Finite Group Databasebeta

← #16933748

Subgroups — #16933748

Order 4352.

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

normal subgroups only

This group has 374 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#14528535C211yesmaximal, characteristic
2#14527547C211yesmaximal, characteristic
3#14527399C212yesmaximal
4#14527296C212yesmaximal
5C17×#3170C211yesmaximal, characteristic
6#11762866C2212yes—
7C2×#10587474C2211yescharacteristic
8#11762655C2211yescharacteristic
9#11762563—21no—
10#11762563—21no—
11#11762562—21no—
12#11762560—21no—
13#11762551—24no—
14#11762546—24no—
15C17×#533C2212yes—
16C17×#500C411yescharacteristic, complemented
17C17×#494C411yescharacteristic, complemented
18C17×#487C412yescomplemented
19C17×#467—21no—
20C17×#467—21no—
21C4×Q16×C17—21no—
22C4×Q16×C17—21no—
23C2×C4×C8×C17C2211yescharacteristic
24C2×Q16×C17—21no—
25C2×Q16×C17—21no—
26C22×C8×C17D411yescharacteristic
27C22×C8×C17D411yescharacteristic
28C17×C4⋊Q8C2×C412yes—
29C17×C4⋊Q8C2×C412yes—
30C17×#125—22no—
31C17×C22⋊Q8—22no—
32C17×C22⋊Q8—22no—
33C4×Q8×C17—22no—
34C4×Q8×C17—22no—
35C17×#117—22no—
36C2×C42×C17C2311yescharacteristic
37C2×C4×Dic17—21no—
38C22×#3622—21no—
39C17×#107—21no—
40C17×#107—21no—
41C17×#103—22no—
42C17×#103—22no—
43C4×C8×C17C2×C411yescharacteristic
44C4×C8×C17C2×C411yescharacteristic
45C4×C8×C17D411yescharacteristic
46C4×C8×C17D411yescharacteristic
47#10587502—21no—
48#10587502D412yes—
49C22⋊#3622—22no—
50C4⋊#3622—22no—
51#10587474—22no—
52#10587474—21no—
53#10587474D412yes—
54C2×Q8×C17—22no—
55C2×Q8×C17—22no—
56C22×C4×C17C2×D411yescharacteristic
57C22×C4×C17C2×D411yescharacteristic
58C22×C4×C17#4111yescharacteristic
59C22×Dic17—22no—
60Q16×C17—42no—
61Q16×C17—42no—
62C2×C8×C17—41no—
63C2×C8×C17—41no—
64C2×C8×C17C22⋊C411yescharacteristic
65C2×C8×C17C22⋊C411yescharacteristic
66C2×C8×C17C22⋊C411yescharacteristic
67C2×C8×C17C22⋊C411yescharacteristic
68C4⋊C4×C17—22no—
69C4⋊C4×C17—22no—
70C4⋊C4×C17—22no—
71C4⋊C4×C17—22no—
72C4⋊C4×C17—42no—
73C4⋊C4×C17—42no—
74C22⋊C4×C17—42no—
75C42×C17—42no—
76C42×C17C22×C411yescharacteristic
77C42×C17C2×D411yescharacteristic
78C42×C17C2×D411yescharacteristic
79C42×C17#4111yescharacteristic
80C4×Dic17—41no—
81C4×Dic17—41no—
82C2×#3622—22no—
83C2×#3622—41no—
84C2×#3622—41no—
85C2×#3622—42no—
86#9765—171nomaximal, Sylow
87C23×C17#12111yescharacteristic
88Q8×C17—42no—
89Q8×C17—42no—
90Q8×C17—82no—
91C2×C4×C17—22no—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—41no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17C2×C22⋊C411yescharacteristic
98C2×C4×C17C4×D411yescharacteristic
99C2×C4×C17C4×D411yescharacteristic
100C2×C4×C17#12111yescharacteristic
101C2×C4×C17#12411yescharacteristic
102C2×C4×C17#12711yescharacteristic
103C2×Dic17—42no—
104C2×Dic17—42no—
105C2×Dic17—42no—
106C8×C17—21no—
107C8×C17—21no—
108C8×C17—21no—
109C8×C17—21no—
110#3622—42no—
111#3622—82no—
112#3170D1711yescharacteristic
113#1746—171no—
114#1568—171no—
115#1531—172no—
116#1506—172no—
117C22×C17—21no—
118C22×C17—21no—
119C22×C17#39011yescharacteristic
120C22×C17#47312yes—
121C4×C17—21no—
122C4×C17—21no—
123C4×C17—21no—
124C4×C17—21no—
125C4×C17—21no—
126C4×C17—82no—
127C4×C17—82no—
128C4×C17#42111yescharacteristic
129C4×C17#42111yescharacteristic
