Finite Group Databasebeta

← #16925684

Subgroups — #16925684

Order 4352.

1174 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 179 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#14528694C212yesmaximal
2C2×#11762535C214yesmaximal, complemented
3C22×C17×C22⋊Q8C211yesmaximal, characteristic
4C2×C22⋊#3622—24no—
5C2×C22⋊#3622C2214yes—
6#11762535—28no—
7C23×Q8×C17C411yescharacteristic
8C24×C4×C17C2211yescharacteristic
9C2×C17×C22⋊Q8—28no—
10C2×C17×C22⋊Q8—24no—
11C22×C4⋊C4×C17C411yescharacteristic
12C22×C4⋊C4×C17C2212yes—
13C22×C22⋊C4×C17C412yes—
14C25×C17C2×C411yescharacteristic
15C22×Q8×C17—24no—
16C22×Q8×C17—22no—
17C22×Q8×C17—44no—
18C23×C4×C17—24no—
19C23×C4×C17—44no—
20C23×C4×C17C2×C412yes—
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17C2×C412yes—
23C23×C4×C17D414yes—
24C23×C4×C17C2311yescharacteristic
25C17×C22⋊Q8—232no—
26C2×C4⋊C4×C17—24no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—24no—
29C2×C4⋊C4×C17—24no—
30C2×C4⋊C4×C17D418yes—
31C2×C22⋊C4×C17—24no—
32C2×C22⋊C4×C17—24no—
33C2×C22⋊C4×C17—24no—
34C22×#3622—24no—
35C22⋊#3622—28no—
36C22⋊#3622—48no—
37C24×C17—24no—
38C24×C17—22no—
39C24×C17—24no—
40C24×C17—42no—
41C24×C17C22⋊C412yes—
42C24×C17C22×C411yescharacteristic
43C2×Q8×C17—28no—
44C2×Q8×C17—416no—
45C2×Q8×C17—44no—
46C2×Q8×C17—44no—
47C22×C4×C17—22no—
48C22×C4×C17—24no—
49C22×C4×C17—22no—
50C22×C4×C17—22no—
51C22×C4×C17—22no—
52C22×C4×C17—22no—
53C22×C4×C17—28no—
54C22×C4×C17—28no—
55C22×C4×C17—28no—
56C22×C4×C17—44no—
57C22×C4×C17—42no—
58C22×C4×C17—416no—
59C22×C4×C17—44no—
60C22×C4×C17C22⋊C414yes—
61C22×C4×C17C22⋊C412yes—
62C22×C4×C17C22⋊C414yes—
63C22×C4×C17C2×D412yes—
64C22×C4×C17C2×D414yes—
65C4⋊C4×C17—28no—
66C4⋊C4×C17—216no—
67C22⋊C4×C17—216no—
68C2×#3622—48no—
69C2×#3622—44no—
70#8308—171nomaximal, Sylow
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—28no—
74C23×C17—22no—
75C23×C17—24no—
76C23×C17—24no—
77C23×C17—22no—
78C23×C17—416no—
79C23×C17—44no—
80C23×C17—44no—
81C23×C17#10114yes—
82C23×C17C2×C22⋊C412yes—
83C23×C17C2×C22⋊C411yescharacteristic
84C23×C17#13114yes—
85Q8×C17—416no—
86C2×C4×C17—28no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—24no—
90C2×C4×C17—24no—
91C2×C4×C17—24no—
92C2×C4×C17—28no—
93C2×C4×C17—44no—
94C2×C4×C17—416no—
95C2×C4×C17—44no—
96C2×C4×C17—44no—
97C2×C4×C17—44no—
98C2×C4×C17—44no—
99C2×C4×C17—44no—
100C2×C4×C17#9914yes—
101C2×C4×C17#12014yes—
102#3622—84no—
103C22×C22⋊Q8D1711yescharacteristic
104#1779—172no—
105C2×#324—174no—
106C22×C17—28no—
107C22×C17—22no—
108C22×C17—22no—
109C22×C17—28no—
110C22×C17—24no—
111C22×C17—416no—
112C22×C17—44no—
113C22×C17—42no—
114C22×C17—42no—
115C22×C17#37911yescharacteristic
116C22×C17#40814yes—
117C22×C17#40812yes—
118C22×C17C2×#9912yes—
119C22×C17C2×#10112yes—
120C4×C17—24no—
121C4×C17—24no—
122C4×C17—44no—
123C4×C17—44no—
124C23×Q8Dic1711yescharacteristic
