Finite Group Databasebeta

← #16926839

Subgroups — #16926839

Order 4352.

690 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 131 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#14528867C211yesmaximal, characteristic
2#14528861C211yesmaximal, characteristic
3#14527317C214yesmaximal
4C2×C17×#541C211yesmaximal, characteristic
5#11762773C2212yes—
6#11762672C2212yes—
7#11762578—22no—
8#11762576—22no—
9#11762546—28no—
10C17×#541—28no—
11C2×C17×C4⋊Q8C411yescharacteristic, complemented
12C2×C17×#126—22no—
13C2×C17×#125—22no—
14C2×C17×#125C411yescharacteristic, complemented
15C2×C17×C22⋊Q8C2212yes—
16C2×C4×Q8×C17C412yescomplemented
17C2×C17×#117C2211yescharacteristic
18C22×Q8×C17C2×C412yes—
19C23×C4×C17C2311yescharacteristic
20C17×C4⋊Q8—24no—
21C17×#126—216no—
22C17×#125—216no—
23C17×#125—24no—
24C17×C22⋊Q8—24no—
25C17×C22⋊Q8D418yes—
26C4×Q8×C17—24no—
27C4×Q8×C17—24no—
28C17×#117—24no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17—22no—
31C2×C4⋊C4×C17—22no—
32C2×C4⋊C4×C17—22no—
33C2×C4⋊C4×C17C2×C412yes—
34C2×C4⋊C4×C17D412yes—
35C2×C22⋊C4×C17—22no—
36C2×C22⋊C4×C17D412yes—
37C2×C42×C17—22no—
38C2×C42×C17C2×C411yescharacteristic
39C2×C42×C17C2×C411yescharacteristic
40C2×C22⋊Dic17—22no—
41C2×#63223—22no—
42#10587502—24no—
43C22⋊#3622—24no—
44C24×C17C2×D411yescharacteristic
45C2×Q8×C17—22no—
46C2×Q8×C17—44no—
47C2×Q8×C17C22⋊C414yes—
48C22×C4×C17—22no—
49C22×C4×C17—22no—
50C22×C4×C17—42no—
51C22×C4×C17C22⋊C412yes—
52C22×C4×C17C22⋊C412yes—
53C22×C4×C17C22×C411yescharacteristic
54C22×C4×C17C2×D411yescharacteristic
55C22×C4×C17C2×D414yes—
56C22×Dic17—42no—
57C4⋊C4×C17—28no—
58C4⋊C4×C17—28no—
59C4⋊C4×C17—28no—
60C4⋊C4×C17—22no—
61C4⋊C4×C17—28no—
62C4⋊C4×C17—24no—
63C4⋊C4×C17C22⋊C414yes—
64C22⋊C4×C17—28no—
65C22⋊C4×C17—24no—
66C42×C17—28no—
67C42×C17—22no—
68C42×C17—22no—
69C22⋊Dic17—44no—
70#63223—44no—
71C2×#3622—42no—
72#8521—171nomaximal, Sylow
73C23×C17—22no—
74C23×C17—42no—
75C23×C17C2×C22⋊C411yescharacteristic
76C23×C17#12012yes—
77Q8×C17—44no—
78Q8×C17—44no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—24no—
84C2×C4×C17—24no—
85C2×C4×C17—44no—
86C2×C4×C17—44no—
87C2×C4×C17—44no—
88C2×C4×C17—42no—
89C2×C4×C17—42no—
90C2×C4×C17C2×C22⋊C412yes—
91C2×C4×C17#12012yes—
92C2×Dic17—82no—
93C2×Dic17—84no—
94#3622—82no—
95C2×#541D1711yescharacteristic
96#1822—171no—
97#1820—171no—
98#1511—174no—
99C22×C17—22no—
100C22×C17—24no—
101C22×C17—22no—
102C22×C17—82no—
103C22×C17#37911yescharacteristic
104C22×C17#41012yes—
105C4×C17—22no—
106C4×C17—22no—
107C4×C17—42no—
108C4×C17—42no—
109C4×C17—44no—
110C4×C17—44no—
111Dic17—162no—
112#541—28no—
