Finite Group Databasebeta

← #17958324

Subgroups — #17958324

Order 4352.

460 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 98 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#14544020C212yesmaximal
2#14543726C212yesmaximal, complemented
3#14543714C212yesmaximal, complemented
4#14531497C211yesmaximal, characteristic
5C2×C22⋊#3624—22no—
6#11763654C2214yes—
7#11763652—22no—
8C2×Q8×Dic17C411yescharacteristic
9C2×D4×Dic17C2211yescharacteristic
10#11762716C2211yescharacteristic
11#11762712C411yescharacteristic
12#11762702—22no—
13#11762684—22no—
14#11762679C2211yescharacteristic
15#11762655C411yescharacteristic
16C2×C17×#124C411yescharacteristic
17C22×Q8×C17C2×C411yescharacteristic
18C22×D4×C17C2×C411yescharacteristic
19C23×Dic17D412yes—
20C22×#3624—42no—
21C17×#124—22no—
22C17×#124C814yescomplemented
23C2×C22⋊C4×C17—22no—
24C2×C42×C17C2×C411yescharacteristic
25C2×C22⋊Dic17—22no—
26C2×C22⋊Dic17D412yes—
27Q8×Dic17—42no—
28D4×Dic17—42no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17—22no—
31C2×C4⋊Dic17C2×C411yescharacteristic
32C2×C4⋊Dic17C2311yescharacteristic
33C2×C4×Dic17—21no—
34C2×C4×Dic17C2×C411yescharacteristic
35C2×C4×Dic17C2×C411yescharacteristic
36C22⋊#3624—24no—
37C22⋊#3624—42no—
38#10587502—21no—
39#10587502—21no—
40#10587502—22no—
41C24×C17C22⋊C412yes—
42C2×Q8×C17—21no—
43C2×Q8×C17—41no—
44C2×Q8×C17—41no—
45C2×Q8×C17C2×C812yescomplemented
46C2×D4×C17—41no—
47C2×D4×C17—41no—
48C2×D4×C17C2×C812yes—
49C2×D4×C17M1612yes—
50C22×C4×C17—21no—
51C22×C4×C17—21no—
52C22×C4×C17C22×C411yescharacteristic
53C22×Dic17—22no—
54C22×Dic17—22no—
55C22×Dic17—21no—
56C22×Dic17—21no—
57C22×Dic17—82no—
58C22×Dic17C22⋊C411yescharacteristic
59C22×Dic17C22⋊C411yescharacteristic
60C22×Dic17C2×D412yes—
61C2×#3624—44no—
62C2×#3624—84no—
63C22⋊C4×C17—28no—
64C42×C17—21no—
65C42×C17C2×C812yescomplemented
66C22⋊Dic17—42no—
67C22⋊Dic17—82no—
68C4⋊Dic17—41no—
69C4⋊Dic17—42no—
70C4⋊Dic17—41no—
71C4⋊Dic17—82no—
72C4×Dic17—22no—
73C4×Dic17—42no—
74C4×Dic17—41no—
75#11481—171nomaximal, Sylow
76C23×C17—22no—
77C23×C17—22no—
78C23×C17—42no—
79C23×C17—42no—
80C23×C17C22⋊C814yes—
81C23×C17C2×C22⋊C411yescharacteristic
82Q8×C17—42no—
83Q8×C17—42no—
84D4×C17—42no—
85D4×C17—42no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—41no—
91C2×C4×C17—41no—
92C2×C4×C17—41no—
93C2×C4×C17—41no—
94C2×C4×C17C22×C811yescharacteristic
95C2×C4×C17C2×M1611yescharacteristic
96C2×Dic17—21no—
97C2×Dic17—22no—
98C2×Dic17—41no—
99C2×Dic17—44no—
100C2×Dic17—42no—
101C2×Dic17—41no—
102C2×Dic17—41no—
103C2×Dic17—42no—
104C2×Dic17—41no—
105C2×Dic17—41no—
106C2×Dic17—41no—
107C2×Dic17—82no—
108C2×Dic17#9912yes—
109#3624—84no—
110#2326—171no—
111#1459—172no—
112#1325—172no—
113#1319—172no—
114C22×C17—21no—
115C22×C17—21no—
116C22×C17—24no—
117C22×C17—44no—
118C22×C17—42no—
119C22×C17—42no—
120C22×C17—82no—
121C22×C17C2×C22⋊C811yescharacteristic
122C22×C17C2×#9911yescharacteristic
123C22×C17#41311yescharacteristic
124C4×C17—21no—
125C4×C17—21no—
126C4×C17—42no—
127C4×C17—42no—