125C24×C4C2×D1711yescharacteristic
126C2×C22⋊Q8—24no—
127C2×C22⋊Q8—28no—
128C22×C4⋊C4Dic1711yescharacteristic
129C22×C4⋊C4C2×D1712yes—
130C22×C22⋊C4Dic1712yes—
131C2×C22⋊C8—174no—
132C2×C22⋊C8—344no—
133#324—348no—
134C2×C17—24no—
135C2×C17—44no—
136C2×C17#147214yes—
137C2×C17#178011yescharacteristic
138C2×C17#178111yescharacteristic
139C2×C17#178411yescharacteristic
140C25C2×Dic1711yescharacteristic
141C22×Q8—24no—
142C22×Q8—22no—
143C22×Q8—44no—
144C23×C4—24no—
145C23×C4—44no—
146C23×C4C2×Dic1712yes—
147C23×C4C2×Dic1711yescharacteristic
148C23×C4C2×Dic1712yes—
149C23×C4C22⋊D1714yes—
150C23×C4C22×D1711yescharacteristic
151C22×C8—344no—
152C22⋊Q8—232no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4C22⋊D1718yes—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C22⋊C8—348no—
162C22⋊C8—688no—
163C17#830811yescharacteristic, complemented, Sylow
164C24—24no—
165C24—22no—
166C24—24no—
167C24—42no—
168C24C22⋊Dic1712yes—
169C24C22×Dic1711yescharacteristic
170C2×Q8—28no—
171C2×Q8—416no—
172C2×Q8—44no—
173C2×Q8—44no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—28no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—28no—
182C22×C4—28no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—416no—
186C22×C4—44no—
187C22×C4C22⋊Dic1714yes—
188C22×C4C22⋊Dic1712yes—
189C22×C4C22⋊Dic1714yes—
190C22×C4C2×C22⋊D1712yes—
191C22×C4C2×C22⋊D1714yes—
192C2×C8—688no—
193C2×C8—684no—
194C4⋊C4—28no—
195C4⋊C4—216no—
196C22⋊C4—216no—
197C23—22no—
198C23—22no—
199C23—28no—
200C23—24no—
201C23—24no—
202C23—22no—
203C23—22no—
204C23—416no—
205C23—44no—
206C23—44no—
207C23#1058750714yes—
208C23#1058763414yes—
209C23C2×C22⋊Dic1712yes—
210C23C2×C22⋊Dic1711yescharacteristic
211Q8—416no—
212C2×C4—28no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—28no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—416no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4#1058750514yes—
227C2×C4#1058763914yes—
228C8—1364no—
229C22—28no—
230C22—22no—
231C22—22no—
232C22—28no—
233C22—24no—
234C22—416no—
235C22—44no—
236C22—42no—
237C22—42no—
238C22#1176266511yescharacteristic
239C22#1176277614yes—
240C22#1176277612yes—
241C22C2×#1058750512yes—
242C22C2×#1058750712yes—
243C4—24no—
244C4—24no—
245C4—44no—
246C4—44no—
247C2—24no—
248C2—44no—
249C2#1452714814yes—
250C2#1452869611yescharacteristic
251C2#1452869911yescharacteristic
252C2#1452870811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic17
commutatorC24×C17C22×C4
FrattiniC23×C4C22×D17
FittingC22×C17×C22⋊Q8C2
socleC23×C17C2×C22⋊C4
prime radicalC25×C17C2×C4
cosocleC23×C4×C17C23
hypercenterC22×C22⋊Q8D17
hypocenterC17#8308
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#11762535C2semidirect
163. C17#8308semidirect

Maximal subgroups

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

subgroupindex
#145286942
C2×#117625352
C22×C17×C22⋊Q82
#830817