Finite Group Databasebeta

← #16923511

Subgroups — #16923511

Order 4352.

478 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 83 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#14535810C211yesmaximal, characteristic
2#14530327C211yesmaximal, characteristic, complemented
3#14530189C212yesmaximal, complemented
4#14526604C212yesmaximal, complemented
5C17×#1272C211yesmaximal, characteristic, complemented
6C2×#10587633—21no—
7C2×#10587479—21no—
8C2×#10587506—21no—
9C2×#10587492—22no—
10#11762726—21no—
11#11762719—22no—
12#11762711C2212yes—
13#11762701—22no—
14#11762679C2211yescharacteristic
15#11762572—22no—
16#11762564C2211yescharacteristic
17C17×#391C2211yescharacteristic
18C17×#336C2212yes—
19C22×Q8×C17D411yescharacteristic
20C22×D68—21no—
21C22×D68—21no—
22C22×D68D412yes—
23C22×C8×C17—22no—
24C2×C4⋊C4×C17D411yescharacteristic
25C2×C42×C17C2311yescharacteristic
26#10587633—42no—
27#10587633—42no—
28C2×#63227—22no—
29C2×#63227—22no—
30C2×#63227—22no—
31C2×#63227—22no—
32C2×C4⋊Dic17—22no—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17D412yes—
35C2×C4×Dic17—21no—
36C22×#3622—21no—
37C17×#95—22no—
38#10587506—42no—
39#10587502—21no—
40#10587492—44no—
41#10587479—42no—
42C23×D17—22no—
43C2×Q8×C17—21no—
44C2×Q8×C17—41no—
45C2×Q8×C17—41no—
46C2×Q8×C17SD1612yes—
47C22×C4×C17—21no—
48C22×C4×C17—21no—
49C22×C4×C17C2×D411yescharacteristic
50C22×C4×C17C2×D412yes—
51C22×Dic17—22no—
52C22×Dic17—41no—
53C2×D68—22no—
54C2×D68—22no—
55C2×D68—42no—
56C2×D68—42no—
57C2×D68—42no—
58C2×D68—81no—
59C2×D68—81no—
60C2×D68—81no—
61C2×D68—82no—
62C2×C8×C17—42no—
63C2×C8×C17—42no—
64C2×C8×C17—42no—
65C4⋊C4×C17—41no—
66C42×C17—41no—
67#63227—44no—
68#63227—42no—
69#63227—42no—
70#63227—42no—
71#63227—42no—
72#63227—82no—
73C4⋊Dic17—42no—
74C4⋊Dic17—41no—
75C4⋊Dic17—41no—
76C4⋊Dic17—42no—
77C4⋊Dic17—42no—
78C4×Dic17—42no—
79C2×#3622—22no—
80C2×#3622—41no—
81C2×#3622—41no—
82#7810—171nomaximal, Sylow
83C23×C17#12011yescharacteristic
84C22×D17—42no—
85C22×D17—42no—
86C22×D17—42no—
87C22×D17—42no—
88C22×D17—42no—
89C22×D17—42no—
90C22×D17—42no—
91Q8×C17—42no—
92Q8×C17—81no—
93C2×C4×C17—21no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—21no—
97C2×C4×C17—22no—
98C2×C4×C17—41no—
99C2×C4×C17—41no—
100C2×C4×C17—41no—
101C2×C4×C17—42no—
102C2×C4×C17C2×SD1611yescharacteristic
103C2×C4×C17#13512yes—
104C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
105C2×C4×C17AGL(1,ℤ/8)12yes—
106C2×Dic17—42no—
107C2×Dic17—42no—
108C2×Dic17—42no—
109C2×Dic17—82no—
110C2×Dic17—81no—
111D68—44no—
112D68—82no—
113D68—84no—
114D68—84no—
115C8×C17—82no—
116#3622—42no—
117#2733—171no—
118#2096—171no—
119#2071—172no—
120#1272D1711yescharacteristic, complemented
121#1270—172no—
122C22×C17#44912yes—
123C22×C17#45011yescharacteristic
124C22×C17#45311yescharacteristic
125C22×C17#45512yes—
126C22×C17#45811yescharacteristic
127C2×D17—82no—
128C2×D17—82no—
129C2×D17—82no—
130C2×D17—82no—
131C2×D17—82no—
132C2×D17—82no—
133C2×D17—82no—
134C4×C17—22no—
135C4×C17—22no—
136C4×C17—21no—
137C4×C17—21no—
138C4×C17—42no—
139C4×C17—81no—
