Finite Group Databasebeta

← #16950593

Subgroups — #16950593

Order 4352.

642 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 107 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#14528712C212yesmaximal
2#14526911C214yesmaximal, complemented
3C17×#2297C211yesmaximal, characteristic
4C2×#10587507C2214yes—
5C2×#10587504—22no—
6#11762773—22no—
7C22×#41×C17C2211yescharacteristic
8C2×C17×#117C411yescharacteristic
9C22×C42×C17C411yescharacteristic
10C17×#391C2212yes—
11C17×#390C412yes—
12C17×#386—22no—
13C4×C22⋊C4×C17—22no—
14C2×#41×C17—24no—
15C2×#41×C17—42no—
16C2×#41×C17—42no—
17C22×Q8×C17C2311yescharacteristic
18C22×D4×C17C2×C411yescharacteristic
19C22×D4×C17D412yes—
20C23×C4×C17—21no—
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17D412yes—
23C17×#117—42no—
24C2×C4⋊C4×C17C2×C412yes—
25C2×C22⋊C4×C17—21no—
26C2×C22⋊C4×C17—24no—
27C2×C42×C17—21no—
28C2×C42×C17—21no—
29C2×C42×C17—42no—
30C2×C42×C17C2×C412yes—
31C2×#63223—24no—
32C17×#95—24no—
33#10587507—28no—
34#10587504—44no—
35C22⋊#3622—44no—
36C24×C17C22⋊C412yes—
37C24×C17C22⋊C411yescharacteristic
38#41×C17—416no—
39C2×Q8×C17—41no—
40C2×Q8×C17—41no—
41C2×Q8×C17#4114yes—
42C2×D4×C17—24no—
43C2×D4×C17—22no—
44C2×D4×C17—24no—
45C2×D4×C17—41no—
46C2×D4×C17—41no—
47C2×D4×C17—42no—
48C22×C4×C17—22no—
49C22×C4×C17—22no—
50C22×C4×C17—22no—
51C22×C4×C17—22no—
52C22×C4×C17—24no—
53C22×C4×C17—24no—
54C22×C4×C17—22no—
55C22×C4×C17—42no—
56C22×C4×C17—44no—
57C22×C4×C17—42no—
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22×C411yescharacteristic
60C22×C4×C17C2×D412yes—
61C4⋊C4×C17—42no—
62C22⋊C4×C17—48no—
63C22⋊C4×C17—42no—
64C42×C17—42no—
65C42×C17—42no—
66C42×C17—42no—
67#63223—44no—
68#63223—44no—
69C2×#3622—44no—
70#12861—171nomaximal, Sylow
71C23×C17—21no—
72C23×C17—21no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—42no—
77C23×C17—44no—
78C23×C17#9912yes—
79C23×C17C2×C22⋊C411yescharacteristic
80Q8×C17—44no—
81D4×C17—48no—
82D4×C17—44no—
83C2×C4×C17—28no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—24no—
91C2×C4×C17—48no—
92C2×C4×C17—42no—
93C2×C4×C17—48no—
94C2×C4×C17—42no—
95C2×C4×C17—44no—
96C2×C4×C17—42no—
97C2×C4×C17#9912yes—
98C2×C4×C17#11712yes—
99C2×C4×C17#12314yes—
100#3622—84no—
101#2297D1711yescharacteristic
102#1785—172no—
103#1394—174no—
104C22×C17—22no—
105C22×C17—22no—
106C22×C17—42no—
107C22×C17—42no—
108C22×C17—44no—
109C22×C17—42no—
110C22×C17—42no—
111C22×C17—44no—
112C22×C17—82no—
113C22×C17#38111yescharacteristic
114C22×C17C2×#9911yescharacteristic
115C22×C17C2×#9911yescharacteristic
116C4×C17—24no—
117C4×C17—22no—
118C4×C17—22no—
119C4×C17—44no—
120C4×C17—82no—
121C22×#41C2×D1711yescharacteristic
122C2×#117Dic1711yescharacteristic
123C22×C42Dic1711yescharacteristic
124C2×#101—174no—
125C2×#100—342no—
126#407—342no—
127#391C2×D1712yes—
128#390Dic1712yes—
129#386—22no—
130C4×C22⋊C4—22no—
131C2×C17—21no—
132C2×C17—21no—
133C2×C17—42no—
134C2×C17—82no—
135C2×C17#181911yescharacteristic
136C2×C17#213212yes—
137C2×#41—24no—
138C2×#41—42no—
139C2×#41—42no—
140C22×Q8C22×D1711yescharacteristic
141C22×D4C2×Dic1711yescharacteristic
142C22×D4C22⋊D1712yes—
143C23×C4—21no—
144C23×C4C2×Dic1711yescharacteristic
145C23×C4C22⋊D1712yes—
146C2×M16—344no—
147#117—42no—
148C2×C4⋊C4C2×Dic1712yes—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—24no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—42no—
154C2×C42C2×Dic1712yes—
155#101—348no—
156#100—684no—
157C22⋊C8—684no—
158#95—24no—
159C17#1286111yescharacteristic, complemented, Sylow
160C24C22⋊Dic1712yes—
161C24C22⋊Dic1711yescharacteristic
162#41—416no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8#6325214yes—
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—24no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—42no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4C22⋊Dic1711yescharacteristic
183C22×C4C22×Dic1711yescharacteristic
184C22×C4C2×C22⋊D1712yes—
185M16—684no—
186M16—684no—
187C2×C8—684no—
188C4⋊C4—42no—
189C22⋊C4—48no—
190C22⋊C4—42no—
191C42—42no—
192C42—42no—
193C42—42no—
194C23—21no—
195C23—21no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—42no—
200C23—44no—
201C23#1058750512yes—
202C23C2×C22⋊Dic1711yescharacteristic
203Q8—44no—
204D4—48no—
205D4—44no—
206C2×C4—28no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—24no—
214C2×C4—48no—
215C2×C4—42no—
216C2×C4—48no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4#1058750512yes—
221C2×C4#1058761212yes—
222C2×C4#1058761514yes—
223C8—1364no—
224C22—22no—
225C22—22no—
226C22—42no—
227C22—42no—
228C22—44no—
229C22—42no—
230C22—42no—
231C22—44no—
232C22—82no—
233C22#1176267211yescharacteristic
234C22C2×#1058750511yescharacteristic
235C22C2×#1058750511yescharacteristic
236C4—24no—
237C4—22no—
238C4—22no—
239C4—44no—
240C4—82no—
241C2—21no—
242C2—21no—
243C2—42no—
244C2—82no—
245C2#1452885811yescharacteristic
246C2#1453046912yes—
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC17×#2297C2
socleC22×C17C2×#99
prime radicalC22×#41×C17C22
cosocleC22×Q8×C17C23
hypercenter#2297D17
hypocenterC17#12861
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. #14526911C2semidirect
159. C17#12861semidirect

Maximal subgroups

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

subgroupindex
#145287122
#145269112
C17×#22972
#1286117