Finite Group Databasebeta

← #17958217

Subgroups — #17958217

Order 4352.

562 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C2×#11763652C212yesmaximal
2#14543728C214yesmaximal, complemented
3#14531398C211yesmaximal, characteristic
4C2×C22⋊#3624—22no—
5#11763652—22no—
6#11763652C2214yes—
7C22×C22⋊Dic17C411yescharacteristic
8C22×C22⋊Dic17C2211yescharacteristic
9#11762684—22no—
10#11762684C412yes—
11#11762684C2212yes—
12#11762669—22no—
13C22×C22⋊C4×C17C411yescharacteristic
14C25×C17C2×C411yescharacteristic
15C23×C4×C17—21no—
16C23×Dic17C2×C412yes—
17C23×Dic17D412yes—
18C22×#3624—42no—
19C2×C22⋊C4×C17—21no—
20C2×C22⋊C4×C17—22no—
21C2×C22⋊C4×C17C814yescomplemented
22C2×C22⋊C4×C17C2×C412yes—
23C2×C22⋊Dic17—21no—
24C2×C22⋊Dic17—21no—
25C2×C22⋊Dic17—42no—
26C2×C22⋊Dic17—42no—
27C2×C22⋊Dic17C2×C411yescharacteristic
28C2×C22⋊Dic17D412yes—
29C2×C22⋊Dic17C2311yescharacteristic
30C2×C4⋊Dic17—22no—
31C2×C4⋊Dic17—22no—
32C22⋊#3624—24no—
33C22⋊#3624—42no—
34#10587502—22no—
35#10587502—22no—
36C24×C17—22no—
37C24×C17—22no—
38C24×C17—82no—
39C24×C17C22⋊C411yescharacteristic
40C24×C17C22⋊C411yescharacteristic
41C24×C17C2×C812yes—
42C24×C17M1612yes—
43C24×C17C22×C411yescharacteristic
44C22×C4×C17—22no—
45C22×C4×C17—24no—
46C22×C4×C17—82no—
47C22×Dic17—22no—
48C22×Dic17—22no—
49C22×Dic17—22no—
50C22×Dic17—42no—
51C22×Dic17—82no—
52C22×Dic17—82no—
53C22×Dic17C22⋊C412yes—
54C22×Dic17C2×D412yes—
55C2×#3624—44no—
56C2×#3624—84no—
57C22⋊C4×C17—24no—
58C22⋊C4×C17—22no—
59C22⋊C4×C17C2×C814yescomplemented
60C22⋊Dic17—41no—
61C22⋊Dic17—42no—
62C22⋊Dic17—41no—
63C22⋊Dic17—42no—
64C22⋊Dic17—42no—
65C4⋊Dic17—82no—
66C4⋊Dic17—82no—
67#11453—171nomaximal, Sylow
68C23×C17—21no—
69C23×C17—21no—
70C23×C17—21no—
71C23×C17—21no—
72C23×C17—24no—
73C23×C17—24no—
74C23×C17—42no—
75C23×C17—42no—
76C23×C17—42no—
77C23×C17—82no—
78C23×C17—82no—
79C23×C17—82no—
80C23×C17—82no—
81C23×C17—84no—
82C23×C17C22⋊C812yes—
83C23×C17C22⋊C812yes—
84C23×C17#9912yes—
85C23×C17#10112yes—
86C23×C17C2×C22⋊C411yescharacteristic
87C23×C17C22×C811yescharacteristic
88C23×C17C2×M1611yescharacteristic
89C2×C4×C17—24no—
90C2×C4×C17—42no—
91C2×C4×C17—42no—
92C2×C4×C17—84no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—44no—
96C2×Dic17—42no—
97C2×Dic17—42no—
98C2×Dic17—42no—
99C2×Dic17—42no—
100C2×Dic17—42no—
101C2×Dic17—82no—
102C2×Dic17—84no—
103#3624—84no—
104#2315—171no—
105C2×#323—172no—
106#1326—174no—
107C22×C17—22no—
108C22×C17—22no—
109C22×C17—22no—
110C22×C17—22no—
111C22×C17—21no—
112C22×C17—21no—
113C22×C17—42no—
114C22×C17—42no—
115C22×C17—42no—
116C22×C17—41no—
117C22×C17—41no—
118C22×C17—42no—
119C22×C17—42no—
120C22×C17—42no—
121C22×C17—42no—
122C22×C17—84no—
123C22×C17—84no—
124C22×C17#32414yes—
125C22×C17C2×C22⋊C811yescharacteristic
126C22×C17C2×#9911yescharacteristic
127C22×C17C2×#10111yescharacteristic
128C4×C17—44no—
129Dic17—42no—
130Dic17—42no—
131Dic17—82no—
132C22×C22⋊C4—171no—
133C22×C22⋊C4—171no—
134C22×C22⋊C4#59511yescharacteristic
135C2×C22⋊C8—342no—