140Dic17—82no—
141Dic17—161no—
142C2×#124—341no—
143C2×#106—341no—
144C2×#103—341no—
145C2×#102—342no—
146#394—341no—
147#392—342no—
148#391C2×D1711yescharacteristic
149#390—172no—
150#388—342no—
151#384—171no—
152#340—342no—
153#336—171no—
154#336C2×D1712yes—
155C2×C17#161212yes—
156C2×C17#161311yescharacteristic
157C2×C17#162411yescharacteristic
158C2×C17#164212yes—
159C2×C17#165911yescharacteristic
160D17—162no—
161C22×Q8D6811yescharacteristic
162C22×D4—172no—
163C22×D4—341no—
164C22×D4—341no—
165C22×C8—22no—
166C22×C8—341no—
167#124—682no—
168#124—682no—
169C2×C4⋊C4—172no—
170C2×C4⋊C4—341no—
171C2×C4⋊C4—342no—
172C2×C4⋊C4D6811yescharacteristic
173C2×C22⋊C4—342no—
174C2×C22⋊C4—342no—
175C2×C22⋊C4—342no—
176C2×C22⋊C4—342no—
177C2×C42—341no—
178C2×C42C22×D1711yescharacteristic
179#106—682no—
180#103—682no—
181#102—684no—
182#95—22no—
183#95—341no—
184C17#781011yescharacteristic, complemented, Sylow
185C24—342no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8#6321912yes—
190C2×D4—342no—
191C2×D4—342no—
192C2×D4—682no—
193C2×D4—682no—
194C2×D4—682no—
195C2×D4—1361no—
196C2×D4—1361no—
197C2×D4—1361no—
198C2×D4—1362no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—342no—
202C22×C4—681no—
203C22×C4C2×D6811yescharacteristic
204C22×C4D4×D1712yes—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—342no—
209C2×C8—681no—
210C2×C8—681no—
211C4⋊C4—41no—
212C4⋊C4—682no—
213C4⋊C4—681no—
214C4⋊C4—681no—
215C4⋊C4—682no—
216C4⋊C4—682no—
217C22⋊C4—684no—
218C22⋊C4—682no—
219C22⋊C4—682no—
220C22⋊C4—682no—
221C22⋊C4—682no—
222C22⋊C4—1362no—
223C42—41no—
224C42—682no—
225C23—682no—
226C23—682no—
227C23—682no—
228C23—682no—
229C23—682no—
230C23—682no—
231C23—682no—
232C23#1058756811yescharacteristic
233Q8—42no—
234Q8—81no—
235D4—684no—
236D4—1362no—
237D4—1364no—
238D4—1364no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—682no—
249C2×C4—682no—
250C2×C4—682no—
251C2×C4—1362no—
252C2×C4—1361no—
253C2×C4C2×#6321911yescharacteristic
254C2×C4#1058759411yescharacteristic
255C2×C4#1058760012yes—
256C2×C4#1058760512yes—
257C8—82no—
258C8—682no—
259C22—1362no—
260C22—1362no—
261C22—1362no—
262C22—1362no—
263C22—1362no—
264C22—1362no—
265C22—1362no—
266C22#1176295712yes—
267C22#1176296211yescharacteristic
268C22#1176297911yescharacteristic
269C22#1176298812yes—
270C22#1176300011yescharacteristic
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—81no—
277C4—2721no—
278C4—1362no—
279C2—2722no—
280C2#1452780112yes—
281C2#1452780611yescharacteristic
282C2#1452786811yescharacteristic
283C2#1452798712yes—
284C2#1452809111yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1272C2
socleC23×C17#120
prime radical#14535810C2
hypercenter#1272D17
hypocenterC17#7810
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #14530327C2semidirect
3. #14530189C2semidirect
4. #14526604C2semidirect
5. C17×#1272C2semidirect
120. #1272D17semidirect
184. C17#7810semidirect

Maximal subgroups

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

subgroupindex
#145358102
#145303272
#145301892
#145266042
C17×#12722
#781017