Finite Group Databasebeta

← #17953609

Subgroups — #17953609

Order 4352.

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

normal subgroups only

This group has 522 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
1#14544017C212yesmaximal
2#14543612C212yesmaximal, complemented
3#14543608C212yesmaximal, complemented
4C4×#10587558C211yesmaximal, characteristic
5#11763825C2214yes—
6#11763655—22no—
7#11763653—22no—
8C2×#10587558C411yescharacteristic
9C2×#10587558C2211yescharacteristic
10C2×C42×D17C411yescharacteristic
11C4×#63225—22no—
12C4×#63227—22no—
13#11762655C411yescharacteristic
14#11762655C2211yescharacteristic
15C42×Dic17C2211yescharacteristic
16C43×C17C411yescharacteristic
17C22×C4×D17—21no—
18C22×C4×D17C2×C411yescharacteristic
19C2×C42×C17—22no—
20C2×C42×C17C2×C411yescharacteristic
21C2×C42×C17C2×C411yescharacteristic
22C2×C42×C17C2×C411yescharacteristic
23C2×#63227—21no—
24C2×#63227D412yes—
25C2×#63225—21no—
26C2×#63225D412yes—
27C2×C4×Dic17—21no—
28C2×C4×Dic17—22no—
29C2×C4×Dic17C2×C411yescharacteristic
30C2×C4×Dic17C2×C411yescharacteristic
31C2×C4×Dic17C2311yescharacteristic
32#10587558—22no—
33#10587558—22no—
34#10587558—22no—
35#10587558—22no—
36C42×D17—22no—
37C42×D17C814yescomplemented
38C22⋊#3624—24no—
39C4⋊#3624—24no—
40C4×#3624—24no—
41#10587502—22no—
42#10587502—22no—
43C23×D17C22⋊C411yescharacteristic
44C22×C4×C17—21no—
45C22×C4×C17—21no—
46C22×C4×C17C22⋊C412yes—
47C22×C4×C17C22×C411yescharacteristic
48C22×Dic17—21no—
49C22×Dic17—21no—
50C22×Dic17C22⋊C411yescharacteristic
51C22×Dic17C2×D412yes—
52C2×C4×D17—21no—
53C2×C4×D17—21no—
54C2×C4×D17—21no—
55C2×C4×D17—21no—
56C2×C4×D17—24no—
57C2×C4×D17—44no—
58C2×C4×D17C2×C812yes—
59C2×C4×D17M1612yes—
60C2×#3624—44no—
61C42×C17—21no—
62C42×C17—21no—
63C42×C17—21no—
64C42×C17—21no—
65C42×C17—44no—
66C42×C17C2×C812yes—
67C42×C17M1612yes—
68#63227—24no—
69#63227—24no—
70#63225—24no—
71#63225—24no—
72C4×Dic17—21no—
73C4×Dic17—21no—
74C4×Dic17—24no—
75C4×Dic17—44no—
76C4×Dic17C2×C812yes—
77C4×Dic17M1612yes—
78C4×Dic17#4112yes—
79C4×Dic17#4112yes—
80#8730—171nomaximal, Sylow
81C23×C17C2×C22⋊C411yescharacteristic
82C22×D17—21no—
83C22×D17—21no—
84C22×D17—42no—
85C22×D17C22⋊C812yes—
86C2×C4×C17—24no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—42no—
91C2×C4×C17—42no—
92C2×C4×C17#11711yescharacteristic
93C2×C4×C17#11711yescharacteristic
94C2×C4×C17C22×C811yescharacteristic
95C2×C4×C17C2×M1611yescharacteristic
96C2×C4×C17C2×M1611yescharacteristic
97C2×C4×C17C2×M1611yescharacteristic
98C2×Dic17—21no—
99C2×Dic17—21no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—42no—
103C2×Dic17—42no—
104C2×Dic17—42no—
105C2×Dic17C22⋊C812yes—
106C2×Dic17#12314yes—
107C4×D17—22no—
108C4×D17—22no—
109C4×D17—22no—
110C4×D17—22no—
111C4×D17—48no—
112#3624—84no—
113C4×#117—171no—
114#1458—172no—
115#1276—172no—
116#1274—172no—
117C22×C17—22no—
118C22×C17#38111yescharacteristic
119C22×C17C2×C22⋊C811yescharacteristic
120C22×C17#40711yescharacteristic
121C2×D17—22no—
122C2×D17—42no—
123C2×D17—42no—
124C2×D17—42no—
125C4×C17—24no—
126C4×C17—42no—
127C4×C17—42no—
128C4×C17#43111yescharacteristic
129C4×C17#43111yescharacteristic
130C4×C17#43211yescharacteristic
131C4×C17#43211yescharacteristic