136#385—172no—
137#385—172no—
138#385—342no—
139#380—342no—
140#323—174no—
141#323—342no—
142C2×C17—21no—
143C2×C17—21no—
144C2×C17—24no—
145C2×C17—42no—
146C2×C17—41no—
147C2×C17—41no—
148C2×C17C2×#32411yescharacteristic
149C2×C17#212112yes—
150C25C2×#59511yescharacteristic
151C23×C4—21no—
152C23×C4—172no—
153C23×C4—172no—
154C22×C8—682no—
155C2×C4⋊C4—342no—
156C2×C4⋊C4—342no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—172no—
160C2×C22⋊C4—171no—
161C2×C22⋊C4—171no—
162C2×C22⋊C4—341no—
163C2×C22⋊C4—341no—
164C2×C22⋊C4—682no—
165C2×C22⋊C4—682no—
166C2×C22⋊C4#362414yescomplemented
167C2×C22⋊C4C2×#59512yes—
168C22⋊C8—344no—
169C22⋊C8—682no—
170#95—342no—
171#95—342no—
172C17#1145311yescharacteristic, complemented, Sylow
173C24—22no—
174C24—22no—
175C24—82no—
176C24C2×#362412yes—
177C24#6324712yes—
178C24C22⋊#59511yescharacteristic
179C24C22⋊#59511yescharacteristic
180C24C22×#59511yescharacteristic
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—82no—
184C22×C4—172no—
185C22×C4—172no—
186C22×C4—342no—
187C22×C4—342no—
188C22×C4—342no—
189C22×C4—682no—
190C22×C4—1362no—
191C22×C4—1362no—
192C2×C8—684no—
193C2×C8—1364no—
194C4⋊C4—1362no—
195C4⋊C4—1362no—
196C22⋊C4—22no—
197C22⋊C4—24no—
198C22⋊C4—681no—
199C22⋊C4—682no—
200C22⋊C4—681no—
201C22⋊C4—682no—
202C22⋊C4—682no—
203C22⋊C4C2×#362414yescomplemented
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—24no—
209C23—24no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—82no—
214C23—82no—
215C23—82no—
216C23—82no—
217C23—84no—
218C23#1058753912yes—
219C23#1058754512yes—
220C23C22⋊#362412yes—
221C23C22⋊#362412yes—
222C23C22×#362411yescharacteristic
223C23C2×#6324711yescharacteristic
224C23C2×C22⋊#59511yescharacteristic
225C2×C4—24no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—84no—
229C2×C4—342no—
230C2×C4—342no—
231C2×C4—684no—
232C2×C4—682no—
233C2×C4—682no—
234C2×C4—682no—
235C2×C4—682no—
236C2×C4—682no—
237C2×C4—1362no—
238C2×C4—1364no—
239C8—1364no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—41no—
250C22—41no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—84no—
256C22—84no—
257C22#1176365414yes—
258C22C2×C22⋊#362411yescharacteristic
259C22C2×#1058753911yescharacteristic
260C22C2×#1058754511yescharacteristic
261C4—44no—
262C4—682no—
263C4—682no—
264C4—1362no—
265C2—21no—
266C2—21no—
267C2—24no—
268C2—41no—
269C2—41no—
270C2—42no—
271C2C2×#1176365411yescharacteristic
272C2#1454504112yes—
subgroupG/H
centerC22C2×#10587539
commutatorC23×C17C22×C8
FrattiniC24C22×#595
FittingC22×C22⋊C4×C17C4
socleC22×C17C2×#99
prime radicalC25×C17C2×C4
cosocleC2×C22⋊Dic17C23
hypercenterC22×C22⋊C4#595
hypocenterC17#11453
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #14543728C2semidirect
21. C2×C22⋊C4×C17C8semidirect
59. C22⋊C4×C17C2×C8semidirect
166. C2×C22⋊C4#3624semidirect
172. C17#11453semidirect
203. C22⋊C4C2×#3624semidirect

Maximal subgroups

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

subgroupindex
C2×#117636522
#145437282
#145313982
#1145317