Finite Group Databasebeta

← #17953315

Subgroups — #17953315

Order 4352.

486 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 96 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#14544663C212yesmaximal, complemented
2#14544041C212yesmaximal
3#14543656C212yesmaximal, complemented
4C2×C8×D68C211yesmaximal, characteristic
5C2×#10587535—22no—
6#11763780C2214yes—
7#11763685—22no—
8#11763654—22no—
9#11763653—22no—
10C22×C8×D17C411yescharacteristic
11C22×C8×D17C2211yescharacteristic
12C2×C4×D68C2211yescharacteristic
13C8×D68—22no—
14C8×D68—22no—
15C2×C4⋊#3622C411yescharacteristic
16C2×#10587491C411yescharacteristic
17C2×#10587491C2211yescharacteristic
18C2×C4×C8×C17C411yescharacteristic
19C22×D68Q811yescharacteristic
20C22×C4×D17C2×C411yescharacteristic
21C22×C4×D17C2311yescharacteristic
22C2×#63243—24no—
23C22×C8×C17C2×C411yescharacteristic
24C22×C8×C17C2×C411yescharacteristic
25C2×C42×C17C2×C411yescharacteristic
26C2×#63227D411yescharacteristic
27C2×#63227Q811yescharacteristic
28C2×C4⋊Dic17D411yescharacteristic
29C22×#3622C2×C411yescharacteristic
30C22×#3622C2×C411yescharacteristic
31C2×C8×D17—21no—
32C2×C8×D17—21no—
33C2×C8×D17—42no—
34C2×C8×D17—42no—
35C2×C8×D17D412yes—
36C2×C8×D17D412yes—
37C4×D68—22no—
38C4×D68—22no—
39C22⋊#3624—24no—
40#10587543—24no—
41#10587535—44no—
42C4×C8×C17—21no—
43C4×C8×C17—21no—
44#10587491—22no—
45#10587491—22no—
46C4⋊#3622—22no—
47C23×D17C4⋊C411yescharacteristic
48C23×D17C2×Q811yescharacteristic
49C22×C4×C17C4⋊C411yescharacteristic
50C22×C4×C17C4⋊C411yescharacteristic
51C22×C4×C17C22×C411yescharacteristic
52C22×Dic17C4⋊C411yescharacteristic
53C22×Dic17C2×D411yescharacteristic
54C2×D68—22no—
55C2×D68—22no—
56C2×D68—22no—
57C2×C4×D17—21no—
58C2×C4×D17—21no—
59C2×C4×D17—41no—
60C2×C4×D17—41no—
61C2×C4×D17—41no—
62C2×C4×D17—41no—
63C2×C4×D17C2×D412yes—
64C2×C4×D17#4112yes—
65C2×#3624—44no—
66#63243—44no—
67#63243—44no—
68C2×C8×C17—22no—
69C2×C8×C17—21no—
70C2×C8×C17—41no—
71C2×C8×C17C22⋊C411yescharacteristic
72C2×C8×C17C22⋊C411yescharacteristic
73C42×C17—21no—
74C42×C17—21no—
75#63227—22no—
76#63227—22no—
77C4⋊Dic17—22no—
78C2×#3622—21no—
79C2×#3622—41no—
80C2×#3622—41no—
81C2×#3622C22⋊C412yes—
82C8×D17—24no—
83C8×D17—42no—
84C8×D17—42no—
85C8×D17—42no—
86#8611—171nomaximal, Sylow
87C23×C17C2×C4⋊C411yescharacteristic
88C22×D17—21no—
89C22×D17—21no—
90C22×D17—21no—
91C22×D17—21no—
92C22×D17—41no—
93C22×D17—41no—
94C22×D17—41no—
95C22×D17—41no—
96C22×D17#10812yescomplemented
97C22×D17C22⋊Q812yes—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—41no—
104C2×C4×C17#10812yescomplemented
105C2×C4×C17C2×C22⋊C411yescharacteristic
106C2×C4×C17#11711yescharacteristic
107C2×Dic17—21no—
108C2×Dic17—41no—
109C2×Dic17—41no—
110C2×Dic17#12112yes—
111D68—28no—
112C4×D17—22no—
113C4×D17—22no—
114C4×D17—42no—
115C4×D17—42no—
116C4×D17—41no—
117C4×D17—41no—
118#3624—84no—
119C8×C17—21no—
120C8×C17—41no—
121C8×C17#9912yes—
122#3622—22no—
123#3622—41no—
124C2×C8×D4—171no—
125#1810—172no—
126#1465—172no—
127#1293—172no—
128C22×C17—21no—
129C22×C17—21no—
130C22×C17#38011yescharacteristic
131C22×C17C2×#10811yescharacteristic
132C22×C17C2×#10811yescharacteristic
133C2×D17—22no—
134C2×D17—22no—
135C2×D17—22no—
136C2×D17—22no—
137C2×D17—42no—
138C2×D17—42no—
139C2×D17—42no—
140C2×D17—41no—
141C2×D17—41no—
142C2×D17—42no—
143C4×C17—21no—
144C4×C17—22no—
145C4×C17—41no—
146C4×C17C2×#9911yescharacteristic
147C4×C17#41011yescharacteristic
148Dic17—22no—
149Dic17—41no—
150C23×C8—171no—
151C23×C8—171no—