128Dic17—22no—
129Dic17—42no—
130Dic17—41no—
131Dic17—81no—
132Dic17—81no—
133C2×#124#59511yescharacteristic
134C2×C4×Q8—171no—
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—44no—
148C2×C17#145911yescharacteristic
149C2×C17#212711yescharacteristic
150C2×C17#213511yescharacteristic
151C22×Q8C2×#59511yescharacteristic
152C22×D4C2×#59511yescharacteristic
153C23×C4—172no—
154C22×C8—682no—
155#124—22no—
156#124#362414yescomplemented
157C4×Q8—682no—
158C4×D4—682no—
159C2×C4⋊C4—171no—
160C2×C4⋊C4—171no—
161C2×C4⋊C4—341no—
162C2×C4⋊C4—342no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—172no—
165C2×C22⋊C4—342no—
166C2×C42—171no—
167C2×C42—171no—
168C2×C42—341no—
169C2×C42C2×#59511yescharacteristic
170C22⋊C8—344no—
171C22⋊C8—682no—
172#95—341no—
173#95—341no—
174#95—342no—
175C17#1148111yescharacteristic, complemented, Sylow
176C24C22⋊#59512yes—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8C2×#362412yescomplemented
181C2×D4—41no—
182C2×D4—41no—
183C2×D4C2×#362412yes—
184C2×D4#6324712yes—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—171no—
188C22×C4—171no—
189C22×C4—172no—
190C22×C4—342no—
191C22×C4—342no—
192C22×C4—341no—
193C22×C4—341no—
194C22×C4—1362no—
195C22×C4C22×#59511yescharacteristic
196C2×C8—684no—
197C2×C8—1364no—
198C4⋊C4—681no—
199C4⋊C4—681no—
200C4⋊C4—682no—
201C4⋊C4—1362no—
202C22⋊C4—28no—
203C22⋊C4—682no—
204C22⋊C4—1362no—
205C42—21no—
206C42—342no—
207C42—681no—
208C42—682no—
209C42C2×#362412yescomplemented
210C23—22no—
211C23—22no—
212C23—42no—
213C23—42no—
214C23C22⋊#362414yes—
215C23C2×C22⋊#59511yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—42no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—172no—
229C2×C4—341no—
230C2×C4—342no—
231C2×C4—681no—
232C2×C4—684no—
233C2×C4—682no—
234C2×C4—681no—
235C2×C4—681no—
236C2×C4—682no—
237C2×C4—681no—
238C2×C4—681no—
239C2×C4—681no—
240C2×C4—1362no—
241C2×C4C22×#362411yescharacteristic
242C2×C4C2×#6324711yescharacteristic
243C8—1364no—
244C22—21no—
245C22—21no—
246C22—24no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22C2×C22⋊#362411yescharacteristic
252C22D17⋊#9911yescharacteristic
253C22#1176379611yescharacteristic
254C4—21no—
255C4—21no—
256C4—42no—
257C4—42no—
258C4—342no—
259C4—682no—
260C4—681no—
261C4—1361no—
262C4—1361no—
263C2—21no—
264C2—21no—
265C2—44no—
266C2#1454402011yescharacteristic
267C2#1454506211yescharacteristic
268C2#1454508711yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C17×#124C4
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C4⋊Dic17C23
hypercenterC2×#124#595
hypocenterC17#11481
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #14543726C2semidirect
3. #14543714C2semidirect
22. C17×#124C8semidirect
45. C2×Q8×C17C2×C8semidirect
65. C42×C17C2×C8semidirect
156. #124#3624semidirect
175. C17#11481semidirect
180. C2×Q8C2×#3624semidirect
209. C42C2×#3624semidirect

Maximal subgroups

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

subgroupindex
#145440202
#145437262
#145437142
#145314972
#1148117