132Dic17—22no—
133Dic17—24no—
134Dic17—42no—
135Dic17—42no—
136C2×#117—171no—
137C2×#117—171no—
138C22×C42—171no—
139#431—174no—
140C4×C4⋊C4—342no—
141C4×C22⋊C4—342no—
142#376—171no—
143#376—171no—
144C43—171no—
145C43#59511yescharacteristic
146#324—342no—
147#323—342no—
148C2×C17—21no—
149C2×C17—21no—
150C2×C17#181711yescharacteristic
151C2×C17#182712yes—
152D17—42no—
153C23×C4—171no—
154C23×C4—341no—
155#117—342no—
156#117—342no—
157#117—342no—
158#117—342no—
159C2×C4⋊C4—172no—
160C2×C4⋊C4—341no—
161C2×C22⋊C4—172no—
162C2×C22⋊C4—341no—
163C2×C42—22no—
164C2×C42—171no—
165C2×C42—171no—
166C2×C42—171no—
167C2×C42—174no—
168C2×C42—342no—
169C2×C42—341no—
170C2×C42—342no—
171C2×C42C2×#59511yescharacteristic
172C2×C42C2×#59511yescharacteristic
173C2×C42C2×#59511yescharacteristic
174C4⋊C8—344no—
175C22⋊C8—344no—
176C4×C8—344no—
177#95—342no—
178#95—342no—
179C17#873011yescharacteristic, complemented, Sylow
180C24—171no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—171no—
184C22×C4—172no—
185C22×C4—172no—
186C22×C4—172no—
187C22×C4—344no—
188C22×C4—341no—
189C22×C4—341no—
190C22×C4—341no—
191C22×C4—341no—
192C22×C4—341no—
193C22×C4—341no—
194C22×C4—684no—
195C22×C4C22⋊#59512yes—
196C22×C4C22×#59511yescharacteristic
197C2×C8—684no—
198C4⋊C4—344no—
199C4⋊C4—344no—
200C22⋊C4—344no—
201C22⋊C4—344no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—44no—
207C42—172no—
208C42—172no—
209C42—172no—
210C42—172no—
211C42—344no—
212C42—341no—
213C42—341no—
214C42—684no—
215C42#6324212yes—
216C42#6324312yes—
217C23—172no—
218C23—341no—
219C23—341no—
220C23—682no—
221C23C2×C22⋊#59511yescharacteristic
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—174no—
229C2×C4—172no—
230C2×C4—341no—
231C2×C4—341no—
232C2×C4—342no—
233C2×C4—342no—
234C2×C4—342no—
235C2×C4—342no—
236C2×C4—342no—
237C2×C4—342no—
238C2×C4—688no—
239C2×C4—682no—
240C2×C4—682no—
241C2×C4—682no—
242C2×C4C2×#6324211yescharacteristic
243C2×C4C2×#6324311yescharacteristic
244C2×C4D17⋊M1611yescharacteristic
245C2×C4D17⋊M1611yescharacteristic
246C2×C4#1058767711yescharacteristic
247C2×C4#1058767711yescharacteristic
248C8—1364no—
249C22—22no—
250C22—342no—
251C22—682no—
252C22—682no—
253C22—682no—
254C22#1176375211yescharacteristic
255C22#1176377711yescharacteristic
256C22#1176378011yescharacteristic
257C4—24no—
258C4—42no—
259C4—42no—
260C4—342no—
261C4—344no—
262C4—682no—
263C4—682no—
264C4#1176382711yescharacteristic
265C4#1176382811yescharacteristic
266C4#1176383111yescharacteristic
267C4#1176383211yescharacteristic
268C2—21no—
269C2—21no—
270C2—682no—
271C2#1454467611yescharacteristic
272C2#1454469312yes—
subgroupG/H
centerC2×C4#10587677
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC43×C17C4
socleC22×C17#381
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC43#595
hypocenterC17#8730
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C43

Complements

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

NH ≅ G/Ntype
2. #14543612C2semidirect
3. #14543608C2semidirect
37. C42×D17C8semidirect
179. C17#8730semidirect

Maximal subgroups

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

subgroupindex
#145440172
#145436122
#145436082
C4×#105875582
#873017