Finite Group Databasebeta

← #17949958

Subgroups — #17949958

Order 4352.

468 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 94 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#14544712C212yesmaximal, complemented
2#14543604C212yesmaximal, complemented
3#14543601C212yesmaximal
4C4×#10587633C211yesmaximal, characteristic
5#11763675—22no—
6#11763675—22no—
7#11763675C2214yes—
8C2×#10587633C411yescharacteristic
9C2×C4×D68C411yescharacteristic
10#11762716C2211yescharacteristic
11#11762710C2211yescharacteristic
12#11762699—22no—
13#11762674C411yescharacteristic
14C4×#63227—22no—
15C42×Dic17C2211yescharacteristic
16C2×C4×Q8×C17C411yescharacteristic
17C22×Q8×C17C2×C411yescharacteristic
18C22×D68C2×C411yescharacteristic
19C22×C4×D17—21no—
20C22×#3624—24no—
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#10587633—22no—
28C2×#63227—21no—
29C2×#63227—22no—
30C2×#63227—22no—
31C2×C4⋊Dic17C2×C411yescharacteristic
32C2×C4×Dic17C2×C411yescharacteristic
33C2×C4×Dic17D412yes—
34C2×C4×Dic17D412yes—
35C2×C4×Dic17C2311yescharacteristic
36C4×D68—42no—
37#10587502—22no—
38#10587502—22no—
39C23×D17—21no—
40C2×Q8×C17—21no—
41C2×Q8×C17—41no—
42C2×Q8×C17—41no—
43C2×Q8×C17M1612yes—
44C22×C4×C17—21no—
45C22×C4×C17—21no—
46C22×C4×C17C22⋊C411yescharacteristic
47C22×C4×C17C22⋊C411yescharacteristic
48C22×C4×C17C22×C411yescharacteristic
49C22×Dic17—21no—
50C22×Dic17C22⋊C412yes—
51C22×Dic17C2×D412yes—
52C2×D68—21no—
53C2×D68—41no—
54C2×D68—41no—
55C2×D68M1612yes—
56C2×C4×D17—44no—
57C2×C4×D17—42no—
58C2×#3624—44no—
59C2×#3624—44no—
60C2×#3624—44no—
61C4⋊C4×C17—41no—
62C4⋊C4×C17—42no—
63C42×C17—22no—
64C42×C17—42no—
65#63227—28no—
66#63227—42no—
67#63227—44no—
68C4⋊Dic17—41no—
69C4×Dic17—21no—
70C4×Dic17—21no—
71C4×Dic17—24no—
72C4×Dic17—42no—
73C4×Dic17—41no—
74#7649—171nomaximal, Sylow
75C23×C17C2×C22⋊C411yescharacteristic
76C22×D17—22no—
77C22×D17—41no—
78C22×D17—41no—
79C22×D17—42no—
80C22×D17—42no—
81Q8×C17—42no—
82Q8×C17—42no—
83C2×C4×C17—21no—
84C2×C4×C17—21no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—21no—
88C2×C4×C17—22no—
89C2×C4×C17—41no—
90C2×C4×C17—41no—
91C2×C4×C17—41no—
92C2×C4×C17—42no—
93C2×C4×C17#10012yes—
94C2×C4×C17C2×M1611yescharacteristic
95C2×C4×C17C2×M1611yescharacteristic
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—41no—
102C2×Dic17—42no—
103C2×Dic17#13514yes—
104D68—42no—
105D68—42no—
106C4×D17—84no—
107#3624—84no—
108C4×#124—171no—
109#1839—172no—
110#1272—172no—
111#1270—172no—
112C22×C17#40711yescharacteristic
113C22×C17#41011yescharacteristic
114C22×C17C2×#10011yescharacteristic
115C22×C17#41612yes—
116C22×C17#42112yes—
117C2×D17—42no—
118C2×D17—44no—
119C2×D17—81no—
120C2×D17—82no—
121C2×D17—81no—
122C4×C17—21no—
123C4×C17—21no—
124C4×C17—22no—
125C4×C17—41no—
126C4×C17—42no—
127C4×C17—81no—
128Dic17—24no—
129Dic17—42no—
130Dic17—81no—
131C2×#124—171no—
132C2×C4×Q8#59511yescharacteristic
