes—
54C2×D68—41no—
55C2×D68—41no—
56C2×D68C2×C812yes—
57C2×D68M1612yes—
58C2×C4×D17—22no—
59C2×C4×D17—22no—
60C2×C4×D17—82no—
61C2×#3624—44no—
62#63242—84no—
63C4⋊C4×C17—41no—
64C4⋊C4×C17—42no—
65C42×C17—22no—
66C42×C17—42no—
67#63227—28no—
68#63227—42no—
69#63227—82no—
70C4⋊Dic17—41no—
71#63225—82no—
72C4×Dic17—21no—
73C4×Dic17—41no—
74C4×Dic17C2×C812yescomplemented
75#11481—171nomaximal, Sylow
76C23×C17C2×C22⋊C411yescharacteristic
77C22×D17—22no—
78C22×D17—22no—
79C22×D17—42no—
80C22×D17—42no—
81C22×D17C22⋊C814yes—
82Q8×C17—42no—
83Q8×C17—42no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—22no—
87C2×C4×C17—21no—
88C2×C4×C17—41no—
89C2×C4×C17—41no—
90C2×C4×C17—41no—
91C2×C4×C17—41no—
92C2×C4×C17—42no—
93C2×C4×C17—41no—
94C2×C4×C17#9912yes—
95C2×C4×C17C22×C811yescharacteristic
96C2×C4×C17C2×M1611yescharacteristic
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—41no—
100C2×Dic17—41no—
101C2×Dic17—42no—
102C2×Dic17—41no—
103C2×Dic17—41no—
104C2×Dic17—41no—
105D68—42no—
106D68—42no—
107C4×D17—44no—
108C4×D17—82no—
109#3624—84no—
110#2326—171no—
111#1459—172no—
112#1325—172no—
113#1319—172no—
114C22×C17—21no—
115C22×C17—21no—
116C22×C17C2×C22⋊C811yescharacteristic
117C22×C17C2×#9911yescharacteristic
118C22×C17#41311yescharacteristic
119C2×D17—24no—
120C2×D17—42no—
121C2×D17—42no—
122C2×D17—44no—
123C2×D17—82no—
124C4×C17—22no—
125C4×C17—21no—
126C4×C17—21no—
127C4×C17—41no—
128C4×C17—42no—
129C4×C17—81no—
130Dic17—42no—
131Dic17—42no—
132Dic17—81no—
133C2×#124—171no—
134C2×C4×Q8#59511yescharacteristic
135C2×C4×D4—171no—
136C2×C22⋊C8—342no—
137#391—171no—
138#390—171no—
139#388—342no—
140#385—342no—
141#384—171no—
142#376—171no—
143#324—174no—
144#323—342no—
145C2×C17—21no—
146C2×C17—21no—
147C2×C17#145911yescharacteristic
148C2×C17#212711yescharacteristic
149C2×C17#213511yescharacteristic
150D17—44no—
151C22×Q8C2×#59511yescharacteristic
152C22×D4—171no—
153C23×C4—172no—
154C22×C8—682no—
155#124—174no—
156#124—342no—
157C4×Q8—42no—
158C4×D4—682no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—171no—
161C2×C4⋊C4—342no—
162C2×C4⋊C4C2×#59511yescharacteristic
163C2×C22⋊C4—172no—
164C2×C22⋊C4—342no—
165C2×C22⋊C4—342no—
166C2×C42—21no—
167C2×C42—171no—
168C2×C42—171no—
169C2×C42C2×#59511yescharacteristic
170C22⋊C8—344no—
171C22⋊C8—682no—
172#95—342no—
173#95—341no—
174#95—341no—
175C17#1148111yescharacteristic, complemented, Sylow
176C24—172no—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8#6324212yescomplemented
181C2×D4—172no—
182C2×D4—172no—
183C2×D4—681no—
184C2×D4—681no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—172no—
188C22×C4—341no—
189C22×C4—342no—
190C22×C4—342no—
191C22×C4—341no—
192C22×C4—1362no—
193C22×C4C22⋊#59511yescharacteristic
194C22×C4C22⋊#59511yescharacteristic
195C22×C4C22×#59511yescharacteristic
196C2×C8—684no—
197C2×C8—1364no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—681no—
201C4⋊C4—1362no—
202C22⋊C4—348no—
203C22⋊C4—682no—
204C22⋊C4—1362no—
205C42—22no—
206C42—42no—
207C42—172no—
208C42—341no—
209C42—681no—
210C23—174no—
211C23—342no—
212C23—342no—
213C23—682no—
214C23—682no—
215C23C2×C22⋊#59511yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—682no—
219D4—682no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—342no—
231C2×C4—342no—
232C2×C4—684no—
233C2×C4—681no—
234C2×C4—681no—
235C2×C4—682no—
236C2×C4—681no—
237C2×C4—681no—
238C2×C4—681no—
239C2×C4—1362no—
240C2×C4#1058755112yes—
241C2×C4C2×#6324211yescharacteristic
242C2×C4D17⋊M1611yescharacteristic
243C8—1364no—
244C22—21no—
245C22—21no—
246C22—344no—
247C22—684no—
248C22—682no—
249C22—682no—
250C22—1362no—
251C22#1176377711yescharacteristic
252C22C2×#1058755111yescharacteristic
253C22#1176379511yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—42no—
258C4—41no—
259C4—81no—
260C4—682no—
261C4—682no—
262C4—1361no—
263C2—21no—
264C2—21no—
265C2—684no—
266C2#1454401911yescharacteristic
267C2#1454505911yescharacteristic
268C2#1454508611yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C4×Q8×C17C4
socleC22×C17C2×#99
prime radicalC22×D68C2×C4
cosocleC2×C4⋊Dic17C23
hypercenterC2×C4×Q8#595
hypocenterC17#11481
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #14543727C2semidirect
3. #14543714C2semidirect
27. #10587633C8semidirect
45. C2×Q8×C17C2×C8semidirect
74. C4×Dic17C2×C8semidirect
175. C17#11481semidirect
180. C2×Q8#63242semidirect

Maximal subgroups

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

subgroupindex
#145440212
#145437272
#145437142
#145314912
#1148117