Finite Group Databasebeta

← #16922613

Subgroups — #16922613

Order 4352.

646 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 163 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#14528764C212yesmaximal
2#14526610C214yesmaximal, complemented
3C2×C17×#391C211yesmaximal, characteristic
4C2×C4⋊#3622—22no—
5C2×C22⋊#3622—22no—
6#11762564C2214yes—
7C23×Q8×C17C411yescharacteristic
8C22×C4⋊C4×C17C411yescharacteristic
9C22×C42×C17C2211yescharacteristic
10C17×#391—22no—
11C17×#391C412yes—
12C17×#391C2212yes—
13C2×C17×#95—22no—
14C22×Q8×C17—24no—
15C22×Q8×C17—21no—
16C22×Q8×C17—42no—
17C22×Q8×C17—42no—
18C22×Q8×C17C2×C412yes—
19C23×C4×C17—21no—
20C23×C4×C17—21no—
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17D412yes—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17—42no—
25C2×C4⋊C4×C17C2×C412yes—
26C2×C42×C17—42no—
27C2×C42×C17C2×C411yescharacteristic
28C2×C42×C17D412yes—
29C2×C42×C17C2311yescharacteristic
30C22×#3622—24no—
31C17×#95—24no—
32C17×#95—24no—
33C22⋊#3622—44no—
34C4⋊#3622—44no—
35C24×C17C22⋊C411yescharacteristic
36C2×Q8×C17—24no—
37C2×Q8×C17—22no—
38C2×Q8×C17—416no—
39C2×Q8×C17—42no—
40C2×Q8×C17—42no—
41C2×Q8×C17—42no—
42C2×Q8×C17—42no—
43C2×Q8×C17M1614yes—
44C22×C4×C17—22no—
45C22×C4×C17—22no—
46C22×C4×C17—22no—
47C22×C4×C17—24no—
48C22×C4×C17—22no—
49C22×C4×C17—22no—
50C22×C4×C17—42no—
51C22×C4×C17—42no—
52C22×C4×C17—42no—
53C22×C4×C17—42no—
54C22×C4×C17—44no—
55C22×C4×C17—42no—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22⋊C412yes—
58C22×C4×C17SD1614yes—
59C22×C4×C17Q1614yes—
60C22×C4×C17C22×C411yescharacteristic
61C22×C4×C17C2×D412yes—
62C4⋊C4×C17—42no—
63C4⋊C4×C17—42no—
64C42×C17—24no—
65C42×C17—41no—
66C42×C17—41no—
67C2×#3622—44no—
68C2×#3622—44no—
69C2×#3622—44no—
70#7643—171nomaximal, Sylow
71C23×C17—21no—
72C23×C17—21no—
73C23×C17—22no—
74C23×C17#10012yes—
75C23×C17#10314yes—
76C23×C17C2×C22⋊C411yescharacteristic
77Q8×C17—44no—
78Q8×C17—48no—
79Q8×C17—44no—
80C2×C4×C17—28no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—21no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—24no—
90C2×C4×C17—44no—
91C2×C4×C17—44no—
92C2×C4×C17—48no—
93C2×C4×C17—42no—
94C2×C4×C17—42no—
95C2×C4×C17—42no—
96C2×C4×C17—44no—
97C2×C4×C17—84no—
98C2×C4×C17#9912yes—
99C2×C4×C17#10314yes—
100C2×C4×C17C2×M1612yes—
101C2×C4×C17C2×SD1612yes—
102C2×C4×C17C2×Q1612yes—
103#3622—84no—
104C2×#391D1711yescharacteristic
105#1797—172no—
106#1272—174no—
107C22×C17—22no—
108C22×C17—24no—
109C22×C17—24no—
110C22×C17—22no—
111C22×C17#33314yes—
112C22×C17#40711yescharacteristic
113C22×C17C2×#9911yescharacteristic
114C22×C17C2×#10011yescharacteristic
115C22×C17C2×#10312yes—
116C22×C17#42112yes—
117C4×C17—22no—
118C4×C17—22no—
119C4×C17—22no—
120C4×C17—24no—
121C4×C17—22no—
122C4×C17—44no—
123C4×C17—82no—
124C23×Q8Dic1711yescharacteristic
125C22×C4⋊C4Dic1711yescharacteristic
126C22×C42C2×D1711yescharacteristic
127C2×C4⋊C8—342no—
128C2×C22⋊C8—342no—
129#391—22no—
130#391Dic1712yes—
131#391C2×D1712yes—
132C2×#95—22no—
133#336—174no—
134C2×C17—24no—
135C2×C17#146211yescharacteristic
136C2×C17#148712yes—
137C2×C17#152012yes—
138C2×C17#153311yescharacteristic
139C2×C17C2×#33311yescharacteristic
140C22×Q8—24no—
141C22×Q8—21no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8C2×Dic1712yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C2×Dic1711yescharacteristic
148C23×C4C22⋊D1712yes—
149C22×C8—344no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4C2×Dic1712yes—
153C2×C42—42no—
154C2×C42C2×Dic1711yescharacteristic
155C2×C42C22⋊D1712yes—
156C2×C42C22×D1711yescharacteristic
157C4⋊C8—684no—
158C22⋊C8—684no—
159#95—24no—
160#95—24no—
161C17#764311yescharacteristic, complemented, Sylow
162C24C22⋊Dic1711yescharacteristic
163C2×Q8—24no—
164C2×Q8—22no—
165C2×Q8—416no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8#6322314yes—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—44no—
182C22×C4—42no—
183C22×C4#6323014yes—
184C22×C4#6323114yes—
185C22×C4C22⋊Dic1711yescharacteristic
186C22×C4C22⋊Dic1712yes—
187C22×C4C22×Dic1711yescharacteristic
188C22×C4C2×C22⋊D1712yes—
189C2×C8—684no—
190C2×C8—684no—
191C2×C8—684no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C42—24no—
195C42—41no—
196C42—41no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23#1058750412yes—
201C23#1058750614yes—
202C23C2×C22⋊Dic1711yescharacteristic
203Q8—44no—
204Q8—48no—
205Q8—44no—
206C2×C4—28no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—48no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—84no—
224C2×C4#1058750512yes—
225C2×C4#1058750614yes—
226C2×C4C2×#6322312yes—
227C2×C4C2×#6323012yes—
228C2×C4C2×#6323112yes—
229C8—1364no—
230C22—24no—
231C22—22no—
232C22—22no—
233C22—24no—
234C22#1176255814yes—
235C22#1176277311yescharacteristic
236C22C2×#1058750511yescharacteristic
237C22C2×#1058750411yescharacteristic
238C22C2×#1058750612yes—
239C22#1176282612yes—
240C4—22no—
241C4—24no—
242C4—22no—
243C4—22no—
244C4—22no—
245C4—44no—
246C4—82no—
247C2—24no—
248C2#1452711111yescharacteristic
249C2#1452721312yes—
250C2#1452735612yes—
251C2#1452740811yescharacteristic
252C2C2×#1176255811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic17
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC2×C17×#391C2
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenterC2×#391D17
hypocenterC17#7643
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. #14526610C2semidirect
161. C17#7643semidirect

Maximal subgroups

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

subgroupindex
#145287642
#145266102
C2×C17×#3912
#764317