130Dic17—82no—
131#533C2×D1712yes—
132#500Dic1711yescharacteristic, complemented
133#494Dic1711yescharacteristic, complemented
134#487Dic1712yescomplemented
135#467—21no—
136#467—21no—
137C4×Q16—21no—
138C4×Q16—21no—
139#432—172no—
140C2×#97—171no—
141C2×C4×C8C2×D1711yescharacteristic
142#376—171no—
143#336—341no—
144#336—341no—
145#335—341no—
146#334—341no—
147#330—344no—
148#328—344no—
149C2×C17—22no—
150C2×C17#196311yescharacteristic
151C2×C17#197611yescharacteristic
152C2×C17#197911yescharacteristic
153C2×Q16—21no—
154C2×Q16—21no—
155C22×C8—341no—
156C22×C8C22⋊D1711yescharacteristic
157C22×C8C22⋊D1711yescharacteristic
158C4⋊Q8C2×Dic1712yes—
159C4⋊Q8C2×Dic1712yes—
160#125—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165#117—22no—
166C2×C42—341no—
167C2×C42C22×D1711yescharacteristic
168#107—21no—
169#107—21no—
170C4⋊C8—342no—
171#103—22no—
172#103—22no—
173C22⋊C8—342no—
174#97—172no—
175#97—342no—
176#97—341no—
177C4×C8C2×Dic1711yescharacteristic
178C4×C8C2×Dic1711yescharacteristic
179C4×C8C22⋊D1711yescharacteristic
180C4×C8C22⋊D1711yescharacteristic
181#95—172no—
182#95—341no—
183C17#976511yescharacteristic, complemented, Sylow
184C2×Q8—22no—
185C2×Q8—22no—
186C22×C4—342no—
187C22×C4D4×D1711yescharacteristic
188C22×C4#6325411yescharacteristic
189C22×C4C2×C22⋊D1711yescharacteristic
190Q16—42no—
191Q16—42no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—342no—
195C2×C8—681no—
196C2×C8—681no—
197C2×C8—682no—
198C2×C8C22⋊Dic1711yescharacteristic
199C2×C8C22⋊Dic1711yescharacteristic
200C2×C8C22⋊Dic1711yescharacteristic
201C2×C8C22⋊Dic1711yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C22⋊C4—42no—
209C42—42no—
210C42—681no—
211C42—681no—
212C42D4×D1711yescharacteristic
213C42#6325411yescharacteristic
214C42C22×Dic1711yescharacteristic
215C42C2×C22⋊D1711yescharacteristic
216C23#1058762411yescharacteristic
217Q8—42no—
218Q8—42no—
219Q8—82no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—682no—
227C2×C4—682no—
228C2×C4—682no—
229C2×C4D4×Dic1711yescharacteristic
230C2×C4D4×Dic1711yescharacteristic
231C2×C4#1058762211yescharacteristic
232C2×C4#1058762411yescharacteristic
233C2×C4#1058762611yescharacteristic
234C2×C4C2×C22⋊Dic1711yescharacteristic
235C8—21no—
236C8—21no—
237C8—21no—
238C8—21no—
239C8—682no—
240C8—1362no—
241C22—21no—
242C22—21no—
243C22#1176271211yescharacteristic
244C22#1176308112yes—
245C4—21no—
246C4—21no—
247C4—21no—
248C4—21no—
249C4—21no—
250C4—82no—
251C4—82no—
252C4—1362no—
253C4#1176282611yescharacteristic
254C4#1176282611yescharacteristic
255C2—22no—
256C2#1452961711yescharacteristic
257C2#1452968311yescharacteristic
258C2#1452970011yescharacteristic
subgroupG/H
centerC22#11762712
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#3170C2
socleC22×C17#390
prime radicalC23×C17#121
cosocleC2×C42×C17C23
hypercenter#3170D17
hypocenterC17#9765
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
16. C17×#500C4semidirect
17. C17×#494C4semidirect
18. C17×#487C4semidirect
132. #500Dic17semidirect
133. #494Dic17semidirect
134. #487Dic17semidirect
183. C17#9765semidirect

Maximal subgroups

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

subgroupindex
#145285352
#145275472
#145273992
#145272962
C17×#31702
#976517