113C2×C4⋊Q8Dic1711yescharacteristic, complemented
114C2×#126—22no—
115C2×#125—22no—
116C2×#125Dic1711yescharacteristic, complemented
117C2×C22⋊Q8C2×D1712yes—
118C2×C4×Q8Dic1712yescomplemented
119C2×#117C2×D1711yescharacteristic
120#407—172no—
121#381—172no—
122#343—342no—
123#342—342no—
124#328—348no—
125C2×C17—22no—
126C2×C17—42no—
127C2×C17#178211yescharacteristic
128C2×C17#179612yes—
129C22×Q8C2×Dic1712yes—
130C23×C4C22×D1711yescharacteristic
131C2×M16—342no—
132C4⋊Q8—24no—
133#126—216no—
134#125—216no—
135#125—24no—
136C22⋊Q8—24no—
137C22⋊Q8C22⋊D1718yes—
138C4×Q8—24no—
139C4×Q8—24no—
140#117—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4C2×Dic1712yes—
146C2×C4⋊C4C22⋊D1712yes—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—342no—
149C2×C22⋊C4C22⋊D1712yes—
150C2×C42—22no—
151C2×C42C2×Dic1711yescharacteristic
152C2×C42C2×Dic1711yescharacteristic
153C22⋊C8—344no—
154#95—344no—
155C17#852111yescharacteristic, complemented, Sylow
156C24C2×C22⋊D1711yescharacteristic
157C2×Q8—22no—
158C2×Q8—44no—
159C2×Q8C22⋊Dic1714yes—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—42no—
163C22×C4—682no—
164C22×C4C22⋊Dic1712yes—
165C22×C4C22⋊Dic1712yes—
166C22×C4C22×Dic1711yescharacteristic
167C22×C4C2×C22⋊D1711yescharacteristic
168C22×C4C2×C22⋊D1714yes—
169M16—684no—
170C2×C8—682no—
171C4⋊C4—28no—
172C4⋊C4—28no—
173C4⋊C4—28no—
174C4⋊C4—22no—
175C4⋊C4—28no—
176C4⋊C4—24no—
177C4⋊C4C22⋊Dic1714yes—
178C22⋊C4—28no—
179C22⋊C4—24no—
180C22⋊C4—684no—
181C42—28no—
182C42—22no—
183C42—22no—
184C23—22no—
185C23—42no—
186C23C2×C22⋊Dic1711yescharacteristic
187C23#1058763912yes—
188Q8—44no—
189Q8—44no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—1362no—
202C2×C4—1364no—
203C2×C4C2×C22⋊Dic1712yes—
204C2×C4#1058763912yes—
205C8—1362no—
206C22—22no—
207C22—22no—
208C22—24no—
209C22—82no—
210C22#1176266511yescharacteristic
211C22#1176278312yes—
212C4—22no—
213C4—22no—
214C4—44no—
215C4—44no—
216C4—42no—
217C4—42no—
218C4—2722no—
219C2—22no—
220C2—42no—
221C2#1452870211yescharacteristic
222C2#1452875912yes—
subgroupG/H
centerC22#11762665
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC2×C17×#541C2
socleC22×C17#379
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenterC2×#541D17
hypocenterC17#8521
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
11. C2×C17×C4⋊Q8C4semidirect
14. C2×C17×#125C4semidirect
16. C2×C4×Q8×C17C4semidirect
113. C2×C4⋊Q8Dic17semidirect
116. C2×#125Dic17semidirect
118. C2×C4×Q8Dic17semidirect
155. C17#8521semidirect

Maximal subgroups

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

subgroupindex
#145288672
#145288612
#145273172
C2×C17×#5412
#852117