133C2×C4×D4—171no—
134#391—171no—
135#390—171no—
136#388—342no—
137#382—171no—
138C4×C22⋊C4—342no—
139C43—171no—
140#336—174no—
141#336—342no—
142#336—342no—
143C2×C17#146611yescharacteristic
144C2×C17#148812yes—
145C2×C17#152112yes—
146C2×C17#153311yescharacteristic
147C2×C17#154311yescharacteristic
148D17—82no—
149C22×Q8C2×#59511yescharacteristic
150C22×D4—171no—
151C23×C4—341no—
152C22×C8—344no—
153#124—342no—
154#124—342no—
155C4×Q8—42no—
156C4×D4—682no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—171no—
159C2×C4⋊C4C2×#59511yescharacteristic
160C2×C22⋊C4—342no—
161C2×C22⋊C4—342no—
162C2×C22⋊C4—341no—
163C2×C42—21no—
164C2×C42—172no—
165C2×C42—171no—
166C2×C42—172no—
167C2×C42—171no—
168C2×C42C2×#59511yescharacteristic
169#95—342no—
170#95—342no—
171C17#764911yescharacteristic, complemented, Sylow
172C24—341no—
173C2×Q8—21no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8#6324312yes—
177C2×D4—172no—
178C2×D4—341no—
179C2×D4—681no—
180C2×D4—681no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—172no—
184C22×C4—172no—
185C22×C4—341no—
186C22×C4—684no—
187C22×C4—682no—
188C22×C4C22⋊#59511yescharacteristic
189C22×C4C22⋊#59511yescharacteristic
190C22×C4C22×#59511yescharacteristic
191C2×C8—684no—
192C2×C8—684no—
193C2×C8—684no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—681no—
197C22⋊C4—348no—
198C22⋊C4—682no—
199C22⋊C4—684no—
200C42—22no—
201C42—42no—
202C42—344no—
203C42—341no—
204C42—341no—
205C42—681no—
206C42—682no—
207C23—342no—
208C23—681no—
209C23—681no—
210C23—682no—
211C23—682no—
212C23C2×C22⋊#59511yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—682no—
216D4—682no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—174no—
228C2×C4—342no—
229C2×C4—342no—
230C2×C4—342no—
231C2×C4—342no—
232C2×C4—342no—
233C2×C4—681no—
234C2×C4—682no—
235C2×C4—1364no—
236C2×C4#1058755312yes—
237C2×C4C2×#6324311yescharacteristic
238C2×C4D17⋊M1611yescharacteristic
239C8—1364no—
240C22—682no—
241C22—684no—
242C22—1362no—
243C22—1361no—
244C22—1361no—
245C22#1176378011yescharacteristic
246C22#1176378811yescharacteristic
247C22C2×#1058755311yescharacteristic
248C22#1176379912yes—
249C22#1176380812yes—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—41no—
254C4—42no—
255C4—81no—
256C4—344no—
257C4—682no—
258C4—1361no—
259C2—1362no—
260C2#1454404311yescharacteristic
261C2#1454411812yes—
262C2#1454422812yes—
263C2#1454426511yescharacteristic
264C2#1454428711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C4×Q8×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC22×D68C2×C4
cosocleC2×C4×Dic17C23
hypercenterC2×C4×Q8#595
hypocenterC17#7649
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #14544712C2semidirect
2. #14543604C2semidirect
171. C17#7649semidirect

Maximal subgroups

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

subgroupindex
#145447122
#145436042
#145436012
C4×#105876332
#764917