152C2×C4×D4—171no—
153C8×D4—342no—
154C8×D4—342no—
155C2×#108—342no—
156C2×C4⋊C8—171no—
157#407—174no—
158C2×C22⋊C8—171no—
159C2×C22⋊C8—171no—
160C2×C4×C8#59511yescharacteristic
161#341—342no—
162#324—342no—
163#323—342no—
164C2×C17—21no—
165C2×C17—21no—
166C2×C17#179911yescharacteristic
167C2×C17#181011yescharacteristic
168C2×C17#181311yescharacteristic
169D17—42no—
170D17—42no—
171C22×D4—171no—
172C23×C4—171no—
173C23×C4—171no—
174C2×M16—344no—
175C22×C8—171no—
176C22×C8—172no—
177C22×C8—172no—
178C22×C8—171no—
179C22×C8—341no—
180C22×C8—341no—
181C22×C8—682no—
182C22×C8—682no—
183C22×C8C2×#59511yescharacteristic
184C22×C8C2×#59511yescharacteristic
185C4×D4—342no—
186C4×D4—342no—
187C2×C4⋊C4—171no—
188C2×C22⋊C4—171no—
189C2×C22⋊C4—171no—
190C2×C42C2×#59511yescharacteristic
191#108—684no—
192C4⋊C8—342no—
193C22⋊C8—342no—
194C22⋊C8—342no—
195C22⋊C8—344no—
196C22⋊C8—344no—
197C4×C8—21no—
198C4×C8—21no—
199C17#861111yescharacteristic, complemented, Sylow
200C24—171no—
201C24—171no—
202C2×D4—342no—
203C2×D4—342no—
204C2×D4—342no—
205C22×C4—171no—
206C22×C4—172no—
207C22×C4—172no—
208C22×C4—171no—
209C22×C4—341no—
210C22×C4—341no—
211C22×C4—681no—
212C22×C4—681no—
213C22×C4—681no—
214C22×C4—681no—
215C22×C4C4⋊#59511yescharacteristic
216C22×C4C4⋊#59511yescharacteristic
217C22×C4C22×#59511yescharacteristic
218M16—684no—
219M16—684no—
220C2×C8—22no—
221C2×C8—21no—
222C2×C8—41no—
223C2×C8—172no—
224C2×C8—344no—
225C2×C8—341no—
226C2×C8—682no—
227C2×C8—682no—
228C2×C8—681no—
229C2×C8—682no—
230C2×C8—681no—
231C2×C8—684no—
232C2×C8C22⋊#59511yescharacteristic
233C2×C8C22⋊#59511yescharacteristic
234C4⋊C4—342no—
235C22⋊C4—342no—
236C22⋊C4—342no—
237C42—21no—
238C42—21no—
239C23—172no—
240C23—172no—
241C23—341no—
242C23—341no—
243C23—341no—
244C23—341no—
245C23—681no—
246C23—681no—
247C23—681no—
248C23—681no—
249C23C2×C4⋊#59511yescharacteristic
250D4—348no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—172no—
258C2×C4—342no—
259C2×C4—342no—
260C2×C4—341no—
261C2×C4—682no—
262C2×C4—682no—
263C2×C4—681no—
264C2×C4—681no—
265C2×C4—681no—
266C2×C4—681no—
267C2×C4#1058753512yescomplemented
268C2×C4#1058767711yescharacteristic
269C2×C4C2×C22⋊#59511yescharacteristic
270C8—21no—
271C8—41no—
272C8—342no—
273C8—681no—
274C8—1364no—
275C8#1058753912yes—
276C22—21no—
277C22—21no—
278C22—342no—
279C22—342no—
280C22—342no—
281C22—342no—
282C22—682no—
283C22—682no—
284C22—682no—
285C22—682no—
286C22—681no—
287C22—681no—
288C22#1176375111yescharacteristic
289C22C2×#1058753511yescharacteristic
290C22D17⋊#10811yescharacteristic
291C4—21no—
292C4—22no—
293C4—41no—
294C4—342no—
295C4—681no—
296C4C2×#1058753911yescharacteristic
297C4#1176378811yescharacteristic
298C2—21no—
299C2—21no—
300C2—682no—
301C2—682no—
302C2#1454464511yescharacteristic
303C2#1454466311yescharacteristic
304C2#1454466811yescharacteristic
subgroupG/H
centerC22#11763751
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C4×C8×C17C4
socleC22×C17#380
prime radicalC22×D68Q8
cosocleC22×C4×D17C23
hypercenterC2×C4×C8#595
hypocenterC17#8611
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8

Complements

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

NH ≅ G/Ntype
1. #14544663C2semidirect
3. #14543656C2semidirect
96. C22×D17#108semidirect
104. C2×C4×C17#108semidirect
199. C17#8611semidirect
267. C2×C4#10587535semidirect

Maximal subgroups

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

subgroupindex
#145446632
#145440412
#145436562
C2×C8×D682
#861117